APPLIED GEOPHYSICS U.S.S.R. APPLIED GEOPHYSICS U. S.S.R. Edited by NICHOLAS RAST, B.Sc, Ph.D., F.G.S. Liverpool University PERGAMON PRESS THE MACMILLAN COMPANY NEW YORK PERGAMON PRESS INC. 122 East 55th Street, New York 22, N.Y. 1404 New York Avenue N. W., Washington 5, D. C. PERGAMON PRESS LTD. Headington Hill Hall, Oxford 4 and 5 Fitzroy Square, London W. 1 PERGAMON PRESS S.A.R.L. 24 Rue des Ecoles, Paris V^ PERGAMON PRESS G.m.b.H. Kaiserstrasse 75, Frankfurt am Main Copyright © 1962 Pergamon Press Ltd. Library of Congress Card Number 60-53385 Printed in Poland to the order of PWN — Polish Scientific Publishers by Drukarnia Narodowa, Cracow /^* 'A 1 WOODS I V HOLE, \ MASS / CONTENTS \»^»^--'^^ ^V i^ ^ .^^^ Page Editor's Foreword 7 Part I — Seismology 1. Intensities of Refracted and Reflected Longitudinal Waves at Angles of Incidence below Critical 11 V. P. Go RB AT OVA 2. Method and Techniques of Using Stereographic Projections for Solving Spatial Problems in Geometrical Seismics 44 E. I. Gal'perin, G. a. Krasil'shchikova, V. I. Mironova and A. V. Frolova 3. Multiple Reflected Waves 75 S. D. Shushakov 4. Diffracted Seismic Waves 99 T. I. Oblogina Part II — Gravimetry 5. The Influence of Disturbing Accelerations when Measuring the Force of Gravity at Sea using a Static Gravimeter 123 K. Ye. Veselov and V. L. Panteleyev 6. Evaluating the Accuracy of a Gravimetric Survey, Selecting the Rational Density of the Observation Network and Cross-sections of Isoanomalies of the Force of Gravity 139 B. V. Kotliarevskii Part III — Electrical Sonde Methods 7. Theoretical Bases of Electrical Probing with an Apparatus Immersed in Water 169 E. I. Terekhin 8. The Use of New Methods of Electrical Exploration in Siberia 196 A. M. Alekseev, M. N. Berdichevskii and A. M. Zagarmistr 9. The Method of Curved Electrical Probes 223 M. N. Berdichevskii 10. The Use of the Loop Method (Spir) in Exploring Buried Structures 241 I. I. Krolenko 11. Allowance for the Influence of Vertical and Inclined Surfaces of Separation when Interpreting Electric Probings 271 V. I. FOMINA 8t2«-j CONTENTS Part IV — Oil Geophysics 12. Some Problems of Gas Logging Estimation of Gas Saturation of Rocks 301 L. A, Galkin 13. Luminescence Logging 315 T. V. Shcherbakova 14. Optical Methods of Bore-Hole Investigation 328 T. V. Shcherbakova 15. Determining the Permeability of Oil-Bearing Strata from the Specific Resistance 349 S. G. KoMAROV and Z. I. Keivsar 16. New Types of Well Resistivity-Meters 384 E, A. POLYAKOV 17. The Use of Accelerators of Cliarged Particles in Investigating Bore-Holes by the Methods of Radioactive Logging .... 397 V. M. Zaporozhetz and E. M. Filippov Author Index 423 Subject Index 426 Publisher's Notice to Readers on the Supply of an English Translation OF ANY Russian Article mentioned Bibliographically or Referred to in this Publication. The Pergamoii Institute has made arrangements with the Institute of Scientific Infor- mation of the U.S.S.R. Academy of Sciences whereby they can obtain rapidly a copy of any article originally pubUshed in the open literature of the U.S.S.R. We are therefore in a position to supply readers with a translation (into Enghsh or any other language that may be needed) of any article referred to in this pubUcation, at a reasonable price under the cost-sharing plan. Readers wishing to avail themselves of this service should address their request to the Administrative Secretary, The Pergamon Institute, at either 122 East 55th Street, New York 22, N.Y., or Headington Hill Hall, Oxford. EDITOR'S FOREWORD Spectacular successes achieved by the Soviet scientists in the field of apphed physics have focused attention on the vigour of scientific research in the U.S.S.R. As a result a concerted attempt is being made to make the extensive Soviet scientific literature available to Western readers. Although, at present, several institutions are concerned wdth translations from Russian only rarely are aspects of applied, as distinct from pure, science given their rightful place. Thus, the attempts of the Pergamon Institute to redress this situation are especially welcome. It must be remembered that in the U.S.S.R. scientific workers often follow what can be called an American tradition in not separating sharply the fundamental research from its technological applications. As a consequence many Russian papers deahng with specific industrial problems contain much of general scientific interest. This is Cbpecially true with respect to geophysics. Every new method of geophysi- cal exploration is valuable since it provides a new possibility of inspecting the imseen parts of the Earth. In any case in a new science practically every investigation is of some significance if only because it adds to the relatively meagre store of factual data. In the U.S.S.R. the methods of geophysical research have been extensively apphed not only in an effort to find useful minerals, but also in order to accumulate information on the geological structure of that vast country. B\nthermore, the accuracy and reproducibility of the geophysical methods has been widely checked with the aid of numerous bore-holes systematically located at critical points. As a result very notable advances have been made in developing the so-called electrical, seismic and gravimetric methods, while the existence of the numerous bore -holes has led to an extensive, application and improvement of the geophysical methods of logging. In the present compendium a selection of papers published in the volumes 18 and 20 of the Soviet journal "Apphed Geophysics" are being presented to the Western scientists. The intention is to illustrate some of the achieve- ments of the Russian apphed geophysicists by translating their recent publi- cations. Although a fairly wide range of topics is being covered there is a bias towards the apphcation of geophysical methods to the search for oil. In this respect the editor, who was responsible for selecting the papers to be translated, followed the tendency discernible in the original journals. Nonetheless the orientation of many of the included articles on seismic, electrical, gravimetric methods and logging techniques is such that the 7 8 Editor's Foreword volume should appeal not only to the oil geophysicists and geologists but to everyone interested in modern developments in geophysical methodology. Since the translation of Soviet scientific hterature is as yet in a pioneering stage it is, perhaps, not inappropriate to add a few remarks of a purely Hnguis- tic nature. The relatively prolonged isolation of Russian scientists from their western colleagues has led to differences in terminology. For instance, the Russian term equatorialnyi sond does not mean an equatorial sonde, but probes with a quadrilateral arrangement, while the word podniatie in various contexts impHes an elevation, a culmination or an upfold. In the present volume, where necessary, the Russian usage is indicated and it is hoped that such editorial remarks will be of use to the translators of scien- tific Russian. Responsibihty for these remarks rests with the editor, but in certain instances Professor R. M. Shackleton and Dr. C. D. V. Wilson of the University of Liverpool were consulted and made suggestions, for which the editor wishes to express his gratitude. Nicholas Rast PART I. SEISMOLOGY Chapter 1 INTENSITIES OF REFRACTED AND REFLECTED LONGITUDINAL WAVES AT ANGLES OF INCIDENCE BELOW CRITICAL V. P. GORBATOVA The dynamic properties of waves can be effectively utilized in interpreting seismic prospecting data, since these properties, in conjunction with the velocity components enable us to recognize the nature of any particular Avave recorded on the seismogram. The solution of problems connected ^vith the dynamic propagation of waves presents difficulties which are well known. While part of the work done in this field by Petrashen' and the team of mathematicians headed by him has already been published, the theory which we have in mind has been fully worked out only for ideally elastic horizontally laminated media. Each of the layers is presumed to be sufficiently "dense", that is to say the travel time of a disturbance in the layer is substantially longer than the duration of the pulse transmitted. The velocity of propagation of longi- tudinal and transverse waves, however, as well as the densities, is constant inside the layer and assume new values on the boundaries of the layers. Quantitative comparisons made up to date have not revealed any sharp discrepancy between theory and experiment. The theoretically discovered qualitative laws also show good agreement with seismic prospecting practice. We suggest that there would be undoubted advantage in introducing the theory, in the form in which it has been worked out to date, into the inter- pretation of field data. A method for calculating the intensities and shapes of seismic traces for different waves propagated in media with plane -parallel boundaries has been worked out in detail in the Leningrad Section of the Institute of Mathematics (Academy of Sciences of the U.S.S.R.) '-^'^K The Section has also compiled tables for fairly accurate calculations. In this paper we offer a number of simpHfied methods for determining the intensities of purely longitudinal waves (in the media referred to) and ■discuss how the different parameters of the medium affect their frequency rate. 11 12 V, P. GORBATOVA In the main these simpHfications mean that the frequency rates of the waves under consideration are determined not by accurate tables but simply by a small number of typical graphs, which we shall give later. Further, the assumption has been made that layers possessing a higher longitudinal velocity have also a higher ratio of transverse velocity to longitudinal veloc- ity and a greater density. This would appear to be true for most real media. The elastic properties of two neighbouring media (the ith and the i+lth) are characterized by the following parameters: a— the ratio of the lower longitudinal velocity to the higher; y— the ratio of the transverse velocity to the longitudinal velocity in the layer where v is the smaller; Zl — the ratio of the transverse velocity to the longitudinal velocity in a layer where- V is the greater; cr— the ratio of the lower density to the higher. If Vi^ p < Vi + i^ p, V; oc = I, p ^i+i, P ^i,s XP A = 'Vj+l, s Vi+l,p Qi If Vi^ p > Vi+i^ p, Vi + l,p %p ^f+1, s Vi+1, p Vi,p a = Qi+i Qi+l That is, y <Zl, a <1,0, a< 1, on all occasions. The parameters a, y, A and a of adjacent layers, on whose boundaries refraction occurs, are chosen within the following Hmits: 0.3 < a < 0.9, 0.3 <y < 0.6, 0.4 <Zl < 0.6, 0.7 <(t < 1.0 But the boundaries of adjacent layers characterized by the parameters y = 0.3, A = 0.6 are excluded. For the ratio of transverse velocities in the boundary media therefore we shall always assume: 0.3<(5<0.9, where a- Vi '' ■' , if ^iV p < ^f +1, p , if % p > ^i+i, p 'Vi+1, s Table 1 shows values for the parameters of adjacent layers on whose bound-^ ary reflection occurs if the reflection takes place from the layer in which the travel velocity of longitudinal waves is higher than in the layer through which the wave has passed. If the reflection occurs from the layer with the lower group velocity the values for the parameters will be found in Table 2. REFRACTED AND REFLECTED LONGITUDINAL WAVES 13 CO o LCl d LO d v/ o d d vo d lO •># d v/ is o d CO d d 3 d o t— 1 1 d d LO d lO d lA co d o d d d vo 3 CO d o d d d OS d q r-H in d lO d ON d 1 d q t-5 d d On d i d q d d CO d q r—J d LO d lO d CO d q 1-4 1 d d d CO d q i-H d ?>- X) s to CO d d 5 vo d 4 d o d d in d LO in CO d q I-H 1 d in d d vo CO d o d d d ON d d q LO d in d Ov d 1 d q vo d vo d ON d d q f-H d d IT- d 1 CO d q 1 ^- d in d in d s d q 1— 1 1 d d S CO d q d ?^ ^ « to 14 V. P, GORBAOVA The angles at which simple and multiple waves strike the reflecting boundaries are assumed to vary from zero to the value of oci where sin oci — 0.75 sin a^^. But in the case of reflection from media where the group velocity wave is travelling, the angles of incidence onto the reflecting bound- ary are treated as having a sine less than or equal to 0.6. In this way then, M^e can investigate the intensity* and shape of vertical displacements of points on a free surface caused by the arrival of purely longitudinal waves. The source of excitation chosen will be a shot fired in the upper layer of the medium. In our analysis of the intensities of the head and reflected waves we shall always assume that the conditions of the excitation remain unchanged and that the velocity of propagation of the longitudinal waves and the density of the medium in the layer where the shot is fired are kept constant. The head waves are examined at a distance from their points of emergence, while the simple reflected waves will be examined at a distance from the points of einergence of the head waves formed at the same interfaces as the simple reflected wave under consideration. By the expression "at a distance from the points of emergence" of a head wave, we mean all points on the ground surface lying at a distance r from the shot point, which satisfy the inequality (1). ro|>0.7 -1 r=0 2hi tan oci hsd tan oCq COS^ CCi COS^ OCq (1) Where r^ is the abscissa of the point of origin of the head wave under consideration : Aj, is the wave-length when the wave is travelhng along the refracting boundary; A^ = v^^T. Here T is the predominant period of the vibration being propagated, and Vj^ is the boundary velocity of the head wave under consideration ; h- is the thickness of the ith layer of the covering medium; a,- is the angle at which the wave under consideration (head or re- flected) passes through the ith layer; hg^ is the shot depth. * By intensity we mean a quantity proportional to the amount of maximum displacement of points on the ground surface caused by the arrival of the wave under consideration. If the same conditions of excitation are maintained, the coefficient of proportionaUty is constant for all primary waves. It is also constant for all reflected waves, but is not the same as for primary waves. REFRACTED AND REFLECTED LONGITUDINAL WAVES IS Multiple reflected waves are examined at a distance from the points of emergence of like reflected -refracted waves. We evaluate the distance with the help of inequalities similar to (1), where only in the term > — ' — - — - ■^-J cos"a/ is each of the items repeated as many times as the multiple wave under consideration passes through the iih layer. METHODS FOR CALCULATING INTENSITIES Head Waves The intensity of a head wave propagated in a medium (Fig, 1) from the n — 1th and nth boundaries of the layers is determined by the expression c,''n\PiPi^^){Pi^^p,)rpp{p) t" = /n\ Jhead— . /^ ^ m/-/ xa; ' ^"^^ where Cq is some multiplier dependent on the properties of the zero layer* r^^ (p) is the coefficient of the head wave formation at the boundary of the Tilth, and nth layers; r is the distance between shot point and the observation point along the free surface; Tq is the abscissa of the point of origin of the head wave under consid- eration ; Aq, and //q are the Lame constants for the layer where the shot is fired; (P-P-^-^ (Pi+iPi) are the coefficients of refraction in the intensities when the wave is passing through the boundary bet\veen the iih and the j+lth layers from above to below and from below to above. n-2 In the product IJ (P^P^j^^ (Pi+i^i) i^ formula (2) the coefficients of i = o refraction of the wave under consideration at all the intermediate interfaces are taken into account. The coefficients of refraction, the coefficients of head wave formation, and also the multipher Cq depend both on the properties of the interfaces themselves and also on the angle at which the primary wave under consider- ation strikes these boundaries. Values for these coefficients are given in the detailed tables compiled by the team of mathematicians headed by Petrashen' at the Leningrad Section of the Institute of Mathematics. Using these tables to determine the quantities mentioned we have compiled graphs * The advisability of using this Hes in the fact that if the calculations are suffciently approximate, we can assume that Cq = 2 when sin OCq < 0.9. 16 V. P. GORBATOVA on which the sines of the angles of incidence with which we are concerned are plotted against the quantities which interest us or quantities which differ from these by having different multiphers. °\ h ' \ / or, \ / ' r / '>^: \ 1 n-l \ / \ / Fig. 1. The process of calculating the intensities of head waves can be sum- marized as follows: 1. Using the law of wave refraction we determine the sines of the angles of incidence of the primary wave under consideration at all the interfaces: sm (Xi == ^n, p 01 2 3 0-'» 0-5 0-6 0-7 0-8 09 10 Fig. 2. Curves for the multiplier Cg. 2. From the ordinates of the curve (Fig, 2) which is characterized by the parameter y^ = -^^ we take the value for the multipHer mth Cq at the ^0, e point X — sin «„ ^0, e REFRACTED AND REFLECTED LONGITUDINAL WAVES 17 3. To determine the value of the coefficient of refraction (P^- Pi^i) {Pi+i Pi) at the point x = sin a- (where a.- is the lesser of the angles a, and a,+i) from the curve in Fig. 3 corresponding to the parameter which is the same as at the boundary of the ith and ilth layers, we take the value of the ordi- ^Qi'^i p Qi + lt^i+l p nate [(PjP,4i) (P,+i i'.)] rel and multiply by. ' ' .3 , which \(Ji ^i, p^ vi+i'^i+i, p) is the product of coefficients of refraction (P,Pj_j.j) (Pj^^ P-), of the plane waves when incidence is vertical. 10 09 ^ 0-. 0-7 0-6 02 Sin a, 0-4 06 08 -^_^_^ i \ \ \ = 0-7 a=0-5 \ 0=0-3. P'iG. 3. Curves for determining the coefficient of refraction [(^j ^j + i) (^/-|-i^0]- The product obtained thus gives the value for the coefficient of refraction (PiPi^x) iPi^iP.)- Vn-l, p 4. The coefficient F^^ (p) is taken at the point x = sin a^ "j p from the curve corresponding to the parameters y, A and a which charac- terize the boundary of the relth and nth layers (Fig. 4). Fig. 4 (a and b) shows the P^^ (p) curves for two values of the parameter a (dotted hnes — cr = 0.7; thick lines -a = 1.0). Applied geophysics 18 V. P. GORBATOVA 5. The values obtained for Cq {P^ -P/+i) [Pi+i -Pj)? ^^p (p) and Tq are inserted in formula (2). Reflected Simple and Multiple Weaves The intensity of a simple wave reflected from the boundary of the nlth and nth layers is determined from formula (3) p sin (Xq (3) 1 / ^ r V^ Attang,- _ hsdtan oco 1 V [ Z-j cos^ Xi cos^ ocq J i=0 Similar formulae are given in Malinovskaia's work <^). The intensities of multiple waves are determined from formulae similar to (3). Here again we must take into account all the refractions and reflec- tions at the intermediate interfaces. Therefore if the multiple wave under consideration intersects the interface between the iJth and i + lth layers while it is travelling from above downwards m times and the same number of times when it is travelling from below upwards, then the multiplier (P^ P,-+i) (P,-^j^Pj) is repeated in the product U (P^Pi^y) {P^_^,■J^P■) also m times. If the wave suffers n reflections from the given interface, the coefficient of reflection corresponding to this boundary is raised to the nth power. We introduce coefficients raised to the appropriate powers for reflection from all the interfaces at wliich the wave under consideration suffers further reflections. We sliall henceforward adopt the following notation : if the reflected wave is travelHng in the mth layer and is reflected from its lower boundary, the coefficient of reflection will be denoted by (P,„ P^) ; but if the reflection takes place from the upper boundary of the mih layer the coefficient of reflection will be denoted by (P„,/ P^>). In the sum > hi L each item is repeated as many times as the multiple -^-1 COS^ CCi wave under consideration passes through the ith layer. If this wave is propagated from the shot point upwards and then is again reflected from the free surface and goes downwards, a further term, 2hsd ^-^ must be cos" a.i introduced into the sum ^2 ^^ tan a,- _ /,^^i^^ in formula (3) ^—1 COS^ (Xi COS^ OCq REFRACTED AND REFLECTED LONGITUDINAL WAVES 19 The method of calculating the intensities of reflected waves can be sum- iiuarized as follows. 1. First we determine the sines of the angles of incidence of the reflected wave under consideration at all the interfaces. These angles will vary from point to point along the ground surface unlike the corresponding angles for the head waves. It is best to have sin oCq Vn.p Fig. 4a. Curves for the coefficient FPP (p) of primary wave formation (A =0.6 and A = 0.4) given and then the angles of incidence of the ray selected will have the same sines at all the interfaces, and these will be determined by expression (4). sm (Xi = — -^ sm (Xn , (4) while the point of emergence of the ray under consideration will lie at a distance r, determined by the formula 20 V. P, GORBATOVA r = 2 2^ j tan a^ — h^^ tan oCq . (5) j=o from the shot pomt along the profile. In a case of multiple waves, the term h^ tan x- in the sum 2 f^i tan a,- hi expression (5) is repeated the same numbers of times as the wave under consideration passes through the ith layer. sinCj^ Fig. 46. Curves for the coefficient /"PP (p) of primary wave formations (zl= 0.5). 2. The multiplier Cq is determined in the same way as for primary waves. Its value is taken at the point A; = sin Uq from the curve (see Fig. 2) cor- responding to the parameter y^ = —2iP^ as for the zero layer of the medium '0, p imder consideration. REFRACTED AND REFLECTED LONGITUDINAL WAVES 21 3. The coefficients of refraction {P^P-_^j) (Pj+jP,) are determined in the same way as for head waves. Only the ordinates of the curves (see Fig. 3) are noAV taken at the points x = sin a, ~ -^^-^ sin a^ where V: ^ is the lesser of the velocities v^ and v-_^-^ , that is at points equal to the sines of the lesser of the angles at which the ray under consideration meets the boundary be- tween the ith and the i + 1th layers. The curves are chosen -with the same parameter as corresponds to the boundary between the iih. and the i+lth layers. Then the ordinates are again multiplied by the quantity, iQii'i,P+ Qi+iVi+i,pf The product also immediately gives us the value of the coefficient of refraction {P,P,.d {P,^iP^)- 4. The coefficients of reflection (P^ P^) and (P,„. P^^.) depend both on the properties of the reflecting boundary itself and also on the angle at which the wave strikes it. To determine these we have constructed graphs (Figs. 5, 6 and 7) based on the tables drawn up by the Petrashen' team; the sines of the angles at which the wave under consideration strikes the given reflecting boundary are plotted against the abscissa, and the parame- ters of the reflecting boundary are used as the parameters of the curves. We shall deal ^vdth each of the following cases separately: (a) Reflection of a wave from a layer having higher velocity of longitudinal wave propagation than the layer in which the incident ray is travelling; (b) Reflection of a wave from a layer having a louver velocity of propagation of longitudinal waves; (c) Reflection of a Avave from the free surface of the medium. Case a. To determine the coefficient (P^ P^) or {P^.P^) from the curve corresponding to the parameter A (see Fig. 5) which characterizes the reflecting interface under consideration, we take the ordinate value (P^ P^Jj.gj, if ^m, p< V+i, p' or (P^, Pnr)rei ^^ ^mp ^^ < %_i, p where the abscissa is equal to the sine of the angle at which the wave under consideration strikes the given reflecting boundary. The value of the ordinate is again multiplied by the coefficient of reflection of plane Avaves when the incidence is vertical. The product gives the value of the co -efficient of reflection. ^m, p ^ ^\n + l, p' that is, if (P p \ ^ _ (p p ^ , QmVm,p — Qm + l'^'m + l, p . Qm '^m, p + Qm + l '^m+1, p 22 V. P. GORBATOVA and if v^ .<Vm-^ 'w, p ^ •^m—1, p' (P ,p \ — — (P , P ,\ , Qmf^m, p — Qm-lVm-l,p 9m'^m, p ''r Qm—l'^m-1, p Cose b. From the curve corresponding to the same parameters y and A as on the reflecting boundary (see Fig. 6) at a point with a reading along the /I- 0-4 C^ QJ 0-9 08 0-7 SinCref 0-2 0- 0-6 , ^ T-- > / / ^ A--Q& j J 06 Fig. 5. Curves for calculating the coefficient of reflection of longitudinal waves from the layer boundary with a higher velocity of propagation than in the layer in which the incident ray is travelling. abscissa equal to the sine of the angle of reflection, we take the ordinate (^m;Pm)rel ifVp>V + l, P' ^r (^,n' ^m')rel' if V p > V-1, p'/^^ "^"1^1- ply it by the coefficient of reflection of planar waves when the incidence i^ vertical. The product gives the coefficient of reflection, that is if '■ m. p -^ m + 1, p then REFRACTED AND REFLECTED LONGITUDINAL WAVES {Pm Pm) — ~ (°m "mjrel Qm l^m, p + Qm + 1 ^m + 1, p and if Vp<^m-i, p' then /p' p' \ _ IP' P' \ , Q'n'^m,P Q m-l'^m-1, p . ^m ^'m, p T^ Qm-1 '^m-l, p 23 SinfS'ret 0-2 0-4 06 1-0 ""^^^^ 1 \i ^ r ^ = 0-4 0-9 \A\\ ffl \v\\ CL \t\ \ S^ Q. 0-8 1 ^\ \>-=o; i;^=04 \ 0-7 \ ^ \ \ m w \ — 06 \ o! \ \ e a \ 1.^=0-5 05 \\ 1 w 1 r -- ^ =0-6 Fig. 6. Curves for calculating the coefficient of reflection of longitudinal waves from the layer boundary- with a lower velocity of propagation than in the layer in which the incident ray is travelling. Case c. The value of the coefficient of reflection (Pq'^o') is taken at the point X = sin ao along the ordinate from the curve (see Fig. 7) corresponding 24 V. P. GORBATOVA to the parameter Yq = ^0, ^0,p characterizing the zero layer of the medium under consideration. The values found for the coefficients of refraction and reflection and also for the multipher Cq are substituted in the appropriate form,ulas. The intensity obtained for the wave relates to a point at a distance r from the shot point along the profile calculated from the formula (5). Fig. 7. Curves for the coefficient of reflection (Pq' Pq') from a free surface. For two -layered media the formula (3) is converted into (6) and (7) for single reflected waves, and into (8) and (9) for waves reflected n times 7ref Cq (Pq Po) s"^ '^o cos Xq Jret = 47r (;io + 2^o) vo, p CoiPoPo) (6) COS'^ OCq 4.71 {Xq + 2/io) t'o, p 2^0 - hga REFRACTED AND REFLECTED LONGITUDINAL WAVES 25 7ref 7ref Co(nn)"(^o'^o')"-^ sinaocosoo 471; (Ao + 2^0) ^0, p r CoiPoPoriPo'Po')"-^ cos^ao 4jr {Xq + 2/i(,) t;o, p 2n/?o - /?sd (8) (9) Here /?q is the thickness of the upper layer. For muki-layered media, the muhipher sin ocq l/'[S hi tan oci COS^ (Xi tan 'sd- COS' aoJ from formula (3) can also be simplified and represented in the form smao when the angles at which the reflected wave under consideration strikes the intermediate interfaces are not too great. The permissible error -with such a substitution lies within the limits shown in Table 3, where Xj is the largest of the angles (x^ (where i = to ?i — 2) at which the ^vave reflected from the n — 1th and nth layers strikes the intermediate interfaces. Table 3 ay- 5° 10° 15° 17°30' 20° 22°30' 25° 30° Error, % 0.3 1.5 3.4 4.5 5.7 7.4 9 12.5 It can be seen from the table that if ocj does not exceed 25° the permissible error mil not exceed 9%. This degree of error will occur if all the angles (X^{i = to n—2) are equal to 25°. If however some of them are smaller than 25° the error will be reduced. Here the true values of the multiplier ^ve are dealing with are lower than the approximate values. If, by analogy nil- 1-11 sin an cos ao . , . with a two-layered medmm, ^\e substitute the value tor tms multiplier, we obtain still better accuracy. If the angles of incidence onto the intermediate interfaces of multi-layered media are not too great, then the multiplier /^&^ sinoco tan oc; ~h. cos' OCi sd- tan (Xq cos^ (Xq 26 v. P. GORBATOVA in the formulas for calculating the intensity of reflected waves (simple and multiple) into (10) sin (Xq cos ocq multiple) can be replaced by . Formula (3) is then converted Co"n (p,p,>i) {Pi+.Pi) (P„-iP„-i) ■^"^ ^ "^"" 47r(Ao+2^o)^o,p "" (W) sincco cos Kg r If we apply eqn. (4) to eqn. (3) and go over to a single independent variable sinag, and then proceed to the limit sin (Zq-^O we shall obtain formula (11) for determining the intensity of waves which reflected once above the shot point when reception is vertical: 2"n'[(P,-P,+i) {Pi+lPi)]o {Pn-lPn-l)v _ i=0 .^^, Jrei-v— ^^3[ » (11) Where [(P^P^-^j) (P^j^^P^]^ is the coefficient of refraction of planar waves when the incidence of the wave is vertical; that is mPi^^){Pi^,Pi)]v- ^^'''''' QiVi,P+Qi+iVi+i,P 2^1+1 ^f+i,P Qi%p+ Qi+i'Vi+i,p (P^_j^P^_j)y is the coefficient of reflection of planar waves when the incidence is vertical on to the reflecting boundary under consideration of the n — 1th and nth layers equal to the following expression / p p >. Qn-lVn-l, p — Qn't^n, p . /-i o\ l^«-l^n-lji; = -; TZrZ, » ^^^' Qn-l^n-l,p'T- fJn^n,p h- is the distance traversed by the wave in the i-th. layer; v^^ is the group velocity of longitudinal waves in the i-th layer. When we determined the intensities of multiple waves above the shot point, formula (11) is transformed in the same way as formula (3). All the coefficients of reflection and refraction of the multiple wave under consider- ation are introduced into the numerator, each of these coefficients being REFRACTED AND REFLECTED LONGITUDINAL WAVES 27 equal to the coefficient of reflection or refraction of plane waves with vertical incidence of the wave on a given interface, and being determined from formulas (12) and (13). The distance h^ travelled by the multiple wave under consideration in the ith layer is introduced into the denominator. The formulas thus obtained add to the knowledge we already have from the theory of plane waves, about the intensitites of simple and multiple reflected waves over a shot point, the possibiUty of taking into account the weakening of reflected waves due to the divergence of a spherical wave. ACCURACY OF THE PROPOSED METHOD OF CALCULATION The method proposed is approximate. We shall now evaluate this method by comparing it with accurate solutions, and with the values obtained when the Leningrad tables and methods were used. The basic formulas are merely another way of writing out the expressions for intensity which were given in the papers (^'^) ; no new errors are therefore introduced. The values obtained from the tables for the multipliers Cq, r^P{p), {Pq Pq) have been plotted in figures 2, 4 and 7. We may regard their graphical values as being determined with a sufficient degree of accuracy. The coefficients of refraction and the coefficients of reflection on the other hand have been found approximately by means of the graphs shown. Let us now evaluate this approximation. The possible error in determining each of the multipliers (Pj-P^-^j) (Pj-^jP^) for interfaces characterized by the quantity lying within the limits 0.7 <a < 0.9 does not exceed 2%; if this quantity lies between the limits 0.5 <a < 0.7 the error is 5% and finally if it Ues within the limits 0.3 < a < 0.5 the error is 10%. This estimate has been made for refracting boundaries with the parameters indicated above. Only for boundaries with y = 0.4 and A = 0.6 are the errors in determining the refraction coefficients slightly higher. The refraction coeffi- cients for such boundaries can nevertheless be calculated by the method referred to, the errors being reduced as the angle of incidence becomes smaller or approximates to critical. For boundaries characterized by the parameters y = A = 0.6, a = 0.7 to 1.0, the error will never exceed 3%. We shall estimate the error entailed in determining the coefficient of reflection separately for the following cases. Reflection from a layer with high acoustic rigidity — When the sine of the angles of incidence on the reflecting boundary is equal to 0.75a (a being the parameter of the reflecting boundary under consideration), the error in determining the coefficient of reflection does not exceed 10%. The degree of accuracy rises rapidly as the angle of incidence becomes smaller. 28 V. P. GORBATOVA The approximation method indicated can be used for reflecting bomidaries mth the parameters shown in Table 1. Reflection from a layer with lower acoustic rigidity — For reflecting boundaries with parameters as shown in Table 2, the magnitude of error in determining the coefficient of reflection is given in Table 4, from which it can be seen that when the angles at which the wave under consideration strikes the reflecting boundary are not too great it is permissible to use our approximate method of calculation. Table 4 Values of parameters Error in determination of coefficient of reflection, % y A sin ocret ^ 0.6 sin «ref < 0.5 sin aref < 0.45 0.6 0.6 5 3 1 0.5 0.5 6 2 1 0.4 0.4 9 5 4 0.5 0.6 33 13 10 0.4 0.5 8 6 5 0.3 0.4 2 2 1.5 ANALYSIS OF THE INTENSITIES OF HEAD (REFRACTED) WAVES A Two -Layered Medium The intensity of head waves in two -layered media is determined from the formula c,rpp{p) 7head = ^^ ^-^ (14) 4:/r(Ao + 2/.o)|//-(r-roF^ The value of the multiplier Cq (see Fig. 2) depends shghtly on the values of the parameter /q and is near to 2 when sin oCq < 0.9. The behaviour of PPP (p) the coefficient of the head wave formation (see Fig. 4) — -will therefore illustrate the dependence of the intensity of head waves in two-layered media on the parameters of the interface at distances r, which are sufficiently far from the point of emergence of the head wave, when we can set /."(r-g'/^^A It can be seen from Fig. 4, moreover, that when y, A and a are fixed, the intensity of the head wave increases in inverse proportion to the difference in the longitudinal velocities at the refracting boundary. At distances r'$>rQ the intensity of a refracted wave increases with reduced sharpness of the refracting boundary just as the coefficient F^^ (p) grows. At distances r comparable wdth Tq, when it is not possible to set yr (/" — /"o) = r^r REFRACTED AND REFLECTED LONGITUDINAL WAVES 29 the increase in intensity of the refracted waves with decrease in the drop in velocities of propagation of the longitudinal waves at the refracting boundary occurs still more rapidly than the growth of the coefficient F^^ {p). Even if the comparison is made at equal distances from the point of emergence, then when r— Tq > O.SAq refracted waves with higher amplitude will correspond to boundaries with less difference in the velocities of propagation, although for these the points of comparison are at a greater distance from the shot point. For boundaries where the difference in propagation velocities is slight, the intensity of a head wave at some distance from its point of origin will be greater than for a sharp interface at the same distance from its point of origin. The curves shown in Fig. 4 show how the intensity of head waves depends on the values of the parameters y, A and a at the refracting boundary. We can however choose these parameters to be such that when the drop in the propagation velocities of longitudinal waves is slight, the head waves -will have a lower intensity than in a case of greater difference in the velocities of propagation at the interface (but with other parameters y and A). It can be seen that the intensity of the head waves increases in direct proportion to A and in inverse proportion to y. The density ratio at the interface also affects the intensity of the head waves. For boundaries A = 0.4 and A = 0.5 the head wave intensity increases as the difference in densities decreases, while for boundaries where A — 0.6 and a ^ 0.35 it decreases. The damping of the head waves with distance is determined by the multi- pHer r~'''(r — Tq)"'^'. The influence of the depth of the refracting boundary on the intensity of the head waves has a substantial effect only at distances r comparable with Tq. If the comparison is made for several two-layered media which differ from one another only by the parameter h^ {r being fixed and the same for all the media), we arrive at what seems to be a contradictory conclusion: namely that the greater the bedding depth of the interface the greater the intensity of the primary waves. If however we compare the intensity of the head waves at uniform distances from their points of origin, everything becomes clear. We find that to get head waves of the same intensity at the same distance from their respective points of origin in the case of much deeper interfaces, a much more violent cause of excitation is required. At distances /" ^ /"o the bedding depth of the interface does not influence the intensity of a head wave. The head waves will dampen with distance as r~^. Multi-layered Media — Of the many problems connected with the origin and propagation of head waves in multi -layered media, we shall here treat only the following: 30 V. P. G6RBATOVA (a) the influence of the velocities at which transverse waves are propagated on the intensity of longitudinal primary waves; • (b) the effect of adding an ith. layer, and changing the longitudinal velocity in it, on the intensity of the head waves excited in layers of greater depth; (c) the effect of a sharp principal refracting boundary on the intensity of a head wave excited in it; (d) the damping of the head waves with distance and the influence of the bedding depth of the main refracting boundary on the intensity of these waves. We shall examine all these questions in order. (a) Formula (2) is used to determine the intensity of the head waves excited in multi -layer media. As has been shown above the multipHer Co as well as the coefficients of refraction (P,Pj.j.j) (P,-^jPj) at all the intermediate interfaces depend only slightly on the parameters y and A, that is on the values of the transverse velocities in the covering layer. The intensity of the the head waves is consequently also only slightly dependent on them. It follows that ignorance or inaccurate knowledge of the transverse velocity values in the covering layer is not an obstacle in the way of calculating the intensities of the longitudinal head waves excited in deep boundaries. In drawing this conclusion we are assuming that the covering layer is charac- terized by the parameters indicated in the introduction. The transverse velocity values in layers directly adjacent to the principal refracting boundary, on the other hand, may well exert a considerable influence on the hitensity of a head wave excited at this interface. The curves in Fig. 4 show at a glance the possible variation in the theoretical intensity of a head wave according to the assumptions we make about the ratio between the transverse and the longitudinal velocities in the adjacent layers, on the boundary of which the primary wave under consideration is formed. It can be seen from the curves that the wave intensity increases in inverse proportion to the parameter A and in direct proportion to the parameter y, which characterize the adjacent layers on the boundary of which the head wave forms. This means that in order to calculate the intensity of the head waves, we must have information about the densities and the values of transverse and longitudinal velocity on both sides of the interface where the head wave is excited, and that we must also know the values of velocity for the longitudinal waves and the densities throughout the covering layer. (b) We shall now see how the intensity of the refracted waves varies if the longitudinal valocity value changes in one of the upper layers, other than the topmost, which is not directly adjacent to the interface at which REFRACTED AND REFLECTED LONGITUDINAL WAVES 31 the head wave is excited. All the other parameters of the medium %vill be regarded as unchanged. We shall find, for example, the variation in the intensity of a head wave excited in the nth. horizon of the medium (see Fig. 1) with v^ = 6000 m/s. if the longitudinal velocity in the ith layer adapts from 1540 to 5150 m/s. When v^^ ^ in formula (2) changes, the quantity Tq (the abscissa of the point of emergence of the head wave under consideration) and the value of the coefficient of refraction at the upper and lower boundaries (that is the ex- pressions (P,._iP,) (P,.P,._i) and (PiP,.+i)(P.+jP.)) of the ith layer will also change. We shall regard the intensities as being determined at such distances r from Tq that we can assume l//-(r— /(,)''' ?« r^. Then, when v^ varies and all the other parameters of the structure of the medium remain unchanged, the intensity of the head waves will vary proportionately to the product of the coefficients of refraction [(P^.^P,-) (P.P._j)] [(P,Pi+i)(P,+iPi)], allowance having been made for the refraction of the wave at the boundaries of the ith layer. The curves in Fig. 8a therefore, where the value is plotted against ''^ illustrate the variations in the intensity of the head V n,p -'i-f 1, p waves with gro'v\1;h of v^ . The curve determined by the parameter b = Vi = 0.35 characterizes the change in intensity of the head wave excited in the ith layer of the medium with v^ = 6000 m/s and f,_j — 1800 m/s when Vj^ ^ varies within the range 1540 — 5150 m/s. The curves for cases where the values of the parameter b = — — — show how the intensity of the head waves excited in the nth layers would vary with variation in v^ , if the medium under consideration were characterized by some other difference in the velocities v,_j and v^+i^p smd by the same value '~^'P = 0.3. If a medium with a different ratio of ^/-i^p/i'n^p were considered then differ- ent curves would be obtained. Fig. 86 shows the l(P,_iPj)(PjP,_i)] [(^,-P,+i) (Pj+iPj)] curves for the same parameters b = f,_jp/z;,.,.j p, but for a ratio «^/-i,p/Vp = 0-175. For a medium which differs from the one discussed above only in having a different value for the velocity y,_j^p (^^i_i^p= 1500 m/s), the intensity of 32 V. P. GORBATOVA the head waves formed in the n-th layer with a travel velocity of 6000 m/s would vary with change in the velocity of the ith layer within the range 1500- 3500 m/s, as is shown by the curve in Fig. 8b which relates to the parameter b = t',_j pli^i+i p = 0.21. The behaviour of all the curves in Fig. 8a and b 1-00 01 0-2 0-3 0-4 0-5 0-6 0-7 08 09 10 1-00 or + or X 0-75 or 01' Ql" X 0-50 q:" T 1 fs} / \\ / / / /i V^ \ V W I 1 V IP I 1 1 \ 1 1 \ \ b = l/0-45 [\ b=0-35 1 ^""^ \"~"Vv^=0 45 1 1 / \ \ b = 21 1 1 \ b = 0-63 1 1 \ b = l-0 (b) 1 1 1 , ! , O'l 0-2 0-3 0-4 0-5 0-6 07 0-8 0-9 10 Vn,p Fig. 8. Curves for the variation in intensity of primary waves if the longitudinal velocity changes in one of the upper layers. reflects the change in intensity of the heat waves from the low-lying horizons Avhen the velocity in the ith layer changes. We cannot, however, compare the ordinates of the various curves, since they give values for the coejfficients of reflection at the boundaries of the ith layer which are proportional but not equal to the intensity values when Vj „ changes. The coefficient of REFRACTED AND REFLECTED LONGITUDINAL WAVES 33 proportionality is constant for each curve but is different for different curves. From an analysis of the curves shown we may conclude that if we take (/•— Tq')''" = (r— Tq)''' where Tq is the abscissa of the point of origin of the head wave under consideration when there is no ith layer, and Tq is the same when there is such a layer, we shall obtain the following. 1. The addition of an ith layer characterized by a velocity v^^ ^ lying within the interval ^j_i n < ^i, n < ^i+i. p' '^l^ cause the intensity of the head waves from the underlying horizons to increase. 2. As v^ increases from v^_^ to v^j^^ the intensity of the head waves at first increases and then diminishes. When v^^ ^ = Vi_^^ ^ or v^j^^^ ^ the intensity of the head waves is of course the same in each case and the same as when there is no ith layer. The bigger the difference between the velocities 'v^_^^ ^ and t'i+i n that is the greater the interval t;j_j ^ < f,- ^ < f,.|.i p given a constant t;,_j Jv^ the greater will be the variations in the intensity values of the head waves excited in the low-lying layers. With a constant value of 6 = r,_j Jvi+i^ „ the intensity of the head waves when v,-^ p changes may show greater variations the higher the value of the parameter ^,_i, pl^n, p* The intensity reaches its peak value at values of v- near to the mean value of the interval Vi_^ ^^i, p ^j+i, p- 3. The addition of an ith layer with a longitudinal wave travel velocity lower than the lower of the velocities v-_^^ p, z;,.,.^^ p, or greater than the greater of the velocities v^_j^ , i;,-+j , leads to a reduction in the intensity of the head waves formed in the low-lying layers. The reason for this is that we now have an even sharper interface than where there was no ith layer. (c) We shall now assume that the value of the longitudinal boundary velocity of the head wave changes, and that the density and velocity throughout the whole covering layer, as well as the density ratio and the density between the transverse and longitudinal velocities on both sides of the interface where the wave is excited remain constant. Under these conditions the angles at which the head wave strikes the intermediate interfaces will vary and this will cause variations in the values of the coefficients of refraction and in the multiplier Cq. Furthermore there mil be a change in the value of the discontinuity in the longitudinal velocities and the principal refracting boundary and on F^^ (p), the coefficient of head wave formation, which depends on it. Lastly the abscissa of the point of emergence of the head wave will also vary. We shall first assume that the intensity of the head waves is being compared at distances from Tq such that we can assume yr (r — Tq)'- = r^. Applied geophysics 3 34 V. P. GORBATOVA The head waves emerge towards points on the ground surface at angles such that shi ocq < 0.9 for which, as can he seen from Fig. 2, we can take Cq !^ 2. Then the variation in the head wave intensity value with variation of v^ will depend solely on the behaviour of the multiphers /"^^(p) and (PiP'i+diPi+iPi)- As v^ diminishes, the discontinuity in the longitudinal velocities at the principal refracting boundary will diminish and the coefficient r^^{p) and the angles of incidence on to the intermediate interfaces mil increase, while all the coefficients of refraction at these interfaces will decrease. We shall now see how the intensity of head waves PqP-J^^P^Pq in a three -layer medium changes as v^ p changes. As can be seen from Fig. 4, the coefficient r^^(p) for boundaries of layers characterized by the parameters y = A = 0.6, a = 1.0 increases least rapidly as the sharpness of the refracting boundary diminishes. Fig. 9 shows curves for the variation in head wave intensity in three- layered media with change in the boundary velocity t^g p ^^ the interface of layers characterized by the parameters y = A = 0.6, a = 1.0. The intermediate interfaces examined have an upper layer to lower layer longitu- dinal wave velocity ratio within the limits 0.3 < agi = — — < 1/0.3 and a density ratio equal to unity. The parameter of each of these curves is the ratio t^o,pK,p = Sr As can be seen from the figure, the intensity of the head wave PqP^PJP^Pq increases as the difference between the velocities v^ „ and v, „ decreases. For three -layered media for which ^o p -^ '^i p "^ ^2 p *^^^ remains true so long as Vq does not approach v^ 'Xh.aX is, so long as v^ <,0.9v^p. With further diminution of v^p the intensity of head waves diminishes despite the concomitant diminution in the values of v^ and v^ . If the adjacent layers are characterized by other parameters, y. A, a, the intensity of a refracted wave formed on their boundary will increase with diminution of the discontinuity in the values of the longitudinal velocities at this boundary more rapidly than is shown in Fig. 9. A variation in a at the intermediate interface does not entail changes in the course of the curves shown but only alters their vertical scale. Accordingly, we may conclude that in three-layered media with the parameters indicated at the start the intensity of the head waves PqPxP^PxPq will increase as the difference in the values of the longitudinal velocities v^p and v^ decreases. Indeed the intensity of such waves will decrease only for media where Vq v-^p when the value oi v^^p approaches VQp, or in other words when the difference between v^ „ and v^ „ decreases. REFRACTED AND REFLECTED LONGITUDINAL WAVES 35 For four -layered media with layer velocities which increase with depth, that is Vq p < ^-'i p < ^^2 p "< %,p' ^^'^ ^^^ ^^^® show that as the difference in the values of V2 „ and v^ decreases the intensity of head waves P0P1P2P3P2P1P0 increases. If in a four -layered medium one or two of the layers have a high speed, that is if the medium is characterized by one of the ratios: ^0, p < ^1, p > ^2, p < ^3, p' ^0, p > ^1, p > ^2, p < ^3, p' ^0,p>''l,p<i2,p<%,p Fig. 9. The increase in intensity of primary waves /head ^ three-layered media as the sharpness of the interface of layers diminishes at which the primary wave originates. %P (but the highest velocity in the overburden does not exceed 0.9 v^^ p) we can then also show as the ratio v^ „/% p increases within the limits 0.3—0.9 the intensity of the head waves P^P-J^^^zP^PiPq^ although it increases, does not do so by a factor of more than 0.7. The greater the number of interfaces with v^ p < ^j+i p which separate the intermediate boundary under consideration from the refracting boundary 36 V. P. GORBATOVA at which the head wave is produced, the more uniformly will it weaken waves from this boundary with differences Vf^. We can therefore assume that in multi-layered media with layered velocities which increase with depth the intensity of the head waves produced at interfaces where the difference in velocity is slight mil be fully comparable Avith the wave intensity which would be found with these velocities and densities of the covering layer, but with a much bigger discontinuity in the velocities of longitudinal Avaves at the interface where the head wave is formed. We consider the para- meters y, A, a at this boundary to be unchanged. If the overburden contains one layer with a high velocity which is less than or equal to 0.9 t;^ then even under the most favourable conditions of the intensity of the primary waves from the underlying layers will diminish by a factor of not inore than 0.45 as the jump in longitudinal velocities in these horizons is reduced (within the limits 0.3—0.9) and provided y, A and a remain unchanged. The conclusion we have reached here has been made on the assumptions indicated at the beginning of this section (that is we are assuming that the intensities of the primary waves are being compared at distances from r^^ such that we can regard K '■('' — /"o)^^* ^ ^^ and that the head waves are emerging towards points on the ground surface at angles such that sincco<0.9). But as we can show, these assumptions are not so important. If for example we make the comparison at a distance r compai'able with Iq then the intensity of the head waves will increase still more rapidly as the discontinuity in longitudinal velocities at the reflecting boundary diminishes than at fairly great distances from Tq. When there are weathered zones present the angles at which the Avaves emerge at the ground surface are small and, of course, satisfy the inequality sincfQ < 0.9. If the longitudinal velocity alone varies in the nth. layer on the boundary of which the wave under consideration is formed, while the transverse velocity remains constant, then, as can be seen from Fig. 4, the diminution in the discontinuity in longitudinal velocities at this boundary will be accompanied by a still more rapid increase in the intensity of the head waves formed at it than in the case we considered above, where we assumed that the ratio v^^ Jv^^ ^^ remained constant. The principal conclusion reached here still holds good. Intensive head waves can be observed coming from refracting boundaries where the difference in velocities is slight. Head waves formed at a weak interface lying above a boundary where the discontinuity in velocities is great can of course be a good deal more intense than waves formed at a sharper boundary lying underneath. It is also possible for refracted waves formed at deeper boundaries, Avhere the difference in velocities is slight, to be more REFRACTED AND REFLECTED LONGITUDINAL WAVES 37 intense than refracted waves formed at higher boundaries where there is a bigger discontinuity in the velocity values. It is a widely known fact in practical seismic prospecting that more intensive head waves can be obtained from an interface where the difference in velocities is sHght ^^K It has been noted in many works that the head waves excited at boundaries where there is a considerable discontinuity in the velocities are less intense than waves formed at boundaries which he higher and in which the difference in velocities is shght (^'*). Our inference that intense head waves, excited at weak interfaces do exist therefore agrees with experimental data. (d) According to the theory head waves suffer damping with distance as r~^- (/■— /■q)"'^" and at greater distances r, as r~^. But at distances r — 10 Tq the replacement of r~''»(r—ro)~''' by /-"^ leads to an error of 15%. At such distances the intensity of the head waves diminishes more quickly than at r~^. The overall depth of the refracting interface and the ratio between the thicknesses of the individual layers influence the intensity of the refracted waves so long as it is not possible to assume r^' {f^^oY^' ~ '■^• Any variation in the geometrical structure of the overburden which leads to an increase in Tq (an increase in the overall depth or thickness of the high velocity layers) also leads to an increase in the intensity of the heads waves at the fixed distance r calculated from the shot point. As r increases this increase falls off, and when r ^ Tq it becomes neghgible. The Form of Refracted Head Waves hi Multi-layered Media All longitudinal head waves excited in multi -layered horizontal layered media as a result of similar shocks have a similar form of trace and phase. The displacements of points on the ground surface repeat the form of the given pulse. Head waves along the profile do not alter the form of the trace. ANALYSIS OF THE INTENSITY OF REFLECTED WAVES A Tivo-Layered A Tedium (a) Single Reflected JFaves — The. intensity of single reflected waves above the shot point is equal to the quantity 2 QoVo,p-QiVi,p 1 47r(Ao + 2//o) ^o^p QoVQ^p+QiVi^p 2hQ-hsd* The index sd signifies the shot depth. 38 V. P. GORBATOVA This increases in proportion to the difference in the acoustic rigidities at the interface, in inverse proportion to the bedding depth h^ and in direct proportion to h^^. The intensity of reflected (longitudinal) waves above the shot point does not depend on the value of the transverse velocities. For media where the difference in acoustic rigidities is such that they are uniform but Vq p<^Vj^ in one while in the other Vq ';> v^ when Vq and ^q coincide in both me- dia, the intensity of the reflected waves above the shot point is equal. 10 ■5 0-5 — «^^ ..,^__^ '^v'^^>:vJ'""-^'>^--"--~'"~'~~>"0'^'^ '°'^ ^^^ ^ ^-A =0-5 ^--A =0-6 0-2 4 0-6 0- 2r/(2ho 3 !-0 1- -^3) 2 1-4 Fig. 10. Curves for the damping of single reflected waves in two-layered media. It diminishes with distance from the observation point along the free sur- face of the medium and is determined by formulas (6) and (7). The rate at which the intensity diminishes ceases to be uniform for media where ^0, p ^ ■^i n ^^^ ^0 p ^ ^1 p ^^^ depends on the transverse velocity values. For media where t;^ < z;.,^ it depends on the ratio between the longitudinal and transverse velocities in a half-space and for media where Vq > v^^ it depends on this value both in a half-space and in a layer. Figure 10 shows curves for the damping of reflected wave intensity ■\\dth distance for media where Vq <i v^ (continuous lines) and for media where Vq '> v^ (dotted lines); it can be seen that if A, the reflected wave will dampen more quickly the greater the parameter. Conversely if Vq^ > ■yj , the intensity will dampen more rapidly the greater the pa- rameter A and the smaller the parameter 7 at a fixed A. Given the same A in media where v^^ > v^^ the reflected waves will dampen with dis- REFRACTED AND REFLECTED LONGITUDINAL WAVES 39 tance slightly more quickly than in media where t'o, p< ^i, p- ^^ ^^^ bedding depth Hq of the boundary increases the waves reflected from it will dampen more slowly with increase of r. At distances r = JiqI^, the intensity of reflected waves when Hq ^ h^^ is not less than 0.8 of the intensity above the shot point when hf^f^hg^ is not less than 0.5. (b) Doubly Reflected Waves — The intensity of doubly reflected waves above the shot point is equal to 2 (oo % p - ^1 ^1, p) ^ 1 47r (Aq + 2//o) Vq^ p {qq vq, p + Qi v^^ p) ^ ^h^ -hgd This does not depend on the value of the transverse -svave velocity and takes on higher values the greater the discontinuity in the acoustic rigidities at the interface, the shallower the depth of the bed ^o ^^^ the greater h^^. The ratio between the intensities of doubly and singly reflected waves above a shot point is equal to y double _ go^'0,p~gl^l, p ^hp — hsd ^ y single ?0 ^'0, p + ^1 ^1, p ^ho-hsd when the shot depth varies from hg^<^hQ to h^^p=ihQ this hes within the limits ^0 ^0, p — Q\ ^1, p ! ydouble }^ Qo^'0,p+Qi^l,p I ysingle 2 ?0^0,p-^l^^l, p Qo^o.p + Qi^hP The doubly reflected wave diminishes more slowly than the single re- flected wave -with growth of ;■. The rate at which it diminishes also depends on Avhether the two -layered medium is characterized by the inequahty Vq < Vj^ or Vq p^- v^^ and on the values of the transverse velocities of the waves. The effect of a change in the transverse velocity on the rate at which the intensity of the doubly reflected waves is damped can be the same as in the case of singly reflected waves. Figure 11 shows curves for the damping of intensity of doubly reflected waves with distance for media where Vq^ p < v^^p (continuous Hues) deter- mined by the parameter A = ^i, s 1% p i^^ 'which the rate of damping also mainly depends) and for media where v^^ p > ^i, p I dotted lines characterized 11 ^1 s 1 ^ ^0, by the parameters y = — '- — and Zi = ^l,P ^0, p; Since doubly reflected waves diminish ^nth. distance more slowly than single echoes, the question arises whether they become more intense, at some distance from the shot point, than such single echoes. We shall not 40 V. P. GORBATOVA discuss all possible two -layered media in this paper but only those which have parameters with the values shown in Tables 1 and 2. We are taking it that in the case of media where Vq <Cv^ the angles at which the wave strikes the reflecting boundary satisfy the ratio sin ocqK 0.75 -AE , and for 'hP media where Vq^ > z;j the ratio sin ocq < 0.6. Given these limitations we see by comparing the curves in Figs. 10 and 11 as well as the intensity values for waves reflected above the shot point, that ^ 0-5 F iG. 11. Curves for the damping of doubly reflected waves in two-layered media. double echoes are always less intense than single ones. The ratio of their intensities is found within the following limits: ^0 ^0, p Qi ^1, p ^ 7double ^ y single go, ^o,p-gi^i,p Qo ^0, p + Qi Vi, p (17) where < h^^ < Aq. The ratio yjjQu]3ie//gjjjgig for media where z;^^ > t;j p will be slightly greater than for media where ^o p '^ ^i p' provided A is the same. This ratio increases as A increases and as y decreases but A does not alter. Multi-layered Media Of the many problems connected with the intensity of reflected waves excited in multi -layered media, we shall here consider only the following: (a) the influence of the values of transverse velocity on the intensity of reflected waves: REFRACTED AND REFLECTED LONGITUDINAL WAVES 41 (b) the influence of the overall depth of the reflecting boundary and the ratio between the thicknesses of the different layers on the intensity of re- flected waves over a shot point; (c) comparison of the intensities of single and double echoes above a shot point. We shall take these problems in the order given. (a) The intensity of reflected waves in multi-layered media is determined from formula (3) and above a shot point from formula (11). Formula (11) —and so also the intensity of reflected waves above a shot point — do not depend on the transverse velocities in the media under consideration. In formula (3) only one coefficient of reflection depends on the transverse velocity values. When the reflection is from an interface with high acoustic rigidity, it depends mainly on the ratio of transverse to longitudinal velocities in the layer from which the reflection takes place; when the reflection is from an interface with lower acoustic rigidity the coefficient of reflection depends on the transverse velocity values on both sides of the reflecting boundary. The curves (see Figs. 5, 6 and 7) show the dependence of the coefficient of reflection (and therefore also the intensity of the echoes) on the trans- verse velocity value at the reflecting boundary. The echoes are more in- tense the smaller the parameter A at the reflecting boundaries, while for waves reflected from media with low travel velocity the intensity of the waves increases (when A is fixed) with increase of v. The influence of the transverse velocities on the intensity of the echoes is greater for multiple echoes. It is becoming clear that to determine the intensities of echoes in multi- layered media we need information only about the values of the transverse velocities at the actual reflecting boundary. If the refle tion occurs from a high velocity layer we must take the value of transverse velocity in the layer under the reflecting horizon; but if reflection takes place from a wea- thered layer we need to know the transverse velocity on both sides of the reflecting boundary. What has been said holds true for media with the above-mentioned parameters when the angles at which the waves strike the reflecting boundaries are lying within the limits indicated. (b) The intensity of reflected waves above a shot point is determined from formula (11). From this formula it is clear that if the depth of the reflecting boundary increases or diminishes n times the thickness ratio for the different layers, and h^^ remaining as before, then the intensity of the echoes above the shot point -will diminish or increase n times. If the velo- city and density-structure of the medium are unchanged and the depth of the reflecting boundary is also unchanged, the intensity of the waves 42 V. P. GORBATOVA reflected from this boundary will be greater in inverse proportion to the thickness of the layers which have the higher velocities and in direct pro- portion to the thickness of the layers which have the lower velocities. The more the velocity of the ith. layer differs from that in the layer where the shot is fired, the greater ^vi\\ be the influence which a change in the thiclaiess of this layer exerts on the intensity of waves reflected from interfaces lying at greater depths. (c) From formula (11) we can easily obtain expressions for the intensity ratio of multiple echoes above a shot point, when the reflection occurs at the boundary of the n — lth. and nth layers and the free surface. n-l 2h; — - — , we shall have the following formula 1=0 for the intensity ratio of waves reflected k and k — 1 times: ^ = -J^ ^n m Pi + l) iPi+1 Pd]v {Pn-lPn-l)v, (18) Jk ^ — -L 1=0 where [{P-P-_^_^) (P^-^^ Pj)]^ and (P„_iP„_i)„ are determined from formulas (12) and (13), and k is the number of times the wave is reflected. As can be seen from (18) the greatest difference in the intensities above the shot point is found to occur between single and double echoes. The greater the number of reflections the smaller the difference between the intensities of waves reflected a neighbouring number of times. For such multiple echoes where the second reflection has taken place from the free surface, we can easily obtain from (18) the following expres- sion: jl {k-l)ik+l) ^k-ijk+l k' (19) where j\ is the intensity above the shot point of a wave reflected k times; //f-i is the intensity above the shot point for a wave reflected k~^ times; Jh^-^ is the intensity above the shot point for a wave reflected k -{- 1 times. For example, for the intensity of a a single ^vaye j\, a double echo yg ^^^ a triple echo 73 we have •'2 ^ r-- = 0.75 7173 The ratio (19) does not depend only on the values of the transverse velo- cities but also on the longitudinal velocities. This is true for any type of effect ^vhen any component of displacerrsgnt is being recorded above the point of disturbance. REFRACTED AND REFLECTED LONGITUDINAL WAVES 43 The intensity of possible interbed echoes above a shot point can also be readily determined by means of formula (11). Comparison of their intensi- ties with the intensity of single echoes from low-lying boundaries can help in recognizing the nature of any particular wave recorded on the seismogram. THE FORM OF SINGLE AND MULTIPLE ECHOES IN MULTI-LAYERED MEDIA We can say the followng about the form of seismic traces for echoes ■excited as a result of shots in multi-layered media up to the points of emer- gence of the corresponding head waves: all reflected waves, whether simple or multiple, have the same trace form when the influences are the same. Their phases cannot be opposite. Displacements of points on the ground surface repeat the form of the given pulse. As to the onset of echoes when the angles at which they strike the reflecting boundaries are small, all the facts that are well known from plane wave theory can be repeated. If the wave is reflected from a layer which has a lower speed than the layer from which the wave has arrived, and if the discontinuity in the transverse velocities at the reflecting layer is greater than that in the longitudinal velo- cities (that is y < A), then when the angle at which the wave strikes such a reflecting boundary increases, the intensity of the echo from it can pass through zero and the wave can change the sign of its onset. For waves reflected from boundaries (with parameters from Table 2) this can occur when the -angles at which the wave strikes the reflecting boundary have a sine greater than 0.6 (not discussed in the present paper). If the reflected wave under consideration nowhere undergoes reflection from such boundaries then, as the angles of incidence increase to critical, it does not change its sign of onset and has the same form as it has above the shot point. What we have said does not refer to the vicinity in •which the corresponding head waves ■emerge. REFERENCES 1. A. M. Epinat "iTLVA, Experimental data on refracted waves in media with poor speed differentiation. Izv. Akad. Nauk SSSR, ser. geofiz. No. 2 (1955). 2. G. I. Petrashf.n', Problems in Dynamic Theory of Seismic Wave Propagation. Coll. 1. Gostoptekhizdat, 1957. 3. G. I. Petrashen', Propagation of elastic waves in layered isotropic media divided by parallel planes. Sci. Rec. Zhdanov State Univ., Leningrad, No. 162, Pt. 26, 1953. 4. D. B. Tal'-Virskii, Tectonics of the Tobolsk Zone from Seismic Prospecting and Deep-drilling Data. Thesis. VNII Geofiz. Foundation. Chapter 2 METHOD AND TECHNIQUES OF USING STEREOGRAPHIC PROJECTIONS FOR SOLVING SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS E. I. Gal'perin, G. A. Krasil'shchikova, V. I. Mironova and A. V. Frolova In seismic prospecting, as in all geophysical methods of prospecting, the solution of linear problems is of great importance for the purpose of analysing data and is an essential step in the working out of methods for interpreting field observations. The solution of ray problems makes it possible to study the shape of surface hodographs for media of various structures, to compare the surface hodographs of different types of waves and discover the possible regions in which they can interfere, to check on the correctness of the con- structions made and estimate the degree of error introduced, to confirm approximate methods of interpretation, to verify the permissibility of any simplified assumptions which have been used in interpreting the seismic data and so forth. Yet it is precisely in seismic prospecting that the solution of linear problems has received comparatively little attention. Until recently opportunities for solving linear problems have been confined to cases where the structure of the medium is very simple, and in the main to two dimen- sional problems, although all the problems in seismic prospecting are by their very nature spatial ones. The reason for this is largely the difficulty of solving spatial problems in geometrical seismics. In those instances when spatial problems have been examined, the examination has been confined as a rule to one interface. The methods used have been both graphical and analytical (I'^'S). An earlier paper <^) describes a method for solving linear spatial problems in geometric seismics for multi-layered media with interfaces of arbitrary shape. The method is based on using stereographic projections which make it possible to determine the direction of rays in space after they have struck the interface. The method is applicable principally to multi-layered media with a cons- tant velocity in each layer, where there is any number of interfaces of arbi- trary shape, and can be used equally well for calculating the seismic fields SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 45 of reflected and of refracted waves. As the graphical constructions are cum- bersome, however, the practical use of the method is confined to three- or four -layered media. It should be stressed that further development both of the graphical and of the analytical methods of solving linear problems in geometrical seisinics is essential. Only by combining both methods of construction is it possible, where necessary, to increase the degree of accuracy by very simple methods. In seismology a Wulfif net is used in processing earthquake records. The construction of spatial fields by means of a Wulff net as apphed to problems in seismology has been described in a thesis by N. Bessonova. The present paper is devoted to a detailed exposition of problems in methodology and technique for the solution of spatial problems in geo- metrical seismics by means of stereographic projections as applied to problems in seismic prospecting. The first part explains the principal properties of stereographic projections, without demonstration, and describes the tech- nique of working with a Wulfif net together with methods for solving problems in geometrical seismics by means of these grids. In the second part we des- cribe the technique for solving linear problems in geometric seismics for multi-layered media with interfaces of arbitrary shape. STEREOGRAPHIC PROJECTIONS Stereographic projections were used in astronomy as far back as over two thousand years ago to represent the surface of the heavenly vault on a plane. Later the method began to be used for the same purpose in map making. At the end of last century stereographic projections began to be used suc- cessfully for studying the angles between lines and planes in space. This use of stereographic projections is of major interest for geometric seismics since it can be used to solve problems connected with the propagation of seismic waves, which by their very nature are spatial. Here we shall not dwell on the theory of stereographic projection, which has been expounded in a number of works <'>, but shall merely describe their main properties which enable them to be used for studying on a drawing the mutual incli- nations of rays in space by first projecting them on to a sphere. 1. The entire upper hemisphere can be represented by a circle. 2. The angles between the rays of the great circles in the sphere are equal to the angles between the arcs of their projections. 3. The arcs of the circumferences of both small and great circles are re- presented in the projection by arcs of circles or in a particular case by straight 46 E. I. Gal'perin et al. lines (in general the latter can be considered as circles of infinitely great radius). Stereographic projection is equiangular projection, that is, the angle be- tween the projections of lines on the sphere is equal to the angle between the lines on the sphere themselves. This property of stereographic projectionsy which is also possessed by certain other projections, is a necessary and sufficient condition for a given projection to be conformal, meaning that figures on the sphere which have infinitely small dimensions in all direc- tions are projected as infinitely similar small figures. A further characteristic of stereographic projections is that the projection of a circle is a circle. The Stereographic Net A projection onto the diametral plane of a sphere divided into degrees is called a stereographic net. Depending on the position of the projection plane a stereographic net can be polar (when the projection plane coincides with the equator, and the observation point with the nadir— the lower pole) or meridianal, when the observation point lies on the equator and the projec- tion plane is a meridian lying at 90° from the point of observation. Fig. 1. Construction of a meridianal stereographic net (after M. K. Razumovskii). For purposes of geometrical seismics the meridional net is the most inte- resting. Let us now look at this in detail. Fig. 1, wh^'ch we have taken from Reference C^^, shows a construction of a meridianal stereographic net. Here the plane of drawing coincides with the meridian ZEZ', and the plane of projection {n) coincides mth the meridian ZMZ' . The point of observation SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 47 lies on the equator EMS at the point S. The equator EMS and the meridian ZEZ' are represented by two mutually perpendicular diameters MTV and ZZ' of the projection circle, the projection of the meridian coinciding with the polar axis of the sphere ZZ'. The remaining meridians are represented by circles passing through the poles Z and Z' and intersecting the equator. Let us consider for example the meridian ZM^Z' for which the angle EOMi is oc°. The projection (wj) of the point at which the equator intersects the meridian (M^) will be distant from an amount Uin^ = K tan — , where R is the radius of the sphere. The parallel* with the co-ordinate q = 15° = ZK lies at this number of degrees from the pole along all the meridians ; we can therefore plot Zat 15° from the pole round the circle of the projections on both sides, and we can also plot the segment ZK =15° along the straight line OZ on the stereo- graphic scale from the point Z. We thus obtain three points belonging to the parallel. These points are sufficient to enable us to construct the whole parallel. The meridianal stereograpliic net constructed through 2° for a sphere R = 10 cm, was first introduced into crystallography by Vul'f (2) and bears his name (normally as Wulff in English) (See Fig. 2). Operations with a Wulff Net The Wulff net is a transparent sheet by which any construction can be transferred to transparent paper (wax paper or ordinary tracing paper) without the use of compasses or a ruler. The tracing paper is centred and a mark is made on it to indicate the end of a meridian which is the point of origin for counting off the azimuths. This fixes the initial position for the tracing paper, and by means of this index the paper can be subsequently brought back into the initial position. We shall now consider problems in geometrical seismics which can be solved by means of a Wulff net. PROBLEMS ENCOUNTERED IN GEOMETRICvy;. SEISMICS Every direction in space can be unambiguously determined by two angles. Let these angles be the azimuth (a), that is the angle counted from the northward direction in the clockwise sense which varies from to 360°, * Throughout the paper the term "parallel" is used as parallel of latitude or small, circle [^Editor's note]. 48 E. I. Gal'perin et al. and the angle 9? with the vertical, which varies from to 180°. The first essential is to learn to plot on the net the diiections of the rays along the co-ordinates so that we can remove the co-ordinates from the net. Let us examine the main methods of working with a WulfF net. 280 270 260 90- a^ 100 250 190 180 '70 Fig. 2. Constructing the direction from given co-ordinates (a = 162°, (p = 54°) by means of a Wulff net. 1. To construct on the grid directions of which the co-ordinates are given. Let the co-ordinates of direction be a =^ 162° and 9? = 54°. In order to construct this direction on the net (see Fig. 2) we must count off an angle equal to the azimuth of our direction on the tracing paper in a clockwise direction round the outer circle of the grid, and mark point oc with a^. Rotating the tracing paper we bring the point which we have just obtained onto one of the diameters of the ])rojection circle. For example, let this be the horizontal SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 49 diameter with the mark 90° (point Oq') and let us plot the angle 9? = 54° (point b') from the centre of the projections to the circumference. When the tracing paper is rotated into the original position (the index coinciding with 0°) the point b' takes up the position b which is also a pro- jection of the trace of the intersection of a direction determined by the given angles a = 162°, (p = 54° with the sphere. It must be noted that all the directions for which q> is equal to or less than 90° will be above the plane of drawing, and in this case they will be marked on the grid by points. If 90° < 93 < 180°, then such directions will be marked on the drawing by means of crosses. If the angle given is not from the vertical but from the horizontal then it will be calculated not from the centre of the projection but from the outer circumference of the grid towards its centre. 2. To determine the co-ordinates of points given on the net (the problem in reverse). To solve this problem we draw a straight Hne from the centre through the given point up to its intersection with the circumference of the net, and we count off on the circumference the azimuth a. We then transfer the point to the equator and count off the angle 9? from the centre. We can now pass immediately to the consideration of problems encountered in geometrical seismics. Problem 1. To determine on the net the plane which includes the direction of the ray and the normal to the boundary. It is known that incident, reflected and refracted rays and the normal to an interface Ue in one plane, which is also the plane of the rays. We shall use this property to find the plane of the rays. Imagine the centre of a stereo- graphic net at the point of incidence of a ray; we now plot on the tracing paper, using the net, the direction of the incident ray and the normal to the interface. We must bear in mind that since the centre of the net is set at the point of incidence of the ray, it will always be essential to take its inverse azimuth when we plot the direction of the incident ray onto the net. Two directions in space have thus been plotted on the tracing paper and the problem is reduced to finding the plane in which both directions he. The meridians of the net correspond to an assembly of circles (planes), differently incHned to the plane of drawing. Consequently, if we rotate the tracing paper until both given directions fall on one and the same meridian of the grid, we shall thereby find the plane in which both given directions He. We produce this meridian and find the pole of arc of the great circle, which will also determine the direction of the normal to the plane of the rays. For this purpose it is sufficient if we count off 90° along the diameter from the arc. Rotating the tracing paper again until it reaches its original Applied geophysics 4 50 E. I. Gal'perin et al. position, we obtain the position of the plane of the rays in space. For purposes of examining these problems the direction of the normal is regarded as given. For example let the direction of a ray incident from a source be represented by co-ordinates 234° and 31° and the direction of the normal to the interface at the point of incidence of the ray by 112° and 14°. Let the imaginary centre of a projection onto the interface be at the point of incidence of the ray and let us plot on the net the direction of this ray and of the normal. It should be kept in mind that the azimuth of the ray will thereby be inverted. 2 70 (b) Fig. 3. a — determination of direction of reflected, refracted and grazing rays; b — towards determination of the direction of the refracted rav. SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 51 In Fig. 3 the point A with the co-ordinates 54° and 31° corresponds to the direction of an incident ray, while the point iV^ (112° and 14°) corresponds to the normal to the interface at the point of incidence. To find the plane of the ray by rotating the tracing paper, we plot both points on the same meridian (points A', iV^) and draw it on the tracing paper (lines produced on the tracing paper are shown on the drawing by a dotted line). We mark on the tracing paper the normal to the plane of the circle (the point N' i). Next we move the tracing paper into its original position and we can determine the position of the plane of the rays in space ; in our case the co-ordinates of the normal to the plane of the rays are determined as 300° and 76°*. If one of the directions is above the plane of drawing and the other is below it then the points will be on symmetrical (equidistant from zero) meridians. This is correct since the lower half of the meridian is projected from the zenith in a symmetrical arc. Problem 2. To determine the angle between two directions in space. Both directions are plotted by means of the net on tracing paper (points A and B Fig. 3), and by rotating the tracing paper the plane in which both points lie is found. The points under consideration, A and B, lake up positions A' and B' respectively. The angle between points A' and B' in this plane are counted off on the net; in our case this is equal to 54°. This angle is also the angle between the given directions in space. Problem 3. Given the direction of an incident ray and of the normal to the interface at the point of incidence, to determine the direction of a reflected, a refracted and a grazing ray. In order to determine the directions of these rays in space, we must first find the plane of the rays, that is we must first solve Problem 1, after which we can construct the directions of the rays in which we are interested in the plane of the rays. Let a point iVj, (see Fig. 3) (112° and 14°) correspond to the direction of the normal to the interface, and a point A (54° and 31°) to the direction of the incident ray (with inverse azimuth). Then the plane of the rays is determined by the points A' and 7V^. We shall consider separately how to determine the direction of each of the rays which interests us (the reflected, the refracted and the grazing rays) in this plane. (a) To determine the direction of the reflected ray. Since the angle of reflection is equal to the angle of incidence, it follows that when we have determined the angle between the incident ray and the normal by means * The Russian original states mistakenly 86°. 52 E. I. Gal'perin et al. of the net (in our case between the points A' and N^ is equal to 27°) we can plot it in the plane of the rays from the normal in the direction opposite to the incident ray (point B'). We next bring the tracing paper into its original position. The point B' is now transferred to point B which also corresponds to the direction of the ray after its reflection. The direction of the reflected ray in our case is determined by the co-ordinates 185° and 30°. (b) To determine the direction of the refracted ray. The method is similar to that used for the reflected ray, the only difference being that in this case instead of plotting the angle of reflection, which is equal to the angle of inci- dence, from the normal we plot the angle of refraction. The angle of refraction is calculated from the angle of incidence and from the ratio of velocities — in the first and in the second layers according to Sn ell's equation: . . v^ i2 1 = arc sm 112—=-. ' ^1 where : i^ i is the angle of refraction and ij g is the angle of incidence. By way of example let — = 1*5 in our case; then when ij^^ = 27° the angle of refraction ig^ will be equal to 43°. Since the refracted ray will be under the plane of the drawing we shall construct on the grid the direction opposite to it. (As indicated above, we shall for the sake of convenience denote the direction in such cases by a cross and not by a point). For this purpose, as can be seen from the ray diagram (see Fig. 3, b) it is sufficient to plot the angle of refraction from the normal in the direction of the incident ray in the plane of the ray. The direction of the refracted ray is denoted in Fig. 3, a by the point C If we rotate the tracing paper into its original position we shall obtain from the point C the co-ordinates of the refracted ray (43° and 47°). We must not forget, however, that here we are dealing with a direction opposite to the direction of the refracted ray, and therefore when we remove the co-ordinates we must take the inverse azimuth. Then the direction of the refracted ray will be 223° and 47°. (c) To determine the direction of a grazing ray. To obtain a grazing ray the angle of refraction must be 90°. To determine the direction of the grazing ray it is sufficient to plot an angle of 90° from the normal in the plane of the ray and having restored the tracing paper to its original position to take the co-ordinates of the grazing ray from the net (point K, Fig. 3, a). SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 53 The problems we have considered are sufficient to enable us to proceed to a description of the methods and techniques used in solving three- dimensional ray problems in geometrical seismics. GENERAL SCHEME OF SOLUTION The solution of three-dimensional ray problems resolves itself into studying seismic fields in space. Seismic fronts are traced along the seismic rays. For this purpose, instead of using individual rays, we find it more convenient to use certain aggregations of rays, for example rays emerging from a source and maldng a known angle with the vertical or intersecting an interface along certain definite lines (isohypses). In the first case these rays will form, in the first medium a conic ray surface, which after striking the very first interface can get deformed. If we take a definite number of such surfaces which differ only in the apical angle of the cone, we can use these to fill in the whole portion of the space which interests us. The behaviour of each such conic ray surface is traced in space along the individual rays forming it, which in the first medium differ from one another only with respect to the azimuth. By tracing consecutively the behaviour of the conic ray surface on all the given interfaces we can find the trace of intersection of the ray surface with the surface or plane of observation. All the constructions are produced on a plan where trace projections of the ray surface intersections with each of the interfaces in turn are drawn. The intersection trace of a ray surface with an interface if the latter has a simple shape can be found by ordinary geometrical constructions, the trace being subsequently projected onto the horizontal observation plane. In cases where the interface is not distinguished by a simple form, the projection of the intersection trace of the ray surface with the interface can be constructed as the geometrical position of the trace projections of the intersection with the interface of the rays forming the ray surface. The point of intersection of the ray with the interface is determined as the point where the interface and the ray have exactly the same depth. Each point of the projection of the ray on the plan corresponds to a particular depth of the ray which can be calculated from the known angle made by the ray with the vertical (the inclination of the ray). Projection points of rays for which the depth along the rays coincide with the interface depths (isohypses) are projections of traces of intersection with the interface of the corresponding rays. The travel times of waves are calculated from the rays separately for each section of the wave path. The length of the section of a ray which is included between interfaces is determined from the angle of inclination of 54 E. I. Gal'perin et al. the ray to the vertical and from the depths of its extreme points. We sum the travel limes of the waves for the different sectors and record the points at which each relevant ray emerges onto the observation surface. An isochro- nous chart is constructed by interpolating the travel time values along each ray. In addition to the time fields we can construct the fields of azimuthal deviations and the angles of emergence (*>. Fig. 4. Construction for radial conic surface formed by rays emerging from source at an angle of 20° with the vertical. a — plan; b — cross-section of structure in vertical plane from 270-90° azimuth. 