Interpretation of geoelectric resistivity curves

-

---

---~-INTERPRETATION OF GEOELECTRIC

RESISTIVITY CURVES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN

DOCTOR IN DE TECHNISCHE WETENSCHAP

AAN DE TECHNISCHE HOGESCHOOL TE

DELFT, OP GEZAG VAN DE RECTOR

MAGNI-FICUS Ir. H. j. DE WIJS, HOOGLERAAR IN DE

AFDELING DER MIJNBOUWKUNDE, VOOR

EEN COMMISSIE UIT DE SENAAT TE

VER-DEDIGEN OP WOENSDAG 10 JUNI 1964,

DES VOORMIDDAGS TE 11 UUR

DOOR

F. G. VAN DER HOEVEN

mijnbouwkundig ingenieur

geboren

te Hilversum

BiBLIOTHEEK

DER

T

ECHNISCHE HOGESCHOOL

DELFT

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR

PROF. O. KOEFOED

CONTENTS

Introduction

....

.

. . .

CHAPTER

I

Problems of solving three layer configurations.

Compar-page

9

ison with computed curves, not always practicabie

14

11

The equivalence concept . . . .

15

111

First method for solving the equivalence problem

16

IV Second method for solving the equivalence problem .

25

V

Application of the second method to computed curves.

31

VI

Electronically computed three layer curves.

34

VII

Errors due to latitude with the first cross

37

Conclusion . . . .

Acknowledgements, References .

Annex, with Algol program and computed curves

Abstract . .

Samenvatting

40

40

42

48

48

/

L

el,

dl,

ea

er,

k

V

r

J

a

8

e2

d2

dr

LIST

'

OF SYMBOLS

Distance between current electrodes in SCHLUMBERGER electrode

con-figuration.

Resistivities of subsurface layers.

Thicknesses of subsurface layers.

Apparent resistivity.

Resistivity and thickness

o~

a fictitious replacement layer.

Resistivity factor, defined as

k

n

=

(en+1-en)

I

(en+1 +en).

Voltage difference between potential electrodes.

Distance from current source.

Current through current electrodes.

INTRODUCTION

The purpose of resistivity measurements is, to jnfer subsurface geology from

surface measurements of electrical quantities.

The applications serve several stratigraphic problems. The existence of a

watertable in fill material or more consolidated rock, can be investigated.

Civil engineering problems using the method, may involve examination of

dam and reservoir sites, location of rock and gravel quarries, tunnel

construc-tion, or shallow depth to bedrock mapping. Resistivity measurements, have

been made on ore bodies, coalbeds and salt domes.

With resistivity measurements, a current is passed through the ground, and

potential drop is measured at the surface with two additional electrodes. The

measurements of current and potential drop, give a figure for apparent

resist-ivity (la, of the subsurface using the formula:

V

=

(la· J/2nr.

The term "apparent resistivity" applies to the hypothetical assumption,

that the subsurface is electrically homogeneous, (which generally it is not).

The term apparent resistivity is also used, since it does not mean the resistivity

of any kind of subsurface material, but depends on the electrical resistance,

offered by various subsurface layers, furthermore depending on the depth of

current penetration. Current penetration, primarily increases with the

dis-tance between current electrodes and depends on resistivity contrasts of the

subsurface layers.

For obtaining information on depth and resistivity of these layers, the

appar-ent resistivity is measured and plotted as a function of electrode distance.

The apparent resistivity

approximates the resistivity value of the surf ace

layer at short electrode distance, (short in relation to the thickness of the

sur-face layer).

A larger electrode spacing gives an apparent resistivity (la, gradually

express-ing more influence of deeper subsurface layers.

In the simple case of a surface layer on a second infinite subsurface layer,

the apparent resistivity graph, has parallel horizontal asymptotes at both ends

of the curve, with values of

and

for short and long electrode distances

respectively.

FIG.

1. For measurements, the potential electrodes are placed on

a straight line, connecting the current electrodes.

The SCHLUMBERGER method, uses potential electrodes, placed close together

in re1ation to the distance between the current electrodes and uses half of this

distance

1/2L,

in its graphical representation. (Using

l/2L

as a parameter,

im-proves the resemblance of the computed curves, and the fieldcurve

measure-ments, made with closely spaced potential e1ectrodes.)

The WENNER method, places the four electrodes an equal distance apart

from each other, and uses this distance "a" in its graphical representation.

As aresult, for any given situation, the WENNER curve is practically the same

as the SCHLUMBERGER curve, except for the figures of its electrode distance

axis, bemg a ratio (about

3/4)

smaller for the WENNER curve than for each

iden-tic al point of the SCHLUMBERGER curve. Two layer curves for the

SCHLUM-BERGER arrangement as computed by CGG [1955] Compagnie Générale de la

Géophysique, are given in FIG. 1, while those for the WENNER arrangement,

as computed electronically, are given in FIG. 25.

The interpretation of the resistivity curves requires comparison with

theoret-ical curves, computed for various assumed subsurface configurations. Such

subsurface configurations can then be inferred if they produce the same

anom-alies as those of the observed curve.

A catalog of computed standard curves such as the CGG collection, of 480

curves, can be used for such comparison purposes.

Additional curves, may have to be computed individually for closer

resem-blance. The calculation of resistivity curves however is such a time consuming

process, th at additional interpretation methods are required for application

in the field office.

