Simulation of full responses of a triaxial induction …Simulation of full responses of a triaxial induction tool in a homogeneous biaxial anisotropic formation Ning Yuan1, Xiao Chun

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GEOPHYSICS, VOL. 75, NO. 2 �MARCH-APRIL 2010�; P. E101–E114, 12 FIGS., 3 TABLES.10.1190/1.3336959

imulation of full responses of a triaxial induction tooln a homogeneous biaxial anisotropic formation

ing Yuan1, Xiao Chun Nie1, Richard Liu1, and Cheng Wei Qiu2

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ABSTRACT

Triaxial induction tools are used to evaluate fractured and low-resistivity reservoirs composed of thinly laminated sand-shalesequences. Thinly laminated and fractured reservoirs demon-strate transversely isotropic or fully anisotropic �biaxial aniso-tropic� electrical properties. Compared to the number of studieson transverse isotropy, relatively little work covers biaxial aniso-tropy because of the mathematical complexity. We have devel-oped a theoretical analysis for the full response of a triaxial in-duction tool in a homogeneous biaxial anisotropic formation.The triaxial tool is composed of three mutually orthogonal trans-mitters and three mutually orthogonal receivers. The buckingcoils are also oriented at three mutually orthogonal directions toremove direct coupling. Starting from the space-domain Max-well’s equations, which the electromagnetic �EM� fields satis-fied, we obtain the spectral-domain Maxwell’s equations by de-fining a Fourier transform pair. Solving the resultant spectral-

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omain vector equation, we can find the spectral-domain solu-ion for the electric field. Then, the magnetic fields can be deter-

ined from a homogeneous form of Maxwell’s equations. Theolution for the EM fields in the space domain can be expressedn terms of inverse Fourier transforms of their spectral-domainounterparts. We use modified Gauss-Laguerre quadrature andontour integration methods to evaluate the inverse Fourierransform efficiently. Our formulations are based on arbitrary rel-tive dipping and azimuthal and tool angles; thus, we obtain theull coupling matrix connecting source excitations to magneticeld response. We have validated our formulas and investigated

he effects of logging responses on factors such as relative dip-ing, azimuthal and tool angles, and frequency using our code.e only consider conductivity anisotropy, not anisotropy in di-

lectric permittivity and magnetic permeability. However, ourethod and formulas are straightforward enough to consider an-

sotropy in dielectric permittivity.

INTRODUCTION

Electrical conductivity/resistivity logs provide valuable informa-ion about the porosity and fluid content of rock near a borehole.onventional electric logs determine apparent scalar �or isotropic�onductivity. The conductivity is actually a symmetric and positiveefinite second-rank tensor �Kunz and Moran, 1958�. In the princi-al axis coordinate system, the conductivity tensor diagonalizes as

�� x 0 0

0 � y 0

0 0 � z� . �1�

For an isotropic medium, the conductivity is a scalar, i.e., � x

� y �� z�a; for a transversely isotropic �TI� medium, two of the

Manuscript received by the Editor 7April 2009; revised manuscript receiv1University of Houston, Department of Electrical and Computer Engineeri2National University of Singapore, Department of Electrical and Compute2010 Society of Exploration Geophysicists.All rights reserved.

hree principal conductivities are equal. On the scale of logging mea-urements, thin-bedded sand-shale sequences frequently exhibitransverse isotropy, i.e., � x�� y. If a layered medium has a fractureattern that cuts across bedding, the conductivity is fully anisotrop-c. Full anisotropy is referred to as biaxial anisotropy in crystals. Inhis case, all three principal conductivities are different, representinghe differences of pore connectivity and conductivity in vertical andateral directions.

Traditional induction tools have only coaxial transmitter-receiv-r coils and measure one magnetic-field component at different re-eiver locations. To characterize anisotropic conductivity, conven-ional induction-logging methods must be extended to providedditional information. Multicomponent induction-logging toolsKriegshäuser et al., 2000; Anderson et al., 2002; Rosthal et al.,003; Zhang et al., 2004; Rabinovich et al., 2006; Wang et al., 2006;

tember 2009; published online 5April 2010.ston, Texas, U.S.A. E-mail: [email protected]; [email protected]; [email protected], Singapore. E-mail: [email protected].

EG license or copyright; see Terms of Use at http://segdl.org/

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abinovich et al., 2007; Davydycheva et al., 2009� are designed tobtain formation anisotropy. The rich information provided by mul-icomponent induction measurements determines complex forma-ions such as biaxial anisotropic media.

In this paper, we consider a triaxial induction tool that includeshree orthogonal transmitters and three orthogonal receivers, ashown in Figure 1 �Anderson et al., 2002�. The study of the impact ofnisotropy on the tool’s response is important for the correct inter-retation of measurements. Among various studies on electrical an-sotropy, most studies assumed a TI or uniaxial medium for conve-ience �Chetaev, 1966; Althausen, 1969; Moran and Gianzero,979, 1982; Chemali et al., 1987; Lüling et al., 1994; Anderson et al.,995; Wang et al., 2003; Zhang et al., 2004; Wang et al., 2006; Leend Teixeira, 2007; Zhong et al., 2008�. Although transverse isotro-y is a reasonable approximation based on stratigraphic geometry,he assumption is made primarily for mathematical convenienceather than for its low frequency of occurrence in nature because

axwell’s equations can be solved analytically, leading to relativelyimple formulas for a TI medium.

The likelihood of encountering biaxial anisotropy in sedimentaryocks has been reported by Sawyer et al. �1971�, Zafran �1981�, andhao et al. �1994�, and interest in studying tool response in a biaxial

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igure 1. �a� Basic structure of a triaxial induction tool and �b� itsquivalent dipole model.


nisotropic medium is increasing. However, because of mathemati-al complexity, there are few theoretical studies on biaxial anisotro-y �Nekut, 1994; Gianzero et al., 2002�. Nekut �1994�, in the firstheoretical work on biaxial anisotropy, considers the response of aypothetical time-domain instrument with a zero-spacing transmit-er and receiver in a biaxial medium. Gianzero et al. �2002� study theesponse of a triaxial induction instrument in a biaxial anisotropicedium. Different from Nekut’s work, Gianzero et al. �2002� con-

ider nonzero spacing between the three orthogonal transmitters andhree orthogonal receivers. Also, their analysis is in the frequencyomain instead of the time domain. The work of Gianzero et al. layshe foundation to the response of triaxial induction logging tools in aiaxial anisotropic formation. However, they only consider a specialase where the instrument is oriented parallel to the principal axes ofbiaxial medium. In practice, the instrument can be oriented arbi-

rarily with respect to the principal axes of the biaxial medium, com-licating the forward-modeling problem.