1 — structure contours of interface (for cylindrical surface taken at every 20 m ; for plane — at every 100 m) ; 2 — trace projection of intersection of radial conic surface with interface ; 3 — geometric locus of points of emergence of rays of 20° conic surface on to observation plane; 4 — ^points of emergence of individual rays; 5 — projection of rays; 6 — azimuthal deviation suffered by the ray on reflection at point c; 7 — line joining cyUndrical surface and plane. In order to solve linear problems, therefore, we must have a plan on the appropriate scale with isohypses for all the interfaces. The plan must show the position of the source of vibrations (shot point). Let us now examine the technique for making the constructions, taking concrete examples first of two layered and then of three layered media. Structure contour [Editor's note]. SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 55 CONSTRUCTING A SURFACE HODOGRAPH FOR REFLECTED RAYS IN THE CASE OF A TWO LAYERED MEDIUM By way of example we shall consider the construction of a surface hodograph or waves reflected from an interface which has a cylindrical form. Formulation of the problem— To solve a linear problem in geometrical seismics (to construct a chart of isochronous lines) for waves reflected from a cylindrical surface with a horizontal axis of infinite extent. The cylindrical surface is joined to planes. A section of the interface across the extent of its vertical plane is shown in Fig. 4, h. The source of vibrations (shot point) is situated above the axis of the cylinder. Description of the constructions —The reflecting boundary has two mutually perpendicular axes of symmetry, the source being above the point of intersection of these axes. In this case we can limit our consideration of the constructions to one -quarter alone, for the field in the remaining quarters will be synunetrical. Taking the direction in which the structure spreads as the initial azimuth reading, we shall make the constructions within the limits to 90°. To solve this problem we shall trace several ray surfaces formed by rays emerging from the source at arbitrarily determined angles (p. The section of these surfaces in the vertical plane is shown in Fig. 8, 6. By way of example we shall consider in detail constructions for any one of these surfaces alone (for example the surface formed by rays which make an angle 99* equal to 20°). For this purpose we shall consider the rays of which the azimuths differ from one another by a definite quantity for example 10°. The projections of the incident rays in the first medium are shown in Fig. 4, a by the lines 5. The constructions for eacla ray surface can be broken down into separate stages : (a) construction of the trace of intersection of the ray surface with the interface on the plane of the projection; (b) determination by Wulff"'s net of the directions (azimuth and angle with vertical) of the reflected rays; (c) construction of the trace of intersection of the ray surface with the observation plane after reflection. Let us examine these stages for each ray surface separately. (a) Construction of the projection of the trace of intersection of the ray surface intersection with the interface. Since the interface is not simple in form we shall find the projection of the trace of intersection of the ray * For the sake of brevity we shall henceforward call such a surface the "9?° ray surface" . 56 E. I. Gal'perin et al. surface with the interface as the geometric locus of the projection of traces of intersection with the interface of the individual rays. Let us take a particular example in order to examine the technique for deter- mining the projection of the points of intersection of the ray with the interface. To do this we shall select from the total number of rays forming the conic surface one ray for example with an azimuth of 70°. Its projection on the horizontal plane is shown in Fig. 5, a, and the section of the vertical plane along the lines /— /, that is in the direction of the ray in Fig. 5, b. Since (0) o , 0, 8 600 1000 1300 c 200 A — \ 400---'! \ 600 800--- ■\ 1000 - - - \ \ 1200 zrr^ 1330 -'- - \ ^ziv-::::: / .._>. '"/ / / / 1 (b) Fig. 5. Projection of point where ray intersects interface, a — plan; h — vertical cross section along direction of incidence of ray; 1 — structure contours of interface; I — I direction of incidence of ray. SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 57 according to the assumption this ray makes an angle of 20° with the vertical, it will be projected onto the observation plane in a section Ax = Ah tan 20°, when it penetrates to a depth Ah. If we now plot on the plan on the scale of the constructions the section Ax in the direction of incidence of the ray, we shall obtain on the projection of the ray a successive series of points corresponding to certain definite depths of the ray: in our case Ah = 100m and the points will be at 100, 200 and 300m. Let us suppose that we strike the point marked 1300m, that is at this point the ray has penetrated to a depth of 1300m; while the structure contour of the interface 1320m passes through this point, it is obvious that the ray has not yet reached the interface. We plot the section once again and obtain a point marked 1400m but the structure contour 1340m passes through this point on the plan and consequently our ray has not yet intersected the interface between the contours 1300 and 1400m, Inter- polating values for the depths between the structure contours 1300 and 1400m and also the depths of the section Ax 1300 — 1400m, we obtain a point c in which the depths both along the structure contours of the interface and along the ray are the same — 1330m. The point c is thus a projection of the trace of intersection of our chosen ray with the interface. Having made such constructions for many rays of the conic surface and having joined the points obtained in a smooth curved line we shall obtain on the plan the trace of intersection between the interface and our ray surface (hne 2 in Fig. 4). This line is also the projection on to the horizontal plane of the hne on which the rejflection of the rays making up our ray surface takes place. (b) Determination of the direction of the reflected rays. Having constructed on the plan projections of the points of incidence of the rays under considera- tion we find the direction of the rays after reflection. This is done by means of a Wulff" net, in the manner indicated above (Problem 3). The centre of the Wulfi" net is placed at the point of incidence of the ray and the direction of the incident rays and the direction of the normal to the interface at the point of incidence are plotted on the Wulff net. The direction of the nor- mal is determined from the structure contour chart of the interface. The azimuth of the normal to the boundary at the point of incidence is determined by the direction of maximum incHnation of the boundary at this point. This direction corresponds on the plan (Fig. 6, a) to the direction of the normal to the structure contour of the interface. The angle of incidence of the normal to the vertical is determined from the value of the angle of incidence of the interface at the point of incidence. The angle made by the normal with the vertical can be determined 58 E. I. Gal'perin et al. graphically; for this purpose we construct, along the direction of the normal azimuth, a section of the interface of the vertical plane and drop a perpendicu- lar to the point of incidence (Fig. 6, 6). In our problem the azimuth of the normal to the interface for all its points is the same, 90°. This is because all the structure contours are rectilinear and have an azimuth of 0°. The angle of incidence of the normal to the interface is constant at 30° in sectors where the interface is represented c b a f e d (b) Fig. 6. Determination of the direction of the normal to the interface, a — determination of the azimuth of the normal at points a, b, c, k, m; b — determination of the angle with the vertical. by planes, while in those sectors where the interface is represented by a cylindrical surface the angle of incidence of the normal to the vertical varies from to 30°. The ray which we are considering which has an azimuth of 70°, strikes the interface at a point which is projected on the plan as point c (Figs. 4 and 5). We place an imaginary centre of the net at the point of incidence and construct on the net the directions of the incident ray ^(70° and 20°) and the normal to the interface (90° and 30°) and then we determine the direction of the reflected ray B (83° and 79°) — Fig. 7. Figure 7 also shows the determination of the reflected ray direction for all the remaining rays of the 20° surface which we are considering. The azimuths of the incident rays are separated from one another by 10° (that is 0°, 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90°). The points are marked on the drawing as 0, 1, 2, 3, 4, and so forth. SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 59 (c) Construction of the trace of the intersection between the observation plane and the ray surface after reflection. For this purpose we plot on the plan projections of the reflected rays according to their known azimuths (these are shown in Fig. 4 by the lines 5). As can be seen from the illustration the direction of the projection of the incident ray coincides with the direction of the projection of the reflected ray only along two azimuths (0° and 90°)^'^\ In neither case is there any azimuthal deviation; in all the remaining cases azimuthal deviations are observed and the projections of the rays are not straight lines but are broken lines consisting of two parts. At the point of reflection then the ray suffers an azimuthal deviation which is easily visible when we project the ray onto the horizontal plane 340 350 10 330 320. 310^ 300^ 290/ 280 J 20 30 40 .50 ,60 JO 270 2601 250> 9«- 6A I <J8- ,80 90 100 240' 230^ 220 210 200 120 130 140 150 160 190 180 l"^0 o -2 * -3 Fig. 7. Determination of the direction of the rays after reflection by means of a Wulff net. 1 — direction of incident rays with azimuths 0-90° at every 10°; 2 — direction of reflected rays; 3 — direction of the normal to the interface at the point of incidence of the corresponding ray. 60 E. I. Galterin et al. o _ • #s ri » <u o % 'S Pi OJ S !-i ft a) to '2 1 "^ 4 2 0) a> rrt OJ a u J o s o Oi ft ^ &i> .S Ti *^ bU Tl J3 a ;3 -^ >. M o o m t4-l t» > UJ o ^ n 3 S ^ o .g n o ft s CvJ % 1 'a ^ a ° ^ o « o> 2 tn O f-i o 1 1 in OD .a .a o o g :5 .1 7 c ^o ■^ o o ft !-i o • #> >. S o % 1 o s o o 1 o tn ^ n s "3 1 CO o in o ^1-1 r ;a ^ 1 o <D ^ ^ I I ^ .a "^ ^ eg O (D cti m a 2 v. •En ^^ § ^ .a 2 » o ^ g ft O t« O O «3 O (D 3 ."S M fl ft .a ® ^ J! go i3 S ■» (L) V bD ,^ SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 61 of observation. The azimuthal deviation for a ray with an azimuth in the first medium of 70° is shown in the illustration by means of arrows and is equal to 15°. We find the intersection trace of the ray surface with the plane of observation as the geometrical position of the points of emergence of rays belonging to one ray surface. These points are calculated for each of the rays of the ray surface from the known depth of the point at which the ray intersects the interface and the angle the ray makes with the vertical after reflection. This angle is taken from the Wulflf net when the direction of the >96 1-00 1-20 I-40 1-60 1-80 2-00 2-20 2-40 2-60 2 80 3-00 sec Fig. 9. Chart of isochronous lines. 1 — points of emergence of rays from which the field was constructed. 90° reflected ray is determined. In the general case we can find this point by the same methods as we use for determining the projection trace of a ray inter- section with an interface. In contrast to the case analysed above what we are here examining is not "descent" of a ray but its "climb". The point on the projection of the reflected ray which corresponds to zero depth will also be the point of emergence of the corresponding ray (see line 4 in Fig. 4). When we have joined all the ray emergence points by a smooth hne we shall obtain the intersection trace of our ray surface with the observation plane. For every point on line 2, then, there is a corresponding point on line 3 (see Fig. 4). Similar constructions can be made for all ray surfaces (Fig. 8), where (a) the lines 4 correspond to radial surfaces of 10°, 13°, 15°, 17°, 20°, 22°, 62 E. I. Gal'perin et al. 25°, 30°, 35°, 40°, 45°, 50°. Using such constructions we can now pass on directly to calculating the travel times along the rays. The length of the ray's segment required for this is determined from the difference in depth AH of the initial and terminal points of the segment of the ray and its angle with the vertical from the formula cos 9? The times for any ray are calculated separately for each segment in the broken line and then the travel times are summed. The travel time obtained is entered on the plan at the emergence point of the appropriate ray. The chart of isochronous lines is obtained by interpolating the travel time values for all emergence points of the rays under consideration. Fig. 9 shows an isochrones chart constructed by the method described. The isochrones are plotted on this chart at 0.05s. As we might have expected the isochrones form smooth hnes extending along the line of the Umits of the structure. We now consider a problem for a three -layered medium. CONSTRUCTING THE SURFACE HODOGRAPH FOR REFLECTED WAVES IN THE CASE OF A THREE-LAYERED MEDIUM By way of example we shall analyse in detail an isochrone chart for waves reflected from a dome, making allowance for intermediate refraction on an incHned plane*. The construction of the medium is shown in Fig. 10, b^ The velocity ratio — in the first and in the second layers is equal to 1.2. The source of excitation is in both cases above the slope of the dome. The general scheme for solving the problem is the same as for problem 1. The difference is that it is necessary to make allowance for an intermediate interface. We do this by constructing projected traces of the intersections of the ray surfaces with the intermediate interface both before and after its reflection when we are tracing the ray surfaces. Let us examine the tracing of several ray surfaces. Fig. 10, a, show.^ constructions for a ray surface formed by rays which emerge from a source and make an angle 9? = 30° with the vertical. It can be seen from the plan that the source is set in relation to the top of the dome. Radial lines emerging from the source are produced at every 10° and correspond to the direction^ (azimuths) of the rays in the first medium which were used in tracing the ray surfaces. * The solution of this problem is given in(^). SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 6a The point from which the azimuths are read off coincides Avith the direction of incidence of the inchned refracting boundary. Line 5 is a trace projection <»f the intersection of the radial surface with the inchned refracting plane. ^•^gxT^ -- 2 — 3 4 — H Fig. 10. Determination of points of emergence of rays onto observation plane (30° ray surface), a — plan; h — cross section in vertical plane containing azimuths and 180° (the generators of the 30° ray surface are shown). 1 — structure contours ol refracting interface; 2 — structure contours of reflecting interface; 3-4-5-6 — trace projections of intersection of 30° ray surface with refracting boundary (3), reflecting boundary (4), refracting boundary after reflection (5), and observation plane (6); 7 — lines of equal, azimuths; 8 — points of emergence of rays; 9 — projections of separate rays. 64 E. I. Gal'perin et at. This projection is found as the geometric locus of the projected traces of the interface with rays with azimuths 10° apart from one another and with projections shown by the radial hnes. The projection of the traces of intersection of the rays with the interface are found as projections of points on the rays and on the interface, then points having the same depth. A ray with its azimuth in the first medium at 300°, intersects the inclined refracting interface at a depth of 356 m. The projection 290 280 ,2 70 260 250 190 igo 170 Fig. il. Example of determination of direction of refracted and reflected rays and of a ray refracted on transition from second medium to first. 1 — plane of rays; the points correspond to directions: A — of a ray incident on a refracting boundary (120 and 30°); Ni 2 — of the normal to the refracting boundary ; B — of a refracted ray (125° and 39°) ; Bi — of a ray striking the reflecting boundary after refraction (125° and 39°); N2 — of the normal to the reflecting boundary (318° and 10°) ; D — of a reflected ray (309° and 59°); E — of a ray striking the refracting boundary after reflection (129° and 59°); A'2 I — of the normal to the refracting boundary (180° and 10°) after reflection; F — of the emergent ray (132° and 50°). SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 65 of this point on to the observation plane is denoted by the letter a. It must be noted, however, that in this case the trace of the projection of the inter- section of the ray surface with the inclined refracting boundary can be found by purely geometrical constructions as an ellipse formed by the intersection of the conical surface with the inclined plane. The line thus constructed is the geometric locus of the points of refrac- tion of rays belonging to the 30° radial surface under consideration. At these points the rays having been refracted pass into the second medium. Let us determine the direction of the rays in the second medium (after they have been refracted). This is easily done with a Wulff net by the me- thod described (Problem 3). Now let us see how to determine the direction of one of the refracted rays, for example a ray with an azimuth in the first medium on emergence from the source equal to 300°. Following the method described above we set the centre of the net at the point of incidence of the ray and plot on the net the direction of the normal to the interface and the direction of the incident ray. Since the refracting interface is a plane the direction of the normal will be the same for all points on this interface. The angle at which the normal inclines to the vertical is equal to the angle of inclination of the plane to the horizontal, namely 10°. The azimuth of the normal is 0°. The direction of the normal (0° and 10°) is shown in Fig. 11 by the letter iVj 2- Since the centre of the projection is placed at the point of incidence, in plotting on the Wulff net the direction of an incident ray emerging from the source with an azimuth of 300° we must take the inverse azimuth, that is 120°. The direction of the incident ray under consideration (120° and 30°) is shown in the illustration by the point A. Using tracing paper we plot points corresponding to both directions on one meridian, and then measure the angle of incidence ij g? which is equal to an angle of 30°. We calculate the angle of refraction from the angle of incidence and the velocity ratio vjv-^ according to the formula . . . . V2 sm ic, 1 = sm 1-, o — , where i^ j^ is the angle of refraction. sin ig^i = 0.588 ij 2 = 0-705; i^^^ = 45°. We plot the angle of refraction we have found in the plane of the ray from the normal in the direction of the incident ray, and mark the point B Applied geophysics 5 66 E. I. Gal'perin et al. obtained by a cross, since the refracted ray lies underneath the observation plane. Rotating the tracing paper to its original position we take the co- ordinates of point B from the net. It must be remembered, however, that the ray is under the observation plane and therefore we must take the in- verse azimuth. The direction of the refracted ray will accordingly be (305° and 39°). Using similar constructions we can find the directions of all the rays under consideration in the second medium. Figure 12 shows such constructions for a 30° ray surface. We now" turn back to Fig. 10. We plot on the plan the directions (azi- 350 10 340 EO 280 270 260 250 240^ 230^ 220^ 210 200 160 190 180 170 .40 ,50 .60 .70 ,80 90 100 no 120 130 150 140 + 2 ® 3 Fig. 12. Determination of the direction of refracted ray for a 30° ray surface. 1 — directions of incident rays having azimuths from 0° to 360° at every 10°; 2 — direc- tions of the corresponding refracted rays; 3 — normal to the interface. SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 67 muths) of the rays in the second medium which we have found, for a ray with an azimuth of 300° and an azimuth of 305°. We then obtain the pro- jection of the ray on to the horizontal plane in the second medium. As can. be seen from the illustration and also from the values of the direction co- ordinates, the ray under consideration undergoes azimuthal deviation of +5° during refraction. Having traced the directions of the rays after refraction in the second medium, we find the trace projection of the intersection of the ray surface 760 750 740 730 720 710 700 690 680 Fig. 13. Determination on plan of projection of point of intersection of a ray with a reflecting interface. 1 — structure contours of reflecting interface; 2 — direction of incidence of ray. with the reflecting boundary. This projection is also found to be the geo- metric locus of the trace projections of the intersection with a reflecting boundary of individual rays of the ray surface, in our case rays with azimuths in the first medium which are 10° from one another. The traces of the rays' intersection with the reflecting interface are found to be points on the ray and the interface which have the same depth. These traces are then projected onto the observation plane. We determine the trace projection of the intersection for a ray with an azimuth of 300°. After refraction on the inclined interface at point a, the ray with the 300° azimuth and the 30° angle with the vertical has changed its direction — the azimuth 68 E. I. Gal'perin et al. is now 305° and the angle with the vertical 39°. In order to make the follow- ing discussion clear we show the ray under consideration and the structure contours of the reflecting interface (Fig. 13) on the plan (constructions for all the rays are usually shown on one drawing). The point a lies in the plane of the inclined interface and has a depth of 356 m. Knowing the angle the ray makes with the vertical (39°) we calculate the amount of projection of the ray onto the observation plane during its penetration at some depth Ah (in our case Ah = 50 m) as Ax =^ 50 tan 39°. We plot the segment Ax from the point a along the direction of incidence of the ray. We now exam- ine the two points on the projection of the ray which we have thus obtained: 706 and 756 m. At the first point the ray has not yet reached the reflecting boundary, since the 742 ra contour passes through this point; at the second point it has intersected the interface since the depth of this point along the 756 m ray is greater than its depth along the 748 m contours. Conse- quently, the projection of the point of incidence of the ray on to the refle- cting boundary lies between the points 706 and 756 m along the ray and between the contours 740 and 750 m and is found by interpolating the depths. Similar constructions can be made for all rays. After joining by a smooth Hne the trace projections of the intersection of the radial surface with the reflecting interface, along which the rays belonging to our radial surface are reflected (Fig. 10, line 4), we must next find the direction of the reflected rays. This again is done by means of a Wulff" net. For this purpose the centre of the net is set at the point of incidence of the ray, and the directions of the incident ray and of the normal to the interface are plotted on the grid. Let us trace the direction, after reflection, of the ray in which we are interested. The ray has emerged from the source with an azimuth of 300° and has been refracted on the inclined interface at a depth of 356 m at a point the pro- jection of which on the plane is denoted by point o, and which has struck the reflecting boundary at a point 746 m deep. The projection of the point of incidence on the observation plane is denoted by the point h. We have already described how to determine the direction of the reflected ray; we shall now briefly recapitulate this method as it applies to our data. We imagine the centre of the net placed at the point of incidence, and plot on it the directions of the incident ray and the normal to the interface. The direction of the incident ray is determined as ^ve have sho^vn above, by the co-ordinates (305° and 39°). The point B (see Fig. 11) mth the re- verse azimuth characterizes the direction of the incident ray. The direction of the normal to the interface is in our case determined by the values (318° SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 69 and 10°), and the position of the normal to the interface on the net is deter- mined by the point A^g- Making both directions coincide with one of the meridians of the net we find the plane of the ray and in this plane we plot from the normal the angle of reflection, which is equal to the angle of incidence. We now obtain the point D. We rotate the tracing paper to its original posi- tion and using the Wulff net we take the co-ordinates (azimuth and angle with the vertical) of the point D. The co-ordinates obtained characterize the direction of the reflected ray. The ray which we are to examine after reflection has a direction deter- mined by the co-ordinates (309° and 59°) and suffers an azimuthal devia- 350 iO 280 270 260 Fig. 14. Determination of the direction of reflected rays for 30° ray surface. 1 — directions of incident rays having azimuths from to 360° at every 10°; 2 — direc- tions of the corresponding reflected rays; 3 — direction of norma] to interface at point of incidence of rays. 70 E. I. Gal'perin et al. tion of +4° on reflection. In this way we determine the directions of the remaining rays after reflection. Having determined the direction of the rays after they have been reflected from the dome and having constructed the projections of these directions on the plane, we can now find the projection of the traces of the intersection of the ray surface under consideration with the inclined refracting boundary. For this purpose we use the methods which have been described to find the projections of the points of intersection of the reflected rays with the refracting interface, and then we join these points on the i3lan by a smooth line. The ray which we are considering, after emerging from the source with an azimuth of 300° and striking the reflecting boundary at a point the projection of which on the plan is point 6, intersects the refracting boundary at a depth of 443 m; the projection of this jDoint on the plan is denoted by the letter c (see Fig. 10). In Fig. 10 the projection of the trace of intersec- tion with the inclined refracting boundary of the ray surface after reflec- tion is shown by the line 5. After being refracted at points on the line of intersection of the ray surface with the refracting boundary, the rays strike the first medium. The directions of the rays in the first medium after refraction are determined from the Wulff net. This is shown in Fig. 11 for the ray under consideration. As we have shown above the ray we are considering has undergone azi- muthal deviation on the reflecting boundary and after reflection has an azimuth of 309° and an angle with the vertical of 59°. It must be remem- bered that the incident ray is plotted with inverse azimuth (309° — 180° = = 129°) and that in determining the direction of the refracted ray we take the inverse azimuth. We mark the direction of the incident ray on the tracing paper by means of the Wulfif net (point E) and also the direction of the normal to the interface N^-^ (180 and 10°), and then make both direc- tions coincide with one of the meridians of the net. In the plane of the rays which we have thus obtained, we measure the angle between the normal and the point E — angle of incidence — and after calculating the angle of refraction according to the formula given above we plot it from the normal in the direction of the incident ray. The point F is marked by a cross since the ray lies under the plane. We now rotate the tracing paper to its origi- nal position and take the co-ordinates of the point F from the net— the azimuth and the angle with the vertical. We thus find that the direction of the ray after refraction is 312° and 50°. Consequently, on refraction the ray undergoes an azimuthal deviation equal to 3°. The Fig. 15 shows how the direction of the refracted rays of the 30° ray surface is determined. The points where the ray emerges are determined SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 71 as points of a ray having zero depth. For the ray under consideration this point is marked on the plan (Fig. 10) by the letter d. After we have deter- mined the points of emergence of all the rays and joined them by a smooth line we obtain the geometrical position of the points of emergence of rays belonging to one ray surface. The intersection trace of the ray surface mth the observation plane is shown by the line 6. We have now consistently traced in space a ray surface (which in the first medium was conic) formed by rays emerging from a source at an angle of 30° with the vertical, and we have obtained for each of the rays under consideration its projection onto the observation plane. The other ray sur- 270 Fig. 15. Determination of direction of rays refracted on plane iaclined interface on transition, from second medium to first (the constructions were made for a 30° ray surface). 1 — directions of incident rays having azimuths from to 360° at every 10°; 2 — directions of refracted rays; 3 — direction of normal to interface. 72 E. I. Gal'perin et al. faces required for solving the problem as a whole can be traced in space in a similar way. The projections of the rays onto the observation plane can be traced on Fig. 10, and the directions (azimuth and angle with tlae vertical) of the rays at any point on the observation plane determined. This could be used to construct the field of lines of equal azimuthal deviations and the field of lines of equal angles of emergence' ' '*\ One tracing paper can be used to determine the direction of the refracted rays in the given case for all the radial surfaces, since the boundary is a plane boundary and the direction of the normal is the same for all points on the boundary. It is convenient to determine the direction of the rays after refraction on a separate sheet of tracing paper for each ray surface, since the direction of the normal varies at different points of the interface. From an examination of the constructions for different ray surfaces, it follows that the azimuthal deviations of rays emerging from a source with the same azimuth increase in inverse proportion to the angle 9? of inclination with the vertical. When cp = 10°, for example, rays with azimuths near to 180° suffer azimuthal deviations of very nearly 180° on reflection. This, in particular, explains the fact that the projections of the intersection traces of the 10° radial sur- face with the reflecting interface and with the refracting boundary intersect after reflection. Calculating the time field— The, time field is constructed by interpola- ting travel time values for each of the rays under consideration. The travel times along the rays are calculated from their various sections in each layer separately, from the length of the section and the velocity value in the layer. The length of the section is determined from the value of its horizontal projection and from the angle of inclination with the vertical as determined from the Wulff net. To determine the travel time of a wave along the ray we sum the values for the travel times along its separate links. The chart of isochrones constructed for our case is shown in Fig. 16. The points of emergence of the ray surfaces which we examined when we were tracing are indicated by small circles. The chart of isochrones is constructed from the values for the travel times at these points. The isochrones are given at 0.05s. intervals. In this paper we have examined the solution for ray problems in seismic prospecting only for reflected waves. The method is equally applicable to head waves (^). The accuracy of the solution depends primarily on two factors— accuracy SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 73 in determining the direction of the ray by means of the Wulff net and the accuracy of the graphical constructions. The former depends primarily on the size of the Wulff net. After some practice of accurate work with a WulfF net 20 cm in diameter, an accuracy of 0.5° can be obtained in determining the direction of the ray. If a higher degree of accuracy is required, the dia- meter of the net can be increased and more divisions can be marked on it. The accuracy of the graphical constructions depends on scale, and is limited by an inherent margin of error which experiment has shown to be 0.2 — 0.3 mm. A transverse scale must therefore be used in constructing the graphs. Scales of 1:10,000 and 1:25,000 may be regarded as convenient. 0° s s o — 180= Fig. 16. a — Chart of isochrones; b — vertical cross section of structure with generators of ray surfaces used in the construction. 1 — shot point; 2 — isochrones; 3— points of em^l^ence of rays from which the fields were calculated. 74 E. I. Gal'perin et al. SUMMARY The method we have described for solving three-dimensional linear problems in geometrical seismics of multi-layered media with interfaces of arbitrary shape is based on the use of stereographic projections. Such projections make it possible to follow the seismic rays in space and thereby open up new possibilities for solving problems in geometrical seismics. The method enables us to study the laws governing surface hodographs for various waves whatever the shape and quantity of the interfaces. REFERENCES 1. I. C. Berson, The space problem of interpretation of the hodographs of refracted waves. Publ. Institute Theor. Geophysics, vol. 2, pt. 2, 1947. .2. G. V. WuLFF, On the methods of projection of crystals in relation to theodolite measure- ments. Proc. Warsaw Univ., 1902. 3. E. I. Gal'perin, The solution of direct space problems of geometric seismics for multi- layered media with arbitrary surfaces of separation. Izv. Akad. Nauk SSSR, ser. geofiz.. No. 9. 4. E. I. Gal'perin, On azimuthal deviations of seismic waves. Izv. Akad. Nauk SSSR, sen. geofiz.. No. 9, 1956. 5. V. D. Tav'yalov, and Yu. V. Timoshin, Hodographs of reflected waves for curved surfaces of separation and their interpretation. Izv. Akad. Nauk SSSR, ser. geofiz., No. 2, 1955. •6. N. N. Puzyrev, On the influence of the curvature of the surfaces of separation and the determination of velocity from hodographs of reflected waves. Collection "Applied Geophysics'" Issue 13 Gostoptekhizdat, 1955. 7. N. K. Razumovskii, Stereographic Projections, 2nd Ed., 1932. Chapter 3 MULTIPLE REFLECTED WAVES S. D. Shushakov In seismic prospecting, the separation and tracing of useful waves may be made difficult by interference from multiple reflected waves. These can also be mistaken for single waves and so give rise to errors in interpretation. Although much has been written on the study of multiple waves the difficulties of recognizing them remained, and in the course of a seismic prospecting operation they are sometimes mistaken for single waves. The present article describes the results obtained fronr modelling multiple reflected waves and gives a number of theoretical calculations performed according to the contour integrals method. The main topic is a description of the longitudinal waves which are commonly encountered in seismic prospecting work. Certain features of multiple reflected waves are noted and practical suggestions are put forward. Multiple reflected waves of one intensity or another probably occur in all, or nearly all regions. In discussing the problems connected with multiple waves the term is frequently applied to those waves which are clearly <listinguishable and consequently possess an intensity commensurate with that of single reflected waves wdtli approximately the same transmission times as the multiple waves. Reviewing the seismological conditions which prevail in regions w"here multiple waves of such intensity are observed, we may conclude that the follo^ving special conditions which favour the formation of such waves are found in these regions: (1) a small number of interfaces characterized by comparatively high reflection coefficients; (2) comparatively poor damping of the waves between these interfaces. In the region Leningrad Station of the Krasnodar district, for example, there is a layer of highly wave -resistant Cretaceous deposits wdiich is a re- flecting horizon for multiple waves. In the outer zone of the Cis-Carpathian depression, a gyp svun -anhydrite horizon from which multiple waves are reflected can be distinguished by its wave resistance. In north-western Ger- many the lower boundary is formed sometimes by the Cretaceous cover, sometimes by a combination of this with one of the intermediate boundaries which has a high coefficient of reflection'^^'. In other regions such bound- 76 S. D. Shushakov aries are formed by the surfaces of the crystaUine basement'^*' or by car- boiiate layers, lava flows and so forth' •'^' ■^^' ■'■^K Some investigators maintain that multiple reflections are found in those regions where surface conditions favour their formation' -'^■'^' ■'■^^ (ground- water level near the surface, this low -velocity zone, level ground surface). This relationship is indeed observed in many areas, but it accounts only for multiple weaves reflected from the Earth's surface. . Various answers are given to the question whether the Earth's surface or the base of the low-velocity zone acts as an upper reflecting boundary, but most investigators think that the base of the low-velocity zone plays the main part here ^^> ^^' ^^K Yepinat'yeva '^' explains this on the grounds that a wave reflected from the surface loses a good deal of energy in the process of being reflected by the bottom surface of the low-velocity zone of absorption in this zone. The influence of the conditions of excitation and reception on the intensity of the recorded multiple waves as yet has not been studied. PoULTER, however, in a paper on the grouping of aerial explosions, notes that after comparing a large number of seismograms obtained from single well shots and from arrays of aerial explosions he is almost completely convinced that no traces of multiple reflections occur on the film when grouping is used. This can be explained in the case of multiple waves reflected first above the excitation point, and also in the case of interfaces Avhich dip, when the apparent velocities of multiple waves near the source may be less than the apparent velocities of single waves, and the loAver these velocities, the greater the number of multiple waves. Conditions which are more favourable to the recording of single rather than multiple waves can therefore be created not merely in grouping the shots but also in grouping the detectors and with other forms of direction finding. This possibility disappears as the angles of gradient of the reflecting boundaries diminishe. KINEMATIC CHARACTERISTICS Various types of multiple reflections can be excited in layered geological media. These waves can be subdivided according to the layers in which they are propagated and according to the layers in which they appear as longitudinal or as transverse waves. The symbols P^ and 5^ are used for longitudinal and transverse waves respectively, the subscripts indicating the layers in which these waves are propagated (Fig. 1). When we are discussing a vuiiform wave we may confine ourselves to indicating merely the particular boundaries which reflect multiple waves. MULTIPLE REFLECTED WAVES 77 These boundaries are usually denoted as follows: the Earth's surface, or the base of the low-velocity zone, by the index zero*, and other boundaries which reflect the multiple wave by the indices 1, 2, ... m, in order from top to bottom. An interbed echo reflected from some boundary denoted by 2, from the base of a weathered zone and from some intermediate boundary 1, is denoted as 201 ; a fullpath echo reflected n times from a lower boundary 1 and 71 — 1 times from the zero boundary is denoted as 10 ... 101. ■!■ ■- ,»-tj;,.-- i,- - i ^ '-- ' Pn ^-'n , / ■' ff I) It n ■ ! Fig. 1. Records of multiple reflections obtained under natural conditions. Different types of multiple reflections have certain different kinematic properties. Full-path multiple waves 10 ... 101 appear as single waves reflected from interfaces which lie deeper than the boundary from which they are in fact reflected. Fig. 2 shows a schematic diagram of multiple wave rays, together with the disposition of the actual reflecting boundary .and a number of fictitious boundaries which could be constructed if one mistook these waves for single ones. If the real boundary 1^ is inclined at an angle y to the zero surface, then a double reflected wave will appear as a single wave reflected from the false boundary Ig, which acts as a mirror image of the zero boundary in boundary 1^ and is inclined at an angle of 2^. A boundary 1„ is inclined towards the zero boundary at an angle n^. If the depth of the boundary Ij along the normal to it below the shot point is equal to H^, then the depth of a boundary along the normal to it below .the same point is given by ^ ^ m\y ^ ' * Henceforth called the zero surface or boundary. 78 S. D. Shushakov Consequently the depth of the fictitious boundary -/„ is given by sm}' The hodograph*' for the niuhiple reflection is given by the equation 1- / . 7-7- o sin^fiv , r^ sin^ n,y „ /qn — 1 / 4^1^ -,-- ^ + 4ili X — ^ + x\ (2) Vj^ |/ sm'^y sm y where n is the number of times the wave is niultipled and x is the distance from the source. Fig. 2. Diagram of multiple reflection rays. r Near the excitation point, when there is a horizontal reflecting boundary^ multiple waves are recorded travelling along a single path normal to this boundary. When the reflecting surface dips, each of these waves is propagated along its own special path (Fig. 3), at the end of which it strikes the zero boundary or boundary I, and after being reflected from this along the same path but in the opposite direction, returns to the detector situated near the point (^' ^^). Odd-number multiples at the end of the path of the incident wave are reflected from the interface I, whereas even-number multiples are reflected from the zero surface. Full-path echoes have the following kinematic characteristics. * A Russian term referring to distance— time curves. [Editor's footnote]. MULTIPLE REFLECTED WAVES 79 1. The travel time is equal to that of a fictitious wave which seems to be reflected from a deeper and more inclined boundary. At short distances from the source and when the dip of the reflecting boundary is slight, the time interval between each arrival of the multiple wave is the same. 2. The effective velocity calculated from the hodograph of a multiple wave which is mistaken for a single one must be near to the effective velocity calculated from the hodograph of a corresponding single wave reflected from the same boundary as that which is reflecting the multiple waves. This effective velocity will generally be lower than the one calculated from SP n— i \ A A V ,\ A A t KJ i A A 1 , cP A A V ^- 4 \ \ V''^ \ /:\ 1 A^^' \ \ y \/°° __^ r -^ 3 i Fig. 3. Diagram of multiple reflection rays recorded near shot-point. the hodograph of a single wave with a travel time near to that of the multiple wave, and lower than the mean velocity obtained from bore-hole measurement data and corresponding to the travel time of the multiple wave. 3. The angles of gradient of the fictitious interfaces constructed from the hodographs of multiple waves which have been mistaken for single ones increase systematically with depth. The angle of gradient of a fictitious boundary constructed from the hodograph of a wave multiplied n times will be n times greater than the angle of gradient of the real boundary reflecting this wave. 4. The number of multiple reflections with a different number of multiples cannot be more than n < 7i\2y^ where y is the angle of gradient of the reflecting boundary, and increases from 1 to cx) as this angle diminishes to zero. 5. When the reflecting boundary dips, the apparent velocities of multiple waves near the shot point diminish as the number of multiples increases. 80 S. D. Shushakov Interbed echoes have different travel paths and are correspondingly different in their kinematic properties, which are determined separately in each case. When the velocity increases monotonically with depth, multiple reflections show the following Idnematic properties (^). 1. Depending on the change of type which the multiple wave undergoes as it passes from one hodograph to another with increase of Iq, (when x = 0), there may be increase or decrease of the effective velocity calculated from hodographs which have been mistaken for the hodographs of single waves. A decrease but not an increase in effective velocity mth growth of Iq can be used for recognizing multiple waves. 2. As the number of multiples increases, these effective velocities may either increase or decrease, depending on whether the greater part of their journey is through a high-velocity layer or a low-velocity one. 3. As a criterion for recognizing multiple waves, it is possible to use the scatter of the effective velocity values corresponding to identical values of t^ and determined in different sectors of the operational area from the hodographs of various multiple waves, or on condition that the various multiple waves predominate in the interference vibrations. A multiple wave of which the first reflection has occurred above the excitation point is distinguished by the fact that the travel time of this wave increases with the increase of the shot depth, whereas the travel time of an ordinary single wave decreases. DYNAMIC CHARACTERISTICS Shape oscillation — \t is not unusual to find the phase of an oscillation changing through 180° when the number of multiplications of the wave alters to an odd number (^^). The explanation is that in such cases, as the wave passes from top to bottom through a lower boundary and from bottom to top through an upper boundary, there is in the one case an increase and in the other a decrease in wave resistance. Fig. 4 shows traces of multiple reflections obtained by modelling. The phase of the vibration can be seen to vary mth the number of multiplications. This effect could not be observed if the wave resistances increased or decreased both when the wave was passing through the lower boundary and when it was passing through the upper one (^). In general the direction of arrival is reversed if the wave is reflected an odd number of times from boundaries in which the wave resistance diminishes as the wave passes through them. Given some information about the structure of the sector under stiidy, we can get a more accurate idea MULTIPLE REFLECTED WAVES 81 of the corresponding properties of the reflecting boundaries by using multiple waves. Conversely, we can determine the arrival direction at the beginning of the path from the arrival directions of waves mukiplied an even and an odd number of times. A more comphcated phase inversion may occur in the propagation of partially multiplied waves because both the lower and the upper reflecting boundaries may be dissimilar as to the direction 28 32->^ 34 36- 4 Fig. 4. Records of multiple reflections obtained by modelling. in which wave resistance alters as the waves pass through them. In such a case one must consider only the number of reflections from boundaries where wave resistance decreases as the wave passes through, and allow for the difference in arrival direction at the beginning and at the end of the path after an odd number of reflections from such boundaries. Some investigators attribute the change in arrival direction which accom- panies a change in the number of multiples to interference reflected from the base and top of a thin layer with a thickness less than half the principal wave length. Applied geophysics 6 82 S. D. Shushakov In general, multiple waves reflected from a thin layer must be character- ized by complex oscillations consisting of vibrations reflected from the base and top of that layer. Reflections from thin layers have been described in works by Gurvich (^) Lyamshev <^) and Ivakin <'>. Fig. 5 shows theoretical hodographs of such waves. At short distances from the source, waves reflected irom the top and base of the layer may pile up one upon another. As the 700 200 Fig. 5. Theoretical hodographs of multiple reflections. distance from the source increases, the phase shifts between the component oscillations change, and head waves appear, superimposed on waves reflected from the base of the layer. At great distances from the source the phase shifts between the oscillations of some reflected waves increase with distance, but the phase shifts between the oscillations of the head waves and of the waves reflected from the base of the layer diminish. Many of these phase shifts increase with the number of multiples. As the number of multiples increases, or as distance from the source increases, the complex oscillations resolve into groups of simpler componental oscillations which are separated in time. Under real conditions it is difficult to distinguish between the multiple and single reflections from the shape of the oscillations. The problem has MULTIPLE REFLECTED WAVES 83 been studied by modelling thin reflecting layers, since such layers are often encountered under natural conditions. It has been shown that if the predominant frequency of single reflections in poorly absorbent media increases with distance from the source as the wave divides into its component parts, the predominant frequencies of multiple waves can increase — some- times after a slight decrease — for the same reasons, but at considerably greater distances from the source. The same thing is observed when the number of multiples increases. At great distances from the source the subdivision into groups of componental oscillations starts further from the source the greater the number of multiples. At first there is a sharp dechne in the predominant frequency, which then rises as soon as the number of multiples increases. In general, the rise in frequency in poorly absorbent media in which the vibrations suffer a change of intensity and shape depending mainly on the divergence of fronts and on interference from these waves is more pronounced when the distance from the source increases at the same time as the number of multiples. This effect becomes neghgible as the wave velocity in the reflecting layer increases. The predominant frequencies of single and multiple waves rise sharply with distance from the source on passing through the points of origin of head waves when the low-frequency components belonging to the head waves separate out from the summed vibrations. Johnson, using data obtained in California, shows that when the number of multiples increases at small distances from the point of origin, the traces of multiple reflections from a thin layer of basalt keep their shape. This agrees only with the results obtained from model experiments or with a rigid reflecting layer, (Fig. 4), an inference which is corroborated in other works (^^); but in recording; waves at high and medium frequency stations it has been noted that multiple waves are more clearly distinguished when they are recorded at medium frequency stations and when their predominant frequencies are consequently lower than those of the single waves. The explanation may well be that real media are more absorbent than models of them, so that as the wave path becomes longer there is a more noticeable enrichment of the vibrations in low-frequency components at the expense of relatively higher -frequency components. In the case of certain thin layers <^^), which possess comparatively slight wave resistance, we do not always find a strict similarity of form between; multiple and single waves. Figure 6 shows theoretical seismograms calculated according to the method of contour integrals for the case of a liquid layer covering a solid halfspace.. 6* 84 S. D. Shush AKOV The shape of the muhiple wave, hke that of the single wave, maintains to a certain distance from the source, but changes as the distance gi'ows further stilh The point at which the shape of the osciUations changes from constant to variable is the point of origua of the head waves. Beyond this point the shape of the multiple wave may not be repeated as the number of multiples increases at any one distance from the source. No such change in form is iound in certain cases of modelhng multiple waves in thm reflecting layers. Water /7| a, H Fig. 6. Theoretical seismograms calculated for large difference between the wave resistances of the media separating the reflecting plane. ^1=1.0, ai=1450m/sec, if=10cm, §2=2,47. ao=5490n/sec, 5=2700 m/sec. This may be because in these cases it is an interference wave consisting of waves reflected from the roof and base of the layer which is recorded. If the wave resistance diminishes as the wave passes through one of these boundaries, there will be no points of origin of head waves and the correspond- ing component Mill keep its form at any distance from the source. This /Component, superimposed on a second component for wliich there are points of origin of head waves, can cause the changes in its shape to be lunnoticeable. Thus, in the case of thin layers it can happen that the changes in shape wliich are chiefly observed are those caused by wave interference of the kind described above. This may explain the fact that, as the distance from the source increases, the single and multiple reflections from many tliin layers are easier to follow than the reflections from tliick layers. The modelhng of multiple waves has also shown that the principal frequencies become considerably higher as the thickness of the thin reflecting layer MULTIPLE REFLECTED WAVES 85 decreases, as the velocity in this layer increases and as the predominant frequencies of the excited oscillations rise. The fact that the predominant frequencies of multiple waves rise as the thickness of the reflecting layer decreases and the velocity in it increases has been confirmed theoretically <^) and from observations under natural conditions. For example, multiple waves reflected from the gypsum-anhydrite layer in the outer zone of the Cis-Carpathian depression, and waves reflected from a thin basalt layer (^*) are of higher frequency than waves reflected from argillaceous sandstone deposits in the same regions. The duration of the oscillations which is expressed by the number of visible periods, noticeably increases as the number of multiples increases (Fig. 4) and the velocities in the reflecting layer diminish. The reason is that the phase shifts between the componental oscillations increase. Oscillation intensity — Since the shape of a reflected wave changes shape with distance from the source, the concept of "intensity" in such cases is an arbitrary one, and we shall often take it as meaning a quantity which is a function of the greatest amplitude of the oscillations, irrespectively of the phase to which this amplitude belongs. In many regions, oscillations caused by multiple reflections are more intensive than many other vibrations. Usually, the multiple waves stand out in sharper relief the greater the wave resistance of the lower reflecting layer and the shallower the depth of absorption. This is particularly noticeable in the outer zone of the Cis-Carpathian depression, where there are sectors close to one another in which multiple waves are found to be reflected from a thin gypsum-anhydrite layer bedded at depths of the order of 500 m and less in some parts and at depths of the order of 1000 m and more in others. In the Ukraine, in the West Siberian plain, in Austria, California and certain other regions <^' ^^» ^'^' ^^) where there is a near-surface crystalline basement records of waves possessing distinctive amplitude and reflected singly and repeatedly from the surface of this basement have been obtained. In neigh- bouring sectors within these regions, where the surface of the crystalHne basement is deep-seated, the waves reflected singly or repeatedly from it are lost among the numerous other waves. The portion of wave energy which is lost through reflection from intermediate boundaries increases as the number of these boundaries increases with greater depth of the crystalline basement surface. Hansen (^*) reports that on apparently good records obtained in the River Salado basin, Argentina, all the distinguishable deep reflections are multiple and so intensive that it is impossible to pick out single reflections among them. Conversely, on seemingly poorer traces, multiple reflection? 86 S. D. Shushakov are not so evident and some singles can be distinguished. Bortfeld^^^) observed the same thing in north-western Germany. Under certain conditions then, mukiple reflections can be more intensive than single ones. Some investigators, comparing the intensities of multiple reflections of different types, point out that full-path echoes are the more intensive <^^). Such waves can in fact be distinguished by their records in the Krasnodar country, in the Ukraine and in many other regions. This can be explained to some extent by the fact that such waves are more noticeable owing to the periodic repetition of their records. Bortfeld, however (^-^J, demonstrates Fig. 7. Asymmetrical paths of wave propagation. that under certain conditions interbed echoes can be more intensive than fuU-path ones. The reason is that an interbed echo does not have a single path along which it is propagated, but when the reflecting boundaries are bedded horizontally, simultaneously arrives at the observation point and there causes co-phasally accumulated vibrations (Fig. 7). With increased angles of gradient of the reflecting boundaries or with increased differences between these angles, the phase displacements of the component vibrations increase. As a result of such interference some multiple waves can be distinguished while others are to some extent suppressed. The intensity of different types of multiple waves and their dependence on seismological conditions can be estimated by using tables compiled according to the method of contour integrals (^''\ The complicated relation- ships obtained by this method have been simpUfied by K. I. Ogurtsov for the following simple conditions : an excitant force applied to a free surface ; a lower reflecting boundary which is the surface of a solid elastic halfspace covered by a medium in which no transverse waves are excited, while the upper reflecting boundary is the surface of this medium parallel to the lower MULTIPLE REFLECTED WAVES 87 boundary. We give below formulas to describe the vertical component of displacement JF at the front of a double reflected wave. U < o < arcsm —^ , ^=ir.^. (3) With angles of incidence B /^\*ii_ ^1 = Q2 A{D^+^BC sin a) ^'y-l -r^ + ^(i)2+45Csin2a) ?2 W.= A^ 2n Ha^^ ^ q^ = E,: (4) Here IV^ is the vertical component of displacement at the front of a single reflected wave; a^, a^, b^ are the velocities of longitudinal and transverse waves in the overlying (1) and underlying (2) media; q^ and Q2 are the densities in these media, and H is the depth of the lower reflecting boundary; ^=|/l-sin2a; B = -Wl Vii)'-'--- sm'^'cc D <h 2 sin^ a. For angles of incidence arcsin — - < a< arcsin -r^ 02 02 W A^ 471 Ha^ |/^i = ]/[Re{E2^)? + [Im{E2^)\^F, (5) where ^2 = B[^V^i-AiD^- 4BCi sin^a) 62/ ^2 i = V — 1; Re and /^ indicate respectively the real and imaginary parts of the quantity £"2^; F expresses the form of the oscillation. 88 S. D. Shushakov For angles of incidence arc sin {ajb^) < a < 7r/2 the quantity W is expressed by the foregoing formula, in which we must replace E^ by ^3-= B l^\ ^^i+A{D^- 45C sin2 oc) \h I Q% g /_^ \ * ^ i _ J (i)2 _ 45C sin2 cj) \ ^2 / Q2 h, Vpi /J, Water M V ■*"~D '~*~~~~~rr"~*T5*~~^p""~~p*'^'p P +P p ~+'-_P •___JTJT^^^"~-^^-— rilllll _ 121 niiai^nzill n^2M IIIII2M2I1I1I nilllll2l "21111111-- , , 10 20 40 50 30 X, cm Fig. 8. Graphs of intensities of single and multiple waves as function of distance from source. Water: ;i=10cm, i;p=1450cm/sec2, ^=1.0; glass: 5=2.47, i;p=5490cm/sec2, i;5=2700cm/sec2. t, - sec 60 lOO Fig, 9. Theoretical seismograms calculated for smaU differences in wave resistances of media separating the reflecting plane. 5=1.0, aj=1450m/sec, //=10cm; ^2=1.46, O2=1710n/sec, B2=380m/sec. MULTIPLE REFLECTED WAVES 89 These formulas do not apply to the vicinity of the points of origin of head waves. Fig. 8 shows graphs representing the dependence of ^on distance from the source for various waves, and Figs. 6 and 9 show the corresponding theoretical Fig, 10a. Graphs showing intensity of double reflections as function of angle of incidence with different wave resistances of a thin reflecting layer, a — predominant frequency of radiated vibrations 60 c/s, b — predominant frequency of radiated vibrations 30 c/s. No Moteriol Thickness Ratio of wave Critical of layer, cm resistance onqle, degrees I Aluminiurn 0-62 9-32 I6''48' I, Aluminium 0-35 9-32 ISMS' n Glass 0-60 9-10 IS" 34' m Texiolite 1-72 3-39 as- 55' rz Organic gloss 1-60 2-26 32° 04' rz Organic gloss 0-50 1-95 38° 1 5" 2 Ebonite l-IO 1 -49 42° 00' vr Rubber 1-0 1 -04 ^0 S. D. Shushakov seismograms. In the calculations no allowance was made for absorption of seismic wave energy (the media are regarded as ideally elastic). In the case under consideration, moreover, we are treating the covering medium, for convenience in calculation, as possessing the properties of a liquid (transverse vibrations are not excited in it), and we do not make allowance for the dependence of the coefficients of reflection on the ratio of longitudinal to transverse wave velocities in the medium covering the lower reflecting layer. However, even in this case the formulas obtained express a compHcated relationship between the intensity of the multiple waves and the ratio between longitudinal and transverse velocities and wave resistances in the media indicated. Fig. 10 shows graphs obtained by modelling seismic waves. These graphs express the relationship between the intensity of waves reflected twice from various thin layers covered by water, and the angle of incidence. From theoretical calculations and experimental observations it follows that with small angles of incidence the intensity of the multiple reflections increases as the absolute values of the difference between wave resistances in the reflecting layer and the covering medium increase. In the case of a thin reflecting layer this intensity depends on the thickness of the reflecting layer to a greater extent, the greater the difference between the wave resistances in this layer and in the underlying medium. Damping — Little work has yet been done on the study of the variation in intensity of multiple reflections as a function of the distance from the source and of the number of multiples. Investigation of this relationship by the modelling method has yielded the following results. The intensity of multiple reflections in the case of thin reflecting layers has a complex relationship with the angle of incidence. Individual intensity peaks stand out against a general background of damping. The greater the differences between the wave resistances in the media separated by the reflecting surfaces, the greater the amount of overall damping and the more strongly do the intensity peaks stand out. A characteristic feature of double reflected waves, for example, is the existence, in many cases, of one extra minimum and one extra maximum at angles of incidence ranging from 10° to 30° (Fig. 10). When the difference between the wave resistance in the reflecting layer and that in the covering medium is not great, the curve representing the damping of the multiple waves distance from the source is found to have a wavehke form (Fig. 10). This is because of the periodicity of the function which expresses the dependence of the reflection coefficient on the angle of incidence <^) and also because of certain other characteristics of reflections MULTIPLE REFLECTED WAVES 91 from thin layers (*). It is also found that in the case of reflection from a thin layer the damping of multiple waves with distance decreases as the difference between the wave resistance in the reflecting layer and the wave resistance in the underlying medium increases. 50 40 £ 30 \ ■^ V^ r - ^ \. "XP y^ ^ \ vn\ \J ^^~- --v^^^ N yiSA ^ - — ^ __^ lU \ \ VI — . rv — 2^ iV ^ ^ ^^vVli ^m^ ^ ^-^ IV VI L, _V1 V *i-' 30 40 c. arad Fig. 10b. 50 m 60 The damping of reflected waves with distance from the source is in inverse ratio to the number of multiples. Fig. 11 shows the results obtained from modelling in the case of a thin reflecting layer. It can be seen that at short distances from the source and when the reflecting boundary is at shallow depths, the intensity of the multiple waves diminishes as the number of multiples increases. Further away from the source double waves become more intensive than single ones (Fig. 4). Still further away, triples are more intensive No Material Thickness of layer, cm Ratio of wave resistances Critical ongle, degrees I Aluminium 0-76 10-18 15° 15 n Glass 0-69 9-35 I5°47' m Textolile 1-56 3-38 24° 23' nz Organic gloss 0-52 3-26 32° 25' 2 Ebonite 0-62 1-57 49°49' a, Rubber (corrug ) 0-62 — — 53 Rubber 093 1-31 83° 13' HI Plastelme 1-06 1 -47 — I, Aluminium 0-35 10-35 15° 00 92 S. D. Shushakov 125 lOO 75 50 25 / ) ; / / ; H/2 1 / / / ; / - ' 22i— 0\ \^ in -^ N>^ ^ / /^^^ i'' / /3H- / ^===: O ( 4H 5H, — — r^^ 100 75 50 25 , H/2 -Q / / ^^5H lo) (b) Fig. 11. Graphs showing intensity A of multiple waves as a function of the number of multiples n at various distances from source; predominant frequency 60c/s; reflectmg layers: a— aluminium 0.62 cm, a^r 16° 48'; {a202)liaiQi) = 9.32; 6— glass —0.6 cm, o^r = 15° 34', (a202)/(aiei) = 9.10; c— organic glass— 1.6 cm, a^r = 32°04', (azQiili^iQi) = 2.26 MULTIPLE REFLECTED WAVES 93 than doubles, and so on. At great distances from the source (for example, four times the depth of the reflecting boundary) there is little difference in intensity between waves repeated a different number of times. In this case the modelHng was done with an emitter and a detector which did not possess acute directional qualities. This is borne out by the theoretical calculations (Figs. 8 and 9) and is explained by the dependence of the reflection coefficient on the angle of incidence, namely, the greater the number of multiples, the slower does the angle of incidence change with distance from the source and the slower does the reflection coefficient change. Modelling multiple waves in a case of a thin reflecting layer shows that when the distance from the source is large, the curve representing the damping of the multiple waves as a function of the number of multiples sometimes has a wavelike form (Fig. 11). Damping increases as the number of multiples increases, in inverse ratio to the difference between the wave resistance in the reflecting layer and the wave resistance in the covering medium (Fig. 11); and in the case of a thin reflecting layer, also in inverse ratio to the difference in wave resistance between the covering medium and the underlying layer. In some cases the refraction coefficient first increases with the angle ol incidence, reaches some maximum and then decreases. This effect not only reduces the damping of the multiple waves in some sectors of the observation hne, but also in certain cases, causes their intensity to increase, with distance from the source. With some slight difference in wave resistance and in the travel velocities of the waves in the media covering and underlying the reflecting boundaries, for example, multiple waves can increase in intensity up to a certain distance from the source. This is implied in the foregoing formulas and is shown in Fig. 9. The relationships we have indicated are to a considerable extent connected with the absorbent properties of the media lying between the reflecting boundaries. Multiple waves were therefore modelled under conditions in which water, marble, paraffin, plastilene, were used as the covering media, and the absorbent properties of real media must he within the range of the absorbent properties of these models. Despite the wide range of the latter, quahtative confirmation of the dependence we have indicated was obtained from the modelling, but the quantitative ratios varied. We may note that the relationship between damping and frequency is connected with the absorbent properties of the media. The damping of multiple waves consequently depends on the prevaihng frequency of the vibrations, and in the case of strongly absorbent media must be more pronounced with relatively high prevailing frequencies than with low ones. 94 S. D. Shushakov In the upper part of Fig. 10 we show the damping of a double wave at higher frequencies than in the lower part of the same figure. The relationships we have indicated are also connected with the depth of the reflecting boundary. If this depth is reduced the path of a multiple reflection is reduced (the more multiples there are, the smaller it becomes) to a greater extent than the path of a single reflection. At the same time, the angles of incidence increase at shallow depths more rapidly with distance from the source than they do at great depths, and become bigger in proportion to the number of multiples. Therefore the intensity of a reflected wave with a large number of multiples grows as the depth diminishes, to a greater extent than does a reflected wave with a small number of multiples. The relationships indicated are more pronounced when the angles of incidence at the lower reflecting boundaries are small in comparison with those at the upper reflecting boundaries, and less pronounced when these angles are large. Moreover, the intensity of a multiple reflection in the direction of a rise of the lower reflecting boundary is characterized by having a more distinct peak than the intensity of a single wave; whereas in the direction of a dip in this boundary the intensity is characterized by monotonic and sharper damping. At the same time, the intensity of waves with a large number of multiples becomes greater in the direction of a dip in the lower reflecting boundary, in inverse ratio to the number of multiples, at considerably greater distances from the source than it does in the direction of a rise. Some Features of Multiple Waves which have their first Reflections above the Excitation point — Records of waves obtained by well-shooting, where the first reflection occurs above the shot point, can be more inten- sive than records of direct waves. The reflection coefficient from the base of the low-velocity zone is roughly equal to 0.6-0.8 (16,5). The amplitude of the vibrations for a spherical direct wave is. A, ^,- 110_ -aSa where Aq is some constant, S^ is the path length and a is the absorption coefficient for one unit of path. For a reflected wave Ar- ^^ e , where K is the reflection coefficient and S^. the length of path. Hence K=4^e'^^'r-s,). (6) MULTIPLE REFLECTED WAVES 95 The amplitudes of multiple waves recorded on the surface, when the first reflection occurs above the shot point, can be less than, equal to or greater than the amphtudes of the corresponding single waves (^). Multiple waves have been found superimposed on single ones. The peak of the combined wave must occur when the shot depth is such that the distance from the excitation point to the upper reflecting boundary is equal to one quarter of the wave length. Furthermore the intensity and shape of the oscillations in a multiple wave depends on the properties of the several boundaries from which it is reflected. This partly explains its greater variability along the observation Hne than that of a corresponding single wave. This intensity and shape of vibration may be different at reciprocal points omng to different conditions of reflection at these points from the upper boundary. REMARKS ON THE USE OF MULTIPLE REFLECTIONS A question which frequently arises in regions where records of multiple reflections predominate in intensity over records of single waves, so that the single waves cannot be distinguished, is whether the multiple reflections can be used. In these regions the follomng problems must first be solved: (1) recognizing multiple reflections (establishing the boundaries from which they are reflected, and determining the types of vibration — longi- tudinal or transverse — with which they travel between these boundaries) ; (2) determining whether it is possible to follow these waves in the region under investigation. To recognize multiple reflections one must use the kinematic and dynamic characteristics of multiple reflections which have been indicated above, and on the basis of appropriate observations show to which type of waves they belong. The second problem is dealt with by trying to follow these waves under the various conditions which exist in the region concerned. For regions where both initial problems can be settled satisfactorily, it only remains to decide whether the corollary task of interpretation can be dealt with. This question can be decided in a first approximation by construc- ting a seismic cross -section with the condition that all the distinguishable waves are regarded as single waves. Then, in the case of full-path reflections, and with a known number of n, one must compute the angle of i ncide nce of the lower reflecting boundary, which is equal to y^^\\ Z'^' ' WOODS HOLE. MASS. 96 S. D. Shushakov where y' is a fictitious angle of incidence computed from the section con- structed. After this formula (1) can be used to calculate the depth H^ along the normal to this boundary and to construct the cross-section for the boundary. For different multiple waves and under different seismic and geological conditions, there may be a different answer to the problem of interpretation. Accuracy may vary and may more or less satisfy practical requirements, just as in the case of single waves. In elaborating methods for using multiple reflections a problem which may arise is how to use these waves to separate, in the cross -section, the reflec- ting boundaries which have the highest reflection coefficients, with the result that multiple waves are reflected from these boundaries. The identification of such boundaries can help to improve the geological interpretation of seismic observations. For this purpose we can use the determination of effective velocities from the hodographs of certain multiple reflections, such as the hodo- graphs of the most distinct full-path echoes. In order to use these velo- cities we must determine the kind of boundaries these waves are reflected from. For purposes of geological interpretation, it is possible to use the relationship ^lescribed in the foregoing sections, between the directivity of vibrations with different numbers of multiples and the combination of properties in the boundaries from which they are reflected; we can also use the shape characteristics of the graphs representing the variation in intensity or form of the record for multiple waves with distance from the source and with increase in the number of multiples. PRINCIPAL CONCLUSIONS Multiple reflections can interfere considerably with the recognition and tracing of single reflections and — under certain conditions — refracted waves. This is particularly true when the interfaces from which these waves are reflected are at shallow depths. Moreover, they can be taken for single reflections and so give rise to errors in interpretation. At the same time, under certain conditions multiple reflections can be used in seismic prospecting, especially in regions where these waves predominate over single waves which have nearly the same travel times as their own. So far, however, insufficient work has been done on elaborating methods for recognizing multiple reflections, Recognition by kinematic indications MULTIPLE REFLECTED WAVES 97 is not sufficiently unambiguous, and must be supplemented by the use of dynamic indications. The first point to be noted is the variation or lack of variation in the direction of the oscillations when the number of multiples changes to an odd number. This depends on the combination of properties of the interfaces from •which the multiples are reflected. The prevailing oscillation frequencies, caused by multiple waves reflected from thin layers and travelling through poorly absorbent media, can under certain conditions rise ■when there is a simultaneous increase in the number of multiples and in the distance from the source. The prevailing frequencies of single reflections usually fall, but in certain cases can also rise, with increase in the depth of the reflecting boundary. Consequently, under certain condi- tions, a frequency analysis of the waves recorded may be an aid to recognizing the multiple waves. Sometimes it may help to compare the duration of the vibrations, expressed by the nmnljer of their periods; for multiple reflections from thin layers this increases with the number of multiples. The damping of multiple reflections with distance from the source is weaker the more multiples there are; at some distances from the source (of the order of the depth of the reflecting boundary) some multiple reflections with a larger number of multiples can be more intensive than multiple reflections with a smaller number of nndtiples. Consequently, at greater distances from the source in comparison with the depth of the principal reflecting boundary, multiple reflections may predominate over single ones with respect to the amplitudes of the traces. Multiple reflections from thin layers may be characterized by a peculiar shape of the curve representing the relationship between a quantity connected with the amplitude of the vibration on the one hand, and the distance from the source and the number of multiples on the other. REFERENCES 1. I. S. Berzon, Hodographs of multiple reflected, reflected-refracted and refracted- reflected waves. Izv. Akad. Nauk SSSR, ser. geogr. i geofiz. No. 6 (1942). 2. I. S. Berzon, Effective velocities and depths determined from hodographs of multiple reflections. Izv. Akad. Nauk, SSSR, ser. geofiz. No. 8 (1956). 3. I. I. GuRviCH, Reflections from thin strata in seismic prospecting. Applied geofiz. Pt. 9. Gostoptekhizdat (1952). 4. A. M. Yepinat'yeva, Some seismic waves with long travel times. Izv. Akad. Nauk SSSR, ser. geofiz., No. 6 (1952). 5. A; M. Yepinat'yeva, Some types of multiple seismic waves. Izv. Akad. Nauk SSSR, ser. geofiz. No. 8 (1956). 6. N. V. ZvoLiNSKii, Multiple reflections of elastic waves in a layer. Tr. Geofiz. in-ta Akad. Nauk SSSR, No. 22 (149) (1954). Applied geophysics 7 98 S. D. Shushakov 7. B. N. IvAKiN, Head, forward and other waves in the case of a thin soUd layer in a hquid. Tr. Geofiz. in-ta Acad. Nauk SSSR No. 35 (162), (1956). 8. I. K. Kupalov-Yaropolk, Multiple reflections. Applied geofiz. Pt. 6, Gostoptekhizdat (1950). 9. L. M. Lyamshev, The Reflection of Sound by Thin Plates and Films in Liquid. Izd. Akad. Nauk SSSR (1955). 10. G. I. Petrashen', Problems in the Dynamic Theory of Seismic Wave Propagation. CoU. papers I, Gostoptekhizdat (1957). 11. R. BoRTFELD, Beobachtimgen multipler Reflexionen ia Nordwestdeutschland. Erdoel- Leitschrift Bd. 72, Nr. 9 (1956). 12. C. H. Dix, The existence of multiple reflections. Geophysics, 13, 1 (1948), 13. J. P. EixswoRTH, Multiple reflections. Geophysics, 13, 1 (1948). 14. R. F. Hansen, Multiple reflections of seismic energy. Geophysics, 13, 1 (1948). 15. J. Sloat, Identification of echo reflections. Geophysics, 13, 1 (1948). 16. F. A. Van Meller and K. R. Weatherburn, Ghost reflections caused by energy initially reflected above the level of the shot. Geophysics, 18, 4 (1953). 17. J. C. Waterman, Midtiple reflection evidence. Geophysics, 13, 1 (1948). Chapter 4 DIFFRACTED SEISMIC WAVES T. I. Oblocina INTRODUCTION In order to solv^e the problems which are encountered in seismic prospecting over areas of complex geological structure it is essential to study the dynamic properties of waves, since kinematic indications are often missing when the wave pattern is of a complicated nature. When waves are diffracted their dynamic properties acquire fundamental importance for determining the different types of waves, and so for a correct geological interpretation of the data obtained in seismic prospecting when tectonic disturbances or steeply dipping and tapering layers are present in the cross-section. Diffraction is generally understood to mean either the curving of waves round obstacles in their path or the scattering of waves on various irregularities in the medium. Diffracting objects in seismic prospecting can be divided into three types: diffracting edges, individual diffracting bodies and multiple diffracting bodies. The first category includes the edges of extended interfaces. Examples of structural features which contain diffracting edges are fault-type dislocations, steeply dipping and tapering strata, the cleges of intrusions and salt domes, and so forth. The second category includes more or less isometrically shaped bodies which are small in comparison with the wave length: ore chutes, caverns and kettle holes. We may note that ore chutes and kettle holes are not objects of study by seismic prospecting methods; at present searches for these are carried out by other geophysical methods. The third category includes a great number of bodies which are small in comparison with the wave length: small inclusions of rocks which are different in their lithological composition from the enclosing rocks, small reflecting surfaces distributed at random in a fractured zone, small irregu- larities in interfaces. Small diffracting bodies make the medium texturally nonhomogeneous ("turbid"). In seismic prospecting this interferes with 100 T. I. Oblogina the isolation of useful ^vaves; the record becomes so difficult to decipher that as a rule the individual waves cannot be distinguished. Of the three types of diffi-acting objects we have mentioned, the one which has greatest practical importance is the diffi-acting edge. Comparatively little ^vork has been done on problems of diffi'action in seismic prospecting. Most of the existing pajjers deal with particular problems in the Idnematics of diffi'acted waves ^^' ^' ^' ^' ^K 0«ly in certain experimental works do we find any indications of the relationship between the amplitudes of diffi-acted waves and the ampUtudes of reflected and refracted waves: some investigators stress the low intensity of diffracted waves as compared with refracted waves (^), while others on the contrary note that diffracted waves may have the same order of intensity as other types of waves (^■'^» ^^K In essence, however, the dynamic properties of waves in a case of diffraction have yet to be examined. This paper explains the kinematic basis for distinguishing diffi'acted waves on seismograms, sets out recent experimental findings on the dynamic properties of these waves and compares these findings with theoretical data. THE KINEMATIC PROBLEM OF DIFFRACTION FROM THE EDGE OF A VERTICAL CONTACT Let us imagine a combination of media consisting of a medium mth a velocity Vq filling an upper half -plane, and media with velocities t\ and v^ filling respectively the left-hand and right-hand quarters of a lower half -plane (Fig. 1). Let the velocities Vq, v-^ and v^ be constants, and let Vq <,v-^'^V2. We shall treat the case of f ^ < '^2 separately from the case of t\ > t'g. The Vq medium is separated from the other two media by a horizontal interface; these two in turn are separated from each other by a vertical boundary. Let a system consisting of a head and a shear wave move along the horizontal interface from left to right. In Fig. 1 (a) the clashed line shows the fronts of these waves at some moment of time f < 0, before diffraction has occurred, and the direction of travel is indicated b}- arrows. It is required to determine the kinematic pattern of Avave propagation after diffraction occurs; that is, we have to find the position of the wave fronts in the plane xy and the form of the wave hodographs* in the plane xt. We choose a system of co-ordinates xy such that its origin lies on a diffracting edge, with the :;c-axis running along the horizontal interface and the r-axis upwards towards the observation line (profile). The contact * A Russian term referring to distance-time curves. [Editor's footnote]. DIFFRACTED SEISMIC WAVES 101 edge— the rectilinear diffracting edge — will then pass through the origin of the co-ordinates normally to the plane of drawing. We shall now examine the case when i\ is greater than ly, that is, when the shear wave passes from the medium with the higher velocity into the medium with the louver velocity. At a mom.ent of time ^ = 0, let the system consisting of the head ^\■ave and the shear wave reach the vertical contact Fig. 1. Fronts and hodographs of a diffracted and a head wave in a case where a shear wave passes from a higher- to a lower-velocity medium. edge 0. According to Huygens's principle, this edge is, as it were, the source of a diffracted wave travelling in all directions. In Fig. 1, (a) shows the position of the wave fronts at a moment of time ^ > 0, when the wave has already encovmtered the contact edge. The front of the diffracted wave in the Vq medium will be the semicircle KCDEF with radius VqI; in the fj medium it will be the quarter circle GA with radius v\t; and in the fg medium it will be the quarter circle MB with radius v^t. The shear wave is partly reflected from the vertical contact and partly refracted by it. The reflected shear wave causes a head wave EG on the path FG, while the refracted shear wave will pull the head wave CB on 102 T. I. Oblogina to the path KB. The fronts of the head waves EG and CB will touch the front of the diffracted wave at E and C respectively. Note that after diffraction there are two head waves for the one head wave before dijffraction (dashed hne): the wave CB which we have referred to above, and a wave DN, the front of which is parallel to the dashed-line front and touches the front of the diffracted wave at D. We may also note the further wave AL excited on the path MA by the front GA. Writing down the equations for the wave hodographs along the longitudinal profile r = H, we obtain ^f + H'; (1) for the diffracted wave KCDEF: 1 H ,_. t-. = — :\- -1 cos oci ; (z) for the head wave DN; for the head wave EG; , 1 H ,„> t-, == X ^ cos oc-i ; (o) V-, Vc to — -X -] COS OCo , {f*>) V2 Vq for the head wave CB where oci = arc sin — , oco — arc sin — — the critical angles. Figure 1, b shows hodographs for Iq, t-^, t^ and t^ waves. In the sector of the profile which lies above the contact we observe a diffracted wave t^, the hyperbolic hodograph of which touches the hodographs of the head waves t-^, t^ and 1^2 (the wave t'-^ has a negative apparent velocity). As we see, when v-^ is greater than v^^^ the diffracted wave emerges into the first onsets in the sector of the profile Ax = H(tdn 0C2 — tan (Xi) = Hvq We shall now examine a case when v^ is smaller than v^ and the system of a head and a shear wave moves as before from left to right; in other DIFFRACTED SEISMIC WAVES 103 words, the shear wave passes from the lower -velocity medium to the higher -velocity medium. By a similar process of reasoning we obtain the arrangement of wave fronts for some moment of time « > in the plane xy (Fig. 2, d) and the position of the wave hodographs in the plane x, t (Fig. 2, h). Fig. 2. Fronts and hodographs of a diffracted and a head wave in a case where a shear wave passes from a lower- to a higher-velocity medium. On the sector of the profile Ax — H{tan oci — tan x^ — Hvq ]Jv^ - V^ Y^ - Vq^ the hodographs of ti{x) and ^2^^) intersect, while the hodograph of the diffracted wave tQ{x) touches these hodographs at the points M(the beginning of the hodograph of t^^ix)) and M (the end of the hodograph of ti{x)). The hodographs of tQ{x), ti{x) and t2{x) form a kind of loop between the points L, M, N. This case differs from the previous one in that the diffracted wave does not emerge into the first onsets. Combining both the cases we have examined, we conclude that the minimum on the direct and inverse hodographs corresponds to one and the same marker peg on the profile ; it lies above the diffracting edge. Here we have a diagnostic 104 T. I. Oblogina Fig 3. Example of a diffracted wave trace ^^^ refracted waves with low apparent velocity; ^^-refracted waves with high apparent velocity; f^ -diffracted wave. DIFFRACTED SEISMIC WAVES 105 character which is pecuhar to a diffracted wave and distinguishes it from all other types of seismic waves. We may further note that in overtaking systems the tangential points of the hodographs of the diffracted and the head waves also correspond to a single marker peg on the profile. FIELD OBSERVATIONS The field observations were made in a region ^vhere the geological cross- section included primarily aranaccous and sandstone deposits of Cretaceous, Ternary and Quaternary ages. The Cretaceous deposits lie directly on a Paleozoic basement at a depth of about 1000 m. Ihe top of this basement contains rocks of different lithological varieties bedded in the form of a block of steeply dipping strata. Diffraction of vertical head waves ^vas observed to occur from the contact edges between these strata, which have different velocities. The observations were made mth a standard detector whose response curve had its maximum at /^ax ^ ^"^ ^h- Kinematic Properties. Three groups of waves -were recorded in the covirse of seismic observations in the sector mentioned: A, B and C. Group A consisted of refracted waves with low apparent velocities (1200-1600 m/sec); group 5 of refracted waves with high apparent velocities (5000-6500 m/sec) ; group C of diffracted waves. In this paper we shall treat in detail only the characteristics of the diffracted waves. The seismograms given below show diffracted ^vaves recorded after the refracted waves ^sith high apparent velocities. The hodo- graphs of these waves invariably have an almost hyperbolic shape. On the seismograms in Fig. 3, group A refracted waves, a group B refracted wave and a group C diffracted wave can be distinguished. The diffracted wave t(^, which has a distinctly curvilinear axis of co-phasality, was followed over a distance of 450 m. Another example of a diffracted wave trace is given in Fig. 4. Here the first waves to arrive are the B group waves t^, t^ and t^. A diffracted wave tf-. is recorded behind the wave t^. This can be seen on the seismogram at a distance of 5100 rn from the shot point and followed up to a distance of 5520 m, where it is already converging -with the wave t^. The diffracted wave has a distinctly curvilinear axis of co-phasality; its intensity is roughly the same as that of the wave t^. 106 T. I. Oblogina Specific observations showed that minima on the opposing hyperbolic hodographs of the diffracted waves corresponded to one and the same marker peg on the profile. Figure 5 shows two seismograms a and b, one of which corresponds to the direct hodograph and the other to its inverse. On both seismograms it can be seen that the minimum travel time of the diffracted wave corresponds to the 5100 marker. Fig. 4. Example of a diffracted wave trace t^, t^, fg-group B refracted waves; f(;;— diffracted wave. Note the following point about the relationship between the intensity of the recorded vertical component of the diffracted wave and the angle at which this w^ave approaches the detectors. For a distance x along the profile from the projection of the diffracting edge on to the plane of the observations, when x is not greater than the depth H of the diffracting edge {x < H), the angle of approach (p of the diffracted wave (that is, the angle between the diffracted ray and the observation line) is not less than 45° (9? ^45°). Within the range of such x distances therefore, depending on the depth H, a fairly intensive vertical component is recorded; whereas for X ^ H the horizontal component will be greater than the vertical. There is a particularly marked variation in the approach angle when shallow depths are being prospected. The only indication that diffracted waves have been recorded during shallow depth prospecting which is to be found in the literature is in Yepinat'yeva's paper (^). Ordinary seismic apparatus DIFFRACTED SEISMIC WAVES 107 Fig. 5. Seismograms obtained from two opposite shot points, showing that the minimum on the direct and inverse hodographs of the diffracted wave correspond to the same profile marker (marker 5100). 