Since the determination of the surf ace layer requires the use of two-layer

standard curves, such additional interpretation methods are based on the

'

same series of two-layer standard curves, as eXplained m the following chapters.

Computing resistivity curves

For theoretical curves, the subsurface is assumed to consist ofhorizontallayers,

each of them electrically homogeneous, on an infinite bottom layer. A

sub-surface of only th ree layers will be subjected to discussion in this paper.

For computing resistivity curves, the electric field of a layered subsurface is

taken to be caused by the current source itself and by its hypothetical mirrored

images

Qn.

Vertically below the current source, these images are taken as

mirrored through each discontinuity between two layers of contrasting

resist-ivity. The strengths of the images

Qn,

are fractions of the strength

'

of the

current source

(Qo

=

1) itself, depending on the resistivity factor

k

n

=

=

Ü?n+1-en)/(en+1 +en)

of the discontinuity, causing the images. The

con-tribution of each image

Qn,

to the complete current pattern, furthermore

decreases with its depth.

The basic formula

V

=

el·

J/2nr.

is consequently multiplied by a function

composed of a series of images, divided by their depth term:

The use of basic formula and image poles is treated in textbooks on geophysical

exploration. HEILAND [1946].

Three calculating methods

The calculating method as described by KOEFOED [1955] has been

pro-grammed for an electronic computer.

FLATHE [1955] gives a method for calculating resistivity curves and a

similar method has been used by CGG [1955]. With these last two methods,

the resistivity curve is obtained as a combination of many simplified three

layer curves ofwhich

d

=

dl

and

=

0

(i.e.

two differently conducting layers

of equal thickness on a perfectly conducting substratum).

Sincc it is customary to express all resistivities as a multiple of

el,

the function

ea/el

=

(I+F(k, dl,

r))

can be plotted on a graph ofwhich the axes represent

ea/el

=

1 and

1/2L/dl

=

1.

This leads to the use of a bilogarithmic scale, having the advantage th at

resistivity curves retain the same shape and size, even if the surface layers of

the two resistivity curves are different in thickness and resistivity. This can be

eXplained as follows:

If

the thickness of all the subsurface layers, (including dl) and the electrode

di stance would be increased with the same percentage, no difference could be

observed on the resistivity curve except for the figures on the electrode di stance

axis being increased with the same percentage. Similarly, if all resistivities

under consideration, would be multiplied with some other factor, this would

change apparent resistivity

ea

in the same proportion. Such a change in both

units of resistivity and electrode distance on bilogarithmic graph paper, would

simply cause a parallel shift of the curve.

(If multiplication, changes the scale

of the curve on normal graph paper, it merely displaces the curve on

log-arithmic paper).

A

logarithmic scale with a modulus of 62.5 mm has been

used throughout this paper, except for FIG. 3 and FIG. 14.

Two layer standard curves are computed for one single surface layer, on an

infinite substratum of varying resistivity contrast. One set of standard curves

on a single page, suffices for all values of dl, el and

e2,

since the curves are

significant as ratios

1/2L/dl

and

e2/el,

and not only for separately variable dl

and

el.

FIG. land FIG. 25.

Determination

rif the surface layer

until it coincides with a twolayer standard curve or an interpolation thereoff.

The selected two layer standard curve must coincide with the field curve over

the greatest possible length. There is little difference between the standard

curves, of slightly different resistivity contrasts, and the accuracy of the method

depends on the similarity of the selected standard curve and the field curve.

A perfect fit of the standard curve on the field curve, means that the ratios

e2/el

and

d

2

/d

l

of the field curve and the standard curve are identical. The

origin of the standard curve's graph where

1/2L

=

ea

=

I can now be marked

on the field curve's graph, as its "first cross" where the values of

L/2

and

ea

are

determined as

dl

and

el.

