Copyright by Guozhong Gao 2005

Copyright

by

Guozhong Gao

2005

The Dissertation Committee for Guozhong Gao Certifies that this is the

approved version of the following dissertation:

SIMULATION OF BOREHOLE ELECTROMAGNETIC

MEASUREMENTS IN DIPPING AND ANISOTROPIC ROCK

FORMATIONS AND INVERSION OF ARRAY INDUCTION

DATA

Committee:

Carlos Torres-Verdín, Supervisor

Kamy Sepehrnoori

Hao Ling

Mary F. Wheeler

Sheng Fang




DATA

by

Guozhong Gao, B.S.; M.S.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

August 2005

Dedication

To my parents

v

Acknowledgements

I would like to express my sincere appreciation to my supervisor, Dr.

Carlos Torres-Verdín, for his support throughout my graduate studies at the

University of Texas at Austin. He is very knowledgeable, nice and patient. I thank

Dr. Torres-Verdín for fully supporting me to do summer internships, through

which I gained lots of knowledge and experience that I could not have otherwise

obtained at the university.

Special thanks go to Dr. Sheng Fang of Baker Atlas, who not only serves

as a member in my supervising committee, but also has inspired me with the

constant interests in electromagnetics (EM) research through the two summer

internships I worked with him. His help was significant toward the completion of

this research.

My gratitude also goes to Dr. Kamy Sepehrnoori, Dr. Hao Ling, and Dr.

Mary F. Wheeler for their comments and suggestions during their busy schedules.

Their excellent lectures in mathematics and electromagnetics greatly helped me

with my research.

I am also grateful to the sponsors of UT Austin’s Research Consortium on

Formation Evaluation: Anadarko Petroleum Corporation, Baker Atlas, BP,

ConocoPhilips, ENI E&P, ExxonMobil, Halliburton, Mexican Institute for

Petroleum, Occidental Petroleum, Petrobras, Precision Energy Services,

vi

Schlumberger, Shell International E&P, Statoil, and TOTAL for their financial

support of this work.

I also would like to express my gratitude to all of my colleagues in the

Formation Evaluation group and all of my friends in the Department of Petroleum

and Geosystems Engineering for their friendship and continuous help in

conducting this research. Finally, I would like to express my gratitude to Dr. Tsili

Wang of Baker Atlas, and Dr. Tom Neville, Dr. Ping Zhang and Dr. Mike Wilt of

Schlumberger for their help during the summer internships.

vii




DATA

Publication No._____________

Guozhong Gao, Ph.D.

The University of Texas at Austin, 2005

Supervisor: Carlos Torres-Verdín

Borehole electromagnetic (EM) measurements play a crucial role in

petroleum exploration. This dissertation develops advanced algorithms for the

numerical simulation of borehole EM measurements acquired in dipping and

anisotropic rock formations. The first technique is a full-wave modeling

technique: the BiCGSTAB(L)-FFT (Bi-Conjugate Gradient STABilized(L)-Fast

Fourier Transform). This technique is efficient both in terms of computational speed [~ ( )2logO N N ] and computer memory storage [~ ( )O N ], where N is the

number of spatial discretization cells. The second technique, referred to as a

“Smooth Approximation (SA),” substantially increases the accuracy of the

viii

simulated EM fields in electrically anisotropic media compared to the Born

approximation and the Extended Born Approximation (EBA). The third

technique, referred to as a “High-order Generalized Extended Born

Approximation (Ho-GEBA),” is developed for further improvement of the

efficiency and accuracy of EM simulation in electrically anisotropic media. These

techniques have been used to simulate tri-axial borehole induction measurements

acquired in dipping and anisotropic rock formations.

Efficient algorithms are also developed for EM modeling in axisymmetric

media. The three full-wave numerical simulation techniques investigated in this

dissertation include the BiCGSTAB(L)-FFT algorithm, the BiCGSTAB(L)-FFHT

(Fast Fourier Hankel Transform) technique, and the finite-difference method. In

addition, two approximation techniques are developed to approach the same

problem: a Preconditioned Extended Born Approximation (PEBA), and the Ho-

GEBA, which includes the PEBA as its first-order term in a series expansion.

These approximations are not only computationally efficient, but easily lend

themselves to developing efficient inversion algorithms.

In addition to forward modeling, inversion algorithms are developed to

estimate spatial distributions of electrical resistivity from array induction

measurements. This dissertation develops two types of inversion algorithms:

Resistivity Imaging (RIM) and Resistivity Inversion (RIN). An inner-loop and

outer-loop optimization technique is developed and used in the RIM. In both

strategies, the Jacobian (or sensitivity) matrix is computed via the PEBA, which

simulates the measurements and computes the Jacobian matrix simultaneously

ix

with only one forward simulation. The RIM assumes a continuous conductivity

distribution, while the RIN assumes a discrete (blocky) conductivity distribution.

Inversion exercises indicate that the RIN is superior to the RIM for the

quantitative evaluation of in-situ hydrocarbon saturation.

x

Table of Contents

Acknowledgements ................................................................................................. v

List of Tables........................................................................................................ xvi

List of Figures ....................................................................................................xviii

Chapter 1: Introduction ........................................................................................... 1 1.1 Problem Statement ................................................................................... 1 1.2 Objective of This Dissertation.................................................................. 6 1.3 Outline of This Dissertation ..................................................................... 6

Chapter 2: Electromagnetic Field Computation by Maxwell's Equations ............. 9 2.1 Maxwell's Equation of Electromagnetism................................................ 9 2.2 Derivation of the Integral Equation from Potentials .............................. 11 2.3 EM Sources in Geophysical Well Logging............................................ 14

2.3.1 Solenoids .................................................................................... 16 2.3.2 Toroids ....................................................................................... 16

2.4 Fields Due to Point Sources in an Unbounded Homogeneous and Isotropic Conductive Medium............................................................. 18

2.5 Explicit Expressions for the Dyadic Green's Functions ......................... 21 2.6 Conclusions ............................................................................................ 21

Chapter 3: Analytical Techniques to Evaluate the Integrals of 3D and 2D Spatial Dyadic Green's Functions ................................................................ 22 3.1 Introduction ............................................................................................ 23 3.2 Integral Equations and the Method of the Moments (MoM) ................ 25 3.3 Evaluation of the Integrals of the Dyadic Green's Functions................. 32

3.3.1 The Principal Volume Method................................................... 32 3.3.1.1 Equivalent Volume Solution for a Single Cell............... 33 3.3.1.2 Geometric Factor Solution for Non-singular Cells ........ 35

3.3.2 A General Intergal Evaluation Technique.................................. 36

xi

3.3.3 Numerical Validation ................................................................. 39 3.4 Conclusions ............................................................................................ 41 Supplement 3A: Derivation of the Expression of the Equivalent Volume

Approximation for a Singular Cell using the Principal Volume Method ................................................................................................ 46

Supplement 3B: Derivation of the Analytical Solution for the Integrals of the Electrical Dyadic Green's Function for a Spherical Volume .... 48 3B.1 Derivation for a Singular Cell .................................................... 49 3B.2 Expression for Non-singular Cells ............................................. 51

Supplement 3C: Derivation of the Analytical Solution for the Volume Integrals of the Electrical Dyadic Green's Function for an Infinite Long Circular Cylinder ....................................................................... 52 3C.1 Evaluation of a Singular Cell...................................................... 53 3C.2 Evaluation of Non-singular Cells ............................................... 54

Supplement 3D: Derivation of the Explicit Expressions for the Integral of the Electrical Dyadic Green's Function over a Spherical Cell from the General Formula ................................................................... 55

Supplement 3E: Derivation of the Explicit Expressions of the Integral of the Electrical Dyadic Green's Function over a General Rectangular Block using the General Formula........................................................ 57

Supplement 3F: Derivation of the Explicit Expressions for the Integral of the Electrical Dyadic Green's Function over a General Rectangular Cell (Rectangular Cylinder) ............................................ 62

Supplement 3G: Derivation of the Explicit Expressions for the Integral of the Magnetic Dyadic Green's Function over a Rectangular Block Cell ...................................................................................................... 65

Chapter 4: Numerical Modeling in Axisymmetric Media .................................... 68 4.1 Introduction ............................................................................................ 68 4.2 Governing Partial Differential Equation for Modeling Axisymmetric

Media................................................................................................... 71 4.3 Governing Integral Equation and Green's Functions ............................. 74 4.4 Full-Wave Modeling Techniques........................................................... 79

4.4.1 The BiCGSTAB(L)-FFT Technique.......................................... 79

xii

4.4.1.1 Computation of the Integrals of the Green's Function ... 80 4.4.1.2 Computation of Background Electric Fields .................. 81 4.4.1.3 Code Development ......................................................... 82

4.4.2 The BiCGSTAB(L)-FFHT Technique ....................................... 82 4.4.3 Finite Differences ....................................................................... 84 4.4.4 Numerical Examples .................................................................. 89

4.4.4.1 Solenoidal Source........................................................... 90 4.4.4.2 Toroidal Source .............................................................. 95

4.5 Approximate Modeling Techniques....................................................... 96 4.5.1 A Preconditioned Extended Born Approximation (PEBA) ....... 97 4.5.2 A High-order Gneralized Extended Born Approximation

(Ho-GEBA) .............................................................................. 100 4.5.2.1 Introduction .................................................................. 100 4.5.2.2 A Generalized Series Expansion of the Electric Field . 102 4.5.2.3 A Generalized Extended Born Approximation

(GEBA) ........................................................................... 103 4.5.2.4 A High-order Generalized Extended Born

Approximation (Ho-GEBA)............................................ 107 4.5.2.5 Numerical Examples .................................................... 109

4.6 Conclusions .......................................................................................... 124 Supplement 4A: Fast Hankel Transform (FHT) ........................................ 125 Supplement 4B: Finite Differencing of the TM Wave Equation ............... 127 Supplement 4C: The Apparent Conductivity and Its Skin-Effect

Correction.......................................................................................... 130

Chapter 5: A BiCGSTAB(L)-FFT Method for Three-Dimensional EM Modeling in Dipping and Anisotropic Media ............................................ 133 5.1 Introduction .......................................................................................... 133 5.2 Electrical Anisotropy............................................................................ 137 5.3 Coordinate System Transformation ..................................................... 142 5.4 Averaging of the Conductivity Tensor................................................. 146

xiii

5.5 Solution of the Linear System of Equations......................................... 147 5.6 The BiCGSTAB(L)-FFT Algorithm .................................................... 148

5.6.1 Toeplitz Matrices...................................................................... 149 5.6.2 Block Toeplitz Matrices ........................................................... 152

5.7 Numerical Examples ............................................................................ 155 5.7.1 1D Anisotropic Rock Formation .............................................. 158 5.7.2 3D Anisotropic Rock Formation .............................................. 162

5.8 Conclusions .......................................................................................... 165 Supplement 5A: Conductivity Tensor Averaging...................................... 166 Supplement 5B: Pseudocode Describing the BiCGSTAB(L) Algorithm .. 177

Chapter 6: A Smooth Approximation Technique for Three-Dimensional EM Modeling in Dipping and Anisotropic Media ............................................ 180 6.1 Introduction .......................................................................................... 181 6.2 Approximations to EM Scattering........................................................ 184

6.2.1 Born Approximation ................................................................ 184 6.2.2 Extended Born Approximation ................................................ 184 6.2.3 Quasi-Linear (QL) Approximation .......................................... 185

6.3 A Smooth EM Approximation ............................................................. 188 6.4 On the Choice of the Background Conductivity .................................. 192 6.5 Sensitivity to the Choice of Spatial Discretization............................... 194 6.6 Assessment of Accuracy with respect to Alternative Approximations 199 6.7 Numerical Examples ............................................................................ 203

6.7.1 1D Anisotropic Rock Formation with Dip=0o ......................... 203 6.7.2 1D Anisotropic Rock Formation with Dip=60o ...................... 206 6.7.3 3D Anisotropic Rock Formation with Dip=0o ........................ 209 6.7.4 3D Anisotropic Rock Formation with Dip=60o ...................... 211

6.8 Conclusions .......................................................................................... 214 Supplement 6A: Algorithmic Implementation of the Smooth EM

Approximation .................................................................................. 216

xiv

Chapter 7: A High-order Generalized Extended Born Approximation for Three-Dimensional EM Modeling in Dipping and Anisotropic Media ..... 220 7.1 Introduction .......................................................................................... 220 7.2 A Generalized Series (GS) Expansion of the Electric Field ................ 223 7.3 The Extended Born Approximation ..................................................... 227 7.4 A Generalized Extended Born Approximation (GEBA)...................... 228 7.5 A High-order Generalized Extended Born Approximation (Ho-

GEBA)............................................................................................... 231 7.6 The Physical Significance of the Ho-GEBA........................................ 233 7.7 Numerical Examples ............................................................................ 234

7.7.1 3D Scatterers ............................................................................ 235 7.7.2 Dipping and Anisotropic Rock Formations ............................ 254

7.7.2.1 1D Anisotropic Rock Formation, Dip Angle=60o ....... 254 7.7.2.2 3D Anisotropic Rock Formation, Dip Angle=60o ....... 258

7.8 Conclusions .......................................................................................... 262 Supplement 7A: Derivation of the New Integral Equation ........................ 263 Supplement 7B: Derivation of the Generalized Series (GS) Expansion

for the Internal Electric Field ............................................................ 267 Supplement 7C: Derivation of the Fundamental Equation of the Ho-

GEBA ................................................................................................ 271 Supplement 7D: Derivation of Special Case No. 2 of the Ho-GEBA........ 272

Chapter 8: Inversion of Multi-frequency Array Induction Measurements.......... 274 8.1 Introduction .......................................................................................... 275 8.2 Two-Dimensional Resistivity Imaging based on an Inner-loop and

Outer-loop Optimization Technique ................................................. 278 8.2.1 Non-Linear Optimization ......................................................... 279 8.2.2 An Inner-loop and Outer-loop Optimization Technique.......... 283 8.2.3 Computation of the Jacobian Matrix Based on the PEBA ....... 285 8.2.4 Resistivity Imaging Examples.................................................. 287

8.2.4.1 One-Dimensional Rock Formation Model ................... 289

xv

8.2.4.2 Two-Dimensional Rock Formation Model .................. 291 8.3 Two-Dimensional Resistivity Inversion for Conductivity Models

with Multi-front Mud-filtrate Invasion ............................................. 308 8.3.1 Constrained Nonlinear Least-Squares Inversion...................... 309 8.3.2 The Computer Code ................................................................. 309 8.3.3 Resistivity Inversion Examples ................................................ 310

8.3.3.1 A 2D Layered Formation with Borehole and No Invasion ........................................................................... 311

8.3.3.2 A 2D Formation that includes Borehole and Mud-filtrate Invasion ............................................................... 312

8.4 Conclusions .......................................................................................... 313

Chapter 9: Summary, Conclusions and Recommendations ................................ 324 9.1 Summary .............................................................................................. 324 9.2 Conclusions .......................................................................................... 327 9.3 Recommendations for Future Work..................................................... 328

Appendix: Selected Publications Completed During the Course of Ph.D. Research ..................................................................................................... 331

Nomenclature ...................................................................................................... 332

Bibliography........................................................................................................ 334

Vita .. ................................................................................................................... 342

xvi

List of Tables

Table 3.1: Matrix filling time and computer storage associated with the

assumption of 1 million discretization cells, and 0.2 CPU

seconds needed to compute 10,000 entries (each entry is a 3 by 3

tensor) of the MoM linear-system matrix . ...................................... 31

Table 3.2: Comparison of integration results obtained with the general

formula and with an external code assuming a rectangular block

of dimensions equal to (0.1, 0.3, 0.5) m. The external code has

been previously validated to render accurate results up to the

second significant digit. Results from two frequencies, i.e., 100

Hz and 1MHz are described in the table. ........................................ 45

Table 4.1: Description of the modified Oklahoma formation model

illustrated in Figure 4.5. .................................................................. 92

Table 7.1: Relationship between the GS and other series expansions of the

internal electric field reported in the open technical literature. ..... 226

Table 8.1: Summary of inversion results for the 2D formation model with

borehole and invasion for different values of noise level added to

the data. One fixed invasion front is assumed for each layer. Odd

numbering is used for the invaded zone, while even numbering is

used for the uninvaded zone within the same layer........................ 322

xvii

Table 8.2: Summary of the inversion results for the 2D formation model

with borehole and invasion for different noise levels added to the

data. One invasion front is assumed for each layer, and the odd

numbering is used for the invaded zone, while the even

numbering is used for the original zone of the same layer. The

invasion fronts and the conductivity of each block are inverted

simultaneously................................................................................ 323

Table 9.1: Comparison of the computer efficiency of the BiCGSTAB(L)-

FFT, the SA and the Ho-GEBA. The number of nodes is equal to

64,000 in all three cases, and the computer platform is a PC that

includes a 3.2 GHz Pentium 4 Intel processor. The number of

blocks for the SA is 2400. .............................................................. 325

xviii

List of Figures

Figure 2.1: Typical EM sources used in Borehole EM logging. (a) solenoid;

(b) toroid........................................................................................... 15

Figure 3.1: Comparison of integration results obtained with the general

formula and the principal-volume approximation assuming a

singular cubic cell. The upper panel shows the amplitude, and the

bottom panel show the phase. In both panels, a is the radius of

the equivalent sphere. ....................................................................... 42


formula and the geometric factor solution assuming a non-

singular cubic cell. Two of the six independent components,

G(1,1) and G(1,2) are shown on the figure including amplitude

and phase. The cell size is (0.2, 0.2, 0.2) m, and the cell is

located at the origin. The observation point is located at (0.2, 0.4,

0.6). In both figures, a is the distance between the cell and the

observation point. ............................................................................. 43

xix





and phase are shown on the Figure. The cell size is (0.2, 0.2, 0.2)

m, and the cell is located at the origin. The observation point is

located at (0.2, 0.4, 0.6). In both figures, a is the distance

between the cell and the observation point. ..................................... 44





and phase. The cell size is (0.2, 0.2, 0.2) m, and the cell is

located at the origin. The observation point is located at (0.2, 0.4,

0.6). In both figures, a is the distance between the cell and the

observation point. ............................................................................. 45

Figure 4.1: Graphical illustration of the borehole logging environment. ........... 69

Figure 4.2: Illustration of a typical annulus invasion profile.............................. 71

Figure 4.3: Illustration of the finite-difference grid used to discretize the TE

wave equation................................................................................... 85

Figure 4.4: Graphical description of the five-point stencil used in the finite-

difference approximation of Maxwell’s equation in axisymmetric

media. ............................................................................................... 87

xx

Figure 4.5: A modified Oklahoma model. Left Panel: Invasion radius versus

depth. Right Panel: Conductivity versus depth. In the

figures, xoσ is the conductivity of the flushed zone, and tσ is the

conductivity of the uninvaded formation. Electrical and

geometrical parameters for this model are given in Table 4.1.. ....... 91

Figure 4.6: Graphical comparison of the three full-wave simulation

techniques applied to the modified Oklahoma model shown in

Figure 4.5. The real part of the magnetic response is shown on

the figure. The tool operates at 10 KHz and consists of one

transmitter and one receiver with a spacing of 0.5 m. On the

figure, “2DIE” designates the BiCGSTAB(L)-FFT; “FFHT”

designates the BiCGSTAB(L)-FFHT; “FD2D” designates the

finite-difference code. ...................................................................... 93

Figure 4.7: Graphical comparison of the three full-wave techniques applied

to the modified Oklahoma model shown in Figure 4.5. The

imaginary part of the magnetic response is shown on the figure.

