1D Sunsurface Em Fields Excited by Energized Steel Casing

1D subsurface electromagnetic fields excited by energizeds

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GEOPHYSICS, VOL. 74, NO. 4 �JULY-AUGUST 2009�; P. E159–E180, 10 FIGS., 1 TABLE.10.1190/1.3131382

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teel casing

ei Yang1, Carlos Torres-Verdín2, Junsheng Hou3, and Zhiyi �Ian� Zhang4

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ABSTRACT

We use numerical simulations to investigate the possibilityof enabling steel-cased wells as galvanic sources to detectand quantify spatial variations of electrical conductivity inthe subsurface. The study assumes a vertical steel-cased wellthat penetrates electrically anisotropic horizontal layers.Simulations include a steel-cased vertical well with a finite-length thin wire of piecewise-constant electric conductivityand magnetic permeability. The steel-cased well is energizedat the surface or within the borehole at an arbitrary depth withan electrode connected to a current source of variable fre-quency. Electromagnetic �EM� fields excited by the ener-gized steel-cased well are simulated with an integral-equa-tion approach. Results confirm the accuracy of the simula-tions when benchmarked against the whole-space solution ofEM fields excited by a vertical electric dipole. Additionalsimulations consider a wide range of frequencies and subsur-face conductivity values for several transmitter-receiver con-figurations, including borehole-to-surface and crosswell.The distribution of electric current along the steel-cased wellis sensitive to vertical variations of electric conductivity inthe host rock. In addition, numerical simulations indicate thatcrosswell and borehole-to-surface receiver configurationscould reliably estimate vertical variations of electric conduc-tivity within radial distances of up to 500 m for frequenciesbelow 100 Hz and for average host rock electric conductivi-ties below 1 S /m.

INTRODUCTION

Beginning with the publication of Kaufman’s seminal workKaufman, 1990; Kaufman and Wightman, 1993�, electromagnetic

Manuscript received by the Editor 12 June 2008; revised manuscript receiv1Formerly The University of Texas atAustin,Austin, Texas, U.S.A.; prese

eotrace.com.2The University of Texas atAustin,Austin, Texas, U.S.A. E-mail: cverdin@3Formerly The University of Texas at Austin, Austin, Texas, U.S.A.; prese

alliburton.com.4Shell International E&P, Houston, Texas, U.S.A. E-mail: ian.zhang@she2009 Society of Exploration Geophysicists.All rights reserved.

E159

EM� borehole measurements acquired in the presence of steel-ased wells have received significant attention because of their prac-ical link with subsurface electric properties. Subsequent technicalevelopments �Beguin et al., 2000; Maurer and Hunziker, 2000�ave led to the manufacture of commercial borehole tools to mea-ure formation resistivity behind casing. In principle, the relativelyarge depth of investigation of through-casing resistivity measure-

ents makes them good candidates to quantify the spatial distribu-ion of electric conductivity between wells.

For the latter case �crosswell EM data acquisition�, prototypeeasurement acquisition systems have been tested with moderate

uccess in shallow reservoirs with well separations of 100 m �Wilt etl., 1995a, 1995b�. However, typical interwell spacing in conven-ional oil fields is approximately 1 km, whereas the maximum inter-ell separation for steel-cased EM surveys is reported as 726 m

Hoversten et al., 2001�. More importantly, the relatively high costf deployment and acquisition, coupled with high levels of measure-ent noise associated with excitation and sensing of EM fields via

nduction systems, has prevented the widespread use of crosswellM data-acquisition systems in conventional oilfield operations.Attempts have been made to develop more economic and efficient

ata-acquisition systems �Schenkel and Morrison, 1990; Nekut,995; Newmark et al., 1999�. One of these acquisition systems con-ists of using the metal casing itself as a current electrode �Takácsnd Hursan, 1998; Newmark et al., 1999�. Understandably, the spa-ial resolution associated with such a method is relatively low atounding frequencies close to DC.Away to improve the spatial reso-ution of cased-well EM acquisition systems is to use higher fre-uencies at the expense of shorter radial lengths of investigationTakács and Hursan, 1998�.

Most studies consider the effect of steel casing on borehole DCndAC induction-measurement systems �e.g., Augustin et al., 1989;chenkel and Morrison, 1990; Wu and Habashy, 1994�. Only a fewublished studies explore and quantify the possibility of using theteel casing itself as the AC source in borehole-to-surface and cross-

ovember 2008; published online 11 June 2009.trace, Houston, Texas, U.S.A. E-mail: [email protected]; wyang@

texas.edu.lliburton Energy Services, Houston, Texas, U.S.A. E-mail: junsheng.hou@

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ell measurement systems �Rocroi and Koulikov, 1985; Takács andursan, 1998�. The numerical and scale-model study by Takács andursan �1998� evaluates a prototype experimental acquisition sys-

em based on steel-casing EM excitation. They conclude that the en-rgized steel-case EM acquisition system is superior to that of a fi-ite electric dipole. However, their study is limited in scope and doesot explore the limits of resolution and detectability of the acquisi-ion system.

Our objective is to develop a numerical method to simulate EMelds excited by energized steel-cased wells to appraise the corre-ponding limits of spatial resolution and radial length of investiga-ion when probing spatial variations of subsurface electric conduc-ivity.After introducing and benchmarking the numerical simulation

ethod, we consider measurements acquired with borehole-to-sur-ace and crosswell acquisition systems. We quantify the differencen electric and magnetic fields excited in a homogeneous wholepace with and without the presence of the steel-cased well. Addi-ional simulation examples explore the possibility of using the sub-urface EM fields excited by steel-cased wells to monitor variations

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igure 1. �a� Vertical grounded steel-cased well energized by a cur-ent source. Current injection occurs at location �0, 0, 0�. The well isligned with the z-axis; length of casing is L meters. When receiversre located on or near the surface ��, the acquisition system is re-erred to as borehole-to-surface configuration. When receivers areocated in another borehole along the vertical direction ��, the ac-uisition system is referred to as crosswell configuration. �b� Hori-ontal multilayer subsurface model assumed in the numerical simu-ation of electric current leakage away from a steel-cased well. Eachayer is assumed homogeneous and isotropic for magnetic perme-bility and dielectric permittivity but transversely isotropic for elec-ric resistivity.

n the geometric and electric properties of deep-target layers usingorehole-to-surface and crosswell acquisition systems. Measure-ent sensitivity to variations of target-layer properties opens the

ossibility of using energized steel-cased wells for time-lapse reser-oir monitoring.

NUMERICAL SIMULATION

Figure 1 shows the geometry of interest and the cylindrical coor-inate system considered in our work to simulate the EM fields ex-ited by an energized vertical steel-cased well. The z-axis coincidesith the axis of the well and points downward. We assume a horizon-

ally stratified subsurface model consisting of M interfaces, wherehe mth layer is homogeneous and exhibits transversely isotropicTI� electric conductivity. The thickness of the mth layer is denotedy hm, and its associated EM properties are denoted by �m, � hm, � vm,nd �m, which are dielectric constant, horizontal and vertical electriconductivities, and magnetic permeability, respectively. The top ofhe mth layer is located at the depth z�zm, and the zeroth and Mthayers are assumed to be semi-infinite �where h0 and hM equal infini-y�. Because the electric conductivity of steel casing is many ordersf magnitude larger than the conductivity of rock formations and hasradius much smaller than the casing length and the lateral extent of

he formation, we approximate the steel-cased well with a metal wiref finite radius ac.