;'Ja/iJl

~~~~~---

--

---i~m---%L

---%

---06

FIG.

I

Two layer standard curves on bilogarithmic scale, for the

electrode

configuration.

A-B electrode distance

L.

m-n potential electrode distance, generally smaller than

l/.L.

12

If

the second layer is sufficiently thick, a single curve solves

dl, el,

and

d

of

the field curve. In this case, the standard curve and the field curve are identica

I

up to the point where the third layer's discontinuity can be noticed at large

electrode distance. A perfect fit of the standard curve on the fieldcurve is

difficult to choose however if the second layer is insufficiently thick. The

determination of

el

and

dl,

is hardly affected then, but the choice of

e2

be-comes doubtful, if the fieldcurve can be matched with different standard

curves. Solving the value of

e2

in such cases is the purpose of the following

chapters.

CHAPTER I

PROBLEMS OF SOLVING THREE LAYER CONFIGURATIONS.

COMPARISON WITH COMPUTED CURVES,

IS NOT AL W AYS PRACTICABLE

Of a two layer configuration, the second layer's resistivity can be determined,

by comparison of the field curve with a series of computed two layer curves,

as explained in the introduction.

With a three layer curve the second layer can not easily be determined in

this manner, since the current pattern largely depends on

the

first and the

third layer, even at an advantageous electrode distance.

Unless the second layer's thickness is more than ten times the thickness of

the surf ace layer, th

.

e field curve does not clearly show the resistivity of the

second layer. In this case, the second layer can only be determined if the

field-curve closely resembles a field-curve that is computed from assumed parameters.

For such close resemblance, the necessary number of computed three layer

curves for comparison purposes becomes prohibitive. A complete catalog of

three layer curves would have to range through all possible value combinations

of

e2/e1;

d2

/d1;

es.

Therefore three layer curves can not always be solved by

comparison with computed curves, and another method is needed.

In order to find an alternative to the use of computed curves, it will be tried

to find a characteristic point of the resistivity curve that is independent of

e3.

These points would be taken from a collection of computed curves, and used

for making a graph or diagram, to solve the fieldcurve or sounding graph,

from which the same characteristic point can be determined.

CAGNIARD [1952] uses the so called replacement value of first and second

layer, as such a characteristic point of the curve. This concept of replacement

value for first and second layer, can be eXplained as follows:

At short electrode distance, the current pattern depends on the contrast

between first and second layer. At large electrode distance, this contrast is no

longer of influence on apparent resistivity, and the two upper layers can be

thought to be combined in a so called replacement layer, with its own

replace-ment resistivity

er,

and replacement thickness

fit..

The three layer curve ean be thought to consist of two smoothly joined two

layer curves, each covering one of the two branches of the three layer curve.

The first branch of the three layer curve is determined by the resistivity

contrast between the surface layer and the subsurface layers. The second

branch of the three layer curve is determined by the contrast between the

third layer and its overburden.

The two values for thickness and the two values for resistivity of the two

surface layers, or over-burden, can be given as one replacement thickness

d

r,

and one replacement resistivity

(lr.

The two replacement values can be

given in the form of a graph, since mathematical representation is too

com-plicated.

The replacement values, can be determined, by finding a two layer standard

curve matching the second branch of a computed three layer resistivity curve.

The two replacement values are then given, by the coordinates of the standard

curve's origin or cross. This point is marked on the curve's graph, as the

"second cross", and characterises the resistivity curve, whereupon these points

can be combined in a graph. This graph can then be used for solving the field

curve, of which the replacement values can be determined by matching its

second branch with a standard curve and marking its cross.

CAGNIARD'S

application of the replacement concept, condenses the complete catalogue of

three layer resistivity curves in a few diagrams; a distinct simplification. Three

separate diagrams are made for different values of

(la.

Since the graphs are

largely coincident however it proves to be possible to combine them into one

generalised

CAGNIARD

graph. The me rits of this simplification have heen

published by

KOEFOED

[1960].

CHAPTER II

THE EQUIV ALENCE CONCEPT

Unequivocal interpretation of resistivity curves is difficult because of the

equivalence phenomenon. This phenomenon causes close resemblance between

resistivity curves of widely different bottom configurations.

This confusing resemblance occurs, when tht; total resistance offered by the

second subsurface layer remains the same because of appropriate variations in

d2

and

(l2.

If

(}2

is smaller than (}l

and

(la,

the total resistanceofthe second layer

remains the same, if

(l2/d2

remains constant. If

is larger than

and

(la,

the

total resistance of the second layer will be as constant as the product

(l2· d2•

Whenever the tota! resistances of the second layers are identical, the resistivity

curves of such configurations are similar, and it

will be hard to distinguish the

right values for

d2

and

(l2,

the more so if the second layer is not thick.

With

CAGNIARD'S

procedure, the second branch of the field curve can be

matched then with several two layer standard curves, and the position of the

"second cross" is ill defined. Each possible position of the second cross, can be

used for a figure for the product

d2·

(!2,

(or the ratio

d2/

(!2)

and it becomes

necessary to find the value for (!2

in a different manner.

In order to obtain this information, the equivalent curves will have to be

studied for slight differences, in spite of their apparent similarity.

