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
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1962
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/^* '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 Crosssections
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 BoreHole Investigation 328
T. V. Shcherbakova
15. Determining the Permeability of OilBearing Strata from the
Specific Resistance 349
S. G. KoMAROV and Z. I. Keivsar
16. New Types of Well ResistivityMeters 384
E, A. POLYAKOV
17. The Use of Accelerators of Cliarged Particles in Investigating
BoreHoles 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.
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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,
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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 boreholes systematically
located at critical points. As a result very notable advances have been made
in developing the socalled 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
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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 wavelength 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;
(PP^^ (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.
n2
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
nl
\ /
\ /
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'» 05 06 07 08 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.
07
06
02
Sin a,
04
06
08
^_^_^
i
\
\
\
= 07
a=05
\
0=03.
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.)
Vnl, 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 [ Zj 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 — QmlVml,p
9m'^m, p ''r Qm—l'^m1, 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 04
C^
QJ
09
08
07
SinCref
02 0
06
, ^ T >
/
/
^
AQ&
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 > V1, 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<^mi, p' then
/p' p' \ _ IP' P' \ , Q'n'^m,P Q ml'^m1, p .
^m ^'m, p T^ Qm1 '^ml, p
23
SinfS'ret
02 04
06
10
""^^^^
1
\i
^
r
^ = 04
09
\A\\
ffl
\v\\
CL
\t\
\
S^
Q. 08
1
^\
\>=o;
i;^=04
\
07
\
^
\
\
m
w
\
— 06
\
o!
\
\
e
a
\
1.^=05
05
\\
1
w
1
r
 ^ =06
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 mukilayered 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 111 sin an cos ao . , .
with a twolayered medmm, ^\e substitute the value tor tms
multiplier, we obtain still better accuracy.
If the angles of incidence onto the intermediate interfaces of multilayered
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 {PnlPnl)v
_ i=0 .^^,
Jreiv— ^^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 >. QnlVnl, p — Qn't^n, p . /i o\
l^«l^nlji; = ; TZrZ, » ^^^'
Qnl^nl,p'T fJn^n,p
h is the distance traversed by the wave in the ith. layer;
v^^ is the group velocity of longitudinal waves in the ith 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 (PjP^^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)//(rroF^
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 twolayered 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
/."(rg'/^^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 twolayered 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~^.
Multilayered 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
'if 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 = 0175.
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 nth 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
100
01 02 03 04 05 06 07 08 09 10
100
or
+
or
X 075
or
01'
Ql"
X 050
q:"
T
1 fs} / \\
/ / / /i V^ \
V
W I 1 V
IP
I 1 1 \
1 1
\ \
b = l/045
[\
b=035
1
^""^
\"~"Vv^=0 45
1
1
/
\ \ b = 21
1
1
\ b = 063
1
1
\ b = l0
(b) 1
1
1 , ! ,
O'l 02 03 04 05 06 07 08 09 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 lowlying 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 lowlying 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 lowlying 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 PqPJ^^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 threelayered 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 ^ threelayered 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^PJ^^^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 multilayered 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 Multilayered 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 TivoLayered A Tedium
(a) Single Reflected JFaves — The. intensity of single reflected waves
above the shot point is equal to the quantity
2 QoVo,pQiVi,p 1
47r(Ao + 2//o) ^o^p QoVQ^p+QiVi^p 2hQhsd*
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 05
— «^^
..,^__^
'^v'^^>:vJ'""^'>^"~'"~'~~>"0'^'^ '°'^
^^^ ^
^A =05
^A =06
02
4 06 0
2r/(2ho
3 !0 1
^3)
2 14
Fig. 10. Curves for the damping of single reflected waves in twolayered 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 halfspace and for media where Vq > v^^
it depends on this value both in a halfspace 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 ^hohsd
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
^ 05
F iG. 11. Curves for the damping of doubly reflected waves in twolayered 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,pgi^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.
Multilayered 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 multilayered 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
abovementioned 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 densitystructure 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.
nl
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 {PnlPnl)v, (18)
Jk ^ — L 1=0
where [{PP_^_^) (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 {kl)ik+l)
^kijk+l
k'
(19)
where j\ is the intensity above the shot point of a wave reflected k times;
//fi 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 lowlying boundaries can help
in recognizing the nature of any particular wave recorded on the seismogram.
THE FORM OF SINGLE AND MULTIPLE ECHOES IN MULTILAYERED MEDIA
We can say the followng about the form of seismic traces for echoes
■excited as a result of shots in multilayered 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
Deepdrilling 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 multilayered 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 multilayered 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
multilayered 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 coordinate 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
coordinates so that we can remove the coordinates 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 coordinates (a = 162°, (p = 54°) by
means of a Wulff net.
1. To construct on the grid directions of which the coordinates are
given. Let the coordinates 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 coordinates 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 coordinates 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 coordinates 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 coordinates 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 coordinates 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
coordinates 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 coordinates 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 coordinates 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 threedimensional 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 — crosssection of structure in vertical plane from 27090° 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 090° 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.
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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
>96 100
120 I40 160 180 200 220 240 260 2 80 300
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 THREELAYERED 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; 3456 — 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 = 0705; 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 coordinates (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 coordinates (azimuth and angle
with the vertical) of the point D. The coordinates obtained characterize
the direction of the reflected ray.
The ray which we are to examine after reflection has a direction deter
mined by the coordinates (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 coordinates 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 threedimensional linear problems
in geometrical seismics of multilayered 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 CisCarpathian
depression, a gyp svun anhydrite horizon from which multiple waves are
reflected can be distinguished by its wave resistance. In northwestern 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 lowvelocity zone acts as an upper reflecting boundary,
but most investigators think that the base of the lowvelocity 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 lowvelocity 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 lowvelocity 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. Fullpath 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 / . 77 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 (^' ^^). Oddnumber multiples at the end of the path of the incident
wave are reflected from the interface I, whereas evennumber multiples
are reflected from the zero surface.
Fullpath 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 shotpoint.
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 borehole 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 highvelocity layer or a lowvelocity 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 lowfrequency 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 lowfrequency 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 gypsumanhydrite layer in the outer zone of the
CisCarpathian 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 CisCarpathian depression, where there are sectors
close to one another in which multiple waves are found to be reflected from
a thin gypsumanhydrite 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 nearsurface 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 deepseated, 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 northwestern 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 fullpath 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
fuUpath 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 cophasally 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)
^'yl 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;
^=/lsin2a; 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^iAiD^ 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
062
932
I6''48'
I,
Aluminium
035
932
ISMS'
n
Glass
060
910
IS" 34'
m
Texiolite
172
339
as 55'
rz
Organic gloss
160
226
32° 04'
rz
Organic gloss
050
195
38° 1 5"
2
Ebonite
lIO
1 49
42° 00'
vr
Rubber
10
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
076
1018
15° 15
n
Glass
069
935
I5°47'
m
Textolile
156
338
24° 23'
nz
Organic gloss
052
326
32° 25'
2
Ebonite
062
157
49°49'
a,
Rubber (corrug )
062
—
—
53
Rubber
093
131
83° 13'
HI
Plastelme
106
1 47
—
I,
Aluminium
035
1035
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 wellshooting,
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 lowvelocity zone is roughly equal to 0.60.8 (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'^^'rs,). (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 fullpath 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 crosssection 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 fullpath 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, reflectedrefracted 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. inta 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. inta Acad. Nauk SSSR No. 35 (162), (1956).
8. I. K. KupalovYaropolk, 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 crosssection.
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 faulttype
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 diffiacting objects we have mentioned, the one which
has greatest practical importance is the diffiacting 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 diffiacted 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 lefthand and righthand 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 coordinates xy such that its origin lies on a
diffracting edge, with the :;caxis running along the horizontal interface and
the raxis upwards towards the observation line (profile). The contact
* A Russian term referring to distancetime curves. [Editor's footnote].
DIFFRACTED SEISMIC WAVES
101
edge— the rectilinear diffracting edge — will then pass through the origin
of the coordinates 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 lowervelocity 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 dashedline 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 oci ; (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 highervelocity 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 (12001600 m/sec); group 5 of refracted waves
with high apparent velocities (50006500 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 cophasality, 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 cophasality; 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^, fggroup 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 lefthand 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 70.9670.45. The smaller the spread, the more marked
is the phase inversion near the point of tangentionality.