The tool operates at 10 KHz and consists of one transmitter and

one receiver with a spacing of 0.5 m. On the figure, “2DIE”

designates the BiCGSTAB(L)-FFT; “FFHT” designates the

BiCGSTAB(L)-FFHT; “FD2D” designates the finite-difference

code. ................................................................................................. 94

xxi

Figure 4.8: Graphical description of the three-layer rock formation model

used to simulate the EM response of a toroidal source. ................... 95

Figure 4.9: Simulation results obtained for the formation model given in

Figure 4.8. The tool operates at 25 KHz and consists of one

transmitter and one receiver spaced at a distance of 0.5 m. The

radius of the toroidal coil in the ρ φ− plane is 0.03 m, and the

radius of the toroidal coil in the zρ − plane is 0.005 m. The left

panel shows the real part of zE , and the right panel shows the

imaginary part of zE . ...................................................................... 96

Figure 4.10: (a) Diagram describing the geometry of a three-layer generic

axisymmetric formation system that includes a borehole and

mud-filtrate invasion. In the figure, wr is the radius of the

wellbore, and xor is the radius of the invaded zone. The zone

where xor r> corresponds to the original (uninvaded) formation.

(b) Spatial distribution of formation resistivity corresponding to

the geometry described in (a), where bR is the mud resistivity in

the well, xoR is the resistivity of the invaded zone, and tR is the

resistivity of the original formation................................................ 110

xxii

Figure 4.11: Three-coil tool configuration. The assumed borehole induction

tool consists of one transmitter and two receivers, with the

spacing between the transmitter and the first receiver ( 1L ) equal

to 0.6 m, and the spacing between the transmitter and the second

receiver ( 2L ) equal to 0.65 m. The measured signal ( zHΔ ) is the

difference between the signal at Receiver 1 ( 1zH ), and the signal

at Receiver 2 ( 2zH ). The operating frequencies are 25 KHz and

100 KHz.. ....................................................................................... 111

Figure 4.12: Formation Model 1. The model consists of a one-layer formation

embedded in a background medium with resistivity equal to that

of the mud in the well, where 1bR m= Ω⋅ , and the background

dielectric constant is 1. The thickness of the layer is 3.2 m, and

0.3xor m= , 0.1wr m= , 0.5xoR m= Ω⋅ , 0.2tR m= Ω⋅ . .................. 113

Figure 4.13: Numerical simulation results for Resistivity Model 1 at 25 KHz.

The left panel shows the real part of zHΔ , and the right panel

shows the imaginary part of zHΔ . The Ho-GEBA (up to the 3rd

order) results are plotted against the accurate solution “2DIE”,

and the solutions obtained with the Born approximation and the

EBA. Note that the GEBA is equivalent to the PEBA for this

case. ................................................................................................ 114

xxiii

Figure 4.14: Numerical simulation results for Resistivity Model 1 at 100

KHz. The left panel shows the real part of zHΔ , and the right

panel shows the imaginary part of zHΔ . The Ho-GEBA (up to

the 3rd order) results are plotted against the accurate solution

“2DIE”, and the solutions obtained with the Born approximation

and the EBA. Note that the GEBA is equivalent to the PEBA for

this case. ......................................................................................... 115

Figure 4.15: Formation Model 2. The model consists of a one-layer formation

embedded in a background medium with electrical resistivity

equal to that of the mud in the well where 1bR m= Ω⋅ , and the

background dielectric constant is 1. The thickness of the layer is

3.2 m, and 0.3xor m= , 0.1wr m= , 2xoR m= Ω⋅ , 10tR m= Ω⋅ . ..... 117

Figure 4.16: Numerical simulation results for Resistivity Model 2 at 25 KHz.

The left panel shows the real part of zHΔ , and the right panel

shows the imaginary part of zHΔ . The Ho-GEBA (up to the 3rd

order) results are plotted against the accurate solution “2DIE”,

and the solutions obtained with the Born approximation and the

EBA. Note that the GEBA is equivalent to the PEBA for this

case.. ............................................................................................... 118

xxiv

Figure 4.17: Numerical simulation results for Resistivity Model 2 at 100

KHz. The left panel shows the real part of zHΔ , and the right

panel shows the imaginary part of zHΔ . The Ho-GEBA (up to

the 3rd order) results are plotted against the accurate solution

“2DIE”, and the solutions obtained with the Born approximation

and the EBA. Note that the GEBA is equivalent to the PEBA for

this case. ......................................................................................... 119

Figure 4.18: Numerical simulation results for the modified Oklahoma model

(described in Table 4.1) at 25 KHz. The left panel shows the real

part of zHΔ , and the right panel shows the imaginary part of

zHΔ . The Ho-GEBA (up to the 3rd order) results are plotted

against the accurate solution “2DIE”, and the solutions obtained

with the Born approximation and the EBA. The GEBA is

equivalent to the PEBA for this case. .......................................... 121

Figure 4.19: Numerical simulation results for the modified Oklahoma

formation model (described in Table 4.1) at 100 KHz. The left

panel shows the real part of zHΔ , and the right panel shows the

imaginary part of zHΔ . The Ho-GEBA (up to the 3rd order)

results are plotted against the accurate solution “2DIE”, and the

solutions obtained with the Born approximation and the EBA.

The GEBA is equivalent to the PEBA for this case. ..................... 122

xxv

Figure 4.20: Comparison of simulation results obtained with three

approximations in the frequency range between 100 Hz and 2

MHz. The formation model considered is Model 1 and the

logging point corresponds to a depth of -2 m. In the figure, the

horizontal axis corresponds to frequency and the vertical axis

describes the percentage errors in amplitude (left panel) and

phase (right panel) of zHΔ ............................................................. 123

Figure 5.1: Example of a typical TI anisotropic rock formation....................... 139

Figure 5.2: Illustration of a generic tri-axial induction tool. The tool consists

of 3 transmitters and 3 receivers oriented along the three

coordinate axes. The transmitters could be deployed at the same

point (collected transmitters). The same is true for the receiver

(collected receivers). ...................................................................... 142

Figure 5.3: Comparison of the convergence behavior of the BiCG and the

BiCGSTAB(L). .............................................................................. 148

Figure 5.4: Structure of block Toeplitz matrix resulting from a 3D EM

problem. Each T is a Toeplitz matrix of size xn , and each entry of

the Toeplitz matrix is 3 by 3 matrix. .............................................. 153

Figure 5.5: Graphical description of the generic 5-layer electrical

conductivity model used in this chapter to test the

BiCGSTAB(L)-FFT algorithm(not to scale).................................. 155

xxvi

Figure 5.6: Graphical description of the assumed double receiver, single

transmitter instrument for borehole induction logging (not to

scale). In general, the transmitter and receivers can be oriented in

the x, y, or z directions. .................................................................. 156

Figure 5.7: Comparison of the Hzz field component simulated with the

BiCGSTAB(L)-FFT algorithm and a 1D code assuming a 1D

formation. The tool and the formation form an angle of 60o.

Results for 20 KHz and 220 KHz are shown on this figure. . ........ 159

Figure 5.8: Comparison of the Hxx field component simulated with the



Results for 20 KHz and 220 KHz are shown on this figure. .......... 160

Figure 5.9: Comparison of the Hyy field component simulated with the



Results for 20 KHz and 220 KHz are shown on this figure. .......... 161

Figure 5.10: Comparison of the Hzz field component simulated with the

BiCGSTAB(L)-FFT algorithm and a 3D FDM code assuming a

3D formation with borehole and mud-filtrate invasion. The tool

and the formation form an angle of 60o. Results for 20 KHz and

220 KHz are shown on this figure. ................................................ 162

xxvii

Figure 5.11: Comparison of the Hxx field component simulated with the




220 KHz are shown on this figure.................................................. 163

Figure 5.12: Comparison of the Hyy field component simulated with the




220 KHz are shown on this figure.................................................. 164

Figure 6.1: Assessment of the accuracy of the integral equation

approximation of Hzz (imaginary part) for a given number of

spatial discretization blocks. The formation dips at an angle of

60o and is modeled in the presence of both a borehole and mud-

filtrate invasion. Simulation results are shown for a probing

frequency of 220 KHz. ................................................................... 196


approximation of Hxx (imaginary part) for a given number of




frequency of 220 KHz. ................................................................... 197

xxviii


approximation of Hyy (imaginary part) for a given number of




frequency of 220 KHz. ................................................................... 198


approximation of Hzz (imaginary part) with respect to alternative

approximation strategies (Born and Extended Born). The

formation dips at an angle of 60o and is modeled in the presence

of both a borehole and mud-filtrate invasion. Simulation results

are shown for a probing frequency of 220 KHz............................. 200


approximation of Hxx (imaginary part) with respect to alternative





xxix


approximation of Hyy (imaginary part) with respect to alternative





Figure 6.7: Comparison of the Hzz field component (imaginary part)

simulated with the SA and a 1D code. In both cases, the

simulations were performed assuming a 1D formation that

exhibits electrical anisotropy. The induction logging tool is

assumed to be oriented perpendicular to the formation.

Simulation results are shown for probing frequencies of 20 KHz

and 220 KHz................................................................................... 204

Figure 6.8: Comparison of the Hxx field component (imaginary part)



exhibits electrical anisotropy. The induction logging tool is

assumed to be oriented perpendicular to the formation.

Simulation results are shown for probing frequencies of 20 KHz

and 220 KHz................................................................................... 205

xxx




exhibits electrical anisotropy and a borehole dipping at an angle

of 60o. Simulation results are shown for probing frequencies of

20 KHz and 220 KHz. .................................................................... 206






20 KHz and 220 KHz. .................................................................... 207

Figure 6.11: Comparison of the Hyy field component (imaginary part)





20 KHz and 220 KHz. .................................................................... 208


simulated with the SA and a 3D FDM code assuming a 3D

formation that includes both a borehole and invasion. The

borehole dips at an angle of 0o. Simulation results are shown for

probing frequencies of 20 KHz and 220 KHz................................ 209

xxxi
















Figure 6.16: Comparison of the Hyy field component (imaginary part)





xxxii

Figure 7.1: Graphical description of the scattering models considered in this

section. The background ohmic resistivity is 10 mΩ⋅ and the

background dielectric constant is 1. One x-directed and one z-

directed magnetic dipole sources with a magnetic moment of 1 2A m⋅ are assumed located at the origin, and 20 receivers are

deployed along the z-axis with a uniform separation of 0.2

meters. No receiver is at the origin. A cubic scatterer with a side

length of 2 m is centered about the x-axis, and is symmetrical

about the y and z axes. Depending on the resistivity, R, of the

scatterer and the distance, L, between the source and the

scatterer, a total of four scattering models are used in the

numerical experiments: Model 1: R=1 mΩ⋅ , L=4.0 m; Model 2:

R=1 mΩ⋅ , L=0.1 m; Model 3: R=100 mΩ⋅ , L=4.0 m; Model

4: R=100 mΩ⋅ , L=0.1 m............................................................... 237

Figure 7.2: Scattered xxH component for Model 1. The left- and right-hand

panels show simulation results for 10 KHz and 200 KHz,

respectively. For each panel, the top figure describes the in-phase

(real) component of xxH , and the bottom figure describes the

quadrature (imaginary) component of xxH . Simulation solutions

from the Born approximation, the EBA, and the Ho-GEBA (up

to the 3rd order) are plotted against the exact solution. .................. 242

xxxiii

Figure 7.3: Scattered zzH component for Model 1. The left- and right-hand



(real) component of zzH , and the bottom figure describes the

quadrature (imaginary) component of zzH . Simulation solutions

















xxxiv






















xxxv








Figure 7.10: Comparison of the EBA, the Born and the EBA over the

frequency range of 10 KHz-2 MHz. The model considered is

Model 2, and the signal is for the receiver at -0.1 m. The left

figure describes the in-phase (real) component of xxH , and the

right figure describes the quadrature (imaginary) component of

xxH . Simulation solutions from the Born approximation, the

EBA, and the Ho-GEBA (up to the 5rd order) are plotted against

the exact solution............................................................................ 250

Figure 7.11: Comparison of the EBA, the Born and the EBA over the

frequency range of 10 KHz-2 MHz. The model considered is

Model 2, and the signal is for the receiver at -0.1 m. The left

figure describes the in-phase (real) component of zzH , and the

right figure describes the quadrature (imaginary) component of

zzH . Simulation solutions from the Born approximation, the

EBA, and the Ho-GEBA (up to the 5rd order) are plotted against

the exact solution............................................................................ 251

xxxvi

Figure 7.12: Graphical comparison of the convergence rate of the Ho-GEBA

and the GS. Model 1 is the assumed scattering and the numerical

simulations correspond to the zzH component. The left-hand

panel shows convergence results for 10 KHz, and the right-hand

panel for 200 KHz. ......................................................................... 252

Figure 7.13: Graphical corroboration of some technical issues associated with

the special case 2 of the Ho-GEBA. Model 2 is the assumed

scattering model and the numerical simulations correspond to the

zzH component. The nomenclature HoGEBAS2-n (n=1, 2, 3)

identifies simulation results associated with the special case 2 of

the Ho-GEBA. The left- and right-hand panels describe the real

and imaginary parts of zzH , respectively....................................... 253

Figure 7.14: Comparison of the xxH field component simulated with the Ho-

GEBA, the Born approximation, the EBA and an analytical 1D

code assuming a 1D anisotropic rock formation. The tool and

the formation form an angle of 60o and the frequency is 220

KHz. ............................................................................................... 255 Figure 7.15: Comparison of the yyH field component simulated with the Ho-




KHz. ............................................................................................... 256

xxxvii

Figure 7.16: Comparison of the zzH field component simulated with the Ho-




KHz. ............................................................................................... 257

Figure 7.17: Comparison of the xxH field component simulated with the Ho-

GEBA, the Born approximation, the EBA and a 3D FDM code

assuming a 3D anisotropic rock formation with borehole and

mud-filtrate invasion. The tool and the formation form an angle

of 60o and the frequency is 220 KHz. ............................................ 259 Figure 7.18: Comparison of the yyH field component simulated with the Ho-




of 60o and the frequency is 220 KHz. ............................................ 260

Figure 7.19: Comparison of the zzH field component simulated with the Ho-




of 60o and the frequency is 220 KHz. ............................................ 261

xxxviii

Figure 7B-1: Rock formation model used to numerically test the convergence

properties of the GS. A conductive cube with a side length of 2

m and a conductivity of 10 S/m is embedded in a background

medium of conductivity equal to 1 S/m. The transmitter and the

receiver are assumed to be vertical magnetic dipoles operating at

20 KHz. The distance between the transmitter and the cube is 0.1

m, and the spacing between the transmitter and receiver is 0.5 m. 269

Figure 7B-2: Graphical comparison of the convergence behavior of the

classical Born series expansion, the GS (starting from the

background field), the EBA series expansion [no contraction

(N.C.)], and the EBA series expansion [with contraction (W.C.)]

for the rock formation model given in figure B-1. The left figure

describes the convergence behavior of both the classical Born

series expansion and the EBA series expansion (N.C.), while the

right figure describes the convergence behavior of the GS and

EBA series expansion (W.C.). The solution line was calculated

using a full-wave 3D integral-equation code (Fang, Gao, and

Torres-Verdín, 2003)...................................................................... 270

Figure 8.1: Flowchart of the inner-loop and outer-loop optimization

algorithm. ....................................................................................... 285

xxxix

Figure 8.2: Schematic of the two array induction tools assumed in this paper.

