electromagnetic modelling of buried line conductors using an

Geophys. J . Int. (1995) 121,203-214

Electromagnetic modelling of buried line conductors using an integral equation

W. Qian and D. E. Boerner Geological Survey of Canada, Continental Geoscience Division, 1 Observatory Crescent. Ottawa, Ont., Canada, K1 A 0Y3

Accepted 1994 August 31. Received 1994 August 31; in original form 1994 March 11

S U M M A R Y An integral equation is derived to represent the electromagnetic response of line conductors buried below the surface of a horizontally stratified earth. By permitting several finite-length line conductors of arbitrary topology, this representation is free of the limitations imposed by analytic solutions. The integral equation is formulated in terms of excess electric current (scattering current) flowing along the line conductor and is solved numerically by dividing the line conductor into many small segments. In general, the excess electric current is controlled by both the internal and external impedance of the line conductor. The internal impedance is the longitudinal resistance of the line conductor. The exterpal impedance is caused by galvanic and inductive coupling between the line conductor and its local environment. The galvanic resistance to current channelling is spatially variable with a minimum in the centre of a uniform line conductor and is determined by conductor geometry and the host conductivity structure. The inductive external impedance is proportional to frequency and quadrature dominant. It is a function of line- conductor geometry, cross-section and burial environment. The inductive impedance effectively reduces the spatial dependence of the external impedance at high frequency by presenting a large reactance (which is uniform at all points along the conductor) to the exciting electric field.

Within the quasi-static limit (i.e. where displacement current can be neglected), electromagnetic excitation by either horizontal electric or vertical magnetic dipoles produces a constant primary electric field at high frequencies (far field). The excess electric current in the line conductor will always be inversely proportional to frequency for these types of sources at sufficiently high frequencies where the inductive external impedance is dominant. Horizontal magnetic dipole and vertical electric dipole sources generate primary electric fields that are proportional to the square root of frequency in the high-frequency limit of the quasi-static domain and the excess electric current excited by such sources will decrease as the inverse of the square root of frequency.

Key words: 1-D integral equation, electromagnetic response, external impedance, internal impedance, line conductor.

INTRODUCTION

The electromagnetic (EM) response of buried line conductors such as cables and pipelines has attracted attention from the EM community for more than a decade (Wynn & Zonge 1975; Campbell 1977,1978,1980; Campbell & Zimmermann 1980; Ogunade 1981, 1982; Chen, Xu & Yang 1991; Zhang & Luo 1991). Indeed, buried cables and pipelines are among the most common cultural conductors

G encountered in EM exploration programs (Nekut & Eaton 1990). Such conductors are effective channels for electric- current flow and thereby distort EM measurements and may completely mask the EM response of a geophysical target (Fitterman 1989; Fitterman et a/. 1991). The need to conduct EM surveys in environments containing buried cables and/or pipelines has been increasing because of a desire to exploit existing infrastructure in resource development and to assess the environmental impact of human activities. It is

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204

therefore essential to have a quantitative understanding of thc E M response and interactions of buried cables and pipelines in different geological settings and for different EM exploration sources. With an understanding of the physics of the problem. optimal methods can be developed to characterize and remove the undesired response from the survey data. Moreover, it may be possible to use the cultural line-conductor response to advantage in the interpretation of the ear th conductivity structure when there is substantial interaction between the line conductor and the host.

The theoretical foundation for calculation of the EM response of an infinite line conductor was derived by Wait (1972). Using Hertz potentials, Wait derived an analytical expression of the E M fields for a straight infinitely long line conductor buried in a homogeneous half-space subject to line-source excitation. Later, Wait (1978), Watts (1978), Wait & Umashankar (1978). Tsubota & Wait (1980), Parra ( 1 984) and Ogunade ( 1986) extended this analytical approach for application to a line conductor buried in a horizontally stratified earth excited with various kinds of sources. However, the elegant mathematical approach of Wait (1972) has limitations. The analytical expression of the EM fields is in a complicated form involving special functions that hinder physical intuition regarding the solution. This kind o f complicated analytical expression is also difficult to implement numerically (Raiche 1994). More importantly, the analytical approach can not be generalized to an arbitrary 3-D earth containing multiple bending conductors of finite-length line.