In the following chapters, two methods will be given for solving those cases,

where several configurations have almost identical resistivity curves.

CHAPTER III

FIRST METHOD FOR SOLVING THE EQUIVALENCE PROBLEM

The two curves of

FIG.

2, represent configurations with wide1y different (but

resistance wise equivalent) second layers.

The first method for solving the equivalence problem, applies to curves that

are shaped like a beU or like an inverted beIl,

((!2

larger than (!l

and (!3;

or (!2

smaller than (!l

and

(!3

respectively).

FIG.

3.

Since the replacement values used by

CAGNIARD,

are found by using the

second branch of the resistivity curve, additional information is likely to be

~==--~---%L

FIG.

2 Two "equivalent" resistivity curves.

((I.

=

_af

!!;

d.

=

dl;

=

and

(I.

=

d.

=

1/.d1;

(I.

with widely different second layers.

The interpretation of these curves faces obvious problems, since the two three layer

configura-tions are different from each other, but their resistivity curves are similar.

%Ll7rJt=r~~

%L

/7a/;01

a

b

FIG.

3 Three layer resistivity curves of which the second layer's resistivity is smaller

(a)

or

larger (b) than the first and third layer's resistivity.

found by using the first branch of the resistivity curve as weIl, this part of the

curve being largely shaped by the second layer. The idea can be practised as

foIlows:

Both branches of the curves are characterised by the tangents through their

inflection point, the part where they have maximum steepness.

FIG.

4.

The intersection of these two tangents is the point that can be used to

characterise the resistivity curve. The points can be joined for making a curve

for constant

and variabie

d

while apparently this curve is independent of

the third layer's resistivity

FIG.

5.

The curve of

FIG.

5, is independent of

the third layer's resistivity, a fact

that ean be demonstrated, by making the same kind of curve in

FIG.

6 for a

I'a/{-ll

Taf7!/ef7l Ihrol/g'h /nf/ecf/on po/ni

of Cl/rves second 6raf7ch

~--1---

--

----~~---~L

\

\

\

\

\

,,/'

x -

/f7ler secf/on of lan9'enl.5

/

\

/

\ Tangenllhrol/!/h inllecl/on po/ni of

\ re.5/sfiv/1y ct/rYcs f/rslbranch

FIG.

4 Three layer resistivity curve

e2

l/lOel;

e.

el;

d

2

dl.

The intersection of the two tangents is the point that can be used to characterise the resistivity

curve.

FIG. 5 A series of three layer curves

(e.

=

l/,rh; e.

=

el;

and varying

d.)

with a curve

joining the intersections of their tangents. The curves ha

v

e been taken from the CGG

collec-tion, [1955].

FIG. 6 A series of three la

y

er curves

(

e2

1/,e1; e.

=

inf

.

and

v

arying

d

₂₎

with a curve

joining their intersections

.

---r---~~----~--~~~~--~--- ~L

~--==---%

---~

FIG.

7 A seri

e

s of three layer curves,

(

(I.

1/,(11;

d.

dl;

and varying

(I

.

)

with the same

curve as

in

5 and 6, joining the intersections of their tangents

.

The three layer curves

have been tak

e

n from "Amt für Bodenforschung" in Hanover.

three layer series with an identical

e2

=

but a different value for

es

(=

infinite).

The two curves of

FIG.

5, and

FIG.

6, are identical with each other. The

curves are also identical with a third one in

FIG.

7, made for a series of three

layer curves with the same

=

1

/

4e1

and varying values of

It

is

demon-strated by

FIG.

5, 6 and 7, that regardless of the values for

d

and

es,

the

inter-section curve for

=

is cdnstant. The same demonstration of identical

curves can be given for other vallles of

e2.

The "intersection" curves in

FIG.

8 and

FIG.

9, are made with resistivity

curves from "Amt für Bodenforschung" in Hanover. The "intersection"

curves are identical and independent of

which makes them suitable for

interpretation of the three layer configuration's resistivity of the second

layer.

In

case the second branch of the resistivity curve dips downwards with a

steepness in excess of 45° (like in

FIG.

8) the method's accuracy improves, if

----+---..~--.._--~,_----~~~---%L

FIG.

8

Two series ofthree layer curves from "Amt für Bodenforschung"

(e.

= 3el; es= ll/.el;

and

e.

3el; es

O.Iel; d./d

= 1,2,3 resp.) with a curve joining the intersections of their

tangents. The tangents ofthe curves with es

O.Iel> are drawn at 45°, instead ofthrough the

inflection points (at a steeper section) ofthe curves.

the tangent through the second branche's inflection point is replaced with a

tangent (at this second branch) of 45° steepness.

The series of computed resistivity curves from CGG are used for making a

graph of such "intersection" curves for different values of second layer's

resistivity

(22

in

FIG.

10. This graph can then be used [or determination of the

field curve af ter its first and second branches have yielded the intersection of

their steepest tangents.

A similar diagram can be made for curves of constant d

and variabie

(22.

~L

FIG. 9 Two series ofthree layer curves from "Amt für Bodenforschung"

(P.

= 3el; ea

=

2el;

d./dl

=

1,2,4 and

e.

=

3el; ea

=

el;

dl/dl

=

1,2,3) with a curve joining the intersection

of their tangents.

10

----~---~--

--

---~---~L

0.1

YJ!J

FIG.

10 A group of curves like the curve in FIG. 5, for different values of resistivity second

layer

et.

FIG.

11. This diagram is somewhat more dependent on es however than the

diagram for constant

e2

and varia bIe

d

₂

in FIG.

10. (In the extreme case where

e2/es

approaches 1, the curves approximate one vertical through

l/2L

=

5.5

for all values of

d

The use of the e2-diagram by itself however already solves

the d

af ter the replacement figure has been taken with CAGNIARD'S

diagram.

---+---+

__

~----~----~~---~L

o

.

z

FIG.

11

A diagram for different but constant values of

d.,

and variabie

!?,

similar to the

diagram

in

FIG. 10.

The ease of the method, (requiring just one extra diagram) suggests its use

as an addition to the method of CAGNIARD.

FIG.

12 demonstrates how the method is used:

lf

the "first cross" of the field

curve is placed on the origin of the diagram, the intersection of its steepest

tangents, determine resistivity and thickness of its second layer.

Application of this diagram to a variety of computed curves (from CGG