Both Tool No. 1 and Tool No. 2 consist of 3 arrays; the

difference being that each array consists of one transmitter and

one receiver for Tool No. 1, and of one transmitter and two

receivers for Tool No. 2. Separations between transmitter and

arrays of receivers are 15 inches, 27 inches, and 72 inches,

respectively. Both tools operate at 25 KHz, 50 KHz, and 100

KHz. ............................................................................................... 293

Figure 8.3: One-dimensional chirp-like formation model used in the

inversion. The widths of the 4 resistive beds are 0.3, 0.6, 1.2, and

2.4 meters, respectively. ................................................................. 294

Figure 8.4: Vertical profiles of electrical conductivity inverted as a function

of the outer-loop iteration number. Inversion results for iterations

1, 2, 3, 4, and 5 are shown on the figure. Each outer-loop

iteration consists of 4 inner-loop iterations. The inversions were

performed using noise-free data “acquired” with Tool 2 at 25

KHz, 50 KHz, and 100 KHz. ......................................................... 295

Figure 8.5: Data misfit for array-2 of Tool No. 2 at 25 KHz (imaginary part)

as a function of the number of outer-loop iterations. Data misfit

results for iterations 1, 2, 3, 4, and 5 are shown on the figure.

Each outer-loop iteration consists of 4 inner-loop iterations. The

inversions were performed using noise-free data “acquired” with

Tool No. 2 at 25 KHz, 50 KHz, and 100 KHz. .............................. 296

xl

Figure 8.6: Plots of data misfit as a function of iteration number for the

inversion results described in Figure 4. The left panel shows

values of data misfit with respect to outer-loop iteration number.

Data misfit values as a function of inner-loop iteration number

are shown in the right panel. .......................................................... 297

Figure 8.7: Vertical profile of electrical conductivity inverted from Tool No.

2 array-induction data simulated for the 1D chirp-like model and

contaminated with zero-mean, 2% random Gaussian additive

noise. The inversion was performed with data acquired at 25

KHz, 50 KHz, and 100 KHz. ......................................................... 298

Figure 8.8: One-and-half (1.5D) electrical conductivity model inverted from

array induction data simulated for the 1D chirp-like formation

model with invasion. Data input to the inversion were simulated

numerically for Tool No. 2 and were subsequently contaminated

with zero-mean, 2% random Gaussian additive noise. Eight fixed

piston-like invasion fronts were assumed in the inversion, with

radii of invasion equal to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9

meters, respectively. The inversion was performed using data

acquired at 25 KHz, 50 KHz, and 100 KHz................................... 299

xli

Figure 8.9: Two-dimensional distribution of electrical conductivity inverted

from array induction data simulated for the 1D chirp-like

formation model with invasion. Data input to the inversion were

simulated numerically for Tool No. 2 and were subsequently



KHz, 50 KHz, and 100 KHz. ......................................................... 300

Figure 8.10: Vertical profile of electrical conductivity inverted from Tool No.

1 array-induction data simulated for the 1D chirp-like model and



KHz, 50 KHz, and 100 KHz. ......................................................... 301

Figure 8.11: Electrical 1.5D conductivity model inverted from array

induction data simulated for the 1D chirp-like formation model

with invasion. Data input to the inversion were simulated

numerically for Tool No. 1, and were subsequently contaminated

with zero-mean, 2% random Gaussian additive noise. Eight fixed

piston-like invasion fronts were assumed in the inversion, with

radii of invasion equal to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9

meters, respectively. The inversion was performed with data

acquired at 25 KHz, 50 KHz, and 100 KHz................................... 302

xlii


from Tool No. 1 array induction data simulated for the 1D chirp-

like formation model with invasion. Data input to the inversion

were simulated numerically and were subsequently contaminated

with zero-mean, 2% random Gaussian additive noise. Inverted

2D conductivity image for the chirp-like 1-D formation model.

2% Gaussian random noise is added. The inversion was

performed with data acquired at 25 KHz, 50 KHz, and 100 KHz. 303

Figure 8.13: Graphical description of the 2D formation model constructed to

test the inversion algorithm. From top to bottom, the thickness of

the 8 layers is 2.1, 2.1, 1.2, 1.8, 0.9, 1.5, 1.8, and 1.2 meters,

respectively. Invasion radii for the 4 invaded layers are 0.6, 0.9,

0.6, and 0.9 meters, respectively, from top to bottom.................... 304

Figure 8.14: One-dimensional profile of electrical conductivity inverted from

array induction data simulated for the 2D formation model

shown in Figure 8.13. Data input to the inversion were


noise. The upper panel shows the conductivity profile inverted

from data “acquired” with Tool No. 2, and the lower panel shows

the conductivity profile inverted from data “acquired” with Tool

No. 1. The inversion was performed with data acquired at 25

KHz, 50 KHz, and 100 KHz. ......................................................... 305

xliii

Figure 8.15: Electrical 1.5D conductivity model inverted from array

induction data simulated for the 2D formation model with

invasion. Data input to the inversion were simulated numerically

for Tool No. 1, and were subsequently contaminated with zero-

mean, 2% random Gaussian additive noise. Twelve fixed piston-

like invasion fronts were assumed in the simulations, with radii

of invasion equal to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1,

1.2, and 1.3 meters, respectively. The inversion was performed

with data acquired at 25 KHz, 50 KHz, and 100 KHz. .................. 306


from array induction data simulated for the 2D formation model

shown in Figure 13. Data input to the inversion were simulated

numerically for Tool No. 1 and were subsequently contaminated

with zero-mean, 2% random Gaussian additive noise. The

inversion was performed with data acquired at 25 KHz, 50 KHz,

and 100 KHz................................................................................... 307

xliv

Figure 8.17: Left Panel: the original 2D conductivity profile; Right Panel:

the inverted 2D conductivity image. Data were generated as a

subset of induction logging tool measurements acquired at 25

KHz, 50 KHz and 100 KHz; 2% additive Gaussian noise was

added to the data before the inversion. From top to bottom, the

thickness of the 8 layers is 2.1, 2.1, 1.2, 1.8, 0.9, 1.5, 1.8, and 1.2

meters, respectively. Invasion radii for the 4 invaded layers are

0.6, 0.9, 0.6, and 0.9 meters, respectively, from top to bottom.

(From Gao and Torres-Verdín, 2003). ........................................... 316

Figure 8.18: Array induction instrument assumed by the numerical examples

considered in section 8.4. The instrument is a subset of the Array

Induction Tool. Sounding frequencies are 25 KHz, 50 KHz, and

100 KHz. ........................................................................................ 316

Figure 8.19: A 1D formation model with borehole and without invasion. The

borehole radius is 0.1 m, and the conductivity of the mud is 0.5

S/m. The shoulders are assumed to have a conductivity of 0.5

S/m.. ............................................................................................... 317

xlv

Figure 8.20: Inversion results and relative error of the inverted conductivities

for a 2D formation model with borehole and without invasion.

The corresponding layer boundaries are assumed known and

fixed. The borehole radius is 0.1 m, and the conductivity of the

mud is 0.5 S/m. The shoulder is assumed to have a conductivity

of 0.5 S/m. The initial guess for the conductivity of each layer is

0.2 S/m. .......................................................................................... 318

Figure 8.21: Inversion results and relative error of the inverted conductivities

for a 2D formation model with borehole and without invasion.

Both layer boundaries and conductivities are inverted

simultaneously. The borehole radius is 0.1 m, and the

conductivity of the mud is 0.5 S/m. The shoulder is assumed to

have a conductivity of 0.5 S/m. Initial boundaries and

conductivities are shown on the left figure. ................................... 319

Figure 8.22: The RMS misfit error versus iteration number for different levels

of noise added to the data. The formation model is shown in

Figure 8.19. The left part of the figure shows the LEVEL 1

inversion results, while the right part of the figure shows the

LEVEL 2 inversion results. ........................................................... 320

xlvi

Figure 8.23: The RMS misfit error versus iteration number for different levels

of noise added to the data. The formation model is shown in

Figure 8.13. The left part of the figure shows the LEVEL 3

inversion results, while the right part of the figure shows the

LEVEL 4 inversion results. ........................................................... 321

1

Chapter 1: Introduction

1.1 PROBLEM STATEMENT

Electromagnetic (EM) methods play an important role in petroleum exploration.

Important hydrocarbon reservoir applications of EM methods are the assessment of fluid

type and in-situ fluid saturation for single-well operations and production monitoring for

both single-well and cross-well operations. In some cases, EM methods are more

sensitive to variations of fluid saturation than seismic methods (Wilt and Alumbaugh,

2002).

In geophysical borehole logging, depending on the type of excitation source, EM

instruments can be classified into galvanic, induction, and propagation types. The main

objective is to accurately estimate the resistivity of the original rock formation that is not

affected by mud-filtrate invasion. Subsequently, hydrocarbon saturation can be estimated

from petrophysical and electrical relationships, such as Archie’s (Archie, 1942) and

Waxman-Smits (Waxman and Smits, 1968) equations.

Numerical simulation plays a crucial role in understanding the physics of EM

instruments and the interaction between EM instruments and rock formations. One of the

main objectives of this dissertation is to develop accurate and efficient numerical

algorithms for simulating borehole EM measurements in complex rock formations.

Emphasis is placed on developing novel numerical algorithms for simulating

measurements acquired with tri-axial induction tools in dipping and anisotropic rock

formations.

2

Several numerical algorithms are developed in this dissertation to simulate the

response of multi-frequency array induction tools in axisymmetric rock formations. For

70 years now, EM modeling in axisymmetric media has been approached by numerous

authors in the petroleum industry. However, new methods are still needed not only for

efficient forward modeling, but also to facilitate data interpretation. In this dissertation,

based on a partial differential formulation of Maxwell’s equations, an algorithm is

developed to model the EM response of axisymmetric media using the finite difference

method. Also, based on an integral equation formulation, two efficient numerical

simulation algorithms are developed and benchmarked: the BiCGSTAB(L)-FFT (Bi-

Conjugate Gradient STABilized (L)-Fast Fourier Transform) and the BiCGSTAB(L)-

FFHT (Bi-Conjugate Gradient STABilized (L)-Fast Fourier and Hankel Transform).

Finally, two accurate and efficient approximate algorithms, the PEBA (Preconditioned

Extended Born Approximation) and the Ho-GEBA (High-order Generalized Extended

Born Approximation), are developed to facilitate the inversion of borehole EM

measurements.

In recent years, electrical anisotropy of rock formation has become increasingly

important, especially in deviated and horizontal wells (Schoen et al., 2000; Yu et al.,

2001; Zhang et al., 2004). The following is an example taken from Wang and Fang

(2001) to emphasize the importance of electrical anisotropy in logging interpretation. The

reservoir under consideration exhibits sand-shale laminations. Electrical resistivity is

assumed transversely isotropic (TI) (details can be found in Chapter 5). In such a

reservoir, the horizontal resistivity, hR , of the laminae is mostly controlled by the

3

resistivity of shale whereas the vertical resistivity, vR , is dictated by that of hydrocarbon-

bearing sands, i.e. (Klein et al., 1997),

11 sh sh

h s sh

V VR R R

−= + , (1.1)

and

( )1v sh s sh shR V R V R= − + , (1.2)

where sR is the resistivity of hydrocarbon-bearing sands, shR is the resistivity of shale,

and shV is the volume fraction of shale. Given hR , vR , and shR , both sR and shV can be

calculated from equations (1.1) and (1.2). If electrical anisotropy is considered, the

corresponding water saturation, aniswS , can be estimated through Archie’s equation (1942)

using the resistivity of sand, i.e.,

1/

φ⎛ ⎞

= ⎜ ⎟⎝ ⎠

nanis ww m

s s

aRSR

, (1.3)

where wR is the electrical resistivity of connate water, sφ is the porosity of sand, m is the

cementation factor, n is the saturation exponent, and a is the tortuosity constant. We

remark that measurements acquired with horizontal coils in a vertical well are sensitive

only to the horizontal resistivity. Use of hR as the formation resistivity in Archie’s

equation yields the water saturation without considering the electrical anisotropy, isowS ,

namely,

1/

φ⎛ ⎞

= ⎜ ⎟⎝ ⎠

niso ww m

h

aRSR

, (1.4)

4

where φ is taken to be the bulk porosity of the formation, which is related to sφ by way

of

( )1s shVφ φ= − . (1.5)

Substitution of equation (1.5) into equations (1.3) and (1.4) yields

( )1/

/1⎛ ⎞

= − ⎜ ⎟⎝ ⎠

nanism nw h

shisow s

S RVS R

. (1.6)

For 10sR m= Ω⋅ , 1shR m= Ω⋅ , 0.2shV = , m n 2= = , and 1a = , one obtains

0.48aniswisow

SS

= . (1.7)

This last result shows that, after considering electrical anisotropy (using the true sand

resistivity value), the estimated water saturation is nearly half of that calculated without

consideration of electrical anisotropy. Such a simple exercise clearly shows the

importance of electrical anisotropy in formation evaluation.

Assessing the effects of electrical anisotropy is important. However, computing

the EM fields in an inhomogeneous 3D anisotropic medium still remains an open

challenge. Such a problem is often solved by means of a finite-difference method (FDM)

(Anderson et al., 2001; Wang and Fang, 2001; Weiss and Newman, 2002; Davydycheva

et al., 2003). However, the integral equation (IE) method provides a flexible and accurate

formulation to solve EM scattering problems (Harrington, 1968). The IE method requires

the construction and solution of a full complex matrix system, which entails

computational difficulties in terms of matrix-filling time, memory storage, and matrix

inversion. In this dissertation, we develop efficient algorithms to circumvent all the

computational difficulties inherent to the IE method. The algorithm, termed

5

BiCGSTAB(L)-FFT, is successfully used to simulate tri-axial induction measurements in

dipping and anisotropic rock formations.

Another advantage of the IE method is that it is suitable for developing efficient

approximate algorithms, such as Born approximation (Born, 1933), extended Born

approximation (Habashy et al., 1993; and Torres-Verdín and Habashy, 1994), and quasi-

linear approximation (Zhdanov and Fang, 1996). However, none of these methods have

been adapted to model the response of anisotropic media. In this dissertation, two novel

and efficient approximate algorithms are developed for modeling the EM response of

electrically anisotropic media: the SA (Smooth Approximation) and the Ho-GEBA.

In well logging, EM instruments measure EM fields or electric voltages instead of

electrical resistivity. The relation between EM response and formation resistivity is

generally nonlinear. However, current procedures used in the industry for the

interpretation of array induction data are based on a linear assumption and a sequence of

corrections and approximations intended to expedite the on-site estimation of apparent

resistivities. The desired commercial product is a set of resistivity curves that exhibit (a)

optimal vertical resolution, (b) minimal shoulder-bed effect, and (c) selective deepening

of the zone of response away from the borehole wall. Rigorous inversion procedures,

however, are needed to properly account for shoulder-bed and invasion effects. A number

of inversion strategies have been advanced thus far, but the challenge is still open to

develop expedient, efficient, and robust algorithms that could possibly be run on-site with

a minimum number of simplifying assumptions. In this dissertation, we develop efficient

algorithms to invert multi-frequency array induction data based on a nonlinear least-

6

squares minimization procedure. An inner-loop and outer-loop minimization technique is

developed to approach this problem.

1.2 OBJECTIVE OF THIS DISSERTATION

The objective of this dissertation is to develop novel and efficient algorithms to

simulate borehole EM measurements acquired in complex rock formations and to

develop rigorous and efficient algorithms for inversion of multi-frequency array

induction measurements. EM tools considered include multi-frequency array induction

and tri-axial induction instruments.

1.3 OUTLINE OF THIS DISSERTATION

The dissertation consists of nine chapters. Chapter 1 describes the motivation of

the dissertation.

Chapter 2 introduces the mathematical background of EM modeling, which

includes Maxwell’s equations, EM field computation, common EM sources used in EM

borehole logging, and EM fields excited by them in an unbounded homogeneous and

isotropic conductive background medium. In addition, dyadic Green’s functions are

introduced, as well as their explicit expressions in Cartesian coordinates for the case of an

unbounded homogeneous and isotropic conductive medium.

Chapter 3 develops analytical techniques to evaluate the integrals of 3D and 2D

spatial dyadic Green’s functions. The theory of the Method of Moments (MoM) and

7

associated computational issues to solve integral equations are also briefly introduced in

this chapter.