To simulate subsurface EM fields excited by a finite-length verti-al wire embedded in a stratified-earth model, we first derive the EMelds excited by a vertical electric dipole �Wait, 1970, 1971, 1982�.ubsequently, EM fields excited by the finite-length wire are calcu-

ated by superimposing the individual EM fields excited by discreteire segments.Maxwell’s equations are the starting point for the simulationethod. Assuming a time harmonic dependence equal to ei�t, whereis radian frequency, t is time, and i��1, Maxwell’s equations

n the frequency domain are written as

�� E�� i��H

� �H�J�Jp� i��E

J��E�, �1�

here E and H are the electric and magnetic vector fields, respec-ively; J is the current-density vector for electric conduction; Jp ishe current-density vector for the impressed current source; � is thelectric conductivity tensor; and � and � are the scalar magnetic per-eability and dielectric permittivity, respectively. For simplicity, we

ocus our attention to the case of TI electric conductivity, where thelectric conductivity tensor is described completely with the valuesf vertical and horizontal conductivity — � h and � v, respectively.

We approach the solution of Maxwell’s equations with a vectorotential formulation. Let � h�� h� i�� and � v�� v� i�� be theorizontal and vertical complex-valued conductivities of the medi-m. The values A and � designate the vector and scalar potentials,espectively, with the corresponding Lorentz gauge condition giveny � ·A�� h�� 0. The EM fields are readily derived from the vec-or potential via the relations �Wait, 1966; Xiong, 1989�

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�H� � �A

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First, we assume that casing penetrates all M layers vertically. Theource is a vertical electric dipole that can be located anywhere with-n the casing segment of each layer. This choice is equivalent to as-uming that each layer could include one source dipole. We have aotal of M dipoles with their moments given by Imdz, where Im is thelectric current of the mth dipole and dz is the differential length ofhe dipole. The coordinates �0,0,zm� � designate the location of the di-ole.

Next, we construct a solution of Maxwell’s equations assumingzimuthal symmetric electric properties of the subsurface about thexis of the steel-cased well. Because of the assumption of azimuthalymmetry around the steel-cased well, it follows that the corre-ponding vector potential exhibits only a z-component, denoted as�Az. Following the principle of superposition, the total vector po-

ential at the depth zm� is the sum of the vector potentials associatedith each of the dipoles:

�m��m1��m2� ¯��m�m�1��mm��m�m�1�

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�i��, and Bm�i�� are determined by enforcing the continu-

ty of tangential EM fields at layer interfaces �see Appendix A forerivation details�. In equations 4 and 5, J0�� is the Bessel func-ion of the first kind of order zero, and Km is the electric anisotropicactor, given by

Km�� hm� i��m

� vm� i��m, �6�

here � , � , and � are horizontal electric conductivity, vertical
hm vm m
lectric conductivity, and dielectric permittivity, respectively, of theth layer. In addition,

um2 ��2Km� m

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m2 � i� hm�m� ��2�m�m. �8�

If we assume a current source in the form of a vertical wire withength L carrying an AC current given by I�z�, it follows from equa-ions 3–5 that the corresponding z-component of the vector potential

z�m��,z�, in the layer where the mth dipole is located, is given by

Az�m��,z��

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Lm�1

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Li�1

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here m�1, 2, . . . , M; i�m,i�1, 2, . . . , m�1; i�m�1, m2, . . . , M. Consequently,

G��,z,z�� Gi��,z,zi��,z��zi�,1� i�m�1

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Gi��,z,zi��,z��zi�,m�1� i�M� .

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Upon calculating Az�m��,z�, we proceed to calculate the distribu-

ion of electric current along the wire and then the corresponding EMelds outside the wire. Appendix B calculates the distribution oflectric current along the wire using the vector potential formula-ion;Appendix C summarizes the numerical procedure used to com-ute the distribution of electric current along the wire via the methodf moments; and Appendix D describes the formulation used to cal-ulate the EM fields excited outside the wire.

NUMERICAL SIMULATION RESULTS

Based on the above derivation, we programmed an algorithm toompute the distribution of electric current along the wire and theorresponding EM fields excited elsewhere in a 1D subsurface medi-m. Unless otherwise noted, we assume the steel-cased well is man-factured with carbon steel �CS� of electric conductivity � c equalo 5.56�106 S /m. Moreover, in all of the simulation examples, un-ess otherwise noted, we adopt the following casing and host-medi-m parameters: length of steel-cased well L�3000 m, number ofqual-length vertical discretization segments N�250, relativeagnetic permeability �RMP�� relative dielectric permittivity

RDP��1, and injected �source� electric current I0�10 A.To verify the accuracy and reliability of the simulation algorithm,

e first benchmark the 1D code against the analytic solution for ahole space of electric conductivity � �0.01 S /m with a current
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ource connected to the wire at the origin. In so doing, we intention-lly set the electric conductivity of all casing segments equal to thelectric conductivity of the host rock �� c�� 1�0.01 S /m� exceptor the first casing segment, which is connected to the input sourceurrent. With this choice, the excited EM fields are equivalent tohose caused by a vertical electric dipole, thereby enabling the com-arison against a well-known analytic solution �Wait, 1970, 1971,982�.

As shown in Figure 2a, the computed electric current along theire is in excellent agreement with the corresponding electric cur-

ent calculated with the analytic solution �equal to the product of theost rock’s electric conductivity, the vertical component of the elec-ric field, and the length of the casing segment�. In addition, Figureb compares the numerical and analytic solutions of the correspond-ng azimuthal magnetic field �amplitude and phase� at receiver loca-ions distributed as in typical crosswell �Figure 2b, radial distanceetween wells equals 250 m� and borehole-to-surface �Figure 2c�easurement acquisitions. The agreement between numerical and

nalytic results is excellent.Additional successful consistency checks �not shown� include

alculating casing current and excited EM fields with a variableumber of casing segments and with different values of frequencynd host rock electric conductivity. In all cases, the agreement be-ween analytic and numerical results is excellent, thereby lendingredence to the accuracy and reliability of the simulation algorithm.

Figure 3a describes the distribution of electric current along theasing for different values of casing electric conductivity in a homo-eneous half-space of electric conductivity equal to 0.1 S /m. Webserve an exponential decrease of current with depth with a locallyaster rate of decay near the surface for decreasing values of casingonductivity. The decay of electric current along the casing is accel-rated with low values of casing conductivity because of increasedurrent leakage into the formation �the difference between casingnd background electric resistivity determines the amount of currenteakage�.

To gain insight into the effect of casing on the electric and magnet-c fields induced in the subsurface, we plot cross sections of the am-litude ratio �base 10 logarithm� of nonzero electric and magneticeld components calculated with and without the presence of casingFigure 3b-d�. The cross sections indicate an enhancement of the ex-ited EM field in the vicinity of the energized well �an enhancementactor as high as five for the case of the azimuthal magnetic field�ith respect to those excited by an electric dipole.In what follows, we focus the simulation cases to the study and in-

erpretation of a wide range of environmental and acquisition vari-bles to evaluate the feasibility of geophysical prospecting with anlectrically energized steel-cased well.

Figure 4a and b shows the effect of different values of whole-pace host rock conductivity on the vertical distribution of electricurrent along the wire at different frequencies. Simulations wereerformed with the set of parameters indicated in the figures. Resultsescribed in Figure 4a and b emphasize the rapid, quasi-exponentialecay of the electric current as a function of depth because of in-reasingly high values of host rock conductivity. The rate of de-rease of electric current increases dramatically for depth segmentsear the lower end of the wire because at those points current leakageccurs not only along the radial direction but also vertically.

Figure 4b and c compares the vertical distribution of electric cur-ent along the wire in the presence of whole-space and half-space ho-ogeneous and isotropic backgrounds, respectively, for different

alues of host rock electric conductivity at 60 Hz. Except near theocation of the source current, where the decrease of wire current isaster for the half-space case than for the whole space model, the dif-erence between the two distributions of electric current is not signif-cant.