Here, we study the response of a triaxial induction sonde in a moreeneral case where the coil axes of the instrument are arbitrarily ro-ated and/or tilted with respect to the conductivity tensor principalxes of the biaxial anisotropic medium. We further extend the meth-d of Gianzero et al. �2002� to derive the formulas for computing theagnetic-field responses. The full coupling matrix connecting the

ource excitations to the magnetic-field response is presented, andhe critical numerical methods are discussed. Numerical examplesre presented to validate formulations and numerical evaluation.he sensitivity of tool responses to factors such as dipping, azimuth-l and tool angles, and frequency are also investigated.

FORMULATION

In this section, we will derive the magnetic-field response of a tri-xial tool in a biaxial anisotropic medium and discuss the evaluationf the inverse Fourier transform.

pectral-domain solution to Maxwell’s equations in aomogeneous biaxial anisotropic medium

A homogeneous, biaxial, unbounded medium can be character-zed by the tensor conductivity defined in equation 1 �expressed inhe principal axis system�.Assuming the harmonic time dependenceo be e�i�t �suppressed throughout our paper�, Maxwell’s equationsor the electric and magnetic fields are

� �H�r�� i��E�r��Js�r�, �2a�

� �E�r�� i��0H�r�� i��0Ms�r�, �2b�

here �0 is the magnetic permeability of the air, r� �x,y,z� is theosition vector, � is the dielectric constant tensor, Ms�r� is the mag-etic-source flux density, and Js�r� is the electric source current den-ity.

For logging devices operating at relatively low frequencies andormations with conductivities greater than 10�4 S /m, we can as-ume that contributions from displacement current determined by�� can be ignored in comparison with �. For the induction loggingroblems considered in this paper, we assume that Js�r��0, whicheans only magnetic dipoles are used to represent induction coils.herefore, Maxwell’s equations are reduced to


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� �H�r��E�r�, �3a�

� �E�r�� i��0H�r�� i��0Ms�r� . �3b�

he solution for equation 3 in the space domain can be expressed inerms of triple Fourier transforms of their spectral-domain counter-arts E�k� and H�k�:

E�r�,H�r��1

�2��3�k

dKeiK·rE�K�,H�K�, �4�

here K� �� ,� ,�� and where

E�K�,H�K��r

dre�iK·rE�r�,H�r� �5�

nd

�k

dKeiK·r��

��

��

d� d�d�ei�� x��y��z�. �6�

hus, for mathematical convenience, we first solve equation 3 in thepectral domain and then use equation 4 to obtain the space-domainolutions from their spectral-domain counterparts.

For the spectral-domain solutions E�K� and H�K�, we can firstliminate the magnetic fields using electric fields in equation 3a andolve for E�K� from equation 3b in the presence of the source Ms�r�.hen, because the source singularity has been totally accounted for

n this solution, the magnetic fields can be determined from a homo-eneous form of Maxwell’s equations.

Substituting equation 3a and 3b results in the following vectorave equation:

� � ·E�r��2E�r��k2E�r�� i�� Ms�r�, �7�

here k2� i��0�. Applying the triple Fourier transform defined inquation 6 and equation 7, we obtain

��K� · E�K�� i�� Ms�K�, �8�

here the coefficient matrix � is given by

��kx2� ��2��2� � � � �

� � ky2� �� 2��2� ��

� � �� kz2� �� 2��2�

� .

�9�

hen, the solutions for the space-domain fields are

E�r��i��

�2��3�K

dKeiK·r��1�K�� Ms�K� . �10�

he inverse matrix ��1 can be computed in terms of its adjoint andeterminant as

��1�

det �. �11�

et �ij�i�1,2,3; j�1,2,3� denote element �i,j� in the inverse ma-rix. The elements are found to be


�11��ky

2� �� 2��2��kz2� �� 2��2��2�2

det �,

�12a�

�12��21�� kz

2� �� 2��2��2��det �

, �12b�

�13��31�� ky

2� �� 2��2��2��det �

, �12c�

�22��kx

2� ��2��2��kz2� �� 2��2�� 2�2

det �,

�12d�

�23��32��kx

2� �� 2��2��2��det �

, �12e�

�33��kx

2� ��2��2��ky2� �� 2��2�� 2�2

det �.

�12f�

he determinant of the coefficient matrix can be written in the fol-owing factored form:

det ��kz2��2��o

2��2��e2�, �13�

here �o and �e are the axial wavenumbers of the ordinary and ex-raordinary modes of propagation. The two distinct modes of propa-ation can be found to be �Appendix A�

�o,e2 �a��b, �14�

here

a�kz

2�kx2�ky

2�� 2�kx2�kz

2��2�ky2�kz

2�2kz

2 �15�

nd

b�a2�� 2��2�kz

2�� 2kx2��2ky

2�kx2ky

2�kz

2 . �16�

he positive and negative square roots correspond to �e and �o, re-pectively. Our derivation �equations 7–16� follows that of Gianzerot al. �2002� but modifies all typographical errors. Detailed deriva-ion of a and b can be found inAppendix A.

Once the space-domain electric field is obtained from equation0, the corresponding magnetic field can be determined from theource-free Maxwell equations.

ull magnetic-field response of triaxial induction sonden biaxial anisotropic medium

In this section, we derive the full magnetic-field response of a tri-xial induction sonde in a biaxial anisotropic medium. The basictructure of a triaxial tool is shown in Figure 1a, consisting of oneroup of transmitter coils, one group of bucking coils, and one groupf receiver coils. All transmitter, bucking, and receiver coils are ori-nted in three mutually orthogonal directions. In the analysis, theoils are assumed to be sufficiently small and replaced by point mag-


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etic dipoles in the modeling. Thus, the magnetic-source excitationf the triaxial tool can be expressed as M� �Mx,My,Mz�� r�, ashown in Figure 1b.

