James R. Wait Departments of Electrical Engineering and Geosciences University of Arizona Tucson, Arizona 1982 ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers New York London Paris San Diego San Francisco Säo Paulo Sydney Tokyo Toronto Copyright © 1982, by Academic Press, Inc. all rights reserved. no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. 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 dimensions 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 coordinates (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 corresponding 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 Â (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.

Geo-Electromagnetism

Similar Geo-Electromagnetism