Figure 5a and b shows the vertical distribution of electric currentlong the wire for different values of frequency in the presence ofomogeneous and isotropic whole spaces of electric conductivityqual to 0.01 and 0.1 S /m, respectively. Remaining simulation pa-ameters are the same as in Figures 4. Simulations indicate that an in-rease of frequency dramatically accelerates the rate of decay oflectric current. The rate of decrease of electric current increasesith an increase in background electric conductivity.Figure 5c shows the vertical distribution of electric current along

he wire for different values of frequency in a homogeneous half-pace of electric conductivity equal to 0.1 S /m. Comparison to sim-lation results shown in Figure 5b indicates that the presence of their-earth interface does not have a significant effect on the distribu-ion of electric current along the wire except at points close to theurrent source, where the decay of casing electric current is faster forhe half-space case than for the whole-space case.

Figure 6 shows the effect of different host rock electric anisotropyactors K, K � �� h1� i��1� / �� v1� i��1�, on the distribution oflectric current along the steel-cased well. The well is located withinhomogeneous half-space, and the operating frequency is 60 Hz,ith the vertical electric conductivity � v1�0.01 S /m. Remaining

imulation parameters are as those in Figure 4. In Figure 6, the calcu-ated values of electric current decrease with an increase of aniso-ropic factor K.

Figure 7a describes a three-layer subsurface model constructed toain additional insight into the behavior of the simulated wire cur-ent and EM fields. The construction of this model is based on geo-etric and electric properties of a practical field example wherein

he hydrocarbon pay zone �conductivity of 0.05 S /m and 50 mhick� is buried 475 m below the surface and is referred to as the tar-et layer. We assume that the location of the contact point for thelectric source �source current�10 A� is 2.5 m below the surface.he length and electric conductivity of the casing are 700 m and05 S /m, respectively.To perform the simulations, we subdivide the casing length into

40 equal-length segments. Figure 7b shows the calculated verticalistribution of electric current along the wire �amplitude and phase�t 1, 10, and 100 Hz. The rate of decrease of the logarithmic currentith depth changes at layer interfaces: The larger the difference of

lectric conductivity, the larger the change.An increase of frequencyurther increases the rate of decrease of logarithmic current andhase shift. Higher frequencies cause more pronounced variations ofhe fields. They also cause a sudden phase change at some layer inter-aces.

Figure 7c and d shows the simulated magnetic fields �amplitudend phase� for crosswell and borehole-to-surface receiver acquisi-ions at 1, 10, and 100 Hz. For crosswell acquisition, the radial dis-ance between the receiver and steel-cased wells is 250 m. The verti-al distribution of magnetic fields simulated with crosswell acquisi-ion is influenced by layer boundaries. For borehole-to-surface re-eiver acquisition, receivers are located at the same depth �2.5 m be-

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Figure 2. �a� Comparison of numerical and analytic solutions of thedistribution of electric current along a vertical steel-cased well. Themagnetic fields �amplitude and phase� measured with �b and c�crosswell �radial distance to the second well�250 m� and �d and e�borehole-to-surface �vertical distance between current source andsurface�6 m� acquisitions in a homogeneous, isotropic wholespace. Conductivity�0.01 S /m. Casing is 3000 m long, dividedinto 250 equal-length segments of conductivity�5.56�106 S /m.The 10-A source current is injected at the midpoint of the first casingsegment �located 6 m below the surface� at 1 Hz frequency. RMPand RDP of host rock equal one.

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10 0

10–2

10–4

10–6

10–8

10–10

10–12

0 500 1000 1500 2000 2500 3000Depth (m)

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a)casing conductivity = 1E–1 S/mcasing conductivity = 1E–2 S/mcasing conductivity = 1E+3 S/mcasing conductivity = 5.556E+6 S/mcasing conductivity = 1E+01 S/m

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Horizontal distance from surface (m)500 1000 1500 2000 2500 3000

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2500

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–6

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igure 3. �a� Distribution of electric current along the vertical steel-cased well calculated for different values of casing electric conductivity in aomogeneous, isotropic half-space. Base-10 logarithm of the amplitude ratio of �b� azimuthal magnetic fields H� simulated with and withoutasing, �c� radial electric fields Er simulated with and without casing, and �d� vertical electric fields Ez simulated with and without casing. Con-uctivity of host rock is 0.1 S /m; casing length is 3000 m, divided into 250 equal-length segments. Conductivity is 5.56�106 S /m. The 10-Aource current is injected at the midpoint of the first casing segment �located 6 m below the surface� with a frequency of 10 Hz. Conductivity ofhe host rock is 0.1 S /m; RMP and RDP equal one.

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b)

c)

igure 4. Distribution of electric current along the vertical steel-ased well calculated for different values of host rock electric con-uctivity in a homogeneous, isotropic whole-space for frequenciesf �a� 10 Hz and �b� 60 Hz and in �c� a homogeneous and isotropicalf-space at 60 Hz. Casing length is 3000 m, divided into 250qual-length segments; conductivity�5.56�106 S /m. The 10-Aource current is injected at the midpoint of the first casing segmentlocated 6 m below the surface�. Conductivity of the host rock is.1 S /m; RMP and RDP equal one.

R

a)

b)

c)

igure 5. Distribution of electric current along a vertical steel-casedell calculated for different values of frequency in a homogeneous

nd isotropic whole space with conductivity of �a� 0.01 S /m and �b�.1 S /m and in �c� a homogeneous and isotropic half-space withonductivity of 0.1 S /m. Casing length is 3000 m, divided into 250qual-length segments; conductivity�5.56�106 S /m. The 10-Aource current is injected at the midpoint of the first casing segmentlocated 6 m below the surface� with different frequencies. RMPandDP of the host rock equal one.

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ow the surface� as the source. Increasing the distance between theource and receivers decreases the amplitude of the magnetic fieldignificantly, and the phase of the same field changes rapidly. A fre-uency increase further emphasizes amplitude and phase changes ofhe magnetic field.

We examined additional variations of the three-layer modelhown in Figure 7a to quantify the sensitivity of the simulated mag-etic fields to perturbations of target-layer resistivity, thickness, andepth of burial. For these simulations, we consider only borehole-to-urface receiver acquisition. Figure 8a shows the simulated magnet-c fields �amplitude and phase� for three values of target-layer resis-ivity �0.1, 20, and 100 ohm-m� at 60 Hz. The simulated magneticelds exhibit variations with respect to the perturbation of target-

ayer resistivity only at radial distances longer than 100 m, evenhough these variations are marginal in amplitude but measurable inhase. Magnetic fields simulated for variations of target-layer thick-ess �30, 50, and 70 m� at 60 Hz �Figure 8b� do not evidence differ-ntial sensitivity to layer thickness. Figure 8c shows the magneticelds simulated for different values of depth of burial of the target

ayer �50, 100, and 200 m� at 60 Hz. For radial distances longer than00 m, simulations indicate measurable sensitivity of the magneticeld to the target layer’s vertical location, especially the phase of theagnetic field.To further quantify the sensitivity of the simulated magnetic fields

o perturbations of the electric conductivity of the target layer, wetudied the dependence of these fields on the location of the point ofontact of the current source along the steel-cased well. Simulationseported here indicate that the location of the current contact point onhe wire has significant influence on the sensitivity of the simulatedelds to perturbations of electric conductivity.For this study, we consider a crosswell acquisition system �Figure

� in which the radial distance between the energized steel-casedell and the receiver well is 250 m and the operating frequency is0 Hz. Figure 9 shows the relative percent difference �real and imag-nary components� of the simulated magnetic fields as a function ofepth in the receiver well because of a change of target-layer con-uctivity from 0.05 to 0.04 S /m. The depth location of the source is

igure 6. Distribution of electric current along a vertical steel-casedell calculated for different values of AK �AK� K� �� h1

i��1� / �� v1� i��1�, host rock electric anisotropy factor� in aomogeneous half-space. The casing length is 3000 m, divided into50 equal-length segments; casing conductivity�5.56�106 S /m.he 10-A electric current at the surface is injected at the midpoint of

he first casing segment �located 6 m below the surface� withrequency�60 Hz. Vertical electric conductivity � v1�0.01 S /m.MP and RDP equal one.