108 T. I. Oblogina was used in the work and the extremely low intensity of these waves was noted. We must also emphasize that the dominant frequency of the diffracted waves is lower than that of the head wave. Therefore an apparatus for which the response curves have a steep left-hand slope and which is designed to record high frequencies produces marked distortions in tracing diffracted waves. Dynamic Characteristics — Examining the diffracted waves on the seismograms we can see that the amplitudes of the vertical component of displacement of a diffracted wave are different at different points on Fig. 6. Seismogram showing growth of diffiacted wave amplitude near point of contact with refracted wave. the profile, and that a diffracted v,'ave has its greatest intensity in the neighbourhood of the point at which is tangential to a refracted wave. Figure 6 shews a seismogram on Avhich a refracted wave is recorded in the first onsets and a diffracted wave in the last onsets.. As we see, the amplitudes of the diffracted wave increase along the profile as the wave approaches the point at which it is tangential to touch the refracted wave. From analysis of the seismograms it follows that in the vicinity of a point of tangentionality the phase of the diffracted wave changes. As a rule this gives DIFFRACTED SEISMIC WAVES 109 Fig. 7. Seismogram showing phase inversion of diffracted wave near point of contact with refracted wave. 110 T. I. Oblogina the impression, that in the vicinity of this point the detectors are connected to the -wTong poles (Fig. 6). In Fig. 7, a and 6, we show a seismogram on which a refracted wave is followed in the first onsets and a diffracted wave in the last. As can be seen, a phase inversion occurs in the region of markers 70.96-70.45. The smaller the spread, the more marked is the phase inversion near the point of tangentionality. 2-2 2-1 2-0 UXLi 0-2 0-4 0-6 0-8 1-0 1-2 K, km Fig. 8. Observed dyuamic hodographs of a diffracLed wave along longitudinal profiles. Dynamic hodographs were constructed for the diffracted waves observed. An example is given in Fig. 8. Here, as usual, distances from the shot point are plotted along the .%-axis and time along the 3'--axis. The amplitudes of the diffracted wave were plotted from the points of the ordinary Idnematic hodograph t = t{x). The dynamic hodographs show that the intensity of the diffracted wave increases as the wave approaches the point of tangentionality we have referred to, and that phase inversion occurs in the neighbourhood of this point. THE DYNAMIC PROBLEM OF DIFFRACTION FROM THE EDGE OF A "TAPERING STRATUM" Formulation of the Problem — • Let us take two media made up of two elastic liquids separated by a plane boundary. The upper medium is characterized by the velocity l/a^ = ]/ (Aq/^q), and the lower one by I/cq = y (^i/^i), where 2.q and /Ij are the elastic constants and ^q, q^ the densities of the media. We shall assume that the velocity in the upper medium is lower than in the medium which is underneath it. DIFFRACTED SEISMIC WAVES 111 We choose a system of co-ordinates so that the ::c-axis runs along the interface and the j-axis up towards the upper medium (we are considering the plane problem). Along the half -line x > 0, y = h let a. cross-section be taken, the edges of which are firmly attached; that is to say, the displacements on them will be zero. Such a cross-section will be an approximate representation of a thin tapering stratum with high velocity in seismic prospecting. Let t <. and let the following wave system n = nfiit - GqX sin ccq - a^y cos ^o), 7o = — • (5) Gq cos (Xq <Pi = A/lit -a^x)~Byf^'{t~a^x}, A = ^^li«_ , B = 2a^. (6) Oo Qx COS Oq i. \ / where Oq = arc sin Oj/ao t>e propagated along the interface in the direction of increase of x. The potentials cpQ and (p-^ satisfy the wave equations in the upper and lower media. As has been shown in (^\ formulas (5) and (6) give the local representation of a head and a shear wave near the interface. At a distance from the boundary we can regard (5) as an ordinary plane wave. We shall give the function f^ in the form \ i when 5 > 0. Since the components of the displacements are expressed by the displace- ment potential as partial derivatives of this potential along the corresponding co-ordinates, the boundary condition on the boundaries of the cross-section X '> 0, y = h \% expressed by the equation Now let the wave (5) meet the edge 0^ of the cross -section at a moment of time f = 0. Fig. 9 shows the wave fronts and the values of the displacement potential in front of and behind the wave fronts at some moment of time i > 0. It is required to find the diffraction disturbance, at a moment of time ^ > 0, which is concentrated inside the region bounded by the contour ABA^CA (Fig. 9). With high values of h, the problem as formulated is in essence the classic problem of the diffraction of an ordinary plane wave from a rectilinear edge 112 T. I. Oblogina for a single ■wave equation. This problem has been solved in several works C^'^). The solutions obtained, however, are unsuitable for studying the dynamic properties of diffracted waves, since the very construction of these solutions makes it difficult to extract an expression for the displacement field near the wave fronts which is suitable for purposes of calculation. The problem of plane -wave diffraction from the edge of a tapering stratum is solved below by the Smirnov-Sobolev method of functional invariants; the dynamic hodographs and theoretical seismograms for diffi-acted waves are calculated. Fig. 9. Diagram to aid formulation of diffraction problem for a plane wave from the edge of a "tapering stratum". Solution. The function 9^0 depends solely on the ratios ;r/i,y/i. As Sob olev shows, by substitution of the variables ^ = xjt t] = yjt the wave equation ° 9t^ 9x^ ' 9y^ is converted into an equation of mixed type, {a,H'-l) n,,+^<hn,n+^< ^'-1) 9'o..+ 2«o^l<. + 2ao2 72990^ = 0. (9) (10) The above equation in the hyperbolic region is reduced by substituting the variables according to the formulas: DIFFRACTED SEISMIC WAVES 113 |2 + ^2' 1 |2_|_,^ to the chord equation In the elHptical region, by substituting the variables according to the formulas _ s ,^^'>/i^v(f:+'/') (13) |2 + ^2 . 1 ^2 + ^2 it is reduced to the Laplace differential '^^"+^-0. (14) 9a^ ' 9tj^ Using this and introducing the complex variable "1" ^ ..2 I /.. 7,\2 ' U"-*-' " .x2 + (^_/j)2 ;»;2 _|. (^ _ /^)2 we shall seek the solution in the region ABA^CO^ A (Fig. 9) which is filled mth a diffracted wave in the form n = Re^{0,)^ (16) where ^{6^ is the analytic function in the region to which the region ABA^CO^A passes; and R^0{6^ is the real part of the function. Let us see what happens to points on the circumference ABA^CO^A after conversion (15). It can be readily seen that O^ = a^ corresponds to the point A {x = t/aQ, y = A} ; ©0 = Oj to the point B {x =^ (^/c^o) ^i^ <^o ' 7 — (*/<^o) ^^^ ^o} '■> 0Q = — Qj to the point A^ {x = —t/aQ, y — h} and 0q = oo to the point Oj_{x=0, y=h]. The lower semicircle is converted into a lower plane, while the whole circle O^ABA^ CO^ passes by means of the conversion indicated into the plane of the complex variable Oq with a cross-section along the real axis Qq^ ~ ciq . The boundary condition (8) is written in the form Re {V a,' -6,^0' (do)} =0. (17) when we use the variable 6q. Applied geophysics 8 114 T. I. Oblogina Making use of the fact that in the region ahead of the front of the plane Avave (5) 99q = 0, in the region ACD (pQ = 2'yQ and in the region to the left of the hne NBA^CDM (p^ = y^, we introduce the boundary conditions for the function 0'{Q^. In the interval Oq < 0q < oo into which the segment 0-^A^ passes, Q^ always satisfies the inequality 0q > Cq. In the boundary equation (17) the radical l/(ao~^o) ^^^ have an imaginary value, and therefore to fulfil this boundary condition the condition ImO' {B^ — must be satisfied. It is sufficient to solve the problem for the upper half -plane. At the point Qq = a^ the function 0'{Q^ will have a pole with its main part — [iyo/^ (0Q — %)]. Let us now construct such a function so that its material part is converted to zero. We multiply the function ^'{d^ by the radical }/{ao~^o)- For the upper half -plane when Qq > a^ we choose a minus sign ; that is, Im]/{aQ-Q^<Q. It will be seen that R^{0' {Q^]f{aQ — d^ =0 on the entire material axis. Making use of this, and also of the fact that at the point ^q = % there will be a pole with the main part we have indicated, we obtain The components of the function required are equal to u = R40'{Q,)^ . = i?.(m)f) (19) Formulas (18) and (19) give the solution to our problem. We thus find the displacements in a region filled with a diffracted wave. The Dynamic Hodographs — To determine the dynamic hodograph of a diffracted wave we must find an expression for the components of the displacement of this wave near its front. First we find asymptotic formulas for the displacement of the diffracted wave in the vicinity of its front for the case of a Dirac pulse as the shape of the incident wave, then we convert to an alternating smooth pulse for the displace- ment. For the region near the front of the diffracted wave when r-> ^/oq? '^^ obtain from (15) and (19) the following asymptotic expressions for the hori- DIFFRACTED SEISMIC WAVES 115 zontal component w^ and the vertical component v^ of the displacement: y2aQ^.x{v-h)Im0'{Oo') 1 k'here "-IV-^' /? '^^ h^f^. ^/t-a^r , ^o^'-r\ v^{x, J, t) f2a,^y-hYlm0'{d,') 1 h^]/to yt — aQT ^o{^^'-<^'). ^0 (20) (21) Here t^ is the time of the first onsets of the diffracted wave. Formulas (20) are valid in the neighbourhood of its front, except for the vicinity of the singular point B (Fig. 9), where they do not apply. We can now give the shape of the incident wave in the form of a smooth pulse G' (0 == 4 h {t-T). (22) To obtain the components of displacement u-^^ and v-^ for such a pulse, we must, as is known, integrate according to a parameter i-^ {t, x,y) = J u{t — r, X, y) dG, (23) where u is the function for the case of a single pulse, which we know already, and T is the parameter. Performing the integration, we get (24) where /{t-a^r) = 4.T^ {At {\^t-]/At-T) - ^ [At^r-~ {At-~T)^I^]] - 116 T. I. Oblogina 12 T{At'- {yjAt -yj At-T) -^At [At^l^-{At~T)^i^ + 1 + — [At^l^-{At-Tyi^]} + 8 {At^ {]/At -YAt-T) - -At^Af^!^-{At-T)^i^] + ^At[At^i^-{At-Tyi^- _^ 1 [Jf7/2 _ (J i_ 2^)7/2] j^ At = t — aQr. To calculate the dynamic hodographs from these formulas we must first find the values of At at which the function /(f — aQ^) has its first ex- treme value. Assuming T — 0.03 sec and calculating values for the function f (t — aQr) at a series of points, we find that when At — 0.01 sec it has its first maximum, at a value of 0.857. We shall further assume that the velocity in the upper medium l/oj = 4 km/sec, and that the bedding depths of the interface and the tapering stratum are y = 1 km and jl — ^^ = 0.5 km respectively. We select observation points on the profile at intervals Ax = 200 m and calculate the first onset times at these points, using the kinematic hodograph equation. For each pair of values of x and ig "^^ compute values for 0q and 0' {0q) according to formulas (21) and (18). We then calculate the functions u^ and v^ according to formula (24) and obtain a value for the wave amplitude at each point on the hodograph. Figure 10 shows the dynamic hodographs for u^^ and Vi calculated for twenty - one observation points on the profile. The following regular features can be noticed: the amphtudes of the diffracted wave are different at different points of the profile; and the nearer the observation point is to the point of contact between the hodographs of the diffracted and the head waves, the greater the amplitudes of the diffracted wave. In the example given, the point of contact has an abscissa x = 0.288 m. The vector of the displace- ment [ui, fj) changes sign near this point of contact. The horizontal com- ponent of displacement u^^ becomes zero at the point on the profile x = 0, which is a projection of a point on the diffracting edge on to the profile. This is obvious even physically, without calculation: the displacement vector runs along the radius of a circle which is the front of the diffracted ■wave; at the point on the profile x = 0, y — 1 km it is perpendicular to the A' — axis, and consequently its horizontal component is nil (Fig. 10). DIFFRACTED SEISMIC WAVES 117 When the diffracting edge hes on the interface, the following may be noted. As can easily be seen, in order to calculate the dynamic hodographs in such a case, we must assume A = and confine ourselves to values of x>0. The shape of the diffracted wave in a case where the incident wave has the shape of a smooth alternating pulse (22) can be found from formula (24). Fio. 10. Calculation dynamic hodographs of a diffracted wave a-horizontal com- ponent of displacement, u^; 6- vertical component of displacement, v^. 118 T. I. Oblogina From this formula we can see that the expression for w^, as well as that for i;i, can be regarded as consisting of two multipliers, one of which is the function /(i — Oor), and the other all the rest of the expression which goes before this function. The function f{t — a^r) itself also gives the variation in time of the components of displacement (that is, the shape of the wave), and the multipher mth it gives the change in the shape of the wave as a func- tion of the position of the observation point % on the profile, the first onset time ^0, the velocity values I/aq and l/aj, the distance on the profile from E -0-2 Fig. 11. Theoretical seismograms for a diffracted wave a-trace of horizontal com- ponent of displacement; 6-trace of vertical component of displacement. DIFFRACTED SEISMIC WAVES 119 the interface and the distance h from the "tapering stratum" to the inter- face. Figure 11 shows two theoretical seismograms of the horizontal and vertical displacements of a diffracted wave in a case where a wave incident on the edge of a "tapering stratum" has the shape of a smooth alternating pvdse. The same values have been taken for all the characteristics of the medium as were taken in calculating the dynamic hodographs. The shape of the diffracted wave can be seen on each trace. The dominant period of this wave {Tp = 0.04 sec) is greater than that of the incident wave {T = 0.03 sec). Moreover, with the given shape of the incident wave, both its extremal values — the first and second— are identical. The value of the first extreme for the diffracted wave, however, is greater than that of the second, so that in the given case this wave is characterized by a less symmetrica] vibration shape. SUMMARY Comparison of the experimental and theoretical findings set out above yields the following conclusions. 1. Minima on the direct and inverse hodographs of a diffracted wave correspond to one and the same marker on the profile. 2. The amplitudes of the diffracted wave increase along the profile as this wave approaches its point of contact with a refracted Avave. 3. The diffracted wave suffers phase inversion in the neighbourhood of this point of contact. REFERENCES 1. I. S. Berzon, Some problems in the kinematics of propagation of diifracted seismic waves. Tr. geofiz. in-ta Akad. Nauk SSSR, No. 9 (1950). 2. G. A. Gamburtsev, et al. The Refracted Wave Correlation Method. Izd. Akad. Nauk SSSR (1952). 3. A. M. Yepinat'yeva, Some types of diffracted waves recorded in the course of seismic observations. Izv. Akad. Nauk SSSR, ser. geogr. i geofiz., 14, 1 (1950). 4. T. I. Oblogina, a local representation of a system consisting of a head wave and a sliding wave. Vestn. MGU {Univ. Moscow), No. 1, (1956). 5. V. I. Smirnov and S. L. Sobolev, Tr. Seistnol. in-ta No. 20, Izd. Akad. Nauk SSSR (1932). 6. P. T. SoKOLOV, Physical and Theoretical Foundations for the Seismic Method of Geophysical Prospecting, GONTI (1933). 7. S. L. Sobolev, Tr. Seismol. in-ta. No. 41, Izd. Akad. Nauk SSSR (1934). 8. G. K. TvALTVADZE, Theory of the seismic method of prospecting for vertical interfaces. Tr. Tbilisskogo geofiz. in-ta, 2 (1937). 120 T.I. Oblogina 9. A. A. Kharkevich, The construction of a qualitative picture of diffraction. Zh. tekh. fiziki, PL 7 (1949). 10. Q. Miller, Fault interpretation from seismic data in southwest Texas. Geophysics. 13, 3 (1950). 11. T. Krey, The significance of diffraction in the investigation of faults. Geophysics, 17, 4 (1952). 12. F. RiEBER, A new reflection system with controlled directional sensitivity. Geophysics 1, 2 (1936). 13. W. B. Robinson, Refraction waves reflected from a fault zone. Geophysics, 10, 4 (1945). PART II. GRAVIMETRY Chapter 5 THE INFLUENCE OF DISTURBING ACCELERATIONS WHEN MEASURING THE FORCE OF GRAVITY AT SEA USING A STATIC GRAVIMETER K. Ye, Veselov and V. L. Panteleyev Gravitational force measurements at sea are of great scientific and practi- cal importance. The perfecting of a method for measuring the force of grav- ity at sea is therefore a pressing problem. Up till now the principal method for measuring the force of gravity at sea has been the fictitious pendulum method, the theory of which was put for- ward by Vening Meinesz <^) and other authors. It has been established that to obtain satisfactory results at sea with a pendulum device it is abso- lutely essential to measure the value of the vertical and horizontal accelera- tions in addition to the inclination of the base. The static method of measuring gravity at sea has been studied for some fe^v years in the Gravimetric Laboratory of the All -Union Research Institute for Geophysical Methods of Prospecting : in particular, they have constructed prototype instruments and tested them at sea. The results of these tests show that gravity can be measured by marine pendulum devices to an accuracy of the order of a few milligals. Some of the basic points in the theory of measurements of this type are examined in paper (1). In the present paper the authors have set themselves the further task of maldng a theoretical examination of certain other points, such as the determination of the equilibrium position of the gravimeter bar from observations taken during its motion, definition of the dynamic coeffi- cient of gravimeters, the effect of inclinations and accelerations on readings, and ways of reducing and compensating these effects. The essence of the static method for measuring gravity lies in the crea- tion of a very powerful damping — several hundred times greater than the critical— in the moving part of the gravimeter, as a result of which high frequency accelerations are filtered off (accelerations due to the vessel's motion) and low frequency accelerations (change of gravity accelerations) remain unaltered. The equation of motion for a system of this type is solved in Veselov's 123 124 K. YE. Veselov and V. L. Panteleyev paper (2) given that only the vertical disturbing acceleration, which is sinu- soidal in character, is acting, that the angles of deviation of the pendulum from the equilibrium position are small and that the damping is propor- tional to the speed at wliich the pendulum moves. In this case the equations for forced and natural motion take the form: ml 6>i = —=r- .:.= sin {pt + d), (1) l/(7Zo2-/>2)_4£2^2 (92=^^-^(aie-«=f-a,e-«^0 (2) where % and a2 are the roots of the characteristic equation {a^ — — e+)/e^ — tZq, a2 = — £ — Ke^ — /?q); Uq is the natural oscillatory frequency of the pendulum without damping ; / is the moment of inertia of the bar in the system ; mla^ cos [pt) is the moment of the disturbing force, on the assumption that the disturbing acceleration is sinusoidal in character; h is the damping coeffi- cient ; K is the torsional rigidity coefficient of the elastic element ; p, a^ are the amphtude and frequency of the disturbing acceleration; — = riQ^, — = 2e. It follows from formula (1) that, under the influence of sinuspidal disturb- ing accelerations, the bar will also accomplish sinusoidal forced vibrations. ml If the value of the static amplitude d = 2 a^ be divided by the value for the dynamic amphtude obtained from equation (2), we obtain the dynamic coefficient (in paper (2) a value which is the reciprocal of the dynamic coefficient is named the coefficient of interference suppression). 1 = , . (3) Where A, the dynamic coefficient, shows the decrease in the oscillatory amplitude of the gravimeter bar under the influence of sinusoidal accelera- tion of a given frequency compared with the constant acceleration of the same frequency. The dynamic coefficient is necessary firstly in order to know the ampli- tude and frequencies of the disturbing acceleration at which measurements can be made, and secondly to find the amphtude of the vertical component THE INFLUENCE OF DISTURBING ACCELERATI ONS 125 of the disturbing acceleration from the oscillatory amphtude of the gravi- meter pendulum. In order to be able to calculate the dynamic coefficient one needs to know the damping coefficient e in addition to the natural frequency of the disturb- ing oscillations. Let us examine equation (2) to find the value of s. Equation (2) indicates that when the gravimeter pendulum is deflected by an angle 6q it \vi\l return to a position of equihbrium exponentially. Let us find approximate values for a^ and ao ; Substituting in equation (2), we obtain -Lilt I 9 2 - /^Of_^'^/\ e.-^oe ^- (l-g-g,a ^--■^). (4) ^2 -(■u--!i)l ... The term — -^ e * is vanishingly small, since in the marine gravimeter the value of e must be greater than 1000 and the value tIq not greater than 10: Let us write the expression (5) for two moments of time and take their ratio. QT-, = e - ^ . (6) Converting into logarithms we obtain Let us rewrite formula (6), replacing 0<^ in it by A^—Aq, and 0o" by A^—Aq, where Aq is the reading on the gravimeter scale when the bar is in a position of equilibrium, Aj^ and A2 are the readings for moments of time ti and t^. rl..z^.,-H' "-'.', (8) whence where B = e 2 _A^Bei^^)-A, 126 K. YE. Veselov and V. L. Panteleyev Let us AvTite the analogous equation for the moments t^ and tj^: From equations (9) and (10) we can find It is not necessary to wait for complete abatement of the gravimeter system at each point to find Aq. A minimum of three readings must be taken at certain points in time. This enables the time taken in observation at the point to be considerably shortened. The accuracy of finding Aq can be raised by increasing the number of readings and averaging the results of the calculations, and also by working out the results of the measurement by the method of least squares. As will be shown below, the gravimeter readings are influenced by the vertical and horizontal accelerations of the motion of the Cardan suspension in addition to the force of gravity. They must be accurately measured to correct the readings for their influence. The amplitude value of the vertical accelerations can be found from the very observations made with the instruments for which it is necessary to divide the oscillatory amplitude of the pendulum by the dynamic coefficient. Consequently, in order to find the value of the vertical accelerations, we have to know the value of the dynamic coefficient: T) 2 or 7 ^0^ w-pr _^^ 4e2;?2 J ' (12) 1 V /i 'V-/^' 2sp \ 8£2p2 Since in gravimeters Hq^ is usually less than 100 radian/sec and p^ is no greater than a few units, formula (12) can be written in the form: A = Ao(l-^v), (13) where Ac, '' ~ 2ep THE INFLUENCE OF DISTURBTNG ACCELERATIONS 127 Experience shows that to measure Ag with sufficient accuracy it must be less than 0.01. Proceeding from this, formula (13) can be rewritten ^vith an accuracy of up to 10~® thus : (14) sep Being in possession of the recorded motion of the gravimeter pendulum when deflected from a state of eqiiilibrium, we can find the value B = e - , o from which — —^ = In B. The dynamic coefficient can be determined in this manner from the formida ?^=-^lnB, (15) if we kno\v Tf^, the period of the vessel's oscillation and are also in possession of the recorded movement of the gravimeter pendulum when deflected from a state of equilibrium. We are now therefore in a position to find the amplitude of the vertical accelerations as we know the dynamic coefficient X and have measured the mean oscillatory amplitude of the pendulum. If it is possible to read off the value Aq, i.e. if the observations close with the pendulum completely at rest, we can obtain from formulas (7) and (14) a formvda for calculating the dynamic coefficient of the gravimeter for two readings, A^ and A2, on the gravimeter scale at points of time ^^ and t^ and for the known position of static equilibrium Aq. X = ^ -A^I^ = 0.3663 log ^f^ -^ . (16) Alt fo — 1\ -^2 ^02 1 The values Aq, A^, A^ can be read ofi" from the curve of change in the reading shown in Fig. 1. For example, let the deviation of the system from a position of equilibrium decrease by 10 times in relation to the initial deviation in a period of 10 minutes, i.e. ^2~^0 10. Let J"^ = 10 sec. Then, by substituting in formula (16) we obtain 10.60 128 K. YE. Veselov and V. L. Panteleyev Let us assume that the ampHtude of the vertical accelerations is 5 gal, and their oscillatory period 10 sec. In this case the gravimeter pendulum will oscillate with an ampUtude of 5x0.0061 = 30 mgal. The above gives rise to certain assertions. Firstly, under conditions of strong damping, , small sinusoidal vertical accelerations do not produce systematic changes in the gravimeter read- ing but on]y forced sinusoidal oscillations in the pendulum, the amp- litude of which is decreased proportionally to the value of A. The ampUtude of the vertical accelerations can be found from the recording of these oscillations. Secondly, it is not necessary to wait for the system to come completely to rest in order to take readings from the instrument. The reading for a posi- tion of static equilibrium of the system may be calculated from the recorded motion of the pendulum, thus considerably reducing the amount of time spent in observations at the point. Thirdly, the dynamic coefficient which, for a given oscillatory frequency of the vessel, is the basic parameter of the gravimeter can be found from the recorded motion of the gravimeter pendulum when deflected from a state of equilibrium. Apart from the methods of measurement and the determination of the instrumental parameters, the problem of the combined effect of vertical and horizontal accelerations and inclination on the instrument readings is of considerable relevance to the static method of measuring gravity at sea. We shall attempt to solve the problem to the first degree of approximation. It is assumed that the instrument is mounted on a Cardan suspension subjected to vertical and horizontal accelerations so that the suspension THE INFLUENCE OF DISTURBING ACCELERATIONS 129 oscillates at natural and induced frequencies. In this case the induced oscillations of the Cardan suspension are mainly due to the horizontal acceleration component of the vessel's motion. Let us introduce a mobile system of co-ordinates IC related to tlie instru- ment. In the absence of vertical accelerations the pendulum of the gravimeter is orientated parallel to the axis 0^. Let us select a further system of co-ordinates xOz in which the direction of the axes is orientated relative to the fixed horizon. Fiu. 2. The transition from one system to the other can be easily achieved by the following scheme X = .Xq + % % ^1 1 cos (p sin (p c sin (p cos (p X = XQ + i cos 9? + C^'m z = ^0 — I sin 9? + C cos (17) The system |04" moves relative to xOz and rotates. The variables in formula (17) are Xq, Zq and cp. $", C" denote projections of the acceleration of the system of co-ordinates |0C translated to the axes C and | via ^qCq^ the co-ordinates of the centre of gravity of the pendulum. It is common knowledge in mechanics that the proiection of the acceleration Applied geophysics 9 130 K. YE. Veselov and V. L. Panteleyev of a point on a moving body onto a co-ordinate consists of the corresponding components of translated, tangential and centrifugal accelerations, i.e. r=Co"-lo'?"-Co(9'r. (18) In order to evaluate the influence on the readings given by the gravi- meter, we need to know the equation of the motion of the instrument itself or the equation of the motion of the Cardan suspension. The Cardan suspension is a damped or undamped pendulum subjected to the disturbing influence of accelerations. The Cardan suspension will have natural and induced oscillations, the period of which will equal that of the horizontal accelerations. Thus, if X = a^ cos (pt^d^), (19) it follows that (p = (Po+ fi cos (%« + dj) + 9?2 cos {pt + ^2)5 (20) where cpQ is the constant component of the angle of inchnation of the instrument's pendulum; 9?^^ is the amphtude of the natural oscillations of the Cardan suspension; re^ is the frequency of the natural oscillations of the Cardan suspension ; 953 is the amphtude of the induced oscillations ; a^ is the amphtude of the horizontal acceleration components ; p is the oscillatory frequency of the horizontal component. Let us now find the values entering equation (18). The influence of the translated acceleration can be written in accordance with (17) as Cq = :tsin 9? + zcos (p = ]/ x^ + z^ cos I 9?— arctan .. | . (21) Taking into account that where ^1 = a, cos (pt + d^), X and also that — is small, we obtain g l/.-^+(., + ^)-^-|/(| + l)Vj=^(l+|^ 2 x^M g g^ g^ 2 _ THE INFLUENCE OF DISTURBING ACCELERATIONS 131 -.(l+f + ^)- (22) We transform the second factor of expression (21). Taking into account (20), we obtain cos I 9? — arctan -tt-I ?«1 — — I99 I = = 1 - 2" U'o + 9^1 co^ {"'1^ + ^1) + ^2 cos (/jf + 62) - --^^-cos(;?i+(5;,)l . (23) Before proceeding to further transformations we wish to interpolate the following comment which mil henceforward apply throughout. We assume that the amplitude of the accelerations does not exceed 20ga] and that the dynamic coefficient is not greater than 0.01. In this case we shall in future disregard all purely sinusoidal components of the second order of smallness by comparison with — 1- in our calculations since their g registration will be reduced by the number of times by which the dynamic coefficient is less than unity. We shall only retain the systematic part of pe- riodic components of the second order of smallness by comparison with — - , i.e. their mean value for a large time interval. Thus we shall write for (21) 8 ^(i+f+-25-)''-(''-f)- j 1 — Y U'o + 9^1 cos {n^t + dj) + (p2 cos (pt +62) cos (pt + dx) I a a ^ 1 1 + -^ cos (pt + 6,) + j^cos^{pt+Sx)} X |l - y [9^0 + (Pi cos (n^t + d^) + aA. (24) 132 K. YE. Veselov and V. L. Panteleyev Here a is the deviation of the Cardan suspension from the direction of the resultant force : the force of gravity plus the inertia! force developed by the horizontal acceleration or the deviation of the Cardan suspension from the direction of the instantaneous vertical, i.e. a"x a == 9?; cos {pt + <5z) — cos [pt + d^^ = Oi^ax cos {pt + S), o where d which depends on d^, (5, is the phase of this deviation; a^^^ is the amplitude of the deviation of the Cardan suspension from the instanta- neous vertical. In marine pendulum devices this angle is registered by damped short period pendulums. Bearing in mind what has been said previously, let us find an approximated . •• . ... (x^^ expression for ^q, disregarding sinusoidal terms of the order of ( — 1 + — COS {pt +^z)+^- j(Po'- jn -J Omax . {2o) THE TANGENTIAL ACCELERATION In accordance with (18) and (20) the tangential acceleration component •on the axis is - lo 9? = ^o[(pin^ cos {n^t + d-i) + (p2P^ cos {dt + d^)] (26) It is evident that this acceleration only yields terms the periods of which are equal to the oscillatory periods of the Cardan suspension and the disturbing accelerations. The periodic terms of frequency n^ can easily be separated from the second term in brackets in expression (26) since the period of the natural oscillations of the Cardan suspension differs from the period of the accelerations. In addition, damping of the Cardan suspension enables one to get rid of the influence of its natural oscillations. On the basis of the above we can replace (26) by - ^0 <? ^ ^oV2P^ cos {pt + 6^). (27) THE INFLUENCE OF DISTURBING ACCELERATIONS 133 THE CENTRIFUGAL ACCELERATION In accordance with (18) and (20) the centrifugal acceleration component is + 992^ sin {pt + d^)^ «:; - :o I y (f'l^n^^ + ^ 993^^ ) > (28) consequently the centrifugal acceleration introduces a systematic error. Thus, for the full component of acceleration in the direction of the axis one can -wTite an approximate equation ^-h^'-h''^ + Oz COS {pt+ (5z) + 2 O-x* <2. max "^4^2 + ^09^2?^ COS {pt + d^) - Co X xfl^'x^nx^ + I^.^VI- (29) The equation for the motion of the gravimeter pendulum can he written down (taking into consideration that lip, the moment of inertia, is small by comparison Avith the remaining terms making up the equation) in the form : /0i" + 2h&^' + KO^ = jnC'l. (30) Let us divide by /. Taldng into consideration the notations in formulas (1) and (2) we obtain ^ ,, ^ ^ . o^ ml ^ ml 0i" + 2e0i'+^2(9___^____^ + ml [ax^ 1 1 2 1 2 + -r\-ir:-ir « max g- tt ng- t ng- ml ia^ _ 1^ 2 / \4^ ^'^ - Co f 2" n^ ^x + 2" ^'^P^ ) + ^2 cos {pt + (3z) + + ^0 ^iP^ cos {pt + ^2)[' (31) Equation 31 contains two periodic terms on the right hand. Let us estimate the value of the second of these terms. 134 K. YE. Veselov and V. L. Panteleyev fla, 271 6.28 Let us assume that a^ ^ 10 gal, (p^ = — ,p = — = = 0.63, /^^ = 0.40. g Tn 10 sec If the distance between the axis of rotation of the Cardan suspension and the axis of rotation of the pendulum does not exceed 1 cm then the last term in formula (31) o is very small by comparison with a^ and can therefore be discarded. The periodic term a^ cos {pt-\-d^ causes the forced oscillations of the system. This phenomenon was analysed in detail in paper (2). If we analyse equation (31) we see that when horizontal accelerations are present the reading of the instrument mil not correspond to the true value for gravity but to a certain value of G and that we shall have to introduce the necessary corrections to obtain a true reading. Thus we have 4^ 2 2 1 9 1 r 1 ' , f- / 2 ^1 2 2\r 9 <Po g- +^r s+^o in ^r + 9^2 />) + j o^max g and further ^x^ , ^ r^ 2 I ^Vax + 9?i ^0 / 2„ 2 , ^ 2 „2\ 9?o H ^ r -—((pi Til +?2 P ) ^g 2 (32) (33) ax" Here is sunply a Brown term, and the expression in square brackets 4g a correction for inclination (deviation from the instantaneous vertical). Formula (33) can also be \^Titten -- = ^- 4^ + 2- [9'o + 2 + -^ \f^ ^ 17]]' where T/^ and Tj, are the oscillatory periods of the vessel and the Cardan suspension respectively. ax For a short period Cardan suspension 992 is normally close to — , and o therefore aa;2 e- r a^max+n^ . 47t^Co /yi^ , ax^ 1 ^ = ^- 4^ + ^r" + — 2— + ^-U? + Fn^ (34) To assess the influence of the acceleration on an instrument mounted on a Cardan suspension with natural oscillations of large period, such as a stabiHzed gyroscopic device, we must return again to expression (26). THE INFLUENCE OF DISTURBING ACCELERATIONS 135 Here a ^ (p2 cos {pt +62) cos {pt + dx)- Taking into consideration that for the given case the phase difference {S^ — d^) is close to 71, we find a ?« — I — + 9^2 1 cos (/)« + dx). 2^^ g (P2 2 (34a) Introducing this expression into formula (33) we obtain More exactly n' + —^ — + -9.2+--C 8 4^%/^ 9?2_ g 2\ n (35) ^+fko^+n^ 992 cos (^2- (5a;) + — 4o Hf^ + where (^2~''^a;) ^^ ^^^ phase difference of the forced oscillation of the Cardan suspension and the disturbing acceleration. Let us examine a few examples: (a) the instrument is mounted on a short period Cardan suspension. 9^2 , C = 0, (Pi = 0, (po=^0, amax = 0, Ox = 20 gal (b) the instrument is mounted on a stabilized gyroscopic device ^vorking with an accuracy of ±2'; 993 = 2', Co = 0. 9^1 = 0» 9^0 = 0, flmax ^^,ax^ 20 gal g Then for case (a) we have Zl^ = ^^105ngal and for case (b) Ag = g w 2 g 6 ngal 136 K. YE. Veselov and V. L. Panteleyev Therefore the gyroscopic stabihzation decreases the correction for the influence of accelerations on the pendulum of the instrument. All the previous conclusions have assumed a single degree of freedom of the Cardan suspension and the presence of a single horizontal component. In actual fact we have to deal with the two horizontal components and two degrees of freedom of the Cardan suspension. Repeating an analogous conclusion for the second degree of freedom we obtain thereby formulas for long and short period Cardan suspensions analogous to formulas (34) and (35) % + f U'Ox + gTL 2 + (pix+ kx bx + -<■ , s + 'w\rou+ + cp\, , ^n^ol. , cLin\ + gTh ny+ r^ n (36) g= G^- —\ 9?gx+ 2 ~ (p2x^o% {Oc^-Ox) + 47r ^ I (pi -^ ^ "1 Tl ' Tf 2 ^2 I yfy+yiy - '^ 'P.; cos (3„„ - ^,) + ^ f„ (^ + f |\] . (37) The quantities denoted by subscript x relate to the degree of freedom of movement of the instrument's pendulum in a plane; the quantities denoted by subscripts relate to the degree of freedom perpendicular to the above. The principal correction term in formula (36) is the Brown term — 03.2+ a, it When the horizontal accelerations are small and the instrument is well adjusted the remaining terms are negHgible. In such a case one can proceed to calculate the accelerations without having recourse to accelerographic devices. One can assume from the theory of trochoidal waves that /^ 2 , /Tl 2 I /-r 2 given that there is a certain degree of approximation in the satisfaction of this equation. THE INFLUENCE OF DISTURBING ACCELERATIONS ]37 It is clear that the Brown correction will equal a^a/^_a/^ (38) ^ 4>g 4>g As was shown above a^ may be recorded by taking readings of the instru- ment which register the amplitude of its forced oscillations. The special reservation should, however, be made that in practice the horizontal and vertical acceleration components on vessels are far from being identical. If the influence of the accelerations on the gravimeter readings is to be more strictly accounted for, an especial measurement of the horizontal accelerations by one of the methods currently in use in pendulum observations becomes necessary. CONCLUSIONS In closing, the main conclusions can be briefly forinulated. 1. It is not necessary to wait for total abatement of the instrument's beam when taking readings with a damped marine gravimeter. The reading for a position of static equilibrium can be calculated from the record of the gravimeter pendulum and this considerably curtails the time required for observations at the point. 2. The basic parameter of the gravimeter (for a given oscillatory frequency of the vessel)— the dynamic coefficient — can be derived from the recorded motion of the instrument's pendulum. 3. When the instrument's pendulum is strongly damped small sinusoidal vertical accelerations only evoke sinusoidal forced oscillations in the pendulum, the amplitude of which is proportional to the value of the dynamic coefficient. The amplitude of the vertical accelerations is arrived at from the record of the oscillations of the gravimeter pendulum. Having allowed for the existence of the relation a^^ = a^-V aJ' we can find the value of the Brown correction from the recorded oscillations of the gravimeter alone. The accelerations and rotatinal movements of the Cardan suspension must be registered if there is uncertainty as to the satisfaction of the equality 4. A period of natural oscillation of the Cardan suspension considerably greater than the period of the vessel's oscillations is selected in order to diminish the influence of the horizontal accelerations. In this connection it is undoubtedly advantageous to use gyroscopic stabilization for the position of the gravimeter. The gyroscopically stabihzed Cardan suspension is treated in this case as a Cardan suspension with a very large period of natural oscillation. It is 138 K. YE, Veselov and V. L. Panteleyev obvious that the use of sufficiently accurate gyroscopic stabilization for a damped gravity measuring device with a horizontal torsion thread makes highly accurate observations possible even for amplitudes of the accelerations of 10-20 gal. This, at the same time, leads to the introduction of a minimum number of corrections arising from the measurements. REFERENCES 1. F. L. Vening Meinesz, Gravimetric observations at sea. Theory and practice. Geo- dezizdat, 1940. 2. K. Ye. Veselov, The static method of measuring the force of gravity at sea. Priklad- naia geofizika, fasc. 15 Gostoptekhizdat (1956). 3. L. V. SoROKiN, Gravimetry and gravimetric prospecting. Gostoptekhizdat, 1951. Chapter 6 EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY, SELECTING THE RATIONAL DENSITY OF THE OBSERVATION NETWORK AND CROSS-SECTIONS OF ISOANOMALIES OF THE FORCE OF GRAVITY B. V. KOTLIAREVSKII In planning gravimetric work, the density of the network of observations is selected with regard to the geological problems and the expected character of the gravimetric field. Normally no calculations are carried out and the selection of the network is mainly based on previous experience. The cross -section of isoanomalies planned in areal surveys is determined as the function of the expected mean square error for the values of the force of gravity at consecutive points. The actual value of this error obtained after carrying out field work is considered as a measure of the accuracy of the survey. Methods of selecting the observation network and evaluating the accuracy of the survey are of course imperfect. They do not make it possible to solve the basic problem, arising in the planning of the work, which is to find the optimal technical and economic solution of the geological problem in the gravimetric survey. The problem consists in predicting the density of the network and the accuracy of the observations, which will ensure the detection of the features of the field with the required accuracy. It is therefore not by accident that the accuracy of the survey and the choice of a rational network of observations have been studied both in this country and abroad. We mention in particular the work of Andreev<1) and LuKAVCHENKO (^), who determined the density of the network as a function of the value and extent of the anomahes, caused by certain geometrically regular bodies; the work of Bulanzhe (2), devoted to the problems of accuracy of survey, selection of the rational section of isoanomalies and the scale of the geophysical map under conditions which permit linear interpolations between the points of observations; finally, a number of recent art- icles by VoLODARSKii <^\ Grushinskii (^), Malovichko (^) and Puzyrev(^) who consider from another point of view the various facets of the problem of the density of the network, the accuracy of constructing from isolines, etc. 139 140 B. V. KOTLIAREVSKII In the present article, an attempt is made to solve these problems in a suffi- ciently general from, taking into account the errors in observation and inter- polation. The proposed method of evaluating the accuracy of a gravimetric survey, of the determining the rational density of the network and of select- ing the cross-section of isoanomalies is basically suitable for any type of survey. However, it has been mainly developed for the commonest types of regional and exploratory surveys. With regard to detailed surveys, another and somewhat different method can be recommended, which owing to the lack of space is not given here. SELECTING THE CRITERIA FOR EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY The accuracy of the observed gravity field depends on the accuracy of determining the values of gravity at the observation points, on the density ©f the network and on the character of the field itself. Furthermore, it depends on the method of interpolating the gravity values in the intervals between the observation points. We have accepted the idea of linear interpolation since it is universally used in gravimetric work. Before proceeding to the development of a method for evaluating the accuracy of a gravimetric survey, it is necessary to select the basic criteria for this evaluation. The main type of gravimetric work is areal survey the results of which are represented by a map of isoanomalies of gravity. The accuracy of the map can be determined from its various features. It is also possible to evaluate the value for the error in the observed field; this value being the mean square error in the value for gravity at a certain arbitrarily selected point on the map. This error does not depend on the cross -section of the isoanomaly and is a function of the errors of the values of gravity at the observation point, the density of the network and the character of the gravity field. For a field along a profile, this error (we will call it s^^ is determined by the follow- ing integral equation: t/ -r / [^ (^) - ^ ((7, o, ^)] 2 dx, (1) where g is the true gravity field (unknown) ; g- is the observed field, which depends on the mean square error a in determining the force of gravity at the observation points, on the average distance a between these points and EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 141 on the character of the true field ^; / is the range of integration on the profile. However, one of the basic problems of a gravimetric survey (if it is carried out for geological purposes) is the study of the changes in the gravity anomaly field from point to point. The pattern of these changes is given by the gravity isoanomalies. It is important to evaluate the accuracy with which these changes are determined by means of the isoanomalies. For this purpose it is necessary to find the mean scjuare value of the error in deter- mining the gravity increment between two neighbouring isoanomalies. Let us find the inathematical expression for this error. iVlong the profile let there be a curve for the true value of gravity g (x) and at the points x^, x^ and x.^ the obtained gravity values g-^^ g^, g^ (Fig. 1). As a result of linear interpolation, we obtain an approximate ex- j)ression of the function g{x) in the interval (v^, Vg) of the profile in the form of a broken LMN. On the profile let us select the points A and B, tt) ^vhich correspond the true values for the force of gravity gj^ and g^ and tJie observed (taken with the broken LMN) values gj^ and g^. Let the points A and B be selected so that gB~SA ^ !'■> "^vhere p is the cross-section of the isoanomalies. Then the expression ^={gB-gA)-(SB-gA) (2) will give the value for the absolute error in deterniining the increase in the force of gravity within the limits of two neighbouring isoanomalies. This value is a function of a, a, g and p. The mean square value (we will call it <5^) of this function, determined for a certain section I of the profile, can be represented by the following integral expression: \j^no.a. 9(v),/>]d.v. (3) As will be showii, the tw^o criteria proposed for evaluating the accuracy of a gravimetric map— from the mean square error in the value of gravity at an arbitrary point of the map (f^) and from the value of the mean square error in determining the gravity increment between limits of two neighbouring isoanomalies (5^^) are in practice sufficient for solving all basic problems connected with an evaluation of the quality of work carried out and for the determination of the necessary parameters of the survey at the planning stage. The indices of accuracy e^^^ and d^ give the absolute value of the errors. However, in practice, anomaly fields with differing intensities are encoun- 142 B. V. KOTLIAREVSKII tered. Therefore, in order to make comparisons of the surveys of such different types of fields, it is necessary to give the field error e^ not in absolute, but in relative magnitudes, for example, in fractions of the mean value of gravity for one gravimetric anomaly (of one extremum*). We will call this value ^^. In this way, the results of different surveys can be repre- sented by maps with differing cross-section of the isoanomalies. In order to compare the accuracy of these maps, it is necessary to determine the error in the gravity increment between two isoanomah'es 6^ in fractions of the cross-section of the isoanomalies (p). For the objective quantity indices of map accuracy, it is convenient to adopt the following expressions: Ern{a,a,g) = -^, (4) Dm{<y,a,g,p) = -^^ (5) where E^^ is the mean square error of the value of gravity at an arbitrary point of the field, expressed as fractions of the mean value of gravity within the limits of one extreme point; D^ is the mean square error in determining the gravity increment be- tween the limits of one cross-section of the isoanomalies, given in fractions of this cross-section. BASIC FORMULAE The study of the expressions (4) and (5) should, strictly speaking, be carried out for the planar field g{x, y), since we are mainly concerned with areal surveys. To faciHtate the task, we will be Umited to a study of these expressions for the field g{x), given along the profile. It can be shown that as a result of this replacement, the values of the E^ and D^ in which we are interested are a little higher. Let us consider the interval X2, x^ on the profile (Fig. 1). The observed values of gravity are determined within the limits of this range by a section of the straight line MN. If the observations at the points X2 and % were absolutely accurate, instead of the section MN it would have been neces- sary to consider the section M'N'. Let us choose a certain point C in the same interval. The true value of gravity at this point is g,. and is obtained from the observations (with subsequent linear interpolation)— g-^. Provided that observations were absolutely accurate, the interpolated value of the force * A Russian word signifying either a maximum or a minimum. Elsewhere throughout this paper the word extreme point is used [Editor's footnote]. EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 14a of gravity at this point would be equal to g^. It can readily be seen that the error in e of the observed (more accurately, interpolated) value for gravity at the point C is composed of the following two errors: « = (S'c - ^c) = iSc - Sc) + i§c - ~Sc) = ^0 + «/' where Eq is the error due to observation errors; £,• is the error due to the non-hnearity of the field. This equation ^vill be justified for any point on the profile. Moreover,, owing to the independence of the signs and the absolute values of the com- ponents Eq and £,- for a multitude of n points on the profile, the following equation will hold or 2 I 2 (6) where e^^ and e,-^ are the mean square errors in the values of gravity at a profile, caused by the corresponding error in the observations and the non-linearity of the field. Similar relationships can readily be established for the value d^: /S2 = A2 . A2 (7) 144 B. V. KOTLIAREVSKII where Sq^ and d^^ are the mean square errors in determining the increment in gravity hetween two isoanomaUes, caused by the error in observations and nondinearity of the field respectively. Comparing the relationships (6) and (7) with (4) and (5), it can be seen that the required indices of accuracy E^ and D^ also split up into two independent terms: F^ - F^ El (8) (9) where •C-Om — fiOr F- — ^Om = ^0/ A> P &m 5 m P Thus, the problem of finding indices of absolute accuracy (e^ and d^) and the relative accuracy {E^ and 2)„^) of a gravimetric map is divided into two independent problems of finding the absolute (eo;„-. ^om) ^^^ ^^^^ relative / g ^ jOm^ 1 errors due to the error in observations, and finding the abso- P -O/n gm lute (f,-,^, (5,-^) and the relative I -i^, — — jerrors, caused by non-linearity \g'n PI of the field. Errors Due to Inaccuracies of the Observations (Com' ^om^ 1. The derivation of the relationship £, Qm -^Oin (a, a). At the points x-^^ and x^ let g^ and g'g t»^ the true values of gravity, and ^1 and g2 the observed values, and a^ and ^2 (Fig. 2) be the errors. -^ qU) I J — »— X, km Fig. 2. EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 145 Within the range x^—Xi = a, we consider that the true values of gravity vary in a hnear manner. Hence we have For a certain point x, within the range {x■^^, x^ the following relationship will hold g (-^) -g {^) = £0 = 0^1 + — ~ {x-Xi) I 1 -^ "^^ -0 Assuming that a;^ = and bearing in mind that the observational errors Gi and (72 are independent and random, after squaring we obtain eQ^ = -^{a^-2ax+2x^). The integral mean of this value within the range < ^ < a has the form : ^Qm Ih Integrating, we obtain p2 _ _ ^2 Thus, Eq^ depends only on the mean square error in the values of gravity at the observation points. 2. The derivation of the relationship ^q^ = Sq^ (a, a, p). As the points x^, x^, x^ let there be true values of gravity g-^, g-g, ^3 and the observed values ^1, ^2? g^ ^^\h errors cTj, a^^ a^ (Fig. 3). Within the ran- ges x^—X2 = ATg— % = a, the change in the force of gravity will be consi- dered linear. Let us select in the range {x^, x^ two such points x^ and % for which the difference in the interpolated values of gravity g^ and g^ is equal to the cross-section of the isoanomahes p, i.e. g^—g^ = />• Assuming x^—x^ = d^a,\fe find the expression ^0 = ig5-gi)-ig5-g4)- Applied geophysics 10 146 B. V. KOTLIAREVSKII There can be two cases for both points (x^ and ^^5) lying in the range {xj^, x^; the point x^ is in the range (x^, x^ and the point % in the range (^2, x^). The values of Sq will be different for these cases. Let us designate these values by {d(^i and (^0)2 respectively. Dispensing with intermediate calculations, we find the expression which does not depend on the abscissa x^ and holds within the range < a;4 < a — (i (the index m for <5o is added so that Sq does not depend on x^, and, consequently, is equal to Sq^). [3^^ = 4" ^' [(3a2-3a(^+ d^) + {3d -6a) x^+ Sx^^] the expression holds in the rartge (a — d) < ^4 < a. Integrating over x^ in the range, we obtain d^ The mean integral value for the required value 5^^ will be equal to dlm = ^[{a-d){dlm)i+d{dlM' Substituting for (^om)i ^^^ (^om)2 ^^e values found for them, we finally obtain The relationship (11) is derived for the case d ^a; the inequafity meaning that between two points of the observations there are not less than two iso- anomahes. In practice, there may be other cases, for example, when between two points of observations there are not less than one and not more than two anomaUes, which is expressed by the inequality a < J < 2a. Without carrying out intermediate calculations, we wiU write for this case the final expression for S^ • Om' d 2— ■ a (12) Formulae (11) and (12) enhance the majority of cases encountered in practice when the isoanomalies (on the average for the whole field) are situated between the observation points. EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 147 Errors Due to Non-Linearity of the Field (e^^, (5,-^) The values e^^ and d^^ are functions of g, a and p. However, we do not know an exact form of the function g — g (x) (to simpHfy we will consider a field along the profile). It remains, therefore, to approximate the field g{x) from its discreet values at the observation points. A check of the various methods of approximation showed that the simplest and most convenient is the parabolic interpolation. X, km The paraboHc interpolation can be carried out by various methods. The most obvious way is to draw a parabola through three observation points. This method leads to very simple expressions for s^^ and 5,-^. However, it necessitates finding the mean square values of the differences of the first and second order for the observed field. ^10 = (82- gi) and A^ = {gs-g2)-ig2-gi)- This involves considerable expense in time. For detailed surveys this method is probably the best. As regards the surveys of a regional and exploratory nature, a much simpler method of parabolic interpretation can be used. The usually observed field consists of a succession of maxima and minima of gravity. Let us assume that by measurements the mean arithmetical values g-Q and Zq have been found, so that 2gQ is the mean value for the differences in gravity of the neighbouring maxima and minima and 21q is the mean distance between them. Given a parabohc law for the change in 148 B. V. KOTLIAREVSKII gravity within the limits of one extreme point, it is possible from the found values of gQ and Iq to reconstruct (on the average) the investigated field. Depending on the number of points used in constructing the parabola, the rule for the change in the force of gravity will be different. We have tested several methods, of which the method of parabolic interpretation with three points would seem to be the best. g, mgaL g 93 ! j \ X X 1 3 1 ><4 ^2 2\ 3 1 / >' 1 y / Fig. 4. Let us construct a parabola through three points (0,0), (Zq, g^ and (2/^, 0). Putting for the sake of simplicity Zq = 1, we obtain an equation for a para- bola of the second power: -g = g^{2x~x\ (13) From this equation we construct the curve of the gravity maximum (Fig. 4). We then draw a parabola through the points (2/q, 0), (S/q,— g^g) (4/q, 0), determining the curve for the gravity minimum. Its equation \iiS\. have the following form: ^ = ^o(8-6% + a;2). (14) For the gravity curve so constructed, we now find the values e^-^ and (5,-,„. 1. Deriving The Relationship £f^ = ^imis^ a)— Let the observations be carried out at the points x^ and ::C2 = x^+a, representing two values ol gravity gi and g2 (remembering that when determining errors due to non- Hnearity of the field, we assume the observational error to be nil). Let us take within the range {xj^, X2) a certain point ^^3, for which the true gravity value will be g^ and the interpolated value g^ (Fig. 4). To find s^ = ^"3— ^3 EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 149 it is necessary to derive an equation for the secant passing through the points (%'^i) ^^^^ (^'2' ^2)' ^^^ to determine g^ from this equation. Then, using the paraboHc equation (13), we substitute in the expression for e,- the magnitudes gj^, g2, g^ by g^ and Iq. As a result we obtain £/ = ^0 [(a^i + -^1^) - (a + 2.Ti) x^ + x^^]. In this expression the variables are the arbitrarily selected points x-^ in the range (0,2) and the point x^, in the range {x^^x^. In order to obtain the value £,-^, it is necessary first of all to find the mean integral value e^ in the range % < ^3 <rv:i+a. Dispensing with laborious calculations, we give the final result e^m = ^g^a\ (15) where a should be measured in fractions of Iq, since in deriving the para- bolic equation (13) it was assumed that Iq = 1. In view of the fact that the results of integration along x^ were not de- pendent on x^, the relationship (15) will also determine the required va- lue of £,-„,. 2. Deriving the Relationship d{^ = ^i„^{g, a,p)—Let two neighbouring isoanomalies of gravity have abscissae x^ and x^ ^ x^ + d (Fig. 4). We find the expression ^i = C?4-^3)-(^4-g'3)' where (gi—gs) is the cross-section of the isoanomalies; igi~gz) is the actual difference in the values of gravity at the points with abscissae corresponding to the two adjacent isoanomalies. Here (as in the derivation of the value Sq^) there can be two cases ; both points (^3 and x^ lie in the range (%, ^2)' the point x^ lies in the range (xi, x^ and the point x^ outside this range. The values of 6^ will be different for these cases. The position of the points x^ and x^ for the first case supposes that < <i < a ; this position is represented in Fig. 4. For this case d,=^gQd[{d-a-2x,) + 2x,]. Squaring both sides of this expression and integrating for x^ in the limits ^1 < ^3 < '^1 + ci—d, we obtain dfm = jgo'dHa-dr. (16) 150 B. V. KOTLIAREVSKIT Here also, the result of integrating for ^3 was not dependent on x-^^; there- fore it is the required value 61^. The position of the points for the second case, which is not given in Fig. 4, presupposes that a < c^ < 2a. For this case the following expression can be obtained for df^: dfm = jgoHa-d)^{2a-dr^ (17) On the right-hand sides of formulae (11), (12), (16) and (17) there is the value d. It can easily be seen that where A^q is the increase in gravity between two neighbouring points of observations. We determine Zl^o ~ ^10 (^' ^)- As above, we will seek this expression for the gravity curve in an appro- xiinate parabola passing through three points. Since the positive branch of the curve g{x), shown in Fig. 4, is identical in form to its negative branch, to find the mean integral value of A^q = (g^—gi)^ it is sufficient to take the range of integration from to 2 (remembering that Iq is taken equal to unity). Here there can be two cases: (a) both points of observation (% and x^ lie Avithin the Umits of the positive branch of the curve g{x) ; (b) the point a;^ is within the limits of the positive branch while the point x^ is in the H- mits of the negative branch (this case is not shown on Fig. 4). The values of ZljQ will be different for these cases. We will designate these values by (z1iq)i and {Aj^q)^, respectively. Dispensing Vith the intermediate calcula- tions, we obtain: for 0<A;i<2-a iAi,),=glan{2-a)-2x,Y, for 2-a<:^i<2 {Al,),=gl[{8-6a+a^)+2{a-^)x,+2xl]^ The required average integral value of the form A^q in the range ^ ,Tj ^ 2 will be equal to half the sum of the following integrals : 2-a 2-a 2-a 2 j[f{^lo)idx,+J{Alo),dx,'^ EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 151 Omitting the laborious calculations, we give the final result: Alo = -^go^a^20-5a^ + a^), (19) where a is given in fractions of Zq, In formula (18), replacing the value of A^q by its mean square value from equation (19), we obtain '- P - ^^ (20) '10 goa]/20-5a^+a^ We now find the value g^ — which is the average value of gravity for one extreme point. Using the parabolic equation (13), we obtain 2 gm = ^j{2x-x^)dx=jgo. (21) From formulae (10) and (21) we find E'om=-^ = 1.5^. (22) gm^ gf. From formulae (15) and (21) we find Efm=^=Om^a\ (23) gm For the error in determining the gravity increments there will be two groups of formulae: (1) < d < a or on the basis of formula (18) </) < /d^Q. From the equations (11) and (20) P' =i(-t)- From the equations (16) and (20) 2 A^=^=fff(i-^y. (25) p2 ^ZlJo (2) zlio<p<2Zlio. From the equations (12) and (20) ^-^iU*- 2 A 3 P (26) 152 B. V. KOTLIAREVSKII The relationship (26) can be represented approximately (with an error <8%) also in the following form: 1.167 P' From formulae (17) and (20) 6q 2^4 A '10 -1 p p A., Finally, substituting in the equation (8) the values E^ formulae (22) and (23), we obtain om and £',.„, from 0.075a* +1.5 In the same way, substituting in equation (9) the values Dq^ and D^^ from the formulae (24) and (25) or (26) and (28), we obtain: for the case </> < ^lo Z)2 = go'a^ '10 1- A., + ^10 A 10 for the case ZIjq </> < 2 A 10 2^4 r,2 _S£^ J-'m n An '10 '10 P 3p^ Ji A 10 (30) (31) In all the formulae, a is given in fractions of Iq. The formulae (29) and (30) or (31) are the most general. They establish the dependence between the seven variable values: the elements of the field (g'o, Zo), the parameters of the survey (o, a,p) and the indices of the relative accuracy of the map (£'^ , D^^). Knomng any five of them, it is possible to find the other two. The most typical in practice can be the following cases. 1. The elements of the field {gQ, Iq) and all the parameters of the survey (a, a, p) are given. It is required to find the indices of accuracy {E^ and D^ of the survey. 2. The elements of the field {g^, Iq), the indices of accuracy of the work (E^, D^) and the value a are given. It is required to find the network density a and the cross-section of the isoanomalies p of the planned survey. 3. The values gQ, Iq, E^, D^ and/> are given. It is required to determine the network density a of the planned investigation and the accuracy of the values of gravity at the consecutive points of observation a. EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 153 To speed up the calculations, graphs and nomograms can be compiled. Some of them are given in this arcticle. We will give a brief description of them. 10 9 8 7 6 5 g 3 2 4 5 Jc' 0-9 0-8 0-7 0-6 0-5 0-4 0-3 0-2 0-1 10 mg.L Fig. 5. Tlie nomogram for the relationship (19) serves to determine A^q as a func- a tion of gQ and the ratio — , which is a variable parameter (F'ig. 5). 'o The nomogram for the relationship (22) serves to determine Eq^ as a func- tion of (7 and gQ, which is a parameter (Fig. 6). 50 40 30 20 10 1 2 / / / / / / / / V / \ <^ y ^ / / /A ^ J ^=^ ^^^ -^7^ 2 4 0-6 0-8 10 I 2 0-2 0-4 0-6 08 iO Fig. 6. Fig. 154 B. V. KOTLIAREVSKII The graph for the relationship (23) makes it possible to determine the a value E^^ as a function of — (Fig. 7). The laomograms for the relationship (24) establishes the connection between Dq^ and — for the variable parameter (Fig. 8). In order to obtain 30 40 50 60 8-0 10 50 40 30 20 1-0 1-2 1-4 1-6 1-8 20 -" // ^ // ^ A ^ z:^ ^ ^ ^ "^yyy. % ^ :d ^ ^ 10 i H i ^ ^ Q\ 2 0-3 4 0-5 6 0-7 08 9 or p Fig. 8. this relationship, the right-hand side of the relationship (24) must be multiplied and divided by p^. The nomograms for the dependences (25) and (28) serve to determine D-, as a function of , for the variable parameter— (Figs. 9 and 10). 40 -^ — - -^ A ^ — 2^ ■ 30 / / ;^ ' _^_ ^^^ 20 ^=S lO "^C^^^^ =1 n ^ ■ — 10 09 0-8 0-7 0-6 05 0-4 0-3 0-2 0-1 10 4q p Fig. 9. EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 155 To derive these relationships in the right-hand sides of formulae (25) and (28), ZljQ must be replaced by its value from formula (19). The nomogram (Fig. 9) is used for </> < A^q, the nomogram (Fig. 10) — for A^q </j < 2z1iq. 1-0 0-8 0-6 0-4 0-2 1 a \ \ \ \ ^-1-^ \^ k\ \ \ / K > \ s \^ ^ [\ / \ s ^ ^\ / \ ^<^\\ \ 10 8 2 0-5 0-6 0-7 0-8 0-9 10 P Fig. 10. To determine the total error E^ as a function of— and — according to formula (29) the nomogram of Fig. 11 is used. To determine the total error D^ for the condition </? < A^q according to formula (30) t-svo nomograms are given (Figs. 12 and 13) : the first for 0-18 0-16 0-14 0-12 010 008 0-06 004 002 "^ ^ Em, % ■"■ V 1 \ \L ^ > ■y^u ^ Nj6 V \ ■^ K Xr\\\ '^ "\ \W\ ■■** 3 N. \i \ \ I x^ k \^V \ ^^ \ \ \ V _4 \ \ \ I \ ' \ \ \ iiit) K \ V _ \ 1 I 1 1 ^ 11 0-1 0-2 0-3 0-4 05 0-6 07 08 09 ^0 Fig. 11. 156 B. V. KOTLIAREVSKII £"„, = 10% and the second for E^ = 15%. To compare these nomograms in formula (30), a should be replaced by its value from formula (29). All the nomograms are drawn for the case — < 1, i.e. supposing that within the limits of one extreme point of gravity there are not less than two points of observation. It was also supposed that — - >2; observations P of this inequality require that each extreme point is mapped by not less than two isoanomalies. 40 1 \ ^ ^ ^ -C Q 20 10 '■••^.^^ ' .. 10 5 -JT 0-2 0-4 0-5 «-0 Fig. 12. It is difficult to find from the nomogram (Fig. 11), for a given E^, values of for large values of -— or small values of — for any values of go 8q It IS also difficult to find from given values of D^ and — a value of — from o / r ''0 the nomograms (Fig. 12) for — > 0.4 and from the nomogram (Fig. 13) for — > 0.5. The reason lies not in any defects of the nomogram or formulae, from wliich they were calculated, but in the very nature of the formation of total errors of the field and errors of the increment of gravity between two isoano- malies. Since the errors E^ and D^^ are made vip of errors in the observations and interpolations for certain relationships between the elements of the field and the parameters of the surveys, the values of the separate parameters EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 157 become variable. For example, with increase in the network density j i.e. with decrease in the ratio — ) the vahie of Dr. increases, at the same time L the value Z),-^ decreases, as a result of which the total error D^ can remain unchanged. The value Eq^^ with decrease in — does not change, and E^^^ 'o decreases. By increasing the value of a, which causes an increase in Ef^^^^ , it 60 40 30 20 > \ %, p 11^ 6 •s ^ N ::::;- ^^ Ciller" 1 4 - k' ~~- — ^ 01 0-2 0-3 0-4 0-5 0-6 07 Fig. 13. is possible to keep tlie total error £■,„ at the previous level. Thus, for certain relationships between the parameters it is possible to find a number of pairs of values for a and — , for which the values of the total errors E^^ 'o and D^ are maintained practically unchanged. Larger values of -— will correspond to the smaller values of a, and vice versa. The corresponding pair of values of a and — is selected in accordance with technical and economic . . ^« considerations. EXPERIMENTAL CHECKS OF FORMULAE The obtained relationships were checked in a number of theoretical examples. We will give the results of checking one of the more complicated examples. 158 B. V. KOTLIAREVSKII The field g{x) -was given along the profile in the form of a smooth curve having maxima and minima. The absolute values of gravity at the maxima and minima, and also the linear dimensions of the latter varied considerably. The length of the profile was 195 km, the total number of extreme points— 50. The obtained values gQ — 4.5 mgal, Zq — 1.95 km. Later, on the profile were fixed the "observation points" with 2 km interval (i.e. for each extreme point there was an average of two observations). A certain error was ascribed to each observation. The values of the errors obeyed the normal law, their distribution between the observation -^gW g, mqaL X, km Fig. 14. points being random. The mean square error in the values of gravity at the observation points was a = ± 2.58 mgal. A Hnear interpolation was caried out between the observation points and the errors were calculated. The errors in the gravity increment between two isoanomalies were calculated for the case p ~ 2. The results of the comparison of the measured values of the errors with their theoretical values, obtained from the formulae, are given in Tables 1 and 2. By way of illustration. Fig. 14 gives a part of the experimental profile. A check was also made of formula (19). Measurements were made along the profile of the first differences zIj^q fo^ different values of a and the Table 1. Mean square error in values of the force of gravity Mean Number of measurements Actual mgal Theoretical quadratic value Actual mgal Formula Error % fOm 297 Sim 297 ^m 297 ±2.19 ± 0.81 ± 2.22 ± 2.11 ± 0.86 ± 2.27 (10) (15) (6) 4 6 2 EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 159 Table 2. Mean square error of the increase in the force of gravity BETWEEN neighbouring ISOANOMALIES (sECTION OF THE ISOANOMALIES 2 mgal) Mean Number of measurements Actual mgal Theoretical quadratic value Actual mgal Formula Error % <5om dim Sm 177 177 ± 0.79 ± 1.38 + 1.36 ± 0.66 ± 1.51 (11) (16) (7) 16 9 mean square values zIjq were determined. The results obtained in com- parison with the theoretical values of A^q, found from formula (19), are given in Table 3. The example considered is extremely unfavourable, since the "observed" field was obtained with a very sparse network and the adopted mean square observational error very large. Table .3. The mean square values of the first differences in Aiq for various distances betv\'een the observation points a, (km) Number of observations Mean quadratic values Actual 1 Theoretical Error 0.5 1.0 1.5 2.0 207 180 117 90 1.37 2.80 4.00 5.00 1.30 2.60 3.70 4.75 5 8 8 5 Despite this, as shown by the data of the sixth columns of Tables 1 and 2 and the fifth column of Table 3, the results of checking the theoretical relation - ships were very good. For a field with a denser network of observations and with a smaller value of the mean square error in the force of gravity for consecutive points, the comparison between theory and practice should give results which, given a sufficient number of tests, will agree still more closely. On the whole, the experimental check showed that the proposed method for determining the errors in the gravity field, from the point of view of the obtained accuracy, is suitable for use in practice. SOME EXAMPLES We give several examples of the use of the derived relationships for determining the accuracy of gravimetric surveys made by production concerns. 160 B. V. KOTLIAREVSKII 1. The gravimetric survey of the Shar'iusk party No. 23/53 in the north- eastern part of the Moscow synchnal basin in 1953 (R. F. Volodarskii). The aim of the work was to study regional tectonics. The area of the survey was 7200 km^, the number of the coordinate points was 674. The points were distributed sufficiently evenly within the limits of the area of work. From the value of the area S and the number of points N, using the formula Vs a = 1.138 ]/N-l (32) which we give without derivation*, we will determine the mean distance between the observation points. It is equal to 3.9 km. The mean square error in the observation a = ± 0.62 mgal. The map with a cross -section of isoanomalies every 2 mgal was drawn to a scale of 1:200,000. Furthermore, in the calculation use was made of a compound map of the area of work by the 20/53, 21/53, and 23/53 parties with the same cross-section of isoanomalies to a scale of 1:500,000. Using this map, which covered a much larger area than the calculation map, we find the average amplitude and the linear dimensions of the anomaly: gQ — 9.5 mgal and Iq = 15.5 km. All further calculations will be made on the assumption that the error in calculation of ±0.62 mgal characterizes the error in the value of gravity at a consecutive point, since for the survey under consideration the remaining errors are small. The results of the calculations are given in Table 4, which also gives calculations for a less dense network of observations: in the second line for o, it is doubled, and in the third line, it is trebled in comparison with its actual value. With increase in a the value Eq^^^ remains constant, as it should do. The values of E^^ and E^^ increase with increase in a. The value Dq^ with increase Table 4. Accuracy of Gravimetric survey Shar'insk party No. 23/53 Given Calculated go ^0 a o P ■^10 Eom Eiin Em Dom Dim Dm 9.5 9.5 9.5 15.5 15.5 15.5 3.9 7.8 11.7 ± 0.62 ± 0.62 ± 0.62 2 2 2 2.7 5.3 7.7 8 8 8 2 8 7 1 11 16 ! 18 25 16 11 4 17 31 25 23 33 * The formula is derived with the assumption that the points are distributed over a square network. When the points are not on a square network, this formula gives results with an error not greater than 5-6%. EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 161 in a decreases rather significantly (from 25 to 11%), the value D^^ increases sharply (from 4 to 31%). Asa result, the value of the total mean square error for gravity increment between neighbouring isoanomalies D^ varies very Httle. It appears that with reduction in the network density by a half the value of E^ increases only by 3% of the actual, and the value D^ becomes even less than the actual. In other words, this reduction in the network had practically no effect on the accuracy of the survey (of the determined value Ejj^ and would somewhat increase the accuracy of the map of the gravity isoanomalies. R. F. Volodarskii, the author of the report considered, also came to the conclusion of the desirability for reducing the network density for this type of survey. He showed this conclusively by reducing the density network of a survey which had already been carried out and by reconstructing the map for the gravity isoanomalies with the new smaller density network. 2. The gravimetric survey of the Atlymsk party No. 7 (55-32) 55-56 in the region of Western Siberia in 1955-1956 (A. A. Serzhant). The aim of this work was to carry out geotectonic mapping of the area and to find local gravitational anomalies within its limits. The author observes that this was the first gravimetric survey to be carried out in this region, there was, therefore, no information on possible dimensions and intensity of local anomahes. The area of the survey was 7100 km^, and there were 1849 coordinate points. Consecutive observation points were placed on profiles, the point - intervals being 1 km. and the profile -intervals 4 km. The mean square error in the values of gravity at the consecutive points was ±0.50 mgal. The map with a cross -section of isoanomalies for every 2 mgal was drawn to a scale of 1:200,000. On the map there were several large anomalies of the force of gravity with value g-Q = 5 mgal and Iq = 10 km. Furthermore, there are doubtful indications of the anomalies of smaller dimensions and ampUtudes for which it can be assumed that g^^ = 2-2.5 mgal and /q ~ ^ ^^^ (values of gQ and Iq in both cases are determined from a very small number of anomalies). Since the point -intervals differ considerably from the profile -intervals, it is desirable to calculate the accuracy of the survey individually for different values of a. In the first line of Table 5 there are the actual values for the relative errors for large anomalies in a direction perpendicular to the profiles, i.e. for a equal to 4 km. In the second and third lines there are the errors for a equal to 2 and 6 km. The data of Table 5 show that with decrease in the point -intervals from 4 to 2 km, the error in the observed field E^ decreases by about 1%. This is Applied geophysics 11 162 B. V. KOTLIAREVSKII explained by the fact that lor the considered field wilh a — 4 km, the error in the linear interpolation E^^^^ is very small. As regards the error Eq^, comiecteil Avith the observational errors, it does not depend on the point - intervals. As a result, the decrease in the value of a has practically no effect on the value of the total error in the field iT^j. The position is diflFerent with errors in the gravity increments. Here, as a result of a certain increase in the error Dq^ , there was also an increase in the total error jD^ (from 24 to 27%). With further decrease in the point -intervals, value of E^ is maintained at practically the same level, and the value of Z)^ can only increase. Table 5. Accuracy of Gravimetric survey Atlymsk party No. 7 (55 — 32) 55 — 56 Given Calculated So ^0 a o P ^10 Edm Eim Eni Dom Dim Dm 5 5 5 10 10 10 4 2 6 ± 0.50 ± 0.50 ± 0.50 2 2 9 - 2.3 1.1 3.4 12 12 12 4 1 10 13 12 16 24 ! 27 17 1 1 12 24 27 21 In the third line (for a — 6 km) the value of E^ increases very little, and Djj^ decreases. A comparison of the figures in the second and third lines shows that to determine the anomalies characterized by the parameters gQ = S and Iq = 10 km, the value a = 6, and not 2 km is the most favourable. The network of observations taken is therefore extremely dense, and this increased density had very little effect on increasing the accuracy of the observed field and at the same time somewhat reduced the accuracy of the determination for the increase in the force of gravity between the neighbouring isoanomalies. In practice, of course, this decrease in accuracy need not happen, since in tracing the isoanomalies it is usual to carry out a certain smoothing of the values for the force of gravity at the points of the observations, as a result of -which the effect of random errors is reduced. However, this smoothing cannot be represented numerically, and therefore we do not take it into account. Table 6. Determining the parameters of the survey' as functions of the elements of the field and accuracy' of the survey Given Calculated go lo P Em Dm a(km) cr(mgal) 2.5 2.5 5 5 0.5 0.5 15 15 29 29 2.6 3.6 ±0.26 ±0.12 EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 163 In tlie report it is mentioned that the anomalies which were very small in their dimensions and amphtudes and which are unreliably indicated on the map, are also of geological interest. Table 6 giv<;s values for a and cr, ensuring a reliable determination of this type of anomaly with parameters i^Q = 2.5 mgal and Iq — 5 km with the condition that the isoanomalies are carried out every 0.5 mgal and the relative errors in the gravity field and gravity increments between the neighbouring isoanomalies are equal to 15 and 29 % respectively. Table 6 gives two pairs of extreme values of a and a, each of which satisfies the requirements. This example shows that the same problem of determining anomalies with a given accuracy and a given cross-secti(m of anomalies in some cases can have a different technical and economic solution. The values of a and a in Table 6 are found from the nomograms (Figs. 11 and 13). Table 7 gives separately the observational and interpolational errors for this case, as found on the appropriate graphs. It follows from the preceding that the proposed method of finding errors from the parameters of an accomplished survey, or of determining the parameters of a plannefl survey from adopted errors, requires in both cases a knowledge of the field ^q and Iq. At the planning stage of an investigation, this requirement can always be fulfilled, since the main problem of gravi- metric exploration is to determine the anomalies. Consequently, their Table 7. Given Calculated ^0 lo a a P ^10 Eom Eiin Dom Dim 2.5 2.5 5 5 2.6 ■^.6 ± 0.26 ± 0.12 0.5 0.5 1.5 2.0 13 6 7 14 23 8 18 28 parameters can be given at the beginning. In the majority of cases, this method can be used to determine the accuracy of the finished work, since the observed field is usually nondinear. Frequently, in regional surveys the field does not have closed anomalies. However, if the curves for the change in the force of gravity along arbitrary profiles cutting the map have even only relative extreme points, this is sufficient for determining g^ and Iq. In connection with this it should be stated that the field factors should be determined along the profiles of the arbitrary directions in those cases where the field is represented by clearly mapped maxima and minima of gravity. ir 164 B. V. KOTLIAREVSKII Much less frequently there are cases of linear or near -linear fields, having points of inflection but no extreme points. For these fields, gQ and Zq cannot be determined. In these cases, keeping to the framework of the proposed method, only negative conclusions can be drawn from the results. Let us assume that the parameters of the survey under consideration, a — 0.6 mgal, a = 6 km and p — 2 mgal from the nomograms (Figs. 11 and 13) we find that such a survey makes it possible to determine anomalies characterized by the values g^ — 11.8 mgal and Zp = 8.5 km with an accuracy of E^= 15% and D^ =30%. However, there are no such anomalies on the map, conse- quently they do not exist in actual fact. Given other indices of accuracy, we find the new values of gQ and Zq, etc. With regard to the actual accuracy of this survey, another method of determination should be used. This method is based on the use of the first and second order differences A^q and ZI2 as gravity field characteristics. SUMMARY In planning the work, a considerable problem is the selection of the values of E^^ and D^ since they determine the parameters of the survey. These values should obviously be different for detailed and regional or exploratory surveys. It is difficult, however, to give actual figures for any of these forms of survey. After studying certain production data, the author came to a preliminary conclusion that for regional and exploratory surveys, the values E^ = 15% and D^ = 25+30% give quite reliable results. However, with respect to the regional and exploratory surveys, these require- ments are too high. Let us consider in particular, the data of the experi- mental example. The survey parameters in this example are exceptionally unfavourable. As a result, the relative errors are very large, being E^ = 50 % and D^ = 69%. Despite this, of the 50 extreme points of the true field, only 5 points were not noticed in the observed field. All the remaining extreme points, although distorted in their shape and amplitude, are reliably diffe- rentiated on the observed curve. Consequently, if the survey is only requi- red to give localization of anomalies, at the planning stage the values of E^ and Dj^ can be considerably increased. To solve this problem finally, it is necessary to analyse a large amount of the actual material, as was done by the author. At the start of the article, a mention was made of the method of evaluating the accuracy of the survey according to the value of the mean square error in the force of gravity at the observation points, and on the method for determining the cross -sections of the isoanomalies according EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 165 to the value of this error. After analysing the errors, it is useful to consider the errors introduced by the classical criteria and to compare them with the criteria E^ and D^ suggested by the author. The value a gives the objective error in accuracy for the whole run of a survey, including both the gravimetric and geodesic measurements, the processing of the observations, the introduction of corrections, etc. However, the value of this error does not make it possible to draw conclusions on the accuracy of the observed field e^^ which, depending on the network density and the character of the field, can have higher or lower values than a. Thus, the "cr criterion" is essential but not sufficient to evaluate the quality of the observed gravity field. It follows from the preceding that the evaluation of the accuracy of the field from the value a is replaced by the evaluation from the value e^ or E^, thus bearing in mind the parameters of the survey a and a, and also the character of the field gQ and Iq. The second criterion, which is determined by the inequality p ^ 2.5 a, does not take into account either the network density or the character of the field, and is not adequate for either of these reasons. We have replaced it by the criterion d^ —D^p. In particular, for the recommended value Dj^ = 30%, we obtain p = 3.3 (5,„. This condition means that the map of the gravity isoanomalies will have practically no sections where the error in the gravity increment between two neighbouring isoanomalies would be equal to the cross-section of the isoanomalies. Although a preKminary test of the proposed method with theoretical examples and with actual material gave sufficiently reliable results, in later practical use of this method, in some cases difficulties might be encountered and individual faults discovered, which have not been mentioned by the author. The future problems will be to check this method and for this pur- pose the cooperation of production workers will be required. The author thanks K. E. Veselov, A. M. Lozinskii, L. V. Petrov and N. N. Puzyrev for valuable advice and comments made during a dis- cussion of the present article. REFERENCES 1. B. A. Andreev, Calculation of a Network of Observations in Gravimetric Work. Documents of Ts. NIGRI. Geofizika, CoUection 5, ONTI NKTP SSSR, (1938). 2. Yu. D. BuLANZHE, Some problems in the method of gravimetric work. Appl. Geo- physics, No. 1, Gostoptekhizdat, (1945). 3. R. F. VoLODARSKli, Some problems of a gravimetric survey. Survey and Industrial Geo- physics, No. 11. Gostoptekhizdat, (1954). 166 B. V. KOTLIAREVSKII 4. N.P. Grushinski, Establishing a rational density of points and the required accuracy in determining the anomalies of the force of gravity in gravimetric surveys. Survey and Industrial Geophysics, No. 15. Gostoptekhizdat, (1956). 5. P. I. LuKAVCHENKO, Determining the density of a survey network in gravimetric and magnetic survey. Survey and Industrial Geophysics, No. 4, Gostoptekhizdat, (1952). 6. A. K. Malovichko, The density and form of the network in an area survey with gravimetry. Survey and Industrial Geophysics, No. 15. Gostoptekhizdat, (1957). 