22
21
20
UXLi
02 04 06 08 10 12
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 coordinates so that the ::caxis runs along the
interface and the jaxis up towards the upper medium (we are considering
the plane problem).
Along the half line x > 0, y = h let a. crosssection be taken, the edges
of which are firmly attached; that is to say, the displacements on them
will be zero. Such a crosssection 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
coordinates, the boundary condition on the boundaries of the crosssection
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 SmirnovSobolev method of functional invariants;
the dynamic hodographs and theoretical seismograms for diffiacted 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 crosssection 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]/{aQQ^<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{vh)Im0'{Oo') 1
k'here
"IV^' /? '^^
h^f^.
^/ta^r
,
^o^'r\
v^{x, J, t)
f2a,^yhYlm0'{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
{tT).
(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
/{ta^r) = 4.T^ {At {\^t]/AtT)  ^ [At^r~ {At~T)^I^]] 
116 T. I. Oblogina
12 T{At' {yjAt yj AtT) ^At [At^l^{At~T)^i^ +
1
+ — [At^l^{AtTyi^]} + 8 {At^ {]/At YAtT) 
At^Af^!^{AtT)^i^] + ^At[At^i^{AtTyi^
_^ 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 ahorizontal 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
02
Fig. 11. Theoretical seismograms for a diffracted wave atrace of horizontal com
ponent of displacement; 6trace 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. inta 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. inta 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. inta. No. 41, Izd. Akad. Nauk SSSR (1934).
8. G. K. TvALTVADZE, Theory of the seismic method of prospecting for vertical interfaces.
Tr. Tbilisskogo geofiz. inta, 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«=fa,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 ^ (lgg,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^
wpr _^^
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 coordinates 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 coordinates 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 coordinates
0C translated to the axes C and  via ^qCq^ the coordinates 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 coordinate 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 ttI ?«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 + di) + (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
Ox* <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 1020 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 CROSSSECTIONS 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 crosssection 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
crosssection of the isoanomalies. Then the expression
^={gBgA)(SBgA) (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 crosssection 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
crosssection 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 crosssection of the isoanomalies, given in fractions
of this crosssection.
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 nonhnearity 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
nonlinearity 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
•COm —
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 nonlinearity
\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 + — ~ {xXi)
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^, gg, ^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 crosssection of the isoanomahes p, i.e. g^—g^ = />•
Assuming x^—x^ = d^a,\fe find the expression
^0 = ig5gi)ig5g4)
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" ^' [(3a23a(^+ 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 = ^[{ad){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 NonLinearity 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^ = {gsg2)ig2gi)
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 gQ 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(86% + 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)(^4g'3)'
where (gi—gs) is the crosssection 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[{da2x,) + 2x,].
Squaring both sides of this expression and integrating for x^ in the limits
^1 < ^3 < '^1 + ci—d, we obtain
dfm = jgo'dHadr. (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 = jgoHad)^{2adr^ (17)
On the righthand 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<2a
iAi,),=glan{2a)2x,Y,
for 2a<:^i<2
{Al,),=gl[{86a+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 :
2a
2a
2a 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^205a^ + 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]/205a^+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{2xx^)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 crosssection 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'
09
08
07
06
05
04
03
02
01
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 06 08 10 I 2
02 04 06 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
80
10
50
40
30
20
10 12
14 16
18
20
"
//
^
//
^
A
^
z:^
^
^
^
"^yyy.
%
^
:d
^
^
10
i
H
i
^
^
Q\ 2 03 4 05 6 07 08 9
or
p
Fig. 8.
this relationship, the righthand 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
08
07
06
05
04
03
02
01
10
4q
p
Fig. 9.
EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY
155
To derive these relationships in the righthand 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.
10
08
06 04 02
1 a
\
\
\
\
^1^
\^
k\
\
\
/
K
>
\
s
\^
^
[\
/
\
s
^
^\
/
\
^<^\\
\
10 8
2 05 06 07 08 09 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) tsvo nomograms are given (Figs. 12 and 13) : the first for
018
016
014
012
010
008
006
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
01 02 03 04 05 06 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
02
04
05
«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
02 03 04 05 06 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
]/Nl
(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 crosssection 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 56%.
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 (5532) 5556 in
the region of Western Siberia in 19551956 (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 gQ = 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^^ = 22.5 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 crosssecti(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 crosssection 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 crosssection 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 twoway dipole axial probes with a distance between
the centres of the dipoles of up to 6 — 8 km, with sea depths of up to 5060 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 seabottom 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 threelayer 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 twolayer curves with ^2/^1 ~ ^^ 2' ^' ^^' ^^ °°' threelayer 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 WeberLipshitz 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^)e2^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 + e2'"''») ^1 A e^'^e^ozo) _
= _ . qH _ Q2mzo\ g2m(hoZo)
__ _J_ n _J_ Q2mZa\ Q2m{hoZa)
5i + Zie2m(h„+ftiz„)_^2_J2e2'"(''o+'^i^«) =
1 ^ 17
B
Qi
j^ Q2m{ho+hiZo
fl
^5.
£»2
\ A^ G^rn{h, + h,Zo) =
^2
1
2m(s/iiZo) _ 2m ( S/JiZo)
^^■=0 ^^ne ^'=0 ^ =0
I 2 hiZo )
( E/lJZo)
^ 1 _ 2'n
Bn1 ^nl e
Qn1 Qn1
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^,
,
, .
..,
,
,
(leD),
Qo
1
Qi
Qi
,
, .
.,
?
1 ,
eL,
 1 ,
e^, .
.,
?
u
1
+ — ,
Qi
Qi
1
^2'
«2
.,
■>
,
,
,
, .
, 1 ,
eQ
■>
e^
,
,
,
, .
1
9nl
1
Qnl
eQ,
'eO
Qn
where D = —'ImliQ',
F = 2m(Ao^o);
L = 2m(/?o+/iio);
Q= 2m( 2^~o
nl
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
— eR,
^1
1
^2'
1
?2
e^.
.,
,
,
,
,
5 •
., 1
e^
eU
,
,
? • •
1
'' Qnl
1
Qnl
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^
'' Qn1
Qnl
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 {Q2mh\
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'« + e2"^''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
nl
S'
i =0
K =PqK hQ+h^ = PiK hQ+h^ + h^ =p^h, ... , 2 /',• =/>„! ^''
where Po,Pi,P2i •••■>Pni ^^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 /e2mnft ^Q^mzo + e^'"^") ^mr) dm, (15)
n = l J
Using the WeberLip 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^{nhZQY
+ 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/72 + 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 waterair 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 4electrode 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 >,^„
72 ■ (72 + 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?nr.
'+(27l/7)2]V=
+
+ Ljin
+
t{ M^' + Hnh+h.^!^ [7•2 + 4(7^/^/^o)^]'/=/J
(19)
Representing p^ — r^
h
+ 4,
, the socalled 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. / ■^(r2^ %)
^ 2 ^"i[r2 + 4V]'/»
+ 2j^"
r^[r^2{nh+h^)^] r^[r2{nhhoy]
(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. + ZJ ^ri{lnpo + 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 multilayered 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
threelayer 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 threelayer 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 seawater.
For a twolayer 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 threelayer 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 AllUnion 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 threelayer 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 threelayer 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^"^^)
^ (?„(e2mh)
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
liPnl + 2)
7=3 ■*
(29)
if p^_j is an even number.