.5, 237.5 �center of the upper layer, Figure 7a�, 500 �center of thearget layer, Figure 7a�, and 575 m �50 m below the lower boundaryf the target layer�. The real component of the percent difference ex-ibits the largest variation among the four cases. Likewise, the larg-st difference in the real component of the percent difference occurshen the source-contact point is within the target layer. This last ob-

ervation is a direct consequence of the fact that the amount of cur-ent leakage through the target layer �the relative proportion of cur-ent leakage is determined by the conductivity of the layer� is condi-ioned by the vertical distance between the source contact point andhe target layer.

Afinal case of study considers a realistic multilayer model associ-ted with a Tertiary clastic sedimentary environment in central Tex-s. The objective is to appraise the use of energized steel-cased wellss sources of electric current for time-lapse monitoring of subsurfaceariations of electric conductivity �such as those resulting from wa-er injection in a hydrocarbon-producing field�. Table 1 describes thehickness and electric resistivity of the 30 layers included in this

odel. All layers have the same dielectric constant and magneticermeability �equal to those of air�. The length of the steel-casedell is 1191 m and its electric conductivity is equal to 5�105 S /m.lectric current �10 A� is injected into the steel-cased well at 2.5 melow the surface. We consider calculations at 1, 10, and 100 Hz ofhe electric current along the wire and of the magnetic field measuredith a crosswell receiver acquisition 250 m radially away from theire.The simulated distribution of electric current along the wire �Fig-

re 10a� indicates a fast decrease of electric current with depth. Sim-lar to simulation results discussed in connection with Figure 10a,he decrease of electric current is faster through conductive layershan resistive layers. Differences between calculated wire currents atand 10 Hz are small; however, differences between the calculatedire currents at 10 and 100 Hz are substantial. An increase of fre-uency improves the vertical resolution, albeit at the expense of fast-r amplitude decay of the fields. This behavior indicates there is aractical trade-off when choosing the sounding frequency betweenertical resolution and radial/vertical probing distance. Simulationsndicate that �a� measurements of casing current could be used to es-imate the electric conductivity behind the casing �similar tohrough-casing resistivity measurements� and �b� vertical variations

a

c

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Figure 7. �a� Three-layer subsurface model; � 0 is air conductivity�10�4 S /m� and � 1, � 2, and � 3 designate the electric conductivitiesof the three layers �0.2, 0.05, and 0.2 S /m, respectively�. Layerthicknesses are 475 m, 50 m, and semi-infinite. The 10-A sourcecurrent is injected to the steel-cased well �aligned with the z-axis�2.5 m below the surface. �b�Amplitude �left� and phase �right� of theelectric current along the steel-cased well. �c� Amplitude �left� andphase �right� of the magnetic field simulated for crosswell acquisi-tion. Magnetic receivers are located 250 m radially away from theenergized steel-cased well. �d� Amplitude �left� and phase �right� ofthe magnetic field, simulated for borehole-to-surface acquisition.Magnetic receivers are located at the same depth as the currentsource. Simulation results are shown at 1, 10, and 100 Hz �solid,dashed, and dotted lines, respectively�.

a

b

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c)

Figure 8. Amplitude �left� and phase �right� of the magnetic fieldsimulated for borehole-to-surface acquisition with perturbations of�a� resistivity and �b� thickness of the target layer described in Figure7a. �c� Magnetic field simulated for different values of target-layerdepth. Frequency�60 Hz, target-layer conductivity�0.05 S /m,and injected current�10 A.

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a)

b)

c)

d)

igure 9. Real and imaginary parts of the percent difference of the simulated magnetic field resulting from a perturbation of target-layer conduc-ivity in the subsurface model described in Figure 7a. Frequency�60 Hz; source current�10 A. The magnetic-field difference is given byH�H�H�, where H is the magnetic field simulated for a target-layer conductivity of 0.05 S /m and H� is the magnetic field for 0.04 S /m.olid and dashed lines describe the values of Re�� H� /Re�H��100 and Im�� H� / Im�H��100, respectively. Receivers are located 250 madially away from the energized steel-cased well. Electric current is injected into the steel-cased well �a� 2.5 m below the surface, �b� at the cen-
er of the upper layer of Figure 7a, �c� at the center of the target layer of Figure 7a, and �d� 50 m below the bottom layer of Figure 7a.

o

ffisctnisctt

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swdwphttto

iwhsttccti

Tt

Ln

0

1

2

3

4

5

6

7

8

9

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

a

FtTetArr

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f casing current could be used to detect layer boundaries.Figure 10b shows the amplitude of the magnetic field simulated

or crosswell receiver acquisition. The amplitude of the magneticeld decreases rapidly with depth. At 100 Hz, the magnitude of theimulated magnetic field rapidly reaches values not detectable withonventional data-acquisition systems. Because measurements areotal fields, the decrease of field magnitude near the source is domi-ated by the casing, with marginal influence of formation conductiv-ty. However, with an increase of vertical distance away from theource, the effect of the casing decreases and the effect of formationonductivity increases. In the latter depth domain, local depth varia-ions of the magnetic field correlate with vertical variations of elec-ric conductivity.

able 1. Geometric and EM properties used in to constructhe multilayer subsurface model.

ayerumber

Layer upperinterfacez-value

�m�Resistivity�ohm-m�

Layerthickness

�m�

�

58 7 33

91 16 29

120 50 33

153 7 42

195 20 20

215 50 98

313 6 52

365 25 28

393 4 11

0 404 40 34

1 438 6 34

2 472 30 31

3 503 5 52

4 555 60 91

5 646 4 27

6 673 45 51

7 724 5 14

8 738 60 29

9 767 5 17

0 784 50 23

1 807 4 35

2 842 50 26

3 868 4 32

4 900 25 70

5 970 4 30

6 1000 25 32

7 1032 3 58

8 1090 40 101

9 1191 4

CONCLUSIONS

We developed and successfully tested an accurate, reliable inte-ral-equation method to simulate EM fields excited by an energizedertical steel-cased well in a horizontal multilayer subsurface medi-m. The presence of the steel-cased well was modeled with a finite-ength thin wire of arbitrary electric conductivity. Validation exer-ises performed against analytic solutions of EM fields excited by aertical electric dipole in a homogeneous and isotropic unboundededium confirmed the accuracy of the simulation algorithm.Simulation examples indicate that energized steel-cased wells

ignificantly enhance the induced EM fields in the vicinity of theell compared to corresponding fields induced by a vertical electricipole. This enhancement increases the radial length of investigationith respect to that of a vertical electric dipole, thereby enabling therobing of subsurface electric conductivity with crosswell or bore-ole-to-surface measurement-acquisition systems. The most impor-ant parameters that control the distribution of electric current alonghe steel-cased well and, hence, the radial length of investigation inhe subsurface are the electric conductivity of the host rock and theperating frequency.

For large values of casing electric conductivity, our simulationsndicate that the distribution of electric current along the steel-casedell is sensitive to vertical variations of electric conductivity in theost rock. In addition, crosswell source-receiver configurations de-igned to measure the induced magnetic field enable reliable estima-ion of vertical variations of electric conductivity within radial dis-ances of 250 m. The specific limit of detectability of percent electriconductivity variations depends on frequency, host rock electriconductivity, and radial distance from the steel-cased well. Simula-ions indicate that measurable magnetic field signals can be detectedn practice to maximum radial distances of approximately 1000 m at

) b)

igure 10. Numerical simulations performed with the multilayer iso-ropic subsurface model described in Table 1 �30 horizontal layers�.he injected current is 10 A, and the electric current source is locat-d 2.5 m below the surface. �a� Distribution of electric current alonghe vertical steel-cased well for different values of frequency. �b�mplitude of the magnetic field simulated with a crosswell configu-

ation at 1, 10, and 100 Hz, with magnetic receivers located 250 madially away from the steel-cased well.

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requencies below 100 Hz for average host rock electric conductivi-ies below 1 S /m.

