Methods of Electrical Profiling and Mapping · 2013. 12. 20. · To illustrate electrical...

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CHAPTER SIX

Methods of Electrical Profilingand Mapping

This chapter briefly describes the use of steady currents to study lateral

changes of resistivity, an application called electrical profiling or mapping.

Unlike the vertical soundings described in previous chapters, which aim to

provide values of electrical resistivity with depth at a specific location,

profiling methods have traditionally been designed to provide only a qual-

itative map of how the apparent resistivity varies over some regions. Mod-

ern equipment of shallow geophysics, however, has made it possible to

collect data in ways that combine profiling and sounding, and thereby,

may allow a quantitative study of electrical resistivity, both laterally and

vertically in the earth. As in the rest of this book, we will focus mainly

on the physical principles that determine the response of electrical profiling

to the resistivity distribution in the earth, as well as on the key model

parameters in different applications.

6.1. ELECTRIC PROFILING

To illustrate electrical profiling, we consider several examples of

geoelectric sections and arrays.

6.1.1. Example 1: Vertical ContactWe start by studying in more detail a model considered earlier of a vertical

interface that intersects the earth’s surface (Fig. 6.1A). The model can be

taken as the simplified representation of a geological fault, juxtaposing

different uniform regions. We will formulate the boundary-value problem

by first assuming that the current electrode eA is placed slightly below

the earth’s surface at depth h, located in a medium with resistivity r1.The charge on the electrode is then eA ¼ e0r1I . We also place the origin

of coordinates x,y,z at the electrode, with the x-axis directed perpendicu-

lar to the vertical contact surface and the z-axis perpendicular to the earth’s

surface and positive downward. As usual, we assume that air is an ideal

Methods in Geochemistry and Geophysics, Volume 44 # 2010 Elsevier B.V.ISSN 0076-6895, DOI: 10.1016/S0076-6895(10)44006-8 All rights reserved.

331


insulator (zero conductivity). Let U1 and U2 be the electric potential in the

two regions separated by the contact and R the distance from an observa-

tion point to the origin. The boundary-value problem for the field has the

following form:

1. At regular points, the potential obeys Laplace’s equation:

r2U1 ¼ r2U2 ¼ 0;

2. Close to the electrode in region 1, the potential approaches the field of

an electrode in the uniform medium:

U1 ! U0 ¼ r1I4pR

, asR ! 0;

3. There is zero current flow across the earth–air interface:

@U1

@z¼ @U2

@z¼ 0, if z ¼ �h;

4. The potential and current are continuous across the vertical contact:

U1 ¼ U2 and g1@U1

@x¼ g2

@U2

@x, if x ¼ d;

5. The potential goes to zero at infinity:

U1 ! 0 and U2 ! 0, if R ! 1.

h

z

xd

eA eA

eA K12eA

K12eA

Air

r1 r2

r1 r2Earth

(a) (b)

r=•

Figure 6.1 (A) Vertical boundary intersecting the earth–air interface. (B) Equivalentmodel with distribution of image charges obtained by the method of mirror reflection.

332 A. A. Kaufman and B. I. Anderson


To solve this boundary-value problem, we will use the method of mirror

reflection and replace the actual model with a fictitious model in which

the charge and medium are reflected across the earth–air interface

(Fig. 6.1B). From the results of Chapter 3, we conclude that both the

source charge eA at z ¼ �h and its mirror image at z ¼ �2h induce sur-

face charge at the vertical interface. As we know, the fields of these

induced charges can be represented by the fields of image charges of

magnitude K12eA, located symmetrically with respect to the vertical plane

x ¼ d (Fig. 6.1B). It is easy to show that this arrangement of four point

charges satisfies all the boundary conditions of the problem, including

the condition of zero current flow across the earth–air interface (condi-

tion 2 above) and the continuity of current across the vertical contact

(condition 4). Inasmuch as we are interested in the case when the current

electrode is located on the earth’s surface, we take the limit h ! 0, which

simply amounts to doubling the electrode charge:

eA ¼ 2e0r1I .

Note that the method of mirror reflection does not allow us to find the

field in the nonconducting region above the earth’s surface. Proceeding

from equations derived in Chapter 3, we have for the potential when

the current electrode is located in the first medium,

U1ðxÞ ¼ r1I2p

1

jxj þK12

2d � x

� �, U2ðxÞ ¼ r1I

2pð1þ K12Þ 1

x, [6.2]

where

K12 ¼ r2 � r1r2 þ r1

.

If the current electrode is located in the second medium, we obtain by

analogy

U1ðxÞ ¼ r2I2p

ð1þ K21Þ 1

jxj , U2ðxÞ ¼ r2I2p

1

jxj þK21

2d þ x

� �. [6.3]

We use these equations to study profiling over a vertical interface with

several different arrays.