For each component of the transmitter moments Mx, My, and Mz,here generally are three components of the induced field at eachoint in the medium. Thus, there are nine field components at eacheceiver location. These field components can be expressed by a ma-

rix representation of a dyadic H as

H��Hxx Hxy Hxz

Hyx Hyy Hyz

Hzx Hzy Hzz�, �17�

here the first subscript corresponds to the transmitter index and theecond corresponds to the receiver index. Therefore, Hij denotes theagnetic field received by the j-directed receiver coil excited by the

-directed transmitter coil.Next, we derive the expressions for the nine magnetic-field com-

onents in a homogeneous biaxial medium.

agnetic-field components generated by a unit x-directedagnetic dipole M� �1,0,0�T

For an x-directed magnetic dipole M� �1,0,0�T located at r�

�x�,y�,z��, equation 8 can be rewritten as

��K� ·�Exx�K�

Exy�K�

Exz�K��0e�i�� x��y��z�� 0

�

�� .

�18�

olving equation 18, we can get

�Exx�K�

Exy�K�

Exz�K��0e�i�� x��y��z��12��13

��22��23

��32��33� .

�19�

he corresponding components of the magnetic field can be deter-ined from the source-free Maxwell equations:

Hx�1

��0�� Ez��Ey�, �20a�

Hy �1

��0��Ex�� Ez�, �20b�

Hz�1

��0�� Ey �� Ex� . �20c�

irect substitution of equation 19 into equation 20 yields


�Hxx�K�

Hxy�K�

Hxz�K��e�i�� x��y��z��

·��32��33��22��23��12��13�� 32��33�� 22��23��12��13�

� .

�21�

hen, the magnetic-field components in the space domain can be ob-ained from their spectral-domain counterparts in equation 21 by ap-lying inverse Fourier transforms defined in equation 6:

xx�r��1

�2��3��

��

��

d� d�d�

· ei� �x�x��ei� �y�y��ei��z�z��2��23��2�33

��2�22�, �22�

xy�r��1

�2��3��

��

��

d� d�d�ei� �x�x��ei��y�y��

· ei��z�z��12��13�� 32��33��,

�23�

xz�r��1

�2��3��

��

��

d� d�d�ei� �x�x��

· ei� �y�y��ei��z�z�� 22��23��12

��13�� . �24�

As can be seen from equations 22–24, triple infinite integrals of x,y, and z are involved in the solution. In the numerical evaluation, aylindrical transformation in the wavenumber space is invoked. Letbe the rotation angle in the � ��-plane, and we have

� �k cos � , �25a�

� �k sin � , �25b�

��

��

��

d� d�d� ��0

2�

d��0

kdk��

d� . �25c�

hus, equations 22–24 can be rewritten as

Hxx�r��1

�2��3�0

2�

d��0

kdk��

d�

· eik cos � �x�x��eik sin � �y�y��ei��z�z��

· �2k� sin � �32�k2 sin2 � �33��2�22�,

�26�


Mm

ff

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Mm

�

T


Hxy�r��1

�2��3�0

2�

d��0

kdk��

d�

· eik cos � �x�x��eik sin � �y�y��ei��z�z��

· ��2�12�k� sin � �13�k� cos � �32

�k2 cos � sin � �33�, �27�

Hxz�r��1

�2��3�0

2�

d��0

dkk2��

d�

· eik cos � �x�x��eik sin � �y�y��ei��z�z�� · �k sin2 � �13

�k sin � cos � �23�� sin � �12�� cos � �22� .

�28�

agnetic-field components generated by a unit y-directedagnetic dipole M� �0,1,0�T

For a y-directed magnetic dipole M� �0,1,0�T at r�� x�,y�,z��,ollowing a similar derivation procedure, we can obtain the solutionor the space-domain magnetic field components as follows:

Hyx�r��1

�2��3��

��

��

d� d�d�

· ei� �x�x��ei� �y�y��ei��z�z�� 33��31�

�� 23��21��, �29�

Hyy�r��1

�2��3��

��

��

d� d�d�

· ei� �x�x��ei� �y�y��ei��z�z��

· �2�� 13��2�11�� 2�33�, �30�

Hyz�r��1

�2��3��

��

��

d� d�d�

· ei� �x�x��ei� �y�y��ei��z�z�� 23��21�

�� 13��11�� . �31�

quations 32–34 are actually used in the numerical evaluation byransforming the Cartesian coordinates into cylindrical coordinates:

Hyx�r��1

�2��3�0

2�

d��0

dkk��

d�eik cos � �x�x��

· eik sin � �y�y��ei��z�z�� · ��2�21�k� sin � �31

�k� cos � �23�k2 cos � sin � �33�, �32�

Hyy�r��1

�2��3�0

2�

d��0

dkk��


· eik sin � �y�y��ei��z�z��2k� cos � �13

�k2 cos2 � � ��2� �, �33�
33 11

Hyz�r��1

�2��3�0

2�

d��0

dkk2��


· eik sin � �y�y��ei��z�z�� · �� sin � �11�� cos � �21

�k cos � sin � �13�k cos2 � �23� . �34�

ote that Hyx has the same expression as Hxy. This is because of theeciprocity of the medium. We see in the following equations that Hxz

nd Hzx as well as Hyz and Hzy have the same expression because ofeciprocity.

agnetic-field components generated by a unit z-directedagnetic dipole M� �0,0,1�T

Similarly, for a z-directed magnetic dipole M� �0,0,1�T at r��x�,y�,z��, the magnetic fields in the space domain are

Hzx�r��1

�2��3��

��

��

d� d�d�

· ei� �x�x��ei� �y�y��ei��z�z��31�� 32�

��21�� 22��, �35�

Hzy�r��1

�2��3��

��

��

d� d�d�ei� �x�x��ei� �y�y��

· ei��z�z��11�� 12�� 31�� 32��,

�36�

Hzz�r��1

�2��3��

��

��

d� d�d�

· ei� �x�x��ei� �y�y��ei��z�z�� 21�� 22�

��11�� 12�� . �37�

ransforming the integral variables � and � into k and � , we have

Hzx�r��1

�2��3�0

2�

d��0

dkk2��


· eik sin � �y�y��ei��z�z�� · �k sin2 � �31

�k sin � cos � �32�� sin � �21�� cos � �22�,

�38�

Hzy�r��1

�2��3�0

2�

d��0

dkk2��


· eik sin � �y�y��ei��z�z�� · �� sin � �11�� cos � �12

�k cos � sin � �31�k cos2 � �32�, �39�

Hzz�r��1

�2��3�0

2�

d��0

dkk3��



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· eik sin � �y�y��ei��z�z�� · �2 sin � cos � �12