Simulation examples performed to quantify the detectability ofeometric and electric properties of a deeply seated target layer withrosswell EM acquisition systems indicate perturbations in the mag-etic-field response of up to 5%. This perturbation is caused by aariation of 20% of the electric conductivity when the target layer isocated 475 m below the surface, the frequency of operation is0 Hz, and the electric conductivity of the overburden layer is.2 S /m. Moreover, the accurate detection of perturbations of tar-et-layer properties �e.g., electric conductivity and thickness� re-uires that the contact point of electric current on the steel-casedell be placed near the target layer. For borehole-to-surface EM ac-uisition systems, perturbations of the same target-layer propertiesause larger variations in the phase than in the amplitude of the totalagnetic field.Detection limits and spatial resolution associated with EM fields

xcited by energized steel-cased wells depend on the specific acqui-ition system, borehole-to-surface or crosswell, the location of thearget layer, and the location of the current source along the steel-ased well. Precise calculation of these parameters also requiresnowledge of frequency range, electric conductivity of the back-round medium, and electric conductivity of the casing. The algo-ithm developed in this study permits the design and appraisal ofource-receiver distances and frequency range specific to a giveneld situation to secure measurable EM fields. Additional work isecessary to validate our conclusions for dipping layers.

ACKNOWLEDGMENTS

The authors express their gratitude to Shell International E&P forponsoring the project and for permission to publish it.Aspecial notef gratitude goes to Jorge Parra, Stan Gianzero, and two anonymouseviewers whose constructive technical and editorial comments im-roved the original manuscript.

APPENDIX A

CALCULATION OF COEFFICIENTSAm„�…, Bm„�…, Am

„i…„�…, and Bm

„i…„�…

The calculation of vector potentials from equations 4 and 5 re-uires prior calculation of the unknown coefficients Am��, Bm��,m�i��, and Bm

�i��. This appendix details the corresponding deriva-ion.

Let a vertical electric dipole be located at �0,0,zi�� in the ith layer.here are i layers above the source layer i and �M � i� layers below

t. In the assumed cylindrical coordinate frame ��,� ,z�, the associat-d vertical components of the vector potentials are given by

�0��,z��Idz

4�0

B0��exp�u0z�J0��d�, �A-1�

� i��,z��Idz

4�0

��

uiexp�uiz�zi��Ai��

� exp��uiz��Bi��exp�uiz��J0��d�

�1 � i � M �1�, �A-2�

�m��,z��Idz

4�0

Am��exp��umz�

�Bm��exp�umz��J0��d�

�m� i,1 � m � i�1,i � m � M �1�, �A-3�

nd

�M��,z��Idz

4�0

AM��exp��uMz�J0��d� .

�A-4�

he corresponding tangential magnetic fields are given by

H��0��

Idz

4�0

B0��exp�u0z��J1��d�, �A-5�

H��i��

Idz

4�0

��

uiexp�uiz�zi��Ai��

� exp��uiz��Bi��exp�uiz��J1��d�,

�A-6�

H��m��

Idz

4�0

Am��exp��umz�

�Bm��exp�umz��J1��d�, �A-7�

nd

H��M��

Idz

4�0

AM��exp��uMz��J1��d� . �A-8�

oundary conditions at z�z1�0 require that H��0��H�

�1�. It thenollows that

0�� i��

u1exp��u1z1��A1��B1��, �A-9�

w

AH

A�

A�

w

H

w

Fc

E

W

W

a

A

w

AE

A�

E172 Yang et al.

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here

� �i��1, i�1

0, i � 1� .

t z�zi�� j�1i�1hj�h1�h2� ¯�hi�1, the condition

��i�1��H�

�i� yields

Ai�1��exp��ui�1zi��Bi�1��exp�ui�1zi�

��

uiexp�uizi�zi��Ai��exp��uizi�

�Bi��exp�uizi� . �A-10�

t z�zi�1�� j�1�i�1��1hj�h1�h2� ¯�h�i�1��1�h1�h2� ¯

hi, the condition H��i��H�

�i�1� yields

�

uiexp�uizi�1�zi��Ai��exp��uizi�1�

�Bi��exp�uizi�1�

�Ai�1��exp��ui�1zi�1��Bi�1��exp�ui�1zi�1� .

�A-11�

t z�zm�� j�1m�1hj�h1�h2� ¯�hm�1, the condition H�

�m�1�

H��m� yields

Am�1��exp��um�1zm��Bm�1��exp�um�1zm�

�Am��exp��umzm��Bm��exp�umzm�, �A-12�

ith

�2�m� i�1� or �i�2�m�M �2�,

�m�2, . . . ,i�1,i�2, . . . ,M �2� .

At z�zM �� j�1M�1hj�h1�h2� . . .�hM�1, the condition

��M�1��H�

�M� yields

� �i��

uM�1exp�uM�1zM �zM�1� ��AM�1��

� exp��uM�1zM��BM�1��exp�uM�1zM�

�AM��exp��uMzM�, �A-13�

here

� �i��1, i�M �1

0, i � M �1� .

or the case of tangential electric fields and corresponding boundaryonditions, it follows that

��0��

1

� 0�

Idz

4�0

B0��u0 exp�u0z��J1��d� . �A-14�

hen z�z��0, for example, z�z , z�z��z�z��, we have
i i i i
E��i��

1

� i�

Idz

4�0

�� exp�uiz�zi��uiAi��

� exp��uiz��uiBi��exp�uiz��J1��d� .

�A-15�

hen z�zi��0, for example, z�zi�1, z�zi��z�zi�, we have

E��i��

1

� i�

Idz

4�0

� exp�uiz�zi��uiAi��

� exp��uiz��uiBi��exp�uiz��

� �J1��d�, �A-16�

E��m��

1

� m�

Idz

4�0

umAm��exp��umz�

�Bm��exp�umz��J1��d�, �A-17�

nd

E��M��

1

� M�

Idz

4�0

uMAM��exp��uMz��J1��d� .

�A-18�

t z�z1�0, the condition E��0��E�

�1� yields

u0

� 0�B0�� i�

�

� 1�exp��u1z��u1A1��u1B1��,

�A-19�

here

� �i��1, i�1

0, i � 1� .

t z�zi�� j�1i�1hj�h1�h2� ¯�hi�1, the condition

��i�1��E�

�i� yields

ui�1

� i�1�Ai�1��exp��ui�1zi��

ui�1

� i�1�Bi�1��exp�ui�1zi��

��

� i�exp�uizi�zi��

ui

� i�Ai��exp��uizi�

�ui

� i�Bi��exp�uizi� . �A-20�

t z�zi�1�� j�1�i�1��1hj�h1�h2� ¯�hi, the condition E�

�i�

E��i�1� yields

�

� i�exp�uizi�1�zi��

ui

� i�Ai��exp��uizi�1�

A�

w

E

w

ScB

B

a

w

a

w


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�ui

� i�Bi��exp�uizi�1�

�ui�1

� i�1�Ai�1��exp��ui�1zi�1�

�ui�1

� i�1�Bi�1��exp�ui�1zi�1� . �A-21�

t z�zm�� j�1�m�1�hj�h1�h2� ¯�hm�1, the condition E�

�m�1�

E��m� yields

um�1

� m�1�Am�1��exp��um�1zm�

�um�1

� m�1�Bm�1��exp�um�1zm�

�um

� m�Am��exp��umzm��

um

� m�Bm��exp�umzm�,

�A-22�

ith

�2 � m � i�1� or �i�2 � m � M �2�,

�m�2, . . . ,i�1,i�2, . . . ,M �2� .