If Pni ^^ ^^ °^^ number, then
(prel+3)
R = 7, e"^""^  V ^ Ve2^^e^^ft (30)
y=3 ^
In these expressions the functions and correspond to
^ \ Q2mh \j^Q2mh ^
a two layer section with ^2 ~ 00 and ^2 ~ ^5 ^^d the functions of the form
1^ .' Q—2mh Q—imh
; correspond to a threelayer section, when the two
l2A;y'e2^'»+e4'"'^ ^ ^
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
^ ' l2A;/e2'"'2+e4'«^
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
\_Q2mh \j^Q2mh '■
form of the series:
00
^2mh ^— ,
, — 2mh
E
\_Q2mh
n=l
2mh
2inh
l_<_Q2mh
^(1)"
/i=l
Let there be a geoelectrical 7ilayer crosssection, where at a relative depth
Pni 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 Q2mh„ /^ (^fnr) dm +2 j B^ /„ {mr) dm +
'"4^L""i'
oo
+ pi(e2'"'" + 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 n1
(pnl+2) 00 . P
4 V bj^ e^'^'''' COS n(p\ loimr) dm +je ""''''' lQ{mr) dm +
+ / &i V* e2mn/i(e2mfto+ e2'n/'<') +
00
{ 5^ V" (  1)" e2""''i ie2mho + e^mho) +
y=3 n=l
5(pnl + 2) 00
+ V 6y V e2"^"'^ COS ;i(^ (e2mho + e^'n'Jo)
y=3 n=l
Using the WeberLipshitz 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^+{2nh2hoY ]/ r"^ + {2nh \ 211^)
+
188
E. I. Terekhin
00
+
i(pnl+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 4electrode 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 [72+ (2Ao)"]
13/2
^^2j \\r'^+{2nh'.
[r2+(27l/i2/2o)2]3/2 [72+ (271/1+ 2/20) T^2
n = l
\{pnl + 2) 00
(l)"r
n r3
(_l)n^3
[r2+(27^/i2/7.o)2]3/2 ' [;2+ (271/2+ 2Ao)2]3/2
+
+
;=3
r"* cos 7Z97y
z'* cos 7799^
^^ \ [72 + (27l/l2/7o)2]3/2 [;2 + (2,z/i + 2h^Y]
3/2
5(P72l+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(pnl + 2) / 00 \ / CO
+ 2 ^ l + 2S/„COS7Z9./ +6J/^„+2(^np.+ ^n+p„)) +
+^2(^p.+S^(lr(/np„+Wp.)]+
(pnl+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
■ {pnl + 2)
Qm^^JSoI 2j ^M ^ + 2 2j ^n COS ncpj' ] +
V
^{pn1+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(^np„+^n+p„)
id 6o
^P„+S(imp.+ ^n+p,)
We will consider in a general form one of the terms in the second sum of
the expression (32):
^Po+ 2 ^^np,+^n+pj cos n9?
/p,+ (L(p„_i)+/(i+p,))cos9P +
+ (/_{p„_2)+ ^2+p.) cos29?+ ...+ (Z_j+ Z2p^_i) cos(/Jol) 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 ^npo + ^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(^npo+^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 (prel+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 (pn1 + 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
(^Pol)+ 27n(^np„+ ^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 threelayer geoelectrical sections and are given
in tables of coordinates of threelayer 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 mth term an error is introduced into the cal
culated value. This error can be expressed by the value
00
Zl5 = 5i52 = y 2j ^nikpo+^n+po^D ■
nm + 1
To evaluate the obtained value, we carry out a series of conversions:
1 v^
^5=2 2j ^ri{lnpo+ in+p,2ln) <
n = m + l
00
^2" Xj ^^"(^"Po+^^+P''^^")
n = m + l
<
^2" 21 k"l^"Po+^«+Po2/„, (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 (^np+ 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:
ilnp,+ In+po^^n) = {lnp,ln){lnhi+p) ^^{AnY.
Putting An = 1, we have
(%Po+^/7+Po ^^n)
dHr
and, consequently,
/ , (^npo+ ^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 + lp„~~^m + l)+ (^m + 2p„~^m + 2) + (^m3po~"^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 + lp„+ ^m + l)+ (^m + 2p„+ ^m + 2) + ••' + \''m~''m+p) ~
Po
^^ 2j \^mpo+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 (^mp,+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 \Kp„'^h+p„ 2Z„— 2 (^npo+ ^n+p„~2/J —
n=/7o^l
no
2 (^npo'^~ ^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 (^np„+^n+Po2U 2 (^np„+^.+po2g.
n = /7o + l
n = m + l
Treating the obtained series exactly the same as for case (a), we ob
tain
oo p„
2j I ^/2po+ ^/i+Po~2i„ I = 2 2j (Zn„po+i~^n„+i)~
n = m + l
P.
~" 2j \''mpa+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 (^nopo+J~"^no+i)"~ Zj (^mp„+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 seabottom 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
115 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 seabottom
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 (1220 km), and power generators,
(1020 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 twoway electrical probing and the method of telluric currents), which
considerably extend the possibihties of electrical exploration, led to the
organization in 195556 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 KolyvanTomsk fold belt and within the hmits of
the Tobol'sk and the VagaiIshim 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 TWOW AY ELECTRICAL PROBING
The theory of twoway 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 twoway
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 fourpole AMNB apparatus. Apart
from the possibility of producing deep probings by using small spacings
of electrodes (12 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
twoway 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 electroprospectors 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 crosssection
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 19551956 in the Western Sibe
rian Lowlands.
An experimental party from VNII Geofizika carried out 40 twoway
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 56% 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,
* iir5Koeffitsient 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 = 1620 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 twoway 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 twoway 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 twoway
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 twoway probing consists in follow
ing the character of the relationships of the positive and negative KS curves
in the profiles (especially their righthand 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 twoway 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 
"AZ55
Suvorovskii
SidorOvKO
Fig. 2. KS map for operating distance i? = 3000 m. 1 — centres of twoway 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 twoway probing on the northwest ends of the
Koly van' Tomsk fold belt (the work of 1955). The surveyed area includes
the preJurassic foundation, covered by MesoCenezoic aranaceous argil
laceous deposits^ and dipping in a northwesterly 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 preJurassic 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 twoway 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 northwesterly direction, corresponding to increase in the thickness of
the friable MesoCenezoic 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.
DZ59
DZ58
DZ50
DZ48 DZ56 DZ57
Fig. 4. Curves for twoway 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 (DZ58, DZ59),
characteristic of a passage across the crest of the structure. On the DZ58
the righthand branch of the positive (southwest) curve, represented by
a continuous line, is placed above the righthand branch of the negative
(northeast) curve, shown by a dotted line, which indicates a rise in the foun
dation in a southwesterly direction.
On the DZ59, reverse relationships were observed, which indicate the
change in direction of dip of the foundation. Very characteristic are the
relationships of the righthand branches of the curves DZ48, DZ^9,
DZ50, showing the presence of a depression in the surface of the founda
tion svith deepest part in the DZ^9 zone, where the righthand bran
ches of the positive and negative curves coincide almost completely.
NEW METHODS OF ELECTRICAL EXPLORATION IN SIBERIA
203
On the northeast section of the profile, the relationships of the right
hand hranches of the twoway probe reflect the rise of the preJurassic
rocks in a northeastern direction (DZ56, DZ57).
The results of the qualitative interpretation facilitate the selection of
parameters for subsequent quantitative calculations and improve their
reliability.
Figure 5 gives a geoelectrical crosssection 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
56 n.xn
50200 /?.m
400
^P— ..
Fig. 5. Geoelectrical crosssection 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 twoway azimuthal
probes in 1956 at one of the sections of the VagaiIshim tectonic zone, where
the rocks of the preJurassic 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 highresistance 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 crosssection 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 MesoCenezoic deposits. The small change in the specific electrical
resistance of the rocks of the MesoCenezoic over a wide range of territory
provides favourable conditions for tracking the relative changes in thickness
of the friable MesoCenezoic 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 ViatkinoKrotovka. 1 — 5 lines;
2 — depth lines; 3 — supposed surface of supporting electrical highresistance 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
A284 AZ83 AZ82 AZBl A280 AZ79 AZ78 Sea level
Krotovskaio structure
Fig. 8. Twoway curves of an azimuthal probe over an upheaval of a support high
resistance level. 1 — centres of twoway electrical probes; 2 — supposed surface of
highly resistant reflecting horizons; 3 — righthand branches of AZ Western curves;
4 — righthand 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 VagaiIshim tectonic zones, the curves of twoway probing
have very well defined qualitative features, associated with the non horizontal
nature of the studied surface. Fig. 8 gives as an example of twoway curves
for the azirauthal probing along the profile intersecting the Krotovskaia struc
ture. It can easily be seen that here the abovedescribed regularities are
maintained in the mutual positions of the righthand 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 DneprovskoDonetsk depression (1952) and the
CisBaltic depression (1954), gave favourable results and made it possible
to change over to the largescale 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 112 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 nonconducting
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 3035 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 5001000 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 socalled 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 largescale 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 crosssection by the downward dip of
the high resistance marker horizon. The regions of increased valwes of E
* Distancetime 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
crosssection 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 1040 sec
and an amplitude of the order of 12 mV/km. The maximum variation as
a rule was observed in the daylight hours (915 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. Lshape arrangement.