To overcome these limitations and to study the importance of line-conductor properties, topology and host environment, we have applied a 1 -D integral-equation modelling method to represent the E M response of line conductors. The integral equation is formulated in terms of excess electric current along the line conductor. The E M response is expressed as the sum of the primary field and the contribution from the excess electric current. By dividing the line conductor into a number of small segments of constant excess current, the integral equation can be discretized. The advantages of the discrete integral equation are its linear form and physically understandable terms.

A convenient method of discussing the physics of line conductors is through the concepts of internal and external impedances. The internal impedance is because of the longitudinal resistance of the line conductor and consequently is a function of the line-conductor conductivity and radius. In cases where the line conductor is coated (insulated from the host), the internal impedance is spatially dispersive, i.e. the internal impedance depends on the spatial wavelength of the primary electric field along the line conductor (Wait & Unmashankar 1978). However, we will only consider uncoated line conductors in this paper.

Since our purpose in this paper is to understand the EM response of finite, bending line conductors, we have limited the host-conductivity structure to be horizontally layered (i.e. 1-D) using the analytical Green's function derived by Weidelt (1975), Ward & Hohmann (1988). Of course, the integral-equation method can be extended to arbitrary 3-D earth structure with the use of appropriate Green's functions (e.g. Adhidjaja & Hohmann 1989; Pridmore, Hohmann & Ward 1981; Wannamaker 1991; Xiong 1992).

W. Qian and D. E. Boerner

1-D I N T E G R A L E Q U A T I O N

Consider the case of line conductors in the vicinity of an EM survey. Let I(!) be the excess electric current flowing along any buried line conductor due to the conductivity differences between the line conductor and the earth. Following Weidelt (IY75), we derive the integral equation for the horizontal electric field in this case. That is,

E(rJ = E ( r J + G(r I ro) - j(r)d3r. 1, where E is the horizontal electric field and E" is the horizontal electric field which would exist in the absence of the line conductor. G(r I ro) is the dyadic Green's function i.e. horizontal electric field at r generated by horizontal electric dipoles at Tor while j(r) = a;,E(r) is the excess electric current density in the line conductor. The integration is to be performed over the volume of the line conductor.

Since the line conductor has a radius that is much smaller than its length, we can assume that the excess electric current j (r) is in the direction of the line conductor and distributed uniformly over the line-conductor cross-section. The volume integral in (1) is thus reduced to a line integral along the path of the line conductor,

In the cross-section o f the line conductor, we can express

the cross-sectional area of the line conductor. Then we can rewrite eq. (2) as

- E(rO) = l (r l l ) / ( smtt ) with go = c T ~ ~ m d u c t o i g'c,irth and being

Partition the entire line conductor into N segments and assume that the current I is constant in each segment to arrive at a discrete form of eq. (3)

(4)

where I,,,, P,,, and Tmk are defined as

r n 2 k = J [ ~ ( r I r m ) - 41 * I,,

and

Ik

P,, = E"(r,,) * I,,,, (5) where ln1 is a unit vector representing the mth segment direction.

If m extends from 1 to N (the number of segments), we obtain the following linear matrix equation

(6) (R - r)I = P, where I and P are N X 1 column vectors having elements /,,, and P,,, respectively. r is an N X N matrix with the element Tmk representing the line integral of the Green's function on the kth line-conductor segment when rO = r,,,. When m = k,

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EM modelling of line conductors 205

the estimation of r,,tk involves principal-value analysis for the treatment of singularities discussed in the Appendix. Clearly the matrix r is diagonally dominant and. if the line conductor is evenly divided. r is also symmetric due to reciprocity. R is a diagonal matrix with R,, representing the internal resistance per unit length of the line conductor at the ith segment i.e. R , = l/(Su,,,). Following Wait (1978), we call r the external impedance.

The solution for the excess electric current in each segment of the divided line conductor is

I = (R - r) 'P. (7)

Eq. (7) is the general form of the excess electric current in a buried uncoated line conductor o f unspecified geometry. I f , for any earth structure, we are able to calculate the integrated Green's function matrix r and the primary electric field (electric field without the presence of line conductor) along the path o f the line conductor, then the excess electric current at all frequencies is readily calculated from (7). Although in principle eq. (7) can be solved for any 3-D earth model, our choice of Green's functions will limit our consideration in this paper to I-D earth structures.