At z�zM �� j�1M�1hj�h1�h2� ¯�hM�1, the condition

��M�1��E�

�M� yields

� �i��

� M�1�exp�uM�1zM �zM�1� �

�uM�1

� M�1�AM�1��exp��uM�1zM�

�BM�1��exp�uM�1zM��

�uM

� M�AM��exp��uMzM�, �A-23�

here

� �i��1, i�M �1

0, i � M �1� .

olving the ensuing system of linear equations yields the coeffi-ients B0��, A1��, B1��, A2��, B2��, . . . ,Am��,m��, . . . ,AM�1��, BM�1��, and AM��.

The corresponding equations are

0�� i��

u1exp��u1z1��A1��B1��, �A-24�

Ai�1��exp��ui�1zi��Bi�1��exp�ui�1zi�

��


�B ��exp�u z �, �A-25�
i i i
�

uiexp�uizi�1�zi��Ai��exp��uizi�1�

�Bi��exp�uizi�1�

�Ai�1��exp��ui�1zi�1��Bi�1��exp�ui�1zi�1�,

�A-26�

Am�1��exp��um�1zm��Bm�1��exp�um�1zm�

�Am��exp��umzm��Bm��exp�umzm�, �A-27�

� �i��

uM�1exp�uM�1�zM �zM�1� ��AM�1��

exp��uM�1zM��BM�1��exp�uM�1zM�


nd

B0�� i��

u1K10 exp��u1z��K10A1��K10B1��,

�A-29�

here K10�K1 /K0. In addition,

Ki�1Ai�1��exp��ui�1zi��Ki�1Bi�1��exp�ui�1zi�

��

uiKi exp�ui�zi�zi��KiAi��exp��uizi�

�KiBi��exp�uizi�, �A-30�

�

uiKi exp�ui�zi�1�zi��KiAi��exp��uizi�1�

�KiBi��exp�uizi�1�

�Ki�1Ai�1��exp��ui�1zi�1�

�Ki�1Bi�1��exp�ui�1zi�1�, �A-31�

Km�1Am�1��exp��um�1zm�

�Km�1Bm�1��exp�um�1zm�

�KmAm��exp��umzm��KmBm��exp�umzm�,

�A-32�

nd

� �i��

uM�1K�M�1�M exp�uM�1�zM �zM�1� ��

�K�M�1�MAM�1��exp��uM�1zM�

�BM�1��exp�uM�1zM��


here K �K /K .
�M�1�M �M�1� M

A

O

w

F

o

w�wi

Z

F

wZK

o

A

FB

w

Fh

L

a

E174 Yang et al.

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From equations A-24 andA-29, we have

1��R10�� i��

u1exp��u1z1��B1�� . �A-34�

n the other hand, from equations A-28 andA-33, we have

BM�1��R�M�1�M��exp��2uM�1zM�

� � �i��

uM�1exp�uM�1zM�1� ��AM�1��,

�A-35�

here

R�M�1�M �K�M�1�M �1

K�M�1�M �1�

KM�1�KM

KM�1�KM

�KM�1�ZM

KM�1�ZM, zM � �

j�1

M�1

hj .

or the ith layer �source layer i�, we have

Ai��Ri�i�1��exp�2uizi�

� � �i��

uiexp��uizi��Bi��, �A-36�

Bi��Ri�i�1��exp��2uizi�1�

� � �i��

uiexp�uizi��Ai�� A-37�

r

Ai��Ri�i�1��exp�2ui �j�1

i�1

hj�� i�

�

uiexp��uizi��Bi��, �A-38�

Bi��Ri�i�1��exp��2ui �j�1

i

hj�� i�

�

uiexp�uizi��Ai��, �A-39�

here Ri�i�1�� Ki�Zi�1� / �Ki�Zi�1�, Ri�i�1�� Ki

Zi�1� / �Ki�Zi�1�, and the quantities Zi�1 and Zi�1 are calculatedith one of two recursion relations. For layers below the source layer

,

j�KjZj�1�Kj tanh�ujhj�Kj�Zj�1 tanh�ujhj�

,

j�M �1,M �2, . . . ,i�1, i�1 � j � M �1. �A-40�

or layers above the source layer i,

Zj�KjZj�1�Kj tanh�ujhj�K �Z tanh�u h �

, j�1,2, . . . ,i�1,
j j�1 j j
1 � j � i�1, �A-41�

here tanh�ujhj�� 1�exp��2ujhj�� / �1�exp��2ujhj��,0�K0� �u0 /� 0��, ZM �KM � �uM /� M� �, andj� �uj /� j��0� j�M�.

By combining equations A-38 andA-39, we obtain

Ai��

uiexp��uizi��Ri�i�1��

�exp�2uizi��Ri�i�1��exp�2ui�hi�zi��

1�Ri�i�1��Ri�i�1��exp��2uihi�,

�A-42�

Bi��

uiexp��uizi��Ri�i�1��exp��2uihi�

�Ri�i�1��exp2ui�zi��zi��

1�Ri�i�1��Ri�i�1��exp��2uihi�

�A-43�

r

i��

uiexp��uizi��Ri�i�1��

�exp�2ui� j�1

i�1hj��Ri�i�1��exp�2ui�hi�zi��

1�Ri�i�1��Ri�i�1��exp��2uihi�,

�A-44�

Bi��

uiexp��uizi��Ri�i�1��exp��2uihi�

�Ri�i�1��exp2ui�zi�� j�1

i�1hj��

1�Ri�i�1��Ri�i�1��exp��2uihi�.

�A-45�

rom Ai��, Bi��, we obtain the corresponding values of A1��,1��, A2��, B2��,. . ., Am��, Bm��,. . ., AM�1��, BM�1��.

From equation A-38, for layers located above the source layer i,e have

Am��Rm�m�1��exp�2umzm�Bm��, 1 � m � i�1.

�A-46�

rom equation A-39, for layers located below the source layer i, weave

Bm��Rm�m�1��exp��2umzm�1�Am��,

�A-47�

i�1 � m � M �1.

et m� i�1. Substituting equation A-46 into equation A-25 gives

Ai�1��R�i�1��i�2��exp�2ui�1zi�1�Bi�1��

�A-48�

nd

T

a

w

L

a

T

B

a

w

td

ftossoffi

We

wt�

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Bi�1��exp�ui�1zi�

R�i�1��i�2��exp�2ui�1zi�1��exp�2ui�1zi�

� ��


�Bi��exp�uizi�� . �A-49�

hus, for upper layers �1 � m � i�1�, we have

Am��Rm�m�1��exp�2umzm�Bm�� A-50�

nd

Bm��exp��umzm�1�

1�Rm�m�1��exp��2umhm�

�� m�1��

um�1exp�um�1zm�1�zm�1� �

�Am�1��exp��um�1zm�1�

�Bm�1��exp�um�1zm�1��, �A-51�

here

� �m�1��1, m�1� i

0, m�1 � i� .

et m� i�1. Substituting equation A-47 into equation A-26 gives

Bi�1��R�i�1��i�2��exp��2ui�1zi�2�Ai�1��

�A-52�

nd

Ai�1��exp�ui�1zi�1�

1�R�i�1��i�2��exp��2ui�1hi�1�

��

uiexp�uizi�1�zi��Ai��

� exp��uizi�1��Bi��exp�uizi�1�� .

�A-53�

hus, for the case of lower layers �i�1 � m � M �1�, we have

m��Rm�m�1��exp��2umzm�1�Am�� A-54�

nd

Am��exp�umzm�

1�Rm�m�1��exp��2umhm�

�� m�1��

uiexp�uizi�1�zi��

�A ��exp��u z �
m�1 m�1 m
�Bm�1��exp�um�1zm��, �A-55�

here

� �m�1��1, m�1� i

0, m�1 � i� .