The measurements were carried out with Lshape arrangements with
measuring lines between 300500 m long (Fig. 9). For synchronization of
the observations a specially developed interference free TB6 Teleswitch
was used in circuit with an RPMS radio station (^). EPO4 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 northwest 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).
preJurassic 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 crosssection, this map shows how preJurassic
rocks dip towards the central regions of the western Siberian Lowlands.
To the south of Kolpashevo, there is a zone of upfolded preJurassic 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 VagaiIshimsk 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 ZavodoukovskVyatkinoVikulovo. 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 <'\amtlc, if ill llic sccoikI hall ol \ii^iil I In varialioiis wil li ainjilil iidf
II I » \i) A inV/kiu wnr coiiliimrd aliiKt^l iiniiil<i riiilrdly 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 crosssection 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 southwest 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 borehole 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 GAZ63
vehicles. The apparatus for the method of telluric currents is in the form of
the EPO4 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 whichERST2356 — is
intended for working in inaccessible country and also in the winter (the
socalled tractor variant); the second — the EPS1656— 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 2356 station is mounted on a trailer and its two PN100 genera
tors are powered by a special gasoline engine ZlS120, 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 ERS2353
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 EPO5 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 50100 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 S80 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^vframes 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 darkroom 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 ERST2356 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 S80 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 cablelaying device, designed for a PUM grade cable. The winch
can hold 68 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 GAZ69
vehicle which can move crosscountry. 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 (PN145)
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 ERS1656 station is mainly intended for dij)ole probes in
Fig. 14. Position of EItST2356 apparatus and equipment in the body of the
trailer. 1— generators; 2 — ZIS120 engine; 3 — control panel; 4 — EPO5 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 ESliO56 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 Tr2 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 vibroconverter.
When the armature of the vibroconverter 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 selfcompensating. 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 vibroconverter is moved to the
right contact, the input of the amplifier is shorted and simultaneously the
circuit of the secondary winding of the Tr2 transformer is disconnected.
In other words, the instrument produces a halfperiod rectification.
Fig. 15. The body of the GAZ69 vehicle with the PN145 generator. 1— PN145 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 2SKU and two
GB225 dry batteries which ensure normal operation of the instrument
for 4050 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
23 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 EShO56 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 GAZ69 motor
car and includes:
1. A 2channel 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 iT* '— 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
TH
'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 liighvokage mput, its circuit being similar to
that of the EShO56 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 TV6 teleswitch used in transmitting and receiving the time signals
by radio (tw^o units weighing 11 kg each). In the teleswitch there is introduced
an interference stable system, which limits the interference level.
Consequently, even with a high level of interference, the TV6 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 EPO4 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 23 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 twoway 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 twoway 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 TV6 teleswitch. Exploratory and Industrial Geophysics,.
No. 17, Gostoptekhizdat (1957).
Chapter 9
THE METHOD OF CURVED ELECTRICAL PROBES
M. N. Berdichevskii
The method of twoway 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 socalled 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 19541956 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 . 103
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 threeelectrode 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 3electrode 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 3electrode 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 = uiFuiF = ^ 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'.emine 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 3electrode 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 crosssection, 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 + iVZi]. (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^{tl) ^
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
rH
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
rH
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
rH
+
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
026
I^ CO 01
(J) CT) CT>
6 6 6
o _
9 9
O
O
o
//
\
\^
\^
S,

/
\
\
022
018
/
/
\
\,
— ■
/
/
1
\
I
/
/
V
014
/
1
\
/
oin
/
—
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
abovedescribed 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 measvuing 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 ocE^ 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 12
Hence
E
MN
IS)
Ee
+ 1 + ^ cos
K
COS A — sin A x
?3
l+2^cos6)
AB
t3
2
. ^{[IABY ^ AB „
ABV , AB „
— +l^cos0
/3
ABV ^ AB „
2^ +1^^0
l2^cos0
AB
2R
sin0
ABV , AB
2R +1 + ^^^
f3
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 11.5°, varies
234
M. N. Berdichevskii
for 70° < < 110° from tenths of a percent to 23 per cent. For angles O
of the order of 60° (120°) or 50° (130°) the error m determining q^ becomes
58% 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 34°, 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 rH
!f CO CO
(>. r^; On
LO LO ■<#
ON CM >0
LO rH '^
t~' r^ NO
c
rH CO C^J O ON 0>
CO CS rH 03 LO CQ
rH ,H rH c4 C>q CS
CO CO CO
P NO CM
LO T}! ^
lH
'e o lo
NO VO LO
d d d
CO CO CO
00 r; NO
r— H rH r— H
O ^ CNl
LO CO ^
CM CM CM
c
CO
CO Cl LO
f \q '^
5.47
5.16
4.71
rH LO ON
CO CO rH
!>: d NO
o
+
1.18
1.11
1.00
CO r t—
NO ■Tj; ^
CO CO' CO
4.91
4.60
4.16
o
rH
+
rH r c^
NO LO LO
d d d
lO LO o
CO !>; NO
r~ rH 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 ci
CI ^ rH
LO CO rH
CO CO Cl3
«
o
o
^ rH LO
LO LO r*
d d d
o cc NO 1 CO NO o
r; NO LO ' CO CM rH
rH rH rH 1 CM CM CM
o
r— 1
+
Cvl CM CM
d d d
ON NO O rH lO r
CO CO CO , CM rH o
d d d ' rH rI 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
rH rH rH 1 r— H 1— H rH
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^
rH C'J CO ^ C^) CO
d d d d d d
rH CM CO
d d d

+
rH
CM
o
o
o
CO
+
■* On t^
NO CNi r^
lH rn' d
1.40
0.35
—1.14
CM 00 r
CNJ CNJ CO
rH 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
rH 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 rH t^
r^ d ri
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 rH 03
d d d
VO NO LO
r CM NO
d d r^
1/3
a
11
CM
CM CO CO
NO •* CM
d d d
CM On CM
LO O LO
d d d
O t NO
LO lH O
d d rH
Co'
K
J
o
o
c^
o
CO
+
0.18
0.1
0.05
rH CM rH
rH O rH
d d d
ON '^ CO
p p CM
d d d
o
CM
+
'' 3 S3
rH O O
d d d
LO rH LO
O O O
d d d
On CM O
O O rH
d d d
o
CO
03 LO
rH rH O
111
rH CM rH
r— 1 O rH
11°
On ^ CO
O O CM
d d d
1
3SS
111
LO rH LO
poo
odd
d d d
o
/ ^
/ (N
lH CM CO
d d d
rH CM CO
d d d
rH CM CO
d dd
O
rH
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)
f3
2
+
sm
ABV ^ AB
__ Ul+^cos0
tz
2
+
ABV , AB
2R +iir^^^^
is
2 ^
/3
1 abV ab
' \
+
A2«)+liJ^°^(^+''>J
1
sin
ABV ^ AB „
t3 f3 •
^ ^^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 were conducted with the ERS23 electrical survey
station, which included two field laboratories, mounted on the GAZ63
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 planetable 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 planetable 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 halfspacing of
the feed electrodes equal to 200600 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 5001000 m.
When transferring from the AMNB arrangement to the quadrilateral or
azimuthal arrangements, the measurements were repeated at 12 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 planetable 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 70110°. 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 12 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 23, 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 5001500 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 11.5°. However, when
obtaining the rising righthand 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 34°, 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
fi562
tv
):

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. Twoway 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 twoway
curved azimuthal probes did not differ from the method of twoway quad
rilateral probes.