In the following we concentrate first on the restricted case of zero frequency (DC) to test the accuracy and precision of the numerical implementation of the integral equation model (the equations are most singular at DC), and to gain intuition about the important physical aspects of external resistance (multiple line-conductor interaction and line- conductor bending). We then extend our study to consider frequency-dependent examples where both external resistance and reactance are important to demonstrate the current channelling characteristics of the conductor as a function of frequency.

NUMERICAL APPROXIMATIONS

To solve numerically the continuous integral equation (3), we make two approximations. One is to represent both the excess current and primary electric field as a constant in each segment of the line conductor ( I ) and the other arises when the Green's function must be integrated numerically (11). The assumption of locally constant electric fields has clear physical ramifications, and the numerical integration of the Green's function must be conducted carefully because of the principal-value analysis required to accommodate the intrinsic singular nature.

To study these approximations, we assume an analytic form for the excess current channelled into the conductor and then derive the analytic expression for the associated primary electric field [E:(x , , ) ] required to drive the fictitious current. Analytic forms for both the current and the driving electric field allow us to examine the performance of the numerical integral equation implementation.

The test model consists of a linear line conductor extending between x = 0 and x = a in a homogeneous half-space. At D C the Green's function is

where u is the half-space conductivity.

The integral equation (3) in this case is therefore

1 a I ( x , , ) - - I" 2PL d x = E',i(.r(l),

JTU () Ix -x(113

where (Y = l/(Su,,). We now assume the current I ( x ) has the form . r3( ( i - x ) ~

from 0 to u. This form of the current is convenient because it mimics our expectations that in a conductive filament of finite length. the excess current will be approximately zero at the two ends of the line conductor and a maximum in the middle. I t also permits analytic evaluation o f thc electric field required t o drive this current in the linc conductor as

To obtain eq. (10). we exclude the interval (x0 - F , ,rll + f) from the integration and use the results of the Appendix to estimate the singular contribution.

Analytic results can now be compared with the numerical evaluation of eq. (9). For2 the test case we assume a = l O - ' n m - ' , p = l / a = l 0 0 0 Q n and u = 3 0 0 0 m . We use a normalization factor of 10'' for both electric field and electric current expressed in eq. (10). We begin with an example where the Green's function has been integrated analytically so that only approximation I is required in the numerical solution. Fig. 1 shows the calculated excess electric current as a function of position for different partitions of the line conductor (SO, 200 and 500). Also shown in Fig. 1 is the incident electric field in eq. (10). As the partition o f the line conductor becomes finer, the calculated current approaches the analytic solution. For a 500 segment partition (6m segments), the largest error in the estimated current is less than 2 per cent. This example provides confidence that the numerical implementation of the integral equation is correct and also demonstrates that the use of constant basis functions to represent the excess current flow requires fine sampling for accurate solutions.

The next test addresses approximation 11. The test is designed to examine the effects of the principal-value analysis when numerically integrating the Green's function. Examining eq. (A12) one sees that at DC the self- interaction term, I',,,,,,, is proportional to - l / L 2 which means that shrinking the volume excluded in the singularity treatment produces larger negative values along the diagonal of the integrated Green's function matrix. These large negative diagonal values are evaluated analytically and must be accurately compensated by the numerically estimated off-diagonal elements, r,,,,,, if the matrix equhtion is to be stable under inversion. Our test consists of the same parameters as described in the previous example except that now we evaluate the Green's function numerically using the trapezoidal rule. The results are shown in Fig. 2 for partitions of 50, 200 and 500 segments respectively. Thick solid lines represent the current calculated using the analytical integration of the Green's function (i.e. equivalent