Case 1. Two-Electrode Array AMConsider first a four-electrode array (AMNB), with electrodes B and N

located far away relative to the distance x between the A and M electrodes,

333Methods of Electrical Profiling and Mapping


so that we can treat it as a two-electrode array. As shown earlier,

the apparent resistivity of such an array is

ra ¼ 2pjxjUðMÞI

. [6.4]

There are three cases to consider: (1) both electrodes are in the first

medium; (2) current electrode is in the first medium with the potential

electrode in the second medium; and (3) both electrodes are in the second

medium. The corresponding apparent resistivities are

rð1Þa ¼ r1 1þ K12jxj2d � x

� �, [6.5]

rð2Þa ¼ r1ð1þ K12Þ,

rð3Þa ¼ r2 1þ K21x

2d þ x

� �. [6.6]

Consider the behavior of the potential and apparent resistivity when this

array moves from left to right over the contact. First, suppose that

r2 > r1, which means that positive charge arises at the contact and at

the earth–air interface. Assume that this array approaches the contact with

the receiver electrode M in front. At very large distances from the contact,

d � jxj, its influence is negligible and

ra ! r1. [6.7]

Since all charges in the model are positive, the presence of the contact

must increase the potential. On approaching the contact, the potential,

along with the apparent resistivity, therefore increases. When the electrode

M touches the contact x ¼ d, the first equation of the set [6.5] gives

rð1Þa ¼ r1ð1þ K12Þ, if x ¼ d. [6.8]

The second equation of [6.5] shows that the value of the apparent resistiv-

ity does not change when the potential electrode enters the second

medium; that is, we again observe continuity of the potential across an

interface. Also interesting is that the potential and apparent resistivity are

constant when the electrodes are located on opposite sides of the contact.

Qualitatively, this behavior can be understood as follows. As the electrode

A approaches the contact, the induced surface charge becomes more con-

centrated near the x-axis, but at the same time, the receiver electrode



moves away from the induced charge, and these two effects compensate

each other completely. When the current electrode A crosses this bound-

ary, the charge induced on the interface becomes negative, but the poten-

tial remains continuous. In fact, we have from Eqs. [6.5] and [6.6],

rð1Þa ¼ r1ð1þ K12Þ ¼ 2r1r2r1 þ r2

¼ r2ð1� K12Þ ¼ rð3Þa .

This result is not surprising since the potential is a continuous function

when the current electrode crosses the interface, even though the sign of

the surface charge changes. As the electrodes move away from the inter-

face into the second medium, the influence of the induced charge

diminishes, and the apparent resistivity tends to the resistivity of the second

medium, ra ! r2 (Fig. 6.2A). Now, suppose that the second medium is

more conductive than the first, r2 < r1. In this case, negative charge

appears at the interface when the current electrode is in the first medium,

and the potential decreases on approaching the interface. When the elec-

trode M touches the interface, the apparent resistivity is still defined by

Eq. [6.8], but now K12 is negative. There is again an interval of constant

potential and constant ra. When the current electrode intersects the con-

tact, the sign of the surface charge changes and, as the distance from the

contact increases, the apparent resistivity gradually decreases and

approaches r2 (Fig. 6.2B).

Case 2. Three-Electrode Array AMNConsider now a three-electrode array with the distance between the volt-

age electrodes MN much less than the distance to the current electrode A:

ra ra

d d

r2

r1 r2

r2r2

r2>r1 r2<r1

r1

r1r1

MA (b)(a)

Figure 6.2 Apparent resistivity curves for a two-electrode array profiling a verticalinterface.



that is, MN << AM . (The second current electrode is assumed to be at

infinity.) The electric field between the receiver electrodes is practically

constant, so

VMN ¼ ExMN .

This allows us to consider an equivalent array AO that measures the elec-

tric field Ex at the midpoint of the receiver electrodes. We first derive

expressions for the electric field when the electrode O is in front of the

current electrode. Using

Ex ¼ � @U

@x,

and the expressions for the potential [Eq. 6.2], we have for the apparent

resistivity in the three cases described above,

rð1Þa ¼ r1 1� K12x2

ð2d � xÞ2" #

, [6.9]

rð2Þa ¼ r1ð1þ K12Þ,

rð3Þa ¼ r2 1þ K21

x2

ð2d þ xÞ2" #

. [6.10]

First, consider the function ra when r2 > r1. As follows from the first

equation of the set [6.9], in approaching a contact, the apparent resistivity

decreases, and when the receiver is about to touch the contact, we have

rð1Þa ¼ r1ð1� K12Þ. [6.11]

This behavior is understandable because the charge of the electrode and

the induced charge at the contact are both positive and the observation

point O is located between them, so the two electric fields oppose each

other. When the receiver O crosses the contact, the second equation of

the set [6.9] gives

rð2Þa ¼ r1ð1þ K12Þ. [6.12]

Thus, the apparent resistivity is discontinuous at the boundary in the ratio

rð2Þa

rð1Þa

¼ 1þ K12

1� K12

¼ r2r1

> 1: [6.13]



This result is expected because the normal component of the current den-

sity jx is continuous, and, therefore, the component Ex has a discontinuity

equal to the ratio of the resistivities. This behavior of the field is useful for

detecting the position of a contact. The apparent resistivity stays constant

when the array straddles the contact, and remains continuous when the

source electrode crosses the interface, even though the induced charge

changes sign. In particular, when the source electrode is in medium 2,

but touching the contact, the apparent resistivity is given by

rð3Þa ¼ r2ð1þ K21Þ ¼ r2ð1� K12Þ,which coincides with Eq. [6.12]. Again, as the electrode array moves away

from the contact in medium 2, the apparent resistivity approaches r2(Fig. 6.3A). Next, suppose that the second medium is more conductive

ðr2 < r1Þ. In this case, negative charge is induced at interfaces when the

current electrode is in the first medium, and the apparent resistivity increases

when approaching the contact, because the x-component of the electric fields

of the electrode and the induced charge reinforce each other. When the

receiver O crosses the contact, these components oppose each other, and

the apparent resistivity abruptly decreases and remains constant at the value

rð2Þa ¼ r2ð1� K12Þ > r2 ðsinceK12 < 0Þ.As the electrode array moves away from the contact in medium 2,

the apparent resistivity gradually decreases and approaches r2. Now,

consider the apparent resistivity curve when the receiver electrode O is

behind the current electrode and r2 > r1. Recall that, by our convention,

O

d d

A(a) (b)