Chapter 4 develops full-wave and approximate modeling techniques for

simulating the response of multi-frequency induction tools in axisymmetric media. Both

the integral equation method and the finite-difference method are considered in this

chapter. Full-wave techniques include the BiCGSTAB(L)-FFT, the BiCGSTAB(L)-

FFHT, and the FDM, whereas approximate techniques include the PEBA and the Ho-

GEBA.

Chapter 5 details a novel BiCGSTAB(L)-FFT algorithm for simulating the

response of tri-axial induction tools in dipping and anisotropic rock formations. The

concepts of electrical anisotropy, coordinate system transformation, conductivity

averaging, and (block) Toeplitz matrix are summarized in this chapter.

Chapter 6 addresses the first novel and efficient approximate scheme – a smooth

approximation – for modeling anisotropic media. Numerical examples for simulating the

response of tri-axial induction tools in anisotropic rock formations are compared against

those obtained from analytical solutions, finite differences, and alternative

approximations.

Chapter 7 unveils the second novel and efficient approximate scheme – a high-

order generalized extended Born approximation (Ho-GEBA) – for EM modeling in

electrically anisotropic media. Numerical examples for simulating the response of tri-

8

axial induction tools in anisotropic rock formations are compared against those obtained

with analytical solutions, finite differences, and alternative approximations.

Chapter 8 focuses on the inversion of multi-frequency array induction

measurements. The theory of nonlinear least-squares inversion is detailed in this chapter.

Numerical inversion examples for multi-front mud-filtrate invasion models are studied

for two types of inversion strategies: “Resistivity Imaging,” and “Resistivity Inversion.”

An inner-loop and outer-loop minimization technique is developed for the inversion of

array induction data in this chapter.

Chapter 9 gives the summary and conclusions stemming from this dissertation

and provides recommendations for future research work.

9

Chapter 2: Electromagnetic Field Computation by Maxwell’s Equations

This chapter is an overview of the mathematical background of electromagnetic

(EM) modeling. We derive electric and magnetic fields using potentials for an

inhomogeneous medium and synthesize them in integral equation forms using the

concepts of dyadic Green’s functions. Also, we review the EM sources commonly used

in geophysical well logging, including solenoidal and toroidal sources, and derive EM

fields excited by them in homogeneous media. Finally, we derive expressions for the

fields excited by a point magnetic dipole and by a point electric dipole in an unbounded

homogeneous and isotropic conductive medium.

2.1 MAXWELL’S EQUATIONS OF ELECTROMAGNETISM

EM modeling consists of solving Maxwell’s equations with proper boundary

conditions using analytical or numerical methods. We shall assume a time harmonic

excitation of the form i te ω− , where ω is angular frequency, t is time, and 1i = − . Also,

we assume that an electric current source EJ and/or a magnetic current source M excites

an electric field E and a magnetic field H within some spatial domain 3τ ⊂ . Here EJ ,

M, E, and H are three dimensional, complex-valued vector fields. Maxwell’s equations

describe the relationships between these fields in terms of the constitutive properties of

the medium. In the frequency domain, the time-harmonic Maxwell’s equations for a

linear medium can be expressed as

iωμ∇× = −E H M , (2.1)

( ) Eiσ ωε′ ′∇× = − +H E J , (2.2)

10

( ) 0μ∇⋅ =H , (2.3)

and

( )ε ρ′∇ ⋅ =E , (2.4)

where σ ′ is ohmic conductivity, ε ′ is the electrical permittivity, and μ is magnetic

permeability. In general, we shall assume that each of these constitutive parameters is

real-valued and strictly positive within τ . Often, the permittivity and permeability are

referenced to those of free space via

0rε ε ε′ ′= , (2.5)

and

0rμ μ μ= . (2.6)

The dimensionless quantities rε ′ and rμ are referred to as, respectively, the

relative permittivity (dielectric constant) and relative permeability. In SI units, the free

space electrical permittivity 0ε and magnetic permeability 0μ are given by

70 4 10 /H mμ π −= × ,

and

120 8.854 10 /F mε −= × .

For convenience, we define a complex conductivity σ and complex permittivity

ε as follows:

iσ σ ωε′ ′= − , (2.7)

and

iσε εω′

′= + . (2.8)

11

In terms of ε , equation (2.2) can be rewritten as

Eiωε∇× = − +H E J . (2.9)

It is convenient to point out that the constitutive quantities could be isotropic or

anisotropic. In the derivations of this chapter, we will not differentiate the isotropy or

anisotropy in the constitutive quantities. However, in the chapters dealing with electrical

anisotropy, the anisotropic quantities will be denoted by “ ”.

In addition, using the concept of equivalent sources, the following relation holds

for the electric current density and magnetic current density:

1E iωμ= − ∇×J M . (2.10)

In the following sections, we shall consider only the case of EM excitation by

either an electrical current source or a magnetic current source.

2.2 DERIVATION OF THE INTEGRAL EQUATIONS FROM POTENTIALS

Potential theory is usually used to solve EM problems. This section is devoted to

deriving the fields excited by a magnetic current source and an electric current source in

an unbounded homogeneous and isotropic conductive medium, in which all the

constitutive quantities are constant. According to Harrington (1961), for an electric

current source the magnetic vector potential A is given by

( ) ( ) ( )0 0 0, Eg dτ

μ= ∫A r r r J r r , (2.11)

where 0( , )g r r is the scalar Green’s function, given by

( )0

00

,4

ikegπ

−

=−

r r

r rr r

, (2.12)

12

which is the solution of

( ) ( ) ( )2 20 0 0, ,g k g δ∇ + = − −r r r r r r , (2.13)

where

2 2k ω με= . (2.14)

According to Harrington (1961), the electric field E and magnetic field H can be

derived in terms of A as

( ) ( ) ( )( )2

1ik

ω ⎡ ⎤= + ∇ ∇⋅⎢ ⎥⎣ ⎦E r A r A r , (2.15)

and

( ) ( )1μ

= ∇×H r A r . (2.16)

Similarly, for a magnetic current source, the electrical vector potential F can be

written as

( ) ( ) ( )0 0 0,g dτ

ε= ∫F r r r M r r . (2.17)

In a similar fashion, the electric and magnetic fields can be written in terms of F

as

( ) ( ) ( )( )2

1ik

ω ⎡ ⎤= + ∇ ∇⋅⎢ ⎥⎣ ⎦H r F r F r , (2.18)

and

( ) ( )1ε

= − ∇×E r F r , (2.19)

respectively.

Substitution of equation (2.11) into equation (2.15) yields

13

( ) ( ) ( ) ( ) ( )0 0 0 0 0 02, ,E Eii g d g dkτ τ

ωμωμ= + ∇∇⋅∫ ∫E r r r J r r r r J r r . (2.20)

Given that

( ) ( )( ) ( )0 0 0 0( , ) ,E Eg g= Ι ⋅r r J r r r J r , (2.21)

one can write

( )( ) ( ) ( ) ( )0 0 0 0 0, ,E Eg g∇ ⋅ Ι ⋅ = −∇ ⋅r r J r r r J r . (2.22)

Thus, equation (2.20) can be written as

( ) ( ) ( )0 0 0,e

Ei G dτ

ωμ= ⋅∫E r r r J r r , (2.23)

where e

G is the electric Dyadic Green’s function, given by

( ) ( )0 02

1, ,e

G gk

⎛ ⎞= Ι + ∇∇⎜ ⎟⎝ ⎠

r r r r . (2.24)

The magnetic field can be expressed as

( ) ( ) ( )0 0 0, Eg dτ

= ∇× ∫H r r r J r r . (2.25)

If we denote the magnetic Dyadic Green’s function as

( ) ( )( )0 0, ,h

G g= ∇× Ιr r r r , (2.26)

equation (2.25) can be written as

( ) ( )0 0 0,h

EG dτ

= ⋅∫H r r J r r . (2.27)

It can be shown that the following relation holds between the electric dyadic

Green’s function and the magnetic dyadic Green’s function:

( ) ( )0 0, ,h e

G G= ∇×r r r r . (2.28)

14

Accordingly, for a magnetic current source,

( ) ( ) ( )0 0 0,e

i G dτ

ωε= ⋅∫H r r r M r r , (2.29)

and

( ) ( ) ( )0 0 0,h

G dτ

= ⋅∫E r r r M r r . (2.30)

Equations (2.29) and (2.30) can also be derived by substituting the equivalent electric

source of M into equations (2.23) and (2.27).

2.3 EM SOURCES IN GEOPHYSICAL WELL LOGGING

Electromagnetic tools are widely used in borehole geophysical logging. The EM

source EJ is classified into inductive or galvanic, depending on whether it is divergence

free or not (Lovell, 1993). The divergence-free source (typical of coils, also termed

inductive), relies on EM inductive coupling to generate the fields, while for the non-zero

divergence source (typical of contact electrodes, also termed galvanic), current will enter

the domain directly through the points of non-zero divergence.

Thus, depending on the type of EM source, borehole EM logging tools are

naturally classified as “Induction Tools”, which work typically at tens of KHz and are

sensitive to electrical conductivity, and “Laterolog Tools”, which work at very low/zero

frequencies and are sensitive to electrical resistivity. A vertical solenoidal source

(typically an electric current loop) is usually employed to simulate the measurements

acquired with an induction tool. In axisymmetric media, this source generates a field

which only contains the azimuthal electric field component, Eφ , the radial magnetic field

component, Hρ , and the vertical magnetic field component, zH , i.e., transverse electric

(TE) fields. However, since a vertical solenoidal source cannot detect electrical

15

anisotropy in a vertical well, tri-axial induction tools have been investigated and

commercialized in recent years (Kriegshauser, 2000; Rosthal et al., 2003). Moreover, a

toroidal source (typically a magnetic current loop) can be used in Measurement-While-

Drilling (MWD) tools (Gianzero et al., 1985). In axisymmetric media, this type of source

generates a field which only contains the azimuthal magnetic field component, Hφ , the

radial electric field component, Eρ , and the vertical electric field component, zE , i.e.,

transverse magnetic (TM) fields. An advantage of the toroidal source is that it is sensitive

to electrical anisotropy in vertical wells (Gianzero, 1999). Electrodes also generate TM

fields; thus, the toroidal source can be used to simulate the response of laterolog tools.

Electrode sources are not the subject of this dissertation. We shall focus our attention to

solenoidal sources only. Toroidal sources are considered for modeling the response of an

induction tool in axisymmetric media.

(a) (b)

Figure 2.1: Typical EM sources used in borehole EM logging. (a) solenoid; (b) toroid.

16

2.3.1 Solenoids

Figure 2.1a shows a typical solenoidal source used in borehole EM logging.

Solenoidal coils are the building blocks of induction devices used in geophysical logging.

A solenoidal source is characterized by its radius, a , number of turns, N, and the

impressed electrical current, EI . For a finite solenoidal source located at ( )s s,zρ in

axisymmetric media, the corresponding current density is expressed as

( ) ( ) ( )ˆ, NE E sz I a z zρ δ ρ δ= − −J z , (2.31)

where δ is the Dirac delta function, and z is the unit vector in the z-direction.

To compute the fields in an unbounded homogeneous and isotropic conductive

medium excited by a solenoidal source, substitution of equation (2.31) into equation

(2.23) yields

( ) ( ), , ; ,cE s sE z i NI ag z zφ ρ ωμ ρ ρ= , (2.32)

where cg is the Green’s function in axisymmetric media associated with an unbounded

homogeneous and isotropic conductive background. Accordingly, the magnetic field can

be computed from either equation (2.25) or equation (2.1).

2.3.2 Toroids

Figure 2.1b shows a typical toroidal source (Amperian loop). It is characterized

by the radius of the coil in the ρ φ− plane, a , and the radius of the coil in the zρ −

plane, tr , the number of turns of the coil, N, and the electrical current supported by the

coil, EI . Following EM induction principles, the electrical current supported by the coil

will induce a magnetic current mI in the φ direction.

17

According to Ampere’s law,

N ESI

∂⋅ =∫ H dl , (2.33)

where S∂ refers to the Amperian loop. For small tr , one has

/ 2EH NI aφ π= . (2.34)

Since the magnetic current density can be written as

( ) ( )ˆm sI a z zδ ρ δ= − −M φ , (2.35)

it follows from equation (2.1) that

ˆ i Hφωμ∇× = = −E Mφ . (2.36)

By combining equations (2.34), (2.35) and (2.36), and by integrating over the

surface of the toroidal coil, one obtains

( )2 N / 2m t EI i r I aωμ π π= − . (2.37)

Now, by substituting equation (2.35) into equation (2.29), one can derive the magnetic

field due to a toroidal source in an unbounded homogeneous and isotropic conductive

medium, namely

( ), ; ,cm s sH i I ag z zφ ωε ρ ρ= , (2.38)

which is a dual expression of equation (2.32). The corresponding electric field can be

computed from either equation (2.9) or equation (2.30).

2.4 FIELDS DUE TO POINT SOURCES IN AN UNBOUNDED HOMOGENEOUS AND

ISOTROPIC CONDUCTIVE MEDIUM

In the previous section, we have shown how to compute the EM fields excited by

a finite source. In the borehole logging industry, a fundamental design consideration

18

when building induction tools is to ensure that the effects of finite-size coils do not

significantly change the response of the tool from that of a pure dipole source. In this

section, we describe how to compute the EM fields in an infinite, uniform conductive

medium excited by point dipole sources.

The current density associated with a point solenoidal source polarized in the u-

direction can be expressed as

( ) ( )ˆ δ= −J r u r rE E sM , (2.39)

where δ is the Dirac delta function, r is the location of the observation point, sr is the

location of the source, EM is the moment of the source, and u is the unit vector in the u-

direction. For a solenoidal source,

( )2E EM NI aπ= . (2.40)


( ) ( ) ( )

( )

0 0 0ˆ,

ˆ, .

τωμ δ

ωμ

= ⋅ −

= ⋅

∫E r r r u r r r

r r u

e

E s

e

E s

i M G d

i M G (2.41)

Similarly,

( ) ( ) ˆ,= ⋅H r r r uh

E sM G . (2.42)

For a point toroidal source with moment MM , one has

( )2

2E t

M

NI rM

aπ

π= . (2.43)

The corresponding magnetic current density can be expressed as

( ) ( )ˆ δ= −M r u r rM sM . (2.44)


19

( ) ( ) ˆ,e

M si M Gωε= ⋅H r r r u . (2.45)

Similarly,

( ) ( ) ˆ,h

M sM G= ⋅E r r r u . (2.46)

In summary, making use of the principle of duality, one has

//

s t E M

s t E M

M MM Mμ ε= ⋅

= ⋅E HH E

, (2.47)

where the subscript s refers to the EM fields excited by a solenoidal source, while the

subscript t refers to the EM fields excited by a toroidal source.

In Cartesian coordinates, assume that ( )x, y, z and ( )0 0 0x , y , z are the observation

and source points, respectively, and x , y , and z are the unit normal vectors. The explicit

expressions of the EM fields due to a point magnetic dipole source in an unbounded

homogeneous and isotropic conductive medium can be written as:

Case 1: x-directed magnetic dipole

0 03ˆ ˆ(1 ) [( ) ( ) ]

4ikREi M ikR z z y y

R eωμπ

= − − − − −E y z , (2.49)

and

( )( )20 00 0 0

1 23 2 2 2

( ) ( )( )ˆ ˆ ˆ ˆ4

ikRE x x y yx x x x z zMR R R Re α α

π⎡ ⎤− −⎛ ⎞− − −

= + + +⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

H x y z x .

(2.50)

Case 2: y-directed magnetic dipole

0 03ˆ ˆ(1 ) [( ) ( ) ]

4ikREi M ikR x x z z

R eωμπ

= − − − − −E z x , (2.51)

and

20

( ) ( ) ( )20 0 0 0 0

1 23 2 2 2

( ) ( )ˆ ˆ ˆ ˆ

4ikRE x x y y y y y y z zM

R R R Re α απ

⎡ ⎤⎛ ⎞− − − − −⎢ ⎥= + + +⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

H x y z y .

(2.52)

Case 3: z-directed magnetic dipole

0 03ˆ ˆ(1 ) [( ) ( ) ]

4ikREi M ikR y y x x

R eωμπ

= − − − − −E x y , (2.53)

and

( )( ) 20 00 0 0

1 23 2 2 2

( )( ) ( )ˆ ˆ ˆ ˆ4

ikRE y y z zx x z z z zMR R R Re α α

π⎡ ⎤− −⎛ ⎞− − −

= + + +⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

H x y z z ,

(2.54)

where

( ) ( ) ( )2 2 220 0 0R x x y y z z= − + − + − , (2.55)

2 21 3 3k R ikRα = − − + , (2.56)

and

2 22 1k R ikRα = + − . (2.57)

The explicit expressions for a point electrical dipole in an unbounded uniform

conductive medium can be derived from equation (2.47).