206 W. Qian and D. E. Boerner

0.1

2 0.05 E

7

i i o aa

0 'E

$ -0.05

-0.1 0 500 1000 1500 2000 2500 3000 rm1

Figure 1. Comparison between the numerically calculated excess electric current and the analytical solution for different partitions of the line conductor a t DC. Analytic expressions are used to calculate thc Green's function matrix r. The thick solid line represents thc analytic current distribution and thc thin lines the calculated distribution for partitions ol' SO, 200 and 500 segments. As the partition o f the line conductor becomes finer, the calculated currcnt approaches thc analytic solution. For a 500 segment partition, the differencc between the two is less than 2 per cent. The primary electric field described in eq. (10) for this case is shown in the lower diagram. The horizontal axis represents the distance in metres along the line conductor.

to cases in Fig. 1). The thin lines represent the current estimated by different spatial sampling rates for integrating the Green's function. In all cases, there is a monotonic decrease in the error of the current estimate with finer subdivision of the segments demonstrating the robustness of the numerical implementation of the integral equation. Fig. 2 also shows that as the partition size is decreased to obtain an accurate estimate of the current, the integrated Green's function must be evaluated more accurately. The two approximations (I and 11) are clearly dependent. Adopting more complicated basis functions for the current distribution and/or using more sophisticated integration rules will certainly improve the numerical performance, but the basic conclusion about the relationship between the two approximations remains, even for higher dimensional modelling (2-D and 3-D).

INTERACTION BETWEEN SEPARATELY BURIED CABLES

To investigate conditions under which the current channelling through separate line conductors interact, we provide a simple theoretical treatment. The model consists of two straight line conductors having the same length a. The two conductors are buried in a homogeneous half-space of conductivity u and have direction vectors denoted by I, and I,. At DC an integral equation is easily developed to

describe the current flow along I , , due to the incident field in the presence of the two conductors.

where a = l / ( S u a ) and G(r I rO) - 1/(na) /r - r,Ip3. Now assume that I,, is coincident with conductor 1. No interaction exists between the conductors when

~ ~ [ G ( ~ ~ ~ ~ ) ) . l , l ~ ~ ~ ~ ( ~ ) ~ ~ ~ >> lL[G(r I ~ 0 ) - l , ] - 1 d ( I ) d I ~ . (12)

Clearly, this condition can only be evaluated exactly when the true geometry and electrical parameters are defined. However, a heuristic analysis of the relative magnitude of the two terms is useful. If we consider that D is the separation between the two parallel straight conductors of length a, we have


20

15

10 - a s? 5

- c c

5

0


(a) 50 segments (b) 200 segments (c) 500 segments

I

10 - 5 c

g 5

5

0

0 3000 0 1000 2000 3000 0 lo00 2000 3000 [ml [ml

Figure 2. Excess electric current calculated for the cases in Fig. I with the Green's function matrix r estimated numerically. (a) is for a 50 segment partition of the linc conductor. Thc thick solid line represents the calculated current with m a t r i x J estimated from an analytical expression. The thin lines are for r estimated numerically by sampling each segment with 31, 51 and 101 points respectively. Similarly, diagram (b) is for a 200 segment partition and diagram (c). a SO0 segment partition. The thin lines in diagram (b) correspond to sampling each segment with 1 0 1 , 151 and 201 points, and i n diagram (c), with 201. 301 and 401 points.

and

where the overline denotes the spatial average o f the quantity. To reach the second equation in (13), we need to exclude the interval (xn - E, xo + E ) from the integration and use the Appendix to take into account the singular contribution. If we assume that the average current flowing in both conductors is approximately equal, conditions (12) and (13) become

There will be n o interaction between the two conductors when the separation of the conductors is much greater than their length. This result implies that the distance from which the line conductor can draw current to channel is comparable to its length. If another line conductor exists within this distance, its effect is to 'scavenge' current which would have been available to the first conductor. A numerical example will suffice to demonstrate this result. Suppose that two straight line conductors each of length 2 km and internal resistance Q m-l are oriented

half-space of resistivity 100 i2 m. The primary electric field along both line conductors is assumed to be 1 mV m-'. Each conductor is divided into 80 segments (segment length 25 m) and Green's function integration is sampled at 500 points per segment. As shown in Fig. 3, the mutual interaction can be safely ignored when the conductor separation is more

0 ' I I I 500 1000 1500 21

lml x)

Figure 3. Mutual interaction for two parallel conductors at DC. The horizontal axis represents the distance in metres along the line conductor. Since the currents in two conductors are identical due to symmetry, only one of them is shown. When the conductor separation is 1 m, the current in each conductor is exactly half that of a single wire of internal resistance 0.5 x 10- 'Rm- ' . For conductors separated by 2000 m, the current distribution is the same as if only one conductor is present. In this case, the mutual

parallel to each other and buried in a homogeneous interaction between the two conductors can be omitted.