�cos2 � �22�sin2 � �11� . �40�

n fact, for the case where the instrument’s transducer axes areligned parallel to the principal axes of the conductivity tensor �i.e.,ll dipping angles, azimuth angles, and tool angles are zero, ��

� �0°�, all cross-coupling terms are zero. However, in general,he axes of the instrument are not parallel to the principal axes of theonductivity tensor, and the nondiagonal terms in the coupling ma-rix are not zero. Therefore, it is necessary to find out the full cou-ling matrix in a more general case.

ull magnetic-field response with arbitrary tool axis

In practice, the orientation of the transmitter and receiver coils arerbitrary with respect to the principal axes of the formation’s con-uctivity tensor. In this section, we consider the magnetic-field re-ponse of a triaxial induction tool in a homogeneous biaxial aniso-ropic medium with an arbitrarily oriented tool axis.

Figure 2 �Zhong et al., 2008� shows the formation coordinate sys-em described by �x,y,z� and a sonde coordinate system described byx�,y�,z��. In Figure 2, , � , and � denote the dipping, azimuthal,nd orientation angles, respectively.Angle is the relative deviationf the instrument axis z� with respect to the z-axis of the conductivityensor.Angle � is the angle between the projection of the instrumentxis z� on the surface of the x-y-plane and the x-axis of the formationoordinate. Angle � represents the rotation of the tool around the�-axis.

The formation bedding �unprimed� frame can be related to theonde �primed� frame by a rotation matrix R, given by

��R11 R12 R13

R21 R22 R23

R31 R32 R33

��cos cos � cos � �sin � sin � �cos cos � sin � �sin � cos � sin cos �

cos sin � cos � �cos � sin � �cos sin � sin � �cos � cos � sin sin �

�sin cos � sin sin � cos � .

�41�

zzy

x

Ry Rz

R x

O y

T yTz

T x

x

�

�

�

igure 2. Aschematic of the formation coordinate system �x,y,z� andonde coordinate system �x ,y ,z �.
� � �

o find the magnetic-field response in the sonde system, the magnet-c moments of the transmitter coils in the sonde coordinates are firstransformed to effective magnetic moments in the formation coordi-ates by the rotation matrix. Then, the magnetic fields in the forma-ion coordinates excited by the magnetic moments M can be ob-ained readily by

H�HM, �42�

here H is the dyadic corresponding to unit dipole source given byquation 17. Once the magnetic fields at the location of the receiveroils in the formation system are determined, the magnetic fields re-eived at the receiver coils in the sonde system can be obtained bypplying the inverse of the rotation matrix �the rotation matrix is or-hogonal; therefore, its inverse is equal to its transpose�. Conse-uently, the coupling between the magnetic-field components andhe magnetic dipoles in the sonde system are given by �Zhdanov etl., 2001�

H��RTHR . �43�

omputing the triple integrals

In the previous section, we obtained the expressions for all nineomponents of the magnetic fields. As can be seen from equations6–28, 32–34, and 38–40, to compute the field quantities, we have toalculate integrals over k, � , and � in the numerical evaluation. Thentegral over � is a definite integral, so any numerical integral meth-d is applicable. For the semi-infinite integral over k, we use a modi-ed Gauss-Laguerre quadrature �Burkardt, 2008�. The order ofauss-Laguerre quadrature is determined mainly by the dipping an-le. Figure 3 shows the relative error of the magnetic field �imagi-ary part� as a function of the Gauss-Laguerre quadrature order forifferent dipping angles. As the dipping angle increases, a larger or-er of Gauss-Laguerre quadrature is required to achieve sufficientccuracy. When the dipping angle is 0°, 60°, 85°, and 89°, Gauss-La-uerre quadrature needs 16, 16, 48, and 90 points to guarantee theelative error smaller than 0.4%.

For the infinite integral of �, the integrands become highly oscilla-ory as � increases, so special integration methods must be consid-red. Here, the integration over � is performed using contour integra-ion.

From equation 13, we can see that the integrands in equations6–28, 32–34, and 38–40 have four poles on the axial wavenumberlane: ��o and ��e. The four poles correspond to the two eigen-odes for forward and backward propagation, describing the two

olarizations of the electromagnetic wave in the anisotropic medi-m. Assume �o

� ��e�� represents the poles between �o ��e� and

�o��e� whose imaginary part is greater than zero. For�z� � 0, one obtains contributions only from two poles at �o

� and

e�; for z�z� � 0, the contributions are from two poles at ��o

� and�e

�.Using the contour integration �Zhang and Qiu, 2001� for �, the re-

ult of the integration over � in equation 26 is


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d�ei��z�z��2k� sin � �32�k2 sin2 � �33��2�22�

�2� i · ei��z�z�� · �2k� sin � �32� �k2 sin2 � �33� ��2�22� �

kz2�� o

�� e�� e

��

��o�

� ei��z�z�� · �2k� sin � �32� �k2 sin2 � �33� ��2�22� �

kz2�� o

�� o�� e

��

��e�� , z�z��0,

�44��

d�ei��z�z��2k� sin � �32�k2 sin2 � �33��2�22�

�2� i · ei��z�z�� · �2k� sin � �32� �k2 sin2 � �33� ��2�22� �

kz2�� o

�� e�� e

��

��o�

� ei��z�z�� · �2k� sin � �32� �k2 sin2 � �33� ��2�22� �

kz2�� o

�� o�� e

��

��e�� , z�z��0,

�45�

here �32� , �33� , and �22� are the numerators of �32, �33, and �22, re-pectively.

Then equation 26 can be rewritten as

Hxx�r��i

�2� �2�

0

2�

d��0

kdk

· ei��z�z�� · �2k� sin � �32� �k2 sin2 � �33� ��2�22� �

kz2�� o

�� e�� e

��

��o�

� ei��z�z�� · �2k� sin � �32� �k2 sin2 � �33� ��2�22� �

kz2�� o

�� o�� e

��

��e�� , z�z��0,

�46�

xx�r��i

�2� �2�

0

2�

d��0

kdk

· ei��z�z�� · �2k� sin � �32� �k2 sin2 � �33� ��2�22� �

kz2�� o

�� e�� e

��

��o�

� ei��z�z�� · �2k� sin � �32� �k2 sin2 � �33� ��2�22� �

kz2�� o

�� o�� e

��

��e�� , z�z� � 0.