2.5 EXPLICIT EXPRESSIONS FOR THE DYADIC GREEN’S FUNCTIONS

In Cartesian coordinates, the explicit expressions for the electrical and magnetic

dyadic Green’s functions can be easily derived as

21

20 0 0 0 0

1 2 1 12 2 2

20 0 0 0 0

1 1 2 13 2 2 2 2

20 0 0 0 0

1 1 1 22 2 2

( ) ( )( ) ( )( )

( )( ) ( ) ( )( )1( , )4

( )( ) ( )( ) ( )

eikR

x x x x y y x x z zR R R

x x y y y y z z y yG eR k R R R

x x z z z z y y z zR R R

α α α α

α α α απ

α α α α

⎡ ⎤− − − − −+⎢ ⎥

⎢ ⎥− − − − −⎢ ⎥

= +⎢ ⎥⎢ ⎥

− − − − −⎢ ⎥+⎢ ⎥⎣ ⎦

0r r ,

(2.58)

and

0 0

0 03

0 0

0 ( )1( , ) (1 ) ( ) 0

4( ) 0

hikR

z z y yG ikR e z z x x

Ry y x x

π

− − −⎡ ⎤⎢ ⎥= − − − − −⎢ ⎥⎢ ⎥− − −⎣ ⎦

0r r , (2.59)

respectively, where R, 1α and 2α are given by equations (2.55) through (2.57).

Equations (2.58) and (2.59) indicate that, in Cartesian coordinates, the dyadic Green’s

functions only depend on the distance between the source and observation points. This

feature is important for designing efficient algorithms to solve large-scale EM problems.

2.6 CONCLUSIONS

This chapter provided an overview of the mathematical background of EM

modeling. It also described EM sources commonly used in the geophysical borehole

logging industry and showed how to compute the corresponding EM fields in an

unbounded uniform background medium using the dyadic Green’s functions. Explicit

expressions in Cartesian coordinates were derived for EM fields excited by point dipole

sources in an unbounded uniform conductive medium.

22

Chapter 3: Analytical Techniques to Evaluate the Integrals of 3D and 2D Spatial Dyadic Green’s Functions

This chapter gives an overview of the Method of Moments (MoM) used to solve

the integral equation of EM scattering. The MoM involves the evaluation of the integrals

of the spatial dyadic Green’s functions, which often requires large computer resources.

We introduce analytical techniques to evaluate the integrals of the spatial dyadic Green’s

functions (Gao, Torres-Verdín, and Habashy, 2005). These techniques not only eliminate

the singularity involved in the evaluation of the integrals, but also substantially reduce

computation times.

The Dyadic Green’s function is in general viewed as a generalized, or distribution

function. A commonly used procedure to evaluate its volume integral is the principal-

volume method, in which an infinitesimal volume around the singularity is excluded from

the integration volume. In this chapter, we develop a general analytical technique to

evaluate the integral of the dyadic Green’s function without the need to specify an

exclusion volume.

The newly derived expressions accurately integrate the singularity of the Green’s

functions and can be used for integration over any shape of spatial discretization cell. We

derive explicit expressions for the integral of the 3D dyadic Green’s function over a

sphere and over a general rectangular block. Similar expressions are obtained for 2D

dyadic Green’s functions over a cylinder and over a general rectangular cell. It is shown

that for spherical/circular cells, simple analytical expressions can be derived, and these

expressions are exactly the same as those obtained using the principal-volume method.

Furthermore, the analytical expressions for the integral of the dyadic Green’s function are

valid regardless of the location of the observation point, both inside and outside the

integration domain. Because the expressions only involve surface integrals/line integrals,

23

their evaluation can be performed very efficiently with a high degree of accuracy. We

compare our expressions against the equivalent volume approximation for a wide range

of frequencies and cell sizes. These comparisons clearly confirm the efficiency and

accuracy of our integration technique.

It is also shown that the cubic cell (3D) and square cell (2D) can be accurately

approximated with an equivalent spherical cell and circular cell, respectively, over a wide

range of frequencies. The approximation can be performed analytically, and the results

can be written as the value of the dyadic Green’s function at the center of the cell

multiplied by a “geometric factor.” We describe analytical procedures to derive the

corresponding geometric factors.

3.1 INTRODUCTION

Integral equations have been widely used to solve EM scattering and related

problems, such as those arising in antenna design (Balanis, 1996), geophysical subsurface

sensing (Fang et al., 2003; and Gao et al., 2003), biomedical engineering (Livesay and

Chen, 1974), and optical scattering (Hoekstra et al., 1998), to name a few. A fundamental

component of integral equations is the dyadic Green’s function, which makes it possible

for the integral equation to exhibit a simple analytical form. This feature is particularly

important for multiple scattering problems, in which the complex physics of a vector field

is properly synthesized by the dyadic Green’s function (Chew, 1989).

The study of dyadic Green’s functions has attracted numerous researchers in the

EM community (Van Bladel, 1961; Harrington, 1968; Livesay and Chen, 1974;

Yaghjian, 1980; Lee et al., 1980; Yaghjian, 1982; Su, 1987; and Chew, 1989). Dyadic

Green’s functions can be classified into a spatial representation, in which the function is

24

written in terms of simple algebraic expressions in the coordinate space r, and an

eigenfunction representation, in which the function is written in terms of vector wave

functions or eigenfunctions suitable for the assumed geometry (Chew, 1989). Chew

(1989) provides a review of these two representations of the dyadic Green’s function and

of their mutual relationships.

This chapter is devoted to the spatial representation of the dyadic Green’s

function in an unbounded homogeneous and isotropic conductive medium. In such a

case, the dyadic Green’s function can be written in closed form using vector and scalar

potential theory (Van Bladel, 1961). A fundamental feature of the dyadic Green’s

function is its singularity in the source region. This feature has been extensively studied

by Van Bladel (1961) and Yaghjian (1980, 1982), among others. Their work has shown

that the dyadic Green’s function can be viewed as a generalized function involving a

Dirac delta function singularity that is only valid in the distribution sense. Its evaluation

is customarily approached using the so-called “Principal Volume Method,” in which an

exclusion volume is specified around the singularity. As for the principal volume

integration, an equivalent volume (a sphere in three dimensions and a circle in two

dimensions) approximation has been frequently used for some special shapes of the

discretization cell given that, for those cases, analytical solutions are available to simplify

the calculation (Livesay and Chen, 1974; Lee, et al., 1980; Su, 1987). This chapter

reviews the derivation of these expressions for the equivalent volume approximation in

the source region and outside the source region using various methods, both for 3D and

2D cases. We remark that the singularity property of the dyadic Green’s function has led

25

to the formulation of the Extended Born Approximation in EM scattering (Habashy et al.,

1993; and Torres-Verdín and Habashy, 1994).

Thus far, the principal volume method is about the only method available to solve

the integral of the dyadic Green’s function in the source region. In this chapter, we show

that the principal volume is actually not necessary. Our derivation is valid for the

integration over any shape of discretization cell and for any spatial location, both for 3D

and 2D domains. We give explicit expressions for the integration over a spherical cell

(circular cell in 2D) and a general rectangular cell (square cell in 2D). It is shown that

the expression for a sphere/circle obtained from our general formula yields the same

solution of the principal volume method specialized for the same cell. Our formula is

also validated by comparing numerical integration results to those obtained from an

already-validated code and the principal volume method for a wide range of frequencies

and cell sizes, both in the source region and outside the source region. For a cubic/square

cell, when the observation points are outside the source region, we derive a geometric

factor solution, which is nothing but the dyadic Green’s function in the geometric center

of the cell multiplied by a geometrical factor. The formulas reported in this chapter have

been used to simulate tri-axial borehole induction tool measurements acquired in

inhomogeneous and electrically anisotropic rock formations (Fang et al., 2003; Gao et

al., 2003; and Gao et al., 2004).

3.2 INTEGRAL EQUATIONS AND THE METHOD OF MOMENTS (MOM)

In Chapter 2, we have described the procedures used to simulate EM scattering

fields using integral equations and dyadic Green’s functions. For an arbitrary

26

inhomogeneous medium, if we consider the medium as the superposition of a background

medium and an anomalous medium, the total electric field can be expressed as the

superposition of the background/incident and scattered fields.

Assume an EM source that exhibits a time harmonic dependence of the type

i te ω− . The magnetic permeability of the medium equals that of free space, 0μ . Thus, the

integral equation for electric and magnetic fields can be written in general as (Van

Bladel, 1961; Hohmann, 1975; and Yaghjian, 1980),

( ) ( ) ( ) ( ) ( )0 0 0 0,e

b G dτ

σ= + ⋅Δ ⋅∫E r E r r r r E r r , (3.1)

and

( ) ( ) ( ) ( ) ( )0 0 0 0,h

b G dτ

σ= + ⋅Δ ⋅∫H r H r r r r E r r , (3.2)

where ( )E r and ( )H r are the electric and magnetic field vectors, respectively, at the

measurement location, r. In the above equations, ( )bE r and ( )bH r are the electric and

magnetic field vectors, respectively, associated with a homogeneous, unbounded, and

isotropic background of dielectric constant rbε and Ohmic conductivity bσ ′ . Accordingly,

the background complex conductivity is given by 0b b rbiσ σ ωε ε′= − , and the

wavenumber, bk , of the background is given by 2 20 0 0 0b b rb bk i iωμ σ ω μ ε ε ωμ σ ′= = + .

Note that the electric dyadic Green’s function is different from that given in

Chapter 2 because the term “ 0iωμ ” is explicitly included here. The magnetic dyadic

Green’s function remains the same. For clarity, the electric dyadic Green’s function

included in equations (3.1) is expressed in a closed form as

27

( ) ( )0 0 02

1, ,e

b

G i gk

ωμ⎛ ⎞

= Ι + ∇∇⎜ ⎟⎝ ⎠

r r r r , (3.3)

where the scalar Green’s function ( )0,rrg satisfies the wave equation

)(),(),( 002

02 rrrrrr −−=+∇ δgkg b , (3.4)

and whose solution can be explicitly written as

( )0

00

,4

bikegπ

−

=−

r r

r rr r

. (3.5)

The electric dyadic Green’s function is the solution of

( ) ( ) ( )20 0 0 0, ,

e e

bG k G iωμ δ∇×∇× − = − Ιr r r r r r . (3.6)

On the other hand, the magnetic dyadic Green’s function is related to the electric

Green’s tensor through the expression

),(1),(0

00 rrrreh

Gi

G ×∇=ωμ

. (3.7)

Finally, the tensor

0b riσ σ σ σ ω ε ε′Δ = − Ι = Δ − Δ Ι , (3.8)

is the complex conductivity contrast within scatterers, with rbrr εεε −=Δ and

Ι′−′=′Δ bσσσ , where Ι is the unity dyad.

In the 2D case, the electric dyadic Green’s function can be expressed as

( ) ( )0 02

1, ,e

b

G i gk

ωμ⎛ ⎞

= Ι + ∇∇⎜ ⎟⎝ ⎠

ρ ρ ρ ρ , (3.9)

where

( ) ( ) ( )10 0 0,

4 big H k= −ρ ρ ρ ρ , (3.10)

28

is the 2D scalar Green’s function, ( ) ( )10H ⋅ is the Hankel function of the first kind and

order zero, and ρ is the location vector in 2D Cartesian coordinates. This chapter is

devoted to equations (3.3) and (3.9) only. The description is focused to the 3D dyadic

Green’s function because the 2D dyadic Green’s function follows the same principles.

Equation (3.1) is a Fredholm integral equation of the second kind. A solution of

this equation can be obtained by the Method of Moments (MoM). The MoM is a

numerical technique that has been used extensively in the solution of EM boundary value

problems. Many excellent texts have been written on this subject (Harrington, 1968). A

characteristic of this technique is that it leads to a full matrix equation which can be

solved by matrix inversion. To solve equation (3.1), two sets of functions are used in the

MoM: basis functions and weighting functions. Such functions are formally defined as

follows:

(1) Basis functions. A set of N basis function, 1 2, , , Nf f f , in the spatial domain

τ is chosen. Then the unknown field ( )E r is expressed as a linear combination of these

basis functions, i.e.,

( ) ( )1

N

n nn

f=

= ∑E r E r . (3.11)

The linear combination of the basis functions should properly represent ( )E r in the

domain. Substitution of equation (3.11) into equation (3.1) yields

( ) ( ) ( ) ( ) ( )0 0 0 01 1

,N N e

n n n n bn n

f G f dτ

σ= =

− ⋅Δ ⋅ =∑ ∑∫E r r r r E r r E r , (3.12)

where the unknown coefficients 1, 2 , , NE E E are to be determined.

(2) Weighting functions (testing functions).

29

A set of N weighting functions ( ) ( ) ( )1 2, , , Nw w wr r r is chosen. Multiplication

of equation (3.12) by ( )mw r and subsequent integration of both sides of the equation

over the spatial domain τ yields

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

0 0 0 01

,N e

n m m n nn

b m

f w d w G f d d

w d

τ τ τ

τ

σ=

⎡ ⎤− ⋅Δ ⋅⎢ ⎥⎣ ⎦

=

∑ ∫ ∫ ∫

∫

r r r r r r r r r r E

E r r r. (3.13)

This last equation can be written in matrix form as

1

N

mn n bmn

G=

⋅ =∑ E E , m = 1, 2, …, N, (3.14)

where

( ) ( ) ( ) ( ) ( ) ( )0 0 0 0,e

mn n m m nG f w d w G f d dτ τ τ

σ= Ι − ⋅Δ∫ ∫ ∫r r r r r r r r r r , (3.15)

and

( ) ( )bm b mw dτ

= ∫E E r r r . (3.16)

Basis functions can use either full-domain functions or subsectional basis

functions. In this dissertation, we choose to use one kind of subsectional basis functions,

i.e., pulse basis function, given by

( )10

nn

iff

otherwiseτ∈⎧

= ⎨⎩

rr , (3.17)

where the domain τ has been divided into N sub-domains nτ , n = 1, 2, …, N.

Weighting functions can use the Galerkin (in which, ( ) ( )n nw f=r r ) or point

matching method. In this dissertation, we choose to use the point matching method

30

because of its simplicity. In the point matching method, the weighting function is written

as

( ) ( )m mw δ= −r r r , (3.18)

where m = 1, 2, …, N, and δ is the Dirac delta function.

Substitution of equations (3.17) and (3.18) into equations (3.15) and (3.16) yields

( ) ( )0 0 0,n

e

mn mn mG G dτ

δ σ= Ι − ⋅Δ∫ r r r r , (3.19)

and

( )bm b m=E E r , (3.20)

where

10mn

m nm n

δ=⎧

= ⎨ ≠⎩. (3.21)

From the computational point of view, a naïve implementation of the MoM

requires extensive computer resources, namely,

(1) Matrix Inversion. In general, the linear system of equations (3.14) embodies

a full complex matrix equation. The solution of a full matrix equation of order N by

matrix inversion, such as LU decomposition, requires ( )3NΟ floating point operations.

This requirement makes the MoM impractical to solve large-scale EM problems.

(2) Computer Memory Storage. The matrix consists of 29N entries. For large-

scale problems, such a condition imposes a large physical memory requirement, in the

order of tens or hundreds of gigabytes. Such a memory requirement cannot be met by

most of the current computer platforms.

31

(3) Matrix-Filling Time. From equation (3.19), one can easily observe that to fill

the matrix, each entry must be calculated using 3D numerical integrations. For large-scale

problems, matrix-filling time can be computationally intensive, of the order of days or

years of CPU time.

For instance, in a numerical modeling excercise involving 1 million discretization

cells, 0.2 CPU seconds are needed to compute 10,000 entries (each entry is a 3 by 3

tensor) of the linear-system matrix. Table 3.1 summarizes two of the most significant

computer requirements associated with this hypothetical example. Quite obviously, such

requirements place rather impractical constraints on most of computer platforms

commercially available today.

Matrix-filling time 231 days Memory storage (single complex precision) 67,054 GigaBytes

In this chapter, we introduce several techniques to expedite matrix filling

operations using efficient formulas to evaluate the integrals of the dyadic Green’s

function. Issues related to memory storage and matrix inversion will be addressed in

Chapter 5.

3.3 EVALUATION OF THE INTEGRALS OF THE DYADIC GREEN’S FUNCTIONS

The most popularly used method to evaluate the integrals of the dyadic Green’s

function is the principal volume method. In addition, in this section we introduce a

general integral evaluation technique that circumvents the principal volume method.

Table 3.1: Matrix-filling time and computer storage associated with the assumption of 1 million discretization cells, and 0.2 CPU seconds needed to compute 10,000 entries (each entry is a 3 by 3 tensor) of the MoM linear-system matrix.

32

3.3.1 The Principal Volume Method

When the source point, 0r , and the observation point, r , coincide, equation (3.1)

yields an improper integral because the double derivatives implicit in the ∇∇ operator

acting on ( )0,rrg in equation (3.3) give rise to a singularity of the type ( )301O −r r as

0→r r .