0.5

0.4

- 9 5 0.3 - L

5 0.2

0.1

0

Conductor 1 - 0.5

0.4

r_ a

s Y -

0.3 L

0.2

0.1

n

Conductor 2

I I 1

0 500 1000 1500 2000 " 0 500 1000 1500 2000 iml lml

Figure 4. Electric current in the two collinear conductors at DC. Thc horizontal axis represents the distance in mctrcs along thc linc conductor.

than half the length of the conductor. This result is in agreement with eq. (14). It is also important to note that as the separation between the conductors is reduced to zero, the current in each conductor is exactly half that of a single wire with 0.5 X 1 0 -' I2 m- I internal resistance.

A revealing test case is to place the two line conductors end t o end. Fig. 4 shows the current in the two collinear conductors. For two conductors separated more than 10 m, the mutual interaction is negligible. This apparent contradiction with eq. (14) is due to the spatial distribution of the current in each conductor. Thus eq. (14). derived assuming a constant average current in each conductor, is no longer valid. When two collinear conductors are close, the current concentrations ('centres of gravity') in the two conductors are not necessarily close, thus the interaction can be weak. The addition of another linear conductor to the end of an existing one does not change the effective volume from which to draw current, it only displaces this volume laterally. Thus, as we have assumed a homogeneous incident electric field, the conductors d o not interact until they are sufficiently close together t o provide a localized path for current leakage between the two conductors. This example serves t o emphasize the importance of geometry in current channelling and also provides intuition regarding estimation of the extent of interactions between multiple elongated conductors.

B E N D I N G LINE CONDUCTOR

Another aspect that has not been addressed in previous studies of line conductors is the effect on current flow of a bent conductor. In comparison to the linear case, there are three major differences.

(1) In general, the distances between sections of the line conductor are reduced by the bending which has significant influence on the mutual interaction terms.

(2) Bending also changes the angle between the different segments of the line conductor, again altering the mutual interaction.

(3) Bending a line conductor changes the coupling with the primary electric field. The effect of ( 1 ) and (2) is to increase the mutual impedance of the whole system while the self-impedance remains unchanged. As the self-impedance is negative and mutual impedance is positive, bending will cause the overall admittance of the system (R - r) I to increase. The effect of (3) is complicated and not easily summarized. It depends on the spatial distribution of the primary electric field and the exact geometry of the line conductor. However, once the admittance of a particular geometry is understood, altering the distribution of the incident field along different parts of the conductor is a minor complication.

To illustrate the above points, assume a line conductor of zero longitudinal resistance is buried near the surface of a homogeneous half-space. The conductor is of length 2L and is bent in the horizontal plane at the middle so the two segments form an angle 8. At DC the Green's function is

To gain insight, consider only the mutual impedance between the two arms which can be derived as

1 2n(r1.7 [4 - (d1.25 - cos 8)' 1 0.5 - cos e - -

For 8 E [O, a], r12 decreases as 8 increases. This means that any bend in a line conductor causes the mutual impedance to increase. If the primary electric field is a unit along these


EM modelling of line conductors 204,

1 0

0 8

__ 0 6 s a, - c 3

04 0

0 2

0 0 1 I 1 1000 0 2WO 0 3000 0 4000 0

lml

Figure 5. Excess electric current distribution along a 40(X) m line conductor o f internal resistance 10 ' ( 2 m ' at I><'. 'l'he conductor is divided into 80 segments and the primary electric lield is assumed to be 1 m V m ' along the conductor. The conductor IS bent at 2000m. Because o f the last spatial decay of the Green's function. the excess current around the two ends of the conductor is independent of the bendins angle.

