Geo-Electromagnetism

271 Pages·1982·3.691 MB·English

by  James Wait (Auth.)

James R. Wait
Departments of Electrical Engineering and Geosciences
University of Arizona
Tucson, Arizona
1982
ACADEMIC PRESS
A Subsidiary of Harcourt Brace Jovanovich, Publishers
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Copyright © 1982, by Academic Press, Inc.
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ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1 7DX
Library of Congress Cataloging in Publication Data
Wa i t, James R.
Geo-e1ect romagnet i sm.
Includes bibliographical references and index.
1. Electric prospecting. I. Title.
TN269.W29 622'.15^ 82-6814
ISBN 0-12-730880-6 AACR2
PRINTED IN THE UNITED STATES OF AMERICA
82 83 84 85 9 8 7 6 5 4 3 2 1
Preface
The purpose of this volume is to provide a survey of the analytical bases
of electrical prospecting methods. Although the emphasis is on theoretical
concepts, many applications are discussed. This book presents derivations
that are for the most part self-contained, at a consistent and uniform level
of difficulty. As a primary prerequisite the reader should have an under-
standing of static and dynamic electricity at the undergraduate level. All
the same, many of the basic electromagnetic concepts and the special
mathematical functions so common in this subject are reviewed in the text.
So that the student will not get lost in mathematical detail, many of the
examples are shown in graphic form. These run the gamut from direct-
current earth testing to radar pulse probing of the earth's surface. A special
effort is made to show the unity of various electrical and electromagnetic
methods of geophysical exploration. Related topics in mining technology
and telecommunications in subsurface environments are also included.
The material is sufficient to occupy two full semesters at the first-year
graduate level, but the book is also suitable for seniors who have taken or
are taking an undergraduate course in fields and waves. The exercises that
are sprinkled throughout the text have been class-tested by seniors and
first-year graduate students in both electrical engineering and geosciences.
In addition, each chapter is supplemented with references to the current
journal literature.
I am grateful to Jeri Bacon and Joann Main for typing the manuscript,
to Bob McDonald and Martha Dempsey for drafting, and to my profes-
sional colleagues Dave Hill, Don Dudley, Allen Howard, John Sumner,
Misac Nabighian, Jeff Hughes, Ken Spies, Charlie Stoyer, Jeff Lytle,
Catherine Levinson, Khalid Nabulsi, Margarete Ralston, David Chang,
and Roy Mattson for help along the way. I also wish to thank Academic
Press for its expeditious handling of the manuscript and for important
editorial suggestions and comments.
ix
Chapter I
Earth Resistivity Principles
INTRODUCTION
The interaction of electromagnetic waves with the earth can be very
complicated, even for fairly simple geometries. Fortunately the situation is
much easier to handle when direct current (dc) excitation is used. This
limiting case of zero frequency is an excellent starting point for our
analysis.
For reasons that are discussed later, we shall deal with a four-electrode
system. That is, the current / is injected into the earth via two electrodes
(e.g., metal stakes) and the resulting fields of the medium are detected by
two adjacent electrodes (e.g., metal pins) (Fig. 1). Normally the voltage
between the two potential electrodes is measured by a detector whose
internal impedance is much greater than the impedance between the
electrode contacts and the earth medium.
The quantity of interest is the transfer impedance, defined as Ζ = Vj I.
In the case of de, Ζ becomes R, the transfer resistance, and has dimen-
sions of ohms. As will be shown below, R is proportional to the apparent
resistivity p of the earth medium. That is, R = Fp , where F is a factor that
a a
has dimensions of length. The name of the game is to interpret the
measured p in terms of the geological structure. Rarely is such an interpre-
a
tation straightforward, but much insight is gained by dealing with idealized
models, at the same time bearing in mind their limitations.
When dealing with static or dc current flow, the resistivity ρ is the
reciprocal of the conductivity σ. In what follows we shall deal with the
latter. For example, we shall specify that the conductivity σ(χ,γ,ζ) is a
function of the coordinates. Furthermore, we shall assume that the earth is
isotropic; then Ohm's law takes the form
J=aE, (1)
1
2 I. Earth Resistivity Principles
Fig. 1. Arrangement for earth resistivity testing.
where J is the vector current density in amperes per square meter and Ε is
the vector electric field in volts per meter.
Now the electric fields can be derived from the gradient of a scalar
potential ψ. That is,
E= -grad^. (2)
Another basic fact is that
divJ = 0, (3)
which holds everywhere except at the source itself.