APPENDIX B

CALCULATING ELECTRIC CURRENT ALONGTHE VERTICAL STEEL-CASED WELL

This appendix derives the expressions used to calculate the dis-ribution of electric current along the steel-cased well. Following theefinition of vector electric potential, one has

Ez�m��,z��i��mAz

�m��1

� hm�

� 2Az�m��,z�� z2

��i��m�1�1

m2

� 2

� z2�Az�m��,z� . �B-1�

Let � �ac, where ac is the radius of the steel-cased well. We en-orce boundary conditions that require the continuity of the tangen-ial components of electric fields at � �ac. From equation B-1, webtain the z-component of the electric field on the surface of theteel-cased well. Because the steel-cased well has a radius muchmaller than its length �ac�L�, we can assume that the z-componentf the electric fields inside the steel-cased well is distributed uni-ormly over its cross section. Thus, the z-component of the electriceld inside the steel-cased well is given by

Ez�m��ac,z�� i�m��1�

1

m2

� 2

� z2��0

L

I�z��G�ac,z,z��dz�.

�B-2�

e now multiply both sides of equation A-2 by � cSa to rewrite thexpression in terms of electric current. This yields

I�z��i�m�� cSa ·�1�1

m2

� 2

� z2��0

L

I�z��G�ac,z,z��dz�,

here I�z� is current intensity, I�z�� cEz�ac,z�Sa, � c is the conduc-ivity of the wire source, and Sa is its cross-sectional area, with Sa

ac2.Algebraic manipulation yields

�0

L

I�z��Gt�ac,z,z��dz�� I�z�, �B-3�

here

a

H

G

G

G

G

G

a

wi

sttamds2

lWlBcvfw

pawcnwzt

wmc

w

Iaft

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Gt�ac,z,z�� i�m�� cSa 1

m2 Gzz�ac,z,z��G�ac,z,z��

�B-4�

nd

Gzz�ac,z,z�� 2G�ac,z,z��

� 2z. �B-5�

ere,

�ac,z,z��

�� Gi�ac,z,zi��,z��zi�,Li � zi� � Li�1,1 � i � m�1

Gm�ac,z,zm� �,z��zm� ,Lm � zm� � Lm�1,i�m

Gi�ac,z,zi��,z��zi�,Li � zi� � Li�1,m�1 � i � M�,

�B-6�

zz�ac,z,z��

�� Gzz�i��ac,z,zi��,z��zi�,Li � zi� � Li�1,1 � i � m�1

Gzz�m��ac,z,zm� �,z��zm� ,Lm � zm� � Lm�1,i�m

Gzz�i��ac,z,zi��,z��zi�,Li � zi� � Li�1,m�1 � i � M

�,

�B-7�

m�ac,z,zm� ��Km

4�0

� �

umexp�umz�zm� �

�Am��exp��umz�

�Bm��exp�umz��J0��d�, �B-8�

zz�m��ac,z,zm� ��

Km

4

� 2

� z2�0

� �

umexp�umz�zm� �

�Am��exp��umz�

�Bm��exp�umz��J0��d�, �B-9�

i�ac,z,zi��Ki

4�0


�i�

� ��exp�umz��J0��d�, �B-10�

nd

Gzz�i��ac,z,zi��

Ki

4

� 2

� z2�0


�i�

� ��exp�u z��J ��d�, �B-11�
m 0
here m�1,2, . . . ,M; i�m,i�1,2, . . . ,m�1;�m�1,m�2, . . . ,M.

Equation B-3 is the integral equation governing the current inten-ity I�z� associated with the wire current source. Inspection of equa-ion B-3 shows that there is only one unknown variable, I�z�. We usehis equation to calculate the vertical distribution of electric currentlong the steel-cased well. To ensure that equation B-3 is well deter-ined, we add one boundary condition by assuming that the current

istribution is known at one of the vertical segments of the wireource �King and Prasad, 1986; Boerner and Qian, 1999; Sadiku,001�.

APPENDIX C

NUMERICAL SOLUTION OF THE DISTRIBUTIONOF ELECTRIC CURRENT ALONG A VERTICAL

STEEL-CASED WELL

This appendix provides technical details about the numerical so-ution of the electric-current distribution along the steel-cased well.

e use the method of moments �MOM; Harrington, 1968� to calcu-ate the vertical distribution of the electric current I�z� from equation-3, assuming knowledge of the electric current along the first verti-al segment of the steel-cased well, I�0�� I0. The calculation in-olves the discretization of the wire source, the selection of the basisunctions used to describe the current, and the definition of theeighting functions.

To segment the steel-cased well, we assume the length of the steelipe is given by L. The corresponding cased-well depth segmentslong the �M �1� horizontal layers are denoted as L1,L2, . . . ,LM�1,ith the corresponding number of depth subintervals along the

ased well given by N1,N2, . . . ,NM�1. Thus, the totalumber of depth subintervals is N�N1�N2� ¯�NM�1,ith the corresponding depth values given by

1,z2, . . . ,zN1�1; zN1�1�1,zN1�1�2, . . . ,zN1�1�N2�1, . . . ,zN�1. We expresshe unknown casing current I�z� as

I�z�� n�1

N

Inun�z�, �C-1�

here N is the total number of basis functions un�z� needed to seg-ent the cased well and In are the corresponding expansion coeffi-

ients. Substituting equation C-1 into equation B-3 gives

�n�1

N

Ingn�ac,z�� I�z�, �C-2�

here gn�ac,z��0Lun�z��Gt�ac,z,z��dz�.

To calculate the unknown current expansion coefficientsn�n�1,2, . . . ,N�, we derive N equations from equation C-2. This ischieved by multiplying equation C-2 by the weighting �or test�unctions Wj�z,zj� � j�1,2, . . . ,N� and by integrating the result overhe cased-well length, i.e.,

�n�1

N

In��0

L

gn�ac,z�Wj�z,zj�dz��0

L

I�z�Wj�z,zj�dz .

�C-3�

Wt

M�

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e select weighting functions Wj�z,zj� equal to a Dirac delta func-ion:

Wj�z,zj�� z�zj��, z�zj

0, z�zj� . �C-4�

oreover, by noting that �0Lgn�ac,z�Wj�z,zj�dz�gn�ac,zj� and

0LIn�z�Wj�z,zj�dz� In�zj�, we obtain

�n�1

N

Ingn�ac,zj�� I�zj�, �C-5�

here j�1,2, . . . ,N �total number of observation points�.For the case of basis functions un�z� equal to pulse functions, i.e.,

un�z��1, zn��z

2�z�zn�

�z

2

0, elsewhere�, �C-6�

t follows that

gn�ac,zj��0

L

un�z��Gt�ac,zj,z��dz�

� �zn��z/2�

zn��z/2�

Gt�ac,zj,z��dz��zB

zA

Gt�ac,zj,z��dz�

r, equivalently,

gn�ac,zj��zB

zA

Gt�ac,zj,z��dz�, �C-7�

here zA�zn� ��z /2� and zB�zn� ��z /2�. Consequently,

gn�ac,zj��zB

zA

Gt�ac,zj,z��dz�

� i�m�� cSa�zB

zA 1

m2 Gzz�ac,zj,z��

�G�ac,zj,z��dz�,

here

gn�ac,zj��i�m�� cSa ·��zB

zA

G�ac,zj,z��dz�

�1

m2 �

zB

zA

Gzz�ac,zj,z��dz�� cSa ·�i�m��

zB

zA


�1

� hm��zB

zA

Gzz�ac,zj,z��dz��, �C-8�

ith

�zB

zA


��zB

zA

Gi�ac,zj,z��dz�,z��zi�,1 � i � m�1

�zB

zA

Gm�ac,zj,z��dz�,z��zm� ,i�m

�zB

zA

Gi�ac,zj,z��dz�,z��zi�,m�1 � i � M��C-9�

nd

�zB

zA

Gzz�ac,zj,z��dz�

��zB

zA

Gzz�i��ac,zj,z��dz�,z��zi�,1 � i � m�1

�zB

zA

Gzz�m��ac,zj,z��dz�,z��zm� ,i�m

�zB

zA

Gzz�i��ac,zj,z��dz�,z��zi�,m�1 � i � M

� .