two wgments, the excess current in the line conductor 1%

From eq. (17). as 6, increases. the exccss current I will decrease. In this model, L = 2000m. u = 10 ' i 2 m ' and v = 10 ~ ' ( Q m ) - ' . We divide the whole conductor into S0m segments. The primary electric field is set to be 1 mV m - ' along the conductor. The estimated current distribution along the line conductor is shown in Fig. 5 for different bending angles. Owing t o the strong spatial decay o f the Green's function, the excess current around the two ends of the conductor is independent of the bending angle. For bending angles 6 > ~ / 2 . the current distribution virtually does not vary with the bendinp angle. F o r bending angles

0 I -

-0 5 i- .A ~ J 1000 0 2000 0 3000 0 4000 0 0 0

Figure 6. Excess electric current disiribution a t DC' along the bending line conductor described in Fig. 5. The primary electric lield is unitorm and in the direction of the first part of the conductor with magnitude 1 niV m I.

6 < n/2, the current close t o the bending point increases marginally. This example is contrived since the electric tield is forced to he uniform along the conductur arld thus must be different for each different bending angle modelled. It is meant only to convey an understanding o f the overall admittance in the line conductor as :I function ot bending angle.

A more realistic model 15 to represent the primarv electric tield as uniform and polarized along one arm o f the conductor. The excess electric current for this situation is shown in Fig. h. A comparison o f F i p 5 and h indicates that the first-order effect o f the conductor bending is caused by the change in coupling with the primiirv electric lield. The increased admittance is only of second-order importance.

F R E Q U E N C Y D E P E N D E N C E

The discussion thus far has been limited to DC where the external reactance is m r o . For non-Lero frequency. the external reactance influences the impedance and. at sufficiently high frequency. dominates the impedance. The external reactance is due almost rntirelv t o the self- interaction of the line conductor and has two effects. One is to eliminate the excess electric current in the line conductor and the other is to generate' uniform ,current ' I cross the line conductor. These two effects can bc demonstrated independently.

Consider a line conductor o l length 1.. iritcrn;il rcsistance u, radius K , buried in a homogeneous half-space medium of resistivity v. The line conductor is excited by a spatially uniform electric field E". We tirst neglect the spatial variation of the excess current. thus treating the whole conductor as one segment. The excess current can be t'ound using eq. (A12) as

At the high-frequency limit where k L >> simplified as

, eq. ( I S ) can be

Assuming E" is independent of frequency. the current will be inversely proportional to frequency when

The smaller the radius o f the line conductor. the larger the external reactance.

Equation ( I Y ) shows that for sufficiently high frequencies. the dominating reactance term -iwpcLI,[3-y - 1 + 2 In ( k K / 2 ) ] / ( 4 n ) is independent of the line-conductor geometry ( L ) . Consequently. the current distribution at these frequencies is mapped directlv from the local electric


210 W. @an and D. E. Boerner

In this regard, the conductor no longer channels current along its length, but rather maps the incident electric field a t each local segment into the excess current through induction. The electric field (primary plus secondary) within the line conductor can be easily derived from eq. (2) for this case as E ( x ) = 2E"(s ) . The secondary electric field outside the line conductor vanishes as l / ( i w p ( , ) .

There are two other high-frequency efiects that we did not consider. One is the skin effect, i.e. the current in the line conductor tends t o flow at the surface. This effect can he represented by a square root of frequency-dependent internal resistance. In general this impedance will be dominated by the external reactance at high frequencies. However. the concentration o f current on the periphery of the conductor can alter the external impedance and this may need t o be considered. For very high frequency, the displacement-current term is no longer negligible and creates an impedance proportional to frequency squared.

The transition from a DC-current-channelling scenario to thc local-induction scenario at high frequency takes place over a very wide frequency band as shown in Fig. 7. The current distribution (real and imaginary parts of current normalized by their value at one conductor end) for uniform electric-field excitation becomes more uniform as frequency increases. This example clcarly demonstrates the transition hctwcen current channelling and induction in terms o f

Real part 50

40

30 - 25 2 c

L

5 20

10

n " 0 1000 2000 3000

[ml

the current distribution along the conductor. A useful mental exercise is to recall that the incident electric field for these examples is uniform and consider that these plots are simply the effective admittance (save for a scale factor) at each point along the conductor.