7. N. N. PuzYREV, The connection between the density of the observation network and the cross-section of the geophysical maps. Applied Geophysics, No. 18. Gostoptek- hizdat, (1957). PART III. ELECTRICAL SONDE METHODS Chapter 7 THEORETICAL BASES OF ELECTRICAL PROBING WITH AN APPARATUS IMMERSED IN WATER E. I, Terekhin Marked successes have been achieved recently in the development of electrical prospecting at sea. A method has been developed for the produc- tion of continuovis two-way dipole axial probes with a distance between the centres of the dipoles of up to 6 — 8 km, with sea depths of up to 50-60 m (0. V. Nazarenko). In this method, during measurements the feeding and measuring dipoles are situated at the bottom of the sea, i.e. at the lower boundary of the first layer. With the same geoelectrical cross -section, the values of the apparent resistance measured with the sea-bottom apparatus differ from the values of apparent resistance measured with the same apparatus on the surface of the water. The solution of the problem of distribution of the field of a point source, within the limits of the first layer, and in particular at its lower bound- ary, is therefore of practical interest. The problem of the distribution of potential for a point source of current at the lower boundary of the first layer of a three-layer horizontally homo- geneous medium was first solved in 1934 by M. Ya. Samoilov. Based on this solution, in the same period in the Geophysical Section of the AzNIl certain theoretical curves were calculated and published later, without a derivation of formulae, in a book by S. Ya. Litvinov^"). These include the two-layer curves with ^2/^1 ~ ^^ 2' ^' -^^' ^^ °°' three-layer curves for the case ^3/^1 = 1.5, qJQi ~ 1' 2, 5, 10, 40, 00 and /12/^h ~ 2, 5, 10, 40. In the general case of a horizontally homogeneous stratified medium, the problem of the distribution of the field of a point source at the boundary of separation of the first and second layers, was solved by L. L. Van'yan(3). In this paper two -layer curves are given for the AMNB apparatus and the dipole axial apparatus and three -layer curves for the AMNB apparatus for the case ^3 = 00, h^ = h-^ and various values of ^2/^1* 170 E. I. Terekhin In the article by V. V. Burgsdorf <^), a solution of the problem of distri- bution of a field for the most general case, with an arbitrary placing of the electrode in a horizontally stratified medium is given in an integral form. Of practical interest for electrical prospecting at sea and in other expanses of water, is the case where the source of current and the measuring apparatus are placed at a certain depth within the limits of the water layer or, in a spe- cial case, at the bottom of the water. The present paper gives a developed solution of the problem of distribu- tion of the field of a point source situated within the limits of the first layer of a horizontally stratified medium and, as a special case, a point source situated at the lower boundary of the first layer. The paper gives expressions for the apparent resistance for a number of instruments placed at the bottom of the water. From these formulae, calcu- lations of the theoretical curve for electrical probing at sea were made. In one of the sections details are given of methods of calculation used for the theoretical curves of probing at sea, and also an evaluation of the accuracy of the calculations. THE FIELD OF A POINT SOURCE AT THE BOUNDARY OF SEPARATION OF THE FIRST AND SECOND LAYERS The main problem in electrical probing is to determine the depth of various layers of the section differing in resistance. One of the main methods of interpreting probe curves is to compare them with specially calculated theoretical curves. To calculate these curves, the character of distribution of the field created by the point source of current should be known. Let us assume a horizontally homogeneous stratified medium. Let 0, 1, 2, 3, ..., n be the orders of the layers (from the top downwards) Iiq, A^, h^, ..., /z„_j be the thicknesses of the layers covering the supporting level and Qq, q^, q^, ..., Q^ be the specific electrical resistances of the various layers. Thus, the upper layer (in this case the water layer) has a zero number and thickness Jiq and specific resistance ^q respectively. The current source A of strength /is placed in the upper layer at a certain depth Zq < Hq from the surface. The potential at the point M, arbitrarily placed in the conducting semi- space at a distance R from the current source, is expressed by the following ELECTRICAL PROBING WITH AN APPARATUS IN WATER 171 functions : U, = ^f^^ +Ui U, = U[, U, = U^, ..., Un = U'n. (1) These functions are partial integrals of the Laplace equation To solve the problem we use the Laplace equation in a cyhndrical system of coordinates. The origin of this system we put at the point A, the axis z is put vertically downwards. The problem has axial symmetry with respect to the z axis, the Laplace differential equation in this case, does not therefore depend on the airgle 99, placed in a horizontal plane, and has the following form: '^2 -U 1 9U 9^U ^ The potential functions Uq, U^, U2, ..., U^, apart from the fact that they should be integrals of the Laplace equation, should satisfy the follow- ing conditions. 1. The fvnictions Uq\ U^, C/g, U^, ..., U^ should be finite at the points situated at a finite distance from the source of the current and change to zero for points an infinite distance away, 2. Wheni?->0 ^» 47r R' 3. For each boundary of separation there should not be jumps in the poten- tial and the normal components of current density should be equal, i.e. Ui = f/j+i and ^- = ^- . Qi 3z Qi+i 3z 4. At the boundary of separation of water — air, the normal component of the current density should be zero, i.e. 1 9U, Qo ^z .= = 0. The solution of the Laplace differential equation is obtained by the known method of dividing variables. A general integral of the equation is the ex- pression CO f/ = / [5e"^^ /o {mr) + Je"'"-' /„ {mr)\ dm. (2) 172 E. I. Terekhin Representing — ^ by q, the potential at any point of the upper layer can be expressed thus: Uo = 2]/r^ + I B^e"'^ Iq (mr) dm + j A^' + I BQe"'^lQ{mr)dm+ l A^e-""' lQ{mr) dm (3) or, using the Weber-Lipshitz formula for negative values of z oo = / e"^^ L (mr) dm, at points in the upper layer above the electrode A, i.e. at points for which z <. 0, we obtain U. z<Q J fe"^^I^{mr)dm+ f B^^ e'"^ lQ{mr) dm+ j B^g^^ Iq {mr) dm + -j_ / Aq&~^^ lQ{mr)dm, (4) and SU. dz oo oo = — g f /n e"'^ Iq (mr) dm + / mB^e'^^ Iq {mr) dm - 2<0 2 ./ J oc mAQe'"^^ Iq {mr)dm. 9U According to the condition (4) 9z and Qq 7^ oo. Consequently, for all values of r = 0, since in general Qq^ oo Hi qe-'^^o + B^e-'^^o -AQe""^" mlQ{mr)dm = 0. ELECTRICAL PROBING WITH AN APPARATUS IN WATER 173 Since m ^ 0, then 1 hence qe-^^" + Bq^-^^^o -AqU"''' = 0, ^o=(^o+Y^)e-2-^X (6) Considering the value of the coefficient Aq, we write the expression for the potential in the upper medium oo At any point lying below the origin of the coordinates, i.e. at any point with 2r ^ 0, to express the potential we have the following formulae: 00 U^ = J (B^e"'' + A^e-""') lQ{mr)dm ; Un~i = f{B,_^c"^' + A^_^e--^'-)Io{mr)dm; 00 U, = f A,e-'-U,{mr)dm. Where B^ = 0, since when z-^ 00 e'"^-> 00, which contradicts the first condition of the problem. To find the values of the factions Uq, U-^, U^, ..., U^, it is necessary to determine the coefficients A: and B;. 174 E. I. Terekhin Using the boundary conditions we ^vl'ite the system of equations: 5o(l + e-2'"''») -^1- A e-^'^e^o-zo) _ = _ . qH -|_ Q-2mzo\ g-2m(ho-Zo) __ _J_ n _J_ Q-2mZa\ Q-2m{ho-Za) 5i + Zie-2m(h„+fti-z„)_^2_J2e-2'"(''o+'^i-^«) = 1 ^ 1-7- B Qi j^ Q-2m{ho+hi-Zo fl ^5. £»2 -\ A^ G-^rn{h, + h,-Zo) = ^2 1 -2m(s/ii-Zo) _ -2m ( S/Ji-Zo) ^^■=0 ^-^ne ^'=0 ^ =0 I 2 hi-Zo ) ( E/lJ-Zo) ^ 1 _ -2'n -Bn-1 -^n-l e Qn-1 Qn-1 1 _ -2m + — ^„e } (8) These expressions form a system of linear heterogeneous equations. The solution of this system of equations with respect to A^ and B^ makes it possible to determine the final expressions for the potential functions. In practice we are interested in the distribution of potential only in the upper medium; it is therefore quite sufficient to find the value of only one coefficient Bq . This system of equations can be solved by the Kramer formulae. In a general form we have B„ M_ J' (9) where A is the determinant composed of coefficients for unknowns {B^, A^), of a system of equations, and Jf is a determinant obtained from A by replacing ELECTRICAL PROBING WITH AN APPARATUS IN WATER 175 the numbers of the first column by terms in the right hand side of the equations. (9a) l + e" , - 1 , -e^, , , . .., , , |(l-eD), Qo 1 Qi Qi , , . ., ? 1 , eL, - 1 , -e^, . ., ? u 1 + — , Qi Qi 1 ^2' «2 ., ■> , , , , . -, 1 , eQ ■> -e^ , , , , . 1 9n-l 1 Qn-l eQ, 'eO Qn where D = —'ImliQ', F = -2m(Ao-^o); L = -2m(/?o+/ii--o); Q= -2m( 2^--~o n-l f=o Removing from the sign of the determinant A the general factor e^'"'-', we obtain: A = e-^"'''''A, (10) where A i 1 + eD - 1 -e^ '• ., , , Qo ''), 1 Qi Qi ? • ., , , 1 e^, - 1 - e«,. ., , , 1 ^1' 1 R — e-R, ^1 1 ^2' 1 ?2 e^. ., , , , , 5 • ., 1 e^ -eU , , ? • • 1 '' Qn-l 1 Qn-l Qn 176 E. I. Terekhin Similarly, for M we obtain: M = ■ (1 + e"^'"^") Q'2'm{n + l]zo]\f^ (11) where M= -9eD -1, -e^ , , ,.. ., , 5 ^-, , ,;. ., , Qo Qi Qi 0,1, e« , -1 , — e , .. ., , , 1 'en, 1 9 O 5 . ., , 5 ' Qi Qi ^2 ^2 , , , , , .. ., 1 , e*^ 5 -e^ , , , , , . 1 1 f.U ^ e^ '' Qn-1 Qn-l e , Qn where i=0 It is known that B^ = ■ — - is a coefficient which enters the expression for the potential function when the current source is at the earth — air boundary (^). Therefore, the desired coefficient, corresponding to an arbitrary placing of the current source within the limits of the first stratum, can be expressed by the expressions for the coefficient B^ (^) which Ave know for any number of layers: 9 B. {l + e^^'o)Bi, (12) p {Q-2mh\ where B^ = -p^—, — ^-^r^r is a fraction, in the numerator and denominator of which are the polynomials of the power Pn_i from the variable e"^'" . Substituting the expression Bq in the formula for the potential with an arbitrary placing of the source and the point of measurement in the upper layer (7), we have oo rr 1 1 1 r *^ ^ 2" ^ i/^ 2 ^ 2" ^ / [e-'"^^"°+") + 5i(e'"^ + e-"'^) + + 5i(e'''(2zo+z) + e-'"^'2zo+z))] ^jnr) dm. (13) ELECTRICAL PROBING WITH AN APPARATUS IN WATER 177 The obtained expression is basic, being the most general integral expression of potential for an arbitrary position of the current source and the point of observation ^vithin the limits of the upper layer. When working with the method of electrical sonde, the apparatus is always placed horizontally, consequently in formula (13) ^ve can put z — 0: oo ^o--^ + j(l I [e-^'""" + 2^1 + 5i (e2'«- + e-2"^''o)] /„ (mrd) m. (14) Let us suppose that the thicknesses of the individual layers Jiq, h^, h^, ..., ^^_j have a general measurement, which we can represent by h such that n-l S' i =0 K =PqK hQ+h^ = PiK hQ+h^ + h^ =p^h, ... , 2 /',• =/>„-! ^'' where Po,Pi,P2i •••■>Pn-i ^^e whole numbers. The function Pnig) ^1 QnigY where g = e~^"^^ is a rational fraction which can be split up into a series in increasing powers of g by dividing the polynomial in the numerator by the polynomial in the denominator. In a general form we have ^1 = 2 ?n n = l " -2mnh where q^ = the coefficients of emission. Substituting the expression for the coefficient 5^ in formula (14) we obtain UQ=-^+-^q j e-^""'" lQ{mr) dm+ qj^ qn e'^""'^ Io{mr) dm + 2r + y ^ V 5n /e-2mnft ^Q^mzo + e-^'"^") ^mr) dm, (15) n = l J Using the Weber-Lip shitz formula, we obtain an expression mth which it is possible to calculate directly the potential at points on one level with Applied geophysics 12 178 E. I. Terekhin the source of the field "within the hmits of the first layer: 47r + Vr^ + W 1 yr^ + 4^{nh-ZQY + 2 + + (16) In a special case for an apparatus placed at the bottom of the water we have Zn = tin and Uo = 1 + + S^" 1 + 2 + E qn 1 + 1/^2 + 4^{nh- ho) 2 j/7-2 + 4 (a/i + ho) 2 (17) If we consider the obtained formula from the point of view of the theory of reflection, then it can be readily seen that when the source of the field is at the boundary of the first and second conducting layers, the distribution of the fictitious reflected sources is complicated in comparison mth the case where the source is at the surface of the conducting semispace. When the source of the field is at a certain depth, the symmetry is lost in the location of the fictitious sources with respect to the real source, reflected sources of equal power appear at different distances {yr'^+{2nzQ)^, yr^ + 4!{nh—z)^ and yr^ + 4i{nh+ZQ)^ from the point of measurement of the potential. The formation of these fictitious sources is connected with the presence of a boundary of separation with a coefficient of reflection i^ = 1 (the water-air boundary), which does not pass through the actual current source. AN EXPRESSION FOR THE APPARENT SPECIFIC RESISTANCE FOR AN APPARATUS AT THE BOTTOM OF THE WATER The apparent resistance is a complex function, depending both on the parameters of the geoelectrical cross -section and on the mutual disposition of the feeding and receiving electrodes of the apparatus. It is calculated from the formula AU Q = K- (17a) ELECTRICAL PROBING WITH AN APPARATUS IN WATER 179 where K is an instrumental coefficient (the apparatus placed at the water surface) determined for the condition that in a homogeneous medium, the value of q is equal to the true specific resistance of this me- dium. If this condition is also taken as a basis of determining the coefficient for instrument at the bottom, then the coefficient of the bottom apparatus will depend not only on the geometrical dimensions of the latter but also on the relative depth of its immiersion. It can readily be shown that in the general case of 4-electrode symmetry of the apparatus, immersed to a depth h, the coefficient K^ can be expressed in the following way: Km = K- t'+* (i)' + 4 Since Kj^j = MK, where M + 4 2(^1+4 depends only on — , the shape of the curves obtained experimentally and h theoretically changes in exactly the same way if, for these and others, the coefficient K is used instead of Kj^. Under working conditions it is more convenient to use the coefficient K instead oiKj^, since the values of iiTfor various types of apparatus are calculat- ed and given in nomograms. In calculating the theoretical curves for sea probing with an apparatus at the bottom, the change in the coefficient of the apparatus on being immersed to the bottom was therefore not taken into consideration. We represent the apparent resistance, measured by the bottom apparatus, as Qj^^ (without correcting the coefficient of the immersed apparatus). The apparent specific resistance of the AMNB apparatus for MN->0 is determined from the formula Q = 27ir^ E 12* 180 E. I. Terekhin We find 5r 47C + 2 >,^„ 7-2 ■ (7-2 + 4V)^/= ^^'"[r^ + {2?ih)^YU_\ + + 2j^" ["on 3 / ~t~ [r2+4(/iA+Ao)']"^= [r2 + 4(7z/z-/io)2]V.J' (18) hence Qm = h^[ ^"^lV2 + 4/l2)V= +22j?n-r. '+(27l/7)2]V= + + Ljin + t{ M^' + Hnh+h.^!^ [7•2 + 4(7^/^-/^o)^]'/=/J (19) Representing -p-^ — r^ h + 4, , the so-called coefficient of recession of the symmetrical apparatus AMNB, by l/^, we finally obtain M 2 \ n=l / ^^o[C+X!^n(C+po+^.-p„)l L n=i -J (20) To obtain an expression for the apparent specific resistance for a bottom radial apparatus, we use the known relationship (^) Qr = Q- r 9q Representing by q^j^j the apparent specific resistance of a radial apparatus at the bottom of the water, without considering the change in the coefficient of the apparatus during immersion, we obtain Qrj^j — Qm 2 * 9r (21) ELECTRICAL PROBING WITH AN APPARATUS IN WATER After substituting and simple conversions we have 181 QrM -JQO n = l ■■ J - 2. / ■^(r-2^ %) ^ 2 ^"i[r2 + 4V]'/» + 2j^" r^[r^-2{nh+h^)^] r^[r-2{nh-hoy-] (22) Representing here = /j'" as the coefficient of re- 7" 2h 3 r / \ 2 2h r 2h. 5/2 cession of the radial apparatus, ^ve obtain a formula similar to the for- mula for the symmetrical AMNB apparatus: ^ru ~ 2^^ ^ + 2 ^ ^"^" + -^ ^0 ^P. + Z-J ^ri{ln-po + in+p,) . (23) It can be readily sho\\aa that for any other arrangement (quadrilateral, per- pendicular, etc.) at the bottom of the water, the general form of the formula for the apparent specific resistance will not differ from the expressions obtain- ed, but only the coefficients of recession will change. Thus, in general we have n = l in = l (24) § Let us consider the asymptotic values of Qj^^. For this we suppose that 7- -> oo, flQ and since Iiq ^ h, then -7- -> 00, in this case (/„_p^ + ^n+p) ~^ 2Z„ , and Ip^ -> 1, 00 and consequently, Qm~^Qo(^'^ ■^^^n^n) ~ Q^ ^•^- ^^^^h sufficiently large n = l operating distances r in comparison with the depth of immersion, the apparent resistance measured with this apparatus will not differ from the value of the apparent resistance which would be obtained if a similar apparatus of these dimensions were placed at the water surface. Assuming that 'T'~^ and also t > 0, we find that all coefficients of 182 E. I. Terekhin recession tend to zero; except ^n_p„, which is 1 when n = Pq. Therefore in this case 1 1 ^M "^ Y ^° "^ 2" ^"^^"' From (') it is Icnown that where k^ is the coefficient of reflection at the first boundary of separation: thn . Substituting the vakie of q ^ in expression Qj^ for -7- -> 0, we find that Thus, on the left part, the probe curves obtained by the apparatus at the bottom have a horizontal asymptote p = — r — •, since the curves obtained by an apparatus at the surface would have an asymptote Qq. Let us consider certain special cases of geoelectrical sections. (a) Under the water layer let there be an electrically homogeneous medium stretching to an infinite depth. In this case />q — 1 and q^ = A;" where k is the coefficient of reflection. The general formula for apparent resistance obtained by an apparatus at the bottom of the water becomes (CO \ / 00 00 \ 1 + 2 ^ kHn\ +\q, ( Zi+ Y, ^"^"-1+ Yj ^"^"+i)' n=i ' ^ n=l n=l After certain simple conversions we finally obtain for a two -layer medium n = l In practice (from the point of view of interpreting field material) this case is not of particular interest although this expression is undoubtedly of theoretical interest. (b) In the case of a multi-layered medium, when the thickness of the layer of water is a general measure of the thicknesses of the underlying ELECTRICAL PROBING WITH AN APPARATUS IN WATER 183 layers (pq = 1), the expression (24) for the apparent resistance for an apparatus at the bottom becomes: (OO V p OO -1 (26) CALCULATIONS OF THE THEORETICAL CURVES OF SEA PROBING With the formulae obtained, a calculation was made for two -layer and three-layer theoretical probe curves with an apparatus at the bottom of the water. The theoretical curves for probing with an apparatus both at the surface and at the bottom of the water correspond to the equation Q rl Qi Q2 K K :f where r is the operating distance of the apparatus; ^1 _ Q2 _ K _ K _ Qq Qq K K are the parameters of the geoelectrical section. The theoretical probing curves are drawn on a double logarithmic scale as the values for the dependence of— on _ - , the remaining para- mo ^0 meters {jx-^, [x^-t •••' '^i' ^2~-") ^^^ eia.c\\ curve are constant values. The para- meters of the calculated curves were selected to correspond with the para- meters of three-layer theoretical curves of the Schlumberger charts and curves calculated at the GSGT, taking into account the features of the section, the upper layer of which is the sea-water. For a two-layer section the following values were selected for [x^: 11/9, 3/2, 13/7, 7/3, 3, 4, 17/3, 7, 9, 19, 39, 99 and oo. The curves with [jl < 1 were not calculated since in practice the water always has a lower resistance than that of the underlying medium. For the three-layer section the following parameters were chosen: ^2 =00, (^0' (" 3/2 (-ri; Qo \e«l \ go \QqI Qi Qo ^^ /2' /s' '*' -'s ^^' 39; h, K = 1, 2, 3, 5, 9, 24. 184 E. I. Terekhin Furthermore, for — = oo curves were calculated with — = 1/5, 1/3, and 1/2. Q2 The curves mth — = were not calculated for a three -layer section since this case is not of practical interest for geoelectrical sections with an upper layer of high conductivity. All curves were calculated both for the symmetrical AMNB apparatus and for the dipole axial apparatus. Thus, the total volume of calculated material was 420 curves combined according to the variable parameter — into 68 graphs. All these graphs were published in 1956 by the All-Union Research Institute for Geophysical Methods of Prospecting (VNII Geofizika), as a collec- tion of theoretical curves for sea probing. According to the new notation of theoretical probing curves adopted in the NIGGR in 1956, all curves corresponding to the bottom position of the apparatus are given the letter M ("morskie" — sea), at the beginning of the number, and the number of curves calculated for a radial (dipole axial) apparatus have the letter P. The variable parameter for which the graph is prepared is designated by the letter c ("Soprotivlenie" — the resistance, of the upper layer under the vvater) or m ("moshchnost" — the thickness, of this layer), and the relative values of the attached parameters of the section are shown by numbers. Thus, the graph M— c — 1/5— c^ means that it contains the theoretical curves of probing with the symmetrical AMNB apparatus at the boundary of the first and second layers; the resistance Qi . . of the second stratum — being different for different curves (it is named for each curve), its thickness constituting 1/5 of the thickness of the upper 'QiV layer, and finally, the resistance of the underlying medium being As is known C^^, the theoretical probe curves in a general form are calculated from the formula _1_ = l + 2^g„Z„, (27) n=l where q^ is a function depending only on the ratios of the resistances and the thicknesses of the various levels and the serial number n; l^ is a function depending only on the size of the apparatus and the serial number n. ELECTRICAL PROBING WITH AN APPARATUS IN WATER l85 For instruments at the boundary of separation of the first and second layers, this formula becomes somewhat complex. Let us consider first a two -layered medium. In this case the problem of the calculation of the theoretical curves is solved most simply. From equation (27) we obtain n=l oo Having substituted the value for the series ^ ^n ^n "^ *^^ expression (25). n = l we obtain the following after simple conversions: M H [A, /< + 1 fjfi -1 \Qq I. (28) [1 — 1 bearing in mind that k — [i + l From this formula, which has the coordinates of the two -layer curves for an apparatus at the surface, it is easy, without additional summation, to calculate the coordinates of the corresponding two -layer curves for an apparatus at the bottom of the -^vater. For a three-layer section, the formulae for the apparent resistance of a bottom apparatus cannot be brought to a form suitable for simple conversion and the calculation of the three-layer curves for a bottom apparatus requires an additional summation of infinite series. A certain part of the curves, i.e. the curves with — =00 and — = 1/5. 1/3, 1/2, 1, 2, 3, 5 were calculated by the method of splitting up the function B into simple fractions. This method was proposed by S. Stefanesko and has been further developed in a number of papers ^^' '' ^> ^\ The essence of this method is as follows. If the underlying medium has a very high (^^ = 00) or a very low (^„ = 0) specific electrical resistance, then the function Pn(e-^"^^) ^ (?„(e-2mh) for such sections can be represented in the form of the sum of elementary fractions. 186 E. I. Terekhin Thus, for Q^ = CO Q—2mh Q—2mh liPn-l + 2) 7=3 ■* (29) if p^_j is an even number. If Pn-i ^^ ^^ °^^ number, then |(pre-l+3) R = 7, e"^""^ - V ^ Ve-2^^-e-^^ft (30) y=3 ^ In these expressions the functions and correspond to ^ \ Q-2mh \j^Q-2mh ^ a two -layer section with ^2 ~ 00 and ^2 ~ ^5 ^^d the functions of the form 1^ .' Q—2mh Q—imh ; correspond to a three-layer section, when the two l-2A;y'e-2^'»+e-4'"'^ ^ ^ layers of equal thickness with resistances ^y and ^y_j (the coefficient of reflec- tion kj') He on the underlying medium of infinite conductivity (^3 = 0). ]y .' Q — 2mh Q—imh From (8) it is known that the function ; can be ^ ' l-2A;/e-2'"'2+e-4'«^ 00 represented in the form of the series 2 e~^'"^ cos ncp, where cp — arc cos k-'. n = l Q—2mh Q—2mh The functions and can also be represented in the \_Q-2mh \j^Q-2mh '■ form of the series: 00 ^-2mh ^— , , — 2mh E \_Q-2mh n=l -2mh -2inh l_<_Q-2mh ^(-1)" /i=l Let there be a geoelectrical 7i-layer cross-section, where at a relative depth Pn-i there is a medium with zero conductivity (q^— 00). If we consider the potential of a point source of current at the boundary of intersection of the first and second layers, then for this boundary of ELECTRICAL PROBING WITH AN APPARATUS IN WATER 187 intersection, according to formula (14), we obtain l/^ = l^\ — + I Q-2mh„ /^ (^fnr) dm +2 j B^ /„ {mr) dm + '"-4-^L""i' oo + pi(e-2'"'" + G^""^") loimr) dm Let/>^_j be an even number. Then, using formula (31) and the correspond- ing expressions for elemental fractions as a series, we obtain: a' \ n=l n-1 |(pn-l+2) 00 . P 4- V bj^ e-^'^'''' COS n(p\ loimr) dm +je- ""''''' lQ{mr) dm + + / &i V* e-2mn/i(e-2mfto+ e2'n/'<') + 00 {- 5^ V" ( - 1)" e-2""''i ie-2mho + e^mho) + y=3 n=l 5(pn-l + 2) 00 + V 6y V e-2"^"'^ COS ;i(^ (e-2mho + e^'n'Jo) y=3 n=l Using the Weber-Lipshitz formula, we obtain lQ{mr) dm ' \ h ^'" + (2n/i)2 ^^ |/r2 + {2nhy |fp7!-l + 2) 00 + 7=3 n=l ' COS n(pj Z^Jf.+ {2nhYl ]/r^+{2KY + 0° + ^^ \ ]/r^+{2nh-2hoY ]/ r"^ + {2nh -\- 211^) + 188 E. I. Terekhin 00 + i(pn-l+2) 00 {2nh - 2 Ao) 2 I//-2 + {2nh + 2 Aq) ■ cos ncpj' cos 7199/ + + In agreement with the obtained expression for the potential, we find an expression for apparent resistance for a symmetrical 4-electrode AMNB or a 3 -electrode AMN apparatus, at the bottom of the water: ^M = 2 ^« 1 + 26 L '"- 2j \r^ + {2r •f&o (_l)n^3 |(p«-l + 2) 00 ET~^ r" cos 7 7Z9?y ;=3 n=l [r2+(27^/7)2]3/2y [7-2+ (2Ao)"] 13/2 ^^2j \\r'^+{2nh-'. [r2+(27l/i-2/2o)2]3/2 [7-2+ (271/1+ 2/20) T^2 n = l \{pn-l + 2) 00 (-l)"r n r3 (_l)n^3 [r2+(27^/i-2/7.o)2]3/2 ' [;-2+ (271/2+ 2Ao)2]3/2 + + ;=3 r"* cos 7Z97y z'* cos 7799^ ^^ \ [7-2 + (27l/l-2/7o)2]3/2 [;-2 + (2,z/i + 2h^Y] 3/2 5(P72-l+2) Since 2 ^y = I5 then, introducing symbols for the coefficients of recession, we have ^M=l^o[6i(l + 2|;^/,j + 6,|l + 2j^(-ir/, 5(pn-l + 2) / 00 \ / CO + 2 ^- l + 2S/„COS7Z9./ +6J/^„+2(^n-p.+ ^n+p„)) + +^2(^p.+S^(-lr(/n-p„+Wp.)]+ |(pn-l+2) / 00 + 2 ^- Ko + 2 cos 779^/ (/„_p^ + /„+ J ;=3 (31) ELECTRICAL PROBING WITH AN APPARATUS IN WATER 189 In a more general form ^ve can write ■| {pn-l + 2) Qm^^JSoI 2j ^M ^ + 2 2j ^n COS ncpj' ] + V ^{pn-1+2] (32) since the terms 6^ 1 + 2 2 ^n ) ^^^ ^2 1 "i" 2 2 ( ~ l)"^/2 1 ^^e partial forms for kj' = + 1 and kj' === — 1. The same can be said for the terms W ^Po+ 2(^n-p„+^n+p„) id 6o ^P„+S(-im-p.+ ^n+p,) We will consider in a general form one of the terms in the second sum of the expression (32): ^Po+ 2 ^^n-p,+^n+pj cos n9? /p,+ (L(p„_i)+/(i+p,))cos9P + + (/_{p„_2)+ ^2+p.) cos29?+ ...+ (Z_j+ Z2p^_i) cos(/Jo-l) 9? + + (^0 + kp) cos Pq(p + (Zi + /gp^+i) cos {pQ+l)(p-\- ... Since the function /,- is an even function ^vith respect to i, then assuming that L — I ;, we find rPo + S ^n-po + ^n+p„) COS nxp = Zo COS po(p + Zi [cos (/>o + 1) 9? + L n = l + cos(/>o — 1)9?] + Z2[co3 {pQ+2)(p+ cos{pQ — 2)g)]+ ...+ + ho [^°^ 2^0 9? + 1] + ^p„+i [cos (2/?o + 1) 9? + cos 9?] + . . . = = cos/>o9)+ 2ZiCos/)q9?cos 9?+ 2Z2C03/>o9'co3 29?+ 3Z3Cosj9o9' cos 39? + + ... + 21 p^cos Pq(p cos Pq(p + 2lp^_^^cos PqCP cos {pq-\- 1) 9?+ ... = = cos PqCP [1 + 2(Zj^cos (p+ Z2Cos29?+ Z3Cos39?+ ...)]. Thus we find that r ^Po+2(^n-po+^n+p„)cosre«; COS Pq(p 1 + 2 2 ^n cos n(p ■ (33) 190 E. I. Terekhin Substituting the obtained value in the expression (32) we obtain I (pre-l+2) Qm^ ^0 y ^ 1+ cos Pq(Pj' ^ ^ ;=i /„ cos n(p I . n=l The first two terms of this sum represent a two -layer medium />q = 1 with. kj' = +1 and kj' = — 1 and, consequently, cos pQ(pj' for these cases is equal to +1 and —1 respectively. Taking this into account and adopting the symbols for apparent resistance according to A. I. Zaborovskii, we finally obtain I (p«-l + 2) Qm^ Qo hQ^ + y=3 7 1 + cos Pq(p/ - , (34) By similar reasoning it can readily be shown that for an odd J5„_j the expression for the apparent resistance for an apparatus at the bottom of the water has the form of ■M Qo Wq' i (pn-1 + 3) V y=3 bj- Po<Pj Qi (34') Thus, sphtting up the coefficient B^ into simple fractions does not depend only on the type of the apparatus (^), but also on its position relative to the section. The coefficients of serial expression 6^, 63, 64, ... and the coefficients of reflection ^'- were found from formulae given by G. D. Tsekov (^). All calculations were carried out mth a projected accuracy of obtaining the coordinates within ±0.5%. Most of the curves were calculated by the method of summation of series. From formula (24), assuming that ? = ^0 1 + 2 2 ^nk n = l we have ^M ^ ^ + 2" ^0 (^Po-l)+ 27n(^n-p„+ ^n+p„-2Zn) (35) In this expression the apparent resistance measured by an apparatus at the bottom of the water, is represented as the sum of the apparent resist- ance, measured by a similar apparatus at the surface of the water, and a certain correction. Since the values of q have been calculated for ELECTRICAL PROBING WITH AN APPARATUS IN WATER 191 a large number of different three-layer geoelectrical sections and are given in tables of coordinates of three-layer probing curves for various types of instruments, only the correction of the expression which is in square brack- ets is subject to calculations. In calculations an important element included in this expression is the series: oo ^1 = 9- 2j 9"(^"-Po+ ^n+p„ — 2/n)- n = l Naturally, in practice, only a certain series 1 "^ n = l can be calculated, the value of which, depending on the value of the number m for which the summation was completed, approximates, with a certain degree of accuracy, to the value of the series Sj. The evaluation of the accuracy of approximation and, consequently, the determination of the necessary number of summation terms m to attain the projected accuracy were carried out according to the method proposed by L. L. Van'yan. In calculating the apparent specific resistance, measured by an instrument at the sea -bottom, with the aid of formula (37), as a result of shortening the svim- mation of the series at the m-th term an error is introduced into the cal- culated value. This error can be expressed by the value 00 Zl5 = |5i-52| = y 2j ^nik-po+^n+po-^D ■ n-m + 1 To evaluate the obtained value, we carry out a series of conversions: 1 v-^ ^5=2- 2j ^ri{ln-po+ in+p,-2ln) < n = m + l 00 ^2" Xj ^^"(^"-Po+^^+P''-^^") n = m + l < ^2" 2-1 k"l^"-Po+^«+Po-2/„|, (36) therefore. n = m + l n = m + l 192 E. I. Terekhin Here ^j^nax ^^ ^^^^ greatest numerical coefficient of emission in the residual series As from ^ = ^ + 1 to cxd. To study the character of the functions oo 2 (^n-p+ In+p-^k) and {l^_p+l^^^-2l^) n = m + l with change in n, we present the second of them as the second derivative of the coefficient of recession l^ for n: iln-p,+ In+po-^^n) = {ln-p,-ln)-{ln-hi+p) ^-^{AnY. Putting An = 1, we have (%-Po+^/7+Po -^^n) dHr and, consequently, / , (^n-po+ ^n+Po *^^n) '^ ~~ For the symmetrical AMNB apparatus l^ din dn entiation we obtain {x^ + 4^2)v. din dn dH 12:^3 {x^ + 4.nyi- 16/z2- — 1 2ir2 dn^ (%2 + 4/z2)'/. • After differ - dn ^„2 Fig. 1. ELECTRICAL PROBING WITH AN APPARATUS IN WATER 193 The function — - has one extreme value for Hq = 0.25a'; for n^>-0 and dn ri -> oo the function — - -> 0, i.e. Hm — - = 0. dn n^o.oo dn Figure 1 shows graphs for the functions — ~ and . dn dn^ A study of these functions shows that in the summation sign in expression (36) in straight brackets, the components can be positive for re > Wq, and negative for n <^ n^. To evaluate the error, these terms should be summed to the absolute value. We will consider two cases. (a) Let m> Hq. In this case, all the components in the summation sign of expression (36) are positive and the error can be evaluated by the inequality 1 I °° "^ n = m + l We will consider the series: oo oo oo n=m+l n=m+l n=m+l ^ V^m + l-p„~~^m + l)+ (^m + 2-p„~^m + 2) + (^m-|-3-po~"^m+3) + •" + "^ v^m + l~^m + l+po/ '^ v^m+2~^m+2+Po^ '^ (^m+3~^m+3+Po) "^ •" ~ ~i^m + l~^/n + l+p„)~(^m + 2~^m + 2+p„)~(^m+3~^m+3+Po)~ •'• ~ ^ (^m + l-p„+ ^m + l)+ (^m + 2-p„+ ^m + 2) + ••' + \''m~''m+p) ~ Po ^^ 2j \^m-po+i~''m+i)' 1 = 1 The magnitude of the error is therefore expressed in this case by the inequality 1 I 1 ^° ^5i <-^ I graax \ 2 (^m-p,+i -^m+z)- (37) ^ 1 = 1 (b) We will now put m <i n^. In this case, the values in the summation sign of expression (36) in straight brackets are both positive and negative; thus it can be written n=m + l n=/7o + l 2 \K-p„'^h+p„ 2Z„|— 2 (^n-po+ ^n+p„~2/J — n=/7o^-l no 2 (^n-po'^~ ^n+Po~'^^n) n=m+l Applied geophysics 13 194 E, I. Terekhin or, representing the second term of the equation as the difference between the two sums, oo oo n = m + l n = no+l -2 2 (^n-p„+^n+Po-2U- 2 (^n-p„+^.+po-2g. n = /7o + l n = m + l Treating the obtained series exactly the same as for case (a), we ob- tain oo p„ 2j I ^/2-po+ ^/i+Po~2i„ I = 2 2j (Zn„-po+i~^n„+i)~ n = m + l P. ~" 2j \''m-pa+i'~^m+i)' 1 = 1 The error in this case is determined by the inequahty ^^2 < 2" I ^max I Po Po "1 2 2j (^no-po+J~"^no+i)"~ Zj (^m-p„+i~^m+i) i=l 1=1 J (38) The relative value of the error can then be found a=^< All geoelectrical sections, for which the curves of apparent resistance were calculated by the shortened method, for a sea-bottom apparatus, had j5q = 1 (the layer lying directly under the water, greater than or equal to the thickness of the Avater layer) the expressions for the errors were therefore considerably simplified : ^S^ ^—\ 5'rnax | (^m ^m + l)? ^•^2 <-2 1 9max I [2 (/n„-/n„+i) -(^m -4l+l)]• Replacing (/„-Z^+i) by Al^, and (/„„-Z„„+i) by Al^^^ we find the expression for the relative error: ELECTRICAL PROBING WITH AN APPARATUS IN WATER 195 for m > Hq - u (39) and for m < Hq ^ = ^^< l + jti 9max I (-^-^'max ^^m) I o - w 39' ^ 1 + /J, Given a certain degree of accuracy (in our case the accuracy taken was d — 0.005, or 0.5%), it is possible to select such a number m of terms of the sum, to satisfy the inequality 100(l + /f) l^maxl or which ensures the given accuracy S. This operation is readily carried out by means of a slide rule and an appropriate table for the function l^. For most of the probe curves for any value of -y- the value //i for an accuracy d = 0.5% lies within the limits of 1-15 and does not exceed 35. REFERENCES 1. L. M. Al'pin, The Theory of Dipole Probes. Gostoptekhizdat, (1950). 2. V. V. BuRGSDORF, Calculation of earthings in heterogeneous grounds. Elektrichestvo No. 1. (1954). 3. L. L. Van'yan, Theoretical Curves for Electrical Sea probing with a sea-bottom apparatus. Applied Geophysics No. 50. Gostoptekhizdat, (1956). 4. V. N. Dakhnov, Electrical Prospecting of Petroleum and Gaseous Deposits. Gos- toptekhizdat, (1953). 5. A. I. Zaborovskii, Electrical Prospecting. Gostoptekhizdat, (1943). 6. S. Ya. LiTviNOV, Electrical Sea Prospecting. Gostoptekhizdat, (1941). 7. R. Maie, Mathematical Basis for Electrical Prospecting with a Direct Current. ONTI. (1935). 8. G. D. TsEKOV, A Method for Calculating Multilayer Curves of Vertical Electrical Probing for a Case where the Underlying Medium is in the Form of Rocks of Very High or Very Low Resistance. Thesis. Documents VNII Geofizika. 9. K. Flathe, a practical method of calculating geoelectrical model graphs for horizon- tally stratified media. Geophysical Prospecting, vol. 3. (1955). 13' ' Chapter 8 THE USE OF NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA A. M. Alekseev, M. N. Berdichevskii and A. M. Zagarmistr The effectiveness of using the method of vertical electrical probing in a number of regions of Siberia and the Far East was shown in the 1930s and 1940's. Electrical studies of the territory of the Western Siberian low- lands, in the regions of Eastern Siberia (especially at Lake Baikal), and on Sakhalin made it possible to obtain valuable geological results. Since then however, electro -prospecting work in Siberia has not received much deve- lopment, since swampy conditions and wooded nature of the territory have presented considerable difficulties in carrying out vertical electrical probing. The necessity of using long feed lines (12-20 km), and power generators, (10-20 kW) have considerably limited the possibilities of using vertical electrical probing in accessible places. Because of this, in some geophysical institutes in Siberia, electrical exploration has been excluded entirely from the exploratory work for oil and gas. The exclusion of electrical exploration from the geophysical studies has necessitated the solution of prospecting and detailed problems of surveying almost exclusively by seismological methods, which are more difficult and expensive in comparison with other methods of petroleum geophysics. The development in the VNII Geofizika of new methods (the method of two-way electrical probing and the method of telluric currents), which considerably extend the possibihties of electrical exploration, led to the organization in 1955-56 of widespread experimental electrical explora- tion in Siberia. The aim of this work was to determine the geological effective- ness of new methods under geoelectrical conditions of the Western Sibe- rian lowlands and to develop a method of measurement in inaccessible re- gions. Experimental work was carried out by parties from the VNII Geofizika (under the direction of V. P. Bordovskii and Yu. N. Popov) on the North West depression of the Kolyvan-Tomsk fold belt and within the hmits of the Tobol'sk and the Vagai-Ishim tectonic zones, and also in the region of .the Berezovka gas deposits. This work showed the desirability for carrying 196 NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 197 out a considerable extension of electrical exploration in the rather in- accessible regions of Siberia. The present article deals with the basic residts of the studies. THE METHOD OF TWO-W AY ELECTRICAL PROBING The theory of two-way electrical probing by dipole instruments was de- veloped by L. M. Al'pin between 1948 and 1950 ^^K In subsequent years, in the VNIIGeofizika a method was developed for the practical use of two-way probes with quadrilateral* arrangements. This has now been put into prac- tice in geophysical studies over the territory of the European part of the USSR (3). A P Fig. 1. Azimuthal arrangement. This method has a number of advantages over the vertical electrical prob- ing carried out with the symmetrical four-pole AMNB apparatus. Apart from the possibility of producing deep probings by using small spacings of electrodes (1-2 km) and the improved quality of the measurement? due to decrease in the harmful effect of leaks, the quadrilateral probings have increased resolving capacity with respect to the inchnation of the reflecting electrical horizon. "j" However, under conditions involving swamps and forests, the method of quadrilateral probings, which requires movements over long straight routes, is of limited application. In this respect, much better possibilities are shown by the method of two-way probings with an azimuthal arrangement (Fig. 1), a special case of * The word equatorial is used in Russian, which however would be misleading in English. [Editor's footnote]. t Electrically reflecting horizon is a horizon of high resistance. This translation is necessary since the same term is used by the Russian geophysicists for a seismically reflecting horizon. 198 A. M. Alekseev et al. which are the quadrilateral probings. When carrying out azimuthal probing, it is by no means necessary to place the centres of the measuring lines on a straight line, directed along the axis of probing. For a number of dipole arrangements including the azimuthal, in a hori- zontally homogeneous medium, the values of KS* do not depend on the an- gle O which makes it possible to use these arrangements in studies on the roads and other curved routes. The curved azimuthal probingsf have all the advantages of the quadrilateral probes but are much more readily ap- plied to the region. The possibility of practical curved azimuthal probes was first shown by the electro-prospectors of VNII Geofizika working (1954) in the Cis -Baltic depression. Three azimuthal probes were carried out with working distances of up to 12 km. The probes were carried out by means of the EPS -23, mobile electrical prospecting station, \\'ith the limiting deviations of angle O from 90° not exceeding ±30°. The curves obtained show sufficiently well the geoelectrical cross-section of the Cis -Baltic depression. The results showed the suitability of a similar arrangement of work for carrying out deep electrical probes and made it possible to plan a programme of further studies on the method of measure- ments with the azimuthal arrangement. The development of the method of curved azimuthal probes was continued in 1955-1956 in the Western Sibe- rian Lowlands. An experimental party from VNII Geofizika carried out 40 two-way curved azimuthal probings in the Kochanevsk region of the Novosibirsk region (1955), where the supporting electrical levels lie at depths of from 200 to 1000 m. The measurements were made along wind- ing roads, running through swampy and wooded territory. The maximum working distance achieved in the majority of cases was 6 km A\dth a length of feed line up to 1 km. As a rule, the length of the measuring line did not exceed 400 m. The topographic basis for the azimuthal probes was worked out by means of a plane table. Nvunerous control measurements showed the good reproducibility of the azimuthal probing results within the limits of 5-6% providing that the angle O did not differ from 90° by more than ±20°. This condition does not seriously limit the possibilities of curved probings, since if necessary it is possible to use 2 feed lines in different directions. In the southern part of the surveyed area, where it was easier to move, * iir5-Koeffitsient soprotivleniya — Coefficient of resistance -resistivity [Editor's note]. t The article by Berdichevskii "The method of curved azimuthal probes" is in the present coUectior. NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 199 several quadrilateral and aziniuthal probes were carried out with coincident centres. The results obtained were sufficiently close to one another. In 1956, curved azimuthal probings were carried out in the Tobol'sk and Vagal' -Ishim tectonic zones (Tyumensk region) in localities characterized by a depth of the reflecting level of 1200 to 2500 m. The studies were car- ried out over a route about 250 km long. The maximum distances between the dipoles of the azimuthal arrangements were between 8 and 10 km. The field laboratories moved along winding roads. Despite the fact that owing to the frequent rains, the road conditions in a number of regions were bad, the party was able to carry out more or less regularly one two- way probing in one working day equivalent to two AMNB probings with maximum spacings of AB = 16-20 kin. The curves of the azimuthal probing agree sufficiently well mth the curves for quadrilateral probing. The oscillo- grams referring to the large dispersions of the azimuthal probing were worked out with a sufficient degree of accuracy. As well as with the azimuthal arrangement tests were made using a radial ar- rangement and a parallel arrangement, with small angle 0, making it possible to obtain results which practically coincided with the results of the radial probes. These studies were also carried out over curved routes. With radial and especially with parallel probes, it was much easier to unreel the wire. However, due to the reduced depth of the study, these methods under conditions involving an electrically reflecting level at a great depth were used in an auxiliary capacity. The work showed the effectiveness of using the irxethod of two-way elec- trical probes for surveying upfolds in the top surface of the marker horizon, which in a number of sections of the Western Siberian Lowlands represents the top surface of the Paleozoic deposits. It was also shown that in zones of sedimentary and volcanic layers of the second structural level the stratigraphic- position of the electrical marker horizons can change depending on the degree of metaraorphism of the sedimentary rocks, and the presence of volcanic formations in them. In this case, the data of the electrical survey reflect the changes in the thickness of the conducting part of the section, which is represented by friable, weakly -metamorphized arana- ceous argillaceous deposits, devoid of widespread intrusive sills. The curves of two-way electrical probing can be interpreted more re- liably than the curves of the vertical electrical probe and give fuller infor- mation about the nature of disposition of rocks. The methods of qualita- tive and quantitative interpretations of electrical probe curves in two-way measurements are supplemented by an analysis of the divergence of posi- tive and negative KS curves. This divergence is caused by the dip of the 200 A. M. Alekseev et al. electrical curves by responsive horizons and by the change in their specific resistance in the region adjoining the probe apparatus. The qualitative analysis of curves for two-way probing consists in follow- ing the character of the relationships of the positive and negative KS curves in the profiles (especially their right-hand branches), and also in construct- ing and studying: graphs of positive and negative KS values {KS lines) for a given dispersion; graphs of positive and negative values of S (5 fines); vectors maps for difference in plus and minus KS values (vectors Aq). In the quantitative interpretation of curves of two-way probing, not only were the average KS curves processed, being largely similar to the curves for vertical electrical probing, but also the plus and minus KS curves, and also the derived KS curves given by the method of 0. V. Nazarenko, V. A. LiPiLiN, et al. The geoelectrical sections were constructed both by using a handbook of theoretical curves, calculated for the AMN arrange - "AZ-55 Suvorovskii SidorOvKO Fig. 2. KS map for operating distance i? = 3000 m. 1 — centres of two-way equatorial and azimuthal probes; 2 — KS isolines. NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 201 ment, and by a number of specially developed methods (method S, the method of giving the value Qi, the method for transforming curves of quad- rilateral azimuthal probing into curves of axial probing). As an example, results are given for studies in two different regions of the Western Siberian Lowlands. First of all, we Avill deal with the material of the qualitative and quanti- tative interpretations of two-way probing on the north-west ends of the Koly van' -Tomsk fold belt (the work of 1955). The surveyed area includes the pre-Jurassic foundation, covered by Meso-Cenezoic aranaceous argil- laceous deposits^ and dipping in a north-westerly direction to depths of the order of 1000 m. The problem of the electrical survey was to study the rehef of the surface of the pre-Jurassic foundation. Troilskoe R Komyshenka Fig. 3. Map for Aq vectors for an operating distance R = 3000 m. 1 — centres of 2 -sided equatorial and azimuthal probes by the VNIlGeofizika party; 2 — vector ofAg; 3 — centres of two-way quadrilateral probes by the Sibneftegeofizika Department. The numerator represents the number of the probe ; the denominator represents the - ? + Q — value for the vector Aq= 100 . 202 A. M. Alekseev et al. Figure 2 shows a KS map drawia from the average KS curves for a worldng distance R = 3,000 m. There is a regular decrease in the KS values in a north-westerly direction, corresponding to increase in the thickness of the friable Meso-Cenezoic formations. On this background to the north of L. Mikhailovskoe a local maximum of KS vakies is identifiable which is an indication that the Paleozoic foundation is lifted up between the points L. Mikhailovskoe and Epifanovsldi. This interpretation of the structure of the Paleozoic rocks is supported by the vectors Aq (Fig. 3) for the same working distance. DZ-59 DZ-58 DZ-50 DZ-48 DZ-56 DZ-57 Fig. 4. Curves for two-way probes along AB profile. As can be seen, the directions of the vectors Aq clearly characterize the basic features of the surface of the Paleozoic foundations (remembering that the vectors Aq indicate the direction in which the KS values decrease). The presence of elevations in the top surface of the foundation are partic- ularly well observed on comparing the plus and minus KS curves (Fig. 4). Thus, on the AB profile, which intersects the zone of increased KS value, there is a reversal of the plus and minus KS curves (DZ-58, DZ-59), characteristic of a passage across the crest of the structure. On the DZ-58 the right-hand branch of the positive (south-west) curve, represented by a continuous line, is placed above the right-hand branch of the negative (north-east) curve, shown by a dotted line, which indicates a rise in the foun- dation in a south-westerly direction. On the DZ-59, reverse relationships were observed, which indicate the change in direction of dip of the foundation. Very characteristic are the relationships of the right-hand branches of the curves DZ-48, DZ^9, DZ-50, showing the presence of a depression in the surface of the founda- tion -svith deepest part in the DZ^9 zone, where the right-hand bran- ches of the positive and negative curves coincide almost completely. NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 203 On the north-east section of the profile, the relationships of the right- hand hranches of the two-way probe reflect the rise of the pre-Jurassic rocks in a north-eastern direction (DZ-56, DZ-57). The results of the qualitative interpretation facilitate the selection of parameters for subsequent quantitative calculations and improve their reliability. Figure 5 gives a geoelectrical cross-section through the profile considered above. As can be seen, the quantitative treatment agrees well with the qual- itative impressions of the surface relief of the Paleozoic foinidation at the 200 . D359 D358 D350 D349 D348 D356 D357 100 20i?.m -^ ^^^ bO/?.m 5-6 n.xn 50-200 /?.m -400 ^P— .. Fig. 5. Geoelectrical cross-section along AB profile. 1 — top surface of the Paleozoic rocks according to data of layer processing of average curves of (juadrilateral and azimuthal probes. investigated section and, furthermore, indicate the structure of the sedi- mentary succession. As well as the layer interpretation with the aid of the- oretical graphs, the average KS curves were also processed by the method of transformation into curves of axial i)robing<^). Fig. 6 compares the results of the quantitative interpretation by botli methods along 40 km of the CD profile. The second example refers to results of the application of two-way azimuthal probes in 1956 at one of the sections of the Vagai-Ishim tectonic zone, where the rocks of the pre-Jurassic foundation lie at depths greater than 1600 m. The curves of the azimuthal probe clearly subdivide the geoelectrical cross- section. They differentiate a thick succession of conducting friable deposits, underlain by the highly resistant rocks which form the electrically reflecting horizon. As shown above, the stratigraphic continuity of the high resistance, electrical marker horizon is apparently not maintained in the zones of development of effusively deposited rocks of the second structural stage. At the present time, due to the absence of geological data, this problem has been insufficiently studied. It can only be supposed that, depending 204 A. M. Alekseev et al. on the degree of metamorphism of the rocks of this stage and the development in them of layers of lava, the surface of the high-resistance electrical marker horizon may be displaced within the limits of the whole volcano -sedi- mentary succession. However, this fact does not prevent the application of electric -prospecting to surveys of elevations in the relief of the bottom of the platform -Mesozoic. The results of electrical well -logging show that volcano -sedimentary strata have a higher resistance, due to which the values of total longitudinal conducti- vity (5), obtained during probing, are mainly determined from the parameters .22 1 22 I v. Vakhrushevo D 6 7 .13 .14 V Novotyryshkinc 10 II 12 13 14 Fig. 6. Geoelectrical cross-section along CD profile. 1^ — top surface of the Paleozoic rocks from data of layer processing of average curves of equatorial and azimuthal probes; 2 — top surface of the Paleozoic rocks from results of transforming average curves of equatorial and azimuthal probes into curves of axial probes. of the Meso-Cenezoic deposits. The small change in the specific electrical resistance of the rocks of the Meso-Cenezoic over a wide range of territory provides favourable conditions for tracking the relative changes in thickness of the friable Meso-Cenezoic succession by the changes in S. This conclusion is confirmed by the results of observations on areas studied by seismic methods^ Figure 7 gives a graph of S compared with the seismic data on the reflecting horizon in the Lower Cretaceous (Valanginian) deposits. The graph of S differentiates two zones of reduced values, associated with the crests of the Viatkinskaia and Krotovskaia structures, which were surveyed by the method of reflected waves. Of interest is the identical behaviour of the graph S and the seismic marker horizon which makes it possible to use the S method with a constant parameter Qi selected from the drilling data of the Vyatkinsk area as one of the methods for the quantitative interpretation. NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 205 £ 400 800 co' 400 ^ 1200 l' 2 000 2800 Vyatkino Structure Sea level Fig. 7. Results of geophysical studies along the profile Viatkino-Krotovka. 1 — 5 lines; 2 — depth lines; 3 — supposed surface of supporting electrical high-resistance level; 4 — surface of seismical supporting levels. The geoelectrical section, denoting the behaviour of the electrically highly resistant horizon level is given in the lower part of the drawing. It follows from the data given that the results of the interpretation of electrical probes make it possible to decide on the tectonics of the Mesozoic deposits. Similar results were obtained on other structures revealed by seismical work in the region between the townships Viatkino and Vikulovo (Dmitriev- skaia and Krutikhinskaia structures) and also on certain areas of the Tobolsk -400 -800 -1200 r -1600 -2000 A2-84 AZ-83 AZ-82 AZ-Bl A2-80 AZ-79 AZ-78 Sea level Krotovskaio structure Fig. 8. Two-way curves of an azimuthal probe over an upheaval of a support high- resistance level. 1 — centres of two-way electrical probes; 2 — supposed surface of highly resistant reflecting horizons; 3 — right-hand branches of AZ Western curves; 4 — right-hand branches of Eastern AZ curves. 206 A. M. Alekseev et al. tectonic zone (Zavodoukovskaia, Komissarovskaia and Kapralikhskaia struc- tures). However, in one case at the section between the township's Bol'shoe Sorokino and Vikulovo an elevation of the electrical marking horizon was observed. The elevation was not reflected in the seismic profile. This fact requires further study. Despite the considerable depths of the electrical marking horizon in the Tobol'sk and Vagai-Ishim tectonic zones, the curves of two-way probing have very well defined qualitative features, associated with the non -horizontal nature of the studied surface. Fig. 8 gives as an example of two-way curves for the azirauthal probing along the profile intersecting the Krotovskaia struc- ture. It can easily be seen that here the above-described regularities are maintained in the mutual positions of the right-hand branches of the plus and minus curves. It is obvious that such clear indications are possible under conditions of comparative constancy of the geoelectrical section with regard to the distances, which is characteristic for the territory being studied. THE METHOD OF TELLURIC CURRENTS The idea of the method of telluric currents was proposed by K. Schlum- BERGER in the thi ties. However, owing to the intensive development in the resistance method, observations on the telluric currents were rarely conducted.. After the war the systematic development of the telluric current method in the USSR was started by the staff of the VNII Geofizika working initially under S. M. Sheinman, and then Alekseev and Berdichevskii (^>. Experi- mental work conducted by jjarties from VNII Geofizika in the Saratovsk Zavolzh'e (1949), the Dneprovsko-Donetsk depression (1952) and the Cis-Baltic depression (1954), gave favourable results and made it possible to change over to the large-scale use of this method. Abroad (^) the method of telluric currents also finds industrial application. The use of the telluric current method involves studies of the average periodic variations of the natural non -stationary electrical field of the earth (field of telluric currents) associated with a certain electrical phenomenon in the ionosphere. Telluric currents, embrace the whole of the globe, forming on its surface regional current whirlpools, and have a pulsating character, changing in time, in value and direction. The maximum intensity of variation of the field of telluric currents is usually observed in the period from 1-12 hr (Greenwich Mean Time) <'^>. The theory of the telluric current method proposes that within the limits of small areas of the earth's surface, the field of telluric currents at any givea instant of time can be considered as the field of a constant current, caused NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 207 by an infinitely long supply line AB. If the medium under investigation is horizontally homogeneous, then this field is always constant in value and direction. On the other hand, on the surface of a horizontally heterogeneous medium, the field shows variations both in value and in direction. These anomalies are associated with features of the geological structure of the medium under investigation, especially with the change in the total longi- tudinal condvictivity of the sedimentary succession lying on a non-conducting foundation. Thus, the field of telluric currents, as distinct from gravitational and magnetic fields of the earth, does not depend on the structure and petro- graphic composition of the rocks underlying the friable deposits, and under favourable conditions reflects the basic, features of the relief of the reflecting highly resistant level. Exploratory observations of telluric currents are made at various points of the studied area at the same time as observations at the stationary base point. The distance between the base and field stations does not usvially exceed 30-35 km. The dimensions of the measuring devices are selected independently of the depth of the electrically reflecting horizon and in most cases are limited to 500-1000 m. This offers possibilities for using the method of telluric current in deep surveys, especially in difficultly accessible locations. By processing the tellurograms obtained simultaneously at the base and field stations, the so-called parameter K associated in a simple way with the ratio of the areas of synchronized closed hodographs* of the field of telluric currents at the base and field points. The parameter K is the ratio of the average field intensities of the telluric currents at the points of observation. Using the value of the parameter K and taking the average intensity E of the field of the telluric current at the base point as being equal to any arbitrary value, it is possible to calculate the value of £" at a number of field points and construct a map of E (a map of the average intensity of the field of the telluric currents) which represents the main results. In large-scale surveys, in addition to the E map, maps are drawn for other values, of auxifiary importance. The geological interpretation of the E map is based on the idea that the decrease in E values is caused by an increase in the total longitudinal conductivity of the layer above the marker horizon and, consequently, under conditions of a constant geoelectrical cross-section by the downward dip of the high -resistance marker horizon. The regions of increased valwes of E * Distance-time curves. 208 A. M. Alekseev et al. are interpreted as zones of reduced total longitudinal conductivity of the succession above the marker horizon and as the zone of elevation of the highly resistant marker horigon. The method of telluric currents doesn ot involve a subdivision of the studied cross-section based on resistance, but gives results somewhat dependent on the influence of the horizontal electrical heterogeneity of the conducting sequence. In this connection, it would be desirable to combine the method of telluric currents ^v'ith electrical resistance probings at various sections of the studied area. A combination of the observations on telluric currents with electrical probes not only increases the reliability of the qualitative conclusions on the geological structure of the region, but also makes it possible to carry out approximate quantitative calculations, necessary for conversion from the map of average field intensity of the telluric currents to a schematic structural map of the reflecting highly resistant horizon. With an equally -spaced network of resistance probes, the quantitative interpre- tation of the data of the telluric current method is made by using the empirical formula H = FE", in which the values of F, oc for the whole area of the survey are determined by comparing the average field intensity E of the telluric currents with the thickness H of the conducting sequence at the reflection points (from the data of electrical probes). Depending on the character of the territorial distribution of the obtained coefficients, they are either averaged or interpolated. In the case of an extremely sparse network of resistance probes, use is made of the simpler formula H = FJE, which, as sho"\vn experimentally, gives less accurate results. It should be mentioned that the quantitative interpretation of data of telluric currents can also be based on the use of values of depths of reflection from the results of seismic exploration under conditions where the seismic and electrical reflecting horizons correspond to the same for- mation. In 1955, exploratory observations of telluric currents were carried out by an experimental party of the Siberian Aerogeophysical Expedition of VNII Geofizika in the Tomsk region in the closed polygon Tomsk — Bakchar — Podgornoe — Chezhemto— Mogochin — Shegarskoe of total length 750 km. The observation points were placed at a distance of 10 km from one another. The transport was provided by vehicles which were particularly suitable for rough ground, and when there was no road at all a helicopter was used for the first time, in electrical exploration work. During the survey, the position of the base station was changed after every 30 km. The values of E at tlie base points were equated by the correlation method and then reduced to NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 209 the original base station which made it possible to construct a combined map of the average intensity of the field of the telluric currents. The field of telluric currents in Western Siberia in the summer of 1955 was characterized by a more or less intense variation with a period of 10-40 sec and an amplitude of the order of 1-2 mV/km. The maximum variation as a rule was observed in the daylight hours (9-15 hr). A periodic weakening was observed in the variations of the field of the telluric currents up to complete extinction (during August, September and October there were 10 such days). Fig. 9. L-shape arrangement. The measurements were carried out with L-shape arrangements with measuring lines between 300-500 m long (Fig. 9). For synchronization of the observations a specially developed interference -free TB-6 Tele-switch was used in circuit with an RPMS radio station (^). EPO-4 oscillographs were used to record the variations in the telluric current fields. Fig. 10 shows the graph for the average field intensity of the telluric currents on the Tomsk -Bakchar profile, constructed on the assumption that the value E at point 24 is equal to one arbitrary unit. The values of E along the profile decrease regularly from Tomsk to Bakchar, changing more than four times. The character of the decrease in the average field intensity of the telluric current agrees well with known geological data and corresponds to a rapid increase in thickness of the sequence at the north-west limb of the Koly van'— Tomsk upfold, there being a clearly defined steplike lowering of the Paleozoic foundation in the immediate vicinity of Tomsk, and also between the villages Markelevo and Plotnikovo. There is an interesting local maximum of average field intensity in the region of the points 15, 16, showing the possible presence here of an elevation of the Paleozoic foundation. It is interesting to notice that in this region, Applied geophysics 14 210 A. M. Alekseev et al. seismological survey had shown an upfold in the Mesozoic deposits (Krasno- Bakcharskaia structure). In view of the lack of a sufficient number of electrical resistance probes over this territory in the process of the geological interpretation of the observations on telluric current it was necessary to deal only with information on the depths of the current of Paleozoic rocks in the region of Kolpashevo (from drilling data) and in the region of Shegarskoe (from the data of vertical electrical probing). On the basis of these data and using the above formula, a schematic map was drawn for the thickness of friable deposits, lying on the Observation points 17 36 37 Fig. 10. Graph of average intensities of the field of teUuric currents along the Tomsk- Bakchar profile (the vertical scale is logarithmic). pre-Jurassic foundation (Fig. 11). Despite the rough diagrammatic construc- tions occasioned by sparse network of observations, and the absence of data on the parameters of the cross-section, this map shows how pre-Jurassic rocks dip towards the central regions of the western Siberian Lowlands. To the south of Kolpashevo, there is a zone of upfolded pre-Jurassic rocks, which are of interest for further study. In 1956 in the Western part of the Lowlands the experimental work of the VNII Geofizika was continued in order to decide on the applicability of the telluric current method for solving regional and survey problems in the zones of development of volcano -sedimentary deposits of the second structural level, and also in the selling of the Berezovka gas deposits. Within the limits of the Tobol'sk and Vagai-Ishimsk tectonic zones, the party together with the department Zapsibneftegeofizika (led by Yu. S. Kopelev) carried out a large scale structural survey of the Zavodoukovsk area and also a route survey along the profile Zavodoukovsk-Vyatkino-Vikulovo. On the Bere- zovka area studies were made in the region of the group of Berezovka upfolds and along a 280 km regional route along the river Severnaya Sos'va. NKW MKTIIODS OK KI.KC'IHTCA I, KX I'l.OHA'I'ION IN SIUKKIA 211 III l*>50 in. Wrslrrii Silirii;! llic pciiods ol iiilciisivr varialioii allciiialcd willi |t('ri(»(ls (»r \\cak<'iiiii^ in the licM iiilnisily. I'Or <'\am|tlc, if ill llic sccoikI hall ol \ii^ii-l I In- varialioiis wil li ainjilil iidf II I » \i) A inV/kiu wnr coiiliimrd aliiKt^l iiniiil<i rii|ilrdly lor a |icii<)d ol M) I2lii', I'M;. II. \la|. ol llii.kiic^s (,r irial.ir. .Irj.o.silrt. I Ivjiial lliirkiirss lino lor iIk; I'rial) flf^posilH. I lull ill S(|)l(iiii)('r, lli(;aiii|)lil iid(' i)f llic varialioiiH was reduced lo 0.5 I inV/km, and I lie diirali(tii. of vaiialions lo f) lir. l"or (^acli iiionlli ol I Ik- lidd s(!asoii oi 1956 thcn^ were, 2 to 6 days whru !li(;n; were, no nicasiirahic varialioiiH. Tlu! av(;raf^(! ral<; of prodiinlion using lli<; l(;]|iiri(; (uriviil mclliod was I point; per i list III ni<;nt. c;liang(; in roiit<; surveys v\illi a 10 .'>0 km step and 2 points in an atcal survey witii a .'i 5 ktii slep. 212 A. M. Alekseev et al. 1200 M50' v/ iiOO'' y\ 1050- ■ -/ ■000- ■s 950-~ lOO^' . ^_^ ' V 900s Pyatkovo. 850 ^ 1 { eo(> \ 1 <■ 1 / \ 1 ' / , Fig. 12. Map of the average intensity of the field of telluric currents (from material of VNII Geofizika and Zapsibenftegeofizika). The value of E at the base point is taken as 100 arbitrary luiits. 1 — structure contours of the seismical reflecting horizon; 2 — iso- lines of average field intensity. Ill the -wooded and swampy Berezovka region, Avork was conducted along the rivers, with sections being landed on the banks and the hnes unreeled in the zone near the banks. Fig. 12 shows the results for the study of telluric currents at the Komissa- rovskaya structure in comparison with the data of seismical surveys. It can be seen that the known Komissarovskaya structure appears clearly on the map of the average field intensity of the telluric currents ; the telluric current method can therefore be recommended for surveys of similar upfolds. Good results were also obtained in the regional studies. Fig. 13 gives a geoelectrical cross-section constructed on the resistance observations of telluric currents, made for a determination of resistance values of the .average field intensity at the base points (step of resistance network 30 km). NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 213 Quantitative calculations were made using the seismic exploration results at the Komissarovskaia structure (65 km to the south-west of the start of the profile) as initial data. A zone of upfolding is observed in the high -resistance level to the west of the Vyatkino well and in the region of the Krotovsk structure. The liehaviour of the electrical marker horizon agrees well with the tectonics of the Mesozoic deposits deduced from the data of seismic explorations. The absolute values of stratum thicknesses of the friable deposits, calculated from the telluric current are close to the bore-hole data in the region of Vyatkino and Vikulovo. 70 50 30 10 400 800 1200 1600 2000 6 Vyatkino well KrotovKa structure Vikulovo well •'>'''''>'WWm; ,,,,,,,,,,„MMW;,.,,yMTMWW/A „ Fig. 13. Geoelectrical section along the Vyatkino- Vikulovo profile. 1 — surface of elec- trically reflecting horizon; 2 — bottom of the friable Mesozoic deposits; 3 — surface of the seismical reflecting horizon at the bottoms of the Mesozoic ; 4 — average field intensity of telluric currents. Also effective was the use of the telluric curreiU method in the Berezov region, where the results of a study by telluric currents make it possible to formulate a general idea of the relief of the Paleozoic foundations. A NEW ELECTRICAL EXPLORATION APPARATUS FOR WORK IN SIBERIA The standard electrical exploration equipment used by the MNP geo- physical parties is not very suitable for working in inaccessible localities. A modern electrical exploration station can be used only in regions with a well developed road system, even when the station is mounted on the GAZ-63 vehicles. The apparatus for the method of telluric currents is in the form of the EPO-4 oscillograph and its power supply is too large for forest parties. For this reason, a number of new designs have been developed in VNIIGeofi- zika to be employed in Siberian settings. In 1956, at the laboratory of electrical exploration and the design bureau 214 A. M. Alekseev et a I. of VNII Geofizika, A. M. Alexeev, N. A. Bulanov and others, designed two new electrical exploration stations, one of which--ERST-23-56 — is intended for working in inaccessible country and also in the winter (the so-called tractor variant); the second — the EPS-16-56— a cross -covnitry dismountable, is only for dipole probing in inaccessible country. The following are the basic features of the new stations which will indicate the possibilities of their use in other problems of surveying under various conditions. The ERST -23-56 station is mounted on a trailer and its two PN-100 genera- tors are powered by a special gasoline engine ZlS-120, the gear box shaft of which is connected by a universal joint to the shaft of the first generator (Fig. 14). The engine and generators are mounted on shock absorbers and since they are connected by a universal joint during operations the power unit does not cause vibration in the body or measuring apparatus. To control the engine, on the control desk there is a special panel which has: a gasoline level indicator, an ammeter to check the operation of the charging generator, an oil manometer included in the lubrication system of the engine, a thermometer to check the cooling system of the engine. Also on the control desk there are buttons for controlling the gas and the air intake, a starter button, and also a clutch pedal and gear shift. With a booster arrangement, the voltage of the two connected generators can be increased to 1000 V providing the resistance of the insulation of the current leads is not less than 5 MQ. The new station has benefited from the experience of the ERS-23-53 stations which are used by the electrical exploration parties. In the station there is a fixed loading resistance, which can be used as a ballast loading to use when operating with the usual arrangement, since in a number of regions in measurements with the doubling arrangement there is distortion in the rectangular nature of the current impulses. Apart from the EPO-5 oscillograph, an extra apparatus is provided to measure A U. Under winter conditions, considerable difficulties are expe- rienced in earthing; the station therefore has a d.c. amjDlifier with an input resistance of the order of several megohms; this makes it possible to carry out measurements of AU in regions with poor earthing conditions of MN, where the intermediate resistance can reach very high values (of the order of 50-100 x 10=^^). In the tractor variant of the station, the generator group and all the measur- ing apparatus are placed on a trailer with caterpillars, which is drawai by an S-80 tractor. On the trailer there is a metal body suitable for operation under winter conditions. Between the two walls of the body there is a heat NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 215 insulating lining, there are double "vvindo^v-frames and to protect the body from cold air there is a tambour. Inside the body there is a stove; for further heating, use can be made of the ballast resistance, the main function of which is to load the generators during the time when their insulation is being dried. The body is di\ided into two parts: in the first there is the operator's section, in the second, the motor section. In the operator's section there is a desk which carries the engine controls, the generators and the measuring apparatus. There is also dark-room for developing the oscillograms. This part is suitable for use as temporary living quarters for technical personnel working at the station. Combining the generator and measuring apparatus in one body makes it possible to use the ERST-23-56 station for probing with the AMNB arrangement with only one operator. The pressure exerted by the tracks of the trailer A\dth the station does not exceed 0.2 kg/cm^. The S-80 tractor intended for operation wdth the station is equipped with a heated body, in which there is a table for the measuring apparatus, if it becomes necessary to carry out dipole probes, and a winch to reel and unreel the supply line when working with the AMNB arrangement. The winch is driven by a shaft from the tractor and is provided with an automatic cable-laying device, designed for a PUM grade cable. The winch can hold 6-8 km of this cable. During reeling and unreeling of the lines, the tractor moves at speeds up to 9 km/hr. The dismountable electro -prospecting station is mounted on the GAZ-69 vehicle which can move cross-country. If the station cannot be moved independently, the generator group can be taken to the place of work on an MI -4 helicopter or on a small barge on the river. The overall layout of the generator group is given in Fig. 15. If necessary, the generator (PN-145) can be removed from the generator group together ^nth the plug board and certain other auxiliary parts. The generator is mounted on a special sledge ^vhich guides it onto the platform of the vehicle. To ease the assembly of the detachable parts there is a special small winch on the vehicle and also screw clamps with which one person without using tools can secure the generator to the vehicle. At the same time as the generator is installed its shaft is connected to the shaft from the gearbox. To reduce the length of the generator group, certain units and components of the motor car body are removed, and the electrical arrangement is simplified as much as possible, even to the exclusion from it of an excitation rheostat. •2](^ A. M. Al.KKSEEV rf (ll. T\\c lalcd |K)\\(M- of tin- j;(MU>nil()r group is 16.5 kW, but as is usual in electrical (^xploraliou slalious, llic power developed can Le booslod to 18 kW with a current strength of up to 40 A, which is sufficient in most cases for carrying out dipole probes. The voltage and the power developed by the generator group are controlled by changhig the number of revolutions in the engine, suicc I he impulse windings arc connected directly to the circuit of the generator's arinaturc. Since the ERS-16-56 station is mainly intended for dij)ole probes in Fig. 14. Position of EItST-23-56 apparatus and equipment in the body of the trailer. 1— generators; 2 — ZIS-120 engine; 3 — control panel; 4 — EPO-5 oscillograph; 5 — high xoltage panel; 6 — low voltage panel; 7 — receiving and transmitting radio [contimicd opposite NEW METHODS OK ELECTRICAL EXPLORATION IN SIREIUA 217 inaccessible regions, to measure zlt/ there arc two iiorlabic lvSli()-56 cleclron loop oscillographs with cinefihu recording. The ESliO-56 electrical exploration loop oscillograpli (Fig. 16) was developed in 1956 at the L'vov Institute of Machines and Automatics at the request of the Electrical Prospecting Laboratory of VNII Geofizika especially for dipole probing in inaccessible country. The oscillograph has an a.c. amplifier with converter, in the form of a VT vibro -converter (Fig. 17). TIk; vibroconverter converts llu' input current into an alU'iiialiug current with a frequency of about 180 c/s and, at the same lime, acts as a mcclumical rectifier for the output of the amplifier. In the output stage of the amplifier there is a Tr-2 tiauslormer, in the secondary winding of which there are connected in series a micro -ammeter, a resistance i?26' one of the resistances R^^-R^^^ (depending on the limit of measurement) and the left contact of the P vibro-converter. When the armature of the vibro-converter is m()v<'(l lo ihc left, the secondary winding of tlu; output transformer is loaded and enrrent llo\\> in it, moving the pointer of the micro -annneter and the mirror of the galvano- meter G. At the same linu' iIk^ voltage of the feedback inpnt ot lh<; amplifier, with a polarity opposite to that fed to the input signal, is taken Irom one of the resistances R^ — R^^, i.e. in the instrument there is a deep (close to 100%) negative feedback with direct current, and the arrangement as a whole (i?00 ^^■''"I'Wy'TsWTa^^rPTr;^^!) S^^"<t"<,' 'ryj- Station; 8 — calling device; 9 — block of contactors; 10 — control panel lor engine opera- tion; 11 — D. C. amplifier; 12 — measurement panel; 13 — output panel; 14 — loading resistance. 218 A. M. Alekseev et al. is self-compensating. The instrument readings do not, therefore, depend on the state of the feeding sources, and there is practically no drifting of the zero point. At the instant when the armature of the vibro-converter is moved to the right contact, the input of the amplifier is shorted and simultaneously the -circuit of the secondary winding of the Tr-2 transformer is disconnected. In other words, the instrument produces a half-period rectification. Fig. 15. The body of the GAZ-69 vehicle with the PN-145 generator. 1— PN-145 gener- ator: 2 — block of contactors; 3 — gearbox; 4 — propeller shaft; 5 — sUdes. The main features of the instrument are as follows: 1. The high sensitivity of the instrument, corresponding to 1.0 mV over the whole width of the film, can be further increased. 2. The record is made on a standard low sensitivity cinefilm. 3. When processing the results of the record, a photomultiplier is used which magnifies the record of up to 10 times linear. 4. The film is moved by a spring motor with a speed of 0.25 mm/sec. 5. The instrument has its own power supply from five 2S-KU and two GB-225 dry batteries which ensure normal operation of the instrument for 40-50 hours. NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 219 6. The temperature range over which the iustruineut operates is deterinined by the caj^acity of the supply sources. 7. The relative error in the camera measurements on the lihn does not exceed 2%. 8. The input resistance of the instrument at the highest sensitivity is 2-3 MD. 9. During recording visual readings can hv taken of a pt»inter. 10. The instrinnent measures 340 x 220 :< 240 mm. 11. The instrument, together with supply sources, does not weigh more than 14 kg. Fig. 16. The EShO-56 oscillograph. Thus, if the generator group is brought to the field location, theu measure- ments can be made in every case, since all the parts of the measuring devices are portable. The device can be used under all earthing conditions of the MN electrodes (sand, frozen earth), and its readings in practice do not depend on the state of the supply sources. Still greater advantages are given by a new electronic measuring apparatus ^dlich operates with the telluric current method. A field station -which is now being Iniilt for the telluric current method is placed on the GAZ-69 motor- car and includes: 1. A 2-channel electron amplifier (weighing about 10 kg together with 220 A. M. Alekseev et al. bC K> O fj_, < o o O > cc; rt cri ^ ^ So o 1^7 pc; ^ t L -^ L rt s a; eel 2 t^ fd Q ^ •z. 1^ Pi o fa cd o fa ..go s ^ g a fa m p::; :: ^ fd fa oo fa U o fc! S O r "^ I ■* „ ^ p^ ... '-'•.> fa ^ <M O O a CO i-T* '— I Pi s § 8 ^ 1 '>^ ^' S oj 1 o 1 ^ 1— 1 l—J pi"=^:^^ II .^^ f^ en w ^ -1 ^2 a Pi ^ epd^ p:; u T-H 'at w to d oil T) 1— 1 fa 1 L 3 O o Pi Pi t_) '« 23 a s NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA 221 the supply sources) with a liigh-vokage mput, its circuit being similar to that of the EShO-56 oscillograph. 2. An oscillograph with a spring motor (weighing not more than 8 kg together with the filament batteries of the illuminator lamp. 3. An RPMS radio station with a power pack, into which is also assembled the TV-6 tele-switch used in transmitting and receiving the time signals by radio (tw^o units weighing 11 kg each). In the tele-switch there is introduced an interference stable system, which limits the interference level. Consequently, even with a high level of interference, the TV-6 does not give false time signals. 4. Non -polarizing portable electrodes; 5. Two cable reels (weighing 2 kg each). The use of the amplifier makes it possible to record also telluric current variations, which due to the insufficient amplitude would be considered unworkable when using an EPO-4 oscillograph. The whole of the apparatus is fixed in the body of the station but if the vehicle is unable to continue, since the arrangement is sufficiently portable the apparatus can be removed in 2-3 min and carried by three of the field - party members to the observation point. CONCLUSIONS The work of the VNII Geofizika has shown the effectiveness of new electrical survey methods under conditions prevalent in the Western Siberian Lowlands. These methods should find applications in solving regional and survey problems of geophysical investigations for petroleum and gas. At the present stage the exploratory observations on telluric currents are best used to solve problems of a regional character (separating out large depressions and culminations, and also detecting upfolds of the second order) in conjunction with gravimetric and aeromagnetic work in tectonic surveys of the Western Siberian territory. The most effective results can be obtained on a scale of 1:1,000,000 with a net work density of 1 point per 100 km^. It is desirable also to carry out separate length profiles along rivers. With a favourable geoelectrical cross- section the telluric current method can also be used to survey local upfolds. The two-way electrical probing should preferably be directed towards the solution of survey and exploration problems, and also for setting up a basic network for the telluric current method in regional studies. The results of experimental work carried out in the Western Siberian Lowlands open up possibilities for a more extensive use of electrical prospect- 222 A. M. Alekseev et al. ing ill Siberia. To introduce electrical prospecting into the geophysical investigations in Siberia, it is essential to speed up the serial production of new equipment. Serious attention should be paid to the further development and improve- ment of electrical prospecting methods, the possibilities of which have not been exhausted by work done up to the present. In particular, the energy of research workers should be concentrated on the creation of methods using natural and artificial variable electromagnet fields. These methods can be used to increase the efficiency of electrical prospecting work and also to extend the sphere of application (studies in the regions of permanent frost) of electrical prospecting. REFERENCES 1. L. M. Al'pin, The Theory of Dipole Probes, Gostoplekliizdat (1950.) 2. A. M. Alekseev, and M. N. Berdichevskii, Electrical prospecting by the telluric current method. Applied Geophysics, No. 8, Gostoptekhizdat (1950). 3. M. N. Berdichevskii, and A. D. Petrovskii, The method of two-way equatorial probes. Applied Geophysics, No. 14, Gostoptekhizdat (1955). 4. M. N. Berdichevskii, Instructions for Processing Electrical Probe Oscillograms. VNII Geofizika (1954). 5. A. M. Zagarmistr, Usmg the Increased Resolving Power of Axial Probe Curves. Applied Geophysics, No. 16, Gostoptekhizdat (1957). 6. L. MiGO, and G. Kuznets, Electrical Prospecting for Petroleum. IVth International Petroleum Congress. Geophysical prospecting methods. Gostoptekhizdat (1956). 7. V. A. Troitskaya, The Earth's currents. Priroda, No. 5 (1955). 8. Yu. V. Khomenyk, The TV-6 tele-switch. Exploratory and Industrial Geophysics,. No. 17, Gostoptekhizdat (1957). Chapter 9 THE METHOD OF CURVED ELECTRICAL PROBES M. N. Berdichevskii The method of two-way quadrilateral probes* has been widely accepted by electrical exploration workers carrying out electrometric studies in regions where field sections can travel along long straight profiles (^). However, when working in country where conditions are diflicult, the use of quadri- lateral probes is often impossible since the swampy or wooded sections of the surveyed area hinder movement of field parties along a straight line. In this case, it is desirable to change from measurements with a quadrilateral setting to those with an azimuthal arrangement, which would make it pos- sible to use the so-called curved probe, i.e. probes with the observation points on curved routes. The curved probes can also be carried out with a radial arrangement and an arbitrary two -component arrangement by means of which measurements are made of the mutually perpendicular components of the field. The properties of an azimuthal arrangement and the method for carrying out curved azimuthal probes (from the results of studies made in the VNII- Geofizika in 1954-1956 by the author together ^nth. T, N. Zavadskaya and V. P. BoRDOvSKii) are given below. AZIMUTHAL ARRANGEMENT The azinuithal arrangement is shown in Fig. 1. Here AB is the supply Hne, MN the measuring line, R the distance between centres and Q of the feed and measuring lines, d is the angle formed by the feed line AB and the line OQ. A basic condition of the azimuthal arrangement is the perpendicvdarity of the measuring line MN to the section OQ, connecting the centers and Q of the feed and measuring lines. It is apparent that the quadrilateral ar- * The quadrilateral [Jiterally equatorial in the Russian edition] probe is one of the modifications of the dipole probes proposed by L. M. Al'pin. He has developed the whole theory of dipole probes^). 223 224 M. N, Berdichevskii rangement is a particular case of an azimuthal arrangement. L. N. Al'pin has shown that the KS value of an azimuthal dipole arrangement* in a horizon- tally homogeneous medium does not depend on the angle and for the same distances R between the centres of the feed and measuring dipoles coincides with the KS value of the quadrilateral dipole arrangement and consequently with the KS value of the limiting AMN arrangement the Fig. 2. length of which is equal to the distance R. This property of an azimuthal dipole arrangement also creates possibilities for carrying out curved probes, since in the transfer from one spacing of azimuthal probing to another it is not necessary to keep the angle unchanged. THE COEFFICIENT OF THE AZIMUTHAL ARRANGEMENT The coefficient of the azimuthal arrangement will be calculated from an approximate formula, the derivation of which is based on the assumption that the length of the measuring line is sufficiently small, and to a sufficient approximation the value of the difference in potential between the poles of the measuring line can be taken as equal to ATTAB _ tpAB n/TAT (1) where: Ej^j^ is the component of the field of the feed line AB along the direction MN. It is obvious that (Fig. 2) E^Pn = Eij^ + E^N = E^ cos ( J^, MTV) + E^ cos (^ MN). The values E and E^ for a homogeneous medium with a specific resist - '^ We will call a dipole arrangement that in which the feed and measuring lines have an infinitely small value. \ THE METHOD OF CURVED ELECTRICAL PROBES 225 ance q are determined by the expression; r9 E^ Am 2oz\R^+— ^+RAB cos e EB = - ^^ JR2 271 1 R^+^ RAB co^ e Bearing in mind that /^4 Ti^ATv ^5 sin 9 cos {E^, MN) = ■'V cos (EB, MN) = R^+^+RABcos0 4 ZB sin V' 21/ R^+-^-RABcos0 we obtain according to (1) , .B lQABMNsm0{[^, AB'- ^—^ ^Y^'' AUmn=~ 7 \\R^+—^+RABcose\ + + \r^ + ^^-RABcos0 \ From the expressions for the difference in potentials we transfer to an expression for the coefficient of the arrangement, representing the latter in the form of the product of two factors K = K*A. (2) The factor K* is calculated from the formula* ABMN ^ ^ and the factor A from the formula 4 = __ __ :^3= ^::^ . (4) sin(9/ AB^ AB A^''' /, AB^ AB ' ~^'" To facilitate the calculations Ave will use a special nomogram for the factor A, shown in Fig. 3. * The ibrmula is given for the case where the current is determined in amperes and the potential difference in millivolts. Applied geophysics 13 226 M. N. Berdichevskii AB On the nomogram values ww ^I'e plotted along the ordinate and the values for the angle along the abscissa. The required value for the factor A should be determined by interpolating bet-\\^een the lines of identical values of the factor A. Example — We mil calculate the coefficient for an azimuthal arrangement, characterized by the following dimensions: AB = 1000 m, MN = 200 m, R = 3000 m, - 75°. From formula (3) we find the value of the factor /C* 30003 . 10-3 1000 • 200 = 135. 