Work by numerous researchers has proved that, although the improper integral in

equation (3.1) does not converge in the classical sense when 0→r r , its principal value

integral does exists. The following form of the integral has been previously suggested

(Van Bladel, 1961; Yaghjian, 1980 and 1982; and Chew, 1989):

( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0 0PV ,

e

bb

LG d

τ

σσ

σ⋅Δ ⋅

= + ⋅Δ ⋅ −∫r E r

E r E r r r r E r r , (3.22)

where [ ] [ ]0 00limPV d d

δδτ τ ττ −→⋅ = ⋅∫ ∫r r stands for the principal volume integral, and δτ is a

small exclusion volume. Because the exclusion volume will cause discontinuous currents

on the surface of the volume, surface charges will accumulate on the surface, which will

be responsible for an electrostatic field inside the volume. This EM field will persist no

matter how small the volume is, and will remain a function of the shape of the volume

(Chew, 1989). The third term in equation (3.22) gives the correction due to the

accumulated charges on the surface, in which L is a tensor that depends on the volume

shape.

When using the method of moments (Harrington, 1968) to solve equation (3.22),

one faces the problem of evaluating the following improper integral in the 3D case:

( ) ( )0 0,e

G G dτ

= ∫r r r r , (3.23)

33

whereas in the 2D case, one has

( ) ( )0 0,e

G G dτ

= ∫ρ ρ ρ ρ . (3.24)

When 0→r r , using the principal volume method, equation (3.23) can be

evaluated as

( ) ( )0 0,e

b

LG PV G dτ σ

= −∫r r r r . (3.25)

Similarly, for the 2D case, when 0→ρ ρ , equation (3.25) can be evaluated as

bS

e LdGPVGσ

200 ),()( −= ∫ ρρρρ , (3.26)

where 2L is a tensor that depends on the shape of the exclusion element.

When /r ρ is outside the source region, no singularity exists, and the integral in

equations (3.23) and (3.24) can be evaluated using a numerical method, or by analytical

means for some special cases.

3.3.1.1 Equivalent Volume Solution for a Singular Cell

To evaluate the principal value integral in equation (3.25), numerical methods

need to be used in general. However, if we take the exclusion volume as a small sphere

(small circle for the 2D case), and approximate the cell using a spherical cell (circular cell

for the 2D case) with the equivalent volume (area for the 2D case), an analytical solution

ensues which has been shown to be a very good approximation for the cubic cell (Livesay

and Chen, 1974; and Lee, et al., 1980) (square cell for the 2D case).

34

In the 3D case, let 1x x= , 2x y= , 3x z= , 1, 2,3p = , and 1,2,3q = . The solution of

equation (3.25) can be written as

( )2 1 33

bpq ik apq b

b

G ik a eδσ

⎡ ⎤= − −⎣ ⎦ , (3.27)

where a is the radius of the equivalent sphere, given by

133

4 la aπ

⎛ ⎞= ⎜ ⎟⎝ ⎠

, (3.28)

and la is the side length of the cubic cell. A simplified derivation procedure is given in

Supplement 3A. A more detailed derivation can be found in Livesay and Chen (1974),

Lee et al. (1980) and Su (1987). Also, an alternative derivation procedure that does not

require the specification of the exclusion volume is given in section 3B.1 of Supplement

3B.

In the 2D case, assume an infinite square cylinder parallel to the z direction. The

solution of equation (3.26) can be written as

( )( ) ( ) ( ) ( )1 11 11 1 ˆ ˆ14 4

b b b b

b b

i k aH k a i k aH k aG

π πσ σ

⎡ ⎤= − + Ι +⎢ ⎥

⎢ ⎥⎣ ⎦ρ zz , (3.29)

where ( ) ( )11H ⋅ is the Hankel function of the first kind and order 1. A derivation of

equation (3.29) is given in section 3C.1 of Supplement 3C.

3.3.1.2 Geometric Factor Solution for Non-Singular Cells

When /r ρ is not in the source region, expressions for the equivalent volume/area

approximation can also be derived analytically. As shown in Supplement 3B and

Supplement 3C, the solution of equations (3.23) and (3.24) can be written as

35

( )3( ) ,e

cG C G=r r r , (3.30)

and

( )2( ) ,e

cG C G=ρ ρ ρ , (3.31)

where cr is the coordinate of the geometric center of the spherical/cubic cell, cρ is the

coordinate of the geometric center of the circular/square cell, and 3C is the 3D

“Geometric Factor” for the integral of the 3D dyadic Green’s function, given by

( ) ( )3 2

sin4 cosbb

b b

k aaC k ak k aπ ⎡ ⎤

= −⎢ ⎥⎣ ⎦

, (3.32)

where a is the radius of the cell for the case of a sphere and is given by equation (3.28)

for the case of a cubic cell.

In equation (3.31), 2C is the 2D “Geometrical Factor” for the integral of the 2D

dyadic Green’s function, and is given by

( )2 12

bb

aC J k akπ ′

′= , (3.33)

where a′ is the radius of the cell for the case of a circular cell and is given by

laaπ

′′ = , (3.34)

for the case of a square cell, where la′ is the side length of the square. Supplement 3B

gives a detailed derivation of equation (3.30) and Supplement 3C gives a detailed

derivation of equation (3.31).

Equations (3.30) and (3.31) are referred to as “Geometric Factor Solutions” in

this dissertation.

36

3.3.2 A General Integral Evaluation Technique

The principal volume method uses an exclusion volume to circumvent the

singularity associated with the dyadic Green’s function in the source region. However, it

can be shown that the exclusion volume is not needed and that the integral in equations

(3.23) and (3.24) can be evaluated in a straightforward manner.

First, substitution of equation (3.3) into equation (3.23) gives

( ) ( )0 2

1

b

G i fk

ωμ⎛ ⎞

= Ι + ∇∇⎜ ⎟⎝ ⎠

r r , (3.35)

where

( ) 0 0( , )f g dτ

= ∫r r r r . (3.36)

From equation (3.4), one obtains

( ) ( ) ( )20 0 02 2

1 1, ,b b

g gk kδ= − − − ∇r r r r r r . (3.37)

Substitution of equation (3.37) into equation (3.36) gives

( )∫ ∫ −−∇−=τ τ

δ 002002

2

1),(1)( rrrrrrr dk

dgk

fbb

. (3.38)

Using the relationship

0

−∇=∇ , (3.39)

where the subscript 0 stands for the derivative with respect to the source coordinates, one

has

( ) ( ) ( )0 0 02 2

1 1,b b

f g d Dk kτ

= − ∇ ⋅ ∇ −∫r r r r r , (3.40)

where

37

( )⎩⎨⎧

=01

rD ττ

∉∈

rr

. (3.41)

Using the theorem,V S

dvψ ψ∇ =∫ ∫ ds , where ψ is an arbitrary scalar function,

one can immediately arrive at

( ) ( ) ( ) ( )0 0 02 2

1 1ˆ,b b

f g ds Dk kτ∂

= ∇ ⋅ −∫r r r n r r , (3.42)

where τ∂ is the closed surface of the integration volumeτ , and n is the outgoing unit

normal vector on the boundary τ∂ .

Following a similar procedure, from equation (3.36) one obtains

( ) ( )0 0 0ˆ, ( )f g dsτ∂

∇∇ = −∇∫r r r n r . (3.43)

Substitution of equation (3.42) and equation (3.43) into equation (3.35) yields

0 0 0

0 0

ˆ( ) ( , ) ( )1( )ˆ( , ) ( )b

D g dsG

g dsτ

τσ

∂

∂

⎡ ⎤− Ι + Ι∇• −⎢ ⎥=⎢ ⎥∇⎢ ⎥⎣ ⎦

∫∫ 0

r r r n rr

r r n r. (3.44)

For the 2D case, the corresponding equation can be derived as

0 0 0

0 0

ˆ( ) ( , ) ( )1( )ˆ( , ) ( )

S

bS

D g dlG

g dlσ∂

∂

⎡ ⎤− Ι + Ι∇•⎢ ⎥=⎢ ⎥−∇⎢ ⎥⎣ ⎦

∫∫ 0

ρ ρ ρ n ρρ

ρ ρ n ρ, (3.45)

where ( )D ρ is given by

( )10

SD

S∈⎧

= ⎨ ∉⎩

ρρ

ρ. (3.46)

For the case of the integral of the magnetic dyadic Green’s function one has

( ) ( )0 0,h h

G G dτ

= ∫r r r r . (3.47)


38

( ) ( ) ( )0 00 0

1 1,h e

G G d Gi iτωμ ωμ

= ∇× = ∇×∫r r r r r . (3.48)

Substitution of equation (3.44) into equation (3.48) together with the property that

the curl of the gradient is zero, one obtains

( ) ( ) ( )0 0 02

1 ˆ,h

b

G g dsk τ∂

⎡ ⎤= ∇×∇⋅ Ι⎢ ⎥⎣ ⎦∫r r r n r . (3.49)

So far, we have transformed the volume/surface integral into surface/line

integrals. For finite-size cells, the distance between the volume surface/ surface boundary

and the cell center will never be zero, which indicates that by making use of equations

(3.44) and (3.45) the singularity has been completely eliminated. Another notable

advantage of equations (3.44) and (3.45) is that they can be used to evaluate the integrals

at any point in space, not only the self-interaction term. These formulas provide a way to

reduce computation times compared to any alternative numerical method because the

surface/line integral evaluation is much more efficient than the volume/surface integral

evaluation.

Equations (3.44), (3.45) and (3.49) are universal for any shape of cell. Depending

on the cell shape, τ∂ / S∂ corresponds to different surfaces/lines, whereby different

explicit expressions can be obtained depending on the shape of the cell. For the 3D case,

Appendices 3D and 3E give the derivations of the explicit expressions for the integration

of the electrical dyadic Green’s function over a spherical cell and over a general

rectangular block cell, respectively. Supplement 3G gives the explicit expressions for the

integration of the magnetic dyadic Green’s function over a general rectangular block cell.

39

As shown in Supplement 3D, the general formula given here leads to exactly the

same solution (equation (3.27)) as the principal volume method for the case of a spherical

cell. This confirms the validity of the general formula.

For the 2D case, Supplement 3F gives the derivation of the explicit expressions

resulting from the integration of the electrical dyadic Green’s function over a general

rectangular cell (rectangular cylinder).

3.3.3 Numerical Validation

For a spherical/circular cell, in Appendices 3B, 3C, and 3D, we show that the new

evaluation method and the principal volume method yield identical analytical

expressions. The validity of the new method becomes apparent for spherical/circular

cells.

For a cubic cell, because the accuracy of the equivalent volume approximation is

relatively high (Livesay and Chen, 1974; and Lee et al., 1980), we choose to compare the

results from the general formula against those obtained from the equivalent volume

approximation for a wide range of frequencies and sizes of discretization cell. The

explicit expressions derived in Supplement 3E are evaluated using a Gauss-Legendre

quadrature integration formula. For all the numerical examples considered in this chapter,

the background Ohmic conductivity bσ is taken to be 0.5 S/m, and the dielectric constant

rbε is taken to be 1. The frequency range considered is up to 1 GHz. Figure 3.1 shows

simulation results versus bk a for a singular cell, where a is the radius of the equivalent

sphere. On that figure, “General Formula” refers to the expressions given in Supplement

3E, while “PV Appr.” refers to equation (3.27). Because for a cubic cell all the diagonal

40

entries are equal, only the first diagonal entry G(1,1) is shown in Figure 3.1. The upper

panel describes the amplitude, while the lower panel describes the phase in radians.

From Figure 3.1, one can easily draw the conclusion that the two results are in good

agreement, even at very high frequencies. This exercise not only validates the general

formula, but also shows that a sphere is truly a good approximation for the case of a cube

with the same volume.

For measurement points located outside the source region, we compare the results

between the geometrical factor solution for the equivalent volume approximation

(equation 2.30) and the exact formula developed in this chapter. We assume a cell with

dimensions equal to dx = 0.2 m, dy = 0.2 m, and dz = 0.2 m, and that the cell is located at

the origin. The observation point is located at (0.2, 0.4, 0.6) m, which is intentionally

chosen to be very close to the cell. Figures 3.2 through 3.4 describe the six independent

components of the integrated tensor. Figure 3.2 shows G(1,1) and G(1,2); Figure 3.3

shows G(1,3) and G(2,2); Figure 3.4 shows G(2,3) and G(3,3). Both the amplitude and

phase are shown in these figures. The values of the integrals of the Green’s function are

plotted against bk a , where a is the distance between the observation point and the

center of the cell. Figures 3.2 through 3.4 confirm the accuracy of the geometric factor

solution. When bk a is very large, the amplitude of the results reaches the truncation

error, and a small discrepancy ensues as shown in Figures 3.2 through 3.4. Because the

geometric factor solution is analytical, it is very efficient from a computational point of

view.

For a general rectangular element, the equivalent volume approximation does not

provide accurate results. Intuitively, for a general rectangular cell the three diagonal

41

entries are not equal, while the equivalent volume approximation always provides equal

diagonal entries. Results from the general formula are compared to those obtained with

an already validated code. The code was developed for the computation of the Green’s

function in a layered medium, and has been optimized to make a compromise between

accuracy and computation speed. It is claimed that the code provides accurate results to

the second significant digit. Such a code is referred to as ‘External Code’ in this chapter.

Table 3.2 compares the results for a cell with dimensions dx=0.1 m, dy=0.3 m, and

dz=0.5 m. Results for 100 Hz and 1 MHz are listed in the table. These results are

matched within 1%. It is believed that the results obtained with the general formula are

superior to those of the external code because there is no approximation involved in the

evaluations other than the numerical integration.

3.4 CONCLUSIONS

This chapter provided an overview of the Method of Moments (MoM) to solve

integral equations of EM scattering. We proposed analytical techniques to accelerate the

evaluations of the integrals of the dyadic Green’s functions (which in general remain

improper integrals).

We have developed a technique for the accurate and efficient evaluation of

integrals of the dyadic Green’s function without the use of an exclusion volume. The

formula developed in this chapter can be used for any cell shape and for any frequency.

Explicit expressions have been derived in three dimensions for the cases of a spherical

cell and a general rectangular block, and in two dimensions for the cases of a circular cell

and a general rectangular cell. Likewise, a geometrical factor solution was derived for the

42

cases of a spherical cell and a cubic cell. The general integration formula developed in

this chapter is universal for any cell shape and frequency.

10-3 10-2 10-1 100 101 102

100

Am

plitu

de o

f G(1

,1)

10-3 10-2 10-1 100 101 102-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

|kba|

Phas

e of

G(1

,1)

General FormulaPV Appr.

General FormulaPV Appr.

Figure 3.1: Comparison of integration results obtained with the general formula and the principal-volume approximation assuming a singular cubic cell. The upper panel shows the amplitude, and the bottom panel shows the phase. In both panels, a is the radius of the equivalent sphere.

43

10-3 10-2 10-1 100 101 10210-20

10-10

100

Ampl

itude

[G(1

,1)]

10-3 10-2 10-1 100 101 102-2

0

2

phas

e[G

(1,1

)]

10-3 10-2 10-1 100 101 10210-20

10-10

100

Ampl

itude

[G(1

,2)]

10-3 10-2 10-1 100 101 102-2

0

2

|kba|

Phas

e[G

(1,2

)]General FormulaGeometric Factor

Figure 3.2: Comparison of integration results obtained with the general formula and the geometric factor solution assuming a non-singular cubic cell. Two of the six independent components, G(1,1) and G(1,2) are shown on the figure,including amplitude and phase. The cell size is (0.2, 0.2, 0.2) m, and the cell is located at the origin. The observation point is located at (0.2, 0.4, 0.6). In both figures, a is the distance between the cell and the observation point.

44

10-3 10-2 10-1 100 101 10210-20

10-10

100

Am

plitu

de[G

(1,3

)]

10-3 10-2 10-1 100 101 102-2

0

2

Phas

e[G

(1,3

)]

10-3 10-2 10-1 100 101 10210-20

10-10

100

Am

plitu

de[G

(2,2

)]

10-3 10-2 10-1 100 101 102-2

0

2

|kba|

Pha

se[G

(2,2

)]

General FormulaGeometric Facor

Figure 3.3: Comparison of integration results obtained with the general formula and the geometric factor solution assuming a non-singular cubic cell. Two of the six independent components, G(1,3) and G(2,2) are shown on the figure, including amplitude and phase. The cell size is (0.2, 0.2, 0.2) m, and the cell is located at the origin. The observation point is located at (0.2, 0.4, 0.6). In both figures, a is the distance between the cell and the observation point.

45

10-3 10-2 10-1 100 101 10210-20

10-10

100

Am

plitu

de[G

(2,3

)]

10-3 10-2 10-1 100 101 102-2

0

2

|kba|

Pha

se[G

(2,3

)]

10-3 10-2 10-1 100 101 10210-20

10-10

100

|kba|

Ampl

itude

[G(3

,3)]

10-3 10-2 10-1 100 101 102-2

0

2

|kba|

Pha

se[G

(3,3

)]General FormulaGeometric Factor

External Code General Formula Freq (Hz) Quantity Real Part Imaginary Part Real Part Imaginary Part

G(1,1) -1.52296340 4.1860558E-06 -1.521676900 4.158624051E-06 G(2,2) -0.34986061 5.1357538E-06 -0.349633068 5.107571269E-06

100

G(3,3) -0.12866613 5.7510060E-06 -0.128690049 5.722195510E-06 G(1,1) -1.52979820 3.3293307E-02 -1.528510330 3.301968426E-02 G(2,2) -0.35686129 4.2770579E-02 -0.356630176 4.248940200E-02

1M

G(3,3) -0.13590108 4.8885193E-02 -0.135921240 4.859771207E-02

Figure 3.4: Comparison of integration results obtained with the general formula and the geometric factor solution assuming a non-singular cubic cell. Two of the six independent components, G(2,3) and G(3,3) are shown on the figure, including amplitude and phase. The cell size is (0.2, 0.2, 0.2) m, and the cell is located at the origin. The observation point is located at (0.2, 0.4, 0.6). In both figures, a is the distance between the cell and the observation point.