The characteristic curves of twoway 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 56%. 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 twoway curved azimuthal probes, there
is nothing which is in principle different from the method of interpreting
twoway 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 19301932 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 MakhachKala (1942), on
Taman' (19501951), in the Turkmenia (19411943 and 19521953), and
was also tried under the conditions prevalent in Western Siberia (19541955)
and UstUrta (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 semiquaUtative 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 MaloBabchikskaia 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 crosscutting 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
ChongelekskoChorelekskaia antichnal zone. This zone, in the limits of
the Kerch peninsula, according to the data of the loop method begins at
the southwest of the Koyashskoe lake in two small anticlines — Chokur
Koyashskaia and Opukskaia, separated by a small saddle, and ends at the
northeast 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 southwest, to the south of the Malaia Babchikskaya anticline
was mapped the YuzhnoBabchikskaia 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 funnelshaped 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 southwestern plain, 24 structures were mapped of which 6 [Yuzhno
Andreevskaia, (Karachskaia) Zhuralevskaia (Chaltemirskaia), KolAlchins
kaia, Ul'yanovskaia (Mangutskaia) YuzhnoMarfovskaia 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 (UzunAyakskaia) 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 lo^
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«ss2
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 (BashKirgizkaia) (^^) on the other have also been
discovered by the loop method.
At the Kharchenkov (ArmaElin) 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
(KoiAsanskaia), which is of interest from the point of view of its possible
petroleum content (^'*). This work in the southwest 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 southeast part of the southwest 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
southwest of the Uzunlar lake, was depicted and parallel to it a small upfold
related to the distant (AtanAlchhiskaia 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 southwestern 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 19491950, using the loop method, studies were made of an area
adjoining the southwest plain from the north and northeast (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 southwest 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 southeastern
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 southwestern plain of the Kerch pensinsula,
have been covered in the postwar 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
southwest 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 lH
%
hT^
WOODS
HOLE,
MASS./
250 1. 1. Krolenko
Experimental seismic studies by the reflected methods were carried out
on a small scale in 19501952 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 cophasal axes and by the socalled
"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
7080°).
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 VladislavovKharchenkovskaia tectonic
zone and in the zone of merging of the Moshkarevskaia and Shirokovskaia
structures.
Seismical studies on these areas, which have no interesting petroleumbea
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 northeast 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 southwest 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 (19501951) showed that within the limits of the studied
area, there was no northwest (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 eastwest 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
NaratTyubinskii range of mountains to the west and south of Makhach
Kala up to the Monas River. On the northwest 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 812 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 MakhachKala 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 MakhachKakhachkalinskian brachyanticline 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 brachyanticline,
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 19521955. 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 19541955 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 largescale 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.
: ^ii \i I/; ; .yV/
I I 1/ /\ \. v/ / /
VI
^ 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 12 km from one
another would probably facihtate surveys of the second order folding on
the background of the large structure of the UstUrt. 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,n3
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 switchingon 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 cinefilm by means of the MPO2 oscillograph.
Visual observations were made with the same arrangement, but in this
case, instead of the oscillograph, the EP1 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 4050 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 southwestern
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 semiquantitative coordinated 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 boreholes, 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 northwestern 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
ArmuDar'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 largescale 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 1053 on
the Territory of the StaroKrymsk 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 SouthWestern Turkmenia. Documents of the Sredneaz. geofiz. tr. Glav
neftegeofizika MNP SSSR (1954).
9. V. N. Dakhnov, The Loop Method. Proceedings of the AllUnion 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. KrolenkoGorshkova, 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. KrolenkoGorshkova, 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 AivazovskoMoshkarevsk Seismic Party
910154 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
MaloBabchik 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. geograogeofizich 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 19411942. 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 NonStabilizing Systems of Electrical Prospecting
Fields in NonHomogeneous 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 (2530 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 crosssection 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 1520° (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.
CV2S and CV3S 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 (CV3S 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 (CV2S 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 crosssection 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 doublelayered 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
CV2S or CV3S reference graphs have been calculated.
A curve is selected from a CV2S or CV3S 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 + LYl^_
(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 SN1 reference
graph for each spacing of the probes ; q^ is the corresponding value of the
ordinate of the curve on a CV2S or CV3S 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.
10
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 CV2S 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 doublelayered medium at the VEP point and a curve
on the reference graph CV25.
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 CV3S 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 nonhorizontal 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 SN20 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 coordinates 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 crosssection 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 crosssection 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 crosssections.
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 SN1 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 CV3s 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 crosssection 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 crosssection 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 lefthand 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 nonhorizontal 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 crosssection consisting of a twolayered 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 CV25 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
• mf* •
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 CV3S 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 1530%. 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
10
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 CV2S 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 crosssection 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
crosssection in the direction of the contact.
H
A double layered curve for Q^ ~ °° ^^^^^ d <^ — is transformed into
a threelayered curve with ^3 < Q2, and for d < ^H into a twolayered 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 threelayered
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 righthand
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 15 2
Profile W
■^ 3py 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 400500 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 nonhorizontal
contact and of the righthand 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 righthand asymptotic branch can be defined as the
mean of the righthand branches of VEP curves with equal .
d
For values of — — > 2, the true position of the righthand 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 crosssection 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
crosssection 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 nonhorizontal 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 righthand asymptotic
branch of VEP No. 466 can therefore be taken to be true. The position of
the righthand 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
crosssection 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
nonhorizontal 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 nonhorizontal contact as a result of which the
righthand 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 electroprospecting 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 righthand 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 righthand
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 nonhorizon
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 nonhorizontal 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
crosssection ^^, 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 crosssections 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
crosssections 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 crosssection. 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 lefthand
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 (lefthand 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 nonhorizontal 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 N35
.\
DQP N36
'■ \
&'
j;^
Ar'W'
<K/
\
<
DQP N37
y
DQP l\J32
B
DQPN26
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 crosssection 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
crosssection 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 nonhorizontal 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 crosssection 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 nonhorizontal 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. CVIS, CV2S and
CV3S reference graphs. GSGT, 1938.
3. E. N. Kalenov, The ClS 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 gasair 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
40 45 50
'■5 20 25 30
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 UralVolga 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 borehole 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 gasbearing 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
Coalbearing
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 mudpump 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 (1520 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
100450 iiA, which corresponds to a drilhng fluid content of 2.613.5 cm^/1.
The gas factor for this case is 315 m^/m^. According to available data the
gas factor of the bed Dj^, when it is oilbearing, 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 curvetotal heavy hydrocarbons). Content of individual
hydrocarbon gases is given from the results of an analysis on a GST2 instrument.
The intervals of increased gas content (12571285 m) detected by means
of the DP degasifier can be described as water bound and oilbearing, since
the gas factor for them is 510 m^/m^, while the oil beds of the coalbearing
sequence and the overlying Tournaisian limestones are characterized by
a gas factor of 1530 m^/m^.
The intervals (12571260 m and 12731283 m) v/ere tested and a flow
of water was obtained. A core sample taken from a depth of 12581259.8 m
consists of quartz sandstone, evenly permeated with oil.
The interval with high gas readings at a depth of 12931303 m, situated
20 m from the top of the Tournaisian stage, has a gas factor of 1520 m^/m^.
This provides a basis for assuming that the given interval is oilbearing,
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 oilbearing
strata which have a small gas factor and are often missed when working
with a floating degasifier.
* Separate analysis was conducted on a GST2 instrument (see B. V. Vladimirov,
The GST2 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 oilbearing 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 13 m of penetration and viewing them in ultra
violet light. For this a luminoscope is used, which consists of a lighttight
chamber equipped with a source of ultraviolet light (UFO2 or PARK4
lamp with a filter passing only idtra violet rays). The sample of drilling fluid,
which is sometimes specially treated, is placed in the ultraviolet 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 svuface 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 ultraviolet radiation and
a photosensitive element placed in a lighttight chamber.
A UFO^A lamp with a UFS3 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 ultraviolet
radiation UF04A; 2 — filter UFS2; 3 — vessel contaiaing fluid; 4 — narrowband
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 ultraviolet rays.
The luminescent emission thus excited is detected by the photosensitive
element, for which an FEU19 photomultipher is used (Fig. 1). The photo
multiplier cathode is protected from the ultraviolet 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 directcurrent 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 fullwave 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 righthand 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 420640 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 AllUnion 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 ultraviolet light. Suspensions
were prepared from two clays (light and dark) and from petroleum with
different intensities of luminescence.