M

r1 r2r1 r2

r1

r2

r1

r2

r2>r1

d

rara

d

AO

N M N

Figure 6.3 Apparent resistivity curves for the arrays AO and OA over a verticalinterface.



the position x of the receiver O is always negative when the receiver trails

the source electrode. In the first medium, we have


x2

ð2d � xÞ2" #

. [6.14]

In this case, both charges are positive, and in approaching the contact,

the apparent resistivity increases. When the source electrode touches the

contact ðd ¼ 0Þ, we have

rð2Þa ¼ r1ð1þ K12Þ > r1. [6.15]

As before, the apparent resistivity is continuous when the source electrode A

crosses the contact, even though the induced charge changes sign.

The apparent resistivity stays constantwhen the array straddles the contact, then

jumps discontinuously when the receiver crosses the interface.When both the

source and the receiver are in medium 2, the apparent resistivity is given by


x2

ð2d þ xÞ2" #

. [6.16]

In particular, at the contact, we have

rð3Þa ¼ r2ð1þ K12Þ > r2.

As the array moves far into medium 2, the apparent resistivity again

asymptotically approaches r2 (Fig. 6.3B). We have considered apparent

resistivity curves over the vertical contact with two- and three-electrode

arrays. Using the principle of superposition, it is a simple matter to obtain

the function ra for four-electrode arrays, such as symmetrical and dipole

arrays. Finally, let us make one obvious comment. If the contact does

not extend to the earth’s surface but is buried beneath an upper layer,

then there is no discontinuity in the electric field, and the apparent

resistivity measured with a three-electrode array is continuous.

6.1.2. Example 2: Profiling Over a Resistive Layer withSymmetrical Array AMNB

Profiling with a symmetrical AMNB array of fixed separation (Fig. 6.4) is

generally effective with simple geoelectric sections. A good example is

when a single object of investigation is surrounded by a relatively uniform

medium and buried under a surface layer (“overburden”) that does not



vary much in its thickness or resistivity. Consider the case shown in

Fig. 6.4, where a tabular resistive body is embedded in a homogenous

medium and covered by a layer of constant thickness. The symmetrical

array can be represented as a sum of the two-electrode arrays: AMN and

MNB. As usual, assume that positive current goes into the medium

through electrode A and, therefore, its charge is positive. Correspond-

ingly, current enters electrode B, which has negative charge. Consider

the apparent resistivity curve as the array moves along the x-axis, that is,

from the left to right. At the beginning, the vertical layer is far away and

the apparent resistivity practically coincides with that for a two-layer

medium, consisting of the overburden and underlying half-space at the

given electrode separation. Near the resistive body, the field of the charges

induced on the resistive body becomes significant. The total field

measured by the receiver line MN is equal to the sum

Ex ¼ E0 þ Es, [6.17]

where E0 and Es are the x-components of the primary and secondary

fields, respectively. In the interval I, the current electrode B is located

closer to the vertical body and its influence on the charge distribution is

stronger (Fig. 6.4B). In this case, both terms in Eq. [6.17] have the same

sign and, correspondingly, the apparent resistivity increases as the array

approaches the body. For the array AMN, the field components E0 and

Es have opposite signs; but its contribution to the response is smaller

because of its greater distance from the body. When the full array AMNB

is above the layer (interval II), the responses of both three-electrode arrays

Zone I

+

+

+

++

++

+

O

O

+

A M N B

r2

r1

ra

+

+

+

+––

–

–

–

–

––

+

++

+

–

Zone II Zone III

Figure 6.4 (A) Symmetrical array. (B) Profiling over the resistive layer.



are positive and reinforce each other, giving a maximum of the apparent

resistivity curve. The roles of A and B are reversed as the array moves to

the right of the body, and the apparent resistivity curve returns to the

background level in interval III.

6.1.3. Example 3: Profiling with a Symmetric Array at TwoElectrode Separations

In rapid profiling with symmetrical arrays, it is useful to repeat the

measurements at two different separations of the current electrodes.

For example, consider the behavior of the apparent resistivity with the

symmetric array AMNB in the presence of the two geoelectric sections

shown in Fig. 6.5. Case (a) corresponds to a basin or “syncline” of resistive

sediment bulging downward into more conductive basement, whereas

case (b) corresponds an arch or “anticline” of resistive basement bulging

upward into a conductive top layer.

Applying the principles described earlier, it is not hard to see that the

apparent resistivity will increase over the anticline, because the more resis-

tive bottom layer gets closer to the surface. But the apparent resistivity will

also increase over the syncline, because in this case, the more resistive top

layer gets thicker. Use of two current electrode separations for each posi-

tion of the potential electrodes, AMNB and A’MNB’, where AB>A’B’,

can help resolve the ambiguity. The array A’MNB’ will have a smaller

depth of investigation at each location and, therefore, be more sensitive

Basin Arch

A�MNB�

M N B� BA

AMNB

A�

A�MNB�

AMNB

r1 r1

r2<r1 r2>r1

A

(b)(a)

Figure 6.5 Electrical profiling with a symmetrical array at two electrode separations.Apparent resistivity curves over a resistive basin and a resistive arch show similarpatterns at one separation, but can be distinguished by changing the depth ofinvestigation.



to the upper layer. Therefore, in case (a), its apparent resistivity curve

raðA0MNB

0 Þ will be located below the curve of the deeper-looking array,

raðAMNBÞ; whereas in case (b), the opposite will occur.