Table 3.2: Comparison of integration results obtained with the general formula and with an external code assuming a rectangular block of dimensions equal to (0.1, 0.3, 0.5)m. The external code has been previously validated to render accurate results up to the second significant digit. Results from two frequencies, i.e., 100 Hz and 1MHz, are described in the table.

46

Supplement 3A: Derivation of the Expression of the Equivalent Volume Approximation for a Singular Cell Using the Principal Volume Method

For a spherical exclusion volume (Chew, 1990), one has

13

L = Ι . (3A-1)

According to the theory of tensor analysis, the operator ∇∇ can be expressed as

23

, 1

ˆ ˆp qp q p qx x=

∂∇∇ =

∂ ∂∑ x x . (3A-2)

Thus, equation (3.3) can be written as

( ) ( )2

0 0 02 0 0

1, ,epq pq

b p q

G i gk x x

ωμ δ⎛ ⎞∂

= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠r r r r . (3A-3)

Note that the derivatives indicated in equation (3A-3) have been written with respect to

those of the source coordinates. By combining equations (3A-1), (3A-2) and (3A-3),

equation (3.23) becomes

( ) ( )0 01PV ,

3e

pq pq pqb

G G dτ

δσ

= −∫r r r r

( ) ( )20

0 0 0 00 0

,1 1PV , PV3pq pq

b p q b

gi g d d

x xτ τωμ δ δ

σ σ∂

= + −∂ ∂∫ ∫

r rr r r r

0 1 21 1

3 pqb b

i D Dωμ δσ σ

= + − . (3A-4)

We consider the following cases of solution:

Case 1: when p q≠ , the first term and the third term on the right-hand side of

equation (3A-4) vanish because of the property of the Kronecker δ function. It can also

be shown that 2D vanishes because the derivative of ( )0,g r r with respect to a particular

47

axis is an odd function about that axis due to symmetry of the coordinates (Su, 1987).

Thus, all the off-diagonal elements vanish.

Case 2: When p=q, because ( )0,g r r is a function of 0−r r only, one can define

a spherical coordinate system centered at r. Thus, without loss of generality, we set =r 0 .

It then follows that

( ) ( )0

0 00

,4

bik reg g rrπ

= =r r . (3A-5)

In a spherical coordinate system, 1D can be written as

( )1 0 0PV , pqD g dτ

δ= ∫ r r r

02

0 0 0 0 00 00

1 lim sin4

ba ik rr e dr d d

π π

ηηφ θ θ

π →= ∫ ∫ ∫

00 00

lim ba ik rr e drηη→

= ∫ . (3A-6)

Integration by parts yields

( )1 2

1 1 1bik ab

b

D ik a ek

⎡ ⎤= − −⎣ ⎦ . (3A-7)

Because of symmetry, 2D remains invariant under a rotation of the Cartesian

coordinates, and this gives (Su, 1987)

2 2 2

2 0 0 02 2 20 0 0

g g gD PV d PV d PV dx y zτ τ τ

∂ ∂ ∂= = =

∂ ∂ ∂∫ ∫ ∫r r r . (3A-8)

Therefore,

( )22 0 0 0

1 ,3

D PV g dτ

= ∇∫ r r r . (3A-9)

In spherical coordinates, one has

48

( ) ( )02 20 0 02

0 0 0

,1,g

g rr r r

∂⎛ ⎞∂∇ = ⎜ ⎟∂ ∂⎝ ⎠

r rr r . (3A-10)

Substitution of equation (3A-10) into equation (3A-9), together with some simple

manipulations yields

( )21 1 1 .3

bik abD ik a e⎡ ⎤= − − −⎣ ⎦ (3A-11)

Finally, substitution of equations (3A-7) and (3A-11) into equation (3A-4) yields

( )2 1 33

bpq ik apq b

b

G ik a eδσ

⎡ ⎤= − −⎣ ⎦ . (3A-12)

Supplement 3B: Derivation of the Analytical Solution for the Integrals of the Electrical Dyadic Green’s Function for a Spherical Volume

To derive a solution of equation (3.35) for a sphere, first ( )0,g r r is expanded in

terms of spherical Bessel and Hankel functions (Habashy et al., 1993), namely,

( ) ( )00

, 2 14

b

n

ikg nπ

∞

=

= +∑r r

( )( ) ( ) ( )∑

= +−

⋅n

m

mn

mnm PP

mnmn

00coscos

!! θθχ

( )[ ] ( ) ( )( )( ) ( )( )⎩

⎨⎧

≤≥

−⋅00

10

10

0cosrrrkhrkjrrrkhrkj

mbnbn

bnbnφφ , (3B-1)

where

r=r , (3B-2)

49

0r=0r , (3B-3)

and

⎩⎨⎧

≠=

=0201

mm

mχ . (3B-4)

Assume that the radius of the sphere is equal to a . For convenience, but without

loss of generality, we set the origin at the center of the sphere.

3B.1. Derivation for a Singular Cell

For a singular cell, r is located within the sphere. Using equation (3B-1), equation

(3.26) can be written using spherical coordinates as

( ) ( ) ( )( ) ( )

0 0

!2 1 cos

4 !

nmb

m nn m

n mikf n Pn m

χ θπ

∞

= =

−= +

+∑ ∑r

( ) ( )2

0 0 0 0 00 0cos cos sinm

nd m d Pπ πφ φ φ θ θ θ⎡ ⎤× −⎣ ⎦∫ ∫

( ) ( ) ( ) ( ) ( ) ( )1 12 20 0 0 0 0 00

a r

n b n b n b n brj k r r h k r dr h k r r j k r dr⎡ ⎤× +⎢ ⎥⎣ ⎦∫ ∫ . (3B-5）

Using the properties of the sinusoidal functions and Legendre functions, one can easily

conclude that in equation (3B-5) only the zero-th order terms of the summation indices m

and n remain. This leads to

( ) ( ) ( ) ( ) ( ) ( ) ( )1 12 20 0 0 0 0 0 0 0 0 00

a r

b b b b brf ik j k r dr r h k r h k r dr r j k r⎡ ⎤= +⎢ ⎥⎣ ⎦∫ ∫r . (3B-6)

Making use of the properties

( )z

zzj sin0 = , (3B-7)

and

50

( )( ) izezizh −=1

0 , (3B-8)

together with some additional algebraic manipulations yields

( ) ( ) ( )2

sin1 1 1 b bik ab

b b

k rf ik a e

k k r⎡ ⎤

= − + −⎢ ⎥⎣ ⎦

r . (3B-9)

To derive the expression for a singular cell, we make use of the power series

expansion of ( )sin bk r , i.e.,

( ) ( ) ( )( )

2 11

1

sin 12 1 !

nn b

bn

k rk r

n

−∞+

=

= −−∑ . (3B-10)

Substitution of equation (3B-10) into equation (3B-9) yields

( ) ( ) ( ) ( )( )

2

21

1 1 1 1 12 1 !

b

nnik a b

bnb

k rf ik a e

k n

∞

=

⎡ ⎤⎛ ⎞⎢ ⎥= − + − + −⎜ ⎟

⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦∑r . (3B-11)

From (3B-11), one obtains

( ) ( )20

1lim 1 1 bik abr

b

f ik a ek→

⎡ ⎤= − + −⎣ ⎦r , (3B-12)

and

( ) ( )0

1lim 13

bik abr

f ik a e→∇∇ = − − Ιr . (3B-13)

Substitution of equations (3B-12) and (3B-13) into equation (3.35) yields

( ) ( )1 21 13

bself

ik ab

b

G ik a eσ

⎡ ⎤= − + − Ι⎢ ⎥⎣ ⎦r . (3B-14)

This last expression is identical to equations (3.27) and (3A-12).

51

3B.2. Expression for Non-Singular Cells

When r lies outside the sphere, r is always greater than 0r . Thus,

( ) ( ) ( ) ( )1 20 0 0 0 00

a

b b bf ik h k r dr r j k r= ∫r . (3B-15)

Using equations (3B-7) and (3B-8), one easily arrives at

( ) 3 4

bik ref Crπ

=r , (3B-16)

where

( ) ( )3 2

sin4 cosbb

b b

k aaC k ak k aπ ⎡ ⎤

= −⎢ ⎥⎣ ⎦

. (3B-17)

Substitution of equation (3B-16) into equation (3.35) yields

3 0 2

1( )4

bik r

b

eG C ik r

ωμπ

⎡ ⎤⎛ ⎞= Ι + ∇∇⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦r . (3B-18)

If the origin is not at the center of the sphere, we assume that the coordinate of the

center of the sphere is cr . Equation (3B-18) then becomes

3 0 2

1( )4

bik

b c

eG C ik

ωμπ

−⎡ ⎤⎛ ⎞= Ι + ∇∇⎢ ⎥⎜ ⎟ −⎝ ⎠⎣ ⎦

cr r

rr r

. (3B-19)

By comparing equations (3B-19) and (3.3) together with equation (3.5), one

arrives at the expression

( )3( ) ,e

cG C G=r r r , (3B-20)

where

0 2

1( , )4

bike

cb c

eG ik

ωμπ

−⎛ ⎞= Ι + ∇∇⎜ ⎟ −⎝ ⎠

cr r

r rr r

, (3B-21)

and 3C is given by equation (3B-17).

52

The interesting feature of equation (3B-20) is that the integral of the dyadic

Green’s function is nothing but the dyadic Green’s function evaluated at the cell’s

geometrical center multiplied by a constant. The constant, 3C , is a function of the

geometry of the cell, and is here referred to as “3D Geometric Factor.” For a sphere, a is

the radius of the sphere, while for a cubic cell, a is given by

133

4 la aπ

⎛ ⎞= ⎜ ⎟⎝ ⎠

, (3B-22)

where la is the side length of the cube.

Supplement 3C: Derivation of the Analytical Solution for the Volume Integrals of the Electrical Dyadic Green’s Function for an Infinitely

Long Circular Cylinder

The integral of the 2D dyadic Green’s function over a cylindrical cross-section S

can be written as

)(1)( 20 ρρ fk

iGb

⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇+Ι= ωμ , (3C-1)

where ( )ρf is given by

( ) ( ) ( )10 0 04 bs

if H k d= −∫ρ ρ ρ ρ . (3C-2)

Assume that the origin is at the center of the cross-section of the cylinder. Using

the addition theorem of Hankel functions (Torres-Verdín and Habashy, 1994), one can

write

53

( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0

0

10 0

10 0

10 0

imm b m b

mb

imm b m b

m

J k H k eH k

J k H k e

φ φ

φ φ

ρ ρ ρ ρ

ρ ρ ρ ρ

+∞−

=−∞

+∞−

=−∞

⎧ ≥⎪⎪− = ⎨⎪ ≤⎪⎩

∑

∑ρ ρ , (3C-3)

where

ρ = ρ , (3C-4)

and

0 0ρ = ρ . (3C-5)

3C.1. Evaluation of a Singular Cell

For a singular cell, equation (3C-2) can be written as

( ) ( ) ( ) ( ) ( ) ( )1 (1)0 0 0 0 0 0 0 0 0 002

a

b b b bif H k J k d J k H k d

ρ

ρ

π ρ ρ ρ ρ ρ ρ ρ ρ⎡ ⎤= +⎢ ⎥⎣ ⎦∫ ∫ρ . (3C-6)

Using the properties of Bessel functions and of their Wronskian, one arrives at

( )( ) ( ) ( )1

102

12

bb

b b

i aH k af J k

k kπ

ρ= − +ρ . (3C-7)

To derive the expressions for a singular cell, which is tantamount to = cρ ρ ,

or 0ρ → , we make use of the series expansion of ( )0 bJ k ρ , i.e.,

( ) ( ) ( )2

00

/ 21

! !

kk b

bk

kJ k

k kρ

ρ∞

=

= −∑ . (3C-8)

It then follows that

( )( ) ( )11

20

1lim2

b

b b

i aH k af

k kρ

π→

= − +ρ , (3C-9)

and

54

( ) ( ) ( )10lim

4b

tbi k af H k a

ρ

π→

−∇∇ = Ι1ρ , (3C-10)

where

ˆ ˆ ˆ ˆtΙ = +xx yy . (3C-11)

Finally, for a singular cell, one has

( )( ) ( ) ( ) ( )1 11 11 1 ˆ ˆ14 4

b b b b

b b

i k aH k a i k aH k aG

π πσ σ

⎡ ⎤= − + Ι +⎢ ⎥

⎢ ⎥⎣ ⎦ρ zz . (3C-12)

3C.2. Evaluation of Non-Singular Cells

For non-singular cells, ρ is always greater than 0ρ . Thus, in a cylindrical

coordinate system, one has

( ) ( ) ( ) ( )10 0 0 0 002

a

b bif H k J k dπ ρ ρ ρ ρ= ∫ρ . (3C-13)

Using the integration formula for Bessel functions, one obtains

( ) ( ) ( ) ( )11 02 b b

b

i af J k a H kkπ ρ=ρ . (3C-14)

If the origin is not at the center of the cross-section of the cylinder, we assume

that cρ is the location of the center. Thus, equation (3C-14) can be written as

( ) ( ) ( ) ( )11 02 b b c

b

i af J k a H kkπ

= −ρ ρ ρ . (3C-15)

Using equation (3.10), it follows that

( ) ( )2 , cf C g=ρ ρ ρ , (3C-16)

where

55

( )2 12

bb

aC J k akπ

= (3C-17)

is the geometrical factor for the 2D Green’s function. Substitution of equation (3C-16)

into equation (3.9) yields

( )2( ) ,e

cG C G=ρ ρ ρ . (3C-18)

This result is analogous to that obtained for the integral of the 3D Green’s function.

In equation (3C-17), a is the radius of the cross-section of the cylinder for the

case of a circular cylinder cell. For a rectangular cylinder, a is given by

laaπ

= , (3C-19)

where la is the side length of the cross-section of the cylinder.

Supplement 3D: Derivation of the Explicit Expressions for the Integral of the Electrical Dyadic Green’s Function over a Spherical Cell from the

General Formula

Assume that the sphere has a radius equal to a . According to equation (3E-26),

only the entries sxxv , s

yyv , and szzv are needed to perform the integration. All the off-diagonal

elements become zero for a spherical cell. Because of symmetry, the following relation

exists for sxxv , s

yyv , and szzv :

s s sxx yy zzv v v= = . (3D-1)

According to equation (3.44), one has

( ) ( )0 0 0ˆ,s s sxx yy zzv v v g ds

τ∂+ + = ∇⋅ ∫ r r n r . (3D-2)

56

Thus, by combining equations (3D-1) and (3D-2) one obtains

( ) ( )0 0 01 ˆ,3

s s sxx yy zzv v v g ds

τ∂= = = ∇ ⋅ ∫ r r n r . (3D-3)

Equation (3D-3) can also be written as

( ) ( )0 0 0 01 ˆ,3

s s sxx yy zzv v v g ds

τ∂= = = − ∇ ⋅∫ r r n r . (3D-4)

In Cartesian coordinates, one has

00 0 0

ˆ ˆ ˆx y z∂ ∂ ∂

∇ = + +∂ ∂ ∂

x y z . (3D-5)

The relations between the orthonormal vectors in Cartesian and spherical coordinates

include

000000cosˆsinsinˆcossinˆˆ θφθφθ zyxr ++= , (3D-6)

and

0 0 0 0 00

sin cos sin sin cosr x y z

θ φ θ φ θ∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂. (3D-7)

Starting from equations (3D-6) and (3D-7), one arrives at

( ) ( ) ( )( )0 0 0 0 0 00

ˆ, ,g ds g dsrτ τ∂ ∂

∂∇ ⋅ =

∂∫ ∫r r n r r r . (3D-8)

Using equation (3A-5), one obtains

( ) ( ) ( ) 00

0 0 20 0 0

1,

4π−∂ ∂

⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦∂ ∂r r

bik rbik r e

g g rr r r

. (3D-9)

Substitution of equation (3D-9) into equation (3D-8) yields

( ) ( ) ( ) ( )0 0 0 0 02

1ˆ, 1

4

b

b

ik ab ik a

b

ik a eg ds ds ik a e

aτ τπ∂ ∂

−∇ ⋅ = = −∫ ∫r r n r . (3D-10)

Moreover, substitution of equation (3D-10) into equation (3D-4) yields

57

( )1 13

bik as s sxx yy zz bv v v ik a e= = = − . (3D-11)

Finally, substitution of equation (3D-11) into equation (3E-26) yields

( )1 2( ) 1 13

bs

ik ab

b

G ik a eσ

⎡ ⎤= − + − Ι⎢ ⎥⎣ ⎦r . (3D-12)

This last expression is identical to equations (3.27) and (3A-12).