" 10
01
' ^
X
y^
10 10"" IQS 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
0.51 cm^ per litre, i.e. approximately as much as can be detected by visual
examination of clay suspension samples in ultraviolet 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 lefthand 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
32
30
26
24
22
20
08
06
02
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 — oilbearing sandstone; 5 — drilling fluid.
Samples of fluid, taken while drilhng an interval of 16001730 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 petroleumbearing bed. A core sample taken from this depth
proved to be oilbearing 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 ultraviolet 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 ultraviolet 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. KonstantinovaShlezinger, 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, Luminescentbituminological 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 BOREHOLE 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 socalled 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 borehole.
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 BOREHOLE 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 CF4 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 300400 ra/j,, and of OS19 glass
for wavelengths of 6001100 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=l02
^
104
^=106 ;k=IIO
^\
. 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
10
01 03 05 07 09 ll 13 15 17 01 03 05 01 03
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
01 03
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 BOREHOLE 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
105
MO
■y
120
Fig. 2. Dependence of attenuation coefficient K. on the specific gravity y of the
suspension and the wavelength }. of the hght.
For the darkcoloured 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
15
10
//
/
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. 13 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 BOREHOLE INVESTIGATION
333
improvement in visibility is observed in the infrared 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 102
^=107
120
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 FM2 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 infrared
(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)
{yl)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 BOREHOLE 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
d1
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 borehole 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 1020 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,
infrared and ultraviolet 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 infrared 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 infrared film sensitive respectively
to rays of wavelengths 840 and 960 m^.
As a result of comparing j^hotographs taken with ilhnnination by ordinary
and infrared rays, the following was established.
* Actually, depending on the method of illuminating the fluid, considerably greater
dilution may be required.
OPTICAL METHODS OF BOREHOLE INVESTIGATION 339
(a) In infrared light, details of the rock are reproduced less clearly than
in ordinary light; so, in infrared 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 infrared
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 infrared 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 mercuryquartz lamp with a filter of Wood
glass was used as the source of ultraviolet rays. The photography was carried
out on isopanchromatic film; a yellow filter, absorbing the ultraviolet 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 ultraviolet rays. Photographs of rock samples exposed to ultraviolet
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'aviolet rays conveys to some extent the structure of the rock. On
photographs of nonbituminous (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 ultraviolet 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, ultraviolet
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 cinefilm 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 threestrand 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 SM29 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 BOREHOLE 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 filmwinding mechanism, are situated in the upper
part. An Industar22 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 filmwinding mechanism, another
carries the supply to the seriesconnected 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
windingon of the film is controlled by the increase in current intensity
associated with this switching.
OPTICAL METHODS OF BOREHOLE 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 finegrain 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 BOREHOLE 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, 13).
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, 68), from those not filled with cement (gaping)
(Fig. 10, 45).
In well 2589 in the Kamenskaia region, having a diameter of 90 mm,
fortyfive 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); coarsegrained 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). 13— brecciated rock
with pieces of darkgrey dolomite and lightgrey limestone; 4 and 5— lightgrey
limestone with open crack and dolomite inclusion; 68 — fissured dolomite, cracks
filled with calcite.
OPTICAL METHODS OF BOREHOLE 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 coarsegrained 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). 12 — coals; 3 — clay rock;
4 — sand rock.
cloudy in sandclay 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 boreholes. 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 OILBEARING
STRATA FROM THE SPECIFIC RESISTANCE
S. G . KOMAROV AND Z. I. KeIVSAR
The method for determining the permeabiHty of oilbearing 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 oilbearing 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 petroleumbearing 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 gasbear
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 petroleumsaturated 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 (waterbearing stratum).
The specific resistance of the oilbearing 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 waterbearing 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 UraloVolzhsk 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 waterbearing 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 waterbearing 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 oilbearing 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 — AllUnion Commission on mineral Resomces.
THE PERMEABILITY OF OILBEARING STRATA
35]
Figure 1 gives tlie curve recommended by Dolina<*) for determining
the permeability of petroleumbearing Devonian sandstone by the specific
resistance. The curve 1 shows the dependence of the coefficient of increase
in resistance Q on the coefficient of oilbearing 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
098 095 09 08 05
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 01
8 10
K^,
Fig. 1. Graphs for determining the permeabihty of the petroleumbearing 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 petroleumbearing 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 petroleumbearing
deposits. This is connected with the fact that close to the contour of the
petroleumbearing 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 petroleumbearing 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 waterbearing 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 OILBEARING STRATA 353
The relative deviation of the accepted value q^ from the actual reaches
a value of the order of 0.10.3, especially in the case when the average
value is taken for the specific resistance of the waterbearing 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. = Q1I, (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 waterbearing 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 waterbearing 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 2025% (*).
THE PERMEABILITY OF OILBEARING 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
xj., 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 170200 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 OILBEARING STRATA
357
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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 waterbearing 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 socalled free water, the quantity of which near the top surface of
petroleumbearing deposits or waterpetroleum contact can be sufficiently
THE PERMEABILITY OF OILBEARING 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 waterpetroleum contact.
MoROZov^^^) considers that the determination of permeabihty by the
specific resistance can be carried out for strata 16 m above the waterpetro
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 waterpetroleum
contact and the characteristics of the stratum.
To establish the effect of the waterpetroleum 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 waterpetroleum contact. This indicates that the effect of the
waterpetroleum 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 oilbearing stratum depends not only on its distance to the
waterpetroleum 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 WATERPETROLEUM 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 OILBEARING 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 OILBEARING 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 ovDobrynin), 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 OILBEARING 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; 0200, 200400, 400600 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 OILBEARING 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 — ■
ltr=
/>*
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 (petroleumbearing
strata of the Groznyi, Turkmen, Baku, Krasnodar and other deposits) led
THE PERMEABILITY OF OILBEARING 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 iLjeiOBTAINED 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 KaraganoChokrakskian
deposits; 8 — NebitDag deposits; Turkmen.
THE PERMEABILITY OF OILBEARING 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 = 0.30.66.
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 (3550%;.
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); 49 — for different deposits in the U.S.A. from the data of Masket'I^);
47 — sandstones; 8 — ^hmestone; 9 — dolomite; 10 — Kartashevo deposit, limestone;
11 — NovoStepanovsk deposit, dolomites (indirect determination); 12, 13 — sandstones,
Tuimazinskian deposit, I and II groups; (1013 according to the data of Zaks^^));
10, 12, 13 — determined by the method of capillary displacement.
THE PERMEABILITY OF OILBEARING 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); 37 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 13 are well known curves of Jones^^',
the curves 49 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 socalled 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 OILBEARING 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 borehole 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 boreholes, (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 oilbearing 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 oilbearing 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 OILBEARING 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 physicochemical 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
petroleumbearing 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 oilbearing 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 OILBEARING 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
petroleumbearing 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 sandclay crosssection (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/IBEARING
STRATA FROM THE SPECIFIC RESISTANCE
1. The method for determining the permeability of petroleumbearing
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 OILBEARING 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, 1020, 2050 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 petroleumbearing
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 waterbearing 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 3050% 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 IndustrialGeophysical 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 Petroleumbearing 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 OILBEARING 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 SaratovVolga 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 RESISTIVITYMETERS
E. A. POLYAKOV
The specific electrical resistance of the drilling fluid (water solution of
salts or a mud suspension ■which fills a borehole) 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 resistivitymeters, 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 RESISTIVITYMETER 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
ivitymeter 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 resistivitymeter coefficient
and therefore errors in the results of measurements. Owing to the shortness
384
NEW TYPES OF WELL RESISTIVITYMETERS
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* resistivitymeters are sometimes influenced
by the surroundings such as the walls of the ^vell (metal of the cylindrical
casing tubes, practically nonconductive 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
electrodebearing 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 electrodebearing 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 resistivitymeter is one in which the probe is not protected by a metallic
or a nonconducting screen, while the influence of the borehole 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 probebearing 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 borehole 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 borehole shaft. As a consequence the usefulness of the instrument is
impaired .
(3) Change of the resistivitymeter 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
itymeters placed in fluids of variable salinity (curves 24).
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 waterfilled 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 RESISTIVITYMETERS
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 0.520 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.