�47�

he integration over � in equations 27, 28, 32–34, and 38–40 can beerformed by following the same procedure. The integrands are noteparable functions of k, � , and �; therefore, the integrals over k, � ,nd � cannot be performed independently.

NUMERICAL RESULTS AND DISCUSSION

In this section, we present numerical examples calculated usinghe Fortran code based on the present theory.

xample 1

First, we compare the apparent resistivity obtained from theresent code with the available data given by Gianzero et al. �2002�.or consistency with the reference, the TRI2C40 triaxial tool �dis-

0 10 20 30 40 50 60 70 80 901E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

Relativeerror

Order of Gauss-Laguerre quadrature

Hxx

Hyy

Hzz

)

0 10 20 30 40 50 60 70 80 90 1001E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

Relativeerror


Hxx

Hyy

Hzz

Hxy

Hxz

Hyz

)

20 30 40 50 60 70 80 90 1001E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

Relativeerror


Hxx

Hyy

Hzz

Hxy

Hxz

Hyz

)

igure 3. Relative errors as a function of Gauss-Laguerre quadraturerder for different dipping angles: �a� �0°, �b� �60°, �c�


tc

tatafifiame

cTas4rfu7sbp

t�

bm

HtCptas

tm

tchtc

ats

E

ea

dqaala

�scgff

E

t3�

T

2

2

4

E108 Yuan et al.

ance between transmitters and receivers is 40 inches; no buckingoils� is used. The operating frequency is assumed to be 20 KHz.

The voltage responses at 20 KHz for one coaxial �2C40zz� andwo mutually perpendicular transverse, coplanar �2C40xx,2C40yy�rrays were converted to units of apparent conductivity.As we know,he quadrature component of the magnetic field �R-signal� is gener-ted by currents in the formation, and the in-phase component of theeld �X-signal� is not easy to observe because of the large primaryeld. So the signal in the air is subtracted from the signal in the biaxi-l formation before being converted to apparent resistivity. The for-ula presented by Wang et al. �2006� is used to calculate the appar-

nt resistivity.In Tables 1–3, a small range of resistivity tensor principal values is

hosen and the corresponding apparent resistivity from theRI2C40 triaxial tool is presented. The first three columns — �x, �y,nd �z — in each table represent the principal components of the re-istivity tensor of the biaxial formation. In Tables 1 and 2, columns–6 present the R-signal apparent resistivity �R, X-signal apparentesistivity �X, and percentage difference between �R and �z obtainedrom our method. The percentage difference %diff �z is computedsing %diff �z� ��R��z� /�z �Gianzero et al., 2002�. In columns–9, the results in Gianzero et al. �2002� are presented for compari-on. The percentage difference between �R and �z should be positiveecause of its definition; therefore, the minus sign in Gianzero’s pa-er is a typo that is corrected here.

In Table 3 for 2C40zz, there is an additional column �column 4�hat gives the geometric mean of the horizontal resistivities �g

��x�y. Columns 5–7 show �R, �X, and the percentage differenceetween �R and �g �%diff �g� ��R��g� /�g� obtained from ourethod. Columns 8–10 give the results in Gianzero et al. �2002�.Comparison of �z and �R in Tables 1 and 2 implies that the Hxx and

yy coupling are primarily sensitive to �z. This is mostly evident inhe highly resistive media where skin effect has minimum impact.omparison of �R with �g in Table 3 indicates that the Hzz coupling isrimarily sensitive to the geometric mean of the horizontal resistivi-ies. It suggests that a biaxial medium characterized by �x and �y ispproximately equivalent to a uniaxial medium characterized by aingle �h��g, confirming Worthington’s �1981� conjecture. Fur-

able 1. Response of a 2C40xx sonde in a biaxial anisotropic f

�x �y �z

Results from prese

�R �X

2 2 8 14.89 22.1

2 4 8 10.51 39.2

4 4 8 10.47 39.6

20 20 80 95.18 590.3

20 40 80 87.40 1091.5

40 40 80 87.32 1105.4

200 200 800 820.85 14,092

200 400 800 806.80 29,377

400 400 800 806.40 29,873

000 2000 8000 8003.41 3,954,041

000 4000 8000 8000.83 1,240,184

000 4000 8000 8000.74 1,221,581


hermore, the R-signal apparent resistivity becomes a better approxi-ation of transverse resistivity as resistivity increases.From these tables, we can see that for formations with low resis-

ivities, the apparent resistivities obtained from our method are verylose to those obtained by Gianzero et al. �2002�; for formations withigh resistivity, the apparent resistivities from our method are closero the expected one. The discrepancies between the methods may beaused by the evaluation of the inverse Fourier transforms.

Gianzero et al. �2002� only consider the case where the instrumentxes are parallel to the principal axes of the conductivity tensor;here are no available data for comparison of mutual coupling re-ponses.

xample 2

To further validate the present method and quantify the numericalrrors of the program, we consider isotropic and TI cases where ex-ct solutions are available.

First, we consider a homogeneous isotropic medium with a con-uctivity of 500 mS /m. The relative dipping angle is 45° and the fre-uency is 20 KHz. We change the spacing between the transmitternd receiver coils from 10 inches to 60 inches. Figure 4a shows thexial component Hzz obtained from the present code and the exact so-ution. Although totally different methods are used, the two resultsre almost the same, validating our method.