As an exercise the reader can easily show that
= 0. (4)
dx2 dy2 dz2 dx dx dy dy dz dz
A case of some importance is that in which the conductivity σ(ζ) varies
only with depth z. Then the terms containing and in (4)
do/dx do/dy
vanish. By introducing cylindrical coordinates (r,z), where r = 2 (x+ y2)l / ,2
we obtain
32Ψ 1 Η 92Ψ , 9Ψ 1
do _
+ + 0 (5)
92 r dr dz2 dz σ dz
r
subject to azimuthal symmetry. This equation may be solved by separation
of variables. To this end, we write Ψ(γ,ζ) = which is a product of
R(r)Z(z),
a function of r and a function of z. Then (5) separates into two equations
that must be satisfied simultaneously,
4 + ί# + λ2Λ = 0
(6)
dr2 r dr
and
dd2Zz2 + Iο ddaz ddZz X2Z = 0, (7)
where λ is the separation constant, which up to this point is arbitrary.
A general solution of (5) can be written in the form
J 00
<* A
(8)
F(X)R(\,r)Z(Xz)d\
ο 9 9
Potentials about a Single Current Electrode 3
where F(X), a function of λ, can be chosen to fit the boundary conditions as
needed. Also, as we shall see later, the contour over λ can be taken along
the real axis.
POTENTIALS ABOUT A SINGLE CURRENT ELECTRODE
FOR A TWO-LAYER EARTH MODEL
As an example of the previous formulation, consider a two-layer earth
model (Fig. 2). The resistivity (σ^ -1 or p is constant for 0 < ζ < h. The
x
current / is injected into this layer at the origin of the cylindrical coordi-
nates (r, φ, z).
To approach the problem straightforwardly, examine the behavior of the
fields that are in close proximity to the current electrode. Assume that the
electrode is a small hemisphere of radius b (Fig. 3). Clearly, the radial
current density J is given by
R
J = I/(2TTR2),
(9)
r
where R = (r2 + z2)1//.2 The corresponding radial electric field E is seen to
R
be
(10)
at least in the immediate vicinity of the current electrode. The correspond-
ing potential can be taken to be
'Pi
2πϋ (11)
I
1
ζ
Fig. 2. Point source current injection into two-layer earth.
Ζ
Fig. 3. Enlarged view of region near current electrode of radius b.
4 I. Earth Resistivity Principles
where the superscript ρ is here used to denote that ψρ is the primary
potential.
We now introduce a well-known integral relation from Bessel function
theory [1],
^ " ϊ τ τ ^ ' Γ ^ ' "^ (,2)
for ζ > 0. Here is the garden variety of the Bessel function of the first
type of order ze0(rX rJo) and λ denotes the integration variable. By no accident, λ
can be related to the separation constant that was introduced earlier.
Equation (12) can be verified by noting that both sides satisfy Laplace's
equation and that the result for the limit r -> 0 is an elementary form.
Returning to (6), note that is really or But since the
latter, the Bessel function of th R(eX,r )second0(X rty)pe J, iY0(sXr )i.nfinite for r = 0, it cannot
appear in the present problem. Thus we are led to write the resultant
potential in the upper layer as
Ψι = ψΡ + J T ^ ^^ + ^(X)eXz]Jo(Xr)dX (13)
for the region 0 < ζ < h. Here we can identify ψ, — ψρ as the secondary
potential ψ J since it vanishes if A-> oo. The right-hand side of (13) clearly
satisfies Laplace's equation
ν2ψ, = 0, (14)
as it should for a position-independent resistivity p over the limits
0 < ζ < h and 0 < r < oo. The two functions Λ (lλ) and Β (λ) are yet to be
determined. Thus, in summary, the desired form for the potential in the
upper layer is
Ψι = fejTU1 (15)
+A(X)]e~X2 +B(X)e^}J(\r)d\,
0
where new dimensionless functions Α (λ) and Β (λ) are used in place of  (λ)
and Β (λ). The integral expression for ψ' is the same as (15) when 1 + A is
replaced by A.
The appropriate form for the potential in the lower layer (i.e., ζ > h) is
easily seen to have the form
Tp2=^f~C(X)e-^J(Xr)d\, (16)
0
where the multiplying factor is chosen for convenience. The function C(X)
is yet to be determined. Note that the f actocanrnot appear if the
potential is to be finite at ζ -» oo. +Xz e
Potentials about a Single Current Electrode 5
Now apply boundary conditions to determine the unknown functions A,
B, and C. First, the vertical current density must be zero at ζ = 0 for
0 < r < oo. Second, the potentials are continuous at ζ = h. Third, the
normal current density is continuous at ζ = h. These conditions in turn
require that
^7=° at z = 0, (17)
ψ, = ψ at z = h, (18)
2
and
p oz p oz ' v
x 2
By applying these to (15) and (16), we obtain
A - B = 0, (20)
(1 + A)e~Xh + BeXh = Ce_AA, (21)
(l/p,)[(l + A)e~Xh - Be™] = (l/p )C*-x\ (22)
2
These are readily solved to yield
A = Β = e~2XhK/(\ - Ke~2Xh), (23)
where
Κ=(Ρ2~Ρύ/(Ρ2 + Ρύ- (24)
Using these values for A and B, the potential \pi(r,z) anywhere in the upper
layer is thus given as an integral with a specified integrand. This solution
was apparently first given by Stefanescu et al. [2]. (For related references,
see [3-10].)