6.1.4. Example 4: Profiling with the Array AMNB–C1Figure 6.6A shows a special type of profiling array designed to detect thin

vertical conductors located close to the surface and surrounded by a more

resistive medium. This is often the configuration of metallic ore bodies

formed near the surface by circulation of hydrothermal fluids through ver-

tical fractures in the surrounding rock. The full array consists of two sepa-

rate four-electrode arrays, AMNC and BMNC, with the common current

electrode C placed far away from the line of profiling in the direction per-

pendicular to the profile. Values of apparent resistivity are measured sepa-

rately for each array at every observation point along the profile.

To understand how the method works, consider first ra measured with

the three-electrode array AMN (which is a good approximation to the full

array AMNC when C is far from the profile line). When the AMN array is

located far to the left of the conductor, negative charge appears on the

conductor’s left flank, with positive charge on the opposite side. In this

configuration, this is because current flows from the more resistive

surrounding rock into the less resistive body through its left flank,

and the contrast coefficient K12 is negative. Current flows out of the body

through its right flank. Because the negative charge is closer to the receiver

MN, the secondary and primary fields reinforce each other: the electric

C

A M N BM

−

−

−

−

−

+

+

+

+

+

A M N

BN

(a)(b)

−

−

−

−

−

−

−

−

−

−

+

+

+

+

+

+

+

+

+

+

ra

Figure 6.6 (A) Combined profiling array with a second current electrode perpendicu-lar to the profile. (B) Schematic profiles over a thin conductor with each array.



field of the positive charge on the electrode and the negative charge on the

left flank of the conductor, both have a positive x-component at the loca-

tion of the receiver. Therefore, the apparent resistivity initially increases on

approaching the body from the left. Closer to the conductor, however, the

relative contribution of the positive charge becomes stronger and acts to

decrease the horizontal component of the secondary electric field. Thus,

a maximum of ra is observed somewhere to the left side of the body. At

some point, the horizontal components of the secondary field caused by

the induced negative and positive charges cancel each other, and the

apparent resistivity becomes equal to the resistivity of the surrounding

medium, r. As the array moves over the body, the influence of positive

charge increases and the value of ra becomes even smaller. Further along

the profile movement of the array, the horizontal component of the sec-

ondary field becomes positive, and the apparent resistivity curve reaches

a minimum value. Then, it returns again to the resistivity of the

surrounding medium as the array moves far to the right of the body.

From the symmetry of the configuration, the apparent resistivity curve

for the array MNB will be a mirror reflection of the curve for AMN,

and the two curves should ideally intersect above the vertical conductor.

This type of survey is useful for detecting a thin conductor, especially in

areas of relatively complicated geology and topography.

6.1.5. Example 5: Method of Middle GradientThe profiling technique called the “method of middle gradient” consists of

a series of current and receiver lines located on the earth’s surface as shown

in Fig. 6.7. The current piece consists of two parallel lines A and B

connected to each other by line C, which has the external current source.

Electric contact with the ground is established by the electrodes placed at

regular intervals along the two lines A and B. Suppose that current flows

into ground from the line A. Then, positive charge arises on each elec-

trode of this line in the amount eA ¼ 2e0rI , where I is current flowing

through the electrode and r is resistivity of the ground surrounding the

electrode. Negative charge of the same magnitude arises at the electrodes

of the negative current line B, if the resistivity of the ground near these

electrodes is the same as that around the line A. (Note that at the surface

of the insulated wire C, there are also electric charges of both signs, but

due to the electrostatic induction, these external charges create no electric

field beneath the earth’s surface.) Thus, we have a system of linear charges



of different signs located along two lines A and B, which create the electric

field in the earth. The direction of the fields is shown in Fig. 6.7A. It is

essential that the sign of horizontal component of the field Ex caused by

both sets of charges is the same. The field of an array of linear charge of

the same sign decreases at a rate that is almost inversely proportional to

the distance from an observation point to the line (provided that this dis-

tance is much smaller than the length of the line). It is obvious, then, that

near the middle part of the area between current lines A and B the tangen-

tial component of the electric field E0 is almost constant. Moreover, this

field practically remains the same at points beneath the earth’s surface if

their distance to the earth’s surface is smaller than that of the current lines.

Thus, with the help of such a system, it is possible to create within some

volume a very simple field, which is usually called the normal or primary

field. This simplifies the detection of conductors located beneath the

earth’s surface. Thus, over a uniform half-space, measuring the voltage

along lines of observation between the two current lines would show a

constant normal field (Fig. 6.7). Next, suppose that there is a conductor

(ore body) beneath the earth’s surface. Since the inhomogeneity is more

B

IC

A + −+

+++

+

−−

−−

−

EB

EA

Ex

E0

E0 r1r2

−−

−

−−

++

++

+

II+ −

++

+++

−−

−−

−

−−

−

−−

++

++

+

(a)

(b)

Figure 6.7 (A) Method of middle gradient and central profiles. (B) Schematic behaviorof the horizontal component of the field in the presence of a conducting body.



conductive, negative and positive charges will appear, respectively, on the

left (A) and right sides (B) of the conductor. These charges are sources of

the secondary field Es. The total horizontal component of the electric field

at each point and, in particular, between receiver electrodes M and N, is

Ex ¼ E0 þ Es. The secondary field contains information about parameters

of the inhomogeneity, such as its position, size, and conductivity.