and "Amt für Bodenforschung in Hanover") has shown the method to be as

accurate as the plotted resistivity curves themselves. This is understandable,

since the diagram is made with tangents to a curve, fitting the points of the

curve themselves.

Latitude with the surface layer's determination of its "first cross", and

there-fore its thickness

dl

and resistivity

el,

influences the accuracy of the second

layer's determination, as treated in the last Chapter.

The diagrams of

FIG.

10, 11 and 12 have been made with resistivity curves

from CGG for the Schlumberger electrode configuration. Resistivity curves

for the Wenner electrode configuration, require a different but similar diagram

that has been made in

FIG.

13.

(Ja/PI

~--+---~~~-+--~----~=---%L

FIG.

12 Diagram for the determination of second

l

ayer's resistivity

I?

and

thickness

d.,

with

a three layer curve, as an example of its use

.

---r---~----~---_r~---a

FIG.

13 An "intersection" curve

(e.

=

S/nel)

from a group of resistivity curves, for the

WENNER

electrode configuration,

1955]. Additional

resistivity curves

[MOONEY

&

1956] having been used to make the other "intersection" curves.

One of the "intersection" curves in this diagram, has been made with

resist-ivity curves from KOEFOED [1955], while the other curves were taken from

MOONEY and WETZEL [1956].

It

will be noted that the resistivity curves

and their intersection curves are

similar for Wenner- or Schlumberger-electrode configuration,

except

for a

ratio of the electrode distance coordinates in the order of3/

The intersection method requires two tangents through the infiection points

of the curve. Not all the resistivity curves are shaped with two infiections

how-ever, like those in

FIG.

3. Since the "intersection" method depends on those

infiection points, another method is given in

the

next Chapter for curves of

a different type.

CHAPTER IV

SECOND METHOD FOR SOLVING THE

EQUIVALENCE PROBLEM

For solving the resistivity curves of which

(23/(22

and

(22/(21

are both larger than

1, and the curves of which

(23/(22

and

(22/(21

are both smaller than 1, FIG. 14,

another method is needed, and has been found to require no extra diagram,

but some careful measurements of the steepness or angle of the field curve.

The steepness of the resistivity curve is a measure for the resistivity contrast

at certain depth, of the total resistance of the layers above this depth and of

part of the layers below thls depth, as deep as the current penetration.

There-fore the resistivity curve's steepness is characteristic for

(22

at small electrode

distances or shallow current penetration, while an increasing part of the third

layer will be of influence at increased electrode distance. The steepness of the

resistivity curve at several points, can be measured by the steepness of its

tangents at those points, or by comparison with standard curves.

The method, using this concept, is based on finding a two layer standard

curve having exactly the same angle as the field curve at an electrode distance

1/2L

that has to be the same for both curves. FIG. 15.

FIG. 15 shows a three layer curve having a steepness of 30

°

at certain

elec-trode distance,

(1/2L

=

1.5). The steepness has a different value at another

electrode distance as shown in fig. 16, where the steepness of the threelayer

curve is 45

°

; (at

1/2L

=

1.9).

~...c~---1---".---

-

~L

a

b

FIG.

14

Three layer resistivity curves

(e./el

smaller than

1

in a;

e./rb

larger than

1

in b)

ofwhich the third layer's resistivity

e3

varies from zero to infinity.

---T~---_T/~O---~L

Two /ayer slal7O'ard cvrye

0.1

Tnree /ayer cvrve;o2

=

~;O/--;03 -

~6;O/

,

-FIG. 15 A three layer curve and a two layer curve having a steepness of 30

°

at an electrode

distance, identical

(1/

2

L

=

P/

2)

for both curves.

;017/;01

--~~~---

____ rm

____________________

~L

0.1

Three layercilrye

P2=)4.P/;P.J=}f6;P/

FIG. 16 A three layer curve and a two layer curve having the same steepness (45

°

) at a

certain electrode distance

(1/2L

=

1.9

)

.

At this point a two layer curve of different

(!2/(!1 (1/6

instead oP/s) is found to

have the same steepness as the three layer curve.

The second layer's resistivity of the selected standard curve, will have some

value between the resistivity of the second, and that of the third layer under

consideration. At short electrode distance

(1/2L

=

1), the

(!2

of the selected

standard curve, approximates the resistivity of the three layer configuration's

second layer, since the current does not penetrate below it yet. At larger

electrode distance, the current reaches greater depth, and the standard curve's

(!2

approximates the three layer configuration's

pa,

(divided by the

replace-ment resistivity of the layers above this depth).

Since the method is intended to determine the resistivity

(!2

of the three

layer configuration's second layer, its steepness at short electrode distance is

most interesting. The matter would be solved if the

(!2

of the standard curve

having the same steepness, can be marked or plotted, at these short electrode

distances

.

The method can be practised by super imposing the three layer

curve on a series of two layer standard curves keeping their l/2L

=

1 axes on

top of each ot her, moving the three layer curve upwards and downwards. At

several electrode distances, the standard curve having the same steepness as

the field curve, is marked by intersecting its

(!

2

with the vertical through the

1/2L

value. The points obtained in this manner can be joined into a curve,

which will then serve to find the

(!2

of the field curve.

FIG.

17.

----r---~w~---

--

---%L

I

I

~

I

~k~---l--

_,

Slandard cl./rve pz

_ _ _ _ _ _

.

_ _ _ _ Slandard cl./rve P2

=

16

p,

'"I

i-~---0.1

.5landardcl./rve

Pi:

= :1ZPI

...

...

_...

,

_...

...

....

....

--

-

Tóree /ayer field wrve

FIG

.

17

Illustration ofa three layer curve

(eo

=

l/.el;

e.

=

l

/

16el;

d

=

dl

)

with several two

I

I

I

,

_,

\

\

\

,oZ=.39(Y1

9

~~

.J

-%

%-*

_Curves

_{COflflecl;fl~Oil7!J}

fhe fwo ldyerJ. 'dl7dard

curve.J havil7g theJame

+'k

.sleepl7l?J.J

~

~

~

~

~

~

ffz

;/,6

%g

FIG.

18 A diagram of

two

layer