Then, we consider a homogeneous TI medium with � x�� y

500 mS /m and � z�125 mS /m. The relative dipping angle istill 45° and the frequency is 20 KHz. Figure 4b shows the axialomponent Hzz obtained from the present code and the exact solutioniven by Moran and Gianzero �1979�. Again, the results obtainedrom the two solutions are almost the same. The discrepancy beginsrom the sixth digit after the decimal point.

xample 3

After validating the derived formulation and the code, we studyhe sensitivity of a real practical three-coil tool that is similar to theD ExplorerSM tool jointly developed by Baker Atlas and ShellKriegshäuser et al., 2000; Rabinovich et al., 2006�. The tool com-

ion.

hod Results in Gianzero et al. �2002�

%diff �z �R �X %diff �z

86.1 14.89 23.34 86.1

31.4 10.51 39.68 31.4

30.9 10.47 40.14 30.9

19.0 93.97 583.84 17.5

9.25 86.6 1108.7 8.25

9.15 86.5 1123.2 8.13

2.61 839.54 17,433 4.94

0.85 819.77 33,754 2.47

0.8 819.49 34,209 2.44

0.043 8118.4 534,262 1.48

0.0104 8059.4 1,040,553 0.74

0.00925 8058.5 1,053,855 0.73

ormat

nt met

7

4

8

3

6

3


pfd

wtt�as

�

iIttTfol

ct

T

2

2

4

T

2

2

4


rises one transmitter and two receivers, respectively, at 1 and 1.5 mrom the transmitter. The tool response �a �apparent conductivity� isefined as

�a�� axx � a

xy � axz

� ayx � a

yy � ayz

� azx � a

zy � azz �, �48�

here � aij is the apparent conductivity of the jth receiver when the ith

ransmitter is excited. The tool response �a is a function of the rela-ive dip angle , azimuth angle � , and conductivity at each direction

x, � y, and � z. The sensitivity of �a to the dip angle, azimuth angle,nd conductivities in each direction are the derivatives of the re-ponse with respect to

��a

� �mS/m/°�,

��a

��mS/m/°�,

able 2. Response of a 2C40yy sonde in a biaxial anisotropic f

�x �y �z

Results from presen

�R �X

2 2 8 14.889 22.17

2 4 8 14.388 23.00

4 4 8 10.471 39.68

20 20 80 95.183 590.57

20 40 80 94.579 616.93

40 40 80 87.319 1106.3

200 200 800 820.85 14,230

200 400 800 818.30 14,842

400 400 800 806.40 30,498

000 2000 8000 8003.4 1,064,692

000 4000 8000 8002.7 979,117

000 4000 8000 8000.7 664,477

able 3. Response of a 2C40zz sonde in a biaxial anisotropic f

�x �y �z �g��x�y

Results from

�R

2 2 8 2.00 2.307

2 4 8 2.83 3.200

4 4 8 4.00 4.422

20 20 80 20.00 20.836

20 40 80 28.28 29.33

40 40 80 40.00 41.272

200 200 800 200.00 203.30 1

200 400 800 282.84 286.40 1

400 400 800 400.00 403.10 2

000 2000 8000 2000.00 2000.89 81

000 4000 8000 2828.43 2829.19 3,56

000 4000 8000 4000.00 4000.45 5,08


��a

�� x,

��a

�� y, and

��a

�� z.

Consider a homogeneous biaxial formation with � x

500 mS /m, � y �250 mS /m, and � z�125 mS /m. The sensitiv-ty of the triaxial tool to , � , � x, � y, and � z is shown in Figures 5–9.n each figure, the horizontal axis is the relative dipping angle andhe vertical axis is the azimuth angle. The color represents the sensi-ivity. Figure 5 shows sensitivity functions for all nine components.he cross pairs xy /yx, xz /zx, and yz /zy have the same sensitivity

unction in a homogeneous formation. Therefore, in Figures 6–9, wenly show six components — xx, xy, xz, yy, yz, and zz — for lengthimitation.

From these figures, we can observe two things. First, apparentonductivity is more sensitive to the dipping and azimuth angle thano formation conductivities. Second, the sensitivity of the cross-cou-

ion.

od Results in Gianzero et al. �2002�

%diff �z �R �X %diff �z

86.1 14.889 22.343 86.1

79.9 14.378 23.185 79.7

30.9 10.470 40.144 30.9

18.9 93.971 583.84 17.5

18.2 93.290 608.32 16.6

9.15 86.502 1123.2 8.13

2.61 839.54 17,433 4.94

2.29 837.81 18,184 4.73

0.8 819.49 34,209 2.44

0.043 8118.4 534,262 1.48

0.034 8113.3 556,384 1.42

0.00875 8058.5 1,053,855 0.73

ion.

nt method Results in Gianzero et al. �2002�

%diff �g �R �X %diff �g

47 15.4 2.308 17.346 15.4

35 13.1 3.200 27.730 13.2

38 10.6 4.419 46.859 10.5

6 4.18 20.888 493.15 4.44

6 3.71 29.37 798.23 3.84

5 3.18 41.241 1375.20 3.10

6 1.65 202.73 15,087.83 1.36

0 1.26 286.19 24,513.56 1.18

7 0.775 403.84 42,482.95 0.961

0.045 2008.61 471,693.0 0.430

0.027 2839.05 759,772.4 0.376

0.0113 4012.26 1,291,113 0.306

ormat

t meth

1

4

3

ormat

prese

�X

17.3

27.8

47.1

509.2

862.2

1571.5

1,540.1

7,539.7

8,849.3

7,171.2

1,097.7

9,149


pp

E

ot�

au2�tscpb

mf

mr

sqwpt�

ppt

a

a

b

Fp�

a

b

Fdf

E110 Yuan et al.

lings xz /zx and yz /zy are comparable to that of the diagonal cou-ling while the cross-couplings xy and yx are less sensitive.

xample 4

Finally, we investigate the effects of frequencies on the responsesf the same three-coil tool in example 3. For clarity, we use resistivi-y instead of conductivity in this example. We consider two cases:1� resistive formation and �2� conductive formation.

For the resistive case, we assume the resistivities of the formationre �x�200 ohm-m, �y �400 ohm-m, and �z�800 ohm-m. Fig-re 10 shows the apparent resistivity as frequency increases from0 to 220 KHz when �� 0°. At a low frequency20 KHz�, the transverse components �a

xx and �ayy are directly propor-

ional to �z and can reproduce �z. Further, �axx and �a

yy exhibit muchtronger skin effect than the conventional coaxial component �a

zz. Toompensate for this effect, measurements at lower frequencies arereferred. For higher frequencies, data at multiple frequencies muste acquired and a multifrequency skin-effect correction technique

10 20 30 40 50 60

0.01

Spacing (inches)

0.1

1

10

Hzz(A/m)

Real (H ), present codezz

Imag (H ), present codezz

Real (H ), exact solutionzz

Imag (H ), exact solutionzz

)

0 10 20 30 40 50 60 70

Spacing (inches)

1E-3

0.01

0.1

1

10

H(A

/m)

zz

Real (H ), present codezz

Imag (H ), present codezz

Real (H ), exact solutionzz

Imag (H ), exact solutionzz

)

igure 4. Comparison of the axial component Hzz obtained from theresent code and the exact solution for �a� an isotropic medium andb� a TI medium.


ust be used. On the other hand, the coaxial component �azz is less af-

ected by the skin effect than �axx and �a

yy; �azz can reflect the geometric

ean of the horizontal resistivities ��x�y� within the frequencyange 20–220 KHz.