60 62 64 66 68 70 72 74 76 78 80 82 86 88 90 Fig. 3. Nomogram of factor A. Frcim the nomogram (see Fig. 3) we determine the value of the factor A AB for the coordinates -gb = 0.167, = 75°. Interpolating between the iso- lines 6.65 and 6.70, we obtain A = 6.68. Thus, the required value for the coefficient K is A^ = 135 • 6.68 = 902. THE METHOD OF CURVED ELECTRICAL PROBES 227 OPERATIONAL DISTANCE AND DIMENSIONS OF THE AZIMUTHAL ARRANGEMENT The operational distance R of the azimuthal arrangement is taken as equal to the length of the three-electrode limiting AMN arrangement which on the surface of a horizontally homogeneous medium gives a KS value, coinciding wth that for an azimuthal arrangement. The KS value for an azimuthal arrangement is determined from the formula Qn = A J , (5) "where: K is tlie coefficient of the azimuthal arrangement. For a limiting 3-electrode AMN arrangement, the KS value will be Q = 2nR^^, (6) where: E is the intensity of the field of the point source; R is the length of the 3-electrode arrangement (the distance from the point source to the center of the measuring dipole). Let R be the operating distance of the azimuthal arrangement. Hence Qa = Q' (7) Starting from equation (7) we will attempt to show the connection be- tween the operating distance of the azimuthal arrangement and its dimen- sions. According to the principle of reciprocity ^ Kl = uiF-uiF = ^ u'Jb-^u^b- (8) Here the lower indices indicate the points and arrangements of the obser- vation, the upper indites the points and arrangements of supply. It is obvious that: B A A where: E^^, E^^ are the components of the field of the point sources M and TV along the direction AB. To de'.e-mine the values of the field E of the point source we return to 15* 228 M. N. Berdichevskii the theoretical curve of q for a hmiting 3-electrode apparatus and we mark out on this curve a certain small section, in the limits of which the depend- ence of the ordinate log q on the abscissa log R can, with sufficient accuracy, be expressed by a linear equation \ogQ = tlo^R + logT,. (10) where: t and T are certain parameters, which depend on the dimensions of the arrangement and the geoelectrical cross-section, t being an angular coefficient of the tangent to the KS curve (in a bi -logarithmic form). Using expression (10) and considering formula (6), we write IT - E = -^Rf-^ Ztc We use this expression to determine the components E"^ and E^^ of the field of the point sources M and N along the direction AB (on the sur- face of a horizontally homogeneous medium). Then integrating (9), we obtain: ^""---Yn T^l Thus, IT 2n{t-\) A IfAB^ = ^ [M5^-i -M^'-i - M'-i + iVZ-i]. (12) Using expressions (11) and (12), we obtain from condition (7) (13) We express the operating distance B by the product of two factors R=pR, (14) where: p is a correction coefficient; R is the distance between the centres of the feed and measuring lines of the azimuthal arrangement. According to (13) and (14), the coefficient p will be equal to ^ [/ 27tR^{t-l) ^ THE METHOD OF CURVED ELECTRICAL PROBES 229 o O CO d 0.9667 0.9736 0.9754 0.9771 0.9632 0.9605 0.9638 0.9555 0.9533 0.9549 0.9485 0.9467 NO ^H NO On d CM d 0.9774 0.9846 0.9866 0.9982 0.9834 0.9793 0.9867 0.9776 0.9752 0.9811 0.9731 0.9703 On d LO d 0.9826 0.9896 0.9920 1.0700 0.9921 0.9879 0.9969 0.9878 0.9851 0.9921 0.9842 0.9826 On ON On d d 0.9864 0.9939 0.9963 1.01302 0.9990 0.9950 1.0053 0.9957 0.9938 1.0017 0.9939 0.9925 CM On ON d o O 0.3 1.0024 1.0097 1.0112 1.0214 1.0073 1.0043 1.0102 1.0020 0.9999 1.0040 0.9979 0.9962 LO q r—l CM d 0.9951 1.0024 1.0043 1.0188 1.0046 1.0059 1.0097 1.0075 0.9986 1.0053 0.9986 0.9965 On rA LO d 0.9926 0.9985 1.00L58 1.0194 1.0048 1.00052 1.0106 1.0016 0.9993 1.0070 0.9996 0.9979 CO B q r—t d 0.9913 0.9986 1.00071 1.0196 1.0049 1.0071 1.0115 1.0024 0.9998 1.0085 1.0095 0.9992 CNl q § CO d 1.0255 1.0318 1.0357 1.0514 1.0393 1.0339 1.0430 1.0356 1.0325 1.0394 1.0336 1.0312 s q 0.1 0.15 0.2 1.0064 1.0135 1.0155 1.0331 1.0193 1.0154 1.0256 1.0169 1.0145 1.0228 1.0158 1.0140 q 0.9993 1.0064 1.0077 1.0274 1.0130 1.0091 1.0198 1.0110 1.0083 1.0170 1.0098 1.0078 <—t rH q rA 0.9956 1.0030 1.0049 1.0216 1.0069 1.0027 1.0149 1.0056 1.0030 1.0126 1.0052 1.0032 NO NO O q r-H o O On CO d 1.0354 1.0417 1.0433 1.0601 1.0480 1.0447 1.0537 1.0464 1.0444 1.0514 1.0459 1.0442 q CM d 1.0105 1.0169 1.0194 1.0380 1.0243 1.0205 1.0310 1.0277 1.0199 1.0285 1.0219 1.0201 CM LO d 1.0018 1.0084 1.0113 1.0302 1.0161 1.0120 1.0230 1.0146 1.0115 1.0205 1.0135 1.0017 NO q r-H d 0.9950 1.0027 1.0048 1.0246 1.0099 1.0054 1.0172 1.0080 1.0053 1.0146 1.0076 1.0052 I © ^ / cc; / 1^ lO CM rH O O d d d LO CNl rH O O d d d LO CM rH O O d d d LO CNl rH O O d d d 1 c. •»-a r-H + rH CNl 1 CO 230 M. N. Berdichevskii We 'ivill study the dependence of the correction coefficient p on the di- mensions of the azimuthal arrangement and the parameter t (Table 1). As can be seen from the Table, the values of /j oscillate around unity and ~iB for fixed ^^^ and O vary within the limits of 3%. In connection with this, 2R it can be concluded that when the conditions are satisfied: ^<0.3,^<0.1,120°<(9<60°. the values of the coefficient p do not depend to any great extent on the shape of the KS curve (on the parameter t). J -30 0-26 I^ CO 01 (J) CT) CT> 6 6 6 o _ 9 9 O O o // \ \^ \^ S, - / \ \ 0-22 018 / / \ \, — ■ / / 1 \ I / / V 0-14 / 1 \ / o-in / — 6 6 4 6 8 7 2 7 6 8 8 4 8 8 Fig. 4. Nomogram of coefficient p. Let us determine the average arithmetic means of p for each vertical column of the table. It can readily be seen that the individual values of p differ from the arithmetic means of p by not more than 1.6%. Let us use the obtained arithmetic means of p to construct a nomogram of the correction coefficient of p shown in Fig. 4. This nomogram gives the required value of the coefficient of p with errors not exceeding 2 % . To find tlie operating distance of the azimuthal arrangement it is neces- sary to determine from its given dimensions — using the above -described nomogram— the value of the correction coefficient p and to calculate the operating distance of the azimuthal arrangement from formula (14). If = 90°, the azimuthal arrangement becomes equatorial, and the operating distance R of such an arrangement can be obtained either by the above-described method or from formulae and nomograms used for quad- THE METHOD OF CURVED ELECTRICAL PROBES 231 rilateral probes. The difference in the vahies of the operating distances obtained by this and other methods does not exceed 2%. Example — Wq will calculate the operating distance of an azimuthal arrange- ment with dimensions ^B = 1000 m, MN = 200 m, R = 3000 m, = 75°. From the nomogram (see Fig. 4) we determine that the value of the AB coefficient p for the coordinates = 0.157 and = 75°. Interpolating 2R between the isolines 1.00 and 1.01, we obtain p = 1.005. Therefore, R = 1.005 • 1000 m = 1.05 m. THE EFFECT OF INACCURACY IN PLACING THE FEED AND MEASURING LINES ON THE RESULTS OF AN AZIMUTHAL PROBE To simpUfy the calculations we will consider an azimuthal arrangement with a feed line AB of finite dimensions and with a limiting small measuring line MN. Since in practice the measuring lines are sufficiently short ( MN^ — R Imn<^r\ it should therefore be expected that with an inaccurate sighting of the lines of the azimuthal arrangement the results obtained here will make it possible to evaluate the order of errors introduced into the KS value. (a) The effect of inaccuracy in position of the measuring line on the KS value — Let us suppose that the measuring line MA^ forms with the tangen- tial direction (here tangential is the direction perpendicular to the radial direction) a certain angle A, measured clockwise and being an angular error in sighting of the measvu-ing line (Fig. 5). The effect of the sighting inaccuracy of the measuring line on the KS value will be represented by the value Qa ^-l]lOO%, (16) where q^ is the KS value for an azimuthal arrangement— obtained with an inaccurate sighting of the measuring lines; q^ is the KS value for the azimuthal arrangement ^obtained with an accurate sighting of the measur- ing line (along the tangential direction). 232 M. N. Berdichevskii To determine the value — ^ we find the component of the field of the feed Qa line AB along the tangential direction 0, the radial direction R and the di- rection of the measuring line MN {Eq, E^ and Ej^j^). Fig. 5. Since in the calculation of q^' and q^ the coefficient of the azimuthal arrangement is taken equal to the same value, then it is obvious that: I.e. fj, Qa Qc E Emn "Ee MN Ee 1 100% (17) From the formula for conversion of coordinates we obtain: Consequently Emn a Ej{ . — = — = cos A — ^^— sm n . t,Q Eq Let us derive expressions for the radial and tangential components Ej^ and Eq of the field of the feed line AB at the observation point: Ep^ = E^-E^^ E^^ cos oc-E^ cos /S = ^ AB ^ ^ AB ^ R+ -^cosO R — — cos6) Ri Ro THE METHOD OF CURVED ELECTRICAL PROBES 233 Eq = E'^ + E^ = E^ sin (x + E^ sin (i AB sin0 AB sin O E^ EB. Here: E^, E^ are the field intensitites of the field line AB at the observation point ; R is the distance from the points of observation to the centre of the feed line AB; i?2 and i?2 are the distances from the point of observation to the poles of the feed line AB; is the angle formed by the feed line AB and the section connecting the middle of the feed line with the observation point. By analogy with the calculations for the previous section we find that ^A IT 1-2 Hence E MN IS) Ee + 1 + ^- cos K COS A — sin A x ?-3 l+2^cos6) AB t-3 2 . ^{[IABY ^ AB „ ABV -, AB „ — +l-^cos0 /-3 ABV ^ AB „ 2^ +1^-^-0 l-2^cos0 AB 2R sin0 ABV , AB 2R +1 + -^-^^ f-3 2 2^ +l-^cos(9 <-3 • 2 1 ( (18) The results of the calculations using formulae (17) and (18), are given in Table 2. As can be seen from this table, the limiting relative error /n for determin- ing the value q^, caused by angular errors A of the order of 1-1.5°, varies 234 M. N. Berdichevskii for 70° < < 110° from tenths of a percent to 2-3 per cent. For angles O of the order of 60° (120°) or 50° (130°) the error m determining q^ becomes 5-8% as shown by calculations. It follows that when carrying out azimuthal probes, it is generally desir- able to use an arrangement with angles from 70 to 110°, and the unreel- ing of the measuring lines should have an accuracy of 1—1.5°. Measurements within the limits of the rising branch of the KS curves, inclined to the axis of the distances at an angle close to 45°, can be conducted with angular errors in the direction of the measuring line, reaching 3-4°, without much effect on the KS values. Fig. 6. (b) The effect of inaccuracy of sighting the feed line on the KS values — We will consider an azimuthal arrangement of which the feed line A'B' is laid with a certain angular error y with respect to the given direction AB (Fig. 6). The error in the KS value, connected with the angular error y will be characterized by the value r = ^^— ^100% = -^-1)100%, Qa \ Qa (19) where : q^' is the KS value obtained with an inaccurate sighting of the feed line; Q^ is the KS value obtained with an accurate sighting of the feed line. It is apparent that THE METHOD OF CURVED ELECTRICAL PROBES 235 °o o T O O CO CS OS !>; C^ ,-H r-H -!f CO CO (>. -r^; On LO LO ■<# ON CM >0 LO r-H '^ t~-' r-^ NO c r-H CO C^J O ON 0> CO CS| rH 03 LO CQ r-H ,-H r-H c4 C>q CS CO CO CO P NO CM LO T}! -^ l-H 'e- o lo NO VO LO d d d CO CO CO 00 r-; NO r— H r-H r— H O ^ CNl LO CO ^ CM CM CM c CO CO C-l LO f- \q '^ 5.47 5.16 4.71 r-H LO ON CO CO r-H !>: d NO o + 1.18 1.11 1.00 CO r- t— NO ■Tj; ^ CO CO' CO 4.91 4.60 4.16 o r-H + rH r- c^ NO LO LO d d d lO LO o CO !>; NO r~ r-H On Tf CO p CM CM CM o CO 1 1.04 0.99 0.91 NO lO f- 03 t^ lO CN] CM •>! O CM 03 CO NO CO CM CM CM CS) NO CO CO NO NO LO dd d CO O 03 CO 03 NO O 00 CM lo CO CNq CN] CM CM H Z w o s 1 CM O t— CO CO CM odd CM On CO ON CO 00 d d d 1.24 1.18 1.10 o CO _1_ r- (M CO I-- r^ NO d d d o t- o lO <* CO oq cvi c-i CI ^ r-H LO CO r-H CO CO Cl3 « o o ^ r-H LO LO LO -r* d d d o cc NO 1 CO NO o r-; NO LO ' CO CM r-H rH r-H r-H 1 CM CM CM o r— 1 + Cvl CM CM d d d ON NO O r-H lO r- CO CO CO , CM r-H o d d d ' r-H r-I i-^ o o O in 0.381 0.381 0.381 CO S S CO CO CO d d d 0.381 0.381 0.381 o CO 0.137 0.137 0.137 t-~ t^ t-- 1 r- r^ t-^ CO CO CO 1 CO CO CO r-H r-H r-H 1 r— H 1— H r-H d d d 1 d d d Cjl NO NO NO o o o d d d \0 *o nO O no nO O O O O o o d d d\ d d d e:; '^ lO LO LO lO LO LO o o o ope o ^ o o o o 0.015 0.015 0.015 LO 0.381 0.381 0.381 0.381 0.381 0.381 0.38] 0.381 0.381 o CO + f~ r- r- CO CO CO ^. ^ '"1 o o o r^ 1^ f- , r^ r~- i~>- CO CO CO 1 CO CO CO d d d\ d d d o NO NO NO o o p d d d sis dd d NO NO NO o o o d d d + 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 o / 1^ r-H C'J CO ^ C^) CO d d d d d d r-H CM CO d d d - + r-H CM o o o CO + ■* On t^ NO CNi r^ l-H rn' d 1.40 0.35 —1.14 CM 00 r- CNJ CNJ CO r-H d CM CM 4- 1.12 0.88 0.53 NO ^ r- On CN] P- d d d 0.84 —0.18 —1.61 7 r-H CO On lO On C^l t- ^ NO -^ CM rj d r-^ '* CM NO rH d CM 7 •^ ra O CM On NO 111 LO O CM p CO CO r-^ d d r- t^ LO CM r-H t-^ r-^ d r-i o O CO o CO CM NO CM t— LO CO d d d 0.60 0.10 —0.61 •^ CM CM d d r-^ h Z < D CM + rH On CM lO CO CM d d d CO I-- CO '^ o >* d d d CNl LO t— CO rH 00 d d d a o CO ON f- ^ 111 CM NO O CC r-H 03 d d d VO NO LO r- CM NO d d r-^ 1/3 a 1-1 CM CM CO CO NO •* CM d d d CM On CM LO O LO d d d O t-- NO LO l-H O d d rH Co' K J o o c^ o CO + 0.18 0.1 0.05 r-H CM r-H r-H O r-H d d d ON '^ CO p p CM d d d o CM + '-' 3 S3 rH O O d d d LO r-H LO O O O d d d On CM O O O rH d d d o CO 03 LO r-H r-H O 111 r-H CM r-H r— 1 O r-H 11° On -^ CO O O CM d d d 1 3SS 111 LO r-H LO poo odd d d d o / ^ / (N l-H CM CO d d d rH CM CO d d d rH CM CO d dd O- r-H 1 1 236 M. N. Berdichevskii where: Eq, is the tangential component of the field of the feed line A' B' E^ is the tangential component of the field of the feed line AB. Thus, E& Ee 1 100%. (20) Eq' In agreement with the previously derived formulae we obtain sin {0+ y) ABV , AB ^^ ^ 2R +l + ^-s(0+y) f-3 2 + sm ABV ^ AB __ Ul+-^cos0 t-z 2 + ABV -, AB 2R +i-ir^^^^ is 2 ^ /-3 1 abV ab ' \ + A2«)+l-iJ^°^(^+''>J 1 sin ABV ^ AB „ t-3 f-3 • ^ ^^ABY , AB (21) + + 1 — =^cos6> Mj +'-ll The results of the calculations for v, obtained from formulae (20) and (21), are given in Table 3. As can be seen, with an angle 0, within the range from 70 to 110°, and a length of feed line up to 0.6 R the angular errors 7, reaching 2°, lead to errors in KS not exceeding 2%. THE PRACTICAL PROCEDURE FOR AZIMUTHAL PROBING The work of experimental parties of VNIIGeofizika made it possible to develop with sufficient completeness a method for azimuthal probes and showed the effectiveness of the described method for measurements under conditions involving difficult transport. The azimuthal probes w-ere conducted with the ERS-23 electrical survey station, which included two field laboratories, mounted on the GAZ-63 vehicle. The probing profiles were curved roads, not particularly suitable for vehicles. Initially, the topographical surveyors marked out the curb of the road into 100 m spacings. The plan of the markings was entered on to a plane-table grid on a scale 1 : 25,000. The markings were fastened with standard pegs carrying the number of the profile and the peg number. Near the pegs there were high m.ounds which were easily recognizable in the THE METHOD OF CURVED ELECTRICAL PROBES 237 -area. The plane-table grid also had the outlmes of the road, diagnostic orien- tation features, trigonometrical points situated within the limits of visibil- ity, etc. At the edges, the plane table grids overlapped by not more than 2 points, with a spacing of over 1000 m. The profiles were connected in accordance with the "Directions for ge- odesical work in geophysical surveys in the petroleum industry". The centres of the azimuthal probes were placed directly on the road. The generator group was placed near the probe centre. To obtain the left branches of the KS curves, an AMNB arrangement was used with feed electrodes placed along the road, and also in quadrilateral arrangemenf. Measurements with the AMNB arrangement were set up to half-spacing of the feed electrodes equal to 200-600 m, and were usually carried out with an electrical prospecting potentiometer. The quadrilateral measurements (under favourable conditions) were made in the range of working distances from 200 to 1000 m. Measurements with the azimuthal arrangement were commenced with a distance between the centres of the feed and measuring lines of 500-1000 m. When transferring from the AMNB arrangement to the quadrilateral or azimuthal arrangements, the measurements were repeated at 1-2 points. When carrying out azimuthal probes, the field laboratories were placed along the road and measurements were made of the difference in potentials at various distances from the probe centres (Fig. 7). These distances were selected the same as with the usual probes (500, 700, 1000, 1300, 1800, 2500 m. etc.). Observations with the azimuthal arrangement were made according to a special procedure, which involved all the necessary information on the direction of the feed and measuring lines, and was arranged as follows: Az. No. Azimuth AB = Marking at the centre No. II/II No. ^ mark T^zimuth J, 1 j^jy Remarks 1 2 3 4 5 6 Here the 2nd column gives the number of the pegs for which the measur- ing lines should be unreeled. The 3rd and 4th columns give the distances R between the centre of the feed line and the centre of the measuring hne and angle between the feed line and the section connecting the centres of the feed and measuring lines. The distances and angles were measured 238 M. N. Berdichevskii graphically on the plane-table by means of a scale rule and a geodesical protractor. The 5th column has the magnetic azimuth of the measuring line calculated from the formula Aj^js^ Aqq 90^ where -^jv/jy ^^ *^® magnetic azimuth of the measuring line; Aqq is the magnetic azimuth of the section OQ connecting the centres of the feed and measuring lines. 1st feed line Fig. 7. In setting up the program of measurements, the azimuth of the feed line was selected so that the angle was between the values 70-110°. Depend- ing on the configuration of the road, the azimuthal probe was cai'ried out with one or several azimuths of the feed line. For example, measurements at the points within the limits of the section 1-2 of the road shown in Fig. 8, were carried out with the direction of the feed line being A^ B^ . To carry out naeasurements at the section 2-3, a second feed line A^B^ was used. On changing from one direction of the feed line to another, the measure- ments at one of the points placed near the point 2 were doubled so that the sections of the KS curve obtained with different direction s of feed line touched one another at one of the spacings. The dimensions of the measuring and feed lines satisfied the conditions MN < — R, AB < 0.6 R. The maximum length of the feed lines was 500-1500 m depending on the conditions of the measurements. In the probe process the length of the feed line changed several times, an attempt being made to keep this number of changes to a minimum. When changing from one feed line to another, the measurements were doubled, Avhich made it jDossible to combine the sections of the KS curve obtained with different feed lines. A compass was used when unreeling the cable of the feed line, the max- THE METHOD OF CURVED ELECTRICAL PROBES 239 imum permissible angular error being about 2°, The wires of the measur- ing hne were unreeled with an accuracy of up to 1-1.5°. However, when obtaining the rising right-hand branch of the KS curves, inclined to the Fig. 8. axis of distances at an angle close to 45°, the accuracy of unreeling of the measuring line cable was sometimes reduced to 3-4°, which was caused by the complex conditions of unreeling in wooded and swampy country. - '•^ ^^ 100 8 / „,._ ^ 5i — - 6 ""T"^ 1 1 -H _j — -f-^-i \ «J fi--562 t-v ): - h \ ' 10 ±jA 1 6 v I " ff^ ^» '"\ [ 1 V. 10 8 ^ 6 ^^;— uTm Qi 10 15 2 4 6 8 10 15 2 4 6 8 100 15 2 R 6 I000'5 2 4 6 10,000 Fig. 9. Two-way curves of azimuthal probes. 1 — AMNB arrangement; 2 — azimulhal arrangement; 3 — middle curve; 4 — western curve; 5 — eastern curve. 240 M. N. Berdichevskii The KS values were calculated from formvila (5) and in the construction of the KS curves were referred to the working distances R, determined according to (14). To find the coefficient of the azimuthal arrangement and the correction factor p, nomograms were used. In other respects, the method of two-way curved azimuthal probes did not differ from the method of two-way quad- rilateral probes. The characteristic curves of two-way curved azimuthal probes are given in Fig. 9. As shown by control measurements, the KS values in azimuthal probing are reproduced with an accuracy up to 5-6%. Under the conditions of a stable geoelectrical cross -section and the more or less gentle slope of the rocks, the curves of the azimuthal probe agree well with curves for the quadrilateral probe carried out in approximately the same direction in which the field laboratory travelled during the azimuthal probing. In the method of interpreting two-way curved azimuthal probes, there is nothing which is in principle different from the method of interpreting two-way quadrilateral probes. When constructing the profiles of the resulting values, the points of the record lay on a straight line by the side of the curved profile of the probe. REFERENCES 1. L. M. Al'pin, The Theory of Dipole Probes. Gostoptekhizdat (1950). 2. M. N. Berdichevskii, and A. D. Petrovskii, The method of quadrilateral probes. Applied Geophysics, No. 14, Gostoptekhizdat (1955). Chapter 10 THE USE OF THE LOOP METHOD (SPIR) IN EXPLORING BURIED STRUCTURES I. I. Krolenko The loop method refers to a group of inductive methods of electrical sur- veying. The first work with this method was carried out in 1930-1932 on the Grozneft regions (by V. N. Dakhnov and then by S. G. Morozov and I. G. Didura). This work was experimental and showed the possibili- ties of the method and the optimum conditions for using it (^^^ From 1937 to 1941 and from 1945 to 1950, work with the loop method was conducted on the Kerch peninsula (I. I. Krolenko), where the physico- geological conditions favoured its extensive use. As a result of this work and the improvements made in 1947 in the tech- niques of field observations, the loop method became an effective, widely re- producible and relatively cheap method. The method was used to study the tectonics of the upper levels of the geological section. Apart from the work on the Kerch peninsula, the loop method was success- fully used on the Caspian shore to the south of Makhach-Kala (1942), on Taman' (1950-1951), in the Turkmenia (1941-1943 and 1952-1953), and was also tried under the conditions prevalent in Western Siberia (1954-1955) and Ust-Urta (1955). In the present article we will consider the basic principles of the method and will not deal with the physical nature and the mathematic basis of the loop method, which are given by Dakhnov (^'^°^ who worked out the theory of this method, and also in the papers of A. I. Zaborovskii, E. N. Kalenov, et al. The loop method is based on the good electrical conductivity of aniso- tropic (banded) deposits along the layers in comparison with the direction normal to the layer. This phenomenon in the case of dipping anisotropic deposits causes an asymmetric magnetic field of the earthed feed loop AB, which for variable fields can easily be set up by means of a closed receiving loop, placed sym- metrically with respect to AB. The e.m.f. values arising in the receiving loop are functions of the 241 Applied geophysics 16 242 1. 1. Krolenko coefficient of anisotropy of the rocks A = "1 / -^ and the angle of inclination V- (X of the studied deposits. In the case of sufficiently homogeneous deposits (A = const.) the observed values, proportional to the e.m.f. with respect to the modulus and direction, characterize the change in a along a certain profile, i.e. the tectonics of the studied deposits. The coefficient of mutual induction between the feed and receiving loops can be changed either by deformation of the wire AB or by means of an induction coil. The value of deformation of the wire, i.e. the area of the constructed triangle of compensation a>, in relation to the area of the whole receiving loop Q, gives a component of the vector of the loop measured for a given . ft) . position AB. The ratio -q is usually small; it is therefore multiplied by 1000 and designated by / = -^ 1000. Thus with observations employing two mutually perpendicular lines AB, the vector loop is measured. The position of this vector which shows the direction of propagation of the current at a given point, i.e. the dip direction of the rocks being studied. By analysing the vector distribution, it is easy to indicate the position of the anticlinal and synclinal hinges. Thus in terms of the isolines, drawn perpendicular to vectors, the upfolds, their sub -divisions, regions of depression, etc. can be mapped. This method of interpreting observations, made "with the aid of a loop, gives a general idea of the structure of the studied area. The method should, therefore, be recommended in reconnaissance or semi-quaUtative surveys, and also in the prehminary processing of data. The observations made in order to obtain details of the structure of a studied feature, should be processed by the integration method. For this purpose, sections intended for detailing should have connecting profiles of the vectors intersecting the basic profiles. This system of profiles forms a closed polygon, in which the integration of vectors can be carried out by the generally known method. The advantages of the mathematical construction of isolines was shown at a number of areas of the Kerch peninsula, Stepnyi Krym* and Turkmenia, where considerable details of their structure were shown. This makes it possible to recommend this form of interpretation for wider use when processing observations by the loop method. An example of comparison of both methods of interpretation is given in Fig. 1. To indicate the importance * The Crimean Steppes [Editor's note]. THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 243 of the loop method in the systems of geological and geophysical surveys of buried structures, one should deal with the geological -geophysical condi- tions necessary for its use, and consider briefly tlie results of work carried out with this method. The most favourable objects for surveying by the loop method are structures made up of sufficiently homogeneous shales or banded sand -clay or clay- marl ' » • • V \- — -^ -7 >• A X ^' A' — V ^^ ^ \ \ — 'n^i r E35 — * %Cx.'4^^' Fig. 1. The tectonic structure of the ]\Iarfov section according to data of integration of the loop vectors and the method of constructing isonormals. 1 — loop vectors; 2 — isonormals to the loop vectors; 3 — isolines from the data of integration of loop vectors; 4 — the axis of the fold according to the data of the loop method. 244 I. I. Krolenko deposits, covered with shallow alluvia. The value of the anisotropy coefficient of such deposits varied between 1.2 and 2.5. The electrical homogeneity over the area is established by parametric profiles of resistances with spacings equal to the length of the AB line adopted for working with the loop method in the given region. The loop method can also be used to determine the direction of dip of the reflecting highly -resistant horizon, covered with electrically homogeneous rocks. The accuracy of the results of surveying by the loop method is also ensured by the accurately defined forms of the buried fold (angle of dip of the studied rocks from 8 to 80°) and the gentle relief of the surface. The presence of these favourable conditions on the areas of the Kerch peninsula determined the possibility of extensive work with the loop method and the effectiveness of the results. The first investigation was carried out in 1937 at the Priozernoe (Chonge- lekskoe) petroleum deposits and then at the Pogranichnaia* (Chorelekskaia) and Malo-Babchikskaia anticlines for determining the tectonic structure of these areas. This work showed the position of the axes of the structures and marked the regions of greatest amplitude. At the Priozernyi deposit the loop method combined with electrical profiling (mapping the strip of the Middle Sarmatian limestone on the flanks of the structure) revealed a number of cross-cutting disturbances. The nature of the relationship between the Chorelekskoe and Opukskoe upfolds and the Malaia Babchikskaia structure with the Katerlezskii dome<^^'^*). In 1950, this work extended the knowledge of the Chongeleksko-Chorelekskaia antichnal zone. This zone, in the limits of the Kerch peninsula, according to the data of the loop method begins at the south-west of the Koyashskoe lake in two small anticlines — Chokur- Koyashskaia and Opukskaia, separated by a small saddle, and ends at the north-east of the Zaozernaia (Tobechikskaia) fold, a considerable part of which is under the water of the Kerch Straits (Fig. 2). Significant corrections were introduced on the basis of the work carried out by the loop method and core drilling in 1950 in the zone of the Malaia Babchikskaya structure, which was coordinated with the deposition of petroleum. As well as the data on this fold and the Katerlezskii dome adjoining it from the south-west, to the south of the Malaia Babchikskaya anticline was mapped the Yuzhno-Babchikskaia fold with a very narrow closure, bent in the form of a crescent (^^' "^K In between the three upfolds there * In order to avoid errors the Russian adjectival form is used for the structures imless ithe correct place name is well known. [Editor's footnote]. THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 245 is a funnel-shaped depression filled with volcanic deposits interstratified with marine sediments ^^K All three folds are represented in the form of periclines, surrounding a large depression, in which many deposits of the Kerch peninsula are found. By means of the geological survey and the loop method, hinges were traced for a number of anticlines of the Zaparpach- skian part of the peninsula <^^)' ^^^K Interesting work was carried out by the loop method in connection with electrical profiling on the Borzovskaia area. This work showed the course of the structure and established the lines of faulting, which conditioned subsequent exploratory drilling. The work carried out on the neighboring area mapped the Glazovskaia structure and established its connection with the Western periclinal closure of the Mayakskaia structure (^^). On the Glazovskaia structure its asymmetry w'as confirmed (wide and gentle southern liml) and a steep northern) and the zone crumpling was shown on the northern limb near the hinge. The latter is apparently the consequence of mud volcanism and diapiric processes ^^K The most effective was the work with the loop method in the flat south- western part of the Kerch peninsula, made up of homogeneous muddy Maikop formation. The lithologically homogeneous sections, devoid of distinct marker horizons and the poor exposure of autochthonous rocks considerably hindered the geological survey of this region and the tectonics of the plain were known only in a very general form ^^\ and only a few anticlines were traced in this area. With the loop method in a short period (4 field seasons) on the territory of the south-western plain, 24 structures were mapped of which 6 [Yuzhno- Andreevskaia, (Karachskaia) Zhuralevskaia (Chaltemirskaia), Kol-Alchins- kaia, Ul'yanovskaia (Mangutskaia) Yuzhno-Marfovskaia and Mar'inskaia] were demonstrated by this type of investigation. Furthermore there were considerable corrections for the majority of the other structures (^^^ According to geophysical data, the Moshkarevskaia (Kerleutskaia) structure is a small fold in the southern part of the western depression of the large Shirokovskaia (Uzun-Ayakskaia) structure, whereas, according to geological ideas, its axis continues further to the east ^^^K The Shirokovskaia struc- ture owing to the axial variation of plunge is subdivided into 3 culmi- nations. Furthermore, it is genetically connected with the Kuibyshevskaia fold, which complicates its northern limb. As well as the convergence of the Moshkarevsk and Shirokovsk anticlines, which localizes the petroleum de- posit, other similar convergences of the Zhuravlevskaia (Chaltemirskaia) and Seleznevskaia (Mamatskaia), on one hand and of Ul'yanovskaia (Mangu- 246 1. 1. Krolenko 9 c>j3 -1 1^ ^\ '^ ■3.3^ b--.s ?.S 1 l-o^ t S I S .'^ " ►? = fl « 3 "^^ .2 I ^ o ? j5 a 1 ^ S > ^ tad ■p s^ I ^ .2 u ^ ^ J '< 2 3 CO 2; s's •■■ S' .2 •- ' sj-^ffl Sfi" T s«|ss-2 a. J; ^ •= a .2 S fi .2 « -o .2 § -s ■- = O n I r! O t(l « Ns ?r . o 0^ rt •- lA -2 > .S -2 S ; ^ >< 3a ' 1^11=^22, s I ^--i -3 3^ S S .2 1 ^ I 3 V *^ I N w)^ ■= ,3 I p I -a .;; ^ S ;P3 .iii.2 to Oi •' c ' .-S -i a'.2 I ■^ -S S .. a" 5 « o t^ " C S '^^ .2 jq -J. A^ 5,^ en S C,-^ u ■« t- ■a cfi > S c -S '~"a 3 6 2^ J, ^>i ti I H > j: ^ S I S3 1 „ 3 8."'" > CO ./-^ 2 o bcS n CO ■* c/: HOC- THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 247 tskaia) and Yarkovskaia (Bash-Kirgizkaia) (^^) on the other have also been discovered by the loop method. At the Kharchenkov (Arma-Elin) area instead of two parallel structures, the loop method showed the presence of one structure of considerable dimensions with a small bulge on the northern limb. The long Vladislavovskaia fold is divided owing to the axial variation of its plunge into two anticlines — Vladislavovskaia and Frontovskaia (Koi-Asanskaia), which is of interest from the point of view of its possible petroleum content (^'*). This work in the south-west plain showed the contin- uation of a number of tectonic zones (Yarkovskaia, Severnaia Belobrodskaia, Dyurmenskaia, Dal'nii, etc.) Work carried out by the loop method on the Uzunlar area was very revealing. The Uzunlar dome is placed in the south-east part of the south-west plain. Before work with the loop method (1949) the Uzunlar dome was an undefined large bulge of Maikop clay. The loop method has been used to study the tectonics of the Maikop core within the limits of which a large Mar'inskaia anticline, which lies to the south-west of the Uzunlar lake, was depicted and parallel to it a small upfold related to the distant (Atan-Alchhiskaia anticline zone (^^) was found. The Dal'naia zone contains the structure of the same name and the geologi- cally known Karangat anticline (^^ which is situated on the Karangat cape. The existence of the Mar'inskaia anticline was shown first by A. D. Arkhan- gel'skaia in the section on the western shore of the Uzunlar lake. The northern side of the Karangat structure and the previously accepted schemes of the tectonic structure of the Kerch peninsula referred to the most southerly anticlinal zone, the remaining parts of which were assumed to be covered by the Black Sea. Geophysical work has shown the course of the Karangat and Dal'naia structures and has produced a more firmly based scheme of tectonics of the southern part of the south-western plain of the Kerch peninsula. The schematic structure of the Uzunlar dome is also supported by the results of a detailed geological survey carried out on this area in 1950 ^^^K In 1949-1950, using the loop method, studies were made of an area adjoining the south-west plain from the north and north-east (Zaparpachskaia part). The structure of this part of the peninsula is represented by core of Maikop clays, which lie in depressions and are surrounded by the deposits of Mediterranean age. They differ in the high complexity of the tectonic structure (Fig. 3). However, under these conditions, the loop method clearly shows the tectonic features of the Maikop cores of the structures (position of the axis, the presence of axial variations of plunge and possible faults), 248 1. 1. Krolenko Avhich appear most distinctly if the obsevations are processed by the integration method <^^\ As a result of a correlation of the above -described investigations, carried out in 1950, a map of the tectonic structure of the south-west plain of the Kerch peninsula (see Fig. 2) was constructed. This map shows a regular pattern of structures, arranged along tectonic lines, and also the region of a sharp change in direction of these lines from close to east -west in the northern and central parts of the plain, to the north- east in the southern and eastern parts (^^' ^^K The change in direction of the structural units of the south-eastern part of the Kerch peninsula in comparison with the central, is of interest since it can provide favourable conditions for accumulating sand facies in the geological section (^^). About 90 % of the structures recorded in the Zaparpachskaia part, by the loop method and up to 60% in the south-western plain of the Kerch pensinsula, have been covered in the post-war years by geological surveying. In the case of many of the structures, core and deep drilling were carried out. On the basis of the materials obtained, composite structural maps were compiled of both the south-west plain and of the Zaparpachskaia part of the Kerch peninsula. A comparison of these maps with similar maps based on the loop method indicates the correctness of the loop method for both the general pat- tern of the tectonics in the Kerch peninsula and of the components of structure of the tectonic zones and individual structures. Geophysical data on many areas not only support the geological survey but in a number of cases considerably improve it with additions and corrections. Oil prospectors on the Kerch peninsula are faced with the problem of finding the relationship between the tectonics of the upper levels, studied by the loop method, and the structure of the lower parts of the Tertiary and Cretaceous systems. To solve this problem use was made of magne- tometry, gravimetry and seismic exploration. The gravitation field of the Kerch peninsula is characterized by a continuous increase in gravity towards the south. Apparently, owing to the presence of a large gravity gradient, local anomalies in the gravitational field appear only in the form of deflections in the isoanomalies. This can be illustrated by the coincidence of the zone of disturbance with a region of gravitational minima, shown on the map by distinct deflections in the isoanomalies. The magnetometric surveys carried out on the Kerch peninsula in 1948 showed an increase in the intensity of the geomagnetic field to the north, which is apparently connected with an increase in this direction of the thickness of deposited formations (^*). THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 249 S . cS J _: o o &, <u o E TO -S S S -5 ^ ;> cc W g <^ <D l-H -% hT^ WOODS HOLE, MASS./ 250 1. 1. Krolenko Experimental seismic studies by the reflected methods were carried out on a small scale in 1950-1952 in the western part of the Kerch peninsula and in territory adjoining from the west to Feodosiya. This work showed that where the anticline crests produced by diapiric and crypto diapiric structures are separated by narrow and deep synclinal depressions, the use of seismic exploration involves considerable difficulties in correlating these reflections. In many profiles the crestal sections are characterized by irregular readings, by different orientation of the co-phasal axes and by the so-called "blind zones" with almost complete absence of reflecting horizons. In the peripheral parts of the structures on the limbs and in the synclines, where the layers slope relatively gently, the record of reflections is perfectly satisfactory. As a result of processing the method of seismic exploration of the western part of the Kerch peninsula in 1954, considerable improvements were obtain- ed in the quality of the recorded seismograms. The Avork showed a certain lack of correspondence of the structural plan of the Tertiary and Meso- zoic deposits and an increase in the angle of dip of rocks with depth. Due to the absence of continuous, correlatable horizons, the construc- tions were made by using two arbitrary seismic horizons: M (stratigra- phically related to the top of the upper Kerleutskian horizon) and C^ (corresponding to the Upper Cretaceous sediments (^^)). The accuracy of the constructions was conditioned mainly by the quality in the initial data. It was, therefore, at its least in the hinges of the anticlines, found by the strongly folded rocks with large angles of dip (up to 70-80°). The seismic profiles showed all the culminations including those which were kno\Mi previously, and those shown by the loop method: Ul'yanovskaia, Shirokovskaia and Yuzhno Andreevskaia^^^). The closures of the structures formed by the arbitrary seismic horizon M in the overwhelming majority of cases coincide "vvith the closures mapped by the loop method or are slightly displaced. Thus, under the conditions existing on the Kerch peninsula on the limbs of the folds there is a correspondence between the structure contours nf the seismic exploration and the isonormals of the loop method. The absence of reflecting areas in the closures of the intensively dislocated folds is easily made up by the data of electrical explorations, i.e. the methods are comphmen- tary to one another, giving a continuous section of observation without "blind zones". This is very important in interpreting geophysical material and should be remembered when planning and carrying out field work. The loop method, as shown above, successfully depicts local cvilminations of axial plunge and secondary complications on the limbs. It explains the THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 251 nature of merging structures and the relationship between the tectonic zones. This material should be used for a rational distribution of seismic profiles. If this is not fully appreciated, errors can be introduced when planning work and eventually errors can arise in the deductions about tlie structure of the area, as was the case in the Vladislavov-Kharchenkovskaia tectonic zone and in the zone of merging of the Moshkarevskaia and Shirokovskaia structures. Seismical studies on these areas, which have no interesting petroleum-bea- ring aspect (although complex in their structure), were carried out by a sparse network of profiles ignoring the details <^^). As a result, the Frontovaia and Kharchenkovskaia structures represented by data of the loop method, of the geological survey and by the subsequent exploratory drilling, as t^vo inde|)endent connected folds, were united by seismic exploration into one wide structure (Fig. 4). A well drilled in the crest of the fold to the south of Lake Parpach gave an oil gusher of several tens of tons. This fact demanded final solution of the problem of the structural position of the well. For this purpose, in the summer of 1956 detailed seismic profiles were made, con- firming that at that section the two structures merged. The gusliing well was situated at a deep western depression of the Kharchenkovskaia brachy- anticline, in connection with which further exploratory drilling was tran- sferred to the Kharchenkovka region. In the light of the preceding, it is necessary to reconsider the schematic structural map of the Moshkarevka area, compiled for the arbitrary seismic horizon M. Surveying by the loop method shows the complex merging of the small, intensively dislocated, Moshkarevskaia fold into the relatively large Shirokovskaia fold. Despite the fact that on the western depression of the Shirokovskaia structure and on the Moshkarevskaia upfold more than 100 wells were drilled, up to the present time opinions about the structure of this region to A\hicli the location of petroleum is related vary. From the seismic data on the arbitrary level M, the anticlinal upfold is drawn on an area 8x4 km^ with an axis striking north-east course ^^^K On comparing contours of structures for various data (Fig. 5) it becomes apparent that the whole eastern periclinal part of the fold according to the data of seismical survey is placed within the limits of the western crest of the Shirokovskaia structiu'e. Confirmation of the fact that the structure contours imited t-\vo different upfolds is provided by the data of the seismic profiles XVIII, XX and XXI, intersecting the Moshkarevskaia anticline, and the XXll profile, cutting the Shirokovskaia structure. The first of them 252 1. 1. Krolenko characterizes the Moshkarevskaia fold with a steep northern hmb and a rather steep south hmb, since on the profile of the XXII closure (Shirokovskaia) of the structure is characterized by small angles of dip on either limb. Apparently under complex seismogeological and structural conditions of this section in the absence of detailed profiles (especially longitudinal) it is difficult to represent the structure of this area correctly. According to the data of the loop method, the main bulk of the drilling- was apparently related not to the Moshkarevskaia fold, but to a wide periclinal nose of the Shirokovskaia structure. Lake Parpoch ->^ -S>t^ / v/ /' ,'^ •""■-^"w / J/ y- -' ' EH' [Z]2 [23^ £^4 ^5 Cl]6 H^ • Fig. 4. The structure of the Kharchenkov area from the data of seismic prospecting and the loop method. 1 — isonormals to the loop vectors; 2 — structure contours; 3 — axis of the fold determined from the data of electrical prospecting ; 4 — • seismic profiles ; 5— well; 6 — Frontovaia structure; 7 — Kharchenkovskaia structure. Similar pecuHar linkages of two structures were often observed mtliin the limits of the south-west plain by surveys with the loop method. The presence of a petroleum deposit on the Moshkarev area may be connected with that type of linkage. Therefore a detailed complex geological geophysical survey of the Moshkarev area can give valuable data for solving the problem of finding oil in a number of other areas of the Kerch peninsula both from the point of view of the method and of the practice. Between 1945 and 1947 the loop method was used in the Crimean Steppes. As a result of these observations, it was found that the physico- geological conditions of this region are vuifavourable for the loop method. The considerable electrical differences of the surface deposits, their weak THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 253 anisotropy, the considerable thicknesses of the alluvia and the extremely gentle angle of dip of the investigated rock led to an irregular distribution of the vectors. However, on the background of even such a vector field it was nevertheless possible to distinguish a number of sections, on which, owing to a more regular disposition of the vectors, a probable presence of anticlinal upfolds <i^) was proposed. To determine the actual nature of the electrical anomaly on three of them, Annovskaia, Severnaia and Yuzhnaia Tslaun'skaia between 1948 and 1949, observations were carried out Mdth the vertical electrical probe method, confirming the data of the loop method. yvm ' Hi H2 [Z]3 Fig. 5. Comparison of the structures of the Moshkarevka area according to the data of seismic prospecting and the loop method. 1 — structure contours; 2 — isonormals to the loop vectors; 3 — axis of folds determined from the electrical prospecting data; 4 — number of seismic profiles. The Annovskaia and Gor'kovskaia upfolds were also proved by the geological survey. On the southern edge of the Priazovskaia depression the variometric survey and the loop method showed the Goncharovskaia (Karagozskaia) structure, confirmed later by geological surveying by deep drilling and by seismic explorations. 254 1. 1. Krolenko After completing the surveys of the Kerch peninsula by the loop method and compiling the tectonic pattern of its structure, a natural continuation was the survey by the loop method of the Taman' pensinsula (^°^ ^^K The geological interpretation of the data of the Taman' peninsula surveyed by the loop method (1950-1951) showed that within the limits of the studied area, there was no north-west (Caucasian) trend of the structures. The extreme northern tectonic line of the Taman' peninsula (anticline folds of the Kamennii Cape and the Peklo Cape) in contrast to the geological ideas (^) was charac- terized by a nearly east-west trend, i.e. corresponding to a trend of the northern tectonic lines of the Kerch peninsula*. This w^ork (see Fig. 2) considerably amplifies the details of a number of structures and their linkages (Fontalov folds, folds of the Mt. Tsimbala, Karbetovka, Bliznets'a Hill, etc.). The structure of the intervening synclines, formed in the weakly anisotropic Pliocene deposits and alluvia of considerable thickness, was not revealed by the loop method. During 1952 the Taman peninsula, except for its w^estern part, was seis- mically surveyed v>?ith the aid of the reflection method ^^^K In the central part of the peninsula, this work covers the areas which have been surveyed by the loop method. A similarity was observed between the isonormals of the loop method with the structure contours of the arbitrary horizons M (the Upper Paleogene deposits) and B (the Upper Neogene deposits) of the seismic survey on the limbs of the fold. The core, composed of intensively faulted Maikop formation, could be studied in detail only with the loop meth- od. These regions on the seismic cross -sections are expressed in the form of zones of almost complete absence of reflecting areas. Thus, on the Taman' peninsula, as on the Kerch peninsula as shown on Fig. 6, both methods successfully complement one another in the preparation of structures for Kreliust drilling. During the second world war, when the prospecting in the Kerch peninsula was temporarily interrupted, the loop method work was transferred to the more eastern regions of the southern USSR. In 1942, a survey was made of a 250 km^ area between the western shore of the Caspian Sea and the Narat-Tyubinskii range of mountains to the west and south of Makhach- Kala up to the Monas River. On the north-west of this area there was the known Makhachkalinsldan brachy -anticline, the crestal part of which was proposed for detailed loop studies. To the south of it was a plain covered * This conclusion is confirmed by the most recent data of structural core drilling on Taman (^^). t Krelius drilling implies the drilling carried out after the preliminary geophysical survey. THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 255 by Quaternary deposits of 8-12 m thickness and descending gently to the sea. For exploratory drilhng it was necessary to obtain reconnaissance informa- tion on the structure of this area. 73.50 62.50 j'je.^^>^iiZ^^^rg,,..^B- '>i.25 -^ i LXXI 2 W\ 3 Fig. 6. A comparison of electrical prospecting and seismic profiles on the up fold of Mt. Yuzhnaya Neftianaia (Taman peninsula). 1 — loop vectors; 2 — number of loop profiles; 3 — number of seismic profiles. Surveying by the loop method gave an accurate structure of the north- western pericline of the Makhachkalinskian fold and determined the position of the axis in the crestal part <^®). The nature of the connection of the south- eastern periclinal nose of the fold with the raonoclinally dipping layers which form the coastal plain was established. According to the data of the 256 I. I. Krolenko reconnaissance surveys of the area to the sovith of Makhach-Kala the strike of the investigated rocks was shown to be uncomphcated by the presence of any new upfolds (Fig. 7). Well drilling, using the results of the loop method, showed the petroleum deposits of the Makhach-Kakhachkalinskian brachy-anticline and the deposit was exploited. Between 1941 and 1943 the loop method was used to map the structures on the upper levels of the section in the western Turkmenia. In the investigated area, the geological structure of which has not been established by the previously conducted gravimetric survey, geological mapping and Krelius drilling, the Bolshaia Tuzluchaiskaia brachy-anticline, complicated by secondary upfolds, was shown by the loop method first in trials (^^) and then in production <^^) work. The results of the observations showed the reason for the lack of success in the preceeding Krelius drilling, determined the further course of investigation and confirmed the possibility of the successful use of the loop method to study structures in the western Turkmenia. Work by the loop method in the western Turkmenia was recommenced in 1952-1955. The survey led to a detailed study of the structure of the Kobek fold, the position of the northern limb being detected as well as the periclinal closures of the structure. On the basis of the studies, further course of development was recommended. By this work, and also by the results of the loop survey of the Chelekenskaia structure, the tracing of faults with the aid of electrical profiling C^' ^) was shown to be possible. The same was found at a number of structures of the Kerch peninsula. In 1954-1955 experimental work was carried ;ut in Western Siberia. The observations were made within the limits of the Chelyabinskii graben, filled with Trias -Jurassic deposits. The Sineglazov area (■'■''') was the most favourable for the application of the loop method. Here, the investigations established the basic elements of the tectonics of the studied sediments. The insufficient volume of experimental work did not permit reliable correc- tion of the results of geological surveying. However, on the resultant map a possible variant of the structure of the Sineglazov area, as deduced from the data of the loop method (Fig. 8), was presented. It is certainly possible to survey by the loop method areas which in their physico -geological characteristics are similar to the Sineglazov region. However, this could not be said of the Miass and Sugovak sections, also situated within the limits of the Chelyabinsk graben. In these areas a clear interpretation of the map of loop vectors was difficult and required the use of geological data(^'^). THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 257 o Z O d 2 bD 3 .s ^ ia 2 ^ J3 ^ 2 Applied geophysics 17 258 1. 1. Krolenko In 1955 small scale trials with the loop method were also carried out on the Ust'-Urt plateau within the limits of the positive gravitation anomaly of the force of gravity, depicted in 1954. Parametric observations showed that the geoelectrical characteristics of the upper part of the section, owing to its electrical heterogeneity, are not favourable for the use of the loop method. However, the existence of large-scale anisotropisra of a sedimentary formation is a factor which makes it possible to recommend working with the loop method provided that the angle of dip of the sediments is sufficient. / I / n/v I ,^w I ' TIT ^ >' ^ ' if / _ / / / r. : ^i-i \i I/; ; .yV/ I I 1/ /\ \. v/ / / V--I-- ^ w Zoozernyi / ' I / [Z]' [Z]2 E35 E3- EE]5 HJe Fig. 8. The structure of the Sineglazovskii area according to the data of the loop method (Chelyabinsk region). Compiled by I. I. Krolenko. 1 — loop vectors {d = 100 m) ; 2 — ^loop vectors (J = 200 m) ; 3 — axis of fold according to the data of electrical prospecting; 4 — axis of fold according to geological data; 5 — hnes normal to the vectors; 6 — number of loop profiles. THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 259 This is confirmed by the results of a reconnaissance profile, carried out by the loop method. Within the limits of the gravitation anomaly, the profile indicated the folding of the strata, which is of practical interest in correlating the surface and the deep structure. The intersecting of the gravitation anomaly, by a number of reconnaissance profiles at distances of 1-2 km from one another would probably facihtate surveys of the second order folding on the background of the large structure of the Ust-Urt. An example of a similar detailed study is the section covered by the loop method in the southern part of the gravitation anomaly (Fig. 9). In 1953 the NIIGR electrical prospecting laboratory carried out experi- mental work to test the possibility of increasing the depth of investiga- tion by the loop method and to increase its efficiency. To solve the first problem observations were made of the final stage of stabihzation of an electromagnetic field in the earth, which characterizes the \/ I y ;/ /\ /;» i [23 1 EI]2 i,n-3 lOO I 2 5 4 500 Fig. 9. The arrangement of loop profiles on the Il'todzhe section (Ust'-Urt). 1 — loop vectors; 2 — lines normal to the loop vectors; 3 — number of profiles. 17* 260 1. 1. Krolenko relatively deeply lying deposits. To solve the second problem use was made of the receiving frame circuit. With the usual loop method, the e.m.f. is measured with a field potentio- meter having a high -inertia galvanometer. In practice, this permits the measurement of only a certain average e.m.f. The value of the measured e.m.f. apparently depends to a considerable extent on its initial value, determined by stabilizing the field in the upper layers of the section. Corresponding to this, the arrangement for the commutation of the AB and MN circuits mth the old method of working on the PU pulsator (in fu- ture it will be referred to as the No. 1 pulsator) provided an observation mainly in the initial stage of stabilization only (Fig, 10). Fig. 10. 1 — Observed stabilizing phase Fig. 11. 1 — recorded phase; 2 — stabi- of field; 2 — stabilizing phase excluded lizing phase excluded by pulsator. by pulsator. For experimental work a No. 2 pulsator, suggested by Yu. A. Dikgof, was made and tested; this pulsator did not transmit the initial phase into the receiving channel, which made it possible to take the e.m.f. cha- racteristic of the structure of the deeper layers (Fig. 11). For this purpose the connection and disconnection of the MTV line somewhat lagged behind those of the AB fine. The quantity characterizing the order of such a shift of collector rings of the No. 2 pulsator Avill later be referred to by the letter a mth positive or negative signs depending on whether the connection of the MTV line lags behind or precedes the connection of the AB line. One of the cases of displacement of the collector rings of the No. 2 pulsator is showir in Fig. 12. The receiving frame circuit is a 2 in square frame with 2500 turns. The frame is tapped off at certain points (500, 600 and 1400 turns). The effective area of this frame is 10,000 m^. Thus, the frame is equivalent to a square loop of 100 m aside. The frame was mounted on a 0.5 m high support; the AB cable was folded on top of the frame and divided it into two symmetrical halves. The position did not change very much when the frame was set up on a two- THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 261 axle trailer, which made it possible to speed up the whole cycle of obser- vations. The increase in the length of the AB line and decrease in the effective value of the e.m.f. received, except for the initial stage of the stabilizing process, made it essential to increase the sensitivity of the measuring instru- ment and to preamplify the e.m.f. received. N >»»> <(<*« M Fig. 12. The mutual disposition of the current and receiving collectors of the No. 2 pulsator with lagging of the start of the e.m.f. record with respect to switching-on of the current (a > 0). 1 — insulating space of the collector MN (receiving); 2 — insulating space of the AB collector (feed); 3 — direction of rotation of the pulsator. With increase in the AB line from 200 to 4000 m, shifts were observed in the phase of the electromagnetic field in the earth with respect to the E and H fields of the feed current in the AB cable. This fact necessitated the use of different methods for cancelling the mutual inductance between both circuits. Compensation of the e.m.f. induced in the receiving circuit was obtained either by moving electrodes perpendicularly to the AB line or by inclina- tion of the frame circuit. The results of field observations (Fig. 13) were re- corded on cine-film by means of the MPO-2 oscillograph. Visual observations were made with the same arrangement, but in this case, instead of the oscillograph, the EP-1 potentiometer was connected. 262 I. I. Krolenko An analysis of the extensive experimental data led to a number of interest- ing conclusions on the dependence of the induced e.m.f. on the number of turns of the receiving circuit, on the AB dimensions, frequency of the pulsating current, the mutual disposition of the collector rings of the pulsator, etc. To study the depth of reach of the method of e.m.f. corresponding to the final stabilizing stage of the field in the earth, tables were compiled giving the ranges of observations for various values of delay time of the start of e.m.f. reception relative to the time of connecting the current for the 1st and 2nd pulsators and also tables of the time for stabilizing the elec- tromagnetic field according to the dimensions of the --45 line ^sere employed. On the basis of these tables a graph was drawn showing the disposition of the intervals of the record on the curves for stabilizing the electrical and magnetic fields for various values of a and AB (Fig. 14), On the graph along the ordinate are plotted the relative deviations of the stabiHzing field {E, H) from the field which is stabilized {Eg^, Hg^), i.e. Ei E — £"00 Hi H—Hoo and along the abscissa axis the ratio of the time of observation t to the so- called constant of time abV ^1" where C is the speed of light; Q is the specific resistance of the investigated rocks. A comparison of the time for stabilizing the field for various values of AB with the time for recording the e. m. f. when working with the No 2. pul- sator led to the conclusion that with the existing design of pulsator, only Fig. 13. A comparison of the e.m.f. curves with various vakies of df^ (single turn contours); ^5=200 m; /=2a; iV =600 rev/min; o=+3 mm; J„ is the distance of earthing from the AB line. THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 263 for AB < 400 m is the e.m.f. received, corresponding to the final stage of stabihzation of the electromagnetic field in the earth. With further increase in AB, the intervals of the observations referred to the initial stage of stabilizing, i.e. the measured e.m.f.s characterized only the upper deposits of the section. Calculation showed the possibility for receiving e.m.f.s corresponding to the final stage of field stabilization for any values of AB and other fixed observational conditions by means of the corresponding change in the pulsator collector. Observations with the frame receiving frame circuit are worthy of atten- tion and are of practical interest. The e.m.f. induced in the receiving frame for each of the two mutually perpendicular positions of the AB line, fed by a low frequency alternating current, are compensated by inclining the frame or can be measured by an appropriate measuring apparatus. The two -component vectors of the loop obtained in this way make it possible to construct a vector at each observation point oriented in the direction of preferential propagation of the current, i.e. along the dip of the rocks. The field of the loop method vectors should characterize the tecto- nics of the studied deposits to a depth corresponding to the depth of penetration of the magnetic field of a given frequency for a known value oiAB. The observations made with a receiving frame circuit even under complex physico -geological conditions, always led to results agreeing with the geologi- cal data. A number of problems arising during the trials on the modernization of the loop method could not be solved due to the low technical level on which they were carried out in 1953. However, this work established the possibility of such a modernization and indicated the future course of development. From this brief review of work carried out by the loop method it can be seen that under favourable physico -geological conditions, the results of the method are not only supported by geological surveying but in a number of cases they add to it and correct it(^^» ^^' ^^K For example, by comparing the character of the field of loop vectors for a number of areas in the Kerch peninsula with the seismic results and elec- tric logging of wells, drilled in these areas, it can be concluded that the structure of the upper 40-50 m of the section (observations by the loop method were carried out mainly with d = 100 m, AB = 200 m) reflects the structure down to a depth of about 500 m. Consequently, in this region the results of surveying by the loop method can orientate the cartographic 264 1. 1. Krolenko THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 265 drilling and, with the established correspondence of deep tectonics to the structure of the upper levels of the section, provide a basis for the laying of deep exploratory wells. By showing the tectonics of the studied area, the results of the loop investigations condition the further geological work, which ensures rational use of materials and time in surveys and explora- tions of petroleum deposits. However, the underestimation of the possibilities of this method, and sometimes the simple lack of inclination to sort out the results of previous studies meant that areas covered by detailed surveys by the loop method were overlapped by detailed geological surveys and cartographic drilling which increased the expense of the exploration, especially bearing in uiind the difficulties of the geological survey under the conditions of the muddy Maikop formations. For example, at the Uzunlar dome of the Kerch peninsula a satisfactory solution of the problems facing the geological survey party was achieved by carrying out a large volume of work. On the surveyed area of 150 km^, 606 prospecting pits were dug to an average depth of 3.4 m, 384 wells -were drilled by hand to an average depth of 12.2 m, 36 ditches were dug and 270 exposures were described. The field work took 9 months (^°). The survey of this area by the loop method took 30 days. During this time, about 1300 loop vectors were measured with an octagonal arrange- ment of diameter d = 100 m. As already mentioned, the results of all these investigations provided the same structural pattern of the studied area. However, owing to the dense network of the observations the pattern is more firmly based on the data of the loop method. To compare the economic and production effectiveness of geological surveying and geophysical work by the loop method another example can be chosen by quoting the survey on the Marforov area (the south-western plain of the Kerch peninsula). To study the structure of this area of 56 m^, 133 prospecting pits were dug to an average depth of 3.5 m, 216 wells were hand -drilled with an average depth of 15 m, and 150 natural exposvires were described. The field period lasted 10 months (^^) and the work cost 183,000 roubles. The survey of this region with the loop method lasted 12 days, 450 loop vectors being measured. The cost of the work was about 30,000 roubles. The results of the surveys are identical. To illustrate the efficiency of a field group using the loop method we give certain figures from the accounts of the group 13/50 of the department Krasnodarneftegeofizika working in 1950 in the Kerch and Taman peninsulas (^^). 266 1. 1. Krolenko A single section group in the course of a field period (lasting 180 working •days) surveyed quantitatively an area of 776 km^ (the distance between the profiles from 600 to 1000 m). Ten structures were mapped in the Kerch and three on the Taman peninsula. 6000 loop vectors were measured mth an octagonal arrangement, d = 100 m, AB = 200 m. The work cost 350,000 roubles. In conclusion, w^e will consider which production problems confront the loop method at the present time and the possibiHties for its development in surveying. The results of studies by the loop method under complex conditions of dislocated, homogeneous and anisotropic deposits, covered by alluvia, were confirmed in most cases by geological surve3dng, by cartographic driUing and partially by seismic exploration. This means that the method can be considered as the first stage in a semi-quantitative co-ordinated exploration of buried structures under the conditions mentioned above. The first objects of exploration by the loop method should be regions which are unfavourable for other methods. In other words, lithologically homogeneous sections, which do not have clear marker horizons and are poorly exposed. When planning subsequent geological surveying and drilhng in these areas, there should be a sparse network of trial shafts and cartogra- phic bore-holes, and the investigation should be concentrated in the zones of local upfolds, variations of axial plunge, secondary com,plications and in zones of merging of the structures, all of which are indicated on the map \\ith the aid of the loop method. If the surface and deep tectonics correspond, a similar method also ap- plies to problems of planning seismic exploration and deep drilling. An im- portant role is then played by the combined use of the loop method and seismic exploration. Carrying out these measures makes it possible to reduce the bulk of expensive geological surveying and seismic exploration, and to improve the extent to which the deposit is studied. A successful application of the loop method in the Crimea, Dagestan, the Taman' peninsula and in the Turkmenia indicate that it was essential to continue this work in these regions. Thus, in the Taman' the loop method only covered the western and central part of the peninsula, in the eastern part and further along the southern edge of the Kuban depression, sur- veying was not carried out, although in combination with seismic explora- tion this work would be of practical interest. In Turkmenia and to the south of Kazakhstan, the loop method could be used for surveying and mapping of buried structures on the western depression of Kopetdag and in north-western Tudakyr and Kaplankyr. THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 267 In this respect also the region of Bukhara and Khiva on the right bank of the Armu-Dar'ia is not without interest. Of considerable help would be the use of the loop method on the exten- sive, but little studied, Ust'-Urt plateau, where the loop method gave favour- able results with deposits showing large-scale anisotropism. The loop method should also be used to a greater extent in tracing faults in disturbed zones. When carrying out experimental work it would be possible to recommend for this purpose objectives such as the Emba salt domes. In connection with the proposed modernization of the method, in partic- ular for extending its depth and effectiveness of study and the use of a small dimension frame instead of a large diameter loop, the loop method can be very helpful in a nuinber of wooded regions in Western Siberia; for example, regions to the east of the Chelyabinsk graben. For the further development of modifications of the loop method, we can also recommend observations at a fixed frequency with increase in the disper- sions of the feed electrodes A and B, for work on a fixed length of AB line with changing frequency of the feed current (1, 2, 3 c/s etc.). This would make it possible to follow the change in vectors of the loops with increase (or decrease) in the depth of the study. At the present time, in view of the absence of a theory for alternating electromagnetic fields in inchned anisotropic media, this suggestion can only be based on an evaluation of the depth of penetration of electromagnetic oscillations at various frequencies in a homogeneous, isotropic unbounded medium. If we assume the resistance of such an idealized medium to be equal to the resistance of the clays of the Maikop formation (^ = 3 . 10^ Q. . cm), then the depth of penetration of the oscillation with a frequency 1 c/s is about 2 km, and for an oscillation with frequency 10 times greater about 640 m. Consequently, the range of depths with change in the frequency from 1 to 10 c/s is sufficiently great to serve as an indication of the possi- bility of increasing the depth of investigation by reducing the frequency of the feed current, as suggested above. However, to interpret field observations it is necessary to have a devel- oped theory for an alternating electromagnetic field in anisotropic media, and in particular a calculation of the actual depth of penetration of the alternating magnetic field of a given (low) frequency in inclined anisotropic deposits. This makes it possible to evaluate in the first place the depth of the proposed modification of the loop method and to calculate the range of depths with change in frequency of the feed current, and in the second place, to establish the resolving power of the method in the presence of a hetero- 268 1. 1. Krolenko geneous section. For this it is necessary to check the influence of the hetero- geneity of the section and the surface heterogeneity on the results of the observations. It is possible that these calculations mil make it possible to extend the area of application of the loop method, using it to detei'mine the angle of inclination of the deposits which are anistropic on a large scale. To interpret the results of observations obtained with change in the coefficient of mutual induction between the circuits by the method of incli- nation of the frame, it is necessary to know the effect of the angle of incli- nation of the anisotropic deposits (a) on the observed angle of inclination of the magnetic field on the earth's surface (q). This effect is given in a gen- eral form by Dakhnov^^**); however, the solution of the integral in the expression for angle /? through a and the coefficient of anisotropy A is given only for two special cases : for points of the observation placed along the dip and for points placed along the strike of the rocks, i.e. along the axes OX and OY. A general solution should be obtained by deriving for- mulae for calculations and establishing graphs from which it would be possible to interpret field observations. Experimental work on the modernization of the loop method showed the way for increasing the efficiency of this method, and also the depth of the study which should extend the region of its appHcation. A number of problems should be solved by field experiment; however, further development and improvement in the loop method is possible only on the basis of more ex- tensive production use, as a cheap and highly efficient method for geophys- ical prospecting. REFERENCES 1. A. D. Arkhangel'skii, et ah, A Brief Report on the Geological Structure and Oil Bearing Deposits of the Kerch Peninsula. Tr. GGRU, No. 13, Gosgeolizdat (1930). 2. P. K. Aleinikova, The Geological Structure of the Glazovsk and Yurkinsk upfolds. Documents of the Krymneftegeologiya Department (1948). 3. P. K. Aleinikova, Report on the Structural Geological Survey of the Malo- Babchinsk Area. Documents of the Krymneftegeologiya Department (1951). 4. V. R. Burs IAN, The Theory of Electromagnetic Fields used in Electrical Prospect- ing. Zhdanov Leningrad Cos. University. 5. V. N. Vaslil'ev, Report of the Work of the Feodosiisk Seismical Party 10-53 on the Territory of the Staro-Krymsk Region of the Crimea. Documents of the Krasno- darsk GSGT Glavneftegeofizika MNP, SSSR (1954). 6. I. M. GuBKiN, M. I. Varentsov, The Geology of Petroleum and Gas Deposits of the Taman Peninsula. Azneftetekhizdat (1934). 7. K. S. GuMAROV, Report of the Work of the Pribalkhansk Electrical Prospecting Party 10152 in the Pribalkhansk Region in 1952. Documents of the Sredneaz. geofiz. tr. Glavneftegeofizika MNP SSSR, (1953). THE LOOP METHOD IN EXPLORING BURIED STRUCTURES 269 8. K. S. GuMAROV, The Results of Work of the Electrical Prospecting Party 7/53 in the South-Western Turkmenia. Documents of the Sredneaz. geofiz. tr. Glav- neftegeofizika MNP SSSR (1954). 9. V. N. Dakhnov, The Loop Method. Proceedings of the All-Union Office for Geo- physical Prospecting of the main control of the petroleum industry. No. 4, ONTI (1935). 10. V, N. Dakhnov, Electrical Prospecting by the Loop Method. Gostoptekhizdat (1947). 11. Yu. A. DiKGOF, Report on the Electrical Prospecting Work on the Tuzluchan and Kobek Sections of the Western Turkmenia. Document of GSGT Glavneftegeofizika MNP SSSR, (1944). 12. N. P. D'yackhov, The Work of the 1950 Kerch Seismical Party 11150 in the Crimea. Documents of the Krasnordarsk GSGT Glavneftegeofizika MNP SSSR, (1951). 13. I. I. Krolenko, The Electrical Prospecting Work on the Areas of the Krymgazneft Trust in 1938. Documents of the Union Office for Geophysical Prospecting. The Central Administration of the Petroleum Industry (1939). 14. I. I. Krolenko, Work by the Loop Method on the Kerch Peninsula in 1939. Documents of the GSGT Central Geological Administration of the NKNP SSSR. 15. I. I. Krolenko, The Work of the Feodosiisk Electrical Prospecting Party 7/46 in the Crimea in 1946. Documents of the TsO GSGT, MNP SSSR. 16. I. I. Krolenko, Electrical Prospecting by the Loop Method on the Turalin Section (Northern Dagestan) . Documents of the GSGT Central geological Administration of the NKNP, SSSR (1943). 17. I. I. Krolenko, Anurov, B. A., The Results of Trials of the Electrical Prospecting Party 13J14 in the Krasnoarmeisk and EtkuVsk Regions of the Chelyabinsk Province in 1954. Documents, Zap. -Sib. Geofiz. tr. Glavneftegeofizika MNP, SSSR, (1955). 18. I. I. ICrolenko, The Work of the Experimental Crimean Electrical Prospecting Party 26153. Documents of the NIIGR MNP SSSR (1954). 19. N. G. Krolenko-Gorshkova, The Subject of the 15148 Correlation of Geophysical Data for the Crimea. Documents of the Krasnodar Geophysical Department of the Glavneftegeofizika MNP SSSR, (1950). 20. I. I. Krolenko, The Work of the Electrical Prospecting Party 9/51 GSKGT on the Taman Peninsula in 1951. Documents of the Krasnodarsk GSGT Glavneftegeofizika MNP SSSR, (1952). 21. N. G. Krolenko-Gorshkova, The Work of the Kerch Electrical Prospecting Party 13J50 KGK on the Kerch Peninsula and the Western Part of the Taman Peninsula in 1950. Documents of the Krasnodar Geophysical Department of the Glavnefte- geofizika MNP SSSR, (1951). 22. A. A. Klimarev, Ya. M. Ryabkin, The Work of the 1952 Taman" Seismic Party 1/52. Documents of the Krasnodar GSGT Glavneftegeofizika MNP, SSSR, (1953). 23. A. A. Klimarev, The Work of the 1954 Aivazovsko-Moshkarevsk Seismic Party 9-10154 in the Crimean Area of the Ukraine SSR. Documents of the Krasnodar GSGT Glavneftegeofizika MNP, SSSR, (1955). 24. A. G. KuRNYSHEV, The Work of the 1948 Magnetometry Party 14148 on the Territory of the Crimea. Documents of the Krasnodar Geophysical Department of the Glav- neftegeofizika MNP, SSSR (1949). 25. G. A. Lychagin, The Conclusions and Possibilities of Petroleum Prospecting in the Crimea. Documents of the Drilling Department of the Krymgaznefterazvedka(1956). 26. G. A. Lychagin, N. V. Furasov, The Structural Mapping Column Drilling at the 270 1. 1. Krolenko Malo-Babchik area in 1941. Documents of the Drilling Department of the Krym- gaznefterazvedka (1952). 27. Z. L. Maimin, Materials for the study of Maikop deposits of the Kerch Penin- sula. VNIGRI, series A, No. 117 GONTI (1939). 28. Z. L. Maimin, The Tertiary Deposits of the Crimea. Tr. VNIGRI, otd. seriya. No. 1 ONTI (1951). 29. G. L. MiSHCHENKO, TheGeologicalStructureof the Maffovsk Area (Kerch Peninsula). Documents of the Krymneftegeologia Department (1951). 30. G. L. MiSHCHENKO, The Structural Geological Survey of the Uzunlarsk area (Kerch Peninsula) . Documents of the Krymneftegeologia Department (1952). 31. M. V. MuRATOV, The Tectonics and History of Development of the Alpine Geo- syncline area of the Southern Eiu'opean area of the USSR. Tr. Inst. Geol. Nauk SSSR, volume II. Izd. Akad. Nauk SSSR (1949). 32. A. N. OsLOPOVSKii, The Correlation of Data Materials of the Geological Investigations in the Crimea. Documents of the Krymneftegeologia Department (1951). 33. M. F. Osipov, The Results of Deep Drilling at the Vladislav area (Kerch Peninsula). Documents of the Drilling Section of the Krymneftegazrazvedka (1953). 34. E. G. Safontsev, The Geological Report on the Results of Core Drilling at the Akhtani- zovsk and Severoakhtanizovsk Area of the Temryukskii Region of the Krasnodar Border. Documents of the GTK Krasnodarnefterazvedka Department (1952). 35. E. G. Safontsev, The Geological Report on Results of Column Drilling at the Fontalovsk Area of the Temryukskii Region of the Krasnodar Border. Documents of the GTK Krasnodarnefterazvedka Department (1955). 36. A. N. TiKHONOV, Stabih'zing the Electrical Field in a Homogeneous Conducting Semispace. Izv. Akad. Nauk SSSR. ser. geograo-geofizich 7, No. 3 (1946). 37. D. P. Fedorova, Yu. S. Kopelev, The Work of the Electrical Prospecting Party 11155 in the Krasnoarmeiskii Region of the Chelyabinsk area. Documents of the Zap. -Sib. Geofiz Glavnefttegeofizika Department MNP SSSR (1955). 38. V. I. Kholmin, Electrical Prospecting Work in the Turkmen in 1941-1942. Documents of the Sredneaz. Section of the GSGT of the Central Geological Administration of the NKNP, (1943). 39. S. M. Sheinman, Establishing Electromagnetic Fields in the Earth. The collection ''''Applied Geophysics", No. 3. Gostoptekhizdat (1947). 40. D. P. Chetaev, The Study of Non-Stabilizing Systems of Electrical Prospecting Fields in Non-Homogeneous Media. The Documents of the Geophysical Institute, Akad. Nauk SSSR (1953). Chapter 11 ALLOWANCE FOR THE INFLUENCE OF VERTICAL AND INCLINED SURFACES OF SEPARATION WHEN INTERPRETING ELECTRIC PROBINGS V. I. FOMINA When interpreting VEP [VES or vertical electrical sonde in Russian; Editor's remark] curves it is assumed that the surface of separation between layers can, in a first approximation, be considered to be horizontal within the limits of each probe point taken separately*. In practice such conditions are encountered very infrequently, especially when the dimensions of the probe feed lines are great (25-30 km). A structure consisting of two or more tectonic blocks, within each of which the condition of horizontal homogeneity of the medium is observed within certain limits, is often the subject of electro -geophysical prospecting. The boundaries of each component of such a structure (as a result of tectonic disturbances) are zones of vertical or inclined contacts between rocks of different specific resistance. In what follows we shall call these zones "zones of vertical or inclined contacts". In this connection it is extremely important to explain the character and magnitude of the distortions introduced into VEP curves by tectonic disturbances in order to be able to estabHsh with greater accuracy the electric cross -section, the depth at which the electrical markerf horizon lies, and the position of the contacts in each component. If the distance from the VEP point to the zone of vertical or inclined contact is sufficiently great by comparison with the maximum spacings of AB, the VEP curve will reflect changes in the electric cross-section at the point under examination with sufficient accuracy and may therefore be interpreted by the ordinary method. If the distance from the VEP point to the contact zone is small, the observed VEP curve will deviate from the curve for a horizontally homogeneous medium. * The use of ordinary comparison curves is permitted if the marker horizon falls within angles that do not exceed 15-20° (sides of the folds). It is assumed that the diurnal surface is also horizontal. t An electrically resistant horizon, through which the current does not penetrate easily. [Editor's footnote]. 271 272 V. I. FoMiNA It is knoAHi that the nature and degree of the observed distortions depend on: . d (1) the size of the ratio — , where d is the distance from the contact to H the VES point; H is the depth at which the reflecting horizon hes; (2) the correlation — between the specific resistances of the media on both sides of the contact; (3) the angle of inclination of the contact surface. CV-2S and CV-3S reference graphs, which are mentioned in some instruc- tion manuals of electro -geophysical prospecting, had been calculated to evaluate the influence of the vertical contact as long ago as 1938 (^^ ^» ^\ An examination of these reference graphs shows that the vertical contact of two media (given that the electrodes MN do not pass through it) brings about specific alterations in the values of q^^. Thus, if the spacings of AB are orientated perpendicular to the vertical contact, the VEP curve at spacings „f ^ close to . WU show a sha, deflee.on of .Ke v.ue of ,, fro. ,„ dependent on the ratio — (increase for Q^^ Qx-, decrease for q^ <. Q^f. Qx . . AB The beginning of these deflections is observed with spacings of an d ' . . ^ AB ' AB ^ order of — . They attain their maximum value at == a; at > d 2 ^ 2 2 AB the deflections are gradually decreasing and at > lOd they are practically imperceptible (CV-3S reference graph). In the case of spacings of AB being orientated parallel to the vertical contact, the VEP curve will, as is known, show an even deflection beginning ,AB with separations of ^ 0.8 — 1.0 c? (increase for Q2> Qi or decrease for Q2 < Qi) in the value of Qf^ from Qi, to an asymptotic value determined by the formula ^\ _ 2g2 Qi AB Q1 + Q2 ^^^ ^2 AB For — =00, the asymptotic value Q/^ = 2^^ ; for < d distortions Qi 2 in the curve are insignificant (CV-2S reference graph). VERTICAL AND INCLINED SURFACES OF SEPARATION 273 In the case of a vertical contact covered by a medium of high specific resi?ta.nce the thickness of which is small by comparison with the dimensions of the apparatus it is assumed that the nature of the distortions described above Avill not be changed. Complex combinations of vertical and horizontal contacts, which are often hidden, are frequently encountered in practice in geophysical investiga- tions, and the influence of the layer which covers the contact can often not be neglected because of its considerable thickness. In certain of the most simple cases an explanation of the nature of the distortions in the VEP curves can be reduced to solution of a known problein concerning the vertical surface of separation between two separate homogeneous media. A M,, N B • *^-^, • />! = I J2. m />2=00 V V V V ^3=/Jj =00 (a) P-^-p^ =1 "O. m l"NB ■Pt^STa V V V V ^3=C0 V (d) Let us consider the case (Fig. 1, a) in which the cross section on one side of a vertical contact (at the VEP points) is represented by a double -layered medium and on the other side by homogeneous deposits. Let us proceed from the hypothesis that a double -layered medium with specific resistivities q-^ and pg can be treated for each VEP cross-section as a homogeneous medium the specific resistivity of which is equal to the apparent resistivity (^i/^)^*. This hypothesis is not open to doubt for the * That this suhstitution might be possible was also suggested by D. M. Srebrodol'skii in a thesis in 1936. Documents of the Leningrad State University. Applied geophysics 18 274 V. I. FOMINA asymptotic portion of the VEP curve, since for sufficiently large spacings over a double-layered medium The approximate construction of VEP curves under these conditions is achieved by graphic summation of the ordinates of two theoretical probe curves, the first of which corresponds to a horizontally layered medium, and the second to cases with a single vertical surface of separation for which CV-2S or CV-3S reference graphs have been calculated. A curve is selected from a CV-2S or CV-3S reference graph which de- pends on the direction of the probe spacings and in which the modulus- h and takes account of various values for ju = — is equal to the ratio Qi iQik,)L the required spacings of AB. This approach to the construction of the curves follows from a considera- tion of known formulas used in electric profiling for calculating ^^ for the case of the vertical contact of two separately homogeneous media: (a) If the electrodes are placed perpendicularly to the contact then Qk {i + A(i._o[. ^^^'+j^'-'-''A]4^T:r /2 (4>x+L)^-P_ ]■ (2) when all the electrodes are on one side of the contact in a medium with specific resistivity q-^, and k 1 L^~l^ {4^x + LY-l^_ (3) if the electrode B is in a medium with specific resistivity ^2? and the electrodes AM and N are in a medium with specific resistivity q-^. (b) If the electrodes are placed parallel to the contact then Qk= QiV- + k y{y+i) I l/4c^2 + j2 j/4^2+(^+/)2_ (4) If the expression in braces is denoted by q^ and if we take the above hypo- thesis into consideration, the value of Qj^ for each of the formulas given above can be expressed as the product Qk=i.Quk}LQk- (^) (?i k)L ^^'^ ^ double -layered medium is expressed in its turn by the known formula VERTICAL AND INCLINED SURFACES OF SEPARATION 275 (6) If the expression in braces is denoted by gj^ and the value (^^ ^)^ is substi- tuted in equation (5) we obtain H^V Qh^iQi.k)LQk -QiQkQk or 6k 6k •> Qi (7) where gj^ is the value of the ordinate of the curve on an SN-1 reference graph for each spacing of the probes ; q^ is the corresponding value of the ordinate of the curve on a CV-2S or CV-3S reference graph. Taking logs of expression (7) we obtain i,(f)=i,rf+i,rf- When, therefore, we are constructing a desired curve on a bilogarithmic paper we sum the ordinates of the curves given above graphically. 1-0 p-\n.m N' /7 :/>, =00(99/?.m) y / / /'.■ ^.mi @ © A -^ f ■' / ■/ / . J/ / ///./> (a) (b) c'ia^Cc) <(d) y / .-/ -^ '/ / / X^ /qnnr).-nnnnnnonO[g^ (T) Values of -p- I 00 I Values of ^ (e) AB 2d Fig. 2. a— approximately constructed VEP curves; b — VEP curves for a horizontally homogeneous medium; c — curves of the CV-2S reference graph; d — curves of the HVC reference graph; e — Hne connecting the points on VEP curves for the maximum separations oi AB used in practical work. 276 V. I. FOMINA Figure 2 shows curves which have been constructed in the manner de- scribed above for case 1, a, Id which the direction of the separations is parallel to the surface of the vertical contact for various values of the ratio — . It H also shows curves for a double-layered medium at the VEP point and a curve on the reference graph CV-25. As may be seen from Fig. 2, approximately constructed curves only give satisfactory results for true transverse conductivities S when d ^ 2H*. oU o 'V I. h,= 'b\p, =1 0.