curves, used for plotting ratio

ea/e.,

in

FIG.

19.

The ratios

ea/e.

are marked for points of the curve having a certain steepness by intersecting

a vertical through

its electrode

distance

I/.L

and a horizontal through

e.,

(resistivity

second

layer) of that standard curve. This is indicated in

FIG

.

18, for a standard curve

(e.

l/SeI)

having a steepness of

30

°

at electrode distance

I/.L

1.75.

The aid curve derived in this manner from the field curve, has a horizontal

asymptote for short electrode di stance

1/2L,

since at these points the second

layer's resistivity of the standard curves, approximate the second layer's

resistivity of the three layer configuration. The aid curve also has a horizontal

asymptote for large electrode distances where the second branch of the field

curve, can be fitted to one and the same standard curve in

CAGNIARD'S

manner.

The aid curve thus derived, having parallel horizontal asymptotes at both

ends, is shaped distinctly similar to the two layer curves. Both having

pa-rallel asymptotes at each end of the curve with a smoothly joining middle

section.

Rigorous proof for the similarity of the two different kind of curves, requiring

their mathematical representation, is beyond the scope of dus paper.

Applica-tion of the "steepness" method, to a number of computed three layer curves,

10

%L

I

FIG.

19 A diagram of

ea/e!

ratios derived from a series of two layer standard curves,

that ....

n.

be used for the interpretation of a three layer curve.

" .

29

i

-_-t---,L-

_ _ _ _ _

_ _ _ _ _ _ _ _

-+'=-o _

_ _ _

_ _

%L

I

~

I

~

I

I

I

I

I

_~

I

"

I

~

",+"

0-"

"

I

I

7,0

'+

"

FIG.

20

Three layer

resistivity

curve

(ll.

=

l/.(h;

e.

=

l/lOel;

d.

=

dl

)

with an aid curve

derived from it for the determination of resistivity second layer

e

•.

however can he taken as an indication of its practicahility. This application

of the "steepness method" will he given in the next Chapter. Once their

sim-ilarity to the aid curve, derived from the three layer curve, is accepted, the

two layer standard curves can he used for comparison purposes. The cross of

the standard curve, matching the aid curve, determines the second layer's

•

resistivity

of the three layer configuration, just like the standard curve's

cross determÏnes resistivity and thickness of the surf ace layer.

Whereas the method is primarily intended for finding

or second layer's

resistivity of the field curve, it yields a curve that can he used to find its

thick-ness

d2

as well.

If

the

ratio of a three layer configuration has a value

he-tween

1/2

and 2, there seems to grow an error in this determination of

d

this

question will he part of

th~ discussion on errors in second layer's resistivity

caused hy errors with the first cross.

Since the method requires considerahle care to select or interpolate the right

standard curve, as far as its steepness is concerned, it may he easier to measure

th~ field curve's

~teepness

directly with a protractor than hy comparison with

by the use of a protractor instead of the standard curve series, since these

measurements are just two ways of putting the original idea into practise.

For this purpose a diagram of "constant angle" curves can be prepared from

the series of standard curves. A "constant angle" curve is made by finding the

ratio between

(la

and

(l2

of all the two layer standard curves where they have

that same angle of steepness. FIG. 18 and FIG. 19.

A diagram is made for curves of 15°,30°,45

°

and 60°, other sequences being

possib1e. The same degrees of steepness can then be measured at certain points

of the field curve. The electrode distance

1/2L

of these points is then used in

the diagram to find the corresponding ratio between field curve and the curve

to be derived from it. FIG. 20, shows the three layer sounding curve

d2

=

1;

(l2

=

1/4;

=

1/16,

from CGG with the aid curve derived from it for the

determination of its second layer's resistivity

(l2

=

1/4(l1.

The ratio A-B for 30

°

, (at

1/2L

=

11/2) has been taken from FIG. 19, the

other ratios having been taken simi1ar1y.

CHAPTER

V

APPLICATION OF THE SECOND METHOD

TO COMPUTED CURVES

Since the "steepness method" applies to a curve type (FIG. 14) of which only

a few computed resistivity curves were available, a number of these curves

had to be computed specifically for a check on the accuracy or usefulness of the

method.

FIGURES 21-24, show a selected number of those computed three 1ayer

curves with their aid curves for the determination of their second 1ayer's

resis-tivity.

The curves have been computed for the WENNER electrode configuration,

requiring a diagram (FIG. 26) a little different from the diagram in FIG. 19,

(for the SCHLUMBERGER configuration).

Points of the curves where their tangents have a certain steepness are marked

as indicated. In all these examples the aid curves have been compared with

two layer standard curves (FIG. 25) for determination of the threelayer

con-figuration's second layer.

This comparison has provided each aid curve with a cross, thus marking

resistivity as weil as thickness ofthe configuration's second layer. The reliability

of the "steepness method" depends on the accuracy of the resistivity curve for

+

---+--+

10 é1

FIG. 21

Three "equivalent" three layer resistivity curves, and

their

interpretation, using the

ea/e.

from FIG. 26.

The curves' parameters are:

e./e1

1,2,3;

d./d

₁

1,2,3;

e.

gel>

resp.

---+-+-+-+-... +'

FIG. 22 Two "equivalent" three layer curves, and their interpretation. Their parameters

are:

e./(h

3,6;

d./d

₁

1/2>

1;

es

ge1'

respectively.

For solving the three layer curve, an aid curve can be made, by "adding"

ea/e.

at several

points

of

measured steepness. The aid curve, solves

e.

and dl of the three layer curve, with its

first "cross".

?a/,/J'

ISO

____

~_4---~~

--

~~---~~~O---a

--+- ...

"

--+----033

---+----

+

Ol

FIG. 23 Three layer curves and their interpretation, using

ea/e.

ratios

from

FIG. 26. The

parameters of the three layer curves have been selected to make the curves

"equivalent"

to

each other. These parameters are: e./el

=

l/S'

1/., 1/0;

d./d

1

=

1,2,3; Ps

=

1/Ue1'

resp.

10

a