006
005
004
003
002
6 8 01
Fig, 1. The dependence of the well resistivitymeter coefficients on the specific electric
resistance of the fluid. 1 — REU type of resistivitymeter; 2 — RA3 type of a serial
resistivity meter; 3— PTL VOGSGT (Ufa) serial resistivity meter ;4GML BGK (Ufa)
serial resistivitymeter.
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 socalled 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 electrolytecontaining tank in testing the
nature of the electric field.
The resistivitymeter 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
ivitymeter 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 resistivitymeter 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 resistivitymeter 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 resistivitymeters, 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 cutouts 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 RESISTIVITYMETERS 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 resistivitymeter coefficient. Thus, in designing a resis
tivitymeter one should start from a sonde corresponding to the electrode
devices 7 and 8.
THE DESIGN OF THE RESISTIVITY METERS
The well resistivity meters REU57 and RSE357* were designed and
manufactured on the basis of the most suitable electrode devices. The
general shape of the resistivitymeter REU57 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 crosssections. 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 resistivitymeter RSE357 adopted for the
borehole 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 resistivitymeter serves as the electrode A^.
To avoid the influence of the terminal parts of the resistivitymeter 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 REU57.
NEW TYPES OF WELL RESLSTIVITY METERS
395
r^
_j
\
J//7
i
Fig. 6. The well resistivity
meter of the type RCE357.
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
resistivitymeter caps have channels through
Avhich — as well as through the median brass
tube — passes the central straird of the cable.
In the resistivity meter RSE357 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 resistivitymeters 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 resistivitymeter 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 RSE357
and REU57 can be used as independent in
struments in the existing logging installations,
as well as parts of complex borehole 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 REU57 in fluids of different
specific resistance. It is clear that the REU57
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 REU57
396 E. A. POLYAKOV
resistivitymeter coefficient changes only by 6 per cent ^vhich is quile
acceptable.
Investigations of the prototypes of the REU57 and RSE357 resis
tivitymeters 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 REU57 and RSE35T
resistivitymeters 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
tivitymeter has no direct relationship to the resistance of the fluid.
(3) In order to achieve a constant resistivitymeter 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 threedimensional
shape.
(4) Based on the electrode arrangement described two borehole resis
tivitymeters REU57 and RSE357 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 BOREHOLES 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 borehole 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 boreholes.
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 borehole 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 borehole 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 electronvolts. 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 borehole investigation.
However, the most important advantage of the accelerators of charged
particles is the possibility of governing their radiation. This allows investiga
tion of boreholes 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 socalled neutron generators
which produce beams of neutrons, is of a paramount interest. The most
promising is the use of a neutron generator in neutrongamma 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 rockforming 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 5003000 /^ sec <i) whereas in water it is 25 ^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. Crosssection 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 101=^ cm.
For the light nuclei — to Ashich category belong most of the rockforming
elements — a lies within the limits of 0.5 to 1.2 barns (Fig. 1) (^). The rela
tively small range of a in the rockforming elements permits consideration of
the inelastic scattering to be approximately equally likely for all of them.
This circumstance precludes the possibility of carrying out neutronneutron
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 rockforming 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^^ (halflife 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 boreholes 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 intermittentlyoperating 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 highvoltage 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 zircontritium target to produce the reaction
H^(c?, n) He*, evolving neutrons, with an energy E^ on average near to
ICT) Cp CO
1
_ ^ .
100200 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 23 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 borehole instruments it is difficult to obtain
potentials higher than 150200 kW. The corresponding deutron energy will
produce a neutron energy of radiation within the limits of 1316MeV<^).
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 {Ed0.096)^'
where £'j£ is expressed in megaelectronvolts 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
Crosssection 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
crosssections
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 singlecomponent 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 68 hour worldng day. This interval of time
is completely sufficient to investigate a borehole. 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 borehole 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
11.7710«
2.1101*
2.1 10"
7.5108
7.5107
7.51010
1.5 109
1
0.009
1.6 10"
1.610s
5.7105
5.7104
5.7 10'
1.15108
101
0.09
1.61010
1.610''
5.7104
5.7103
5.710«
1.15105
102
0.9
1.610s
1.610*
5.7103
5.7102
5.7105
1.15104
103
9
1.6108
1.6105
5.7102
5.7101
5.7104
1.15103
104
90
1.610''
1.610*
5.7101
5.7 —
5.7103
1.15102
105
900
1.610«
1.6103
5.7 —
57 —
5.7102
1.15101
io«
9000
1.6105
1.6102
57 —
570 —
5.7101
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 boreholes 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 gammagamma 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 borehole investigations with the aid of the standard
apparatus RARK (containing one discharge counter of the type VS9)
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 2.42.8 g cm^)
no thicker than 57 cm ^^' ^°) in a borehole. Consequently, the presence
of a mud crust on the borehole .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 23 g/cm^ increases four to fivefold
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 electronposition 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 (0.050.07 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.
GAMMANEUTRON LOGGING (GNL)
To study the geological sections of boreholes 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 borehole 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 photoneutrons, 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 gammaquanta 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 oilbearing strata from a higher content of deuterium in oil
(1.53 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
borehole 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
Gammaquantum
energy in MeV
1
4
13
47
74
H
Be
Al
Ag
W
2
9
27
109
9
7 • 10*
3 • 10*
3 • 103
1.7 • 102
2 • 102
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 oilbearing 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 108 X 1.1 X 10* X 103 = 1.3 X 109
The effectiveness of the fast neutron scintillation counters (diameter
4 cm; length 10 cm) used in borehole investigations is approximately
0.1 per cent. Consequently, the probability of registering one neutron will
be Pg = 1.3 X 1012.
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 crosssection of 100 cm^ the natural neutron background I^^ affecting
the readings in GNL the neutron stream should exceed severalfold 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 gammaneutron
log for the following purposes.
1. Prospecting for berylliumbearing 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 borehole 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 gammaneutron logging
in bombarding the deuterium of a wateroil 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 11.5 MeV. By a proper
selection of the displacement potential in the network of the amplifiers
input cascade it is possible to succeed having the noncorresponding 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 106 X 105 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 noncorresponding
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 • 1011
1.7 • 101"
6 • 10'
Fi9(/),a,y)Oi6
CaFa, NaF
0.338
6.3
1.74 • 108
5 • 106
Li'(p,y)Be^
Li (metal)
0.460
17.6
14.8
1.9 • 108
0.95 • 108
8 • 105
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 • 1011
~4 • 108
Na23(p,a)Ne2i
0.2870.539
—
2.0 • 1012
—
Mg2*(p,y)AP5
—
0.226
0.418
—
7 • 1011
2 • 101°
~2 • 10'^
3 • 10^
Mg25(p,y)A126
—
0.317
0.391
0.496
—
3 • 1011
1 • 1010
1 • 10'
3 • 10'
H3(p,7)He*
—
0.16.2
20
7 • 1011
2 • 10'
H3(a,7)He5
H3(a,n)He*
—
0.16
0.16
—
2 • 1012
5 • 109
* 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 BOREHOLE 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 sourcepower necessary for GGK in curies
Gammaquanta
Length of the probe in cm
energy in MeV
20
30
40
50
60
70
10.0
17.6
2.7 • 105
2.9 • 105
8.2 • 105
8.6 . 105
1.6 • 10*
2.0 • 10*
4.3 • 10*
4.0 • 10*
7.9 • 10*
6.8 • 10*
1.4 103
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 (110 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 79.
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 borehole.
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 gammagamma 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 wellreactor.
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 Gammagamma logging Method
(GGK). Applied Geophysics, No. 18, Gustoptekhisdat, (1957).