^p-^-p^ 00 jy jC3_ ® ® o ©@ C'/.o) V \ X V ■ \ / -X /^^/ X '' , ■ — ' X J3 q^lXi Q O O O f (a) (b) -(c) Q Voiues of [oo] Values of — AB Fig. 3. a — VEP curves for a horizontally homogeneous medium: b — approximately constructed VEP curves; c — curves of the CV-3S reference graph. H When d < — the value of S obtained is half the true value, and when 2 2H > J > iH, S takes on a value intermediate between S and 5 The curves obtained are compared in Fig. 2 with curves calculated by M. N. Berdichevskii (HVC reference graph) for the case when two contact- ing media lie on a third medium with a specific resistivity of ^3 = 00, where we have taken ^2 = 99^^ instead of Q2 = 00 (Fig. 1, e). The curves obtained practically coincide with the HVC curves. On this basis we consider that the accuracy of the approximate con- struction of curves by the methods described is fully adequate for practical * Given a cross section of this type, VEP curves may usefuUy be extended as far as the cross sections bounded on Fig. 2 by the horizontal line C. VERTICAL AND INCLINED SURFACES OF SEPARATION 277 purposes and the curves obtained can be used to explain the nature of distor- tions in probings arising from, the influence of non-horizontal boundaries. The approximate construction of curves distorted by the influence of a horizontal boundary can be extended to the more complex case shown in Fig. 1, b, when the medium at the VEP point is a triple -layered cross- section. Fig. 3 shows curves obtained for this case: ^0 = 9^1' ^^0 = 1^1- In this case gjf has already been determined on an SN-20 triple -layered comparison curve for each separation taken; the value of S has been approx- imately determined with the same degree of error as in the double -layered case cited above. The ordinate of the minimum point is increased under the influence of the vertical contact. Its abscissa remains practically cons- taut. In this case the variation in the co-ordinates of the minimum point d . and the value of S in relation to the distance — • is the same as would obtain H if instead of sudden increase in the resistivity at the contact there were to be a gradual increase in the resistivity of the conducting horizon. In connection with this, it is obvious that a fictitious rise of the marker horizon may be obtained in interpreting VEP curves when fixing the para- meter which corresponds to the conducting horizon above the marker horizon. For VEP curves parallel to the contact, the position of this contact may be defined either from the beginning of the deviation of Qj^ from q^^ j^ or from the alteration in the type of curve as the VEP point approaches the contact (transition from a double -layered curve to a triple -layered curve of type A), while for a triple -layered cross-section it may be defined from the increase in the ordinates of the minimum points. It is obvious that these criteria may not be of use in all cases as the electric cross -section at the VEP point is not sufficiently studied in the majority of cases. A more accu- rate determination of the position of the vertical contact may be made from VEP curves in which the spacings are perpendicular to the contact. By using the approximate method described above for constructing curves for VEP in which the spacings are orientated perpendicular to the surface of the contact, let us make a preliminary comparison of certain of the curves obtained with the curves of the HVC reference graphs. The comparison shows that the calculated curves coincide with the approximated curves for a double -layered cross -section. Approximated curves for a triple -layered cross -section of type H, are shown in Fig. 4 and for a double -layered cross-section in Fig. 5. A value 278 V. I. FOMINA for S determined from the first rise in the VEP curves is close to its true value (when d > 2H) for both double and triple -layered cross-sections. When d < 2H the first rise of the asymptotic branch of the VEP curves is brought about by the influence of the vertical contact and the value of S determined from this rise is considerably less than the true transverse con- ductivity, this divergence being the greater for the SN-1 type of VEP the smaller is the ratio H (-0 ^ ^ 9^ 1. ^ V V V V vv] I \ \ \ \ \ y \ X w X y V / V \ / ■ / \ ,^^ . , , E] Values of -^ -(a) b c — ^1 f ; Fig. 4. a — VEP curves for a horizontally homogeneous medium; b — approximately L i ^|f constructed VEP curves; c — curves of the CV-3s reference graph. If the values of S are determined for the terminal branches of observed VEP curves, these values will be very slightly lowered and the lowering will decrease as the ratio — • decreases. H The maximum increase of Qf^ in relation to {q-^ j^) is observed for separa- tions in which the feed electrode intersects the surface of separation be- tween the media. We may therefore establish the position of the vertical contact by plotting the abscissae of the maximum increase of Qf^ in relation to (Qi,k)L ^long the profile from the centre of each VEP. In the case of a triple -layered cross-section a considerable deviation of Qi^ from (^j j^)^ at the minimum of the curve is to be observed. The greatest deviation in the minimum is to be observed in VEP curves obtained when d is close to the abscissa of the minimum. VERTICAL AND INCLINED SURFACES OF SEPARATION 279 Fig 5. a — approximately constructed curves for p2 ^ (&i fc)L' ^ — approximately constructed curves for q^ < (?i \ilL.'^ '^ — VEP curves for a horizontally homogeneous medium; d — line connecting the points on VEP curves for the maximum separations of AB used in practical work. The deviation of q^ ^^^ from (^^ ^^ of the minimum is insignificant for distances d being greater or smaller than the abscissa of the minimum. The maximum deviation on the VEP curves in the cross-section under consideration is observed when d — H. The observed deviation can be largely utilized for qualitative evaluation of the depth at which a reflecting horizon is lying. Fig 6. a — approximately constructed J_ VEP curves ; 6 — approximately constructed 1 1 VEP curves; c — VEP curves for a horizontally homogeneous medium. 280 V. I. FoMiNA When interpreting the distorted VEP by using parameters of the conduct- ing horizon the deviation in the ordinate of the minimum and the value of S already noted may considerably influence the accuracy with which the depth at which the electrically reflecting horizon lies can be determined. In this case the influence of the vertical contact may lead to a fictitious rise in the reflecting horizon, the axis of the rise being situated at a distance d from the vertical contact. Calculation of the influence of the contact is facilitated if cross VEP are made with spacings parallel and vertical to the contact. Such investigations give us two curves J_ VEP and VEP. The approximately constructed d cross VEP curves for various values of — are compared in Fig. 6. This comparison yields the following diagnostic features for the curves when Q2^ Qi- (1) The magnitude of S (determined from the VEP curve 11 for c? < 2H) is approximately twice less than the analogous value obtained from the VEP curve JL, and the value obtained from the latter is closer to the true value. (2) The abscissa of the sharp maximum increase of Qj^ on the _L VEP curve relative to Qj^ on the j VEP curve is equal to d. (3) Both curves intersect to the right of the line d. The abscissa of the point of intersection is approximately equal to 2d. Fig. 7 shows VEP curves obtained from a drill hole (Pribel'skaia zone, Alakaevskii sector) near the contact \d = — H\ where the angle of its dip exceeds 45°. VEP Nos. 67 and 324 were carried out with the spacings of AB respectively orientated parallel and perpendicular to the contact. The approximately constructed ±VEP and 1] VEP curves for case 1, a( when d l\ — = — I are introduced in the lower part of Fig. 7. Comparison shows that H 5 / the left-hand branches of these curves and the other curves are the same in type. A certain diflerence is to be observed in the form of the initial bran- ches of the VEP curves and this is, no doubt, caused in the upper part by the complex geological cross section. The characteristic differences between _L and || VEP curves are revealed with sufficient clarity here, and this enables one to establish values of d and S {d =^ 23 m, S = Qm). We shall now clarify the nature of the distortions brought about by the influence of the vertical contact when the cross VEP are arranged over VERTICAL AND INCLINED SURFACES OF SEPARATION 281 a medium with resistivity {Qi^h)L^ Q2 (Fig- l, candd). A similar case may be encountered when working over an antichnal fold, the core of which is composed of rocks of high resistivity while the overlaying strata of anticlinal crest and limbs are of reduced resistivity. Use is also made in this case of approximately constructed curves and the error in construction for the case ^2 < (Qi, i^l ^^ preliminarily estimated by comparison of the approximated curve with an HVC reference graph. Fig. 7. o— approximately constructed || VEP curve; 6— approximately constructed X VEP curve; c — H line; d — position of the non-horizontal contacts; e — VEP curve for a homo- geneous horizontal medium. Let us examine the special case of the HVC reference graph when q-^ = 99^2 (Fig. 1, d). This case may be identified with the case which we adduce in Fig. 1, c, if one allows in the latter case that the thickness of the layer with a resistivity q-^ at the VEP point is small in comparison with the dimensions of the arrangement. A cross-section consisting of a two-layered medium disposed on both sides of the vertical contact is taken for the HVC reference graph (Fig. 1, e). It is necessary, when constructing approximated curves, to determine the 282 V. I. FOMINA ratio — , which we adopt for the case given in Fig. 1, e provided that (Qi. k)L the foUomng assumptions are made {Q2, k)L Q2 {Ql,k)L {Ql,k)L' since for small spacings of the probes it is possible to treat the medium ^2 as being of infinite thickness. 1 r\j4 1 -d— 1 ,A \ A® ^' /°, = ^9/Oj|/°2 V V V J 'if- - \ ^^ d X \ X ----(a) AB 2d Fig. 8. a — VEP curves for a horizontally homogeneous medium; h — HVC comparison curves; c — approximately constructed curves for — =1/99; d — approximately Pi ' constructed curves for = — ^^^ Q\ (^1, ft)L The equation (?2, k) g2 1 99 is adduced for the spacings which correspond to the asymptotic part of the curve. Both variants of an approximately constructed curve for the case d (1 /7 1 \ ratios — =2 and — = — j and spacings oi AB parallel to the contact are given in Fig. 8. VERTICAL AND INCLINED SURFACES OF SEPARATION 283 A curve constructed for the ratio — '- = , coincides with the calcu- {Qi.k)L 99' lated curve in its initial and final parts. In the central portions the minimum of the constructed curve rises higher {Q2, kJL Q2 We were unable to construct a curve for the ratio foi {Qi, k)L {Qi, kJL sufficiently large spacings of AB, owing to the absence of curves with mod- Q2 < 0.01 on the CV-25 reference graph. ulus (Qi, k)L In the part for which the construction was made, the least divergence from HVC curve was observed for - — = 2, while for — = — complete coin- cidence with the HVC curve was observed. If the distance of the separations of AB is increased the approximately constructed curve for the condition (^2, k)L Q2 deviates considerably, as would be expected, from the (Qi, k)L (Qh k)l AB HVC curve, since (^2 k)L ~^ °°' ^°^ ~^ °°- When the spacings of AB are orientated perpendicular to the plane of the contact the approximately constructed curve is observed to coincide completely with the calculated curves owing to the fact that the influence of the vertical contact diminishes as the spacings of AB increase. A M N B • m-f* • n" n " o^o V |h,=5ho /f=IJ2.m V V V V V \r^ />=00 -(a) (b) ■o-'^(c) ® Values of^ \oo] Values of ^ Fig. 9. a — VEP curves for a horizontally homogeneous medium; b — approximately constructed VEP curves; c — curves of the CV-3S reference graph. 284 V. I. FOMINA Basing ourselves on the comparison of calculated and approximately constructed VEP curves for q^ < (^i /c)l ^^'® ^^^ therefore consider that up to the limits of spacings AB used in practice the accuracy of the approxi- mate construction of curves is also fully adequate to enable us to explain relative variations in the form of VEP graphs. Curves for perpendicular orientation of the AB for spacings the cases c d , . and d (Fig. 1) and various values of — are given in Figs. 9 and 5. A satisfac- H tory value of S may be obtained by analogy with the previous cases from the first asymptote of the VEP curves when d > 2H. For d < 2H up to <i = \H, values of S obtained from the final branch of the curve will be increased in relation to their true value by anything from 15-30%. For d <.^j^H lh.e values of S obtained will fall within the limits of accuracy of the observations. Here, as in the case of a vertical contact for {q^ j^j^ <C Q^-, the abscissa of maximum deviation of q^^ is equal to d. The position of the vertical contact may therefore be obtained and the depth at which the reference horizon lies be estimated by profiling the abscissae of the maximum deviation of 1-0 -y^ v; -■;l- >...>'. y:^T^ \' \ \ V V AB 26 Fig. 10. a — VEP curves for a horizontal homogeneous medium; b — approximately constructed VEP curves; c — curves of the CV-2S reference graph. VERTICAL AND INCLINED SURFACES OF SEPARATION 285 Qf. from {qi f^)^ from the VEP points, since this sharp deviation of Q/^ from (Qi k)L should be lacking for curves in which d < 2H. When the separations are orientated parallel to the plane of contact, the VEP curves for cases c and d on Fig. 1 are given for a doubledayered cross- section in Fig. 11 and for a tripledayered cross-section j Qq = 9qi ; Aq ~ — ^i in Fig. 10. The VES curves are so distorted that a change in the type of curve . d is observed as the ratio — decreases. The observed deformation in the H curves is smooth and comes about as a result of variation in the electric cross-section in the direction of the contact. H A double -layered curve for Q^ ~ °° ^^^^^ d <^ — is transformed into a three-layered curve with ^3 < Q2, and for d < ^H into a two-layered curve with terminal restivity Q2 — ^Qi- A three -layered curve of type H with g^ = <x> for d > ^H is transformed into a four -layered curve with Q^ <. ^3, and for d << ^H into a three-layered curve of type H but with terminal resistivity ^3 = 2qi. The nature of the deformations on VEP curves for (p^ [^)j^ >- ^2 described above corresponds to the case when the medium Q2 ^^ homogeneous. In practice one has to deal with contacts with inhomogeneous media on both sides, and, in particular, with a two -layered medium. p. Fig. U. a — approximately constructed curves for q^ > (g^ ^)^; h — approximately constructed curves for q^ < (g^ ^)j ; c — VEP curves for a horizontally homogeneous medium. 286 V. I. FOMINA <\- — (T)values of h (a) (b)- — (c) Fig. 12. a — approximately constructed J_ VEP curves; b — approximately constructed II VEP curves; c — VEP curves for a horizontally homogeneous medium. In this case it is clearly possible to consider the medium as being homo- geneous up to the spacings used in practice and to make the construction for the ratio . {Qx, kJL Thus, the following characteristic changes are to be observed in cross VEP curves for the case {q^ ^j^ > ^2' made near the vertical contact (Fig. 12): (1) the absence of an asymptotic branch ascending at an angle of 45° in the || VEP curve: (2) the abscissa of the maximum diminution of q^ on the asymptotic branch of the J_ VEP curves is equal to d; (3) The abscissa of the point of intersection of the ± and VEP curves (or of their convergence for small values of d) is approximately equal to 2 d. Let us turn our attention to the observed deviations in the right-hand parts of the experimental VEP curves Nos 324 and 67 (Fig. 7), which are similar to the deviations obtained on approximately constructed curves for / d the case {q^^ ^^ > ^2 I for '^ =" 5 A somewhat different correlation between q^^ for experimental and approxi- mately constructed VEP cross curves will evidently be produced by the in- fluence of a preceded non -horizontal boundary with {Qi,k)i >1. A second non -horizontal surface of separation of the media with VERTICAL AND INCLINED SURFACES OF SEPARATION 287 AB 2 km 5 10 1-5 2 Profile W ■^ 3p-y 4p. ip 457 458^462^5:=^^^ g^-460jg3g|^3| |^4e2>i46^^^ ^p „465_,^ 466 468 Ip oTo) A(b) — (c) i^ra(d) C=l(e) ==^(f) (g) (h) (i) ^(j) — (k)---(L) Fig. 13. a — VEP points; b — deep drilling holes; c — line of equal 5;^; d — areas of in crease in Qj^; e^areas of decrease in Qi^',f- — line of true values of 5; g — line of reduced values of 5; h — line of increased values of S due to the influence of the western non- horizontal boundary ; i — line of increased values of S due to the influence of the eastern n on -horizontal boundary; j — position of the non -horizontal boundaries; k — abscissae of the maximum decrease in qj^; I — abscissae of the maximum increase in Qj^. (Qi k)L > ?2' ^t a distance approximately 400-500 m from the VEP point can be assumed on the basis of the diagnostic features of the cross VEP curve. We can obtain an idea of the relative variations in the forms of VEP curves along the profile by comparing the approximately constructed curves for cases a and c with respect to h and d for various values of — — u (Fig. 1.) and this enables us to determine the position of the non-horizontal contact and of the right-hand asymptotic branches of the VEP curves (cf. Fig. 5)*. * It is here assumed that the thickness of the layer h^ in the raised block (case c in Fig. 1) is sufficiently small by comparison with the thickness of this layer in the lowered block (case a in Fig. 1). 288 V. I. FoMiNA It is obvious from Fig. 5 that when the AB are spacings orientated per- pendicular to the plane of contact on the VEP curve for all values of — , H the true position of the right-hand asymptotic branch can be defined as the mean of the right-hand branches of VEP curves with equal . d For values of — — > 2, the true position of the right-hand asymptotic H branch can be defined by the line running at an angle of 45° from the point . d of divergence of curves wdth equal — . The position of the contact is fixed by the abscissae of the maximum devia- tions of Qj^ in opposed directions. When the separations of AB are orientated parallel to the plane of the ^ d contact for — > 2, the true position of the asymptotic branch running at H d an angle of 45° from the point of divergence of curves with equal — is H depicted on Fig. 11. If all the characteristic variations in VEP curves along profiles men- tioned above are used in the interpretation of VEP curves, one can indicate zones in which the horizontal homogeneity is disturbed and the correspond- ing tectonic dislocations and also establish the true position of the asymptotic branch of the VEP curve. An example of the interpretation of VEP curves in one of the regions is given in Fig. 13. The VEP profile is disposed transversely to the direction of the anticlinal fold axis. As has been showai above, the position of the non -horizontal bound- aries is determined by the analysis of the distortions in VEP curves. Despite the fact that there are two non -horizontal boundaries in the given case, the total influence of which creates a much more complicated picture in the distortions on the VEP curves true values of S were deter- mined by painstaking analysis of all the VEP curves along the profile and the depth at which the reflecting horizon lay was established. These results were subsequently confirmed by drilling (the drilling points are plotted on the profile). The abscissae of the maximum deviations of Qj^ from (o^ j^j^ are indic- ated (by arrows) in the upper part of this profile. In sectors where the abscissae of ^^ converge wth deviations of different direction one can assume the presence of a non -horizontal contact. It is not difficult in this case to show that the smaller the area of convergence of VERTICAL AND INCLINED SURFACES OF SEPARATION 289 the abscissae the greater is the angle of inchnation of the contact to the diurnal surface (when the strata overlaying the contact are of equal thickness) . In the given case the western limb of the anticline dips at a greater angle than the eastern. At the same time the latter obviously exhibits a step struc- ture. The vertical cross-section of the resistivities constructed from the VEP curves gives a qualitative confirmation of the above hypothesis (Fig. 13). Here the region of increased values corresponds to the anticlinal upfold. On the western and eastern flanks of the anticline the character of the be- haviour of the isolines varies. One might consider that the vertical electric cross-section gave a sufficiently good qualitative indication (by the transi- tion from low to high resistivity) of the position of non -horizontally incHned strata and that there was no necessity for detailed analysis of the abscissae of the maximum deviations of Q/^ from {q^ y.)^. Determination of the position of the inchned contact by means of the ver- tical cross -section is, however, not always sufficiently precise. In addition the value and direction of the anomalous variation in Q/^ can be evaluated as a result of analysis of the abscissae of the maximum deviations of Q)^. In practice the points of maximum deviation in ^^, the abscissae of which are depicted above the profile, do not ahvays correspond to a sufficiently clearly defined anomalous break in the isolines. In VEP No. 466 (Fig. 14), for example, an area of anomalous increase in Q[^ is to be noted. Its abscissa is equal to 5 km, which corresponds to the distance from VEP No. 466 to the eastern non-horizontal contact (Fig. 13). There is, however, no ano- malous behaviour of the ohm isolines to be observed in the vertical cross- section at VEP point 466 and the position of the right-hand asymptotic branch of VEP No. 466 can therefore be taken to be true. The position of the right-hand asymptotic branch of VEP curve No. 466 should also be determined from the analysis of the abscissae of the maximum deviations of Qf^, as is shown in Fig. 14. At VEP point No. 460, situated at the peak of an anticline (Fig. 13), the anomalous lowering of ^^ is denoted by the course of the isohnes for 16Qm. and above. This lowering can be treated as an alteration in the electric cross-section at the area of VEP No. 460 (inclusion of an additional conduct- ing horizon). This lowering is, however, conditioned by the influence of the western and eastern non -horizontal boundaries (as may be seen from the disposition of the abscissae of the maximum deviations of Qf^ over the profile). The corrections indicated in Fig. 14 must be introduced into the VEP curve on this basis. In VEP No. 464 (Fig. 14), the point of maximum decrease in ^^ is given Applied geophysics 19 290 V. I. FOMINA VES N466 \ VES N462\ // /.- \i y VES M464 VES N460 \ \ \ VES N461 ^Ntt. L...!j(a) ;\. \ W AV- \X. \ W "V ._.y^ \ \ \ \ /' ^^ ' / / ](b) (c) -(d) (e) Fig. 14. a — section of the VEP curve with increased values of q^\ b — section of the VEP curve with decreased vahies of Qj^ ; c — VEP curve for a horizontally homogeneous medium; d — ^VEP curve in a lowered block; e — VEP curve in a raised block. by the abscissa equal to 5 km which is due to the influence of the western non-horizontal contact. At the same time, the degree of variation in the configuration of the 12 and 16i3m isolines brought about by this lowering beneath VEP point No. 464 may be mistakenly ascribed to the result of the influence of the eastern non-horizontal contact as a result of which the right-hand branch of the VEP curve should be corrected in the direction of a decrease in q^^. The analysis of the maximum deviations of Qi^ of VEP VERTICAL AND INCLINED SURFACES OF SEPARATION 291 curves along the profile allows us to determine the true position of the asymptotic branch of VEP curve No. 464 as depicted in Fig. 14. The first asymptotic rise is, in all probability, due to the influence of the eastern contact which increases the value of Qf^ The further lowering in the values of Qi^ with increase in the separations of AB is due to the influence of the western contact. 458 o 460-(o) A2p-(b) ^4p-(c) 1(d) E^(e) ^B(f) vvv(g) Fig. 15. a — ^VEP points; b — drill holes in existence at the moment when the materials were being studied; c — drillings made after the materials had been studied; d — Upper Cretaceous deposits C^25 ^ — lower Cretaceous deposits C^^; / — Paleozoic deposits P^; g — top of the reflecting horizon of infinitely high resistivity from electro-prospecting data. We obtain confirmation of this hypothesis by examining VEP point No. 462, which comes next on the profile, since the sharp increase in the ordinate of the minimum which is to be observed on the curve points to the siting of the point close to the edge of the eastern contact. The distance d for this VEP point is close to the depth H. On the other hand the abscissa of the maximum decrease in Qf^, which denotes the western contact, is decreased in passage between points 464 and 462 by comparison with point 464. Its right-hand asymptotic branch should be corrected in con- formity with this, as is shown in Fig. 14. The analysis of distortions in VEP curves thus carried out enables us to determine the position of the non -horizontal contact and the right-hand branch of the VEP curve with greater reliabihty and from this, the value of S. Possible values of S obtained from VEP curves are given in the lower portion of the drawing. The most reliable values established by the analysis of the distorted curves are joined by a continuous line. Fig. 14 gives an example of the comparison of VEP curves sited on both d sides of the contact for approximately equal values of — . This type of H juxtaposition of the curves enables one to determine the true path of the asymptotic branch as a hne running at an angle of 45° from the point of 19* 292 V. I. FOMINA divergence of the curves. The resuhs obtained by interpretation were confirmed by drilhng. The method for analysing distortions in VEP curves which has been examined can also be apphed to dipole quadrilateral probing (DQP)*, for which the characteristic distortions from non -horizontal boundaries will be more sharply reflected. We shall examine distortions in DQP curves brought about by passage <a) 37 5000 ■(b) (c) 10,000 (d)= (e) - ^ yi ' 15,000 <g).'r.-- .(hH„ 20,000 (i) ■(j> - 25,000 Fig. 16. a — VEP points; b — ohm isolines of qj^; c — ohm isoUnes of qj^ compensated for the influence of surface distortions; d — Hnes of maximum decrease in Qf^ from deep non -horizontal boundaries ; e — Unes of maximum increases in Qi^ from deep non-horizon- tal boundaries; / — lines of minima of q/^ from surface non -horizontal boundaries; g — Unes of the maxima of Qf^ from surface non -horizontal boimdaries; h — position of the dipole MN for maximum deviations of qi^ from (oj ^)^ — influences of surface non -horizontal boundaries ; i — abscissa of the maximum decrease in Qj^ from the influence of deep non-horizontal boundaries; j — abscissa of the maximum increase in g;^ from the influence of deep non -horizontal boimdaries. Literally dipole equatorial Bonding or (DES) in Russian. [Editor's footnote]. VERTICAL AND INCLINED SURFACES OF SEPARATION 293 of the dipole MN across the contact, the position of the dipole AB being considered as constant. In analysing the distortions of DQP curves we make use of a vertical cross-section ^^, which is constructed in the following manner; the abscissa of the values of ^^ taken from the plus and minus DQP curves are plotted on an arithmetical scale on lines orientated vertically from the centres of the corresponding MN dipoles. When the same vertical and horizontal scales are used, the values of Qj^ of DQP curves are distributed along lines at an angle of 45° to the line of the profile. If we make separate use of the plus and minus branches of the DQP curves "\ve obtain vertical cross -sections from which we can obtain a qualitative idea of the character of the variations in the values of Qj^ along the profile in both the vertical and the horizontal directions. If there is a non -horizontal boundary along the profile, we shall not observe a zone of increase or decrease in the values of ^^^ [relative to [q^ ^)^ of the horizontally layered medium] in constructing the vertical cross -section in this manner, at the site of the non -horizontal contact. In this case the line Avhich connects maximum deviations of Qj^ wth the same sign will coincide approximately with the position of the surface of the non -horizontal boundary. Reciprocal signs of the deviations of Qj^ will be observed at the site of the non -horizontal contact in vertical cross -sections constructed from the plus and minus branches of DQP curves. Vertical cross-sections for the plus and minus branches of DQP curves are given in Fig. 16 for the region in which an electric prospecting party of the Spetsneftgeofizika Office (Penza province) carried out investigations. A region of decrease in Qj^ is to be seen quite clearly in the eastern part of the upper profile for the DQP branches. The characteristic DQP curve No. +27 given in Fig. 17 shows an area in which Qj^ is decreased. The abscissae of the maximum decreases in Qj^ are plotted over the vertical cross-sections as in VEP studies*. In that part of the profile which is being examined there are two further sections with anomalous deviations of p^, initially increasing and subsequently decreasing, and consequently giving rise to subsidiary maxima and minima. The lines which connect the abscissae of these points form vertical zones * It is not always essential to depict the abscissae of the deviations of q^^ from {o■^ j^j^ when analysing distortions of DQP, since in the majority of cases the anomalous region of deviation on the cross -section manifests itself with sufficient clarity and is always sited in the immediate vicinity of the vertical contact owing to the principle upon which the profile is constructed. 294 V.I. FoMiNA within the limits of the measurement interval. The positions of the dipoles MN have been plotted in pairs over the profile for the maximum and minimum deviations of Qj^. DQP curves No. +32; +27; +30 are typical examples of distortions of this type. The sharp deviations in the magnitudes of q^^ which are obtained are (as is obvious from the vertical cross section given in Fig. 16) brought about by the non -horizontal boundaries in the uppermost part of the cross -section (they may possibly be revealed in the diurnal surface)*. When the position of the surface non -horizontal boundaries has been established (from analysis of the DQP curves), these distortions can be removed from the DQP curves, as is shown in Figs. 16 and 17. Areas in which Qj^ is increased on account of the buried contact can then be isolated. An anomalous region of increase in ^^ (on the minus branches of the DQP) will correspond to an anomalous region of decrease in Qj^ (on the plus branches of the DQP) in the vertical cross-section. Sections in which Qj^ is increased are indicated on DQP curves No. —35; —36; —37 (Fig. 17). An anomalous region of decrease in Qj^ is also to be observed in the left-hand part of the profile from the minus branches of the DQP. This decrease is to be seen quite clearly in DQP curve No. —32 (Fig. 17) while on DQP curve No. —26 it is complicated by surface distortions. On the plus DQP given above the profile a region of increase in Qj^ should correspond to this region of decrease in Qj^. A region of this type (left-hand side of the upper profile) cannot be isolated out from the ohm isoline configuration, as was possible in the previous case; it merges with a region of high values in q^^ which correspond to the high resistivity horizon of a layered medium in a raised block. An anomalous increase in Qj^ due to the influence of a western non -horizontal contact is, however, revealed with sufficient clarity on the DQP curves and may be isolated by joining the abscissae of the maximum deviation of Qj^ in the DQP along the profile (Fig. 17, DQP curve No. +30). There is no justification for ascribing the increases in Qj^ in this instance to surface distortions, since deviations in ^^ on the side of decrease are possible from the nature of the behaviour of the ohm isolines in the upper part of the cross -section. Having thus isolated distortions in the magnitudes of Qj^ due to buried non -horizontal boundaries, it is possible to establish the position of the latter as the mean the lines of maxima and minima. In the DQP profile which we have examined, the position of the eastern non-horizontal boundary * In what follows these distortions will be called surface distortions. I J VERTICAL AND INCLINED SURFACES OF SEPARATION 295 should lie between DQP points 37 and 33, and of the western between DQP points 29 and 28. The configuration of the ohm isolines should be the same for the plus DQP N+27 DQPN + 32 DQP N-35 .\ DQP N-36 '■ \ -&' j;^ A-r--'W' <-K/ \ < DQP N-37 y DQP l\J-32 B DQPN-26 DQPN + 30 •v, 4- Lir.KalCZDtb), l.(c)\ \^-_-^- FiG. 17. a— area of a DQP curve with increased values of qj^ ; 6— area of a DQP curve with decreased values of p/, ; c— DQP curve corrected for the influence of surface non- horizontal boimdaries. 296 V. I. FOMINA and minus branches of the DQP in the absence of distortions from buried contacts. This is not to be observed on the profiles under consideration, owng to considerable distortions. In calculating the influence of distortions from buried contacts we make use of the fact that the deviations Qj^ from {q^ f^)j^ are of reciprocal sign for the plus and minus branches. 12 3 5 o(32al .MOW (d (d) le) S(fl ==(g) .--"-IM (■l==(i)-.. .(«} E IJ Fig. 18. a — DQP points; b — values of qi^; c — ^Unes of equal values of Qf^; d — lines of equal values of O/j (variant); e — lines of equal values oi Qj^, the configuration of which is conditioned by the influence of buried non -horizontal boimdaries;/ — the position of the buried non -horizontal boundaries ; g — the Hne qj^ j^^jq of VEP curves of a horizon- tally homogeneous medium ; h — lines of the maximum values of Qj^ relative to (g j i^j^ ; i — hues of origin of the increase of g^jjjjjj relative to {q-^ ^)j^; j — Unes of maxima in the increases of g;^ relative to (o, }^l', k — line of origin of the decrease of g^ relative to (g»j Ji)i,'i I — regions of distortions of Qj^. We compile a single vertical cross-section of the plus and minus DQP branches for surface inhomogeneities of the strata. In this case the abscissae of the values of q^^ are laid off along the vertical lines from the central point between the dipoles AB and MN. If the vertical cross -section is constructed in this way, the anomalous / ^\ values of p^ I beginning with values of R close to — I will be found in a zone in which the Hne of the maximum deviations of Qj^ from {q-^ j^j^ is inclined to the line of the profile at an angle the tangent of which equals the ratio between the vertical and horizontal scales, while the line of origin of the deviation of q^^ lies along the bisectrix of this angle. The values of Qi^ from the various DQP branches will fail in varying degrees to coincide in this region (Fig. 18). The line of origin of the distortions of Qf^ divides the vertical cross-section into 5 regions in relation to the degree and direction of the deviation of q,^, as follows: VERTICAL AND INCLINED SURFACES OF SEPARATION 297 I — Distortions of Qj^ are lacking for any directions of the DQP branches. II — For one direction of the DQP branches a deviation of Qf^ on the side of increase is observed; values of Qf^ on the opposite branches of the DQP are not distorted. III — For one direction of the DQP branches a deviation of Qi^ on the side of increase is observed; values of ^^ on the opposite branches of the DQP are not distorted: IV— A deviation of Qj^ on the side of decrease is observed for any directions of the DQP branches; deviations from the opposite DQP branches are equal in the central part of the region: V— A deviation of Qj^ on the side of decrease is observed for one direction of the DQP branches and on the side of increase for the other direction (in the central part of the region the deviations are of equal magnitude but of opposed sign). In addition, there is a very shght decrease in Qi^ in this region due to the other non-horizontal contact which can clearly be ignored. If we take a graphic mean of ^^ and allow for the possible level of its deviation from (^j j^)j^, we obtain the behaviour of the isolines in the lower part of the cross-section which describes the relative variation in the value of Qi^ along the profile. It is not possible to obtain more or less reliable values of Qf^ in region IVowing to the fact that the directions of the deviations of Qi^ here coincide on the opposite DQP branches because of the presence of two non-horizontal contacts. The isohnes of Qj^ are therefore given as a dashed hne (two variants) in region IV. These variants do not, however, significantly alter our general view of the qualitative features of the change in the electric cross -section along the line of the profile. The method for calculating distortions in DQP therefore enables us to establish the position of the buried contacts and to obtain values for the total conductivity which are closer to the true values from the distorted curves and thus to improve the quantitative interpretation of DQP curves. REFERENCES 1. V. N. Dakhnov, Electric Prospecting for Oil and Gas Deposits. Gostoptekhizdat (1953). 2. ''Elkageer,'' 1/2, 1938. Articles on materials from the firm SPE. CV-IS, CV-2S and CV-3S reference graphs. GSGT, 1938. 3. E. N. Kalenov, The C-l-S reference graph. ''Elkageer," 3/4, GSGT, 1938. 4. A. M. Pylaev, a Guide to the Interpretation of VEP. Gosgeolizdat (1948). PART IV. OIL GEOPHYSICS Chapter 12 SOME PROBLEMS OF GAS LOGGING ESTIMATION OF GAS SATURATION OF ROCKS L. A. Calkin The object of gas logging is the detection of productive beds in the section ■of a well. However, the currently used methods of gas logging do not permit rehable evaluation of a bed. This is associated with the fact that, as a resuh of gas logging, one determines the content of hydrocarbon gases in a gas- air mixture, obtained by degasifying the drihing fluid. This content largely depends on the method of degasification and the properties of the drilling fluid being degasified. Thus, the degree of extraction of hydrocarbon gases from the drilhng fluid will oscillate within wide limits, depending on the viscosity, static shear stress, temperature and salinity of the fluid. There- fore a direct relationship is not always observed between the gas measure- ments on the gas log and the concentration of hydrocarbon gases in the fluid. In its turn, the gas saturation of the drilling fluid depends not only on the gas content of the bed but also on many other factors, the main ones being rate of boring, rate of circulation of the drilling fluid and the nature of the bed. It is quite clear that exact evaluation of the bed from the gas log data can be given only when the above factors are taken into consideration. In order to establish the relationship bet^veen the readings obtained in gas logging and the actual gas saturation* of the drilling fluid, we analysed the results of gas logging conducted in various regions, by the simultaneous appUcation of a floating degasifier and a TVD degasifier, whose degree of degasification is about 100 per cent. It was established that in most cases a direct correspondence is observed between the actual gas saturation of the drilling fluid in cm^/1. (data of TVD instrument) and the readings of a sta- tion, the reading being obtained in working with a floating degasifier. How- ever, in some cases this regularity is not maintained, and quite different readings are obtained for the same gas saturation, the readings being lower * By actual gas saturation we understand the amount of gas (in cm^) contained in 11. of drilling fluid. 301 F = io/<:-^, (1) 302 L. A. Galkin at low temperatvire for slightly saline fluids than for less viscous and more saline fluids, and for high temperature fluids. It is obvious that, in order to allow for the degree of degasification of the degasifier used, and to exclude the eSect of drilHng fluid properties on the results of the gas logging, it is essential to perform a calibration — a deter- mination by experimental means of the dependence of gas readings on the gas content of the drilHng fluid. Such a caHbration should be carried out before each operation and on every occasion in case of changes in the prop- erties of the drilling fluid. In calibration, a sample of drilhng fluid is taken, it is degasified by means of a TVD instrument, and its content of hydrocarbon gases is determined. Then the gas saturation of the driUing fluid in cm^/1. is determined from the formula C_ Q' where K is amount of gas-air mixture in the gas container of the TVD degasifier in cm^; C is the concentration of combustible gases in the gas- air mixture in per cent; Q is volume of the degasified solution in cm^. At the same time as the sample is taken, the readings of the gas analyser of the station are noted. The results of the calibration are plotted on a graph with percentage gas indications as abscissae and the gas saturation i% obtained by the method described above, as ordinate (both after allowance has been made for back- ground values). By drawing a straight line through the origin of coordinates and the point obtained on the graph, we obtain the calibration curve. By making use of this curve it is possible to arrive from the gas indications in per cent, obtained in the gas logging (after allowance for background values), at the actual gas saturation of the solution. In practice it is recommended to choose a calibration curve from those drawn on Fig. 1 according to the results of comparison of gas readings with data on the actual gas saturation of the drilling fluid, obtained by the method described above. Using data on the actual gas saturation of the fluid one can make a quaU- tative estimation of the gas contained in a bed, which is marked on the gas log by high gas readings. The gas content of a bed can be expressed by the gas factor— the content of gas in 1 m^ per volume of porous space in 1 m^. The mechanism whereby hydrocarbon gases enter the drilling fluid has hitherto not been studied. Various points of view are held on this problem. Most investigators consider that gas and oil are transferred to the drilling fluid principally from the pores of rock which has been drilled through; PROBLEMS OF GAS LOGGING ESTIMATION 303 4-0 4-5 5-0 '■5 2-0 2-5 3-0 Gas readings, % Fig. 1. Calibration curves for determination of gas saturation of the drilling fluid from readings obtained in gas logging with a floating degasifier. the amount of gas entering the drilhng fluid as a result of diffusion through the walls of the well is very small and has no practical significance and in most cases infiltration does not occur since the pressure in the bed is below that of the column of drilhng fluid. In contradiction to this, E. M. Geller considers that gas enrichment of the drilhng fluid is largely caused by infiltration of oil or gas from the bed, arising as a result of a reduction of pressure in the well during rota- tion of the drill bit*. According to Geller, a favourable condition for this * E. M. Geller. On the conditions of passage of gas and ofl into clay solution of weUs during drilling. Sb. "Geokhiraicheskie metody poiskov nefti i gaza" Geochemical methods of prospecting for oil and gas 2nd ed. Gostoptekhizdat, 1954. 304 L. A. Galkin is a high rate of circulation of the drilhng fluid at rates of boring not exceed- ing 5 m/hr. However, this point of view is incorrect ; this follows if only from the fact that in regions of the Ural-Volga province productive beds of the Naryshevskian Beds and sediments of the Upper Givetian substage, despite the occurrence of the favourable conditions indicated above, are quite often not shown up by gas sampling. Neither is confirmation obtained for the possibility of creating the pressure difference— necessary for filtra- tion—between the bed and the bore-hole during rotation of the bit. Assuming that gas and oil enter the drilling fluid from the rock that has been drilled out, the following formula can be written for the gas factor a: Vnm VnTn where: m is the coefficient of porosity; Q^ is the pump capacity; V^ is the volume of rock drilled out in the time t\ Q is the consumption of drilling fluid in the time t. This formula can be used to determine the gas factor. In calculating the gas factor it is convenient to use Table 1, which has been constructed from formula (2) for the most probable values of effective porosity {m = 0.2) and well diameter (llf in.). In order to find the gas factor, using Table 1, for a selected portion of the gas log showing high gas readings, the consumption Q of drilHng fluid corresponding to the drilling time for an interval of 0.5 m is determined; against the value found for Q and the actvial gas saturation of the drilling fluid, found as described above, the gas factor is obtained from the top Une*. If the consumption of drilling fluid is kno^vn for a penetration of 0.25 or 1 m, then the same course is followed as in the previous case, but the result ob- tained is respectively multiplied or divided by two. If the bit diameter is different from llf in. (the diameter for which the table is calculated) then we obtain the gas factor by multiplying the value obtained from the table by a coefficient K, which has the following value '^. Drilling bit 73/, in. 93/4 in. IOV4 in- 133/4 in. K 0.46 0.72 0.84 1.44 * In calculations the coefficient of oil and gas saturation is taken as unity, and therefore the gas factor obtained must be multiplied by the coefficient of oil saturation. + The correction factor for a core drill is not derived: it is the same as for an ordinary drill. PROBLEMS OF GAS LOGGING ESTIMATION 305 If the actual value of the gas saturation exceeds those given in Table 1, then it must be reduced by a factor of five or ten and the result obtained must be increased by the same factor. Example —The consumption Q of drilling fluid during the time taken to penetrate 0.5 m through the bed is 40,800 1. ; the gas saturation of the drilling fluid is 21 cm^/1.; the porosity of the bed is 20 per cent; the diameter of the drill bit is llf in. It is required to find the gas factor for the bed. Because a gas saturation of 21 cm^/1. does not appear in Table 1, we reduce it ten times; for a gas saturation of 2.1 cm^/1. and the nearest value of Q to the actual consumption of 40,800 1. the gas factor will be between 10 and 15 (more precisely 12.4) ; thus the gas factor of the bed will be 124 m^/m^. Table 1 is calculated for a bed porosity of 20 per cent. If the porosity of the bed has some other value, the gas factor is calculated from the formida jna where: a is the gas factor obtained from the tables; % is the required gas factor; m is the porosity (per cent) of the rocks for the given bed. Conclusions as to the nature of the bed are reached by comparison of the gas factor obtained with values of the gas factor for loiown beds of the given and neighbouring localities. Usually gas factors below 3 m^/m^ correspond to waterbearing strata, containing dissolved hydrocarbon gases; beds with residual gas and oil saturation have gas factors roughly three to four times less than productive beds. The gas factor for gas-bearing beds is numerically equal to the bed pressure. In practice it is not rare to find cases where waterbearing beds with resid- ual gas and oil saturation have the same gas factor as productive strata. In order to avoid errors in such cases, it is necessary to know the composi- tion of the gas or the quality of the bitumen. On the basis of the foregoing it can be concluded that interpretation using the calibration curves and Table 1 permits the effect of physicochem- ical properties of the drilling fluid, and also the effect of drilling condi- tions on the results of gas logging to be excluded; it also allows a more cor- rect result for the gas saturation of the rocks which are being drilled. GAS ENRICHMENT AND DEGASIFICATION OF THE DRILLING FLUID The gas saturation of the drilling fluid during boring of a productive bed depends very much on the rate of boring. Table 2 shows rates of boring Applied geophysics 20 306 L. A. Galkin Table 1. ^'alue of gas saturation F (c 7I) OF DRILLING FLUID FOR VARIOUS GAS FACTORS (m/^] VARIOUS a3). 113/^ fact 10 15 20 25 30 40 200000 140000 100000 72000 69600 67200 64800 62400 60000 57800 55200 52800 50400 48000 45600 43200 40800 38400 36000 33600 31200 28800 26400 24000 21600 20000 19200 18800 18400 18000 17600 17200 16800 16400 16000 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.8 0.9 0.9 0.1 0.1 0.2 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.9 0.9 1.0 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.3 1.3 0.1 0.2 0.3 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1.1 1.2 1.3 1.4 1.5 1.5 1.5 1.6 1.6 1.6 1.7 1.7 1.7 0.2 0.2 0.3 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.9 0.9 0.9 1.0 1.1 1.2 1.3 1.5 1.6 1.7 1.8 1.9 1.9 1.9 1.9 2.0 2.1 2.1 2.2 0.3 0.5 0.7 0.9 1.0 1.0 1.0 1.1 1.2 1.2 1.3 1.3 1.4 1.5 1.5 1.6 1.7 1.8 1.9 2.1 2.2 2.4 2.6 2.9 3.2 3.5 3.6 3.7 3.8 3.9 3.9 4.1 4.2 4.3 4.4 0.5 0.7 1.0 1.5 1.5 1.6 1.6 1.7 1.8 1.8 1.9 1.9 2.1 2.2 2.3 2.4 2.6 2.7 2.9 3.1 3.4 3.6 3.9 4.4 4.8 5.2 5.5 5.6 5.7 5.8 5.9 6.1 6.2 6.4 6.5 0.7 0.9 1.4 1.9 2.0 2.0 2.1 2.2 2.3 I 2.4 2.5 2.6 2.8 2.9 3.1 3.2 3.4 3.6 3.9 4.2 4.5 4.8 5.3 5.8 6.5 6.9 7.3 7.4 7.6 7.8 7.8 8.1 8.3 8.5 8.7 0.9 1.2 1.7 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.3 3,5 3.6 3,8 4.1 4.3 4.5 4.9 5.2 5.6 6.1 6.6 7.3 8.1 8.7 9.1 9.3 9.5 9.7 9.9 10.2 10.4 10.6 10.9 1.0 1.2 1.4 1.5 1.8 1.9 2.0 2.4 2.8 2.9 3.4 3.9 3.0 3.5 4.0 3.1 3.6 4.2 3.2 3.7 4.3 3.4 3.9 4.5 i 3.5 4,2 4.7 3.6 4.2 4.8 3.8 4.4 5.1 3.9 4.6 5,3 4.1 4.8 5.5 4.4 5.0 5.8 4.6 5.3 6.1 4.9 5.7 6.5 5.1 5.9 6.8 5.4 6.3 7.2 5.8 6.8 7.8 6.2 7.3 8.3 6.7 7.8 8.9 7,3 8.5 9.7 7.9 9.2 10.6 8.7 10.2 11.6 9.7 11.3 12.9 10.5 12.2 13.9 10.9 12.7 14.6 11.1 13.0 14.9 11.4 13.3 15.2 11.6 13.6 15.5 11.8 13.9 16.0 12.2 14.2 16.2 12.5 14.6 16.6 12.3 14.9 17.0 13.1 15.3 17.5 PROBLEMS OF GAS LOGGING ESTIMATION 307 CONSUMPTIONS Q (1) OF DRILLING IN. BIT, 20% EFFECTIVE POROSITY FLUID DURING 0.5 m PENETRATION AND FOR G as t a c t r 45 50 55 60 65 70 75 80 85 90 95 100 1.6 1.7 1.9 2.1 2.3 2.5 2.6 2.8 2.9 3.1 3.3 3.5 2.2 2.5 2.7 2.9 3.2 3.5 3.7 3.9 4.2 4.5 4.7 4.9 3.1 3.5 3.8 4.2 4.5 4.9 5.2 5.6 5.9 6.3 6.6 6.9 4.3 4.8 5.3 5.8 6.2 6.7 7.3 7.8 8.2 8.6 9.1 9.7 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.3 8.5 9.0 9.5 10.0 4.7 5.2 5.7 6.2 6.8 7.3 7.8 8.3 8.8 9.4 9.9 10.4 4.8 5.3 5.8 6.7 6.9 7.4 7.9 8.6 9.0 9.5 10.1 10.7 5.0 5.6 6.2 6.7 7.3 7.8 8.4 8.9 9.5 10.1 10.6 11.2 5.2 5.8 6.4 7.0 7.5 8.3 8.7 9.4 9.9 10.4 11.0 11.7 5.4 6.0 6.6 7.3 7.8 8.4 9.0 9.7 10.2 10.8 11.4 12.1 5.7 6.3 6.9 7.6 8.2 8.8 9.5 10.2 10.7 11.3 11.9 12.7 5.9 6.6 7.3 7.9 8.6 9.2 9.9 10.6 11.2 11.9 12.5 13.2 6.2 6.9 7.6 8.5 8.9 9.7 10.4 11.0 11.7 12.4 13.1 13.8 6.5 7.2 7.9 8.7 9.4 10.1 10.8 11.6 12.2 12.9 13.7 14.5 6.8 7.6 8.4 9.2 9.9 10.6 11.4 12.2 12.9 13.7 14.4 15.3 7.3 8.1 8.9 9.7 10.5 11.3 12.2 12.9 13.8 14.6 15.4 16.2 7.7 8.5 9.4 10.3 11.1 11.9 12.8 13.7 14.5 15.3 16.2 17.1 8.1 9.6 9.9 10.9 11.7 12.6 13.5 14.5 15.3 16.2 17.1 18.1 8.7 9.7 10.7 11.6 12.6 13.6 14.6 15.5 16.5 17.5 18.4 19.4 9.4 10.4 11.4 12.5 13.5 14.6 15.6 16.6 17.7 18.7 19.8 20.8 10.1 11.2 12.3 13.4 14.6 15.7 16.8 17.9 19.0 20.2 21.3 22.4 10.9 12.1 13.3 14.5 15.7 16.9 18.2 19.4 20.6 21.8 22.9 24.2 11.9 13.2 14.5 15.8 17.2 18.5 19.8 21.1 22.4 23.8 25.1 26.4 13.1 14.5 15.9 17.5 18.1 20.3 21.8 23.3 24.7 26.1 27.6 29.1 14.5 16.1 17.7 19.4 20.9 22.5 24.2 25.8 27.4 28.9 30.6 32.3 16.0 17.5 19.1 20.9 22.6 24.5 25.2 27.9 29.6 31.4 33.1 34.9 16.4 18.2 19.9 21.8 23.5 25.5 27.2 29.1 30.8 32.8 34.4 36.4 16.7 18.6 20.4 22.2 24.1 26.0 27.8 29.7 31.5 33.4 35.2 37.1 17.1 18.9 20.8 22.7 24.6 26.3 28.5 30.3 32.1 34.1 35.9 37.9 17.5 19.4 21.3 23.3 25.2 27.2 29.1 31.0 32.9 34.9 36.9 38.8 17.9 19.8 21.8 23.5 25.7 27.9 29.8 31.4 33.7 35.3 37.6 39.2 18.3 20.3 22.3 24.4 26.4 28.4 30.8 32.5 34.5 36.5 38.6 40.6 18.7 20.8 22.8 24.9 26.9 29.1 31.2 33.3 35.2 37.4 38.9 41.6 19.2 21.3 23.3 25.6 27.6 29.8 31.9 34.1 36.0 38.3 40.3 42.6 19.6 21.8 23.9 26.2 28.3 30.5 32.7 34.9 37.1 39.2 41.4 43.6 308 Table 1 (continued) L. A. Galkin Q G a s f actor 1 2 3 4 5 10 15 20 25 30 35 : 40 i 15600 0.4 0.9 1.3 1.7 2.2 4.5 6.7 8.9 11.1 13.4 15.5 17.9 15200 0.5 0.9 1.4 1.8 2.3 4.6 6.9 9.2 11.5 13.8 16.0 18.4 14800 0.5 0.9 1.4 1.9 2.4 4.7 7.1 9.4 11.8 14.2 16.5 18.9 14400 0.5 0.9 1.5 1.9 2.4 4.9 7.3 9.7 12.1 14.6 16.9 19.4 14000 0.5 0.9 1.5 1.5 2.5 4.9 7.5 9.9 12.5 14.9 17.5 19.9 13600 0.5 1.0 1.5 2.1 2.6 5.1 7.7 10.3 12.9 15.4 18.1 20.5 13200 0.5 1.1 1.6 2.1 2.6 5.3 7.9 10.6 13.2 15.9 18.5 21.2 12800 0.5 1.1 1.6 2.2 2.7 5.5 8.2 10.9 13.6 16.4 19.1 21.8 12400 0.6 1.1 1.7 2.3 2.8 5.6 8.4 11.3 14.1 16.9 19.7 22.5 12000 0.6 1.2 1.7 2.3 2.9 5.8 8.7 11.6 14.6 17.5 20.4 23.3 11600 0.6 1.2 1.8 2.4 3.0 6.0 9.0 12.0 15.1 18.1 21.7 24.1 11200 0.6 1.2 1.9 2.5 3.1 6.2 9.4 12.5 15.6 18.7 21.8 24.9 10800 0.6 1.3 1.9 2.6 3.2 6.5 9.7 12.9 16.2 19.4 22.6 25.8 10400 0.7 1.3 2.0 2.7 3.4 6.7 10.1 13.4 16.8 20.2 23.5 26.9 10000 0.7 1.4 2.1 2.8 3.5 6.9 10.5 13.9 17.5 20.9 24.4 27.9 9600 0.7 1.5 2.2 2.9 3.6 7.3 10.9 14.6 18.2 21.8 25.5 29.1 9200 0.8 1.5 2.3 3.0 3.8 7.6 11.4 15.2 18.9 22.8 26.6 30.4 8800 0.8 1.6 2.4 3.2 3.9 7.9 11.9 15.9 19.8 23.8 27.8 31.7 8400 0.8 1.7 2.5 3.3 4.2 8.3 12.5 16.6 20.8 24.9 29.1 33.3 8000 0.9 1.7 2.6 3.5 4.4 8.7 13.1 17.4 21.8 26.2 30.6 34.9 7600 0.9 1.8 2.8 3.7 4.6 9.2 13.8 18.4 22.9 27.6 32.1 36.8 7200 0.9 1.9 2.9 3.9 4.9 9.7 14.6 19.4 24.3 29.1 34.0 38.8 6800 1.0 2.1 3.1 4.2 5.1 10.3 15.4 20.6 25.7 30.8 35.9 41.1 6400 1.1 2.2 3.3 4.4 5.5 10.9 16.4 21.8 27.3 32.8 38.2 43.6 6000 1.2 2.3 3.5 4.7 5.8 11.6 17.5 23.3 29.1 34.9 40.7 46.6 5600 1.2 2.5 3.7 4.9 6.2 12.5 18.7 24.9 31.2 37.4 43.7 49.9 5200 1.3 2.7 4.0 5.4 6.7 13.4 20.2 26.9 33.6 40.3 46.9 53.7 4800 1.5 2.9 4.4 5.8 7.3 14.6 21.8 29.2 36.4 43.7 50.9 58.2 4400 1.6 3.2 4.8 6.3 7.9 15.9 23.8 31.8 39.6 47.6 55.4 63.4 4000 1.7 3.5 5.2 6.9 8.7 17.5 26.2 34.9 43.7 52.4 61.1 69.8 3600 1.9 3.9 5.8 7.8 9.6 19.4 29.1 38.8 47.9 58.2 67.1 77.6 3200 2.2 4.4 6.6 8.7 10.9 21.9 32.8 43.7 54.6 65.5 76.4 87.3 2800 2.5 4.9 7.5 9.9 12.5 24.9 37.4 49.9 62.4 74.9 87.3 99.9 2400 2.9 5.8 8.7 11.7 14.6 29.1 43.7 58.2 72.8 87.3 101.9 116,6 2000 3.5 6.9 10.5 13.9 17.5 34.9 52.4 69.9 97.3 104.8 122.2 139.7 1600 4.4 8,7 13.1 17.5 21.8 43.7 65.5 87.4 109.1 131.0 152.7 174.6 1200 5.8 11.6 17.5 23.3 29.1 58.2 87.4 116.5 145.5 174.7 203.7 232.8 800 8.7 17.5 26.2 34.9 43.7 87.4 131.0 174.7 218.3 262.1 305.2 349.2 400 17.5 34.9 52.4 69.8 87.3 174.7 262.1 349.4 436.5 524.2 581.1 698.4 PROBLEMS OF GAS LOGGING ESTIMATION 309 Gas factor 45 ! 50 55 60 65 70 75 80 85 90 95 100 20.2 22.1 24.3 26.9 28.7 30.9 33.2 35.8 37.6 40.3 41.9 44.8 20.7 22.9 25.2 27.5 29.8 32.0 34.4 36.7 38.9 41.3 43.5 45.9 21.5 23.6 25.9 28.3 30.1 33.0 35.4 37.8 39.9 42.5 44.7 47.2 21.8 24.3 26.6 29.1 31.5 33.9 36.4 38.8 41.1 43.7 45.9 48.5 22.5 24.9 27.4 29.9 32.4 35.0 37.4 39.9 42.3 44.9 47.3 49.9 23.1 25.8 28.3 30.8 33.4 36.1 38.6 41.0 44.7 46.2 48.8 51.3 23.8 26.5 29.0 31.7 34.3 36.9 39.7 42.3 44.9 47.6 50.2 52.9 24.6 27.3 29.9 32.8 35.9 38.2 40.9 43.7 46.2 49.1 51.7 54.6 25.3 28.2 30.9 33.8 36.5 42.2 45.0 45.0 47.8 50.7 53.4 56.3 26.2 29.1 32.0 34.9 37.9 40.7 43.7 46.6 49.5 52.4 55.3 58.2 27.1 30.1 33.1 36.1 39.1 42.1 45.2 48.2 51.2 54.2 57.2 60.2 28.1 31.2 34.2 37.4 40.4 43.7 46.8 49.9 52.9 56.2 59.1 62.4 29.2 32.3 35.5 38.8 41.9 45.2 48.5 51.8 54.9 58.2 61.4 64.7 30.2 33.6 36.8 40.3 43.6 47.0 50.4 53.8 56.9 60.5 63.7 67.2 31.5 34.9 38.4 41.9 45.4 48.9 52.4 55.8 59.3 62.8 66.3 69.8 32.7 36.4 39.9 43.7 47.2 50.9 54.5 58.2 61.7 65.5 68.9 72.8 34.2 37.9 41.7 45.5 49.3 53.2 56.9 60.7 64.4 68.3 72.0 75.9 35.7 39.7 43.6 47.6 51.5 55.6 59.5 63.5 67.3 71.5 75.2 79.4 37.4 41.5 45.7 49.9 53.9 58.1 62.3 66.6 70.6 74.9 78.9 83.2 39.3 43.7 47.9 52.3 56.7 61.2 65.3 69.8 74.1 78.5 82.8 87.2 41.4 45.9 50.5 55.1 59.7 64.3 68.9 73.5 78.0 82.7 87.2 91.9 43.7 48.6 53.5 58.2 63.2 68.0 72.9 77.6 82.6 87.3 92.3 97.0 46.2 51.4 56.4 61.7 66.7 71.8 76.9 82.2 87.2 92.5 97.5 102.8 49.1 54.6 59.9 65.5 70.8 76.4 81.8 87.4 92.6 98.3 103.4 109.2 52.4 58.2 64.0 69.0 75.7 81.5 87.3 93.1 98.9 104.8 110.6 116.4 56.8 62.4 68.5 74.9 80.9 87.4 93.5 99.8 105.9 112.3 118.4 124.8 60.4 67.2 73.8 80.6 87.2 93.9 100.7 107.5 114.1 120.9 127.5 134.4 65.5 71.8 79.9 87.4 94.5 101.8 109.1 116.1 123.6 131.0 138.1 145.6 71.4 79.2 87.0 95.3 102.8 110.9 118.7 127.0 134.5 142.9 150.3 158.8 78.6 87.3 96.0 104.8 112.5 122.2 130.9 139.8 148.4 157.2 165.9 174.7 87.3 95.9 105.4 116.4 124.5 134.3 143.8 153.3 162.9 174.7 182.0 194.1 98.2 109.1 120.0 131.0 143.8 152.7 163.7 174.7 185.5 196.6 207.3 218.4 112.2 124.7 137.8 149.8 162.1 174.6 187.1 199.7 211.9 224.6 236.9 249.6 130.9 145.5 160.0 174.6 189.2 203.7 218.3 232.8 247.4 261.9 276.5 291.0 157.1 174.6 192.1 209.6 226.9 244.4 261.9 279.5 296.9 314.5 331.7 349.4 196.4 218.3 240.0 262.1 283.7 305.5 327.4 349.4 370.9 393.1 414.6 436.8 261.9 291.0 320.1 349.4 378.3 407.4 436.5 465.9 494.7 524.2 552.9 582.4 392.8 436.5 480.5 524.2 567.7 610.4 654.8 698.9 742.6 786.2 829.9 873.6 785.7 873.0 960.9 1048.32 1135.7 1162.2 1309.5 1397.8 1485.1 1572.5 1659.8 1747.2 310 L. A. Galkin of productive strata of the coal bearing sequence of the Tournaisian stage, of the Naryshevskian Beds and of the Upper Givetian substage of the Bash- kiriyan strata (Shkapov and Stakhanov areas). It can be seen that the pene- tration rate varies within wide hmits. Table 2. Rates of boring in productive beds in certain wells a Coal-bearing sequence Tournaisian stage Naryshevskian Beds Upper Givetian substage 6 Rate of boring (m/br) min av max min av max min av max min av max 7 9 10 13 15 18 29 31 3 4 11 15 0.8 Shkapovka area — _ 1.2 3.8 7.6 0.3 1.0 2.0 0.5 3.2 7.6 0.6 2.5 5.5 _ — — 0.3 1.4 1.2 2.5 5 1.6 7.4 14.5 1.2 4.2 7.8 1.2 5 1 4 7.4 0.6 7.5 15 1 4.1 7.5 0.5 5 2 7 13.2 1.5 6.8 13.6 0.3 3 6.2 0.5 2.5 1.2 8 14.6 1 7.5 15 0.4 5.6 11.2 0.4 1.4 0.8 3 6 2 11 22.8 1 6.2 13 — — 1.8 1.5 15 1.2 3.6 7.4 0.4 2 4.6 0.4 1.2 Stakhanovka area 0.5 4.5 10 0.5 5 10 0.4 4 10 0.4 7.6 . — — — 0.8 6.2 13.8 0.4 0.8 1.6 0.4 0.5 0.5 1.7 3.4 0.8 3.8 7.6 0.6 3.4 7.7 0.5 2.2 0.6 3.2 6.7 1 2 4 1.2 6.4 13.6 1.2 4 Average of Shkapovka and Stakhanovka areas 8.2 I 1 , I 5.2 I 10.7 I 0.6 I 3.6 I 7.8 I 0.5 3.1 2.8 11.6 11.2 5 2.8 2.4 15 1 4.5 9.4 6.8 It follows from formula (2) that the gas saturation of the drilling fluid is inversely proportional to the duration of boring and is therefore directly proportional to the rate of boring. In correspendence with the wide range of variation of the boring rate, the gas saturation of the drilling fluid can also vary within very wide limits — by ten times or more. It varies within narrower limits with pump output, which varies from 20 to 60 1/sec during the boring of productive beds. Table 3 gives calculated valvies of gas saturation of the drilling fluid for some values of the rate of boring and pump output, illustrating the consider- able variation of the gas saturation of the drilling fluid with rate of boring. It should be noted that, as experience of gas logging shows, actual values for the gas saturation of the drilling fluid often lie below the calculated PROBLEMS OF GAS LOGGING ESTIMATION 311 Table Value of gas saturation of the fluid at various rates OF boring and mud-pump capacities Rate of boring Mud -pump 1/sec Gas saturation (cni"/l.) of drillinc; fluid at gasifactor (m3/ra3) of:' m/hr 30 40 50 60 70 1 80 90 100 1 20 40 60 2.9 1.3 0.9 3.8 1.8 1.2 4.7 2.2 1.5 5.2 2.7 1.8 5.6 3.1 2.1 7.6 3.6 2.4 8.5 4.05 2.7 9.5 4.5 3.0 5 20 40 60 14.5 7.2 4.8 19.4 8.6 6.4 24.2 12 8 29.1 14.4 9.6 33.9 16.8 11.2 38.8 19.2 12.8 43.6 21.6 14.4 48.5 24.0 16.0 10 20 40 60 29.0 13.5 9 38 18 12 47.5 22.5 15 52 27 18 56.5 31.5 21 76 36 24 85.5 40.5 27 95 45 30 15 20 40 60 40.5 21 14.5 54 28 19.4 67.5 35 24.2 81 42 33.9 94.5 46 29.1 108 56 38.8 126.5 63 43.6 135 70 48.05 values given in Table 3. This is evidently explained by the driving of the gas back into the bed during the process of drilling. With a knowledge of the gas factor of the productive beds, it is possible to form a preliminary idea of the possibility of detecting them by gas logging under the drilling conditions that are used in boring productive beds. As folloAvs from Tables 2 and 3, for some regions of Bashkiria the gas saturation of the drilling fluid in exploring the productive deposits will often be less than 16 cm^/1. This is below the sensitivity possessed by a gas- logging station with a floating PG degasifier. As a result of this, a whole series of productive beds is not indicated on the gas logs. In order to record productive beds in the Bashkirian regions, the sensi- tivity of the apparatus employed must be increased between seven and ten times. This can be done by increasing the degree of degasification of the drilling fluid. In this connection the DP* degasifier, with a higher degree of degasification than the floating degasifier, was developed and tested. The DP degasifier consists of a metal cylinder, inside which is fixed a hollow drum together with a wire brush. The drum is rotated by means of an electric motor fixed above the degasifier. The latter is placed in the * The degasifier was put forward by B. V. Vladimirov and L. A. Calkin (inventor's certificate No. 90747 of 28 June 1950). 312 L. A. Galkin trough containing the drilhng fluid (as close as possible to the bore hole opening) on two floats which hold the degasifier in the same place relative to the level of the drilling fluid. Forced agitation of the drilling fluid stream and splashing of the fluid by the wire brush bring about a considerable increase in the degree of degasi- fication of the drilhng fluid and the release not only of the free, but also of the dissolved gas. Changing the rate of rotation of the rod permits some measure of variation in the degree of extraction of gas from the drilling fluid / 120 / / lie 100 / / 450 500 550 600 Reading of gas analyser, //A Fig. 2. Dependence of gas analyser readings on the voltage FsuppUed to the electric motor in the DP degasifier. by the degasifier. This is illustrated in Fig. 2, which shows the increase in gas indications when the voltage appKed to the electric motor is increased, thus producing an increase in speed. In this case the gas saturation of the fluid remained constant, this being controlled by means of a floating degasi- fier. Changing the degree of degasification by adjusting the speed of the brush is of great value because it allows the amount of degasification to be suit- ably selected for the given geological conditions (different gas factors and reservoir properties of the rocks) and the drilling fluid used. We will examine the results of gas logging by means of the DP degasifier, conducted in order to show the possibility of using gas logging in detect- ing strata with a low gas factor (15-20 m^/m^). PROBLEMS OF GAS LOGGING ESTIMATION 3B Fig. 3. Results of gas logging by means of DP and PG degasifiers (total content of hydrocarbon gases is given). Case i— Fig, 3 shows the results of gas logging with DP and PG degasi- fiers against productive Devonian beds in one of the wells. According to the curve of total gas content, obtained with the PG degasi- fier, the whole interval is characterized by low gas indications (background values). When using the DP gasifier, the bed D^ is marked by readings of 100-450 iiA, which corresponds to a drilhng fluid content of 2.6-13.5 cm^/1. The gas factor for this case is 3-15 m^/m^. According to available data the gas factor of the bed Dj^, when it is oil-bearing, is equal to 35^5 m^/ra^, and consequently, in the given case the bed can be described as water- bearing, containing residual petroleum. The bed D^ on the gas log obtained with the DP degasifier is shown by individual peaks, in quite good agreement with the mechanical log and the deviations of the curve PS. Even the thin layers, separated by sections of clay, are distinguished. Case 2 {Fig. 4)— A well was developed in the Tournaisian deposits. On the total gas content curve, obtained when using the DP degasifier, a series of intervals with a high content of combustibles can be detected. At the same time on the corresponding curve obtained by using the floating PG degasi- fier, only small increases are shown against these intervals. The whole curve is very smooth. 314 L. A. Galkin Fig. 4. Results of gas logging by means of DP and PG degasifiers (continuous curve- total hydrocarbon gases; broken curve-total heavy hydrocarbons). Content of individual hydrocarbon gases is given from the results of an analysis on a GST-2 instrument. The intervals of increased gas content (1257-1285 m) detected by means of the DP degasifier can be described as water -bound and oil-bearing, since the gas factor for them is 5-10 m^/m^, while the oil beds of the coal-bearing sequence and the overlying Tournaisian limestones are characterized by a gas factor of 15-30 m^/m^. The intervals (1257-1260 m and 1273-1283 m) v/ere tested and a flow of water was obtained. A core sample taken from a depth of 1258-1259.8 m consists of quartz sandstone, evenly permeated with oil. The interval with high gas readings at a depth of 1293-1303 m, situated 20 m from the top of the Tournaisian stage, has a gas factor of 15-20 m^/m^. This provides a basis for assuming that the given interval is oil-bearing, which is also supported by analysis of the gas composition, which differs markedly from the gas composition in the intervals mentioned above*. The following conclusions may be reached on the basis of the foregoing: (1) the DP degasifier possesses a considerably higher degree of degasifi- cation than the floating degasifier; (2) in working with the DP degasifier it is possible to detect oil-bearing strata which have a small gas factor and are often missed when working with a floating degasifier. * Separate analysis was conducted on a GST-2 instrument (see B. V. Vladimirov, The GST-2 gas analyser for determining gas composition during gas logging. Neft. Khoz. No. 8, (1956). Chapter 13 LUMINESCENCE LOGGING T. V. Shcherbakova Bitumens, including petroleums and petroleum products, possess the abil- ity to luminesce. This property of bitumens is used to determine the content of petroleum in rock samples and the drilling fluid. Systematic luminescence analysis of the drilling fluid during ^\ell drilling, carried out with the object of singling out oil-bearing beds penetrated by the well, is called lumines- cence logging. At the present time, luminescence logging is carried out by taking samples of the drilling fluid every 1-3 m of penetration and viewing them in ultra- violet light. For this a luminoscope is used, which consists of a light-tight chamber equipped with a source of ultra-violet light (UFO-2 or PARK-4 lamp with a filter passing only idtra -violet rays). The sample of drilling fluid, which is sometimes specially treated, is placed in the ultra-violet light, while the luminescent glow is observed through a window in the chamber (the viewing window). The luminescence is arbitrarily characterized by some relative quantity, for exanrple by the fraction of the svu-face area of the sample occupied by luminescent spots. Together with this, during the examination of the drilling fluid samples, the colour of the luminescent radiation is de- termined. Curves, showing the variation in luminescence capacity of the drilling fluid with variation in depth of the well drilling, are constructed from the data obtained, and constitute a luminescence log. These curves, to greater or lesser extent, represent the content of oil in beds penetrated by the well. The main drawback of luminescence logging is the subjectiveness of the evaluation of the intensity and colour of the drilling fluid luminescence, which precludes the possibility of a reliable determination of the content of oil in the drilling fluid from the data obtained in luminescence logging. Moreover, the method employed for observing the luminescence makes automatization of luminescence logging difficult. In this connection, work was carried out to elucidate the possibility of evaluating luminescence objectively and, on the basis of this, to determine from luminescence analysis data the oil content in a drilling fluid. 3] 5 316 T. V. Shcherbakova APPARATUS FOR OBSERVING LUMINESCENT RADIATION Observations on the intensity and spectral constitution of the lumines- cent radiation from the driUing fluid were conducted on an apparatus devel- oped by N. 0. Chechik. The apparatus consists of a measuring and a power unit. The measuring unit consists of a source of ultra-violet radiation and a photosensitive element placed in a light-tight chamber. A UFO^A lamp with a UFS-3 filter, transmitting rays with wavelengths less than 420 m/^ (maximum about 365 mju), is used as the source of ultra violet radiation. In order to ensure constant illumination, the supply to the lamp is stabilized by a baretter. ^' r- Fig. 1. Optical scheme of photoelectric luminoscope. 1 — source of ultra-violet radiation UF0-4A; 2 — filter UFS-2; 3 — vessel contaiaing fluid; 4 — narrow-band filter; 5 — ^prism; 6 — filter; 7 — ^photomultipUer. The sample of hquid under investigation is placed in a quartz cell which is put into the chamber so that its bottom is irradiated by ultra-violet rays. The luminescent emission thus excited is detected by the photosensitive element, for which an FEU-19 photomultipher is used (Fig. 1). The photo- multiplier cathode is protected from the ultra-violet radiation by a yellow- green filter, which stops rays with wavelengths less than 420 m^ and trans- mits radiation in the visible region of the spectrum. The radiation received by the photomultipher is converted into an electric current, which is amplified by a direct-current amplifier (Fig. 2). The direct -current amplifier consists of a bridge, two arms of which are the internal resistances of the anode circuit of the valve L^ and the resis- tances i?oK — i?oq and the two other arms are the anode resistances R "-39 27 and i?28- The current from the output of the photomultipher varies the LUMINESCENCE LOGGING 317 H --r f^" ! 318 T. V. Shcherbakova voltage on the grid of one of the triodes. A 300 [aK microammeter (MKA) is connected across the measuring diagonal of the bridge. The switch P per- mits selection of one of three values for the amplification factor of the ampli- fier — 1100, 5500 and 27,500 (positions 1,2 and 3 respectively). The power unit contains sources of supply for the photomultiplier and the amplifier. A voltage of 725 V, obtained from a twin rectifier assembled about the two valves L^ and L^ (type 2Ts2S), is used for the photomultiplier supply. The voltage is stabilized by an electronic stabilizer assembled about the 6P6S valve L-^^- The comparatively low supply voltage for the photo- multiplier is chosen with the object of obtaining a low value for the intensity of the dark current (of the order 10~^°A). The supply to the anode circuits of the direct -current amplifier is obtained from a rectifier, assembled as a full-wave rectification circuit about the 5Ts4S valve L^. Just before the measurement, with a completely darkened photomultiplier, the current intensity in the anode circuit of the right-hand half of the 6N8S valve is adjusted by means of the rheostats iJgg and R^^ so that the measur- ing instrument indicates zero. If light falls on the cathode of the photo- multiplier, the balance of the bridge is upset and a current begins to flow through the MKA instrument. The instrument readings /„ will be propor- tional to the intensity of the luminescent radiation reaching the cathode of the photomultiplier. In order to ensure that it is possible to compare the readings obtained at different times for different settings of the instrument, with the readings /„ due to luminescent radiation from the sample of petroleum under inves- tigation, the readings /^ from a standard are determined. The standard used is a mixture of luminophors giving out a bluish -white light. The ratio — ^ is taken as the quantity defining the relative intensity of the luminescent radiation. By putting into the light path a filter transmitting a narrow region of the visible spectrum (in the range 420-640 m/^), it is possible to determine the intensity of the radiation in any part of the spectrum. Thus, by using a green filter, it is possible to measure the relative intensity —of the green light {X = 530 vafi) in the luininescent radiation. By means of a set of filters the complete spectral characteristics of the radiation (the dependence of intensity of the radiation on wave length) from the sample under investigation can be obtained. When using light filters, tlie absorption coefficient of the filter must be LUMINESCENCE LOGGING 319 allowed for; the value of / obtained from measurements must be multiplied by the absorption coefficient K. The coefficients K of the filters were determined by comparison of four petroleum samples with the results of direct luminescence measurements conducted in the spectral laboratory of the All-Union Scientific Research Institute for Petroleum and Geological Survey under the direction of A. A. Il'ina. In testing the apparatus described above, the following data were ob- tained : (a) a threshold sensitivity of the order 1.3 xlO"-'^*' lumen; (b) a mean error of 2.5 per cent in ten measurements, the greatest being 8 per cent; (c) radiation intensity, as determined by the apparatus, proportional to the actual intensity of the luminescent radiation. The last of these Avas established by comparing the readings of the instru- ment with the resvdts of luminescent radiation intensity determinations made in the laboratory with a visual photometer (see Table). Petroleum samples Relative intensity of green light, foimd on instrument Intensity of green light, found with visual photometer 0.31 0.78 1.04 1.2 1.82 2.6 0.9 3.0 4.3 5.3 8.0 11.5 Luminescence of Petroleum and Drilling Fluids. Presented below are the results of an investigation of the luminescence from petroleum, solutions of petroleum in chloroform, and drilling fluid containing petroleum, using the apparatus described above. Figure 3 gives the results of measurements of the luminescent emission from samples of various petroleums. It can be seen that the lower the specific gravity of the petroleum, and consequently the higher the gum content, the greater is the intensity of the luminescence. With an increase in specific gravity and reduction in gum content, the maximum of the luminescence spectrum is displaced towards longer wavelengths, i. e. towards the yellow region of the spectrum. These regularities are well known from the practice of luminescense — bitumino- logical analysis of petroleum. 320 T. V. Shcherbakova 400 440 480 520 560 600 Fig. 3. Luminescence spectra of petroleums of different specific gravity: 1 — 0.797; 2—0.801; 3—0.813; 4—0.820; 5—0.831; 6—0.853; 7—0.861. Figure 4 shows the results of determining the luminescence of solutions of petroleum in chloroform. As can be seen, the luminescence spectrum of petroleum changes Httle with addition of chloroform, but the intensity of the radiation increases considerably. The intensity of the green radiation changes in proportion to the concentration of petroleum in the chloroform up to one hundredth of one per cent (Fig. 5). With further increase of the amount of petroleum m the chloroform the intensity of the luminescence changes little and even begins to decrease (region of concentration extinction [1, 3, 6]). Figure 6 shows the dependence of instrumental readings (in ^A) on the LUMINESCENCE LOGGING 321 content of different petroleums in a clay suspension. The clay suspension was of such viscosity that oil drops did not move about in it. The petroleum was carefully mixed with the clay suspension until a stable emulsion was 600 Fig. 4. Luminescence spectra. 1 — solution of petroleum in chloroform with a concen- tration of 8 X 10"^ % ; 2 — the same with a concentration of 2 X 10"^ % ; 3 — undUuted petroleum. formed, this being checked visually under ultra-violet light. Suspensions were prepared from two clays (light and dark) and from petroleum with different intensities of luminescence. -" 1-0 0-1 ' ^ X y^ 10 10"" IQ-S IQ-^ II C, % Fig. 5. Intensity of luminescence of a solution of petroleum in chloroform as a function of concentration. Applied geophysics 21 322 T. V. Shcherbakova It can be seen that, by means of the apparatus described above, it is possible to detect petroleum in the clay suspension down to a content of 0.5-1 cm^ per litre, i.e. approximately as much as can be detected by visual examination of clay suspension samples in ultra-violet light in the lumi- noscope. 150 iQG ^ 50 2 / ^--•'■■4 10 20 30 C. cm^/L Fig. 6. Intensity of luminescence of petroleum in drilling fluid as a fmiction of concentration, 1 — petroleum with intensity Iqll^ = 30.0, dark clay suspension; 2 — ^petroleum with intensity Iqll^ = 2.62, same suspension; 3 — ^petroleum with inten- sity IgjI^Q = 1.74, light clay suspension; 4 — the same petroleum, dark clay suspension. For the concenlrations of petroleum in clay suspensions which we have examined, the intensity of the luminescent emission is proportional to the amount of petroleum and no concentration extinction has been noticed. The coefficient of proportionality depends strongly on the nature of the petroleum and the clay suspension. However, it can be expected that, for a given type of clay suspension and petroleum, it is possible to obtain a quan- titative estimation of the amount of petroleum in the clay suspension from the instrument readings. SPECTRAL CHARACTERISTICS OF PETROLEUM AND PETROLEUiM PRODUCTS In luminescence logging it becomes necessary to differentiate between the luminescence of explored petroleum and. the luminescence of petroleum products that have entered the drilling fluid in the process of drilling. It is LUMINESCENCE LOGGING 323 proposed that this can he done by studying the spectral characteristics of the luminescent radiation. In order to check this, the spectral characteris- tics of petroleum and petroleum products were determined and are shown in Figs. 7 and 8. As can he seen, the principal petroleum products, which can enter the drilling fluid (kd^ricating materials — motor oil, solidol, machine oil, graph- ite grease), have a sharp maximum in the left-hand part of the spectral characteristic (wavelength about 480 m^), which differs markedly from the spectral luminescence characteristics of heavy petroleums. 'O / WOODS HOLE, MASS. 600 Fig. 7. Luminescence spectra. 1 — motor oil No. 13; 2 — motor oil No. 6; 3 — macliine oil; 4- — solidol; 5 — graphite grease; 6 — fuel oil No. 350, Figure 8 shows the spectral characteristics of luminescence from a clay suspension, to which petroleum and lubricating materials had been added. It can be seen that the spectral characteristics of the luminescence from the clay suspension correspond to the spectral characteristics of bitumen contained in the clay suspension. When motor oil is added to the clay suspension, the maximum shifts to the left and, as the concentration of motor oil is increased, approaches more and more closely to the motor oil characteristic. Thus, in the case when the explored petroleum is heavy and contains a fair amount of resinous components, it can be distinguished from lubri- 324 T. V. Shcherbakova eating materials which have entered the drilhng fluid from the surface. To this end it is sufficient in many cases to conduct observations with a filter, which absorbs the blue and transmits the yellow region of the spectrum, i.e. to determine mainly the luminescence of only the resinous components 3-2 3-0 2-6 2-4 2-2 2-0 0-8 0-6 0-2 400 1 -■ •«x — V 480 A, m/z 560 Tig. 8. Luminescence spectra. 1 — drilling fluid; 2 — drilling fluid with 10% petro- leum; 3 — drilling fluid with 10% petroleum and 5% motor oil No. 6; 4. — drilling fluid with 10% petroleum and 25% motor oil No. 6; 5— drilling fluid with 10% petroleum and 50% motor oil; 6 — petroleum. Continuous lines — Tuimazin petroleum. Broken lines — Kued petroleum. LUMINESCENCE LOGGING 325 of the petroleum. However, it is difficult to distinguish on spectral character- istics between the explored petroleum, containing a large quantity of light fractions or lubricating oil components, and petroleum products which have entered the drilling fluid. Figure 9 shows results of luminescence analysis of drilling fluid sample:^ taken while drilling one of the wells of the Aleksandrov area (Bashkiria). 05 440 600 Fig. 520 9. Luminescence spectra. 1 and 2 — drilling fluid with addition of graphite grease; 3 — drilling fluid with petroleum; 4 — oil-bearing sandstone; 5 — drilling fluid. Samples of fluid, taken while drilhng an interval of 1600-1730 m, give fairly intense luminescence; the spectral characteristics 1 and 2 have a maximum about 480 m/ii, i.e. at the same place as for lubricating materials. This indi- cates that the luminescence of the given samples of drilling fluid is due to lubricating materials which have fallen in. The spectral characteristic of drilling fluid sample taken at a depth of drilling greater than 1750 m has a maximvim in the region of 540 nijLi, ^vhich corresponds to the maximum in the spectral characteristic of petroleum in the bed. This indicates the discovery of a petroleum-bearing bed. A core sample taken from this depth proved to be oil-bearing sandstone. The spectral characteristic of this sand- stone is the same as that of the drilling fluid, sampled during the drilling of this interval. 326 T. V. Shcherbakova VIEWING WINDOW To provide for continuous observation of the luminescence of the drilUng fluid coming from the well, the apparatus must be equipped with a viewing window immersed in the drilling fluid. To avoid distortion of the results of the observations it is essential that the petroleum from the drilling fluid should not stick to the viewing window and remain on it. Thus, the surface of the viemng window must be hydrophylic. Trials of various materials showed that it is best to use glass* with a de- greased surface for the viewing window. The drilling fluid easily washes oil drops from the surface, and the petroleum does not leave traces on it. This was checked by the following method. The usual clay suspension ^vas poured into a narrow bath 1 m long; in one part this "was replaced by a fluid containing petroleum, the boundary between the ordinary clay suspension and the clay suspension containing petroleum being made as sharp as possible. Observations were conducted with the aid of an optical arrangement containing a suorce of ultra-violet light and a photoelement ; optical contact with the clay suspension was achieved by means of a window, covered in one case by glass and in another by Plexiglas, iQO 90 80 70 < 60 50 ^ 40 30 20 10 O I , 25 100 L, cm Fig. 10. Record of the luminescence intensity when glass is moved through drilling fluid. 1 — drilling fluid; 2 — drilling fluid with petroleum. I — measurement with de- greased glass; II — measurement with organic glass. * The glass partially absorhs ultra-violet rays, and this leads to a reduction in the sensitivity of the apparatus. However, this does not create any fundamental difficulty in the operation of the apparatus. LUMINESCENCE LOGGING 327 Observation consisted in immersing the window in the clay suspension and slowly moving the whole apparatus along the bath; every 5 to 10 cm the instrument was stopped and the intensity of the luminescence was measured. Values of intensity were obtained as a result of the measurements (Fig. 10). It can be seen that, in going from the clay suspension containing petroleum to the ordinary suspension, the petroleum ceases being detected at a distance of roughly 10 cm; with the Plexiglas window, on further displacement of the apparatus, the petroleum is washed from the window very poorly. CONCLUSIONS 1. An apparatus for studying the luminescence of samples of drilling fluid allows petroleum present in the fluid to be detected at concentrations down to 0.5 cm^/1. Thus device may serve as a basis for automatizing tlie process of lumines- cence logging according to the drilling fluid. 2. Study of the spectral characteristics of the luminescence makes it possible in most cases to distinguish between the luminescence arising from the petroleum in the bed and the luminescence arising from lubricating materials that have fallen into the drilling fluid. REFERENCES 1. A. A. Il'ina, Study of dispersed and concentrated types of bitumens by spectral and chemical methods. Report on topic No. 50. Documents VNIGNI (1954). 2. M. Kh. Kleinaian, Distinguishing petroleum from oils and resins and determining the content of bitumen in a core sample and in drilling fluid. Razvedka nedr. No. 6, (1941). 3. M. A. Konstantinova-Shlezinger, Luminescence analysis. Izd. Akad. Nauk SSSR. (1948). 4. P. A. Levshunov, Oil and gas logging. Exploratory and production geophysics, Ed. 2. Gostoptekhizdat (1950). 5. V. N. Florovskaya, An introduction to luminescence bituminology. Gostop- tekhizdat (1946). 6. V. N. Florovskaya, Luminescent-bituminological method of studying and prospecting for petroleum deposits. Gostojitekhizdat (1954). 7. F. M. Efendiev and E. I. Mamedov, Spectroscopic investigation of the luminescence of petroleum. Izd. Akad. Nauk Azerbaidzhan SSR, No. 1, (1952). Chapter 14 OPTICAL METHODS OF BORE-HOLE INVESTIGATION T. V. Shcherbakova In order to determine the sequence of stratification and the nature of beds penetrated by a well use is currently made of geophysical methods, of bore hole investigation, the chief of which are electric and radioactivity logging. However, in many cases the data from electric and radioactivity logging do not give a sufficiently complete representation of the rocks penetrated by the well, either because of unfavourable conditions of logging (for example, strongly saline clay suspensions) or because certain lithological and petro- graphical features of the beds, important for geologists, are not indicated by electric and radioactivity logging. In this connection it is necessary to work out new methods for studying the geological sections of wells. The so-called optical methods of photographing the rocks along the well walls, and possibly also direct examination by means of television deserve much attention as possible methods of studying the lithological and petro- graphical nature of beds penetrated by a bore-hole. Some problems connected with the photography of rocks along the wall of a well are reviewed below, and the results of Avork carried out along these lines are presented. OPTICAL PROPERTIES OF DRILLING FLUID The bore -hole is filled with a liquid (the drilling fluid), which may be clay suspension, very muddy water or, in extremely rare cases, clear water. When the wall of a well is being photographed there will be between it and the objective lens of the camera a layer of drilling fluid which impairs visi- bility. Therefore in examining the problem of photography in a well it is first of all necessary to find tlie effects of the layer of clay suspension or muddy water on the image of the well wall. For this it is necessary to know certain optical properties of clay suspensions. Clay suspensions used in drilling consist of solid mineral particles sus- pended in water. According to the degree of division of the particles, clay suspensions can be referred to as polydisperse solutions containing particles 328 OPTICAL METHODS OF BORE-HOLE INVESTIGATION 329 of different dimensions — from fractions of a micron to tenths of a millimetre witli particles from 0.1 to 10 m/< being most freqnently encountered. Clay suspensions are classed as very turbid media. When light passes through such a medium it is \\eakencd, mainly because of scattering and to a negligible extent because of absorption. In this connection the attenuation of a light iDeam by the clay suspension layers of various thickness was deter- mined. Measurements were conducted on an CF-4 specrophotometer and consisted of the following. An empty cylinder was placed in the path of the light beam coming from the monochromator and the light intensity was noted. Then in the path of the beam, in place of the empty cylinder, was placed a cylinder with a layer of clay suspension of the required thickness, and the width of the slit limiting the light beam was adjusted so that the light intensity Avas the same as in the first case. The ratio of the slit width iii the absence of the clay suspension to the slit width in the presence of the layer of suspension, equal to the ratio of the strength of the light beam in air to the light beam which has passed through the suspension, gives the quantity T called tlie transmittance (a quantity the inverse of attenuation). The transmit - tance is usually expressed in per cent. To obtain a layer of clay suspension of the necessary thickness (from 0.03 to 1 mm) cylinders from 4.03 to 5 mm in length were chosen. Part of the cylinder was filled with a quartz plug of length 4 mm and the rest with clay suspension forming the thin layer. Measurements were carried out with reference to an incandescent lamp used in the spectrophotometer, and possessing wavelengths from 320 to 1100 m/Lt. The wavelength ■was determined from the position of the mono- chromator prism. To improve the monochromatism there were fitted filters of UFS glass for wavelengths of 300-400 ra/j,, and of OS-19 glass for wavelengths of 600-1100 m//. Clay suspensions of different specific gravity {y — 1.2 — 1.02), obtained by means of diluting the initial clay suspensions, were subjected to the investigation. The investigations ^vere mainly carried out with clay suspensions prepared from dark Buguruslanskian clay Avith the addition of 1 per cent NaoCOg*. Data for this svispension are: viscosity 30 sec; specific gravity 1.2; water yield 9 cm^; thickness of the crust 3 mm. The fractional composition of the clay suspension is given in Table 1. * The sample of clay suspension was given by the clay suspensions laboratory of the- Gubkin Moscow Petroleum Institute. 330 T. V. SHCHERBA.KOVA Table 1. Particle composition of clay suspension in per cent Particle diameter (/u) Determination 1 1—2 2—3 3^4 4—5 5—6 Above 7 Sum First Second 50 49 26 24 7.5 8 4.5 3 3 5 4 2 6 8 100 100 Average 49.5 25 7.25 4,75 3.5 2.5 7 100 In addition, investigations were carried out with clay suspensions prepared from light clay (kaolin). Measurements were carried out with a light beam intensity equal to 2.5 Ivimens; for samples of clay suspensions with y = 1.013, 1.026 and 1.045 measurements were made with different light intensities. Variation of the light intensity was achieved by interposing a photographic film ^vith different degrees of blackening between the light source and the sample. The attenuation coefficient of the film had been determined beforehand. From the results of the measurements curves of the value of the transmit - tance T as a function of the thickness h of the layer of clay suspension were constructed for various wavelengths A of light (Fig. 1). 1- 100 7-=l-02 ^ 1-04 ^=1-06 ;k=I-IO ^\ . s \ k y \, s kN V/n ^1 ^ -f- fo s: <?- b"> ^ y \^^ ^ N' s v^,^ \ N N — k \^o \ s, N N ^o K \, ' \ N ^ K N \l N ^ N ^ s <j k N — y \ i V c^ i!\\^ Vo 1 *t 1 — — fe 11 u 1-0 0-1 0-3 0-5 0-7 0-9 l-l 1-3 1-5 1-7 0-1 0-3 0-5 0-1 0-3 h. mm h, mm h. mm Fig. 1. Dependence of attenuation of light on specific gravity y of the suspi thickness h of the layer and wavelength ?i of light. 1 h- 1 , 1 ' — 1 R 1 Loo- CsoO- poo ;■; 1 : V 0-1 0-3 h, mm ;nsion, It can be seen that clay suspensions are indeed very turbid media and even in thin layers transmit very little light. Thus, even when the thickness of the layer of clay suspension is only some tenths of a millimetre the trans- mittance of the light is only about 1 per cent. OPTICAL METHODS OF BORE-HOLE LNVESTIGATION 331 The value of the transiuiLtancc T falls sharply as the specific gravity of the drilling fluid increases, and becomes a little lower as the wavelength is reduced. For not very large thicknesses of the layer of fluid the connection between the transmittance T and the thickness of the layer h of the <la\ suspension can be expressed approximatelv by the formula T = 100 -Ml (1) where k is a certain coefficient., called the attenuation coetficient, rejnesenting the rate of reduction of light transmission wdth increase in layer thickness. For monochromatic light the attenuation coefficient (see Fig. 2) is: A=.