A more realistic example involving an electic dipole source is shown in Fig. 8. The electric dipole is placed 5 km away and collinear with the line conductor. As frequency increases, the current distribution approaches the primary electric-field distribution. The real part of the current distribution flattens at a lower frequency than the imaginary part. I f we set the source dipole moment to be 10' A m. Fig. 9 shows the actual temporal variations o f the current in Fig. X for six individual segments. The rapid current variation at frequencies of between 10 Hz and 100 Hz corresponds to the transition between the near and far field of the source. Beyond 100 Hz, both real and imaginary parts of the current decay as the inverse of frequency.

Our final example involves a 20 km long conductor o f radius 0.01 m and internal resistance R m ' buried in a homogeneous half-space of resistivity 100 R m. A unit moment electric dipole source in the line conductor direction is placed 5 km away from the line conductor as shown in Fig. 10. The line conductor is partitioned to 100 segments. For a receiver S00m away from the line conductor on the other side, the EM fields are shown in Fig. 11. Owing to the symmetry, only E x , H,. and H3 are non-zero. Solid lines represent the field without the presence of the line conductor while dashed lines represent the EM field with line conductor distortion. 'R' denotes real parts and

Imaginary part 60

50

40

- 5 @ 30 c

L z 20

10

n " 0 1000 2000 3000

[ml

Figure 7. Relative current distribution (real and imaginary parts of current normalized by its value at one conductor end) for a 3 km long line conductor embedded in a homogeneous half-space of (+ = m I . The radius of the conductor is 0.01 m and the internal resistance is

R rn-'. The conductor is partitioned into 300 segments. As frequencies increase from 10'. Id , lo', lo4, lo5 to lo6 Hz, the current distribution becomes more uniform. Note that for 10 and 100 Hz, the relative current distributions for the imaginary parts is the same.



Real part

\ 1 \

0 1000 2000 3 [ml

0

Imaginary part I I I

40

30 - 5 c

g s

20

10

0 3 0 1000 2000 3000

[ml

Figure 8. Relative current distribution for the line conductor described in Fig. 8 excited by an electric dipolp source 5 kni away and collinear with the line conductor. As frequencies increase, the current distribution approaches the primary electric field distribution.

0.02

0.01

0

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Segment 1

1 10 100 1000 10000 Frequency [Hz]

Segment 180 0.6 I I ,,,,,,,, , , ,rml

z c 0.3 Bkj- E

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Segment 120

, . - - * . 8 _ ,

1 10 100 1000 10000 Frequency [Hz]

Segment 300

_“Jt 1

1 10 100 1000 10000 Frequency [Hz]

Figure 9. Temporal variation of the excess electric current at six individual segments of the line conductor described in Fig. 8. The rapid variation between 10 H z and 100 Hz is due to the near and far field transition of the source. Beyond 100 Hz, both real and imaginary parts (solid and dashed lines respectively) decay as the inverse of frequency.



X

L,

Source I, 5 km

Figure 10. A 2 0 km line conductor of radius 0.01 m and internal resistance 10 ’ Cl ni ’ is buried in a homogcncous halGspace with resistivity I00OL1 m. A unit moment electric dipole sourcc is located 5 km away from the line conductor. A receiver is 500 m away from the line conductor on the other side.

Ex [vlm]

2e-09

1 e-09

0

-1e-09 1 10 100

Frequency (Hz]

‘I’ denotes imaginary parts. The line-conductor distortion of the EM fields can be 20-30 pcr cent from this example.

CONCLUSIONS

In this paper, we have developed a 1-D integral equation to model the EM response of line conductors. This approach is more general than previously derived analytical solutions and allows the line conductor to be of finite length, have arbitrary topology, and to be excited by an arbitrary source. The solution of the 1-D integral equation is expressed in terms of internal and external impedances of the line conductor. The internal impedance is the longitudinal resistance o f the line conductor and depends on the conductivity and cross-section of the line conductor. The external impedance is due to the EM interaction between the line conductor and its host environment, i.e. conductivity structure of the earth. I t has two parts, galvanic and inductive. The galvanic part is due to the current channelling into the line conductor. This part of the external impedance is resistive and has a strong spatial variation which causes the excess current distribution peaks at the central part of the conductor. Inductive external impedance (the reactance) is due to the inductive coupling between the line conductor and its host environment. At sufficiently high frequency, the reactance will dominate the E M behaviour of the line conductor. The general role of the reactance is to eliminate the excess current in the conductor. It also tends to make the current distribution uniform along the conductor. The line conductor modelling approach developed here is useful to assess the response of elongated conductors in an EM survey. Such conductors can be either natural or artificial (buried cables, pipelines, powerlines, etc.).