~~~--+-~~--~--~~---~---+

0.2

---+----+

short electrode distances.

If

the points of the aid curve, can not be joined with

a smooth curve, this would indicate an unlikely section of the resistivity curve.

Such sections can be eliminated by measuring more suitable parts of the

curve with tangents for different steepness values. Generally the precision of

the curve improves proportionally with the thickness of the second layer.

Inaccuracy due to latitude with the "first cross" will be treated in the last

Chapter.

CHAPTER VI

ELECTRONICALL Y COMPUTED THREE LAYER CURVES

In order to obtain more evidence for the use of the "steepness method" more

curves were needed as a check.

About two hundred three layer curves have been computed with an

elec-tronic computer for this purpose. The method as described by

KOEFOED,

O.

[1955] has been used for making the "Algol" program for the electronic

computer (TR 4). The image pole series has been continued in this program

beyond a certain Ininimum number, until reaching an image pole with a

contribution of less than O.OOl12a.

The curves have been selected to provide a variety of equivalent cases, for

123/122

and 122/121 both larger than 1; or both ratios smaller than

l.

Six different values of 123, ranging from 123

=

lh6121

to 123

=

16121,

have been

selected, each with about five different 122 (resistivity second layer) and seven

different

d

(thickness second layer) values.

The curves as well as their "Algol" programIning, are given in Annex A.

The "Algol" program can repeatedly be used for computing all possible

three layer configurations, for seven different

d

's

on an infinite bottornlayer.

The Wenner electrode configuration has been used for all the electronically

computed curves.

Since the two layer standard curves for the Wenner electrode configuration

are different from those computed for the Schlumberger configuration,

FIG.

25

and

FIG

.

26 respectively show these standard curves and their diagram of

12a/122

ratios.

It

will be noted that the curves are almost identical, except for

a shift of the electrode distance axis in the order of

3/4.

/0

8

_ _

- - . 3

FIG.

25 Two layer standard curves

for

electrode configuration.

A-B current electrode distance 3a.

m-n potential electrode distance a.

350 ~Oo ,350 20° /SO 2 10 0

a

.30°

FIG.

26 Aid curves derived from

two la

y

er standard curves for the interpretation

of three layer curves.

CHAPTER VII

ERRORS DUE TO LATITUDE WITH THE FIRST CROSS

As described in the introduction, the first cross of the three layer curve is taken

from the two layer standard curve, having the c10sest resembling shape over

the greatest possible length. There is some latitude with the choice of the

standard curve however, creating a possible error with the first cross and the

resulting first layer's thickness dl and resistivity el of the three layer curve.

This error influences the second layer's

d

z

and

e2

as weil, since these are

expressed in el and dl as a ratio. Apart from this ratio, there is another re1ation

ship, which causes a more complicated error with CAGNIARD'S method as weil

as with the steepness method. In view of this possible error both methods are

compared herewith:

Tr{/f!

pOJ/'lOn

TirJ! cross

field

curve

Correcf

pOJtiion

2 nd cross

Shifled

pos/f/on

sec on

if

cross

FIG. 27 Interpretation of three layer (or field) curve, (Is

=

es =

l/au(!1;

dl

=

dl>

with

diagram

(e.

0).

Latitude with the first cross, leads to an error

in

the interpretation of resistivity and thickness

second layer.

\

Correcf

Znd cros",

Shifted

2 ndcro",,,

r----r---+-____

FIG.

28 Interpretation of a three layer curve with

diagram.

A vertical shift

with the

first cross,

causes

an

error

in the determination of the thickness and

resistivity of the three layer curve's second layer.

Using

CAGNIARD'S

method, the field curve's first cross is placed on the origin

of

CAGNIARD'S

graph, whereupon second layer's thickness and resistivity are

determined by the position of the second cross. Latitude with the first cross

therefore causes the same latitude with the relative position of the second cross.

Depending on the value of

{!2,

and the spacing between the

{!2

lines of

CAG-NIARD'S

network, this causes errors of varying importance.

If

the first cross is shifted too much to the left (right) this not only decreases

(increases) dl, but also increases, (decreases) the ratio

{!2/{!1

and d

/d

If

the

first cross is shifted too high,

(low)

this not only increases

{!l,

but also decreases,

(increases) the ratio

{!2/{!1,

in a somewhat larger degree than the shift of the

first cross.