(y^l) (2) 200 150 lOO 50 V / / ^^ y^ > j0^ ^ lO 1-05 MO ■y 1-20 Fig. 2. Dependence of attenuation coefficient K. on the specific gravity y of the suspension and the wavelength }. of the hght. For the dark-coloured clay suspensions investigated a = 1.4 and n — 1.5. A suspeitsion prepared from the light clay transmits light better than a suspension from the dark clay, the average difference being 50 per cent (see Fig. 3). The difference in the transmission of light by suspensions 332 T. V. Shcherbakova (a) A=500m// \ \\ \\ w \ i \ \ 2l \ \\ V k s \ ^ ^ ^-^ _ 40 30 20 15 10 (b) h =0*1 mm ^^ / ^ / / 1 y - 8 / ^^,00"^ / ^^^ 6 / ^ — ' ^y^ 4 y^ 2 y^ 2 1-5 1-0 // / 300 500 \ 700 900 IIOO Fig. 3. Dependence of attenuation of light, for light and dark driUing fluid, on the thickness h of the layer of fluid and the wavelength A of the light. prepared from light and dark clays is apparently associated with the fact that the particles of the two clays have different reflection coefficients. The values for the light transmittance shown in Figs. 1-3 were obtained for the case when the angle 99 between the principal optical axis of the photometer and the direction of the light beam was equal to zero. Scattering of light at angles 99 differing from zero was investigated by Timofeeva on particles from 3 to 20 /^ in milk solutions of low concentration and in rosin suspensions (^). Figure 4 shows the curves obtained by Timofeeva for the ratio of the intensity / of the light passing through the layer of suspension to the intensity I^ of the light in air (i.e. values of the transmittance of the hght) as a function of the thickness of the layer of suspension for various 9?. As can be seen, for 99 = and at not very large layer thickness h the value of the transmittance decreases according to an exponential law, i.e. just as in the analogous case for a clay suspension. At angles 99 differing from zero the value of the transmittance varies according to another law. In this case for small h values of ///q are considerably less than for 99 = 0; the curve has a maximum which is particularly sharply defined for 99 = 30 — 60°. It should be noted that in clay suspensions attenuation of the light with change of wavelength is observed to a much smaller extent than in the usual turbid media with small particle dimensions, in which attenuation according to the Rayleigh law is inversely proportional to the fourth power of the wavelength of the light (*>. Thus, in a clay suspension no significant OPTICAL METHODS OF BORE-HOLE INVESTIGATION 333 improvement in visibility is observed in the infra-red region, such as takes place, for example, in fogs. This has already been theoretically predicted by Shuleikin and Ambartsumyan (^' "^K Fig. 4. Dependence of relative intensity of light passing through a suspension, on the thickness h of the layer of suspension and on the angle g). Together with attenuation of light in a turbid medium the phenomenon called loss of contrast, which consists of variation of the ratio between the brightness of parts of the object examined is also observed. Loss of contrast is characterized by a coefficient K which is the relative reduction in the ratio of the brightness of the white and dark backgrounds in the presence of a. turbid medium. K Brl' B , 100% (3) 334 T. V. Shcherbakova where B^ and B^ are the vahies of the brightness of the white and dark backgrounds in air; B^. and B^, are the values of the brightness of the white and dark oroimrls in a turbid medium. 100 K 10 )/-= I- 01 y- 1-02 ^=1-07 1-20 h, cm Fig, 5. Dependence of loss of contrast on the thickness of the layer of suspension, the specific gravity y of the suspension, and the wavelength X of the light. To determine the reduction in degree of contrast in drilhng fluid, coefficients- of loss of contrast were determined for layers of clay suspension of various thickness. Determinations were made with an FM-2 photometer in the following manner. With the aid of the photometer the brightness of the white ground (a sheet of white paper) B^^, and B^ in the presence of a layer of clay suspension and without it was compared in turn with the brightness of the dark ground (black paper) B^. and 5^ with and without a layer of clay suspension in front of it. In each of these cases the brightness of the white and dark grounds was obtained in units of light flux from a standard, for which a baryta disk (reflection coefficient 91 per cent) was used. OPTICAL METHODS OF BORE -HOLE INVESTIGATION •'•■mimm: 33S Fig. 6. Photographs of rock samples. 1 — sandstone in ordinary (left) and infra-red (right) light; 2 — dolomite (same conditions of photography); 3 — sandstone without magnification (left) and ten -fold magnification; 4 — sample of dolomite with lateral illumination (left) and direct illumination (right). 336 T. V. Shcherbakova The coefficient of loss of contrast was determined from formula (3). Determination of the coefficient of loss of contrast was conducted with the same samples of clay suspension with which the transmission of light was determined; the layer of clay suspension was obtained in the same way. Measurements were made with monochromatic light of wavelength varying within the limits of 430 to 730 m//. Curves shomng the dependence of the coefficient of loss of contrast K on the thickness of the layer of suspension h for various wavelengths A of light were constructed from the results of the measurements (Fig. 5). As can be seen from Fig. 5, the contrast falls sharply v,dth increase in thickness of the layer and in specific gravity of the clay suspension. The contrast also decreases, although comparatively slightly, "with decrease in the wavelength of light. In practice variation of contrast with variation of wavelength of the light can be neglected over the visible region of the spectrum. Curves of the dependence of K on h, when both quantities are plotted on logarithmic scales, are quite close to straight lines and, consequently, the dependence of K on the thickness of the layer h can be represented in the following manner: K = Ch-"" (4) The coefficient m is approximately constant and equal to 0.54; the coefficient C depends on the specific gravity of the clay suspension samples C=-7-^ (5) {y-l)P For the clay suspensions, on which the investigations were carried out, p = 0.7 and D = 2.5. For clear reproduction of an object it is most important in photography to convey its contrasts. However, in the passage of light rays through the layer of clay suspension a reduction of contrast is observed. Loss of contrast is the main factor obstructing the possibility of photography. Therefore in examining the possibility of photography in a clay suspension it is first of all necessary to get an idea of the loss of contrast. Starting from the data obtained, one must determine the permissible thickness of the layer of drilling fluid between the viewing window of the camera and the wall of the well, and the permissible turbidity of the drilling fluid (Table 2). The main factor in the sharpness of the image is the resolving power of the system, which specifies the number of lines in 1 mm that can be distinguished on the photographic film. OPTICAL METHODS OF BORE-HOLE INVESTIGATION 337 In the case of photography of rocks in a well the resolving power must be such that it is possible to discern the grain of sandstones having grain dimensions of 0.2 mm; for this it is obviously necessary to have a resolving power of 5 lines per mm. It is well known (^) that resolving power is equal to N = Nr, where if is the coefficient of reduction (loss) of contrast ; -/Vmax ^^ ^^^^ resolving power for K = 1. We obtain the permissible coefficient of loss of contrast K by putting into the formula N = 5 and N^g^^ = 30 lines per mm (the most probable value of A^jnax ^^^ the Industar 22 objective). Hence we obtain K = 0.05 = 5% It is obvious that the permissible thickness of the layer of drilling fluid Amax between the viewing window of the well camera and the wall of the well is equal to that at which the coefficient of loss of contrast has a value less than K = 0.05. The value of Amax ^^^ ^^ obtained from Fig. 5. Table 2. Specific gravity of Turbidity Maximum suspension % thickness (mm) 1.01 1.6 27 1.025 4 13 1.05 8 5 1.07 11 2.5 1.15 20 0.8 1.2 32 0.6 In evaluating the properties of clay suspensions it is best to use, instead of the specific gravity, the content of solid particles in the solution, expressed in weight per cent. This quantity 7], which may be called the turbidity of the suspension, is connected with the specific gravity of the drilling fluid by the relationship V d-1 100%, (6) where d is the mineralogical density, which can be taken as equal to 2.65. As can be seen, even a thin layer or film of clay suspension excludes the possibility of photographing the bore-hole walls. Consequently, for photo - Applied geophysics 22 338 T. V. Shcherbakova graphy in a well, the clay suspension or turbid drilling fluid must be re- placed by clear water. In well cameras the viewing window is put as close as possible to the wall of the well, but for one reason or another (constructional shortcomings of the instrument, unevenness of the walls of the well) the distance from the viewing window to the rock in many cases will be of the order 10-20 mm. Hence it follows that, to provide for photography of the well, the liquid filling it must have a turbidity not greater than 4 per cent. If the well is filled with ordinary drilling fluid of specific gravity 1.2, then before photo- graphy this clay suspension must be diluted to more than ten times with clean water*. In photography the necessity to have the barrel of the well filled with comparatively clear water is a real disadvantage of the method, but this does not exclude its application where it is necessary to study the geological section of a well. SOME PROBLEMS OF PHOTOGRAPHING ROCK SAMPLES In order to investigate some problems connected with photographing rocks along the walls of a well, photography of some samples was carried out at the surface. 1. Light of different spectral composition has different absorptive and reflective capacity. In connection with this photographs of rock samples will differ somewhat, depending on the light in which the photography takes place. To find out the difference in the nature of photographs taken in different light, photographs of rock samples were taken in ordinary light, infra-red and ultra-violet rays. Samples of sandstones, clays, siltstones, gypsum, dolomite and anhydrite were photographed; some of the samples which were photographed in ultra- violet illumination contained petroleum. The source of infra-red rays in photographing the rock samples was an ordinary 500 W incandescent lamp with a dark -red filter. The films used in this case were two types, produced at the Scientific Research Institute for Cinematography and Photography, of infra-red film sensitive respectively to rays of wavelengths 840 and 960 m^. As a result of comparing j^hotographs taken with ilhnnination by ordinary and infra-red rays, the following was established. * Actually, depending on the method of illuminating the fluid, considerably greater dilution may be required. OPTICAL METHODS OF BORE-HOLE INVESTIGATION 339 (a) In infra-red light, details of the rock are reproduced less clearly than in ordinary light; so, in infra-red light the clarity with which grains of sandstone and lamination of clays (Fig. 6) are determined is less. (b) No additional data are detected on photographs obtained in infra-red light. As was shown above, increasing the wavelength of the light improves the transmittance a little, but the contrast is practically unchanged. Thus a large effect in going to long waves in not observed. Considering what has been stated above, and also that photography with infra-red rays is technically more difficult, the conclusion can be reached that photography of rocks along the walls of a well should be carried out in ordinary light. In photographing the samples a mercury-quartz lamp with a filter of Wood glass was used as the source of ultra-violet rays. The photography was carried out on isopanchromatic film; a yellow filter, absorbing the ultra-violet rays, being placed in front of the objective lens. In connection with this, lumines- cent radiation was mainly recorded on the photographs of rock samples in ultra-violet rays. Photographs of rock samples exposed to ultra-violet radiation indicate the presence of bitumens and their distribution through the rock. Since the distribution of bitumen in the rock depends to some extent on the structure of the rock, photography of bituminous rock conducted in ulti'a-violet rays conveys to some extent the structure of the rock. On photographs of non-bituminous (for example, extracted) rocks, the structure of the rock is not shown up clearly enough. If the question of studying the bituminosity of rock is excluded from the investigation, illumination by ultra-violet rays in photography of rocks does not offer any advantages over illumination by ordinary hght in the study of rocks by photographs, because, owing to wide scattering, ultra-violet rays are transmitted Avorse by the drilHng fluid and possess a greater capacity for loss of contrast. It is obvious that there is no sense in photographing rock in order to study the bituminosity; for this it is sufficient to record the total intensity of luminescent radiation. 2, To find out how the position of the source of illumination affects the clarity of the rock image, some rock samples were photographed with illumination by direct and oblique (at an angle of 45%) rays. Comparison of the photographs obtained shows that with illumination of the rock sample by direct rays the photographs are "blank" (see Fig. 6); the grain of the rock is more poorly defined, and the rock projections are less noticeable. Therefore photography of rock should as far as possible be conducted with side illumination. 340 T. V. Shcherbakova THE WELL CAMERA Photography of wells was first carried out by D. G. Atwood in 1907 (^). Ho^v"ever, this work had no great practical significance. In 1924 Reinhold in Holland made a camera for examining the walls of artesian wells of large diameter (-). The camera permits several hundred photographs to be obtained. Simultaneously Avith photography of the well wall the position of a compass is fixed, permitting determination of the elements of the bed sequence. The instrument is lowered in the tubes which are used to replace the drilling fluid by clear Avater. In reference (9) there is a description of a well caxnera used in the U.S.A. The camera consists of two parts. One of them is filled with clear water and contains an electric lamp to illuminate the object, an inclined mirror and a viewing window; the actual camera and a winding mechanism are located in the other part. Photography is carried out with a 16 mm cine-film and 450 shots can be taken. The camera is lowered into the well on a cable; a spring presses the side of the instrument where the viewing window is situated to the Avail of the well. In the Soviet Union development of a Avell camera was undertaken in 1935 at the Central Scientific Research Institute of Geological Survey but was not concluded. Since 1954 in investigations of shallow dry wells horoscopes (^) have been vised, consisting of a light source and mirror directing the image of the Avell Avail to an observer situated at the mouth of the well. In 1955 at the All -Union Scientific Research Institute for Geophysics an experimental model of a well camera Avas designed and constructed. Its technical featiu'es are the folloAving: (1) The camera is lowered on a three-strand coring cable. (2) The winding mechanism of a photo -inclinometer is vised as the Avinding mechanism; the Avidth of the film is 35 mm, frame dimension 18 mm by 13 mm, and the number of frames about 120. (3) The size of the viewing windoAv is 5.7 cm X 4 cm. (4) The size of instrument is: length 1.5 m, diameter (Avithout spring) 70 mm, Aveight 50 kg. Fig. 7 shoAvs a general view of the well camera, in the loAver part of Avhich the light source, mirror and AdcAving AvindoAv are located. The light source consists of three type SM-29 bulbs (the light flux from each lamp is 28 lumens) equipped with hoods to protect the camera, situated in the upper part, from the direct light of the bulbs. Light from the bulbs passes by way of the mirror and viewing Avindow on to the Avail of the Avell. OPTICAL METHODS OF BORE-HOLE INVESTIGATION 341 342 T. V. Shchebbakova 0^\ing to constructional considerations it was not possible to provide oblique illumination to the wall of the well. The rock is transmitted to the cameira of the upper part of the instrument through the same window. The position of the mirror can be varied within certain limits; being chosen so that light flashes, reflected from various parts of the camera, do not fall on to the photographic film. The light source and mirror are put into a glass cylinder; in the casing opposite the mirror there is an opening 5.4 cm x7.8 cm forming the viewing window. The lower part of the instrument is filled "with distilled water; a pressure compensator serves to equalize the pressures outside and inside the chamber. The lower part of the instrument is separated from the upper part by thick glass; hermetic sealing is provided by rubber packing. The objective lens and film-winding mechanism, are situated in the upper part. An Industar-22 lens (focal length 52.4 mm, aperture 1:3.5) was chosen as the objective lens. Fig. 8. Path of rays in well camera. 1 — driUing fluid; 2 — viewing glass {n = 1.46); 3 — ^water (re = 1.33) filling part of instrument; 4 — ^protective glass (re = 1.5); 5 — air (re = 1) ; HH' — principal planes of objective ; 00' — axis of optical system ; F^^ , F^ — forward and rear foci of objective. The instrument is enclosed in steel casing; in the lower part the casing of the compensator with a head for the load is screwed in, and above it there is a light bridge with the three lamp leads. On the instrument are fixed springs which press the side of the instrument on which the viewing window is located to the wall of the well. The instrument is lowered on a three -strand cable. One strand carries the supply for the electric motor of the film-winding mechanism, another carries the supply to the series-connected lamps of the light source, and the third strand is common to both, circuits. The contact located on the film- winding mechanism switches a certain resistance into parallel with the electric motor when the film is wound -on the length of one frame. The winding-on of the film is controlled by the increase in current intensity associated with this switching. OPTICAL METHODS OF BORE-HOLE INVESTIGATION 343 Fig. 8 sho^vs the paths of rays in the well camera, obtained by means of graphical construction; the ray paths and image of AB are shown by a broken line for the case when the medium is wholly air, and the actual path is shov,ii by continuous lines, where account is taken of the fact that part of the medium is water and glass, which have refractive indices different from unity. It is evident that this circumstance leads to displacement of the image by B'B" = 5 mm relative to that observed in air. The image is produced with fourfold reduction. A very important quantity for the camera is the depth of field which is the distance within the limits of which objects at different distances from the objective lens will appear sufficiently sharply on the image. It is obvious that, in order to obtain a sharp image of the usually uneven surface of a well wall, it is necessary to have as large a depth of field as possible. Increase in depth of field can be obtained by increasing the distance from the objective lens to the subject of the photograph and reducing the effective aperture of the lens. Fig. 9. Test plate. The distance from the subject of the photograph to the objective lens is taken as equal to 267 mm; further increase in the distance is undesirable, since this would lead to considerable lengthening of the camera. In choosing the effective aperture of the objective lens it should be remembered that reducing it leads to a sharp increase in exposure time. In the well camera the relative aperture of the objective lens is taken as 1:5.6, calculated for a depth of field of about 2 cm; in the majority of cases the distance from the viewing window to the rock wall lies within the limits of this distance. 344 T, V. Shcherbakova For the lamps installed in the camera (the power of a single lamp is 5 candle power) and the distance of 20 cm from them to the subject of the photograph, the illumination of the subject (allowing for absorption) is about 6 lux. For this ilhmiination the exposure time A\dth a film speed of 45 GOST units equals 7 sec. Test photographs of plates showed that the resolving poAver of the well camera mth the lower part of the instrument not filled with water is about 30 lines per mm, and when filled with water about 10 lines per mm. This resolving power ensures that rock grains of size 0.1 mm and greater can be distinguished. The reduction of resolving power when the lower part of the instrument is filled is associated with the scattering of light in the water filhng the lower part of the instrument. Figure 9 shows a photograph by the well camera of a test plate (photograph taken in water). Corrosion of components in the lower part of the instrument filled with water caused great difficulties. To avoid this and to increase the resolving power it is proposed in the future not to fill this part of the instrument with water but to fix thick glass into the viewng window and provide it with reHable hermetic seahng. PHOTOGRAPHY OF ROCKS ALONG THE WALL OF A WELL The well camera described above was used to take photographs in wells 8,9 and 10 of one of the sites of the South Kazakhstan region and in well 2089 in part of the work of the Kamenskaia geological survey party in Donbass. The wells were filled with clear water. In wells of a site in the South Kazakhstan region the well diameter is 90 mm. Photographs were taken at depths from 40 to 150 m in wells 8 and 9, and from 23 to 80 m in well 10; in all, one hundred photographs of the rock were obtained. The section consists mainly of dolomites and argilHtes. From the photographs it is usually possible to establish the lithological nature of the rock. Dolomites are distinguished by nodular irregularities with sharp contours, associated with the special nature of the fractures of this rock. In some cases the fine-grain structure of dolomite is over- looked. The argillites are distinguished by the presence of erosion, marked by shadows and breaks in the sharpness of the image. Brecciated rocks are clearly detected ; on a photograph the rock fragments OPTICAL METHODS OF BORE-HOLE INVESTIGATION 345 (dark grey dolomite, light grey limestone) constituting the breccia are easily visible; the dimensions, shape and mutual disposition of these fragments can easily be discerned (Fig. 10, 1-3). In the photographs the stratification, presence of veins and seams in the rock, and the fissuring of the rock can be easily seen. Here it is possible to distinguish the cracks filled with some cementing material, in the given case white calcite (Fig. 10, 6-8), from those not filled with cement (gaping) (Fig. 10, 4-5). In well 2589 in the Kamenskaia region, having a diameter of 90 mm, forty-five photographs were taken at depths from 42 to 90 m. In these photographs it is also possible to establish the lithological character of the bed. Sandstones are distinguished by their clearly visible individual grains (Fig. 11, 4); coarse-grained sandstones being distinguishable from fine- grained sandstones. The greater the grain size, then, naturally, the more accurately is the grain structure of the sandstone recorded and the more distinctly can the individual grains be distinguished. Clayey rocks are marked by irregularities in the wall, which are due to erosion of the rock (Fig. 11, 3), and by stratification. Coals (in the given region close to anthracites) are well distinguished by alternation of dark and light regions which are due to the presence of sharp boundaries at the fracture of specimens; often these regions are extended in one direction, corresponding to the stratification of the rock (Fig. 11, 1 and 2). On the basis of an analysis of the photographs the following conclusions can be reached. 1. From the photographs of the rock it is possible to form an idea — al- though not always unambiguous — of the lithological character of the rock. 2. Photography of the rock permits the petrographic character of the rock — structure, grain, presence of cementation, etc. — to be defiiaed more accurately. As a result of photography of the rocks along the walls of wells, problems of elucidating the structural features of the rock (the presence of inclusions, stratifications, fissures) are solved most successfully. It should be noted that geophysical methods at present employed for well surveying do not permit detection of the degree of fissuring of rock, while such fissuring is a very important property of the rock. In a series of reservoirs the reservoir properties of the beds are connected with the fissuring. :346 T. V. Shcherbakova Fig. 10. Photographs of well walls (South Kazakhstan region). 1-3— brecciated rock with pieces of dark-grey dolomite and light-grey limestone; 4 and 5— light-grey limestone with open crack and dolomite inclusion; 6-8 — fissured dolomite, cracks filled with calcite. OPTICAL METHODS OF BORE-HOLE LNVESTIGATION 347 CONCLUSION In a well with a sand- clay section (or in part of a well penetrating sand- 'clay deposits), from photographs of the well walls it is possible to disting- uish approximately between clays and sand beds and for the latter to iden- tify medium and coarse-grained sands and sandstones. However, generally speaking, this is not of particular interest. These problems are mpre suc- cessfully solved by other geophysical methods of investigation or by sam- pling ground by a lateral core -lifter. At the same time the drilling fluid is Fig. 11. Photographs of well walls (Donbass region). 1-2 — coals; 3 — clay rock; 4 — sand rock. cloudy in sand-clay deposits, and on the walls of the well there is a thick clay crust which is difficult to remove. Therefore at the present time optical methods of surveying wells can be recommended for application in those wells where the section consists of dense, finely grained rock; dense sand- stones, carbonate rocks and hydrochemical sediments. In such wells the drilling fluid is often transparent; if necessary, it can easily be made trans- parent. From the photographs in such wells it is possible to identify fissured reservoir rock and to determine the nature and structure of various beds. To solve these problems by geophysical means is difficult and sometimes even impossible. 348 T. V. Shcherbakova REFERENCES 1. A.-M. ViCTOROV, Well Borescopes. Gosgeoltekhizdat, (1954). 2. I. B. Vil'ter, Photographs of bore-holes. Mastering the technique of the coal and slate industry. No. 6 (1932). 3. A. A. Lapauri, Photographic Optics. Iskusstvo, (1955). 4. J. W. Street, The wave theory of light. Gostekhteoretizdat, 1940. 5. V. A. TiMOFEEVA, Multiple scattering of light in turbid media. Trudy Morskogo Gidrofizicheskogo Instituta Akad. Nauk SSSR, Vol. 3 (1953). 6. V. V. Shuleikin, The physics of the Sea. Izd. Akad. Nauk SSSR, (1953). 7. K. 8. Shifrin, Scattering of Light in a Turbid Medium. GostekhteoTetizdat, (1951). 8. I. 0. Yacobi, Methods, Instruments and the Work of Bore Hole Surveying. United Scientific and Technical Press, 1938. 9. 0. E. Barstow, C. M. Bryant, Deep well camera. Oil Weekly, 5 May, 1947. Chapter 15 DETERMINING THE PERMEABILITY OF OIL-BEARING STRATA FROM THE SPECIFIC RESISTANCE S. G . KOMAROV AND Z. I. KeIVSAR The method for determining the permeabiHty of oil-bearing strata from the specific resistance was proposed in 1947 by G. S. Morozov. On the basis of experimental studies at the Research Institute for Geophysical Survey Methods, he established a relationship between the coefficient of increase in resistance and the permeability. This relationship was recommended for the determination of permeability. A number of papers and in particular those of Dolina^^'^) have been devoted to the determination of permeability of oil-bearing rocks from the specific resistance. As a result of this work, Dolina developed a procedure for determining from their specific resistance (^), the permeability of the oil- bearing Devonian sandstones. We give here an account of the method for determining the permeability of the Devonian petroleum-bearing sandstones from their specific resistance. THE METHOD FOR DETERMINING THE PERMEABILITY OF THE PETROLEUM - BEARING DEVONL\N SANDSTONES FROM THEIR SPECIFIC RESISTANCE To determine the permeability of the stratum, at first a determination is made of the coefficient of water saturation of the petroleum and gas-bear- ing stratum. The coefficient of water saturation refers to the ratio K^^ of a part of the volume of the pores, filled with water, to the total volume of the pores. Let us note that with the coefficient of petroleum and gas saturation (or, having in mind only the petroleum-saturated strata, the coefficient of petro- leum saturation) K refers to the ratio of the volume of pores filled with petroleum to the total volume of pores. Obviously, The coefficient of water saturation is determined by the coefficient of in- "Crease in resistance '? = -^. (1) 349 350 S. G. KoMAROv AND Z. I. Keivsar where q^ is the specific resistance of the stratum; Q^is the specific resistance of this stratum Avith 100% filhng of itp> pores by water (water-bearing stratum). The specific resistance of the oil-bearing stratum q^, necessary for calcul- ating the coefficient of increase in resistance Q, is determined from the data, of VKZ*. The specific resistance of the water-bearing stratum is calculated for the- formula 9w=^Qs.w (2> \vhere q^ ^ is the resistance of the stratum Avater, at the temperature of the- stratum ; F is the relative resistance of the rock. The stratum water of the Devonian sandstones of the Uralo-Volzhsk pro- vince has about 260 g/1. of salts; in agreement with this its specific resis- tance can be taken as being equal to 0.034 D/m at a temperature of 30'^ (according to Morozov 0.03 O/m^^^), according to Dolina 0.05 Q/m(^*)). To determine the relative resistance of the Devonian sandstones according to their porosity Dolina recommends (^) the use of the relationship ob- tained from the results in determining porosity of pores and the VKZ data in structure contoured wells; F = 0.65m-^'^\ (3> where m is the coefficient of porosity in relative units. For this purpose, it is also possible to use other results (^'■'^°). For comparatively homogeneous strata in the absence of results on the porosity of the stratum, the specific resistance of the water-bearing stratum. Q^j^ is taken as the average value of the resistance of the stratum for the struc- ture contoured wells. Thus, Dolina, at one of the Tataria deposits, from the VKZ results for 154 structure contoured wells obtained an average value for the resistance- of the water-bearing Devonian sandstone of 0.7 Q/m. This value was recommended for use in calculating the coefficient of increase in resistance. The coefficient of water saturation is determined from the coefficient of increase in resistance Q from the curve Q ^f{K^), obtained on the basis- of experimental studies of rock samples. It is assumed that the water contained in an oil-bearing stratum cannot take part in the general movement of the liquid in the stratum and is inter- stitial (connate). In this case, the water saturation determined from the curve Q =f{K^) is the residual interstitial water saturation -K^^. id- * VKZ — All-Union Commission on mineral Resom-ces. THE PERMEABILITY OF OIL-BEARING STRATA 35] Figure 1 gives tlie curve recommended by Dolina<*) for determining the permeability of petroleum-bearing Devonian sandstone by the specific resistance. The curve 1 shows the dependence of the coefficient of increase in resistance Q on the coefficient of oil-bearing Kp and water saturation K,^; curve 2 shows the dependence of the residual interstitial water saturation on the permeability. Both curves are drawn from the results of experimental studies on rock samples carried out by MoROZOv. The curve 3 expresses the 0-98 0-95 0-9 0-8 0-5 1000 ^ ,v ' ! 1 KDer=f(Kw) 1 C nK,e,) 1 1 1 ' / 1 , \ f \ k / \ / L V A \ \ f \ \ \? i J / \J \ / \, \ / / ^ / \ '' \ \ \ / Q=f (K.) ^ / / / V / \ n -6-- ? ^- \ 'e 1 ' \ 1 _I_Ll Kpe , ^ \ \ \ \ \ \ > v ^ 8 0-1 8 1-0 K^, Fig. 1. Graphs for determining the permeabihty of the petroleum-bearing Devonian sandstones according to their specific resistance (according to Dolina). 1 — dependence of the coefficient of increase of resistance on the coefficient of water saturation K^^ or the coefficient of petroleum saturation ^p; 2 — the dependence of the coefficient water satu- ration K„^ on the permeability K-, 3 — dependence of the coefficient of increase in resistance Q on the permeability i^pgr- 352 S. G. KoMAROv and Z. I. Keivsar direct connection between the coefficient of increase in resistance Q and the permeabihty K p^^- To determine the permeabihty of the petroleum-bearing Devonian sand- stone the appropriate value of permeability K , after having determined its coefficient of increase in resistance Q, should be determined from curve 3 (Fig. 1). The method of determining permeability from the specific resistance in principle is applicable only in those cases where the studied petroleum- bearing stratum is at a distance from the contour of petroleum-bearing deposits. This is connected with the fact that close to the contour of the petroleum-bearing deposits or the intermediate zone in the stratum the water content will exceed the residual water saturation. In thick strata containing bottom water, the method is apphcable only to the upper part of the stratum at a considerable (for example, greater than 10 m) distance along the vertical from the surface of the water, or if the petroleum-bearing part of the stratum is separated from the water- bearing part by a layer of clay. AN ANALYSIS OF THE REASONS FOR ERRORS IN THE DETERMINATION OF PERMEABILITY BY SPECIFIC RESISTANCE The error in the determination of the permeability of the petroleum- bearing strata by the specific resistance is due to the eiTor in determining the coefficient of the interstitial Avater saturation and the insufficiently close connection between the coefficient of the interstitial water saturation and the permeability of the stratum. The coefficient of the interstitial water satura- tion is determined from the coefficient of increase in resistance Q. The relative error dQjQ in determining the coefficient of increase in resistance is equal to the sum of the relative error dgjQg, in determining the specific resistance of the stratum and the relative deviation dq^JQ^, taken for calculations of the value of the resistance of the water-bearing stratum from its actual value. The error in determining the specific resistance from the results from the data of lateral electrical logging is caused by the inaccuracy of the results of the measurements of the apparent resistance, by the difference between the actual probe curve and the calculated curve chosen for it and the insuffi- ciently accurate consideration of the various factors (the well diameter, the zone of penetration, heterogeneity of the stratum). On the whole, this error is appreciable and can reach up to 30% {dg J q^ =0.3). THE PERMEABILITY OF OIL-BEARING STRATA 353 The relative deviation of the accepted value q^ from the actual reaches a value of the order of 0.1-0.3, especially in the case when the average value is taken for the specific resistance of the water-bearing stratum. The connection between the coefficient of increase in resistance and the coefficient of water saturation K^^ is approximately expressed by the formulae (3,10) Q = -^ or Kr.u. = Q-1I-, (4) where ;i is a certain power close to two. If dQ is the error in determining the coefficient of increase in resistance, and dn the deviation of the value for the power from its actual value, the error in determining the coefficient of residual water saturation will be* --n -i -^ -1 dKr.u, = ^-%^dQ+%—dn = -Q " dQ+-\Q "inQdn. 9Q dn n n In accordance with this the relative error in the coefficient of residual water saturation will be dKr.u, IdQ I —f? — = -pr-\ ^ in (2dn. Kr.w n {J w^ If we assume that n = 2, the deviation of the actual value from that taken dn = 0.1, the coefficient of increase in resistance Q = 100 and the relative . . , . . dQ error m its determination — = 0.3, then we obtain Q dKr.u, 0.3 , 0.1 In 100 ^^, As can be seen, despite the considerable errors in the original data, the coefficient of the residual water saturation (providing there are no large errors) is determined comparatively accurately}". To determine the closer connection of the coefficient of the residual water saturation with the permeabihty of the strata let us compare the results in determining permeability with the data of permeability established from measurements on samples. * We will only take the value of error without considering its sign. t If re differs considerably from 2, then the error in determining Kj. ^ is greater. Applied geophysics 23 354 S. G. KoMAROv and Z. I. Keivsar A COMPARISON OF THE RESULTS IN DETERMINING THE PERMEABILITY BY THE SPECIFIC RESISTANCE WITH DATA FOR CORE ANALYSIS For a number of strata of Bashkiria, Tataria and Nizhnii Povolzh'e de- posits the values of permeabiHty of the strata determined, according to the instructions (*), from their specific resistance were compared with data on the permeabihty of strata estabhshed by measurements on samples. The permeabihty of strata of the Tuimazinskian and Romashkinskian deposits was deterinined by Dolina<^'^)*. The specific resistance of the waterbearing stratum, was taken, as shown above, to be 0.7 O/m. Guzanova determined the permeabihty of the strata for the Shkapovskian deposit (^^). The specific resistance of the water-bearing stratum was also taken as being 0.7 Q/m. For the Nizhnii Povolzh'e the permeability of the strata was determined by SusLOVA (^^) ; the specific resistance of the water-bearing stratum was determined from the porosity of the strata; the resistance of the stratum water was taken as 0.04 O/m. The permeability of the samples was taken from the data of the appropriate laboratories. The permeability from the specific resistance was determined for strata removed from the water— petroleum contact. In comparing the results of determining permeability from the specific resistance with the results of core analysis, all the wells, in which samples were taken from the stratum and its specific resistance measured, were se- lected. For each well: (a) KpQj. tbe permeability derived from the specific resistance according to the curve 3 of Fig. 1 was determined; (b) The mean arithmetic value of the permeabihty of the stratum was determined from measuremerxts on samples taken from the sanie well, K ; (c) The absolute value for the difference was found X = \ K^ ~K^ \ ■^ 1 ^^per ^^per \ Since the values of the permeabihty K^^^ from the large number of determinations are more accurate than the values obtained from a small number of determinations then the "weight" of the difierence x for a differ- ent number of determinations of permeability from the cores will be different. We considered that for four or more determinations, the weight of the * From these data the conclusion was previously made that the mean error in deter- mining the permeabihty is close to 20-25% (*). THE PERMEABILITY OF OIL-BEARING STRATA 355 t3 23* 356 S. G. KoMAROv and Z. I. Keivsar difiference x is equal to unity. For three, two and one determination the weight of the difference x is equal to 0.86, 0.71 and 0.5 respectively. In all the wells for a given stratum, according to the obtained differences x-j., x^, •■■,x^, in the values of the permeability, found from the specific resistance and cores, and the weights of these differences P^, P2, ...,P„, each of which has one of the above four values, the following calculations were made (") : (a) the mean error in a separate determination _ %°i + ^2 °2 + • • • + x^Pn ^ /--v "^ P, + P, + ...+P, ' ^""^ (b) the mean square error of a separate determination [/ ni + 0.8( ~t~ ^2 -^2 ~l" • • • + ^fi Pn 86n. + 0.71/IO + 0.5/14 - 1 ' (7) where n-^, n2, n^, n^ are the numbers of wells in which the difference x has the weight of 1.0; 0.86, 0.71, 0.5 respectively. The error was expressed in millidarcies and relative units in percentages with respect to the value K^^^. As can be seen from Table 1, the error in the determination of perme- ability from the specific resistance is rather high, being 170-200 milhdarcies or about 50%. This indicates the presence of errors in the method for determin- ing the permeability by the specific resistance or the insufficiently close connection between the residual water saturation and permeability. The strata are characterized by a comparatively high degree of homoge- neity: the permeabiHty /L^g^obtained from the cores from separate wells is close to the average value for the permeabiHty for the stratum as a whole. In connection with this, determinations were made of: (a) The actual mean permeability of the stratum j^k ^K^P^ + K^P^+^^^ + K^^ '^'"- P, + P2+... + Pn ' where K![, K^, •••, Kn ^^^ ^^^^ values for the permeability of the stratum, obtained from cores for separate wells; P-^, P^, •••,Pn the weights of the separate values of the permeability taken for four or more, three, two and one determination respectively equal to 1, 0.86, 0.71 and 0.5; (b) K\-K^^^,K\-K^^, ...,/C;-ir^„are the deviations of the values for the permeability of the stratum obtained from the cores of separate wells, from the actual mean value of the permeability for the stratum K^^', THE PERMEABILITY OF OIL-BEARING STRATA 357 w w AOUBJ[ng 1— 1 ON CO so S in so in CO o CM AOZOIOJ^ CO 00 o CO OS o so CO ^ 03 ^ o^ (ajBjnooB) o o OS CO t^ '^ o o ir- so so Os CO 3 '"' CM Bin]OQ in 1— 1 l-H o CO o O CO CO l-H CO AOire^pg LO OS in sO r— ( o in CM a OJ <u l-H CM CO in ID o AOZOIOJ^ £3 ^ s o in § o Os c3 -a CO CM CO CM CO ij" (ajBinooB) o o SO o o CM § BUT[OQ CSI l-H CM CM so CM CO CO Os BUqOQ o in T— ( On g in Cvl CO t~- C^J CM <M CM CO SO AOOBjpig tr- io ^ in CO I-- CM Os AOZOIOJ^ lO CO in CM so ^ CM CO CO ^ o^ (aiBinooB) o in in in r- ON ^ ^ ^ CM (>■ 2 o l-H '^ o CM in (-~ g Bin|OQ ^ in SO "* CO l-H S 1 — 1 f- CO SO t^ CM •^ 0) <u OS r— ( CO l-H CM o CM Os l-H l-H in Q in o o o AOZOJOJ\[ CM CM l-H CM c^ o CM ^ (gjBJnooB) in in o in CO o i-H o l-H ;s^ ^ BinjOQ l-H I— 1 l-H CM CM so o in o O in CO BUTJOQ o CM in i-H OS l-H (M r-H CM § AOlIBipg so o l-H CM 00 1 l-H f£ « CO in O o O CM 1 ID S 'o 13 O SO S^ CO i^ in CM (gjBinooB) O ^ o in o CO o CM i-H o ID 3 1=) BUTJOQ rfi Tf CO •* ■* m in in o o in > 1—1 C^ CO l-H in ^ s in o o o o CO V S9I00 UIOIj^ CO c> in o CO s ■* ^ r? in so t~ snaAv JO -oisj Os OS CO r-H CO CO o CO CO CM ^^ -1 X5 o > ranjBijg C) C5 Q C5 Q Q % a +. .3 ^ -T^ -^ < .S •S to > -a o > c 1 S ^ -« s S 3 rO o o "o H H tn eci Ph CTj 358 S. G. KoMAROv and Z. I. Keivsar (c) The mean and mean square values of these deviations, calculated from formulae similar to (6) and (7), These mean and mean square deviations characterize the error introduced when, instead of the actual value of permeability for the stratum, we take the mean value of permeabihty for the stratum. There are small differences between the errors in determining the perme- ability of the stratum from the specific resistance and the errors which are obtained when, for the permeability of the stratum of a given section, we take the mean value of permeability for the stratum. Thus, the use of the method for determining permeability from the specific resistance, in accordance with the existing instructions, does not lead to a noticeable improvement in accuracy of the data on stratum perme- abihty. The stratum permeabihties obtained by averaging the values are used more effectively since it is not necessary to process a large amount of materia;!. Thus, the method for determining permeabihty by the specific resistance in the variant recommended for the Devonian sandstones of the Bashkiria, Takaria and Nizhnii Povolzh'e deposits lead to large errors. CERTAIN PROBLEMS IN THE METHOD FOR DETERMINING PERMEABILITY FROM THE SPECIFIC RESISTANCE It was assumed that the large error in determining permeabihty from the specific resistance for the Tuimazinskian, Shkapovskian and Romashlcinskian deposits is connected with the incorrect choice of the same specific resistance of the water-bearing stratum for all cases (0.7 O/m). Consequently for the deposits mentioned, a second determination of the permeability from the specific resistance was conducted; the specific resistance of the water- bearing stratum being taken as its value obtained for each separate well. Calculations showed that the values of the errors are about the same (see Table 2, corrected by Do una results). Apparently, the main reason for the error is not the initial geophysical data, but the poor connection between the interstitial water saturation and the permeability or the error in deter- mining it. The determination of permeability of petroleum strata by the specific resistance is based on the assumption that in the petroleum stratum there is only interstitial (connate) water. Apparently, apart from interstitial water in the petroleum stratum there is also the so-called free water, the quantity of which near the top surface of petroleum-bearing deposits or water-petroleum contact can be sufficiently THE PERMEABILITY OF OIL-BEARING STRATA 359 large, and as a consequence the water saturation of the stratum will not correspond to its permeability. The method for determining permeability by the specific resistance is therefore applicable only to a stratum (or part of a stratum), which is much higher than the water-petroleum contact. MoROZov^^^) considers that the determination of permeabihty by the specific resistance can be carried out for strata 16 m above the water-petro- leum contact, since only then is the water saturation of the petroleum stra- tum interstitial. This condition represents a very great limitation of the method. Evidently, depending on the surface properties of the particles making up the stratum, the properties of the stratvim water and petroleum this distance will be different. According to Dolina^^' the possibility of deter- mining the permeability from the specific resistance should be estabhshed in each individual case, depending on the position of the water-petroleum contact and the characteristics of the stratum. To establish the effect of the water-petroleum contact on the accuracy in determining permeability from the specific resistance the above mentioned determinations were made for strata at varying distances from the water- petroleum contact. The results are given in Table 3. As follows from Table 3, there is no systematic reduction in permeabi- lity and no large error in its determination at a small distance of the stratum from the water-petroleum contact. This indicates that the effect of the water-petroleum contact on the results of determining permeability is small. MoROZOV^^^) considers it possible to determine the permeability at a distance of less than 16 m from the stratum to the water petroleum con - tact. He recommends that the obtained values Kp^^ should be multiplied by a correction coefficient " = !¥• <«> where h is the distance along the vertical from the middle of the studied zone to the water -petroleum contact. However, the use of this correction is clearly undesirable for the following reasons. 1. It is insufficiently well established* and it is obviously faulty. The water saturation of an oil-bearing stratum depends not only on its distance to the water-petroleum interface, but also on a large number of other values : the density of the petroleum and water, their capillary properties, the character * Morozov does not give the basis of the correction coefficient //. 360 S. G. KoMARov AND Z. L Keivsar Table 3. The values of the errors in determining the permeability from THE SPECIFIC resistance FOR VARIOUS DISTANCES ALONG THE VERTICAL FROM THE MIDDLE OF THE STRATUM TO THE WATER-PETROLEUM CONTACT (VNK)* Mean value of permeability, Mean error Mean quadra- tion erroT- a 1 CO o is to Ph d millidarcie Deposit o o s o u o a .2 o .2 ^ o^ Tuimazinsk Di up to 10 15 516 376 252 69 840 80 >16 7 519 517 317 86 312 118 RomasUdnsk Dia 0—16 3 410 533 219 37 370 50 16—25 9 289 536 335 156 396 220 >25 2 295 395 175 72 175 72 Dih 0—16 20 416 424 274 65 342 103 16—25 7 288 338 213 37 278 52 >25 3 681 704 166 73 224 78 Die 0—16 30 531 517 351 92 465 168 16—25 2 1030 762 274 25 542 51 >25 2 542 505 107 20 160 28 Sokolovgorsk Dy 0—10 6 953 499 543 85 782 217 10—16 4 1014 541 539 43 861 63 16—25 9 787 593 312 38 551 61 >25 16 830 674 663 87 840 122 of the surface of the sohd phase-^' ^^^^ ^^®\ The simplified formula (8), which does not give this fact, will not correspond in the majority of cases to the actual state of affairs. 2. Due to the heterogeneity of the stratum, the distribution of water will deviate considerably from an even course, determined by the coefficient jjl. As recommended by Dolina it is much better to use stratum characteris- tics, and to exclude from consideration all doubtful cases. As follows from Table 3, the usual method can give results free from the effect of the water -petroleum contact even at a distance of less than 16 m from it. CURVES FOR THE DEPENDENCE OF THE COEFFICIENT OF INCREASE IN RESISTANCE ON PERMEABILITY The Dolina curve (i on Fig. 2) can be expressed approximately by the formula Q = lxlO-^K];^^, (9) where K is the permeability in millidarcies. * a Russian transliteration is used [Editor's note]. THE PERMEABILITY OF OIL-BEARING STRATA 561 As well as this curve, Dakhnov recommended for the Devonian sand- stones the curve Q = /{K^^^ (Fig. 2, curve 2) which "was drawn by the MoROZOV method"'^' P^^®*^*'. Since this curve differs httle from the DoLiNA curve, the determination of permeability from it was not carried out and the errors in the determinations of the permeabihty were not calcul- ated. According to MoROZOv'-^^' p^^*^^^' the connection between the coeffi- cient of increase in resistance ^ and the reservoir properties of the strata (permeabihty K^^j., porosity m) for the Devonian sandstones is expressed as shown by dotted lines in Fig. 3 (Morozov does not give the basis for the curve in this figure). 1000 2 / 8 100 8 1000 m darcies Fig. 2. Various curves for the dependence of the coefficient of increase of resistance Q on the permeability J^^yer ■'■ — according to Dolina(*); 2 — according to Dakhnov(3, page 424). 3_according to Wyllie and Rose for c = 10("). Using these curves, the permeability of a number of strata was determined by employing the same values of the coefficient for the increase in resistance, that serve for the calculation of permeabihty by the Dohna curve (the accurate curve). 362 S. G. KoMAROv and Z. I. Keivsar Table 2 shows the mean and mean square errors for the separate determina- tions of permeabiHty from the specific resistance by means of the MoROZOV curves. These errors were calculated by the already mentioned method from formulae (6) and (7). It can be seen that the use of the relationship between the coefficient for increase in resistance and the reservoir properties as proposed by Morozov, means that in some cases (for example, the D^^ and D^^ strata of the Romash- kinsldan deposit) the error in the results decreases, in the other cases (for example, the Z)j, Djj strata of the Tuimazinsldan deposit) it increases in comparison with that observed in determining permeability by the Dohna curve. The Morozov curves do not give favourable results, the error in deterinining permeability by the specific resistance being rather large. Considering the curve suggested by Morozov Q =f{Kp^^^j^), we find that they are obtained on the assumption that the interstitial water saturation and consequently the coefficient of increase in resistance depend not only on the permeabiHty of the rock, but also on their porosity, depending to a much greater extent on porosity than on permeability. In fact, the Morozov curves Q "^fiKp^^^^) are represented approximately by the formula Q-'-^-^lr (10) It follows that the relative change in the coefficient of increase in resistance - — is connected with the relative changes in permeability dK jK and dm porosity in the following way (neglecting the sign of the error) : m ^^2^ + 10^. (11) Q Kper m Any relative change in porosity causes a 5 times greater change in the coefficient of increase in resistance than the same relative change in permea- bility. Therefore, despite the comparatively large changes in permeability, the porosity will have a much greater effect on the coefficient of increase in resistance, than the permeability. Thus, if the curves proposed by Morozov are correct then in view of the large effect of porosity and the impossibility of an exact allowance for its effect, we would still be unable to obtain from the specific resistance sufficiently accurate values of permeability. The unusually high power -index of porosity and the shape of the curves THE PERMEABILITY OF OIL-BEARING STRATA 363 at their extreme parts cast doubts on their vahdity. According to the Morozov's curves, in a low porosity rock in a number of cases there will be very small values for the interstitial water saturation and very large values for the coefficient of increase in resistance; for example, for m = 0.1 and K = 10 milhdarcies we should have Q pn 10 and an interstitial water saturation of only 10%. However, this is not very probable. Sultan ov and Dobrynin (^'*) propose for the determination of permeability of the Devonian sandstone from the specific resistance the use of the relation- ship between the coefficient of increase in resistance and the reservoir properties of the strata, represented by continuous curves on Fig. 3. 100 / 1 f. ( 7 lll\ V ; 1 1 f\f\ 1 if ^ / 1 1 1 n fllf\ ' / 1 / / /^ m / 1 f 1 J ' I 1 1 1 1 ) f 1 f / 1 1 / 1 1 / / (1 /// if'/ 1 / / f / / 1 / / ■'/' ■71 m ^/' / t 1 f 1 '// 1/ //m 1 o 6 6 3 o"| d" o P o 1 OIC ^ i .9 f 1 ] r \r / / /l / fit 1^ i 1 1 / / y^ n « t Jj'l 1 / / / f / // '/ / 1 / o>V // ///(//\ / 1 / / / ' / / //l//j ' j ' t- / / / ■!, /' //\> /////( / / / / / / / / ^ i / /, /// ^A V// '/,' ' 1 1 1 / ,6'' / ^ / / '/ // ^ ?l, 11 / / / yf / / /l / '//. 'A / / / / 1 / 1 1 1 Fig. 3. Curves for the dependence of the coefficient of increase in resistance Q on the permeability Kp^^ for sandstones of the Povolzh'yc Paleozoic. Cnrve symbols- porosity m. Dashed curves — according to MoROZOvd^, Fig. 6). continuous curves- according to SuLTANOv and Dobrynin'^*). Table 2 gives the mean and mean square errors for individual determinations of permeability by specific resistance using the Sultan ov- DoBRYNiN curves. The coefficient of increase in resistance was represented by those values for which determinations were made of the permeability according to the Dolina curve (corrected). The values for the coefficient of increase in resistance were obtained from the value of resistance of the stratum, corrected by the use of the porosity data of a stratum with 100% filling of its pores with water. 364 S. G. KoMAROV and Z. I. Keivsar As can be seen from Table 2, the use of the curves suggested by Sultanov and DoBRYNiN Q ^/(Kp^^^^) does not lead to a noticeable reduction in the errors; therefore, these curves have no advantages over the Dolina and MoROZOV curves. The justice of this conclusion will be apparent if we study carefully the curves, which were suggested by Sultanov and Dobrynin, and which express the relationship between the coefficient of increase in resistance and the per- meabihty and porosity. As can readily be seen, this relationship can be approximately represented by the formula The possibility of the connection of such an artificial character is doubtful. Tiks'e,* Wyllie and Rose (^^^ i^), starting from theoretical considerations, give the following expression for the interstitial water saturation K^.^ : TTl Kr.w. =^ C .-— , (13) where K. is the permeability in millidarcies : m is the porosity in relative units; C is a constant usually taken as ten. It is shown that in the general case, C is the following function of the porosity: w? (072)' C-^^-rn^,- (14) Assuming that Q = K^^ , we obtain C = #^. (15) This dependence for C == 10 and m = 0.2 is shown by the curve 3 on Fig. 2. The formula was not checked. However, starting from the fact that the corresponding curve has the same character as the other curve (Dolina, Dakhnov, Morozov, Sultan ov-Dobrynin), it can be considered that the use of the formula of Wyllie and Rose will not lead to an improvement in the results for determining permeabihty from the specific resistance. FURTHER ANALYSIS OF THE CAUSES OF ERRORS IN THE DETERMINATION OF PERMEABILITY FROM THE SPECIFIC RESISTANCE The large error in the determination of permeability by the specific resistance can be connected with a systematic error due to the fact that the * Possibly a transliteration of a French name. THE PERMEABILITY OF OIL-BEARING STRATA 365 curves for Q ==f(K) diflfer considerably from the positions which they should in fact have. From this point of view, it is of considerable interest to compare the average values for the permeability of a stratum, determined from the specific resistance (i^^„) and cores (X'ay). These data are given in Table 2. As follows from Table 2, the average value of the permeabiUty of a stratum determined from the specific resistance is sufficiently close to the average o o JUU \ A 8 j^ _/y 6 y\,'' ^ ^^^^^^ 4 / / 100 / / Di / / / /' 1 ~\ \0]- 100 1000 1000 100 / / / ^ • 1 «:ir??C— -- • •cp^,*^ ^ — ^» ^ ■-•-;/ o^ ,-^^ / /' -— Dib / / y / /^^^ / (b) iOO 1000 Ko. Fig. 4. A comparison of the mean values of permeability (in miUidarcies), determined from the specific resistance (K°y) and from the core {K^.^. (a) — Tuimazinskian deposit (strata — D J and T>y\)\ (b) — Romashkinskian deposit (strata — Z)j)3, DjJ; the black permeability determinea from the Dolina curve; the open points — from the points — Morozov curves. 366 S, G. KoMAROv and Z, I. Keivsar value for the permeability of a stratum, determined from a core; while in some cases (the strata D^ and Djj of the Tuimazinskian deposit, Dj^^ of the Romashkiusldan deposit) they practically coincide. However, the difference between the values of the permeabiUty determined from the specific resistance and from cores, increases sharply, if we compare them separately for large and small permeabilities of the stratum. This is illustrated in Fig. 4, where the average values of the permeability K^^, determined from the specific resistance, are compared with the average value for the permeability K^^, obtained from cores for different ranges of permeability values; 0-200, 200-400, 400-600 and above 600 millidarcies. The comparison was carried out for the Dj and Z)jj strata of the Tuimazinskian deposit and the Dj^, and Dj^ strata of the Romashkinskian deposit ; the original data are the same as those from which the mean values of permeability were calculated in Table 1. Evidently only for a permeability close to the average value for the stratum does the method for determining permeabihty by the specific resistance, based on the use of the above curves Q =f{K), give favourable results. For large and sm.all permeabihties, a large systematic error is observed — increasing for the small permeabilities, and decreasing for the large permeabihties. There is a sort of levelHng out of the readings. This error is very dangerous, since in an individual well it can lead to a distortion of even the qualitative idea of the permeability of the strata. The possibility of the method for determining permeability by the specific resistance can be shown by comparing directly the coefficient of increase in resistance with the actual value of the permeability. Fig. 5 gives the results of these comparisons for the Dj stratum of the Tuimazinskian deposit. On the graph they are points, the abscissae of which are the values of the permeability Kpgr-> determined from the core, and the ordinates — the corresponding values for the coefficient of increase in resistance Q. For the values of Q were taken those values wliich were used in calculating the errors given in Table 1. The points are divided into two groups; one group (marked on the graphs with circles) includes points for which the values of the permeability were not very accurate (small number of cores, for which the permeability was measured; a large difference in the results) ; the other group (black) includes points for which the values of the permeabihty were determined with a sufficiently high degree of reliability. On each of the graphs an "empirical" curve for the dependence of the coefficient of increase in resistance Q on the permeability Kp^^ was drawn through the points. This is done in the following way: the abscissa is divided into a number of separate intervals and for each of them THE PERMEABILITY OF OIL-BEARING STRATA 367 the most probable value for the coefficients of increase in resistance (the value with respect to which a half of the points has the greatest, and a half, the least value of Q) is determined. Through the points corresponding to these average values is drawn, with a certain amount of averaging, the curve Q=f{K,er)- ' On the graphs, the dotted curves give the probable deviations* ("probable error") of the coefficient of increase in resistance from its value corresponding Q 100 8 100 Kper, 01 darcies Fig. 5. Comparison of the ceofficient of increase in resistance Q with the values of the permeabiUty Kp^j. Tuimazinsk deposit, stratum Dj. to a given permeability according to the curve Q ^/{K^^^. These curves are drawn so that between each of them and the curve Q =f{Kp^j) there is a quarter of all the points on the graph. Similar comparisons were carried out for the Djj stratum of the Tuima- zinskian deposit and also for the D^, and D^^ strata of the Romashkin- skian deposit and the stratum Dy of the Sokolovogorsk location. The actual curves for the dependence of the coefficient of the increase in resistance of the permeabihty for the given regions are shown in Fig. 6. * The probable deviation is conditional by the fact that for a half of the cases the devia- tion is greater, and for the other half smaller than the probable. 368 S. G. KOMAROV AND Z. I. KeIVSAR An opinion is widely held (^^' i*' ^^\ that the residual water saturation, and consequently the coejEficient of increase in resistance depend on the porosity of the stratum. In connection with this it is interesting to compare the values for the coefficient of increase in resistance against the perme- abiUty and also the porosity of the stratum. However, a comparatively small number of points, related to a small number of wells, where the cores were selected, and the wide scatter of the points does not allow this. It should be noted that in view of the comparative homogeneity of the strata and the .- __,-- .^^ ^ ^' -- 1 !=> ^. ,^- '' •■• / / 1/ 4' '>4 > / ^ 2 ^^ ^ r / ^^ >?• 3 — ■ l-tr= />* 2" / > J^' 1- '■ .■ ••6 Kpj,, m darcies Fig. 6. Empirical curves for the dependence of the coefficient of increase in resist- ance Q on the permeability Kp^^.. 1 — Tuimazinskian deposit, stratum i)j; 2 — Tuima- ztaskian deposit, stratum i)jj;3 — Romashkinskian deposit, stratum Di^; 4— Roma- shkinskian deposit, stratum Dj^. ; 5— Sokolovogorsk deposit, stratum D^; 6 — Dolina curve. small change in porosity, a large change in the coefficient of increase in resistance on the porosity cannot be expected in practice. This is illustrated in Fig. 7 where for the D^ stratum of the Tuimazinskian deposits, points are given separately for the large and small values of porosity. In both cases, the regularity of distribution of points differs little from that given in Fig. 5 where points are given -without considering the porosity of the stratum. The method for determining permeability by the specific resistance, developed for the Devonian sandstones, of other regions (petroleum-bearing strata of the Groznyi, Turkmen, Baku, Krasnodar and other deposits) led THE PERMEABILITY OF OIL-BEARING STRATA 369 to values which did not correspond to the actual values for the permeabiHty (Table 4). Tliis is due to the fact that the most probable values for the coefficient of increase in resistance of a number of deposits is much less than the most probable value of the coefficient of the increase in resistance for the Devonian sandstone of the Bashldrian and Tatarian deposits (Fig. 8). lOOO I — 1 — rrr 1 1 — m 1 1 • — o • » •y • •\ 100 o • • I''' <T /L 1 M i. J \ ^ - T. • » ■ ; M 1. o .*' • M ,^ o " % r ° o lO 1 " 9 100 « 6 e 000 ? Fig. 7. A comparison of the values of the coefficient for the increase in resistance with permeability for different porosities. Tuimazinskian deposit, stratum i)j. Circles — porosity up to 19%; black points — porosity above 22%; the curve corresponds to the dependence Q = /{Kp^j) for the Z)j stratum (according to Fig. 5). The followng conclusions can be drawn from a consideration of Figs. 6 and 8. 1. For different strata there are different curves for the dependence of the coefficient of increase in resistance on the permeability. Even for the Devonian sandstone, and in the same oil field the difference between the curves Q = f{K ) is so great that for each stratum, it is necessary to use its own curve for the dependence of the coefficient of increase in resistance on the permeability. The reasons for the difference between the curves Q ^f{Kp^^) for different strata are: (a) The difference in the values for the interstitial water saturation of separate strata and in the character for the change in the interstitial water saturation on the permeability; (b) The absence of a unique dependence of the coefficient of increase Applied geophysics 24 370 S. G. KOMAROV AND Z. I. KeIVSAR Table 4. A comparison of the most probable values for the permeability ^pgj., determined from the specific resistance, with the average values of the permeability iLjei-OBTAINED FROM THE CORE (FROM EXISTING DATA) Probable values for the ^av f^o'" ^av from Deposit Stratum coefficient of increase Dolina in resistance curve cores Tuimazinskian Di 80 500 510 Tuimazinskian Dii 90 380 420 Shkapovskian Di 60 395 490 Romashkinskian Dih 85 580 550 Romashkinskian Die 110 480 585 Sokolovogorsk Di 500 635 738 Turkmenia — 5 70 — Grozny! — 4 50 150 Baku — 8 80 300 Perm — 10 100 350 in resistance on the interstitial water saturation, as a result of which, for approximately the same value of water saturation, there are different values for the coefficient of increase in resistance. The latter is confirmed by the sharp change in the most probable values Fig. 8. Curves of the distribution of coefficients of increase in resistance Q. {N — per- centage of values of Q corresponding to the interval of changes). 1 — Tuimazinskian deposit, strata DjandDjj; 2 — Romashuiskian deposit, stratum Dj; 3 — Sokolovogorsk deposit, stratum D^; 4 — Perm Priural'e; 5 — Shkapovskian deposit, strata Z)j andDy; 6 — Baku, productive stratum; 7 — Groznyi, sandstones of the Karagano-Chokrakskian deposits; 8 — Nebit-Dag deposits; Turkmen. THE PERMEABILITY OF OIL-BEARING STRATA 371 of the coefficient of increase in resistance for petroleum strata of different deposits (see Fig. 8). 2. The curves proposed for the Devonian sandstones for the dependence of the coefficient of increase in resistance on the permeabihty <^' *> ^^' i^> differ considerably from the actual curves (Fig. 6). As a result, in the determination of permeability from the specific resistance using the Dolina curve or similar curves (see Figs. 2 and 3) there is a large systematic error, considerably distorting the results (see Fig. 4). The main error which is introduced in estabhshing a connection between the coefficient of increase in resistance and the permeability, consists of the fact that the effect of the permeability on K^. ^, and consequently, on Q; is over -read. In other words, it is assumed that the coefficient of increase in resistance is connected with the permeability by the relationship: Q = ^Kler where ^ is a very large value. Thus, according to Dakhnov ^ ^^ 1, according to Dolina q = 1.56, according to Morozov ^ = 2. In actual fact, as follows from the curves given in Fig. 6, q = 0.3-0.66. Of the curves Q =f{Kp^j) suggested for the Devonian sandstones the closest to actuality of the empirical curves is the Dakhnov curve (2 on Fig. 2), although it also differs considerably from them. 3. It can be seen from Fig. 5 that the connection between the coefficient of increase in the resistance and the permeability are insufficiently close. Therefore, even with the help of the curve Q =^f(K) there is a considerable error in determining the permeability from the specific resistance. Table 5 gives the probable and mean square errors in the separate determinations of permeability from the specific resistance using empirical curves Q =f{K) for a number of strata. The probable error is calculated directly from Fig. 5 and from similar curves for other regions (the distance along the abscissa between the dotted and continuous curves), the mean square error was taken as 1.48 of proba- bility(^7,pagel80)_ The error in determining the permeability from the specific resistance even in the most favourable case is high (35-50%;. In some cases, for example, when the coefficient of increase in resistance changes little with variations in permeability (a small slope of the curve Q =f{Kper) relative to the abscissa axis), the error is so great that the determi- nation of permeability from the specific resistance loses all meaning; as for example, for the D^ stratum of the Sokolovogorsk deposit. 24* 372 S. G, KOMAROV AND Z. I. KeIVSAR Table 5. Errors in the determination of permeability from empirical, CURVES Q = f{Kp^r) Deposit Stratum Errors as fractions of the true value of permeability Errors obtained in the case when the permeabiUty is taken as the mean value for the stratum Probable Mean square Probable Mean square Tmma2inskian Tuimazinskian Romashkinskian Romashkinskian Sokolovogorsk Di Dii Dih Die Dv 1.3—0.5 1.5 1.2—0.4 0.35 1.8 1.9_0.74 2.2 1.8—0.6 0.50 2.7 0.4 0.29 0.37 0.3 0.48 0.6 0.43 0.55 0.45 0.71 A comparison of the error in determining the permeabihty of the stratum from the specific resistance, using the empirical curves Q =f{Kp^^) with the errors which are obtained when, for the permeabiUty of the stratum in a given well, the mean value of the permeability for the stratum (see Table 5) is taken, shows that the method of determining permeability from the specific resistance is insufficiently accurate and not very effective. 60 1 ^ \ 50 40 S? 30 ^^ 20 ID 1 1 ""^ -^ ^v^ XT =0^ \^ \ \ ^ ^ ^::^ \3 \ '" — ^ S;^;- /^ 3 \ \ \i. "~-!^ -~^;i?t>^ ^ ____9___^ ■^N 10 2 ■; 100 2 5 1000 2 Kpe„ mdarcies Fig. 9. Interstitial -water content K,.^ in the rocks of different permeabihty Kp^,. from the results of studies of samples. 1, 2, 3 — the proposed value for fine grain sandstones, medium grain sandstones, dolomites and limestones, respectively, according to JoNEs(7); 4-9 — for different deposits in the U.S.A. from the data of Masket'I^); 4-7 — sandstones; 8 — ^hmestone; 9 — dolomite; 10 — Kartashevo deposit, limestone; 11 — Novo-Stepanovsk deposit, dolomites (indirect determination); 12, 13 — sandstones, Tuimazinskian deposit, I and II groups; (10-13 according to the data of Zaks^^)); 10, 12, 13 — determined by the method of capillary displacement. THE PERMEABILITY OF OIL-BEARING STRATA 373 4. Starting from erroneous notions on the high accuracy of determining permeabihty from the specific resistance of the Devonian sandstone, incorrect conclusions were drawn as to the most favourable possibilities of this method for the Devonian sandstones in comparison with other sediments. It is obvious however, that the method for determining permeabihty from the specific resistance in other deposits will give the same results as in the Devonian sandstones, and that there is apparently no reason to limit the field of application of the method. In any case, this problem should always be solved separately for each stratum and independently of the type of deposit and the region. Kpj, m darcies Fig. 10. Curves for the dependence of the coefficient of residual water saturation Kj. ^ on the permeability K^^j.. 1 — according to the Dolina curve (1 — curve from Fig. 2); 2 — from the Dakhnov curve (2 — curve from Fig. 2); 3-7 from the empirical curves of the dependence of Q on the permeability Kp^j. (Fig. 6) ; 3 — Tuimazinsk deposit, stratum Dj; 4 — Tuimazinskian deposit, stratum -Dij, 5— Romashkinskian deposit, stratum D^Yy ; 6 — Romashkinskian deposit, stratum D^^; 7 — Sokolovogorsk deposit, stra- tum i)y ; 8 — from experimental data by Morozov, obtained from samples of the Devonian sandstone (results of separate measurements are given by points); 9 — from the curves of Morozov for porosity m = 0.2(^^). ESTABLISHING THE CONNECTION BETWEEN THE COEFFICIENT OF INCREASE IN RESISTANCE AND THE PERMEABILITY To use the method of determining permeability from the specific resistance, it is essential first of all to establish the connection between the interstitial water saturation and the permeability. Fig. 9 gives some results on the interstitial water saturation of rocks of 374 S. G. KoMAROv and Z. I. Keivsar different permeabilities. The curves 1-3 are well -known curves of Jones^^', the curves 4-9 were constructed by Masket (^^), the other curves were taken from the work of Zaks(^). Fig. 10 shows the curves for the dependence of the residual water saturation on the permeability, drawn from the curves for the dependence of the coeffi- cient of increase in resistance on the permeabiUty. The curves are drawn from the data of Figs. 2 and 3 and the actual curves Q=fiKper) (Fig. 6). The value of thecoefficient of interstitial water saturation K^ ^ was determined from the coefficient of increase in resistance Q from the formula For the curves 1 and 2 of Fig. 2 the interstitial water saturation was determined from the curves proposed by Dakhnov and Dolina A comparison of the curves on Figs. 9 and 10 shows that the values obtained when studying samples for the interstitial water saturation (Fig. 9) differ considerably from the so-called empirical value of the interstitial water saturation, with which we start in determining permeability from the specific resistance (Fig. 10)*. Let us note that methods of determining the interstitial water saturation lead to different results. The data for the interstitial water saturation obtained by analysing cores, cannot be used to estabhsh the dependence of the coefficient of increase in resistance on the permeabilityt. It follows from Fig, 10, that the residual water saturation for different rocks has a different value. This is in full agreement with our conclusion reached above that each stratum should have its own curve Q —f{K). The curves for the dependence of the coefficient of increase in resistance on the permeability, which are necessary to determine the permeability from the specific resistance, should be constructed for each stratum from the results of comparing the actual coefficient of increase in resistance with the values of permeabihty obtained in the study of cores, as was done for example, in Fig. 5. Since the connection between these values is insufficiently close, it is necessary to have a sufficiently large number of results (large number of points), at the same time bearing in mind the porosity of the strata. * An exception is provided by the experimental results of MoROZOV (8 on Fig. 10), casting some doubt in this connection. + Furthermore, in deducing from the interstitial water saturation the coefficient for increase in resistance an extra error would be introduced which would lead to a decrease in the accuracy of the curve Q = /{Kp^j). THE PERMEABILITY OF OIL-BEARING STRATA 375 The obtained curve Q =f{K) should be considered from the point of view of the possibility of using it to evaluate the permeability from the specific resistance, since there can be cases when, due to the wide scatter of the points or the small inclination of the curve Q =f{K), it cannot be used to evaluate the permeability from the specific resistance with accept- able accuracy. On the basis of what has been said above, we conclude that the determination of the permeability of the stratum from its specific resistance in any single bore-hole is undesirable for the following reasons: (a) A preHminary curve for the dependence Q =f{K) is necessary and can only be obtained if, after the appropriate treatment there is sufficient material on the permeability of the stratum for individual bore -holes. (b) The results may contain considerable errors, due to the incorrect calculation of the coefficient for the increase in resistance (errors in the determination of the specific resistance of the stratum and the stratum water, the evaluation of porosity of the stratum and the relative resistance), as well as errors connected with the influence of the water -petroleum contact. There is a probability of a considerable increase in the number of incorrect conclusions for individual bore-holes, (at the present time, generally speaking, comparatively small) due to a large error in the results for determining permeability. It is apparent, however, that in the interpretation of material for geophysical studies of single wells, it is useful to compare the obtained value of the coefficient of the increase in resistance with its values for the wells where the permeability of the stratum is known and to consider it in the interpretation. The method of evaluating permeability from the specific resistance is applicable in determining the mean value of permeability for the whole stratum. The above conclusions are based on a comparison of data obtained from the material of geophysical investigations of wells with the results of core analysis. Opinion exists that derived comparison of the results for the determinations of the reservoir properties of the strata obtained in the covirse of geophysical studies, with the data of core analysis are not convincing and do not make it possible to define the accuracy of the determinations of reservoir properties of the strata from geophysical results, since cores do not characterize a stratum as a whole, and their properties can differ considerably from those of the stratum. In actual fact, the accuracy of the determination of the reservoir properties of the strata from measurements on samples is relatively high and even with a small number of cores from the stratum they give a more accurate idea of the properties of the stratum than geophysical data. Furthermore, 376 S. G. KoMAROV and Z. I. Keivsar it should be remembered that the methods for determining reservoir properties of the strata from geophysical data, on the whole or for separate stages, are based on a comparison of the geophysical data with the results of core analysis. Thus, from the very start we assume that the true properties of the strata are characterized by the core. Evidently the data of core analysis characterize the actual permeabiHty of the stratum sufficiently well and therefore the result of comparing them to the permeability obtained from the specific resistance is very important. This is confirmed by the following: (a) the small difference in the values of the mean and mean square errors in determining the permeability from the specific resistance for a different number of cores from the stratum (Table 6); (b) the more or less identical mean values for the permeability of the stratum obtained from the data of core analysis and determinations of specific resistance. The permeability of a stratum can be determined from the results of logging. It is necessary to compare the results for determinations of permeabiHty from the specific resistance with values for the permeability obtained from the logging data. THE BASES OF THE METHOD FOR DETERMINING THE PERMEABILITY FROM THE SPECIFIC RESISTANCE Opinion exists that separate strata (for example the Devonian sandstones) have a close (approaching the functional) connection between the interstitial water saturation of the oil-bearing layers and their permeability. A conclusion is therefore drawn suggesting the possibility of the method of determining permeability from the specific resistance (3, 4, 13). However, of the actual data confirming the existence of a close connection of the interstitial water saturation with the permeability, there are only the experimental data of Morozov (see Fig. 10) which, however, are clearly insufficient to prove this affirmation. More widespread is the point of view, according to which the interstitial water saturation is a complex function of a number of factors and that therefore a close connection cannot be expected between the interstitial water saturation and permeability. Thus, according to Kotyakhov ^^' ^^^^ i^°), although a general tendency is observed of the increase in Avater saturation in the oil-bearing strata with the decrease in their permeability, there is no single and universal connection between the residual water saturation and permeability of the rocks. Further- THE PERMEABILITY OF OIL-BEARING STRATA 377 Table 6. A comparison of the results obtained for different numbers of CORES FROM THE STRATUM Tuimazinskian deposit Romashkinskian deposit Di Dib No. of cores No. of cores Indices CO c CO a o 1 2 3 o 1 1 2 3 o M No. of wells 16 16 19 46 97 6 3 4 20 33 Data of core analysis: mean value of permeability, (mUlidarcies) 610 400 390 510 485 350 625 460 590 550 mean deviation, (millidarcies) 300 285 185 185 210 110 200 145 250 270 % 50 70 47 36 44 36 30 35 50 45 mean square deviation. (millidarcies) 390 370 240 240 270 170 270 184 310 275 % 64 90 60 48 59 50 43 42 65 57 The values of permeability deter- mined from the specific resist- ance (according to Dolina): mean value, (millidarcies) 615 455 440 450 475 530 400 480 520 510 mean error, (millidarcies) 210 230 205 185 127 250 250 165 200 200 0/ /o 60 120 100 40 65 90 29 40 40 46 mean square deviation, (miUidarcies) 800 310 290 240 260 285 495 235 290 240 /o 110 170 180 60 117 122 55 60 65 70 more, even in rocks of one category, there is no single dependence between the water saturation and perraeabihty. According to Kotyakhov, the content of buried water in the strata depends not only on the physical properties of the rocks, but also on other factors, the most important being the "conditions of expulsion of water from the reservoirs by petroleum and gases, and also the physical and physico-chemical properties of the petroleum, water and gas". In this connection, Kotyakhov considers that "it is incorrect to determine the permeability of the rocks from the data of water saturation and specific electrical resistance" (^). In Baku in recent years a considerable amount of work has been done on the study of interstitial water saturation. These results are given more fully by Babalyan <^).This work indicates that the interstitial water saturation increases with the decrease in permeabiHty. However, the degree of activity of the petroleum and the type of stratum water has a considerable effect on the 378 S. G. KOMAROV AND Z. I. KeIVSAR value of the interstitial water saturation. Analogous opinions are also held by Avanesov(^): "The interstitial water saturation can be different both in the type of water and the quantity of the pore content, which is conditioned by the capillary and adsorption phenomena, and also by the physical properties of the liquids, the rocks and the structural featurss of the reservoirs". It follows from these remarks that the structural features of the reservoirs, including the permeability, usually play a less important role than the properties of the petroleum, the stratum water and the adsorption pheno- mena. This is illustrated by results on the residual water saturation (Fig. 11), recommended by Babalyan for practical use. Fig. 11. Curves for the dependence of coefficient of residual water saturation iC^_ j^ on the permeability -Kpe^ for sandstone (from the data of (2)). The rock was saturated with fresh (tap water) T, distilled D, laboratory alkaline A and stratum alkaline (water of the PK suite) S water; the extraction was carried out with inactive, i, low active 1, active a and highly active h petroleum (petroleum with different content of polar impurities). Jones <^) and Masket (^^), indicating the decrease in water saturation of petroleum-bearing strata with increase in their permeability, observed that this connection does not have a general character. According to Masket *'the value of the water saturation in the oil-bearing rocks changes within wide hmits for different reservoirs, even if their physical characteristics (for example, porosity and permeability) are close to one another". However, for individual rock types Masket admits the possibility of a closer connec- tion between the interstitial water saturation and the permeability and also THE PERMEABILITY OF OIL-BEARING STRATA 379 the possibility of some practical use for it. For example, this can be done in evaluating the total mean water saturation of a reservoir under the condi- tion of established distribution of the permeability. Wyllie and Rose^^^) suggested the determination of permeabihty of petroleum-bearing strata from the specific resistance and gave formula (■'^^) which expresses the dependence of the residual water saturation on the permeabihty. However, this method has not been used. Walstrom (^^) justifies this in the following way: "It is possible to evaluate the lower limit of the mean permeability from the interstitial water saturation and the relative resistance, obtained from data of electrical logging. In exploratory drilHng, however, the relative resistance is determined from the core taken from the productive sandstones, in which the permeability can be meas- ured directly. Consequently, it is not necessary to try to determine perme- ability from the data of electrical logging, since cores must be used for this. "Tiks'e shows that at the present time there is no method of logging which would make it possible to determine the permeabihty of rocks cut in a bore- hole" (16- P^°^ 349) _ The data given fully agreed with the conclusion which we obtained in testing the method for determining permeability from the specific resist- ance. Let us consider the possibility of determining permeability from the specific resistance from another point of view. It is generally known that the permeability varies considerably along the stratum and over wide limits. This is readily supported by the large differ- ences in the values of permeability of separate cores from the stratum. Fig. 12 shows a core diagram illustrating the sharp change in permeabihty along the stratum. In contrast to this, the curve for the resistance against the petroleum- bearing stratum differs comparatively gently; it does not have any sharp changes of apparent and specific resistances, typical of changes in perme- ability. The deviations of the curve of resistance against the petroleum- bearing stratum are caused by the form of the resistance curve against the high resistance strata. This serves as an indirect indication of the fact that the change in permeability of the stratum is not reflected in the variations of the specific resistance and that between the permeability, the interstitial water saturation and the coefficient of increase in resistance there is no direct unambiguous connection. Thus, the shape of the resistance curves indicates unfavourable conditions for using the specific resistance method in measuring the permeability. 380 S. G. KOMAROV AND Z. I. KeIVSAR 'SP \ F^r iX 2260 . > K c" ■ ^ H CI ^ 2270 ■l^ ;;■;:;; b 2280 / d^ :=) 2290 ^ §i '--'{'■ V / on % 10 20 30 10 ^Der, ^ darcies 5 100 5 1000 Fig. 12. Core diagram of a part of a typical sand-clay cross-section (ChLli)(20i page 97)^ 1 — clay; 2 — sandstones. It should be noted that if from the data of electrical logging, it would be possible to detcfmine a certain mean permeability of the stratum, it cannot necessarily be considered sufficiently characteristic of the stratum. Thus neither fractures nor the very thin beds, including some which are very permeable, are depicted on a log. On the other hand, the clayey beds show a greater effect on the resistance curve than on the permeability data of the stratum. Therefore, the obtained mean value of permeability gives an in- sufficiently accurate idea on the permeability of the stratum, which is deter- mined by a few very permeable members. As a result of what has gone before a method can be recommended for determining the permeability from the specific resistance in accordance with instructions which are given below, instead of conclusions. METHOD FOR DETERMINING PERMEABILITY OF PETROLEm/I-BEARING STRATA FROM THE SPECIFIC RESISTANCE 1. The method for determining the permeability of petroleum-bearing strata from the specific resistance is based on the presence of a connection between the interstitial water saturation and the permeability. It is also assumed that the water saturation of the petroleum -bearing stratum, deter- mined from the data of an electrical log, corresponds to the interstitial water saturation . 2. To use the method of determining permeability from the specific re- sistance for any bed, it is necessary to construct for this stratum the curve- THE PERMEABILITY OF OIL-BEARING STRATA 381 of the dependence of the coefficient of increase in resistance Q on the perme- abihty K . This is done in the following way: (a) For all Avells in which cores of the given stratum were selected and the permeability determined, the mean values of the permeabihty and the coef- ficient of increase in resistance are determined. (b) Points, the coordinates of which correspond to the average value of the permeabihty K and the coefficient of increase in resistance Q ob- tained from each of the wells are plotted on logarithmic paper. (c) The axis which has the values of permeabihty, is split up into inter- vals (for example, 10-20, 20-50 etc.) and for each interval the most probable value of Q is found (that value, with respect to which a half of the points in the given interval is above, and half below); this value is carried on to the middle of the interval; (d) From the points obtained in this way and with some averaging (to obtain a smooth curve) a curve is drawn for the dependence Q = /{K^^^. In order to obtain the curve Q = f{K) with sufficient accuracy, the number of points on the graph should be sufficiently large (not less than 15). On both sides of the main curve Q = f{Kper) auxiliary curves are drawn so that between them and the main curve there is a quarter of the total number of points on the graph. The distance from the main curve to the auxiliary curve in a direction parallel to the K axis gives the probable error in the results of determining the permeability. 