Figure 11 shows the apparent resistivity of the same tool for theame resistive formation at �75°, � �30°, and � �0° as fre-uency increases from 20 to 220 KHz. Comparison of Figure 11ith Figure 10 shows that, in this case, the diagonal and cross com-onents of the apparent resistivity are less sensitive to frequencyhan with zero dipping and azimuthal angles. Also, the cross terms

axy and �a

xz are negative. The apparent resistivities are inversely pro-ortional to the induced magnetic field, so negative cross terms im-ly that the induced magnetic field is phase shifted 180° with respecto the transmitter current.

For a relative conductive case, the resistivities of the form-tion are supposed to be �x�2 ohm-m, �y �4 ohm-m, and

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

360

320

280

240

200

160

120

80

40

00 20 40 6080

Azi

mut

h(°

)

xx

360

320

280

240

200

160

120

80

40

00 20 40 6080

Azi

mut

h(°

)

Dip (°)

0.80.60.40.20

–0.2–0.4–0.6–0.8–1–1.2

yx

0.80.60.40.20

–0.2–0.4–0.6–0.8–1–1.2

360

320

280

240

200

160

120

80

40

00 20 40 6080

xy

360

320

280

240

200

160

120

80

40

00 20 40 6080

Dip (°)

6

5

4

3

2

1

yy

360

320

280

240

200

160

120

80

40

00 20 40 6080

10

8

6

4

2

0

–2

–4

–6

–8

xz

1.5

1

0.5

0

–0.5

–1

–1.5

–2

360

320

280

240

200

160

120

80

40

00 20 40 6080

Dip (°)

yz

360

320

280

240

200

160

120

80

40

00 20 40 6080

Azi

mut

h(°

)

10

8

6

4

2

0

–2

–4

–6

–8

1.5

1

0.5

0

–0.5

–1

–1.5

–2

360

320

280

240

200

160

120

80

40

00 20 40 6080

–0.6

–0.8

–1

–1.2

–1.4

–1.6

–1.8

–2

–2.2

–2.4

360

320

280

240

200

160

120

80

40

00 20 40 6080

Dip (°) Dip (°) Dip (°)

zx zy zz

)

)

igure 5. Sensitivity of a three-coil triaxial tool with respect to theip angle �� a /� �mS /m / ° � in a homogeneous biaxial anisotropicormation.


�sfe

f��tp

Fdf

F�

F�

F�


z�8 ohm-m. Figure 12 shows the apparent resistivity of theame tool for �60°, � �30°, and � �0° as frequency increasesrom 20 to 220 KHz. All of the diagonal components of the appar-nt resistivity �a

xx, �ayy, and �a

zz increase as the frequency increases.As

360

320

280

240

200

160

120

80

40

0020 40 6080

Azi

mut

h(°

)

3

2

1

0

–1

–2

–3

–4

360

320

280

240

200

160

120

80

40

0020 40 6080

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

–1

–1.2

360

320

280

240

200

160

120

80

40

0020 40 6080

2

1

0

–1

–2

–3

xx xy xz

yy yz zz360

320

280

240

200

160

120

80

40

0020 40 6080

Azi

mut

h(°

)

2

1

0

–1

–2

–3

Dip (°)

360

320

280

240

200

160

120

80

40

0020 40 6080

5

4

3

2

1

0

–1

–2

–3

–4

Dip (°)

360

320

280

240

200

160

120

80

40

0020 40 6080

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

Dip (°)

igure 6. Sensitivity of a three-coil triaxial tool with respect to theip angle �� a /�� mS /m / ° � in a homogeneous biaxial anisotropicormation.

360

320

280

240

200

160

120

80

40

0020 40 6080

Azi

mut

h(°

)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

–0.1

xx

020 40 6080

350

300

250

200

150

100

50

0

0.080.060.040.0200.020.040.060.08

–0.1–0.12

xy360

320

280

240

200

160

120

80

40

0020 40 6080

–0.15–0.2–0.25–0.3–0.35–0.4–0.45–0.5–0.55–0.6–0.65

xz

360

320

280

240

200

160

120

80

40

0020 40 6080

Azi

mut

h(°

)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

–0.1

Dip (°)

yy360

320

280

240

200

160

120

80

40

0020 40 6080

0.4

0.3

0.2

0.1

0

–0.1

–0.2

–0.3

–0.4

–0.5

Dip (°)

yz360

320

280

240

200

160

120

80

40

0020 40 6080

0.2

0.15

0.1

0.05

Dip (°)

zz

igure 7. Sensitivity of a three-coil triaxial tool with respect to�� /�� in a homogeneous biaxial anisotropic formation.
x a x

or the cross components of the apparent resistivity, the amplitudedespite the phase shift with respect to the transmitter current� of

axz and �a

yz increases as the frequency increases, whereas the ampli-ude of �a

xy decreases as the frequency increases. This rule also ap-lies to the resistive case, as we can see from Figure 11.

360

320

280

240

200

160

120

80

40

00 20 40 6080

Azi

mut

h(°

)

0.6

0.5

0.4

0.3

0.2

0.1

0

–0.1

–0.2

xx360

320

280

240

200

160

120

80

40

00 20 40 6080

0.08

0.06

0.04

0.02

0

–0.02

–0.04

–0.06

–0.08

–0.1

xy360

320

280

240

200

160

120

80

40

00 20 40 6080

–0.2

–0.3

–0.4

–0.5

–0.6

–0.7

–0.8

–0.9

xz

360

320

280

240

200

160

120

80

40

00 20 40 6080

Azi

mut

h(°

)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

–0.1

Dip (°)

yy360

320

280

240

200

160

120

80

40

00 20 40 6080

0.4

0.3

0.2

0.1

0

–0.1

–0.2

–0.3

–0.4

–0.5

Dip (°)

yz360

320

280

240

200

160

120

80

40

00 20 40 6080

0.2

0.15

0.1

0.05

Dip (°)

zz

igure 8. Sensitivity of a three-coil triaxial tool with respect toy�� a /�� y� in a homogeneous biaxial anisotropic formation.