Using

CAGNIARD'S

network, th ere is no tight rule for the precise results of

errors with the first cross, since these results depend too much on the value of

second layer's resistivity and thickness. Generally, the errors loose (gain)

importance with increasing (decreasing) thickness second layer.

The diagram used for the "intersection method"

FIG.

12, causes similar

inaccuracy as

CAGNIARD'S

diagram, due to latitude with the first cross.

Using the "steepness" method, latitude with the first cross, leads to a similar

{!2/{!1

error as with

CAGNIARD'S

method, in a different manner:

38

If

the first cross is shifted too far left, (right) this not only decreases

(in-creases)

dl,

but also decreases, (increases) the steepness of the curve's tangent

at each electrode distance

1/2L.

This in turn, increases the ratio

(22/(21,

see

FIG.

29. As aresult, resistivity second layer

(22,

would be deterrninèd too large,

(for those cases where

(21/22

is smaller than 1).

This left ward shift of the first cross represents a subsurface configuration

with a second layer of which the resistivity

(22

=

(21.

This extra layer should

have been included in the surf ace layer if the first cross had been determined

in the proper position. This resistivity

(22

=

(21

can be accurately deterrnined

unless the shift of the curve is quite small.

If

resistivity of the second layer

(22,

approximates resistivity of the third

layer

(23,

the accuracy of the method improves, since such a curve resembles

one of the standard curves on which the method is based.

If

the cross is shifted too high, (low) this increases (decreases)

(21,

but there

is no further change in

(22,

since the steepness of the resistivity curve at each

electrode distance value

1/2L

remains unaffected.

If

compared, the errors from a horizontal shift of the first cross, with

GAG-

"-"

"-"

"--

...

"-,

"

" ,

"-"

,

_,

"

,

,

_,

,

"-"

_"

...

"

,

"-" "-"

FIG.

29

Interpretation of a three layer curve

Ü?2

=

l/,(!1; es

=

l/leel;

d.

=

dl)

with

"steep-ness" method.

An

erroneous horizontal shift of the first cross, causes an error in the interpretation of resistivity

second layer

el'

NIARD'S

method and with the "steepness" method, show similar results. A

vertical shift of the first cross, involves a larger error with the CAGNIARD

method, than with the steepness method.

Considering the shape of the curves in FIG. 19 and FIG. 26, it is tempting

to formulate them with mathematical expressions. The expression

eah!l

=

=

_(1/2

L-0.8)

W/

2

L

-

0.3)

fits the curve for 15

°

quite accurately (for the curves

of 30, 45 and 60

°

the constants 0.8 and 0.3 are 1.1, 0.3; 1.5, 0.5; and 2.0, 0.5;

respectively and have to he multiplied with 0.75 if the WENNER electrode

configuration is used).

The whole standard curve series

(e2lel

smaller than 1

)

is represented with these

simple formulas, and the standard curves can actually he constructed, using

these formulas to give steepness at each point of the curve. The mathematical

procedure in case of a three layer curve would be too complicated however to

he of advantage over the graphical method for solving them.

CONCLUSION

The equivalence phenomenon causes latitude in the interpretation of

geo-electric resistivity curves, in terms of thickness and resistivity of the suhsurface

layers.

An effort has been made to find the characteristics of the field curve leading

to better accuracy.

As aresult, this paper gives two simple methods, and illustrates their use

with several kinds of three layer resistivity curves .

.

These methods can usefully be employed as an addition to the earlier known

procedures for solving three layer resistivity curves.

ACKNOWLEDGEMENTS

The au thor is indebted to "Wiskundige dienst" (Mathematical dept) of T.H.

Delft, and specifically to Mr. L.

B.

D. HARKEMA, for his assistance with the

programming for the electronic computer TR 4.

REFERENCES

KOEFOED,

0.,

1960, A generalised Cagniard graph for the interpretation of geoelectric

sound-ing data. Geophysical Prospectsound-ing. Vol.

111, 1960

(

3

)

.

LAsFARGUES, P.,

1957,

Prospection

électrique

par courants continus, Masson &

Cie., Paris.

MOONEY, H. M. &

W. W. WETZEL,

1956,

The potentials about a point electrode and apparent

resistivity curves for a two-, three- and four layer earth.

KOEFOED, 0.,

1955,

Resistivity curves for a conducting layer of finite thickness embedded in

an otherwise homogeneous and less conducting earth. Geophysical Prospecting. Vol.lll (3).

-

Compagnie Générale de Géophysique,

1955,

Abaques de sondages électrique Geophysical

prospecting, Vol.

111

suppl. no. 3.

FLATHE, H.,

1955,

A practical method of calculating geoelectrical model graphs for

horizon-tally stratified media, Geophysical Prospecting,

111,

p

.

268-294.

-

CAGNIARD,

L., 1952,

La prospection géophysique des eaux sousterraines, congres sur

l'hydro-logie des zones arides, UNESCO, Ankara

.

]AKOSKY,].].,

1950,

Exploration geophysics. Los Angeles.

ANNEX

The following flow diagram of the "Algol" program illustrates computing method of three

layer resistivity curves on an eIectronic comput

e

r. The program

is

made for thicknesses of the

second layer of d

₂

1J., 1J., 1J2>

1

,

2, 3, 5, dl'