3. To determine the permeability from the specific resistance the coeffi- cient of increase in resistance, Q, is found, and then from the curve Q=/{K ) for a given stratum, the corresponding value of permeability K is calcu- lated. 4. In principle the method for determining permeability from the specific resistance is applicable only to those cases where the petroleum-bearing stratum is at a distance from the vipper surface of the petroleum reservoir and where this stratum has no bottom water or an intermediate zone (the transition zone from the purely petroleum to the purely water part of the stratum). The method of determining permeability from the specific resistance of the strata containing an intermediate zone in the upper part is applicable only to thick strata provided: (a) It is separated from the water-bearing part by a bed of clay; (b) It is at a considerable (for example, more than 10 m) distance along the vertical from the upper surface of the water. Usually the error in the determination of permeability from the specific resistance is 30-50% and greater. For some strata the error is so great that 382 S. G. KoMAROv and Z. I. Keivsar the values obtained for the permeability are of no practical interest. As- a rule, the values for the permeability from the specific resistance are ob- tained with less accuracy than the values selected using the mean values of permeability for the strata. The main area of application of this method is in studying the distribu- tion of permeabiHty over a stratum as a whole. The preHminary processing of the material makes it possible in this case to improve the accuracy of certain original assumptions (the resistance of the stratum with 100% filling of its pores with water, the position of the water — petroleum contact, etc.) and to exclude the possibility of large errors arising due to this. In process- ing the data for separate wells, deviations appear between the obtained val- ues of permeability and the actual value of permeability. In view of the insufficiently high accuracy of the results, it is not advis- able to use this method in individual wells when they are in the process of being drilled. REFERENCES 1. E. T. A VANES ov, The Role of water saturation in the Mechanism of Reservoir Yield. Trudy AzNII DN No. 2, Aznefteizdat, (1955). 2. G. L. Babalyan, Problems of the Mechanism of Petroleum Yield. Aznefteizdat, (1956). 3. V. N. Dakhnov, The Interpretation of Results of Geophysical Studies of Wells. Gostoptekhizdat, (1955). 4. L. P. DoLiNA, The Determination of Petroleum Saturation and Permeability of Devonian Reservoir, from the Specific Resistance. Temporary instruction. Docu- ments of the VNII geofizika, (1955). 5. L. P. DoLiNA, The Study of Reservoir Properties of Strata from the Data of Elec- trical Logging. Report on the topic No. 421. Documents of the VNII geofizika (1954). 6. L. P. DoLiNA, Improving the Method of Studying the Reservoir Properties of Pro- ductive Strata from the Data of Industrial-Geophysical Studies. Report on the topic No. 531. Documents of the VNII geofizika, (1955). 7. P. D. Jones, Mechanics of a Petroleum Stratum. Gostoptekhizdat, (1947). 8. S. O. Zaks, The method of studying the connate waters in the oil reservoirs. Proceed- ings of the Conference on the Development of Research in the Field of Secondary Methods of Petroleum Exploitation. Izv. Akad. Nauk AzSSR, (1953). 9. F. I. KoTYAKHOV, The Principles of the Physics of a Petroleum-bearing Stratum. Gostoptekhizdat, (1956). 10. S. G. KoMAROV, Determining the porosity specific resistance. Appl. Geophysics No. 14, Gostoptekhizdat (1955). 11. S. G. KoMAROV, et al.. Improving and Introducing Methods for Determining the Reservoir Properties of Strata from Geophysical Data. Account on the Topic No. 132. Documents of VNIIgeofizika, (1956). 12. M. Masket, Principles of the Technology of Petroleum Exploitation. Gostoptekhizdat, (1953). THE PERMEABILITY OF OIL-BEARING STRATA 385 13. G. S. MoROZOV, Methods for studying the reservoir properties of the Devonian sand- stones from the data of electrical logging. Uch. zap. Kaz. gos. universiteta im, Ul'yanov- Lenin. Geology, Volume 114, Book 7 (1954). 14. S. A. SuLTANOV, The Analysis of Data of Industrial and Geophysical Studies of Well Sections in Order to Study the Reservoir Properties and the Petroleum and Water Saturation of Productive Deposits in the Devonian of the Western Bashkiria and Eastern Tataria. Thesis. Documents of the Moscow Petroleum Institute (1956). 15. L. L. SuSLOVA, The Study of Reservoir Properties of Productive Horizons in the Devonian of Saratov-Volga Basin from the Data of Industrial and Geophysical Studies. Report of a party on the topic No. 132. Documents of VNII geofizika, (1956). 16. M. P. Tiks'e, The Determination of Permeability of Rocks Revealed in Well Drilling. IV International Petroleum Congress, Vol. 2, Geophysical Methods of Prospecting. Gostoptekhizdat (1956). 17. K. P. Yakolev, Mathematical Treatment of the Results of Measurements, Gostop- tekhizdat (1953). 18. J. E. Walstrom, The quantitative aspects of electric log interpretation, Transactions AIME, Vol. 195 (1952). 19. M. R. J. Wyllie, W. D. Rose, Some theoretical considerations related to the quantitative evaluation of the physical characteristics of reservoir rock from electrical log data. Petroleum development and technology, Apr., Vol. 189 (1950). 20. A. I. Levorsen, Geology of Petroleum. Freeman Co., San Francisco, (1956). Chapter 16 NEW TYPES OF WELL RESISTIVITY-METERS E. A. POLYAKOV The specific electrical resistance of the drilling fluid (water solution of salts or a mud suspension ■which fills a bore-hole) is normally determined with the aid of a well resistivity -meter, of which the principal part consists of a small logging probe. The measurements are made according to a pro- cedure analogous to that used in the electrical resistance logging. The specific electrical resistance q of the drilling fluid is determined according to the following formula Q = K^, (1) Where / is the current strength, passing through the feeding electrode A ; Av is the potential measured between the electrodes M and N and K is the resistivity -meter coefficient. It is usually assumed that the resistivity -meter coefficient remains con- stant, which is not true in the existing resistivity-meters, since their probe coefficient varies depending on the specific resistance of the driUing fluid, and on the conditions of measurement, and is generally unstable. In determining the specific resistance of the drilling fluid, this circumstance leads to in- accuracies, often reaching 50 per cent. CAUSES OF VARIATIONS IN THE RESISTIVITY-METER COEFFICIENT Variations in the resistivity -meter coefficient can be evoked by the follow- ing causes. (1) Changes in the mutual positions of the constituent parts of the resist- ivity-meter and in the dimensions of the probe. It is obvious that any change in the mutual positions of the constituent parts of the instrument and of the distances between them leads to a redis- tribution of the electric field produced by the current electrodes of the probe, consequently causing a change in the resistivity-meter coefficient and therefore errors in the results of measurements. Owing to the shortness 384 NEW TYPES OF WELL RESISTIVITY-METERS 385 of the sonde, changes in the shape of the electrodes and in the distance be- tween them have an especially considerable influence. In order to ensure the stability of the resistivity -meter coefficient the device should be sturdily constructed and the junctions — and in particular the probe electrodes — should not change their respective positions or size when the apparatus is being used. (2) Influence of the surrounding medium. The readings of the open* resistivity-meters are sometimes influenced by the surroundings such as the walls of the ^vell (metal of the cylindrical casing tubes, practically non-conductive rocks of a high resistance, etc.). Obviously the shorter the probe the less is the influence exerted by the surrounding medium and the possibility of distortion of the results of the measurements. The following experiments were made in order to determine the length of the probe so that an open resistivity -meter would not reflect the influence of the well walls. A probe with its electrodes on one side of an insulating disc was immersed in a relatively large tank filled with water, then the electrode-bearing surface of the disc was moved towards the water surface, which represented an insulating screen. When the disc is at some considerable distance away from the screen the latter, naturally, has no influence on the measurements, but as the electrode- bearing disc approaches the screen the readings begin to rise, becoming Table 1. Percentage deviations of the readings with a screen from the readings with a distant screen. L — the length of the probe; H — the distance from the disc on which the electrodes are placed to the screen (the screen is parallel to the disc). The insulator electrode-bearing disc has a size of 10x20 cm; AM = 5 cm, MN = 5 cm, the diameter of the electrodes 1 cm; the screen 150x150 cm. LjH 0.5 0.6 0.7 0.8 0.9 1 1.2 1.5 2 Deviation with the screen made of insul- ating material 1.1 2.6 5.2 8.1 11.5 20.7 38.8 74 Deviation with the screen made of condu- cting material 0.74 1.8 3.3 5.2 7.4 12.2 19.6 33 1. Readings increase 2. Readings decrease * The open resistivity-meter is one in which the probe is not protected by a metallic or a non-conducting screen, while the influence of the bore-hole wall is minimized by the use of a short probe situated along the axis of the device. Applied geophysics 25 386 E. A. POLYAKOV progressively higher as the movement towards the screen (Table 1) pro- ceeds. Similar observations were made on moving the probe-bearing disc towards a conductive screen. In this case, however, the influence of the screen is demonstrated by the decline of the readings. From Table I it follows that, in order to prevent the bore-hole walls having any influence on the results of the measurements, the probe should be separated from the walls by not less than 1.25 times its size. In such a case the error in the results does not exceed 5 per cent. The length of the probe in the open resistivity -meter of the type GML BGK (Oufa) equals 25 mm. The container of this resistivity -meter consists of three tension shackles, which have an external diameter of about 60 mm and the probe electrodes can approach the bore -hole walls up to a distance of 25 mm. Consequently, in this resistivity -meter — < 1.25 the influence of the walls is likely, which is actually recorded in practice. In order to remove the influence of the well walls the resistivity -meter probe is commonly placed inside a flat cylinder made of a good conductor or insulating material. This, however, interferes with the drilling fluid circulating through the resistivity -meter, when the latter is moved along the bore-hole shaft. As a consequence the usefulness of the instrument is impaired . (3) Change of the resistivity-meter coefficient depending on the salinity of the drilling fluid. The coefficients of the bore -hole resistivity -meters in current usage change strongly depending on the degree of salinity of the fluid. This is represented in Fig. 1, Avhich shows the values of the coefficients of certain serial resistiv- ity-meters placed in fluids of variable salinity (curves 2-4). Obviously, in this case, the reason for the changes in the resistivity- meter coefficient is due to a rearrangement of the electric field, produced by the changes in the salinity of the ffuid. To verify this a water-filled Plexi- glas tank of variable resistance was used. The bottom of the tank was cov- ered with a sheet of zinc -plated iron. Observations on the electric field were made with the aid of two small gradient probes, which had common current electrodes (Fig. 2) and were pointing in mutually perpendicular directions ;v and z. As a result the following equation was determined. where: Av^ is the potential difference between the electrodes M^ and N^ of the probe placed along the z axis. NEW TYPES OF WELL RESISTIVITY-METERS 387 Av^ is the potential difference between the electrodes M^ and A^^ of the probe placed along the x axis. (The same current was used). x\s a consequence of these observations it was established that the value of y changes from 7 to 10 with the change of the specific resistance of the fluid from 0.5-20 D m, confirming the notion that a rearrangement of the electric field is caused by changes in the salinity of the fluid. In the quoted example, judging from the nature of the change of y, it follows that, as salinity of the fluid decreases the current -density along the x axis decreases in comparison ^vi^h the current density along the z axis. 0-06 005 004 003 0-02 6 8 0-1 Fig, 1. The dependence of the well resistivity-meter coefficients on the specific electric resistance of the fluid. 1 — REU type of resistivity-meter; 2 — RA-3 type of a serial resistivity -meter; 3— PTL VOGSGT (Ufa) serial resistivity -meter ;4-GML BGK (Ufa) serial resistivity-meter. The reason for the rearrangement of the field with the change in the sahn- ity of the fluid should, evidently, be sought in the phenomena which occur at the electrolite -metal interface, and in particular in the so-called contact resistance of this boundary. The contact resistance is not proportional to the specific resistance of the electrolite and with the decrease in the salinity of the fluid the contact resistance increases much less than the specific resistance. It is this that causes a rearrangement of the electric field consequent upon a change in the specific resistance of the fluid. 25* E. A. POLYAKOV Fig. 2. A scheme of placing of probes in an electrolyte-containing tank in testing the nature of the electric field. The resistivity-meter and many of its constituent parts are enclosed in a metallic container. Furthermore, certain of the metallic parts are situated in the immediate vicinity of the probe electrodes. As the specific resistance of the drilling fluid alters, the ratio of the contact resistance of the metallic parts to the resistance of the fluid changes as well, bringing about a re- arrangement in the electric field of the probe, since with the decrease in the salinity of the drilling fluid the intensity of the electric field grows in the direction of the shortest distance from the metallic parts of the resistivity- meter and vice versa. The rearrangement of the field leads to a change in the relative current density and correspondingly to a change in the resist- ivity-meter coefficient. It follows from (1) that: Jg_ Av ' K (3) The potential difference Zly is proportional to the specific resistance q of the fluid, to the distance / between the potential electrodes and to the modulus j being the vector component of the current density at the middle point between the measuring electrodes. Consequently equation (3) can be wTitten as follows: K = K' _1_ y'o (3a) / where /q = — is the relative current density. NEW TYPES OF WELL RESISTIVITY -METERS 389 K' is a certain constant determined by the geometric parameters of the resistivity -meter. It follows that with a rearrangement of the electric field a change in the relative current density the resistivity -meter coefficient also changes. The rearrangement of the electric field in a well resistivity-meter with a change in the salinity of the fluid is shown in Fig. 3, ^vhere there are shown diagrammatically the lines of force in a resistivity -meter GML BGK placed in a strongly and a weakly saline fluid, respectively. Fig. 3. The distribution of the Hnes of force in an electric field depending on the salinity of the fluid in a well resistivity-meter GML BGK (diagrammatic plan). 1 — in a strongly saline fluid; 2 — in a weakly saline fluid. In a strongly saline fluid the lines of force of the electric field are distrib- uted relatively uniformly. In changing over to the weakly saline fluid, when its specific resistance grows much faster than the contact resistance, the lines of force of the probes electric field become denser in the direction of the shortest distance to the metallic surfaces. This leads to a decrease in current density in the region of the measuring electrodes and consequently to the lowering of the potential difference being measured, thus causing the growth of the coefficient. The considerable increase of the GML BGK resistivity -meter 390 E. A. POLYAKOV coefficient with the growth of the fluids resistance is a fact illustrated by the data shown in Fig. 1. It is necessary to point out that the contact resistance also depends on the purity of the metallic surface concerned. Oxidation and pollution of the metallic parts as well as their corrosion can lead to a rearrangement of the electric field and to a change of the coefficient of the device. This explana- tion evidently applies to the majority of sharp oscillations of the resistivity- meter coefficient such as are observed in practice. In order to achieve a con- stant coefficient it is necessary to remove the influence of the instrumental metallic parts on the probe. SELECTION OF THE ELECTRICAL ARRANGEMENT. Various probes, differing in their sizes, shapes and mutual positions of the electrodes, will obviously have different degrees of coefficiental inconst- ancy. By using a suitable probe one can attain a constant resistivity meter- coefficient, and ensure the possibility of getting accurate results in meas- uring the specific resistance of the drilling fluid. In order to select the most suitable probe a series of electrical devices involving probes with different shapes of electrodes were constructed and investigated. The probe electrodes were mounted on ebonite discs. Thus, as is usually done in resistivity-meters, a gradient -probe was used in every instance. Fig. 4 shows some typical electrode devices, which can be divided into three groups. 1. The electrodes are discs in a linear arrangement (i in Fig. 4). 2. The measuring electrodes are concentrically arranged rings {2 and 3 in Fig. 4). 3. The measuring electrodes are rings with loops. It can be considered that a spherical surface with large cut-outs is adopted for one (A^) or both {N and M) of the potential electrodes. Consequently, the electrodes of the latter group of devices (4—8 in Fig. 4) can be called three dimensional. With the exception of the first device the current electrode is placed in the centre of the rings, which are the potential electrodes of the probe represented by the spheres formed from the loops. Using these electrode arrangements measurements were made in a tank having (1) a metallic body as the second current electrode B, (2) a body made of insulating material with the second current electrode being repres- ented by a metallic disc at the bottom of the tank. The tank was filled with sodium chloride solutions of varying concentrations. The electrode device NEW TYPES OF WELL RESISTIVITY -METERS 391 , . to-" ■;."! «'■ -• ■ '•■•'; •' * • •• >M4" ' • >^«^1,-.-. Fig. 4. Diverse types of electrical devices used in investigating their properties. 392 E. A. POLYAKOV Avas placed on the surface of the solution, so that, the probe electrodes were in the liquid, while their leads were in the air. The results of measuring the electrode coeflEcients in solutions of different concentrations are given in Table 2. The coefficient for the arrangements 7 and 8 is evidently less than for the other devices. Table 2. Percentage increase of the coefficients of The electrical devices WHEN THE specific RESISTANCE IS MEASURED FROM 0.1 TO 20 D, m Number of the electrical device In the tank 1; the coefficient is fixed at 100 per cent at 2050 m In the tank 2; the coefficient is fixed at 100 per cent at 0.1 ni 1 2 3 \ 4 5 6 7 8 40 34 ' 20 50 25 31 7 5 '^ 18.5 % ^# 20 11,5 3 2 To determine the influence of the external medium, changes in the read- ings of the electrode devices, on approach to a screen of insulating or conduct- ing material, were taken. The results of these observations are given in Table 3. Table 3. The least distance from the electrical device to the screen when THE influence OF THE LATTER IS NOT NOTICED L — the length of the probe; /• — the radius of the outer loop. Number of the electrical device Conductive screen Screen of insulating material Tank 1 Tank 2 - Tank 1 Tank 2 1 2 5 6 7 8 0.9 L 1.3 L r*- r r r 1.35 L 2.0 L 1.25 r 1.45 r r r 1.15 i 1.6 L 1.1 r 1.2 r r r 1.75 L 2.5 Z 1.55 r 1.6 r r r (*) The screen is brought up to the outer loop. Table 3 shows that the arrangements 7 and 8 are again the least influenced by the external medium, so that even when the screen touches the loops the readings of these devices do not change. Obviously 7 and 8 are the most useful of the electrode devices shown on Fig. 4. NEW TYPES OF WELL RESISTIVITY-METERS 393 The constancy of the coefficient and the small influence of the external medium on these devices are due to the mutual positions of the electrodes A and M (the electrode M is sitviated on the body of the current electrode) and especially due to the shape of the electrode N. Consequent upon the shape of the electrode N the electric field inside the sphere (outlined by the electrode loops) is stable and is not influenced much by changes in resistance and rearrangements in the electric field outside the sphere. It may appear that the electrodes A and M could be placed at the same point. In such a case, however, the measured potential difference would include the poten- tial drop produced by the contact resistance of the electrode A. This would lead to a large error in the determination of the specific resistance, affecting the constancy of the resistivity-meter coefficient. Thus, in designing a resis- tivity-meter one should start from a sonde corresponding to the electrode devices 7 and 8. THE DESIGN OF THE RESISTIVITY -METERS The well resistivity -meters REU-57 and RSE-3-57* were designed and manufactured on the basis of the most suitable electrode devices. The general shape of the resistivity-meter REU-57 is shown in Fig. 5. The principal part of this resistivity -meter, which is open, is an electrode device built analogously to the electrode arrangement 8. The body of the resistivity -meter consists of an upper (5) and a lower (3) parts connected by three tension shackles {4). The electrode device (i) is tightened in a holder (2) -which is welded to the middle of two tension shackles (9) with circular cross-sections. In the upper part of the resistivity meter there is a plug bridge (7) protected by a cylinder (6) and a hood (S). The general shape of the resistivity-meter RSE-3-57 adopted for the bore-hole gear OKS is shown in Fig. 6. The resistivity -meter consists of an upper (4) and a lower (5) cap a cont- ainer mth large (75°) longitudinal windows, and a brass tube (i) passing through the centre. The brass tube with the exception of its middle parts is insulated and protected from the body by a rubber insulator (3), while its bare part serves as the electrode A. Along a spiral groove of the middle part of the tube lies a live wire (5) which is insulated from the tube and serves as a measuring electrode M. The enclosure of the resistivity-meter serves as the electrode A^. To avoid the influence of the terminal parts of the resistivity-meter on * The construction of the resistivity -meters has been carried out by Ya. A. Magnoush- evskaya and L. O. Globus. 394 E. A. POLYAKOV WOODS HOLE, Fig. 5. The well resistivity -meter of the type REU-57. NEW TYPES OF WELL RESLSTIVITY -METERS 395 r^ _j \ J//7 i Fig. 6. The well resistivity- meter of the type RCE-3-57. the electrode device the latter is placed away from the teriniiiations for a distance approxi- mately equal to three diameters of the container. The upper part of the cap (4) has an affixed standard cable head, while the lower cap (5) is connected to the probe by a special joint. The resistivity-meter caps have channels through Avhich — as well as through the median brass tube — passes the central straird of the cable. In the resistivity -meter RSE-3-57 the elec- trode N again has a three dimensional shape and encloses the electrodes A and M, which are near to each other. As a result the measuring device of this resistivity -meter does not, in prin- ciple, differ from the arrangement 8, although superficially they appear to be very different. In order to avoid dirtying the resistivity -meter by grease aird to simplify the circulation of the drilling fluid near the electrode device all the parts of both resistivity-meters are streamlined and any corners which may preclude the motion of the drilling fluid have been eliminated. To stabilize the influence of the metallic parts of the resistivity-meter oil the probe the con- tainers and the terminal parts of the instruments are made of stainless steel, and all the instru- mental parts are nickel plated to decrease sticking of clay particles. The resistivity -meters RSE-3-57 and REU-57 can be used as independent in- struments in the existing logging installations, as well as parts of complex bore-hole devices. In certain cases it is necessary to alter the terminal parts of their bodies. Figure 1 shows the results of determining the coefficient of REU-57 in fluids of different specific resistance. It is clear that the REU-57 resistivity -meter coefficient changes much less than the coefficients of the existing resistivity- meters. With the increase in the specific resis- tance of the fluid from 0.1 to 30 the REU-57 396 E. A. POLYAKOV resistivity-meter coefficient changes only by 6 per cent ^vhich is quile acceptable. Investigations of the prototypes of the REU-57 and RSE-3-57 resis- tivity-meters in bore holes have given good results, the specific resistance of the drilling fluid being accurately determined, the walls not affecting: the instrumental readings and the instruments staying clean and unconta- minated by the drilling fluid or grease. Thus the REU-57 and RSE-3-5T resistivity-meters can be recommended for serial reproduction. CONCLUSIONS (1) The existing resistivity -meters have an inconstant coefficient, which causes errors in the residts of measurements. (2) The causes of alterations of the resistivity -meter coefficient K are: (a) Mutual displacement of constituent parts of the resistivity -meter and changes in the size of the probe. (b) The influence of the surroundings. (c) Rearrangements of the electric field owing to changes in the fluids- salinity, since the contact resistance of the metallic parts of the resis- tivity-meter has no direct relationship to the resistance of the fluid. (3) In order to achieve a constant resistivity-meter coefficient its elec- trode arrangeinent must have the electrodes A and M near to each other (the electrode M is placed on the current electrode), while the electrode N must enclose the electrodes A and M and must have a three-dimensional shape. (4) Based on the electrode arrangement described two bore-hole resis- tivity-meters REU-57 and RSE-3-57 have been constructed. These achieve an accurate deterinination of the specific resistance of drilling fluid. Chapter 17 THE USE OF ACCELERATORS OF CHARGED PARTICLES IN INVESTIGATING BORE-HOLES BY THE METHODS OF RADIOACTIVE LOGGING V. M. Zaporozhetz and E. M. Filippov Recently an increasing number of announcements of research, involving the use of radioactive logging ^^^' ^*) in bore-hole investigations, have been appearing in the overseas periodicals. With us similar studies are also being conducted. Hence, it is timely to discuss certain problems of utilizing the well accelei'ators and to consider what new techniques and methods can be introduced into the radioactive logging (RL) of bore-holes. Of all the varieties of radiation, which can be obtained by using accelera- tors, there is no sense considering any but the strongly penetrative rays, since the instrument to be lowered into a bore-hole must be securely safe- guarded in a strong container; implying that the emergence of proton, deutron, alpha particles radiation, etc., out of the instrument will be in practice impossible. Consequently ^ve will consider the use of the well accelerators as possible sources of neutrons, gamma rays and electrons. Before going on to consider the use of individual types of accelerators let us determine their general superiority over natural sources of radiation, which are commonly used in contemporary radioactive logging. The most obvious, although not the most important advantage of accele- rators rests in the possibility of obtaining with their aid a much stronger radiation than is possible with natural sources, whose strength for safety considerations has to be limited. An accelei'ator radiates only on being switched on and therefore can be rendered safe when it is brought up to the surface. Moreover, a considerable strength of the source radiation ^vill allow a sharp increase in the speed of the radioactive investigations of bore- holes. In order to avoid inaccuracies in a case where natural sources are being used this speed has to be very small. This advantage can in principal be very important, and has a significant bearing on production; thus, justi- fying the use of bore-hole accelerators. No less important an advantage is the possibility of obtaining considerable energy from the radiation acceler- ators, whereas in the case of the natural sources such energy is always limited 397 398 V. M. Zaporozhetz and E. M. Filippov to a relatively small magnitude. Thus, for instance, the maximum energy of the gamma rays from natural sources does not exceed 3 MeV, whereas with the aid of a betatron with a heavy element target one can obtain gamma quanta with an energ}' of tens of millions of electron-volts. The energy of most neutrons produced by the commonly used Po+Be or Ra+Be sources does not exceed 8 to 9 MeV whereas with the use of a neutron generator it is not difficult to obtain neutrons with an energy of the order of 14 MeV. The third feature of the accelerators, which is advantageous in radio- active logging, consists of the possibility of obtaining with their aid, a mono- energetic radiation, or at any rate a radiation with a distinctly limited maxi- mum energy. This provides a basis for the development of new methods of bore-hole investigation. However, the most important advantage of the accelerators of charged particles is the possibility of governing their radiation. This allows investiga- tion of bore-holes using a variable intensity of the source and the Avorking- out of a vast variety of intermittent methods of investigation. NEUTRON GENERATOR In investigating bore -holes the use of the so-called neutron generators- which produce beams of neutrons, is of a paramount interest. The most promising is the use of a neutron generator in neutron-gamma logging (NGL). In interactions involving the high energy neutrons and the nuclei of the rock forming atoms the principal role is played by the inelastic collisions which produce gamma radiation. A fast neutron as a result of inelastic collisions loses a considerable part of its energy and thereafter, owing to a large number of elastic collisions, it is slowed down to a thermal state. Afterwards, it gets captured by some atomic nucleus, frecjuently involving the emission of one or several gamma quanta. The gamma rays produced by the initial inelastic collision of the neutrons with the nuclei of rock-forming atoms can be considered to come into exist - ance instantaneously. Retardation of the neutrons to the thermal state is- a relatively lengthy process. Its continuity depends on the nature of the decelerator and is inversely proportional to the hydrogen content of the latter. For instance, in lead the average life of the neutrons with an energy of 14 MeV is 500-3000 /^ sec <i) whereas in water it is 2-5 ^sec^^). In consequence, the gamma rays produced by the capture of the slow neutrons are separated from the gamma rays produced by inelastic collisions with atomic nuclei by relatively long intervals of time. Thus, if during NGL the rocks are subjected to short duration (of the order of microseconds) spurts THE USE OF ACCELERATORS OF CHARGED PARTICLES 399 of neutron radiation, separated by intervals of time (several hundreds or tens of microseconds) sufficient for the capture of all the generated neutrons, and the scintillometer is smtched on for a short period after each spurt then it is possible to register only the gamma rays produced by the initial inelastic colHsions of neutrons \\ith atomic nuclei. ~^ 1 1 Ai, ^.^ B^ ^ . J h*n ^ U Crt ^ ^ Yi '' , ■^ --' i" y " - AL y A y' c 'r/ h ) 50 iOO 150 A Fig. 1. Cross-section of an inelastic scattering of the neutrons with an energy of 14 MeV. The cross -section g of the inelastic scattering of the neutrons with an energy of 14 MeV increases with the growth of the naass number A of the scattering target. Experimental studies (^) show that the relationship be- tween these quairtities is well approximated by the follo^ving equation. where : R = (2.5 + \.\A"'-) X 10-1=^ cm. For the light nuclei — -to Ashich category belong most of the rock-forming elements — a lies within the limits of 0.5 to 1.2 barns (Fig. 1) (^). The rela- tively small range of a in the rock-forming elements permits consideration of the inelastic scattering to be approximately equally likely for all of them. This circumstance precludes the possibility of carrying out neutron-neutron logging ^dth the aid of high energy neutrons. The situation is different in the case of neutron -gamma logging. Experiments, carried out on specimens of pure elements and rocks, have shown that the intensity and the energy of the gamma rays spectrum, produced by the initial inelastic collisions of neutrons mth atomic nuclei of various rock -forming elements have characteristic features (^^). This is obvious in 400 V. M. Zaporozhetz and E. M. Filippov Fig. 2 where there are shown the characteristic spectra of gamma rays pro- duced during the irradiation of specimens of various elements by neutrons with an energy of 14 MeV. There is also shown the background curve (8) ob- tained in the absence of the scattering target. The study of hydrogen and sulphur has revealed somewhat weak gamma rays with the corresponding energies of 2.2 and 2.3 MeV. 700 600 500 400 300 200 100 ■ ■ 2 1 • 1 Vi n V'j 1 <l > I Y \ V 45MeV 1 \ 20 40 60 80 100 V i ''^ e^ 20 40 60 80 100 V Fig. 2. Differential spectra of the gamma radiation produced in inelastic collisions of the neutrons with an energy of 14 MeV with the atomic nuclei of diverse substances. 1 — graphite; 2 — water; 3 — aluminium; 4 — magnesium; 5 — iron; 6 — calcium; 7 — silicon; 8 — background curve, obtained in the absence of the scattering agent. Of all the rock-forming elements only carbon (4.5 MeV) and oxygen (6.5 MeV) produce considerable gamma radiation in inelastic collisions. This allows us to anticipate the possibility of direct detection of carbon, and consequently under favourable geological conditions of petroleum, with the aid of the NGL method involving a neutron generator. The capture of the fast neutrons occurs at the same time as their elastic and inelastic scattering. Of a neutron with an energy of 14 MeV approximately 0.1 part is captured in collisions with nuclei, giving rise to radioactive nuclei. Experiments with activation of rock specimens by fast neutrons have shown that their radioactivity is connected with the formation of the radio- active isotopes 0^^ (half-life of 27.5 sec and the gamma rays energy of 1.2 and 1.6 MeV), AP^ (half -life of 2.3 min and the gamma rays energy 1.8 and 2.3 MeV) and Na^* (half -life of 15 hr and the gamma rays energy of 1.4 and 2.76 MeV). Al is formed as a result of activation of silicon, Na of alumin- THE USE OF ACCELERATORS OF CHARGED PARTICLES 401 ium, etc. It is established that a few minutes after the end of irradiation by fast neutrons the gamma rays activity in sandstones is already considerably higher than in shales, limestones, etc. Since under the conditions prevalent in bore-holes the activation of rocks will happen both due to the fast and to the retarded neutrons, in order to develop the fast neutron rock activation method of investigation it is ne- cessary to use an intermittently-operating neutron source. PRINCIPAL CONSTRUCTION PROBLEMS OF A WELL NEUTRON GENERATOR A normal well neutron generator is shown diagrammatically in Fig, 3. Its princij)al constituent parts consist of an ion source, an acceleration tube, a generator of high potential and a target. The ions which form in the ion generator fall into the acceleration tube. Here they are accelerated in the electric field, produced by the high-voltage generator, and bombard the target. Depending on the nature of the target and the ionized gas selected a certain nuclear reaction occurs, which results in the production of a specific radiation. The most widely used neutron generators are those in which ions of deuterium bombard a zircon-tritium target to produce the reaction H^(c?, n) He*, evolving neutrons, with an energy E^ on average near to ICT) Cp CO 1 _ ^ . 100-200 kV Fig. 3. A basic construction scheme of a neutron generator. 1 — an ion source with gas pressure of 10~^-10~^ mm Hg; 2 — an acceleration tube with gas pressure of 10"^ -10"^ mm Hg; 3 — a generator of high potential; 4 — zirconium — tritium target; 5 — - vacuum pump; 6 — a tritium filled cavity, connected via a palladium ventilator, with the ionization chamber; 7 — ionizer. Applied geophysics 26 402 V. M. Zaporozhetz and M. M. Filippov 14 MeV. The high efficiency of the reaction H^((i, n) He* and the high energy of the resultant neutrons, as well as the small values of the accelerat- ing potential necessary for it, make this reaction the most useful among all the reactions known to be used to generate neutrons for radioactive log- ging. For the sake of comparison let us consider the reaction lP{d, n) He* produced by bombarding a deuterium target. This reaction previously has been very widely used to generate neutrons, but the quantity of neutrons formed in this reaction is much less than in the previous reaction. The neutron energy is also much less. For instance, when E^ = 0.2 MeV the value of £"„, which depends on the angle of motion of the neutrons, varies within the limits of 2-3 MeV. The neutron energy associated with the reaction }i^{d,n) H*, in general, depends on the angle of emission of the neutrons out of the target, but this dependence weakens as the energy E^ of the accelerated ions of deuterium,^ or in other words the accelerating potential, decreases. With the present day techniques of electrification of bore-hole instruments it is difficult to obtain potentials higher than 150-200 kW. The corresponding deutron energy will produce a neutron energy of radiation within the limits of 13-16MeV<^). If the deutron energy is < 0.5 MeV the cross -section of the reaction H*(cf, 7i)He* is determined by the following formuJa^^); 58exp[-1.72£d~°-^] a = Ed[l + 5705 {Ed-0.096)^' where £'j£ is expressed in mega-electron-volts and a in barns. Table 1 shows the dependence of the cross -section of this reaction on the energy of the deutrons. The reaction has a sharply developed maximum o" = 4 bams when E^ ^ 0.26 MeV. Since in the target a neutron is relatively rapidly slowed down, it would be sensible to accelerate the deutrons up to energies higher than those which would promote the maximum efficiency of the re- action. This, however, is precluded by difficulties of insulating the high -voltage electrical mechanism in the well apparatus. This circumstance hmits the highest possible potential to the maximum of 250 kW. Let us consider the requirements to be satisfied in the construction of a well neutron generator. The well generator has limited dimensions. At any rate its diameter must not exceed 15 cm. The planning of a generator within these limits is a diffi- cult technical problem. In the first place it is difficult to ensure dependable insulation of high -voltage links; and a high voltage is necessary for the work- ing of the acceleration tube. Even a greater difficulty appears owing to the THE USE OF ACCELERATORS OF CHARGED PARTICLES 403 necessity to maintain vacuum in the acceleration tube (10~^ to 10~^ mm Hg). To this end the following methods can be employed. Table 1. Comparison of the effectiveness of the reactions H^ {d,n) He^ AND K^{d,n) He* The energy of a deutron in MeV. Cross section of the reaction E.^{d,n)}ie^ in bams. 0.005 0.01 0.02 0.04 0.05 0.07 0.09 0.105 0.105 0.1 Cross-section of the reaction H3(c?,re)He* in bams. 0.3 0.7 4 3.3 2 0.75 ~0.2 ~0.2 ~0.2 ~0.2 Ratio of the reaction cross-sections H3(f,ra)He* and H2(d.re)He3 0.025 0.05 0.1 0.2 0.3 0.5 1 1.5 2 0.005 0.01 0.02 0.04 0.05 0.07 0.09 0.105 0.105 0.3 0.7 4 3.3 2 0.75 -0.2 -0.2 ~0.2 60 70 200 82 40 10.7 ~2 ~2 -2 1. Pumping out the gas using a vacuum pump The use of an ordinary — for instance a steam oil — vacuum pump in a well generator is unsuitable for the following reasons: (a) the need for a vacuum cyhnder which w^ould occupy a lot of space in the well instrument; (b) the difficulty of cooling the pump. Since, as a rule, the drilling fluid in the bore- hole has an elevated temperature it would be necessary either to introduce a cooling device (e.g. a semiconductor) or to introduce a cold substance (solid carbon dioxide, ice, etc,). However, the use of cooHng devices which are inefficient demands a considerable power transmission through the cable of the apparatus, whereas the use of a cold substance involves operational invonvenience, such as a frequent necessity of taking apart and reassembling the instrument, etc. Consequently, it is advisable to find other less cumber- some means of maintaining vacuum in the acceleration tube. 2. The use of a sealed tube It is possible to avoid the necessity of pumping the gas out if a sealed acceleration tube — necessary for the acceleration process — is filled with a suitably rarified gas. The construction scheme of such a tube is shown in Fig. 4(14). Positive ions, which spontaneously form around the electrode 4, are accelerated by the potential difference E^ existing between the electrode and the grid 3 and enter the acceleration interval and then the target. As the target 1 is being bombarded by the accelerated neutrons nuclear reactions- occur. These depend on the nature of the target and the gas with which. 26* 404 V. M. Zaporozhetz and E. M. Filippov the tube is filled. The grid 2 prevents the electrons ejected from the target 1 to fall on to the grid 3. The distance between the electrodes and the grid is chosen so as to exceed considerably the average free path of the gas ions under a particular pres- sure. As a result on the way from the electrode to the grid the ions undergo many collisions with the gas molecules, thus evoking its additional ioniz- ation. The distance between the grid and the target is chosen so that it is smaller than the free path of an ion. The difficulty of producing such a tube lies in the fact that with the mini- mum gas pressure, which is necessary to ensure a sufficiently energetic run- ning of the process of ionization (10~^ to 10~^ mm Hg), the average free path E3 + - X E, Fig. 4. A basic plan of a sealed acceleration tube with equal pressures at the ion source and the tube. of the ions becomes smaller and the target 1 and the grid 3 of the tube have to be so near to each other that it is difficult to insulate them from each other owing to the high potential necessary for the effective running of the nuclear reactions. Another disadvantage of such a tube is its lower efficiency compared with the tubes of normal type. Whereas in the normal tubes the ratio of the intensity of neutron radiation to the current strength of the accelerated ions is usually equal to 10^ n/sec fxK in tubes with deficient discharge it does not exceed 10^ n/sec /^ A. It is necessary to point out that if the sealed tube is filled with deuterium THE USE OF ACCELERATORS OF CHARGED PARTICLES 405 and a tritium, target is being used, as a result of the change in the gas content in the tube, conditioned by the diffusion exchange of the target and the tube hydrogen, there arises a certain degree of instabihty in the work- ing conditions of the tube. This can be avoided if a mixture of deuterium and tritium is introduced into the tube. In such a case, when small accelerating potentials are employed, the principal role is played by the reaction H*(c?, re) He* since on collision vdih deutrons the cross -section of this reaction is considerably smaller. Evidently the filling of the tube ^vith a mixture of H^ and H^ lessens the neutron emission by approximately half, per unit of strength of the ion current, in comparison with values obtained when the tube is filled with a single-component gas, since in the former case nearly half of the collided hydrogen atoms would be used up in useless reactions H^ {d, n) He^ and H^ {d, n) H^. On the whole we consider the testing of a sealed tube in a well neutron generator to be important. 3. Use of absorbents It is possible to avoid the installation of a vacuum pump in a well neutron generator by placing an absorbent in the acceleration tube. Titanium, tanta- Uum, zirconium and certain other substances can be used as absorbents of hydrogen. For instance, it is known that under a pressure of ~10~* mm Hg titanium on heating up to 200 °C (optimal temperature of absorption) is capable of absorbing up to 1700 cm^/g of hydrogen. As a result of experiments conducted by G. D. Glebova^*) it follows that at a temperature of about 40°C the velocity of absorption of hydrogen by metallic titanium is approximately equal to 60 cm^H/cm^ Ti sec. Obviously the velocity of absorption of deuterium would be 1/2 times less. Let us determine the probable velocity of arrival of hydrogen — derived from an ionic source — into the acceleration tube. Experience of using neutron generators shows that the relationship between the intensity of the neutron radiation and the current strength of accelerated ions is usually equal to lO^n/sec ^aA. Consequently, to obtain a ra- diation of lO^n/sec a current of 10 //A is necessary in which case 6.25 — 10^^ ions 10 n/sec per second are transported. Thus, — XlOO% = 1.5X10^ 6.25 X 10^^ ions/sec per cent ions of deuterium react with the tritium of the target. Adopting the ratio of the atomic to the molecular ions to be equal to 0.5 and an ionic source as given above, we obtain the velocity of arrival of the gas into the acceleration tube to be 3.6 x lO^^D/sec or 1.7 x 10^ cm^/sec under atmospheric pressure. This implies that the increase in the gas content of the acceleration tube in operating a neutron generator is so small that with the aid of an 406 V. M. Zaporozhetz and E. M. Filippov absorbent it is possible to rely on the possibility of supporting the necessary vacuum in the tube during a 6-8 hour worldng day. This interval of time is completely sufficient to investigate a bore-hole. Afterwards the absorbing substance can be changed. It is completely sufficient to maintain pressures of the order of 10~* mm Hg in an acceleration tube. In order to ensure the absorption of the gas flowing into the tube under this pressure the necessary surface of the absorber 1 7 X 10"^ ' t— should be of an order of — x7.6x 10^x1/2 r^S cm^ . Considering 60 the unavoidable contamination of the constituent parts of the tube and also its imperfect hermetic qualities this figure should evidently be increased several times. An absorber possessing such a surface can be used in a bore-hole device. It is considered quite desirable to test the use of absorbers. 4. The use of an ion pump with an absorber In order to maintain vacuum in an acceleration tube it is also possible to use an ion pump with an absorber. This is a device in which gas molecules are ionized in an electric field and are directed towards an absorbing surface. However, such a method of maintaining vacuum is evidently only slightly more effective than the use of an absorber by itself (^^). 5. The use of an ion pump to extract the gas Figures 5 and 6 show two types of sealed acceleration tubes. Fig. 5. A basic construction scheme of a sealed acceleration tube with different pres- sures at the ion source and the tube. 1 — target; 3 — an electrode extracting the ions; 4 — an electrode; 5 — a device ensuring the pressure difference between the chambers 6 and 7 ; other notations are the same as in Fig. 4. THE USE OF ACCELERATORS OF CHARGED PARTICLES 407 In the tube shown in Fig. 5 there are two chambers (6 and 7) with different concentrations of gas in them. The necessary difference of pressure between the two chambers is maintained by the device 5, which is an ion pump with a cathode which does not absorb the ionized gas. Experimentation with an ion pump of this type has shown that with a relatively small expenditure of effective power it can ensure a pressure difference of 10"^ to 10~^ mm Hg C^). Table 2. Expenditure of deuterium in the acceleration tube Molecular free path in cm Expenditure Df deuterium Pres- sure To form 10« neu- trons Source power when is 10^ n/sec working The quantity of the gas used in 10 hrs m mm Hg 1 sec 1 hour 10 hours In the accelera- tion tube In the ion source (11) (0.51) 760 11.77-10-« 2.1-10-1* 2.1 -10-" 7.5-10-8 7.5-10-7 7.5-10-10 1.5 -10-9 1 0.009 1.6- 10-" 1.6-10-s 5.7-10-5 5.7-10-4 5.7- 10-' 1.15-10-8 10-1 0.09 1.6-10-10 1.6-10-'' 5.7-10-4 5.7-10-3 5.7-10-« 1.15-10-5 10-2 0.9 1.6-10-s 1.6-10-* 5.7-10-3 5.7-10-2 5.7-10-5 1.15-10-4 10-3 9 1.6-10-8 1.6-10-5 5.7-10-2 5.7-10-1 5.7-10-4 1.15-10-3 10-4 90 1.6-10-'' 1.6-10-* 5.7-10-1 5.7 — 5.7-10-3 1.15-10-2 10-5 900 1.6-10-« 1.6-10-3 5.7 — 57 — 5.7-10-2 1.15-10-1 io-« 9000 1.6-10-5 1.6-10-2 57 — 570 — 5.7-10-1 1.15 — Figure 6 shows an interesting variety of sealed acceleration tube in which a single device performs the functions of an ion pump and an ion source for the acceleration tube. In Table 2 there are presented the results of an estimate of expenditure r 3 — + Fig. 6. A basic construction scheme of a sealed acceleration tube in which the ion pump 5 also serves as an ion source. IV — power source of the ion pump; other notations are the same as in Figs. 4 and 5. 408 V. M. Zaporozhetz and E. M. Filippov of deuterium in an acceleration tube. The following factors were considered in this calculation. (a) The average molecular free path in a deuterium or hydrogen atmosphere is equal to 11.77 X 10~^ cm at 760 mm Hg pressure. (b) There are 6 x 10^^ diatomic molecules in 22 htres of deuterium at temperature and 760 mm Hg pressure. (c) The volume of an acceleration tube in a well neutron generator equals 1000 cm^; the volume of an ion source is 500 cm^ and the operational temperature 40 °C, while the duration of an uninterrupted run of the mecha- nism (until it is filled with gas) is 10 hours. The neutron emission is lO^n/sec, namely 100 times more than is normally obtained in logging with Po +Be sources. As is evident from the Table 2 the expenditure of deuterium in a nuclear reaction of the type H^(i, ra)He* is very small. Even if the source produces lO^n/sec in a sealed acceleration tube which is shoAvn diagrammatically in Figs. 5 and 6, the expenditure of deuterium, with the gas pressure at the ion source being not less than 10~^-10'* mm Hg, does not exceed 1 per cent per eighteen working hours. Thus, the tube is capable of working for a long time without renewal of the gas that fills it. GAMMA QUANTA GENERATOR Well accelerators are used as sources of gamma particles. First let us consider what new^ data can be obtained by using these accelera- tors in gamma -gamma logging. The contemporary demands of safety in Avorking on bore-holes do not al- low the use of natural uninterrupted radiating sources stronger than 30 m Cu. The relative safety of working with an accelerator allows the use of much more powerful gamma quanta in gamma-gamma logging. This makes it possible to increase the velocity of running GGL without lowering its accuracy and to conduct investigations involving considerable distances (provided the distance between the mid -points of the source and the indicator is large). A source of gamma quanta no stronger than 30 mCu allows us to conduct the GGL method of bore-hole investigations with the aid of the standard apparatus RARK (containing one discharge counter of the type VS-9) with an additional device which screens the counter from the radiation scattered by the drilling fluid. In such a case the deflection of the registered GGL curve from statistical fluctuations wiU be no more than 1 % if the device is raised at a velocity no higher than 500 m/hr. The use of powerful accelerators of gamma quanta presents a possibihty THE USE OF ACCELERATORS OF CHARGED PARTICLES 409 of increasing the speed of measurement and of making measurements with large sondes. This, as is known from theory and from experimental bore -hole investigations (^» ^"^ involves a diminution in the influence of the bore- hole, leading to an increase in the sensitivity of the GGL method to the density variations of rocks. The second circumstance favourable for GGL is the possibility of obtaining a high energy with the aid of gamma ray accelerators. The desirabihty of using high energy sources during GGL is obvious from the followdng considerations. The use of cobalt sources of gamma quanta with an energy of 1.25 MeV in GGL allows the investigation of a rock stratum (density 2.4-2.8 g cm^) no thicker than 5-7 cm ^^' ^°) in a bore-hole. Consequently, the presence of a mud crust on the bore-hole .wall and a layer of drilhng fluid between the crust and the body of the apparatus leads to a considerable diminution in the accuracy of determining the density of the rock. The increase in the gamma quanta energy increases the depth of investiga- tion* and considerably diminishes the influence of the above mentioned factors on the results of investigations. Calculations have shown that if the energy of the gamma quanta source is increased from 1 to 15 MeV the average free path of a gamma quantum in the rocks possessing a density of 2-3 g/cm^ increases four to five-fold and correspondingly the depth of investigation also increases. The increase in the source energy of the gamma quanta to over 15 MeV is useless since in conjunction with such an increase the phenomenon of formation of electron-position pairs <^°^ also increases and the penetration depth does not grow much. In order to obtain more accurate information on the rocks investigated in bore -holes by the GGL method with a high energy gamma quanta generator it is necessary to register the soft part of the spectrum of the scattered gamma ray (0.05-0.07 MeV) by using, for instance, a differential gamma spectrometer. This helps to increase the depth of investigating the rocks. This is connected with the fact that the soft gamma radiation reaching the gamma quanta recorder arrives from those rock strata which are the most distant from the instrument. GAMMA-NEUTRON LOGGING (GNL) To study the geological sections of bore-holes and to determine the constitution of certain elements in the rocks gamma neutron logging may * In this context implies the power of penetration of the rays. [Editor's footnote] 410 V. M, Zaporozhetz and E, M. Filippov Table 3. Photonuclear reactions* Periodic Elemental symbol Atomic Abundance Reaction Maximal number of weight of in percents threshold Reaction the element the element {y, n) MeV output (y, n) 1 H 2 0.015 2.23 2 He 4 100 20.6 24.0 3 Li 6 7.52 5.35 — 7 92.47 7.15 17.5 4 Be 9 100 1.67 30 5 B 10 18.45 8.55 — 11 81.35 11.50 — 6 C 12 98.89 18.7 22.4 7 N 14 99.64 10.5 22.5 8 16 99.76 15.5 22.5 9 F 19 100 10.4 22.2 10 Ne 20 90.92 16.9 21.5 22 8.82 10.4 — 11 Na 23 100 12.05 12 Mg 24 78.60 16.4 — 25 10.11 7.25 — 26 11.29 11.15 — 13 Al 27 100 12.75 19.6 14 Si 28 92.27 16.8 — 29 4.68 8.45 — 15 P 31 100 12.05 17.19 16 S 32 95.1 14.8 — 34 4.2 10.85 — 17 CI 35 75.4 9.95 19.0 37 24.6 9.5 — 18 Ar 40 99.6 9.8 20.0 19 K 39 93.08 13.2 — 20 Ca 40 96.97 15.9 19.6 22 Ti 46 7.95 13.3 — 48 73.45 11.6 — 49 5.51 8.7 — 23 V 51 99.76 11.15 17.7 24 Cr 50 4.31 13.40 19.0 52 83.76 11.80 17.5 i 53 9.55 7.75 19.7 * This table is composed principally from the data given by Segre (^ ) with certain other additions (11, 16, 17, 19, 21). THE USE OF ACCELERATORS OF CHARGED PARTICLES 411 Table 3 continued. Periodic number of Elemental symbol Atomic weight of Abundance Reaction threshold Maximal reaction the element the element in percents (y, n) MeV output (y, n) 25 Mn 55 100 10.1 18.4 26 Fe 54 5.84 13.8 18.3 56 91.68 11.15 57 2.17 7.75 — 27 Co 59 100 10.25 17.3 28 Ni 58 67.75 11.17 18.5 60 26.16 16.0 61 1.25 7.5 — 29 Cu 63 69.1 11.0 17.5 65 30.9 10.0 19.0 30 Zn 64 48.98 11.65 18.5 66 27.81 11.15 — 67 4.11 7.0 — 68 18.56 10.15 — 70 0.62 9.2 — 31 Ga 69 60.2 10.1 71 39.8 9.05 — 33 As 75 100 10.2 17.3 34 Se 82 9.19 9.8 — 9 — 7.3 — ? — 9.35 — 35 Br 79 50.52 10.65 16.0 81 49.48 10.1 16.0 37 Rb 87 27.85 9.3 17.5 38 Sr 86 9.86 9.5 — 87 7.02 8.4 — 88 82.56 11.15 — 40 Zr 90 51.46 12.2 18.0 91 11.23 7.2 — 41 Nb 93 100 8.7 17.0 42 Mo 92 15.86 13.28 18.7 97 9.45 7.1 — — — 6.75 — — — 7.95 — 44 Ru ? 7.05 ? — 9.50 — 45 Rh 103 100 9.35 16.5 46 Rd ? — 7.05 — •? — 9.35 — 47 Ag 107 51.35 9.5 16.0 109 48.65 9.05 16.0 412 V. M. Zaporozhetz and E. M. Filippov Table 3 continued. Periodic, number of Elemental symbol Atomic weight of Abimdance Reaction threshold Maximal Reaction the element the element in percents (y, n) MeV output (y, n} 48 Cd 113 12.26 6.5 49 In 115 95.77 9.05 — 50 Sn 118 24.01 9.10 — 119 8.58 6.55 — 124 5.98 8.50 — 51 Sb 121 57.25 9.25 14.5 123 42.75 9.3 14.5 52 Te 9 6.5 ? — 8.55 — 53 I 127 100 9.3 15.S 55 Cs 133 100 9.05 — 56 Ba •? 6.80 9 — 8.55 — 57 La 138 0.09 15.5 139 99.01 8.8 13.a 58 Ge 140 88.48 9.05 142 11.07 7.15 — • 59 Pr 141 100 9.4 _ 60 Nd 150 5.6 7.4 — 73 Ta 181 100 7.6 13.5 74 W ? 6.25 — 9 — 7.15 — ■75 Re 187 62.93 7.8 77 Ir 193 61.5 7.8 — 78 Pt 194 32.8 9.5 — 195 33.7 6.1 — 196 25.4 8.2 — 79 Au 197 100 8.0 13.5 80 Hg 201 13.22 6.25 6.60 81 Tl 203 29.5 8.8 — 205 70.5 7.5 — 82 Pb 206 23.6 8.25 207 22.6 6.88 — 208 52.3 7.40 — 83 Bi 209 100 7.40 13.3 90 Th 232 100 6.35 — 92 U 238 99.27 5.97 13.9 THE USE OF ACCELERATORS OF CHARGED PARTICLES 413 be found suitable. In this method the bore-hole wall is irradiated by a powerful beam of gamma quanta of high energy while photo -neutrons, which are knocked out by the gamma quanta from the atomic nuclei, are registered •with the aid of a neutron recorder. Table 3 gives a list of rock -forming elements, which participate in photo - nuclear reactions, the threshold values of the gamma quanta energy which may lead to the appearance of photo-neutrons, and also the maximal energy of the gamma quanta which produce the maximum emission of neutrons. K IS 10-^ _-• — "—— ,•' y / / / / 1* ^ / • / / / 10-^ , / ' .^ • 1 60 80 100 120 140 160 180 200 A Fig. 7. The dependence of the neutron output per one gamma quantum on the elemental atomic weight A for thick targets of these elements. The gamma-quanta energy of the low atomic weight elements (up to 20) is 2.76 MeV and for the other elements is 17.6 MeV. Table 3 shows that the values for the energy of the gamma quanta — ■which cause the knocking of neutrons out of the elemental nuclei — is least for beryllium (1.67 MeV) and deuterium (2.23 MeV). For all the other elements the value of this energy is no less than 6 MeV. Table 4 and Fig. 7 show the emission of neutrons for 1 gamma quantum. Evidently with the increase in the atomic weight of an element the production of neutrons per one gamma quantum at first grows and then becomes constant and approximately equal to 2 x 10~^ neutrons per 1 gamma quantum. The diminution of the gamma quantum energy leads to the diminution of the production of neutrons per 1 gamma quantum. Thus, for silver the diminution of the gamma quanta energy from 17.6 to 10 MeV leads to a 70 times lower neutron production per 1 gamma quantum <^^ 414 V. M. Zaporozhetz and E, M. Filippov The most promising outlook for the GNL method exists in the possibihty of discovering oil-bearing strata from a higher content of deuterium in oil (1.5-3 times exceeding that in water (^)). Since the threshold energy of photo- spHtting of deuterium is not high (2.23 MeV) an accelerator is used which has a relatively low energy of gamma quanta. The building of such an instru- ment will not involve unsurpassable difficulties. In particular, in such a case, no confusion will arise as a result of the formation of photo -neutrons in the material from which the instrument is constructed or the material of the bore-hole casing tubes for which the threshold energy exceeds 6 MeV. Table 4. Neutron output per gamma quantum Periodic number of the element Elemental symbol Atomic weight of the element Neutron output per gamma quantum Gamma-quantum energy in MeV 1 4 13 47 74 H Be Al Ag W 2 9 27 109 9 7 • 10-* 3 • 10-* 3 • 10-3 1.7 • 10-2 2 • 10-2 2.76 2.76 17.6 17.6 17.6 The measurement of the density of photo -neutrons can be made on the background formed by the natural neutron radiation of the rocks. However, the intensity of the latter is very small and it will not preclude the detection of photo -neutrons. To confirm let us make the following estimate. In petroleum there i& 13 per cent of hydrogen; in which the average content of deuterium will be 4 X 10~^ per cent (^). If we accept the porosity of sandstones to be 30 per cent we wiU see that deuterium forms 4 xlO~^ x0.13 xO.3 = 1.6 xlO~^ per cent of the mass of the oil-bearing stratum. Since the production of neutrons per one gamma quantum is of the order of 7.4 X 10~* (with a thick target) for deuterium, the probability of obtaining photo -neutrons in an oil bearing layer will be 7.4 x 10"* x 1.6 x 10~^ x 10"^ = 1.2 x 10~^ per gamma quantum. Since the investigation depth in the bore -hole rocks is of the order of 10 cm the probability of the photo -neutrons, which are formed in the stratum and have an energy of about 2 MeV, to reach the recorder will conform to the following rule. P- 4nr^ where : X is the average free path of the fast neutrons in the substance under investigation. THE USE OF ACCELERATORS OF CHARGED PARTICLES 415 r is the average distance of the centre of formation of photo -neutrons from the recorder. Accepting according to S. A. Kant OR (Private communication) A = 5 cm and r = 10 cm we obtain P = 1.1 xlO"*. The volume of the rock with which the gamma radiation (in the instance of a beam of gamma quanta) will interact, will be no less than 1000 cm*. Consequently, the probability of the photo -neutrons reaching the recorder of the fast neutrons mil be: 1.2 X 10-8 X 1.1 X 10-* X 103 = 1.3 X 10-9 The effectiveness of the fast neutron scintillation counters (diameter 4 cm; length 10 cm) used in bore-hole investigations is approximately 0.1 per cent. Consequently, the probability of registering one neutron will be Pg = 1.3 X 10-12. The following values of intensity of natural neutron radiation are known: sedimentary rocks in the presence of the cosmic background —230; extrusive rocks (at depth) — 610 ; pegmatites — 3000 n x cm-^ x day-^. Based on these data it is possible to consider that in oil wells the natural neutron back- ground will not exceed 0.003 n X cm-^ x sec"!, whereas in a rock layer with a cross-section of 100 cm^ the natural neutron background I^^ affecting the readings in GNL the neutron stream should exceed several-fold the natural neutron background. For this it is necessary to have a source possessing the following power where: m is the demanded increase of the induced neutron radiation over the natural; M is the power of the source measured in gamma quanta per second. Substituting the previously quoted values for I^^ and Pg and assuming m = 20, we obtain M = 4.6 x lO^^. It is interesting to examine the possibility of using the gamma-neutron log for the following purposes. 1. Prospecting for beryllium-bearing rocks. Using a method similar to the one used above it is possible to show that to distinguish rocks containing up to 5 per cent of beryllium a source with a power of the order of lO^i gamma quanta per sec is necessary. In this connection it is necessary to notice that V. N. Dakhnov's suggestion about beryllium prospecting using the GNL method with an antimony source of gamma quanta from the point of view of safety of bore -hole working is not easily realizable. 2. Recognition of sandstone layers from their content of the isotope of 416 V. M. Zaporozhetz and E. M. Filippov silicon with an atomic w^eight of 29. In this case a gamma quanta source of vip to 10^^ gamma quanta per second is necessary. 3. Recognition of rocks containing rare elements with the atomic weight of an order of 200 in a bore-hole section. In order to recognize the rare elements of such a high atomic weight when their content in the rock varies from tenths to hundredths of one per cent it is necessary to use a power source of an order of 10^° gamma c[uanta per second. Scintillation neutron counters used in radioactive logging, besides the neutrons, register also gamma quanta. Let us consider how the scattering of the gamma ray will influence the results of gamma-neutron logging in bombarding the deuterium of a water-oil contact. The effectiveness of the scintillation neutron indicators using the gamma quanta with energy of an order of 1 MeV is approximately 100 times less than their effectiveness in using neutrons. Let us assume that the generator produces a directed beam of gamma quanta with an energy of 3 MeV. In reaction wth the electrons of the rock atoms this radiation as a result of one act of scattering will lose an average of 50 per cent of energy (^°), i.e. it will diminish to 1-1.5 MeV. By a proper selection of the displacement potential in the network of the amplifiers input cascade it is possible to succeed having the non-corresponding impulses unrecorded by the registering instrument. However, owing to the great power of the gamma quanta source there will be a certain number of correspond- ing impulses which would influence the readings. If the source power is 4.6 X 10^2 gamma quanta per sec, and the resolution time of the electronic device of the apparatus used in radioactive logging of an order of 10 /t/sec, and the effectiveness of the recorder towards the gamma quanta is of an order of 10~^, then the number of corresponding impulses is of an order of 10 X 10-6 X 10-5 x4.6 X 1012 _ 460 per sec. When the power of the gamma quanta source is 4.6 x 10^^ gamma quanta per sec then m — 20. The speed of the photo -neutron count wll be equal to 0.3 n/sec x20 = 6n/sec. Thus the influence of the synchronously corresponding impulses of the scattered gamma radiation will considerably exceed the effect measured by GNL. To surpass such an influence it is necessary to increase the network displacement of the spectral input into the amplifier so as to distinguish completely between the synchronously corresponding impulses. Tlais is done by increasing the potential twice in comparison with such as is necessary to suppress the non-corresponding impulses. This leads to a slight, but only slight, diminution in the speed of the neutron count. THE USE OF ACCELERATORS OF CHARGED PARTICLES 417 Table 5. Nuclear reactions, used in obtaining gamma quanta* Nature of the reaction Nature of the target Resonance energy of protons and alpha particles Gamma- quanta energy in MeV Gamma- quanta output per proton Maximal available power of the source in Cuf B^Hp,y)0' B4C 0.162 16.3 11.8 4.2 • 10-11 1.7 • 10-1" 6 • 10-' Fi9(/),a,y)Oi6 CaFa, NaF 0.338 6.3 1.74 • 10-8 5 • 10-6 Li'(p,y)Be^ Li (metal) 0.460 17.6 14.8 1.9 • 10-8 0.95 • 10-8 8 • 10-5 Si29(;j,y)P3° Si^'Ogis 0.326 0.414 5.86 5.27 0.7 — — Na23(j5,y)Mg2* NaCl 0.310 4.24 1.3 • 10-11 ~4 • 10-8 Na23(p,a)Ne2i 0.287-0.539 — 2.0 • 10-12 — Mg2*(p,y)AP5 — 0.226 0.418 — 7 • 10-11 2 • 10-1° ~2 • 10-'^ 3 • 10-^ Mg25(p,y)A126 — 0.317 0.391 0.496 — 3 • 10-11 1 • 10-10 1 • 10-' 3 • 10-' H3(p,7)He* — 0.1-6.2 20 7 • 10-11 2 • 10-' H3(a,7)He5 H3(a,n)He* — 0.16 0.16 — 2 • 10-12 5 • 10-9 * Composed from the data in t^^, ^3, 20, 2)_ t Number of gamma quanta equals the strength in Curies multiplied by S.7 X IQi". BASIC PROBLEMS IN CONSTRUCTION OF A BORE-HOLE GAMMA QUANTA GENERATOR A normal acceleration tube or a betatron is used as a gamma quanta generator. The construction of a gamma quanta generator using a normal acceleration tube in principle does not in any way differ from the construction of a neutron generator which is shown diagrammatically on Fig. 3. However, in order to obtain gamma quanta unstead of neutrons it will be necessary to use other targets and ionized gas. Nuclear reactions which can be used to obtain gamma quanta are given in Table 5. The data on the possible power of a gamma quanta source, which are given in Table 5 are based on the following deductions. From the data on Applied geophysics 27 418 V. M. Zaporozhetz and E. M. Filippov the well neutron generators it is known that currents of charged particles in such generators cannot exceed 10 ^A, which corresponds to -^10^* protons per sec. Knowing the relative output of gamma quanta per proton it is easy to obtain the power of the gamma quanta per second. Knowing the relative output of gamma quanta per proton it is easy to obtain the power of the gamma quanta source. From the Table 5 it is evident that from the point of view of a maximum output of gamma quanta and of obtaining the necessary energy for this the best targets are of Uthium and fluor. The use of the former target allows obtaining gamma quanta energy up to 17.6 MeV; while the use of the latter up to 6.3 MeV. Let us consider whether these sources are sufficiently powerful for GGL and GNL. For GGL with scintillation counters of gamma quanta it is necessary to have the activity of the gamma quanta source in the proximity of the counter of not less than 10~^Cu. The distribution of the scattered gamma radiation in a homogeneous medium is expressed by the formula /- Qe- 47ll^ where : Q is the power of the gamma quanta source; t is the average coefficient of gamma radiation weakening in the rock. I is the sonde length. 6 is a coefficient which is a function of the rock density. Table 6. The source-power necessary for GGK in curies Gamma-quanta Length of the probe in cm energy in MeV 20 30 40 50 60 70 10.0 17.6 2.7 • 10-5 2.9 • 10-5 8.2 • 10-5 8.6 . 10-5 1.6 • 10-* 2.0 • 10-* 4.3 • 10-* 4.0 • 10-* 7.9 • 10-* 6.8 • 10-* 1.4 -10-3 1.14 • 10-' Table 6 shows the power of the gamma quanta source wth energies of 10 and 17 MeV calculated for different lengths of the sonde according to the above mentioned formula. In the table the quantities of t^^ and b are adopted from(^). Comparing the data of this table with the data of the Table 5 it is evident that the gamma quanta generator with an acceleration tube can be used only with GGL probe of no more than 40 cms. As has been shown already, to carry out GNL gamma radiation sources with a power of lO^^^-lO^^ gamma quanta per second (1-10 Cu) are necessary. This exceeds the permitted power of a gamma quanta generator with an acceleration tvibe. THE USE OF ACCELERATORS OF CHARGED PARTICLES 419 Consequently, at present, charged particle accelerators which can accelerate protons up to energies of 0.5 MeV, cannot be used in the construction of a gamma quanta generator useful for GGL and GNL. This can be achieved only when it becomes technically possible to accelerate protons up to 1 MeV and more, within the framework of the deep RL apparatus, which circumstance wall lead to a sharp increase of neutron output per 1 gamma quantum (^^). To obtain powerful sources of gamma quanta the well betatrons can be used. Using such betatrons it is possible to produce impulses of an order of 10^^-10^'' gamma quanta with a duration of an order of 1 ^sec with the upper limit of the gamma quanta energy higher than the natural gamma radiations. The gamma quanta flow per second in such betatrons depends on the current frequency. For instance, when the current frequency is 300 c/s it is possible to generate an output of 10^° to 3 x 10^^ gamma quanta per sec, which is quite sufficient for operating GGL as well as GNL. The relationship between the kinetic energy W^g^^ (in eV) of accelerated electrons, caused by an inductive force B (in gauss) and orbital radius (in centimetres) can be expressed by the following formula: r^3, = SOOBr. The size of the orbit r will determine the dimensions of the well betatron. The diminution of the orbital radius, necessary in construction, will demand an increase of the magnetic field so that the radiation energy is maintained. For instance, if the orbital radius is taken to be 10 cm, then in order to obtain gamma quanta with an energy of ^^ax =^ ^ ^^^ ^^ is necessary to generate a field with an inductive force of 1000 G, and for a radius of 5 cm 2000 G, which is quite possible. ELECTRON LOGGING Having obtained an electron beam from a betatron it is possible to build an electron log, which involves irradiating the bore -hole walls by a powerful beam of electrons, while the reactive gamma radiation of the rocks is registered. The energy E and the electron path R in a substance of density q are connected (for E > 0.8 MeV) by the formula qR = 0.542£'- 0.133. The connection between the energy spent by the electrons on the reaction radiation, and the parameters characterizing the medium (density q and effective atomic number Z^f) can be seen from the following relationship. 420 V. M. Zaporozhetz and E. M. Filippov where: E is the energy of an electron; R is the penetration path of the electron in a substance. Table 7. Penetration of electrons with different energy in substances OF different densities (cm) Ee^^^^^ MeV ^^ 2.0 2.6 3.0 5 10 1.3 2.65 1.0 2.0 0.86 1.8 Table 8. The energy used by an electron (with E^ energy) to produce reaction radiation over an interval R cm (MeV). MeV ^^^ 2 • 12.7 2.6 • 13 3 • 15.7 5 10 0.25 1.3 0.15 0.6 0.1 0.4 Table 9. The energy Ey, used by an electron (with E^ energy to produce reaction radiation over an interval 7-<;i? (MeV) Ee MeV r, cm ^""^■\, 2 • 12.7 2.6 • 13 3 • 15.7 5 10 0.5 1.0 0.1 0.4 0.008 0.2 0.05 0.2 The penetration paths of electrons and the energy used by electrons over different path -intervals in the rock, as calculated in terms of these formu- lae, are shown in the Tables 7-9. These tables show the following: (a) The increase in the density of the rock leads to the diminution in the penetration path of an electron in it. (b) The increase in the electron energy leads to the increase of its penetra- tion path in rocks. (c) The increase in the density of a rock and of its effective atomic number leads to the generation of a harder reaction radiation. The reaction radiation, produced in the irradiated medium has its beam near to the direction of the electron beam. At the same time the higher the THE USE OF ACCELERATORS OF CHARGED PARTICLES 421 electron energy the smaller is the deviation of the reaction radiation, generated in the irradiated medium, from the direction of movement of electrons. Hence in order to distinguish the rocks, their mineral composition and density, with the aid of electron logging, it is necessary to lodge the betatron in the core device so that the hard radiation produced in the rock would reach the gamma quanta counter directly or after the least number of scatterings. The possible use of electron logging is in dry bore -holes. In bore -holes filled with drilhng fluid this method cannot be used owing to its small penetra- tive power and technical considerations, such as that an electron beam emerges from the vacuum chamber of the betatron via a thin slit which can be easily closed by the liquid filling the bore-hole. CONCLUSION The use of the neutron -generators and the gamma quanta in radioactive logging allows a considerable improvement in the effectiveness of neutron logging and gamma-gamma logging, and will also bring about new methods of gamma -neutron and electron types of logging, which will widen considerably the circle of problems solved with the aid of radioactive logging. The possibility of radioactive methods of investigation can also be considerably widened when a powerful controllable neutron source is construc- ted as a small size well-reactor. REFERENCES 1. A. A. Bergman, et al.. Physical Studies (A collection of papers). Izd. Akad. Nauk SSSR, (1956). 2. N. A. Vlasov, et al., Zh. eksp. tear, fiz., (1955). 3. N. A. Vlasov, Neutrons, Gosteortizdat, (1955). 4. G. D. Glebova, An investigation of the Kinetics of Absorption of Hydrogen by Barium and Other Metals. Candidate thesis, Lenin State Library, (1955). 5. V. N. Dakhnov, Contemporary Position and the Outlook for Further Development of Well Radiometry. Exploration and Conservation of Minerals, No. 6, (1956). 6. Isotopes in Geology (A collection of papers). Foreign Literature Publishing House, (1954). 7. E. M. Reikhrudel, G. V. Smirnitskaia, A. I. Borisyenko, Radiotechnics and Electronics, No. 12, (1956). 8. E. Segre, Experimental Nuclear Physics, Vols. I and II, Foreign Literature PubUshing House, (1955). 9. E. M. FiLippov, A Contribution to the Theory of the Gamma-gamma logging Method (GGK). Applied Geophysics, No. 18, Gustoptekhisdat, (1957). 10. E. M. FiLippov, Gamma-gamma Logging. The Use of Radioactive Isotopes in the Oil Industry. Gostoptekhisdat, 1957. 422 V. M. Zaporozhetz and E. M. Filippov 11. Photonuclear Reactions (A collection of papers), part 1, Foreign Literatxure Publishing House, 1953. 12. R. L. CoLDWELL, World Petroleum., 27 (1956). 13. P. M. Endt, Phys. Rev., 95, 580 (1954). 14. R. E. Fearon, I. M. Thayer, United States Patents, No. 2712, 0.81, (1955). 15. G. D. FsRGussoN, Phys. Rev. 95, 776 (1954). 16. I. GoLDENBERG, L. Katz, Canud. Journ. Phys., 32, 49 (1954) ; Phys. Rev., 95, 464 (1954). 17. H. E. JoHNES, et al., Phys Rev., 80, 1062, (1950). 18. L. Katz, Phys. Rev., 82, 271, (1591). 19. R. NoTHENS, J. MoLJERU, Phys. Rev., 33, 437, (1954). 20. J. E. Perry, S. J. Bame, Phys. Rev., 99, 1368, (1955). 21. R. Shar, Phys. Rev., 84, 387, (1951). 22. V. Stont, M. Giffous, Appl. Journ. Phys. (1955). AUTHOR INDEX Aleinikova, p. K. 268 Alekseev, a. M. 196, 206, 214 Al'pin, L.M. 195, 197, 222, 223, 240 AMBARTSUMTi'AN, 333 Andreev, B. A. 139, 165 Anurov, B. a. 269 Arkhangel'skaia, a. D. 247, 268 Atwood, D. G. 340 AvANESov, E. T. 378, 382 B Babalyan, G. L. 377, 378, 382 Bame, S. J. 422 Barstow, 0. E. 348 Berdichevskii, M. N. 196, 206, 222, 223 Bergman, A. A. 421 Berzon, I. S. 97, 119 Bessonova, N. 45 BoRDOVSKir, V. P. 196, 223 Borisyenko, a. I. 421 BORTFELD, R. 86, 98 Bryant, C. M. 348 Bulanov, N. a. 214 BuLANZHE, Yu. D. 139, 165 BURGSDORF, V. V. 170, 195 BuRsiAN, V. R. 268 Chechik, N. O. 316 Chetaev, D. p. 270 COLDWELL, R. L. 422 D Dakhnov, V. N. 195, 241, 269, 361, 371, 373, 374, 382, 412, 421 DiDURA, I. G. 241 DiKGOF, Yu. A. 269 364, Dix, C. H. 98 Dobrynin, 363, 364 DoLiNA, L. P. 349, 350, 351, 354, 358, 359, 360, 361, 362, 363, 364, 368, 371, 373, 374, 382 D'YACKHOV, N. P. 269 E Efendiev, F. M. 327 Ellsworth, J. P. 98 Endt, p. M. 422 EPINAT'-i'EVA, A. M. 43 Fearon, R. E. 422 Fedorova, D. p. 270 Fergusson, G. D. 422 FiLippOY, E. M. 421 Flathe, K. 195 Florovskaya, V. N. 327 FOMINA, V. I. 271 Frolova, a. V. 44 Galkin, L. a. 301, 311 Gal'perin, E. I. 44 Gamburtsev, G. a. 119 Geller, E.M. 303 GiFFous, M. 422 Glebova, G. D. 405, 421 Goldenberg, I. 422 Gorbatova, V. P. 11 Grushinskii, N. P. 139, 166 Gubkin, I. M. 268 Gumarov, K. S. 268 GuRviCH, 1. 1. 82, 97 Guzanova, 354 423 424 Author index H Hansen, R. F. 85, Il'ina, a. a. 319, 327 IvANKiN, B. N. 82, 98 Johnson, 83 Jones, P. D. 374, 378, 382 K Kalenov, E. N. 241, 297 Kantor, S. a. 415 Katz, L. 422 Keivsar, Z. I. 349 Kharklvich, a. a. 120 Khomenyk, Yu. V. 222 Kleinman, M. K. 327 Klimarev, a. a. 269 KoMARov, S. G. 349 Konstantinova-Schlezinger, M. a. 327 Kopelev, Yu. S. 210, 270 Kotliarevskii, B. V. 139 KoTYAKHOv, F. I. 376, 377, 382 Krasil'shchikova, G. a. 44 Krey, T. 120 Krolengo, 1. 1. 241, 258, 269 Krolenko-Gorshkova, N. G. 269 Kupalov-Yaropolk I. K. 98 Kurnyshev, a. G. 269 KuzNETS, G. 222 Lapauri, a. a. 348 Levorsen, a. I. 383 Levshunov, p. a. 327 LiPiLiN, V. A. 200 LiTviNov, S. Ya. 169, 195 lukavchenko, p. i. 139, 166 Lyamshev, L. M. 82, 98 Lychagin, G. a. 269 M Mate, R. 195 Maimin, Z. L. 270 Malinoskaia 18 Malovichko, a. K. 139, 166 Mamedov, E. I. 327 Masket, M. 372, 374, 378, 382 MiGO, L. 222 Miller, Q. 120 MiRONOVA, V. I. 44 Mishchenko, G. L. 270 MoLJERU, J. 422 MoROzov, G. S. 241, 350, 351, 359, 361, 362, 363, 364, 365, 371, 373, 374, 376, 383 MuRATOv, M. V. 270 N Nazarenko, 0. V. 169, 200 Nothens, R. 422 O Oblogina, T. I. 99 Ogurtsov, K, I. 86 Osipov, M. F. 270 OsLOPOvsKii, A. N. 270 Panteleyev, V. L. 123 Perry, J. E. 422 Petrashen', G.I. 11, 15, 21, 43, Petrovskii, a. D. 222, 240 POLYAKOV, E.A. 384 Popov, Yu. N. 196 POULTER, 76 Puzyrev, N. N. 139, 166 Pylaev, a. M. 297 R Reikhrudel, E. M. 421 Reinhold, 340 RiEBER, F. 120 Robinson, W. B. 120 Rose, W. D. 361, 364, 379, 383 Author index 425 Safontsev, E. G. 270 Samoilov, M. Ya. 169 schlumberger, k. 206 Segre, E. 410, 421 Serzhant, a. a. 161 Shar, R. 422 Shcherbakova, T. V. 315, 328 Sheinman, S. M. 206, 270 Shifrin, K. S. 348 Shuleikin, V. V. 333, 348 Shushakov, S. D. 75 Sloat, J. 98 Smirnitskaia, G. V. 421 Smirnov, V. I. 112, 119 SoBOLEV, S. L. 112, 119 SoKOLOV, p. T. 119 Sorokin, L. V. 138 Srebrodol'skii, D. M. 213 Stefanesko, S. 185 Stont, V. 422 Street, J. W. 348 Sultanov, S.A. 363, 364, 383 SusLOVA, L. L. 354, 383 Tal'Virskii, D. B. 43 Terekhin, E. I. 169 Thayer, I. M. 422 TiKHONOv, A. N. 270 TiKS'E, M. P. 364, 379, 383 Timofeeva, V. A. 322, 348 Troitskaya, v. a. 222 TsEKov, G. D. 190, 195 Tvaltvadze, G. K. 119 Van Meller, F. A. 98 Van'yan, L. L. 169, 191, 195 Varentsov, M. K. I. 268 Vaslil'ev, v. N. 268 Vening Meinisz, F. L. 123, 138 Veselov, K.Y. 123, 138 ViCTORov, A. M. 348 ViL'TER, I. B. 348 Vladimirov, B. v. 311, 314 Vlasov, N. a. 421 Volodarskii, R. F. 139, 160, 165 VUL'F 47 W Walstrom, J. E. 379, 383 Waterman, J. C. 98 Weatherburn, K. R. 98 Wyllie, M.R.J. 361, 364, 379, 383 Yacobi, I. 0. 348 Yakolev, K. p. 383 Yepinat'yeva, a. M. 76, 97, 106, 119 Zaborovskii, a. I. 190, 195, 241 Zagarminster, a. M. 196 Zaks, S. 0. 372, 374, 382 Zaporozhetz, v. M. 397 Zavadskaya, T. N. 223 ZvoLiNSKii, N. V. 97 4 . "* ^'■' >--. SUBJECT INDEX Absorbancy of media, effect on seismic wave intensity, 93 Acoustic rigidity, effect on seismic wave reflection, 27-8, 38, 41 Anisotrophy coefficient, of sedimentary deposits, 244 Anticlinal folds, effect on electrical probing, 289-91 Azimuthal-dipole probes, 198-206, 223^0 arrangement, 223 errors in, 231-6 practical procedure, 236^0 theory of, 224-31 B Berezovka gas deposits, 210 Beryllium, detection by radiation, 415 Boreholes, camera for, 340-4 optical studies of, 328-48 Boring rate, effect on gas saturation of drilling fluid, 311 variation in, 310 Camera, for boreholes, 340^ Caspian Sea area, geology of, 254-6 Cis-Baltic depression, geoelectric survey, 198 Cis-Carpathian depression, seismic wave reflection in, 75, 85 Clay suspensions, effect on oil detection, 320-2 optical properties, 328-38 Cretaceous deposits, and multiple wave reflection, 75 Crimean steppes, geology of, 252 D Damping of seismic waves, with distance, 29 37, 38-9, 90-4 Density, effect on seismic wave intensity, 30 Devonian sandstones, oil-bearing, permeability, 349 specific resistance, 335, 350, 354, 361, 363 Diffracted seismic waves, 99-119 dynamic properties, 108-110 kinematic properties, 100-8 Diffraction, structural formation causing, 99 from tapering strata, 110-9 Dipole probes, azimuthal, 197, 223-40 effect of non-horizontal rock boundaries on, 292-7 for inaccessible country, 214 practical appUcation, 196-206 theory of, 169-95 DP degasifier, for drilling fluid analysis, 311-2 DrilUng fluid, degasification, 311 gas saturation, estimation, 302 factors affecting, 302, 305, 310 luminescence of, 289-92, 319-22 luminescent spectral properties, 322-5 optical properties, 308, 328-38 E Echoes, single/double comparison, 40, 42 Elastic properties of media, 12 Electrical probing, distortion by non-horizontal rocks, 271- -97 in inaccessible country, 214, 223 Western Siberia, 196-222 strata relief studies, 201-6 426 Subject index 427 at sea, theoretical curves for, 183-95 at sea, theory of, 170-8 two-way, 196-206 Electrical surveys, by loop method (see Loop method) mobile units for, 214-21 Electron logging, of rock strata, 419-21 Kaolin, optical properties, 330 Kerch peninsula, geophysical data, 241-2, 244-5, 247, 263 gravitational field, 248 seismic exploration, 250 Gamma-gamma logging, of rock strata, 408-9 Gamma-neutron logging, 381-9, 409-17 for oil, 414 Gamma quanta generator, construction of, 417-9 Gas factor, of oil-bearing strata, 313 Gas logging, factors affecting, 301-5 Gas saturation, of driUing fluid, effect of boring rate on, 280, 305, 310 estimation of, 302 fluid, factors affecting, 302 of rocks, estimation, 302-5 Gravimeter, static, for use at sea, 123-38 Gravimetric surveys, accuracy of, 140-2 of Kerch peninsular, 248 observational errors, 144-6 of Russian basins, 160-1 source of errors in, 144-7 Grazing rays, spatial direction, 52 Gypsum-anhydrite, and multiple wave reflec- tion, 85 H Head waves, intensity, 28-30, 33-35 Hodographs, construction for reflected waves, 55-73 of diffracted waves, 101, 114-7 theoretical, for multiple reflections, 82 Infra-red photography, of rocks, 338 Intensity of seismic waves, 14--43 Isochrone chart, for waves reflected from dome, 62 Lame constants, 15 Leningrad tables, 27 Light transmittance, of clay suspensions, 331 Logging, gamma-gamma, 408-9 gamma neutron, 409-17 gas, 84, 301-14 luminescence, 315-27 radioactive, 397-421 Longitudinal seismic waves, intensity, 11^3 Loop method, developments in, 244-63 economics of, 265 errors in, 251 geological surveys by, 244^59 principles of, 241 uses of, 242-4, 251, 266 Luminescence, of drilling fluid, 319 of oil, 319-20 of oil-chloroform solutions, 320-1 oil deposit logging by, 285-97, 315-27 Luminoscope, photoelectric, design of, 286-9, 316-319, 326 M Mesozoic deposits, 201, 204, 210, 250 MobUe units, for electrical surveys, 214-21 Moscow syncUnal basin, gravimetric survey, 160-1 Multi -layered media, reflected wave intensity in, 40 refracted primary waves in, 40 surface hodograph for reflected waves in, 62-73 428 Subject index N Neutron-gamma logging in, 398-401 Neutron generators, for borehole logging, 398^01 construction of, 373-80, 401-8 theory of, 401-2 Neutron radiation, of rock strata, 415 Oil, -chloroform solutions, luminescence of, 320-1 luminescence of, 298-90 318-20 spectral characteristics, 322-5 Oil-bearing strata, detection by gas logging, 283-4, 313-4 by loop method, 244, 248, 252, 256 luminescence logging, 285-97 315-27 permeability, 352^, 380-2 radiation, 414 seismical data, 251 permeability, determination, 321-5, 349-53 Oscillations, of multiple reflections, 82^ Oscillograph, for electrical exploration 217-9 Paleozoic deposits, electrical surveys of, 202-4, 209-10, 213 Permeability, determination, errors in, 324-5, 330-2, 352-3, 358-60, 364-73 from core analysis, 354-8, 365, 375-7 from specific resistance, 349-52, 352-8, 360, 365, 376-82 of Devonian sandstones, 349-52, 376 of oil-bearing strata, 349-52, 380 -specific resistance increase, 332-6, 340-8 360-4, 368-76 -water saturation, of oil strata, 353, 358-60 Photography, of rock samples, 338, 344 of wells. 336-8. 340-7 Porosity, -specific resistance of rock strata, 362, 368 Primary seismic waves, intensity of, 15-18, 28-37 in multi-layered media, 29-37 in two-layered media, 28-29 R Radioactive logging, of boreholes, 397-421 Reflected multiple waves, 75-97 damping of, 90^ dynamic properties, 80-95 intensity of, 18-27 interference with single waves, 75-95 kinematic properties, 76-80 oscillation intensity, 85-90 Reflected single waves, intensity of, 18-27, 37^3 interference by multiple waves, 75-95 in multi-layered media, 40-3 in two-layered media, 37-9 Reflection coefficients, calculation, 18, 21-4, 26-7 Reflective rays, spatial direction, 51 Refracted rays, spatial direction, 52 Refracted waves, 15-18, 28-37 Refraction coefficients, 15, 17, 31 calculation, 17, 21, 26-7 Resistivity meters, coefficient variation, 384-90 design of, 393-6 electrode design, 390-3 Rock density, effect on seismic wave intensity, 30 Rock strata, disturbance, effect on electrical probes, 271-97 identification by gamma radiation, 400-1, 409 by photography, 344-6 natural neutron radiation values, 415 Salinity, effect on resistivity measurements. 358-62, 386-90 Subject index 429 Sandstone, detection by radiation, 415 Sea probes, electrical apparatus for, 169-83 Seismic rays, spatial direction, 44-73 Seismic waves, damping with distance, 29, 37-9 diffracted, dynamic properties, 108-110 kinematic properties, practical, 105-8 properties, theoretical, 100-5 doubly reflected, 39-10, 91 intensity calculation, 15-27 longitudinal, intensity, 11-43 multiple reflected, damping with distance, 90-4 reflected, dynamic properties, 80-95 intensity, 18, 20, 85-90 interference by, 75 kinematic properties, 76-80 primary, in multi-layered media, 28-37 factors affecting, 30-7 intensity of, 15-8, 28-37 reflected, hodograph construction for, 55-73 intensity of, 18-27, 37-43 in multi-layered media, 40, 55-73 refracted, intensity of, 28-37 simple reflected, 18 Seismical studies, comparison with loop method, 250-4 with telluric currents, 212 of Kerch peninsular, 250 Siberia, west, electro -geological survey, 196-222, 256 gravimetric survey, 161-3 telluric current fields in, 209 Specific gravity of clay suspensions, effect on light transmittance, 331 Specific resistance, of Devonian sandstones, 350, 354, 361, 363 drilhng fluid, errors in measurement, 384-90 marine electrical probe instruments, 178-83 Mezo-Cenezoic deposits, 204 oil-bearing strata, 349-53 measurement (see Resistivity meters) -permeabihty of rock strata, 360-4, 368-82 -porosity of rock strata, 368-73 Spectral characteristics, of oil products, 323 Stereographic nets, construction of, 46-7 Stereographic projections, 44-73 Taman peninsula, geology of, 254 Telluric currents, electronic measuring apparatus, 219-21 in geological surveys, 206-13 in Western Siberia, 209 Transverse waves, effect on longitudinal wave intensity, 30, 39^0 Turkmenia, geology, 256 Two-layered media, wave intensity in, 28-9, 37 U Ust-Urt plateau, geology of, 258, 267 Volcano -sedimentary strata, effect on electrical surveys, 199, 204, 210 W Water saturation, -permeability of rock strata, 373-4, 377-^, 380-2 -specific resistance of oil strata 351, 353, 358-60 theoretical value, 364 Well camera, design of, 340-4. Wulir net, 45, 55, 57 accuracy of, 73 uses of, 47-53, 65, 68-9