Measurements of the voltage along parallel survey lines are usually

represented in the form of a system of curves ExðxÞ; a typical curve is

shown schematically in Fig. 6.7B. The behavior of such curves can be eas-

ily explained by reference to the position of the induced charges. At points

of part I of the profile, negative charges on a conductor create a horizontal

component of the same sign as the normal field, whereas positive charges

create a horizontal component of the opposite sign. Since the sum of the

positive and negative charges is zero, and the negative charges are located

closer, we conclude that at this part of the survey line, the field increases,

Ex > E0. Near the middle of the profile (part II), located above the conduc-

tor, both positive and negative charges create a horizontal component of

negative sign for the secondary field, giving Ex < E0. Finally, within part

III of the profile, the horizontal components of the normal and secondary

fields have the same sign. Since the positive charge is located closer to obser-

vation points than the negative charge, the total field again increases. A full

set of such curves covering the area between the two current lines provides

a qualitative map of the structure, and can also enable quantitative estimates

to be made of some parameters. In particular, the distance between the

maxima and their magnitudes can often be used to characterize the size of

a buried conductor. The separation between observation points and survey

lines is chosen mainly from available geological information.

6.1.6. Example 6: Profiling and Sounding, ElectricalResistivity “Tomography”

A modern survey technique for shallow geophysics made possible by

advances in electronics is to lay out an array of electrodes on a dense reg-

ular grid covering the survey area. A computer can control the electrode

array injecting current between any two electrodes and measuring the

electrical potential at all the other electrodes (with respect to a common

reference), or more simply between pairs of electrodes. The measurement

sequence can then be repeated with two different electrodes serving as the

current electrodes, etc. The complete set of data can then be plotted in

various formats to give a qualitative picture of the subsurface geology or



analyzed quantitatively by inversion software that systematically adjusts the

parameters of a model to fit the data. This type of data collection and anal-

ysis is sometimes called “electrical resistivity tomography.”

6.2. THE CHARGED-BODY OR MISE-À LA-MASSE METHOD

One of the oldest and most effective methods for electrical mapping is called

the “charged-body” or the “mise-a la-masse” method, in which a current

electrode is placed into direct contact with a large conductor, usually ametallic

ore body. The next sections briefly describe the applications of this method.

6.2.1. Isometric Ore BodyAssume that the current electrode A is connected directly to a large buried

conductor, such as an ore body, while the other electrode B is removed far

enough away to neglect its influence. The current from the electrode A

flows through the ore body into the surrounding rock, causing positive

electric charge to appear on its surface (Fig. 6.8). These charges are sources

of the electric field inside and outside the body. If the body has much

higher conductivity than the surrounding medium, these surface charges

will create a nearly equipotential surface over the body, which is the most

A

MN

B

++

+

++

+

+ +

+

U2

U1

E

Surface equipotentials

Ore body

Profile line

++

+

++

+

+ +

+

Figure 6.8 Charged-body (mise-à la-masse) method in which electric current isinjected directly into a buried conductor, with schematic electric field and equipoten-tial lines.



important feature of the method. (The same conclusion can be reached by

applying Ohm’s law to the current flow within the highly conducting

body.) Since the surface of the body is almost equipotential, the vector

lines of the electric field will be perpendicular to the body and the

surrounding equipotential surfaces will also have practically the same shape

as that of the conductor. Of course, far away from the body, the equipotential

surfaces will tend to a spherical shape. In effect, the entire conducting body

plays the role of the current electrode in this method. It is also obvious that

the intersection of the earth’s surface with the equipotential ones forms closed

lines of equal potential (equipotential lines) that approximately map the pro-

jection of the body’s shape onto the surface. Measuring the voltage on a grid

on the earth’s surface thus gives information about the dimensions and shape

of the conductor (Fig. 6.8). There are two main approaches allowing one to

study the field on the earth’s surface. In the first, the receiver electrode N is

placed in some point located relatively far away from the electrode A, while

the second receiver electrodeM roams over the surface above the body.Mea-

suring the voltage difference between electrodeM and the fixed referenceN is

equivalent to measuring the potential over the body (which is in fact defined

only to within an arbitrary constant). Correspondingly, these data allow us

to plot a map of equipotential lines on the earth’s surface (Fig. 6.8).

A second approach (Fig. 6.9) is based on measuring voltage differences

along a profile or a set of profiles. The behavior of the voltage can be easily

explained using the principles described earlier.

6.2.2. Elongated Conducting BodyGeological structures are often elongated in one direction, usually called

the “strike” direction. If an ore body is highly elongated, it may be neces-

sary to take into account its finite resistivity, which will eventually cause

the potential to drop along the current flow. At the same time, the fact

that the conductor is elongated, means that positive charges at its surface

VMN I

MN IAB

Figure 6.9 Profile of a two-electrode array over a conductor with the charged-bodymethod.



act to direct the flow mainly along the body, so that leakage into a more

resistive medium is minimized. (The current wants to follow the path of

least resistance.) This suggests that the density of charge in the charged-

body method decreases relatively slowly along the axis of elongation of a

conductor, and outside the electric field, is mainly directed perpendicular

to this direction. Measurements along profiles perpendicular to the strike

direction generally have a characteristic shape (Fig. 6.10) that is easy to

explain from the distribution of charge. With measurements VMN made

between closely spaced electrodes, there is a point, usually directed over

the body, where the voltage difference is equal to zero. A system of such

profiles will often define the strike of the conductor and even its shape.