Hz [A.m]

3

2e-09

1e-09

0

- 1 e-09

-2e-09

-3e-09

Frequency [Hz] Frequency [Hz]

Figure 11. EM field distortion by the line conductor in Fig. 10. Solid lines represent the undistorted responses and dashed lines distorted ones. ‘R’ denotes rcal parts and ‘I’ imaginary parts.


EM modelling o,f line conductors 213

ACKNOWLEDGMENTS

Jim Craven and Ron Kurtz are acknowledged for their valuable criticism and comments during an internal review of this manuscript. Tilman Hanstein and an anonymous referee are thanked for their important contributions. Geological Survey of Canada Contribution No. 53393.

REFERENCES

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APPENDIX: SINGULAPITY TREATMENT

The deriviation of the singular term I-,,,,,, closely follows that of Qian & Boerner (1994). From Weidelt (1075), we write the homogeneous full space Grecn's function iis

h l r ril

G(r 1 r0) = -(k21 - "'7). ( A l l 47ro /r - ro/ '

where I is an identity matrix and o is the conductivity of the whole space. For our purposes 0 can be taken to be the conductivity of the layer in which the cable is buried. The propagation constant is given by k 2 = i w p o o , where o is the angular frequency and p0 the free-space magnetic permeability.

Since we are primarily interested in line conductors buried more shallowly than the EM skin depth. the Green's function can be approximated as

G(r I ro) = GTF(r I r0) + GTM(r 1 ro)

where

G."'-(r I ro) = - k 2 g ( r I ro)l

Gm(r 1 r,,) = 2VV * g(r 1 ro) e-klr-rol

= 4 m lr - ro/ . Compared to the homogeneous full-space Green's

function, we can see the TM part is doubled due to the air-earth interface. Eq. (A2) approximates the homogeneous half-space Green's function to the second order [O(k21r-r,12)] for k ) r -rJ<< 1, when both r and r,) are close to the air-earth interface. The TM part gives the dominant contribution to the singularity. We first integrate the TM part over a thin conducting cylindrical cable of radius R. Implicit to this calculation is the assumption that all current in the small segment of the conductor under consideration flows parallel to the cable. Assuming the



current density is uniform across a cylindrical volume, we find

where e , is along the cable direction and R is the cable radius.

If R << L , we have the expansion

- h l

[l + k L - O(R’)]. - -~ - IKTL’

The remaining TE part of the contribution to the singularity can be derived as

We now concentrate our attention on the first term in (AS), we have

From Gradshteyn & Ryzhik (1980, p. 343, 3.479.2, note the

typographic error in their expression)

d X

Setting v = 1 and taking the derivative with respect to p we obtain

Substituting equation (A8) into (A6) and expanding H\”(ikR) to the order of ikR as lkRl<< 1 we find

iakR’ ( 1 - 2 y ) - -

4 4 ’ kR2 _ _

where y = 0.57722 (Euler’s constant).

approximated as When R / L < < l , the second term in (A5) can be

where ‘Ei’ is the Exponential-Integral function (Gradshteyn & Ryzhik 1980).

Substituting eq. (A9) and (A10) into (A5), and using k 2 = iwpou, we finally reach

Summing the TM and TE contributions we have

k R r,,,,,, = !!!!!I [ 2y - 1 + 2 In (2) - 2 Ei ( - k L ) ] 4a

e - k l -

(1 + kL) . -~

nuLZ

If we let L - x , the external impedance per unit length in (A12) approaches to the form quoted by Campbell (1978) from Wait (1978). This result is also in agreement with eq. (44) derived by Watts (1978). Note, however, that Watts (1978) used e-Iw‘ time dependence rather than e””‘.


electromagnetic modelling of buried line conductors using an

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