350

300

250

200

150

100

50

00 20 40 6080

Azi

mut

h(°

)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

xx

360

320

280

240

200

160

120

80

40

00 20 40 6080

Azi

mut

h(°

)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

–0.1

–0.2

Dip (°)

yy

360

320

280

240

200

160

120

80

40

00 20 40 6080

0.06

0.04

0.02

0

–0.02

–0.04

–0.06

–0.08

xy

360

320

280

240

200

160

120

80

40

00 20 40 6080

0.2

0.1

0

–0.1

–0.2

–0.3

yz

Dip (°)

360

320

280

240

200

160

120

80

40

00 20 40 6080

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

xz

360

320

280

240

200

160

120

80

40

00 20 40 6080

0.7

0.6

0.5

0.4

0.3

0.2

0.1

zz

Dip (°)

igure 9. Sensitivity of a three-coil triaxial tool with respect to�� /�� in a homogeneous biaxial anisotropic formation.
z a z

Tcat

amtembd

t

Fin a resistive formation � �� 0° �.

a

b

Fit

a

b

Fint

E112 Yuan et al.


Overall, our method is very efficient because it is quasi-analytical.he actual CPU time per transmitter varies with dipping angle be-ause different numbers of integral points are required for sufficientccuracy. The general CPU time is around 10�2 to 10�1 s on a Pen-ium 2.4-GHz PC.

CONCLUSIONS

We have presented a theoretical analysis of the response of a tri-xial induction logging tool in a homogeneous biaxial anisotropicedium. The logging tool is composed of three mutually orthogonal

ransmitter and receiver pairs. The axes of the transmitter and receiv-r can be oriented arbitrarily with respect to the principal axes of theedium’s conductivity tensor. A Fortran code has been developed

ased on the theory, and the results are compared with the publishedata.

The formulas we develop are capable of analyzing cross-couplingerms, which contain information on relative deviation and relative

20 40 60 80 100 120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20

22

Apparentresistivity(�m)

Frequency (KHz)

�

�

�

�

�

�

xx

ayy

azz

a

z

y

x

)

Frequency (KHz)

20

15

10

5

0

5

10

15

20

25

30

35

4020 40 60 80 100 120 140 160 180 200 220

–

–

–

–

–

–

–

–

App

aren

tre

sist

ivity

(�m

)

�

�

�

a

a

a

xz

yz

xy

)

igure 12. Frequency effect on responses of a three-coil triaxial tooln a conductive formation � �60° ,� �30° ,� �0° �. �a� Diago-al term of apparent resistivity. �b� Cross terms of apparent resistivi-y.

020 40 60 80 100 120 140 160 180 200 220

200

400

600

800

1000

�z

xx

ayy

azz y

a

x

�

�

Apparentresistivity(�m)

Frequency (KHz)

�

�

�

igure 10. Frequency effect on responses of a three-coil triaxial tool

Frequency (KHz)

1100

1000

900

800

700

600

500

400

300

200

100

020 40 60 80 100 120 140 160 180 200 220

App

aren

tres

istiv

ity(�

m)

� ��

� �

�

xx

a zyy

azz

a y

x

)

1000

0

1000–

2000–

3000–

4000–

5000–

6000–

20 40 60 80 100 120 140 160 180 200 220

Frequency (KHz)

App

aren

tres

istiv

ity( �

m)

�

�

�

xz

ayz

axy

a

)

igure 11. Frequency effect on responses of a three-coil triaxial tooln a resistive formation � �75° ,� �30° ,� �0° �. �a� Diagonalerm of apparent resistivity. �b� Cross terms of apparent resistivity.


bettttr

fLv

e

Bm

O

C

T

a

A

A

A

B

C

C

D

G

K

K

L

L

M

—

N

R

R

R

S

W

W

W

Z

Z

Z


earing. Therefore, our formulas are more useful and practical thanxisting formulas because cross-coupling terms help determine thehree principal components of the conductivity tensor for an arbi-rarily oriented transceiver system. However, before any interpreta-ion method can be practical, borehole effects must be corrected. Fu-ure work includes investigation of the effects of the borehole envi-onment, such as borehole fluid, invasion, and tool eccentricity.

ACKNOWLEDGMENTS

The authors would like to acknowledge S. Gianzero for his fruit-ul discussion. We acknowledge the financial support of the Wellogging Industrial Consortium, which is composed of 13 oil and ser-ice companies.

APPENDIX A

DERIVATION OF PARAMETERS a AND bIN EQUATIONS 14–16

The determinant of the coefficient matrix � can be written fromquation 9 directly:

det �� kx2� ��2��2��ky

2� �� 2��2��kz2� �� 2��2��

�2� 2�2�2�� 2�2�ky2� �� 2��2��2�2�kx

2

� ��2��2�� 2�2�kz2� �� 2��2�� . �A-1�

y arranging the right-hand side of equation A-1 according to theean of �, it can be rewritten as

det ��kz2�4��2�kz

2�kx2�ky

2�� 2�kx2�kz

2��2�ky2

�kz2�� 2��2�kz

2�� 2kx2��2ky

2�kx2ky

2� .

�A-2�

n the other hand, from equation 13, we obtain

det ��kz2�4�2kz

2�2a�a2�b . �A-3�

omparing the coefficients of � in equations A-2 andA-3 yields

�2kz2a�� kz

2�kx2�ky

2�� 2�kx2�kz

2��2�ky2�kz

2��,

�A-4�

a2�b� �� 2��2�kz2�� 2kx

2��2ky2�kx

2ky2� . �A-5�

herefore, the parameters a and b are

a�kz

2�kx2�ky

2�� 2�kx2�kz

2��2�ky2�kz

2�2kz

2 �A-6�

nd

b�a2�� 2��2�kz

2�� 2kx2��2ky

2�kx2ky

2�kz

2 .

�A-7�


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Documents

Simulation of full responses of a triaxial induction …Simulation of full responses of a triaxial induction tool in a homogeneous biaxial anisotropic formation Ning Yuan1, Xiao Chun