Often, instead of measuring the voltage between receiver lines, a coil is

used to measure the vertical and horizontal components of the magnetic

field on the earth’s surface caused by the low-frequency current flowing

+

B

+

+

+

+

+

+

+

+

+

B

B

j

Magnetic field line

VMN

U

Potential

+A

+

+

+

+

+

+

+

+

+

Figure 6.10 Behavior of the voltage, potential difference measurements, and mag-netic field over an elongated conducting body in the charged-body method.



through the body. If the body is highly elongated, the magnetic field res-

embles that from a linear current filament and has a particular simple

behavior: the components Bx and Bz have profiles of the same shape as

the potential and electric field, respectively: that is, Bx peaks over the

body, whereas Bz changes sign (Fig. 6.10).

6.2.3. Correlation of Conducting Layers Between BoreholesThe charged-body method can be used to correlate geoelectric sections in

neighboring boreholes, or to outline the contours of a conducting body

crossed by one of the holes. Suppose that each borehole crosses two con-

ducting layers at different depths (Fig. 6.11). Our goal is to correlate the

layers labeled 1 and 2 in the first borehole with layers 3 and 4 in the second

borehole. Note that in the presence of complex geology, including faults

and overturned structures, any type of correlation is possible, including a

“reverse” correlation in which layers 1 and 4, and 2 and 3 match up, or

no correlation at all. With this in mind, we place electrode A in contact

with layer 1 and measure the potential in the second borehole. If the

intervals 1 and 3 represent the same geological layer, then the maximum

1 1

2 2

3

4 4

1 1A j MVMN / IAB

N

Borehole 1 Borehole 2

3

B

3

Figure 6.11 Correlations between layers with the charged-body method.



of potential in the second borehole will be observed near layer 3, because

positive charge will appear at the top and bottom of the layer, as current

injected in the first borehole flows along the layer and leaks out into the

surrounding medium. If there is no pronounced maximum of the potential

in the second borehole near layer 3, it is likely that these conducting zones

are not connected. Similarly, placing the current electrode A in layer 2 can

determine whether layers 2 and 4 are connected. The results of interpreta-

tion become more reliable if the measurements are repeated with the roles

of the two boreholes reversed: that is, with the current electrode in layers

3 and 4 and potential measurements made in the first borehole. In general,

the profile of electric potential in the measurement borehole—the positions

of maxima and minima of the potential and any sign changes—gives useful

information about the geometry of the body between the holes.

6.2.4. Determination of the Velocity and Directionof the Groundwater Flow

A modification of the charged-body method can be used to determine the

direction and velocity of underground water flow. This application works

as follows. The current electrode A is placed inside a porous bag of salt

(NaCl) and lowered into the borehole to the depth of the water-bearing

formation. The second current electrode B is located on the earth’s

surface far from the borehole (Fig. 6.12A). At the start of the experiment,

the charge on the electrode A dominates the electric field, whose

A*A

B

vc

t

bt

(a)

(b)

Figure 6.12 (A) Method of charged body for determination of the direction andvelocity of water flow. (B) Dependence of the velocity of the center of the equipoten-tial line on time.



equipotential surfaces are nearly spherical (if the resistivities near the bore-

hole do not vary much). The corresponding equipotential lines measured

on the surface are approximately circles centered on the borehole axis.

As the salt dissolves in the water layer, the region with higher concentra-

tion of salt—and, therefore, higher conductivity—expands away from the

borehole in the direction of the water flow. Surface measurements of the

equipotential lines at different times (usually intervals of several hours or

even days) will show elongation of the equipotential lines in the direction

of the flow. An approximate method can be derived for calculating the

velocity of the flow. Let us assume that with time, the center of the zone

with high concentration of the salt can be treated as the point electrode

A�. This model has two current electrodes A and A�, with charges

eA ¼ e0raI and eA� ¼ e0rð1� aÞI . [6.18]

The value of a varies between 0 and 1, and characterizes the distribution

of current between the electrodes. If we assume that the potential is

caused by only these charges, which are located at a distance bt from each

other at time t, then at any given time, the potential on the earth’s

surface is

UðrÞ ¼ rI4p

a1

ðr2 þ h2Þ1=2þ rI4p

ð1� aÞ 1

½ðr � btÞ2 þ h2�1=2, [6.19]

where h is the depth of the electrodes and r is the resistivity of the

surrounding medium.

Note that this model assumes that the depth of water flow does not

vary laterally. The maximum of the potential as a function of radial dis-

tance from the borehole is where the derivative of the potential with

respect to r equals zero:

ar

ðr2 þ h2Þ3=2þ ð1� aÞ ðr � bÞ

½ðr � btÞ2 þ h2�3=2¼ 0: [6.20]

The solution of this equation depends on a. Assuming that current is

equally split between A and A� so that a ¼ 1=2, we obtain rmax ¼ bt=2.The displacement of the maximum of potential (the center of the equipo-

tential curves) is therefore one-half of the displacement of the center of the

fictitious electrode A�, which measures the average displacement of the

mass of dissolved salt. In other words, the velocity of displacement of this

center is twice smaller than the velocity of the water flow. If all current



goes through the electrode A�ða ¼ 0Þ, then the coordinate of the center of

equipotential line is equal to b. In this case, the velocity of this point

coincides with that of the water flow. These considerations can be used

to put bounds on the velocity. Note that the velocity of the center of

equipotentials gradually approaches that of the water flux (Fig. 6.12), and is

related to the fact that the development of the salt halo from the borehole

requires some time.