10. E. M. FiLippov, Gammagamma 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
KonstantinovaSchlezinger, 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
KrolenkoGorshkova, N. G. 269
KupalovYaropolk 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, 278, 38, 41
Anisotrophy coefficient, of sedimentary
deposits, 244
Anticlinal folds, effect on electrical probing,
28991
Azimuthaldipole probes, 198206, 223^0
arrangement, 223
errors in, 2316
practical procedure, 236^0
theory of, 22431
B
Berezovka gas deposits, 210
Beryllium, detection by radiation, 415
Boreholes,
camera for, 3404
optical studies of, 32848
Boring rate,
effect on gas saturation of drilling fluid,
311
variation in, 310
Camera, for boreholes, 340^
Caspian Sea area, geology of, 2546
CisBaltic depression, geoelectric survey, 198
CisCarpathian depression, seismic wave
reflection in, 75, 85
Clay suspensions,
effect on oil detection, 3202
optical properties, 32838
Cretaceous deposits, and multiple wave
reflection, 75
Crimean steppes, geology of, 252
D
Damping of seismic waves, with distance, 29
37, 389, 904
Density, effect on seismic wave intensity, 30
Devonian sandstones,
oilbearing, permeability, 349
specific resistance, 335, 350, 354, 361,
363
Diffracted seismic waves, 99119
dynamic properties, 108110
kinematic properties, 1008
Diffraction,
structural formation causing, 99
from tapering strata, 1109
Dipole probes,
azimuthal, 197, 22340
effect of nonhorizontal rock boundaries
on, 2927
for inaccessible country, 214
practical appUcation, 196206
theory of, 16995
DP degasifier, for drilling fluid analysis, 3112
DrilUng fluid,
degasification, 311
gas saturation,
estimation, 302
factors affecting, 302, 305, 310
luminescence of, 28992, 31922
luminescent spectral properties, 3225
optical properties, 308, 32838
E
Echoes, single/double comparison, 40, 42
Elastic properties of media, 12
Electrical probing,
distortion by nonhorizontal rocks, 271
97
in inaccessible country, 214, 223
Western Siberia, 196222
strata relief studies, 2016
426
Subject index
427
at sea, theoretical curves for, 18395
at sea, theory of, 1708
twoway, 196206
Electrical surveys,
by loop method (see Loop method)
mobile units for, 21421
Electron logging,
of rock strata, 41921
Kaolin, optical properties, 330
Kerch peninsula,
geophysical data, 2412, 2445, 247, 263
gravitational field, 248
seismic exploration, 250
Gammagamma logging, of rock strata, 4089
Gammaneutron logging, 3819, 40917
for oil, 414
Gamma quanta generator, construction of,
4179
Gas factor, of oilbearing strata, 313
Gas logging, factors affecting, 3015
Gas saturation,
of driUing fluid, effect of boring rate on,
280, 305, 310
estimation of, 302
fluid, factors affecting, 302
of rocks, estimation, 3025
Gravimeter, static, for use at sea, 12338
Gravimetric surveys,
accuracy of, 1402
of Kerch peninsular, 248
observational errors, 1446
of Russian basins, 1601
source of errors in, 1447
Grazing rays, spatial direction, 52
Gypsumanhydrite, and multiple wave reflec
tion, 85
H
Head waves, intensity, 2830, 3335
Hodographs,
construction for reflected waves, 5573
of diffracted waves, 101, 1147
theoretical, for multiple reflections, 82
Infrared photography, of rocks, 338
Intensity of seismic waves, 1443
Isochrone chart, for waves reflected from
dome, 62
Lame constants, 15
Leningrad tables, 27
Light transmittance, of clay suspensions,
331
Logging,
gammagamma, 4089
gamma neutron, 40917
gas, 84, 30114
luminescence, 31527
radioactive, 397421
Longitudinal seismic waves, intensity, 11^3
Loop method,
developments in, 24463
economics of, 265
errors in, 251
geological surveys by, 244^59
principles of, 241
uses of, 2424, 251, 266
Luminescence,
of drilling fluid, 319
of oil, 31920
of oilchloroform solutions, 3201
oil deposit logging by, 28597, 31527
Luminoscope, photoelectric, design of,
2869, 316319, 326
M
Mesozoic deposits, 201, 204, 210, 250
MobUe units, for electrical surveys, 21421
Moscow syncUnal basin, gravimetric survey,
1601
Multi layered media,
reflected wave intensity in, 40
refracted primary waves in, 40
surface hodograph for reflected waves
in, 6273
428
Subject index
N
Neutrongamma logging in, 398401
Neutron generators,
for borehole logging, 398^01
construction of, 37380, 4018
theory of, 4012
Neutron radiation, of rock strata, 415
Oil,
chloroform solutions, luminescence of,
3201
luminescence of, 29890 31820
spectral characteristics, 3225
Oilbearing strata,
detection by gas logging, 2834, 3134
by loop method, 244, 248, 252, 256
luminescence logging, 28597
31527
permeability, 352^, 3802
radiation, 414
seismical data, 251
permeability, determination, 3215,
34953
Oscillations, of multiple reflections, 82^
Oscillograph, for electrical exploration 2179
Paleozoic deposits, electrical surveys of,
2024, 20910, 213
Permeability,
determination, errors in, 3245, 3302,
3523, 35860, 36473
from core analysis, 3548, 365, 3757
from specific resistance, 34952,
3528, 360, 365, 37682
of Devonian sandstones, 34952, 376
of oilbearing strata, 34952, 380
specific resistance increase, 3326, 3408
3604, 36876
water saturation, of oil strata, 353, 35860
Photography,
of rock samples, 338, 344
of wells. 3368. 3407
Porosity, specific resistance of rock strata,
362, 368
Primary seismic waves,
intensity of, 1518, 2837
in multilayered media, 2937
in twolayered media, 2829
R
Radioactive logging, of boreholes, 397421
Reflected multiple waves, 7597
damping of, 90^
dynamic properties, 8095
intensity of, 1827
interference with single waves, 7595
kinematic properties, 7680
oscillation intensity, 8590
Reflected single waves,
intensity of, 1827, 37^3
interference by multiple waves, 7595
in multilayered media, 403
in twolayered media, 379
Reflection coefficients, calculation, 18, 214,
267
Reflective rays, spatial direction, 51
Refracted rays, spatial direction, 52
Refracted waves, 1518, 2837
Refraction coefficients, 15, 17, 31
calculation, 17, 21, 267
Resistivity meters,
coefficient variation, 38490
design of, 3936
electrode design, 3903
Rock density, effect on seismic wave intensity,
30
Rock strata,
disturbance, effect on electrical probes,
27197
identification by gamma radiation, 4001,
409
by photography, 3446
natural neutron radiation values, 415
Salinity, effect on resistivity measurements.
35862, 38690
Subject index
429
Sandstone, detection by radiation, 415
Sea probes, electrical apparatus for, 16983
Seismic rays, spatial direction, 4473
Seismic waves,
damping with distance, 29, 379
diffracted, dynamic properties, 108110
kinematic properties, practical, 1058
properties, theoretical, 1005
doubly reflected, 3910, 91
intensity calculation, 1527
longitudinal, intensity, 1143
multiple reflected, damping with distance,
904
reflected, dynamic properties, 8095
intensity, 18, 20, 8590
interference by, 75
kinematic properties, 7680
primary, in multilayered media, 2837
factors affecting, 307
intensity of, 158, 2837
reflected, hodograph construction for,
5573
intensity of, 1827, 3743
in multilayered media, 40, 5573
refracted, intensity of, 2837
simple reflected, 18
Seismical studies,
comparison with loop method, 2504
with telluric currents, 212
of Kerch peninsular, 250
Siberia, west,
electro geological survey, 196222, 256
gravimetric survey, 1613
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,
38490
marine electrical probe instruments,
17883
MezoCenezoic deposits, 204
oilbearing strata, 34953
measurement (see Resistivity meters)
permeabihty of rock strata, 3604,
36882
porosity of rock strata, 36873
Spectral characteristics,
of oil products, 323
Stereographic nets, construction of, 467
Stereographic projections, 4473
Taman peninsula,
geology of, 254
Telluric currents,
electronic measuring apparatus, 21921
in geological surveys, 20613
in Western Siberia, 209
Transverse waves, effect on longitudinal wave
intensity, 30, 39^0
Turkmenia, geology, 256
Twolayered media, wave intensity in, 289,
37
U
UstUrt plateau,
geology of, 258, 267
Volcano sedimentary strata,
effect on electrical surveys, 199, 204, 210
W
Water saturation,
permeability of rock strata, 3734,
377^, 3802
specific resistance of oil strata 351, 353,
35860
theoretical value, 364
Well camera,
design of, 3404.
Wulir net, 45, 55, 57
accuracy of, 73
uses of, 4753, 65, 689