An important special case is that in which the observer is on the earth's
surface (i.e., ζ = 0). Then
^(r,Q) = (I /2irr)G(r,K), (25)
9x
where
G{r,K) = 1 + 2Kr£ -^-^J(Xr)d\. (26)
Q
Clearly, if the thickness of the upper layer becomes sufficiently large (i.e.,
h » r), then G(r, K)-^ 1. Another special case of some importance is that in
which the upper-layer thickness is very small (i.e., h < r). Then
G(r, Κ) -> 1 + -j^L J^°%(Ar) d\. (27)
6 I. Earth Resistivity Principles
The integral here is equal to 1/r. Thus
1 + p / . (28)
G(r,K)^> 2K/(\ -K) =
The potential is thus given by 2 Pwl hich is the appropriate
value for a ho m\p(ro,0g)eneous half-space o2/f2i rrre, sIipstivity p . Although the integral
in (26) can be evaluated by numerical means, i2t is useful to obtain a series
form. To this end, we expand the factor in the integrand as
(1 - Ke~2Xh)~l= 1 + Ke~2Xh + K2e~4Xh + K3e~6Xh + · · ·
= Σ Kne~2nXh (29)
« = o
Then (26) can be written as
00
G(r,K) = 1 + 2Kr 2 ΚΊ, (30)
η
n = 0
where
/„ = r e - V ' + W j ^ d X- ! —- . (31)
2
j° [r^ + ^+OlA]2]7
In obtaining this result we have interchanged the order of integration and
summation. This is permissible because (29) is an absolutely convergent
series.
The series form for the potential has a clear physical interpretation. As
indicated in Fig. 4, the potential at results from the total series of
image sources located at ζ = (η -h 1)2A P,(r ,a0n) d the strength of each of these is
2KA\ n alternative and possibly more general image picture is to locate the
images of strength Kn both above and below the earth's surface. In this
z=0-
case, the observer at could be anywhere within the layer 0 < ζ <
The proof of thisz =sthat eP-m(r,ze)nt is left to the reader. h.
2JKI
z=2h
z=4h-2lK2I
Fig. 4. The image representation for a single current point source at the surface of a
two-layer earth when the observer is also on the surface.
General Four-Electrode Array 7
GENERAL FOUR-ELECTRODE ARRAY
In a practical situation, as we indicated in the introduction, the current
source of +/ amperes must be accompanied by a current sink of — /
amperes. Also, we need two potential electrodes, as indicated in Fig. 1. The
voltmeter then measures the difference V between the potentials [4,5,11].
To allow some generality, let the current and potential electrodes be
located arbitrarily as indicated in Fig. 5, in which a plan view of the surface
ζ = 0 is shown. The resultant potential ψ, at P is
x
G(r,K) G(r ,tf)
ul 2il (32)
1,1
where /·,, = (x2 + /2)l/2 and r , = [(x _ Χ\Ϋ +/?]' /2- The resultant poten-
2 0
tial ψ ι at Ρ2 is
G(r ,Ä-) G(r , ,tf)
u 2 2 (33)
Ψι(*2·Λ)= 2^
'2,2
where r = {x\ + / 2)l/2 and = [(x - xf + /2]l/2.
I>2 2 0 2 2
The voltage V between the electrodes is now given by
(34)
Now, if the ground were fully homogeneous (i.e., p = Pi and/or h -> oo),
2
clearly the Gs in (32) and (33) would be 1. In this case
/p. ' J_ _ J_ __ _L 1
(35)
+
2w r2,\ rl,2 '2,2
But if the ground is inhomogeneous, we can define an apparent resistivity
p by writing
a
'Pa ' J_ . _ _L_. J_ JL
(36)
+
2m r2,l rl,2 r2,2
•I -I
C,(0,0) C(x,0)
2 0
·Ρ|(Χ|ιΥ|)
y
Fig. 5. Plan view ot the ground surface showing the locations of the current electrodes C,
and C2 and the potential electrodes P x and P2.