6.3. SELF-POTENTIAL METHOD

We have so far described electric methods involving steady currents that are

injected into the earth with an external source. There are also steady

currents that occur naturally in the earth as a consequence of electromotive

forces of different origins, including currents driven by the natural under-

ground flow of conducting water through porous rocks. In surface electrical

methods, these forces are mainly related to the effects of electrochemical and

electrokinetic origin which generate a steady electric field.

6.3.1. Self-Potential of Electrochemical Originin Mineral Exploration

Suppose that an ore body with electronic conductivity is surrounded by

sedimentary rock whose conductivity is dominated by ions in its water-

filled pores, and that part of the body extends above the water table.

The surface of the ore body below the water table is a boundary between

media with different mechanisms of conductivity (electronic and ionic),

and a natural electrochemical reaction takes place generating an external

non-Coulomb electromotive force in the form of a double layer of charge:

net positive charge arises on the external surface of the ore body and is bal-

anced by negative charge of equal magnitude on its internal surface

(Fig. 6.13). Above the water table, the process of oxidation plays an

important role and reverses the double layer: net negative charge appears

on the external surface, balanced by positive charge on the internal surface.

The ore body, thus, acts as a natural battery. The strength of the double

layer below the water table depends on the mineral content of the ore

body and on the concentration of ions in the surrounding fluid-filled

rocks. The difference of potential across the layer can reach several hun-

dred millivolts. Two conditions must be met to generate a steady electric

field outside the body. The first is a variation of the dipole density along

the double layer (a uniform double layer does not create an external



electric field). This first condition is the reason that massive sulfide ore

bodies, which usually contain different types of minerals, cause a strong

self-potential field. The second condition for generating a steady external

field is a process that continuously renews the double layer by removing

stray ions created by electrochemical processes near the contact that act to

neutralize the surface charge. Often, the flow of the groundwater performs

this function by carrying oxygen to the region. The self-potential field can

be studied either by measuring the potential at observation points with

respect to a fixed reference electrode, or by measuring potential differences

between a moving pair of electrodes. In the first case, the reference elec-

trode N is usually located at a large distance from the survey grid occupied

by the second electrode M. This method gives a direct map of the self-

potential field. Inasmuch as the dipole moment of the double layer in the

upper part of the ore body is directed downward, the potential usually has

a minimum directly above the body (Fig. 6.13A). A map of equipotential

lines on the earth’s surface will have a similar depression. The second

+

+

+

+

−

−

−

−

−

−

−

−

+

+

+

+

+

+

+ +

+

−

−

−

−

Water table

Ore body

+

−

U

−

Figure 6.13 Distribution of charge and self-potential around an ore body that extendsabove the water table.



approach is based on measuring of the voltage difference between observa-

tion points along survey lines. In this case, the difference of potentials

between two points 1 and 2 can be written as

rU1 ¼ rUMN þ e1 � e2,

where rUMN is the voltage between electrodes M and N caused by the

self-potential field, and e1 and e2 are the electrode potentials. Their effect

can be eliminated by interchanging the electrodes and repeating the mea-

surement, giving

rV2 ¼ rUMN þ e2 � e1.

The average of the two measurements is the desired quantity. (With an

artificial current source, the influence of electrode potentials can be simi-

larly “averaged out” by repeating the measurement after reversing the

direction of the current.) Over a massive ore body, the potential can be

relatively large, reaching several hundred millivolts. When the mineraliza-

tion is disseminated through the rock, each clump of mineralized rock

acquires a double layer, but the net effect is usually small because of ran-

dom orientations.

6.3.2. Self-potential of Electrokinetic OriginA self-potential field also arises from underground electrokinetic processes

that accompany filtration of groundwater through pores of geological

formations. This phenomenon is used to study groundwater and civil engi-

neering problems. The mechanism for generating electrokinetic potentials

is complicated but can be summarized as follows. At the interface between

solid rock and pore fluids in underground rocks, there is often a net absorp-

tion of dissolved ions of one charge by minerals lining the pores. For exam-

ple, clay minerals in the pore walls of sedimentary rocks generally absorb

negative ions from the pore fluids. This process changes the electrochemical

equilibrium of the solution (which acquires an excess of positive ions), but

the total charge in any macroscopic elementary volume is still zero. In fact,

the net positive ions in solution are balanced by net negative charge on the

pore walls; that is, the macroscopic charge density vanishes,

d ¼ 0, [6.21]

and despite the redistribution of ions, there is no macroscopic electric

field. The double layer that arises in the pore space is, however, asymmet-

ric in the sense that the negative ions are more or less rigidly attached to



the solid rock while the positive ions are free to move in the pore fluid. In

reality, the positive ions accumulate in a thin diffuse layer of fluid lining

the pore walls. Next, suppose that there is mechanical movement of the

groundwater. Some of the positive charges in the diffuse layer can be

dragged along by the flow, which is equivalent to a net electric current

of mechanical origin. The presence of this “convection current” itself

is not sufficient to generate a macroscopic electric field, since the macro-

scopic charge density may still vanish everywhere. Of course, there will

always be a magnetic field associated with the convection current. To

understand how a steady electric potential—in this case, called a “stream-

ing potential”—can be generated by electrokinetic processes, we must find

places where electric charge can arise from the convection current. To

solve this problem, we first consider the forces acting on each elementary

volume of moving water. In general, there are three forces: the Coulomb

force caused by charges, the mechanical force due to a change of pressure,

and finally, the force of resistance of the porous medium. When a sum of

these three forces is equal to zero, we observe a movement of ions with a

constant velocity. The relation between the current density and the first

two forces for an elementary volume has the form

j ¼ gE� Lrp. [6.22]