�C-10�

urther algebraic manipulation yields

�

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a

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I

a

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zB

zA

Gm�ac,zj,z��dz�

�Km

4�0

� �

umQ01�QA1�QB1�J0��d�

�Km

4�0

� �

um�zB

zA

exp�umz�zm� �dz�

�exp��umz��zB

zA

Am��dz�

�exp�umz��zB

zA

Bm��dz��J0��d�, �C-11�

here

Q01��zB

zA

exp�umz�zm� �dz�,

QA1�exp��umz��zB

zA

Am��dz�,

QB1�exp�umz��zB

zA

Bm��dz� �C-12�

nd

�zB

zA

Gzz�m��ac,zj,z��dz�

�� 2

� z2�zB

zA

Gm�ac,z,z��dz�z

�zj�

Km

4

� 2

� z2�0

� �

um�zB

zA

exp�umz�zm� �dz�


zA

Am��dz�


zA

Bm��dz��J0��d�

�Km

4�0

� �

umQ02�QA2�QB2�J0��d�

�1

4�0

� �

umQ02�um

2 · QA1�QB1��J0��d�,

�C-13�

here

Q02�� 2

� z2�zB

zA

exp�umz�zm� �dz�, �C-14�

QA2�� 2

� z2 exp��umz��zB

zA

Am��dz��um2 ·QA1,

QB2�um2 ·QB1.

n a similar manner,

�zB

zA

Gi�ac,zj,z��dz��Ki

4�0


zA

Am�i��dz�


zA

Bm�i��dz��J0��d�

�Ki

4�0

�QA�QB�J0��d� �C-15�

nd

�zB

zA

Gzz�i��ac,zj,z��dz��

� 2

� z2��zB

zA

Gi�ac,z,z��dz��z�zj

�Ki

4�0

um2 �exp��umz��

zB

zA

Am�i��dz�


zA

Bm�i��dz��J0��d�

�Ki

4�0

um2 · �QA�QB�J0��d�,

�C-16�

here

ci

w

w

o

T

w

a

Ws

fit

Fl

a

Te

a

F


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QA�exp��umz��zB

zA

Am�i��dz�, �C-17�

QB�exp�umz��zB

zA

Bm�i��dz�.

The above infinite integrals are calculated with zero-order dis-rete Hankel transforms �Anderson, 1979�; the remaining definitentegrals are computed with standard numerical methods.

By enforcing the known condition of the injected electric current,e obtain

I�zj�z�zjo� I�zjo�� Ijo� I0,

here I0 is the known injected current. Consequently,

�n�1

N

Ingn�ac,zj�� I�zj�� n�1

N

Inun�zj�,

j�1,2,3, . . . ,N,

j� jo,

r

Ijo� I0,

�n�1

N

Ingn�ac,zj��un�zj��0,

j�1,2,3, . . . ,N, j� jo .

his equation can be written in matrix form as

A ·X�B, �C-18�

here

A� �ajn�N�N��a11 a12 . . . a1N

a21 a22 . . . a2N

. . . . . . . . . . . . . . .

aN1 aN2 . . . aNN

�,

X� �xn�N�1��I1

I2

. . .

IN

�,

B� �bj�N�1��0

0

. . .

0�, bj�bjo� I0,

nd

ajn�� un�zj�, j� jo

gn�ac,zj��un�zj�, j�1,2,3,. . . ,N,j� jo� .

e obtain the current distribution along the cased well by solving theystem of linear algebraic equations C-18.

APPENDIX D

ELECTROMAGNETIC FIELDS EXCITEDOUTSIDE THE STEEL-CASED WELL

This appendix derives the expressions used to calculate the EMelds excited by the energized steel-cased well at arbitrary points in

he subsurface. Because of axial symmetry, one has

E��m��0, H�

�m��0, Hz�m��0, m�0,1, . . . ,M .

rom the relationship between vector potential and EM fields, it fol-ows that

H��m��,z��

�Az�m��,z��

,

E��m��,z��

1

� hm�

� 2Az�m��,z�

�� z,

nd

Ez�m��,z��i��mAz

�m��,z��1

� hm�

� 2Az�m��,z�� z2 .

he relationships between components of electric current and totallectric field are given by

J��m��,z��E�

�m��,z�� hm,

Jz�m��,z��Ez

�m��,z�� vm,

nd

J��,z��J��m��2� �Jz

�m��2.

or the case of the magnetic field, one has

H��m��,z��

�Az�m��,z��

��

��0

L

I�z�� ·G��,z,z��dz�

��l�1

N

I�zl��

��zl

zl�1

G��,z,z��dz�

��l�1

N

I�zl� ·G��,z,zl� . �D-1�

F

a

A

A

B

B

H

H

K

K

K

M

N

N

R

S

S

T

W

———W

W

W

X

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or the case of the electric field, one has

E��m��,z��

1

� hm�

� 2Az�m��,z�

�� z

�1

� hm�

� 2

�� z�0

L

I�z��G��,z,z��dz�

�1

� hm��l�1

N

I�zl�� 2

�� z�zl

zl�1


�1

� hm��l�1

N

I�zl�G�z��,z,zl�, �D-2�

J��m��,z��E�

�m��,z�� hm�� hm1

� hm��l�1

N

I�zl��G�z��,z,zl�,

Ez�m��,z��i�m��1�

1

m2

� 2

� z2��0

L

I�z��G��,z,z��dz�

��i�m��1�1

m2

� 2

� z2��l�1

N

I�zl�

� �zl

zl�1


� �l�1

N

I�zl��Gt��,z,zl�, �D-3�

nd

Jz�m��,z��Ez

�m��,z�� vm�� vm�l�1

N

I�zl�Gt��,z,zl� .

�D-4�

REFERENCES

nderson, W. L., 1979, Computer program: Numerical integration of relatedHankel transforms of orders 0 and 1 by adaptive digital filtering: Geophys-ics, 44, 1287–1305.

ugustin, A. M., W. D. Kennedy, H. F. Morrison, and K. H. Lee, 1989, Athe-oretical study of surface-to-borehole electromagnetic logging in casedholes: Geophysics, 54, 90–99.

eguin, P., D. Benimeli, A. Boyd, I. Dubourg, A. Ferreira, A. McDougall, G.Rouault, and P. van der Wal, 2000, Recent progress on formation resistivi-ty measurement through casing: 41st Annual Symposium, Society ofPetrophysicists and Well LogAnalysts, Transactions, Paper CC.

oerner, D. E., and W. Qian, 1999, Physical expansion functions for electro-magnetic integral-equation modeling, in M. Oristaglio, and B. Spies, eds.,Three-dimensional electromagnetics: SEG, 59–75.

arrington, R., 1968, Field computation by moment method: MacmillanPubl. Co.

oversten, G. M., G. A. Newman, H. F. Morrison, E. Gasperikova, and J.-I.Berg, 2001, Reservoir characterization using crosswell electromagneticinversion: A feasibility study for the Snorre field, North Sea: Geophysics,66, 1177–1189.

aufman, A., 1990, The electrical field in a borehole with casing: Geophys-ics, 55, 29–38.

aufman, A., and W. E. Wightman, 1993, A transmission-line model forelectrical logging through casing: Geophysics, 58, 1739–1747.

ing, W. P., and S. Prasad, 1986, Fundamental electromagnetic theory andapplications: Prentice-Hall, Inc.aurer, H. M., and J. Hunziker, 2000, Early results of through casing resis-tivity field tests: 41st Annual Symposium, Society of Petrophysicists andWell LogAnalysts, Transactions, Paper DD.

ekut, A. G., 1995, Crosswell electromagnetic tomography in steel-casedwells: Geophysics, 60, 912–920.

ewmark, R. L., D. William, and A. Rodriguez, 1999, Electrical resistancetomography using steel cased boreholes as electrodes: 69th Annual Inter-national Meeting, SEG, ExpandedAbstracts, 18, 327–330.

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1D Sunsurface Em Fields Excited by Energized Steel Casing

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