Here, E is the Coulomb electric field, p is the fluid pressure, g is the electricalconductivity, and L is a “coupling coefficient” between mechanical and elec-

trical effects. L generally increases with increasing permeability of the porous

medium. The negative sign in front of the second term arises because the

motion of the fluid is directed toward smaller values of pressure. The following

notations are usually used: j ¼ jt is the total current density, jcond ¼ gE is the

conduction current density, and jconv ¼ �Lrp is the convection current den-

sity. Correspondingly, the total current density is the sum:

jt ¼ jcond þ jconv. [6.23]

The total current density in the steady state has one important feature,

namely, conservation of charge:

div jt ¼ 0, [6.24]

that is, its flux through a closed surface surrounding an elementary volume

is equal to zero, otherwise, an accumulation of charges would take place

generating a time-varying electric field by Coulomb’s law. We therefore

have



div jcond þ div jconv ¼ 0:

In particular, at points where div jconv ¼ 0, we also have div jcond ¼ 0.

Expanding the individual terms gives

div jcond ¼ divðgEÞ ¼ g divEþ E �rg ¼ gde0þ E �rg, [6.25]

where d is the volume density of the electric charge, and

div jconv ¼ � divðLrpÞ ¼ �L�r2p�rL �rp. [6.26]

Let us represent Eq. [6.22] in the known form of the generalized Ohm’s

law:

j ¼ gðEþ EextÞ. [6.27]

Here, E is the field-caused charges, while Enc is an extraneous “non-

Coulomb” force. Comparing Eqs. [6.22] and [6.27], it is clear that

Enc ¼ �rLrp, [6.28]

where r is a resistivity and C ¼ rL is called the “streaming current coeffi-

cient.” We therefore see that a non-Coulomb force arises at every point

where groundwater moves through a resistive porous medium. We are

now ready to find the distribution of macroscopic electric charge associated

with this motion. Using Eqs. [6.22], [6.24], and [6.25], we obtain

0 ¼ ge0dþ E �rgþ div jconv,

or

d ¼ �e0rE � gradg� e0r div jconv. [6.29]

The first term,

d1 ¼ �e0rE �rg,

shows that charge appears where the electric field is aligned with changes

in conductivity, that is, where E �rg differs from zero. This type of charge

was studied in detail earlier. In particular, in the vicinity of a point where a

medium is uniform by conductivity d1 is equal to zero. The second type of

charge is caused by the divergence of the convection current density,

d2 ¼ �e0r div jconv. [6.30]



This charge arises because of the motion of groundwater and is the ulti-

mate driving force of the electric field. For instance, we may observe the

appearance of this type of charge (and its electric field) even when

d1 ¼ 0. But, in the absence of the convection current density and its

charge d2, charges of the first kind d1 do not arise because we are dealing

with a natural field with no extraneous sources. In other words, charges

with density d2 are “primary” sources of the electric field of electrochemi-

cal origin. Nevertheless, since the measured electric field is caused by

charges of both types, it may happen that the maximum (or minimum)

of the potential on the earth’s surface is not located over “primary”

sources. Next, consider a contact between media with different values of

resistivity and parameter L. From continuity of the total current density,

we have for its component normal to the interface:

jt,2n � jt,1n ¼ g2E2n � g1E1n þ jconv,2n � jconv,1n ¼ 0,

or

12½ðg2 � g1ÞðE2n þ E1nÞ þ ðg2 þ g1ÞðE2n � E1nÞ� þ jconv,2n � jconv,1n ¼ 0:

[6.31]

Taking into account that

E2n � E1n ¼ se0,

we have for the surface charge density s:

s ¼ 2e0g1 � g2g1 þ g2

Eavn þ e0

gavðjconv,1n � jconv,2nÞ, [6.32]

where g1,E1n and g2,E2n are conductivity and normal components of the

electric field at both sides of the interface, respectively. Also

gav ¼ g1 þ g22

, Eavn ¼ E1n þ E2n

2,

and jconv,1n, jconv,2n are the normal components of the convection current

density. In accordance with Eq. [6.32], we can define two types of surface

charge:

s1 ¼ 2e0K12Eavn and s2 ¼ e0

gavð jconv,1n � jconv,2nÞ, [6.33]



where

K12 ¼ r2 � r1r2 þ r1

.

The first type of surface charge, s1, is caused by a change of resistivity in

the direction of current flow and has been studied in detail in earlier

chapters. The second type, s2, appears at points of the interface where

the density of convection current has discontinuity. For example, positive

charge appears at the interface if jconv,1n > jconv,2n, and vice versa. The sec-

ond equation in the set [6.33] can also be written as

s2 ¼ e0gav

L1

@p1@n

� L2

@p2@n

� �, [6.34]

where L1,p1 and L2,p2 are values of the coupling parameter L and pressure

p on the two sides of the interface.

The configuration shown in Fig. 6.14 illustrates a case where the self-

potential method can detect leakage of groundwater from a near-surface

aquifer. If some of the water flowing through the top layer is diverted into

a vertical layer with higher permeability, then negative charge arises at the

top of the vertical layer (see Eq. [6.33]), and a negative self-potential

anomaly will appear at the surface.

Water flow

Highpermeability

U

Figure 6.14 Leakage of groundwater flow into a vertical layer of high permeabilityand associated self-potential anomaly.


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