Supplement 3E: Derivation of the Explicit Expressions of the Integral of the Electrical Dyadic Green’s Function over a General Rectangular

Block using the General Formula

Assume a Cartesian coordinate system, in which the center of a rectangular cell is

located at ( ), ,c c cx y z and the observation point is located at ( ), ,x y z . The side lengths of

the cell in the x, y, and z directions are 2a , 2b, and 2c, respectively. Equation (3.44) can

be written as

1 ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ( ) ( ) ( ) ) ( ( ) ( ) ( ) )x y z x y zb

G D l l l l l lσ

⎡ ⎤= − Ι + Ι∇• + + −∇ + +⎢ ⎥⎣ ⎦r r r x r y r z r x r y r z ,

(3E-1)

where

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∫ ∫

+

−

+

−

by

by

cz

czx

Rik

x

Rik

xc

c

c

c

xbxb

dydzR

eR

el 0021

21

41π

r , (3E-2)

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∫ ∫

+

−

+

−

ax

ax

cz

czy

Rik

y

Rik

yc

c

c

c

ybyb

dxdzR

eR

el 0021

21

41π

r , (3E-3)

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∫ ∫

+

−

+

−

ax

ax

by

byz

Rik

z

Rik

zc

c

c

c

zbzb

dxdyR

eR

el 0021

21

41π

r , (3E-4)

58

[ ] 2/120

20

21 )()()( zzyyaxxR cx −+−+−−= , (3E-5)

[ ] 2/120

20

22 )()()( zzyyaxxR cx −+−++−= , (3E-6)

[ ] 2/120

20

21 )()()( zzxxbyyR cy −+−+−−= , (3E-7)

[ ] 2/120

20

22 )()()( zzxxbyyR cy −+−++−= , (3E-8)

[ ] 2/120

20

21 )()()( xxyyczzR cz −+−+−−= , (3E-9)

and

[ ] 2/120

20

22 )()()( xxyyczzR cz −+−++−= . (3E-10)

By making use of the general expressions for the gradient and divergence,

equation (3E-1) can be recast as

( )( ) ( ) ( ) ( )

1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

xx yy zz

xx xy xz yxb

yy yz zx zy zz

D v v v

G v v v v

v v v v vσ

⎧ ⎫⎡ ⎤− + + + Ι −⎣ ⎦⎪ ⎪⎪ ⎪= − − − −⎨ ⎬⎪ ⎪− − − −⎪ ⎪⎩ ⎭

r r r r

r r xx r xy r xz yx

yy yz zx zy zz

, (3E-11)

where the unit dyad Ι can be written in terms of three orthonormal vectors, i.e.,

ˆ ˆ ˆ ˆ ˆ ˆΙ = + +xx yy zz . (3E-12)

Accordingly, equation (3E-11) can be written as

( )ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )

1 ˆ ˆ ˆ ˆ ˆ ˆ[ ( ) ( ( )]

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )

yy zz xy xz

yx xx zz yzb

zx zy xx yy

D v v v v

G v D v v v

v v D v vσ

⎧ ⎫⎡ ⎤− + + − −⎣ ⎦⎪ ⎪⎪ ⎪= − + − + + −⎨ ⎬⎪ ⎪

⎡ ⎤− − + − + +⎪ ⎪⎣ ⎦⎩ ⎭

r r r xx r xy r xz

r yx r r) r yy yz

zx zy r r r zz

. (3E-13)

Using matrix notation,

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++−−−−++−−−−++−

=)()()()()(

)()()()()()()()()()(

1)(rrrrr

rrrrrrrrrr

r

yyxxzyzx

yzzzxxyx

xzxyzzyy

b vvDvvvvvDvvvvvD

Gσ

,

59

(3E-14)

where

[ ]1 2

1 20 03 3

1 2

( ) ( )

( ) ( 1) ( ) ( 1)14

b x b xc c

c c

xx x

ik R ik Ry b z c c b x c b xy b z c

x x

v lx

x x a e ik R x x a e ik R dz dyR Rπ

+ +

− −

∂=∂

⎡ ⎤⎛ ⎞− − − − + −= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-15)

1 21 2

0 03 31 2

( ) ( )

( ) ( 1) ( ) ( 1)14

b y b yc c

c c

yy y

ik R ik Rx a z c c b y c b y

x a z cy y

v ly

y y b e ik R y y b e ik Rdz dx

R Rπ+ +

− −

∂ ⎡ ⎤= ⎣ ⎦∂

⎡ ⎤⎛ ⎞− − − − + −⎢ ⎥= −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-16)

[ ]1 2

1 20 03 3

1 2

( ) ( )

( ) ( 1) ( ) ( 1)14

b z b zc c

c c

zz z

ik R ik Rx a y b c b z c b zx a y b

z z

v lz

z z c e ik R z z c e ik R dy dxR Rπ

+ +

− −

∂=∂

⎡ ⎤⎛ ⎞− − − − + −= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-17)

[ ]

( )1 2

1 20 0 03 3

1 2

( ) ( )

( 1) ( 1)1 ,4

b x b xc c

c c

yx x

ik R ik Ry b z c b x b xy b z c

x x

v ly

e ik R e ik Ry y dz dyR Rπ

+ +

− −

∂=∂

⎡ ⎤⎛ ⎞− −= − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-18)

[ ]

( )1 2

1 20 0 03 3

1 2

( ) ( )

( 1) ( 1)1 ,4

b x b xc c

c c

zx x

ik R ik Ry b z c b x b xy b z c

x x

v lz

e ik R e ik Rz z dz dyR Rπ

+ +

− −

∂=∂

⎡ ⎤⎛ ⎞− −= − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-19)

60

( )1 2

1 20 0 03 3

1 2

( ) ( )

( 1) ( 1)1 ,4

b y b yc c

c c

xy y

ik R ik Rx a z c b y b y

x a z cy y

v lx

e ik R e ik Rx x dz dx

R Rπ+ +

− −

∂ ⎡ ⎤= ⎣ ⎦∂⎡ ⎤⎛ ⎞− −⎢ ⎥= − −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-20)

( )1 2

1 20 0 03 3

1 2

( ) ( )

( 1) ( 1)1 ,4

b y b yc c

c c

zy y

ik R ik Rx a z c b y b y

x a z cy y

v lz

e ik R e ik Rz z dz dx

R Rπ+ +

− −

∂ ⎡ ⎤= ⎣ ⎦∂⎡ ⎤⎛ ⎞− −⎢ ⎥= − −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-21)

[ ]

( )1 2

1 20 0 03 3

1 2

( ) ( )

( 1) ( 1)1 ,4

b z b zc c

c c

xz z

ik R ik Rx a y b b z b zx a y b

z z

v lx

e ik R e ik Rx x dy dxR Rπ

+ +

− −

∂=∂

⎡ ⎤⎛ ⎞− −= − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-22)

[ ]

( )1 2

1 20 0 03 3

1 2

( ) ( )

( 1) ( 1)1 ,4

b z b zc c

c c

yz z

ik R ik Rx a y b b z b zx a y b

z z

v ly

e ik R e ik Ry y dy dxR Rπ

+ +

− −

∂=∂

⎡ ⎤⎛ ⎞− −= − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫

r r

(3E-23)

and ( )D r is given by equation (3.41).

It can be easily shown that

zyyz

zxxz

yxxy

vvvv

vv

==

=

. (3E-24)

61

Equation (3E-14) is valid for any observation-point location. When 0=r r , one

obtains 1 2x xR R= , 1 2y yR R= , and 1 2z zR R= , whereupon

0xy xz yzv v v= = = , (3E-25)

which is equivalent to the conclusion drawn from Supplement 3A when p q≠ .

Therefore, for a singular cell one obtains

( )( ) ( )

( ) ( )( ) ( )

1 0 01 0 1 0

0 0 1

s syy zzs

s sxx zz

b s sxx yy

v vG v v

v vσ

⎡ ⎤− + +⎢ ⎥= − + +⎢ ⎥⎢ ⎥− + +⎣ ⎦

r rr r r

r r,

(3E-26)

where

0 02

( 1)( )2

b xik Rb cs b xxx b c

x

e ik Rav dz dyRπ −

⎡ ⎤⎛ ⎞−= − ⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫r , (3E-27)

0 02

( 1)( )

2

b yik Ra c b ys

yy a cy

e ik Rbv dz dxRπ − −

⎡ ⎤⎛ ⎞−⎢ ⎥= − ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∫ ∫r , (3E-28)

0 02

( 1)( )2

b zik Ra bs b zzz a b

z

e ik Rcv dy dxRπ − −

⎡ ⎤⎛ ⎞−= − ⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫r , (3E-29)

1/ 22 2 20 0xR a y z⎡ ⎤= + +⎣ ⎦ , (3E-30)

1/ 22 2 20 0yR b x z⎡ ⎤= + +⎣ ⎦ , (3E-31)

and

1/ 22 2 20 0zR c x y⎡ ⎤= + +⎣ ⎦ . (3E-32)

62

Supplement 3F: Derivation of the Explicit Expressions for the Integral of the Electrical Dyadic Green’s Function over a General Rectangular

Cell (Rectangular Cylinder)

Using vector and tensor analysis techniques, equation (3.45) can be written as

( ) ( ) ( ) ( ){ } ( ) ( ){ }1 ˆ ˆ ˆ ˆx y x yb

G D l l l lσ

⎡ ⎤ ⎡ ⎤= − +∇⋅ + Ι −∇ +⎣ ⎦ ⎣ ⎦ρ ρ ρ x ρ y ρ x ρ y . (3F-1)

Assume that the location of the center of the rectangular cell is given by

ˆ ˆc c cx y= +ρ x y , (3F-2)

and that the side lengths of the rectangular cell are 2a and 2b in the x and y directions,

respectively. It then follows that

( ) ( ) ( ) ( ) ( )( )1 10 1 0 2 04

c

c

y b

x b x b xy b

il H k H k dyρ ρ+

−

⎡ ⎤= −⎢ ⎥⎣ ⎦∫ρ , (3F-3)

and

( ) ( ) ( ) ( ) ( )( )1 10 1 0 2 04

c

c

x a

y b y b yx a

il H k H k dxρ ρ+

−

⎡ ⎤= −⎢ ⎥⎣ ⎦∫ρ , (3F-4)

where

( ) ( )2 21 0x cx x a y yρ = − − + − , (3F-5)

( ) ( )2 22 0x cx x a y yρ = − + + − , (3F-6)

( ) ( )2 21 0y cy y b x xρ = − − + − , (3F-7)

and

( ) ( )2 22 0y cy y b x xρ = − + + − . (3F-8)

Substitution of these last expressions into equation (3F-1) yields

63

( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

ˆ ˆ ˆ ˆ1 ˆ ˆ ˆ ˆ

ˆ ˆ

yy xy

yx xxb

xx yy

D v v

G v D v

D v vσ

⎧ ⎫⎡ ⎤− + − −⎣ ⎦⎪ ⎪⎪ ⎪= + − + +⎡ ⎤⎨ ⎬⎣ ⎦⎪ ⎪⎡ ⎤− + +⎪ ⎪⎣ ⎦⎩ ⎭

ρ ρ xx ρ xy

ρ ρ yx ρ ρ yy

ρ ρ ρ zz

, (3F-9)

or, in matrix notation,

( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

01 0

0 0

yy xy

yx xxb

xx yy

D v vG v D v

D v vσ

⎛ ⎞− + −⎜ ⎟

= − − +⎜ ⎟⎜ ⎟− + +⎝ ⎠

ρ ρ ρρ ρ ρ ρ

ρ ρ ρ,

(3F-10)

where

( ) ( )( )xx xv lx∂

=∂

ρ ρ

( ) ( ) ( ) ( ) ( ) ( )1 11 1 1 2

01 24

c

c

y b c b x c b xby b

x x

x x a H k x x a H kik dyρ ρ

ρ ρ+

−

⎧ ⎫⎡ ⎤− − − +⎪ ⎪= − −⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

∫ ,

(3F-11)

( ) ( )( )yy yv ly∂

=∂

ρ ρ

( ) ( ) ( ) ( ) ( ) ( )1 1

1 1 1 20

1 24c

c

x a c b y c b ybx a

y y

y y b H k y y b H kik dxρ ρ

ρ ρ+

−

⎧ ⎫⎡ ⎤− − − +⎪ ⎪⎢ ⎥= − −⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

∫ ,

(3F-12)

and

( ) ( ) ( )( )xy yx yv v lx∂

= =∂

ρ ρ ρ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }1 1 1 10 1 0 2 0 3 0 44 b xy b xy b xy b xy

i H k H k H k H kρ ρ ρ ρ= − − + − ,

(3F-13)

64

where

( ) ( )2 21xy c cx x a y y bρ = − − + − − , (3F-14)

( ) ( )2 22xy c cx x a y y bρ = − − + − + , (3F-15)

( ) ( )2 23xy c cx x a y y bρ = − + + − + , (3F-16)

and

( ) ( )2 24xy c cx x a y y bρ = − + + − − , (3F-17)

In equation (3F-10),

( )10

self cellD

otherwise⎧

= ⎨⎩

ρ . (3F-18)

For a self–cell, one has

( )( )

( )( ) ( )

1 0 01 0 1 0

0 0 1

syys

sxx

b s sxx yy

vG v

v vσ

⎛ ⎞− +⎜ ⎟

= − +⎜ ⎟⎜ ⎟− + +⎝ ⎠

ρρ ρ

ρ ρ, (3F-19)

where

( )( ) ( )11

02b b xs b

xx bx

H kik av dyρ

ρ−

⎧ ⎫⎡ ⎤⎪ ⎪= ⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

∫ρ , (3F-20)

( )( ) ( )11

02a b ys b

yy ay

H kik bv dxρ

ρ−

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥= ⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

∫ρ , (3F-21)

2 20x a yρ = + , (3F-22)

and

2 20y b xρ = + . (3F-23)

65

Supplement 3G: Derivation of the Explicit Expressions for the Integral of the Magnetic Dyadic Green’s Function over a Rectangular Block Cell

Starting from equations (3.49) and (3E-11), one can easily show that

( ) ( )( )2

1 ˆ ˆ ˆ ˆ ˆ ˆh

xx yy zzb

G v v vk

⎡ ⎤= ∇× + + + +⎣ ⎦r xx yy zz . (3G-1)

Further manipulation of the above equation yields

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

00

01)( 2

yzxz

yzxy

xzxy

b

h

wwwwww

krG , (3G-2)

where

)( zzyyxxxy vvvz

w ++∂∂

−= , (3G-3)

)( zzyyxxxz vvvy

w ++∂∂

= , (3G-4)

and

)( zzyyxxyz vvvx

w ++∂∂

−= . (3G-5)

After some algebraic manipulations, one obtains

1 21 2

0 0 05 51 2

( ) ( )1( ) ( )4

b x b xc c

c c

ik R ik Ry b z c x c x cxx y b z c

x x

x x a e x x a ev z z dz dyz R R

α απ

+ +

− −

⎡ ⎤− − − +∂= − −⎢ ⎥∂ ⎣ ⎦

∫ ∫ ,

(3G-6)

66

1 2

1 20 0 05 5

1 2

( ) ( )1( ) ( )4

b x b xc c

c c

ik R ik Ry b z c x c x cxx y b z c

x x

x x a e x x a ev y y dz dyy R R

α απ

+ +

− −

⎡ ⎤− − − +∂= − −⎢ ⎥∂ ⎣ ⎦

∫ ∫ ,

(3G-7)

1 21 2

0 05 51 2

1( )4

b x b xc c

c c

ik R ik Ry b z c x xxx y b z c

x x

e ev dz dyx R R

β βπ

+ +

− −

⎡ ⎤∂= −⎢ ⎥∂ ⎣ ⎦

∫ ∫ , (3G-8)

1 21 2

0 0 05 51 2

( ) ( )1( ) ( )4

b y b yc c

c c

ik R ik Rx a z c y c y c

yy x a z cy y

y y b e y y b ev z z dz dx

z R Rα α

π+ +

− −

⎡ ⎤− − − +∂= − −⎢ ⎥

∂ ⎢ ⎥⎣ ⎦∫ ∫ ,

(3G-9)

1 21 2

0 0 05 51 2

( ) ( )1( ) ( )4

b y b yc c

c c

ik R ik Rx a z c y c y c

yy x a z cy y

y y b e y y b ev x x dz dx

x R Rα α

π+ +

− −

⎡ ⎤− − − +∂= − −⎢ ⎥

∂ ⎢ ⎥⎣ ⎦∫ ∫ ,

(3G-10)

1 21 2

0 05 51 2

1( )4

b y b yc c

c c

ik R ik Rx a z c y y

yy x a z cy y

e ev dz dx

y R Rβ β

π+ +

− −

⎡ ⎤∂= −⎢ ⎥

∂ ⎢ ⎥⎣ ⎦∫ ∫ , (3G-11)

11 2

0 0 05 51 2

( ) ( )1( ) ( )4

b z b zxc c

c c

ik R ik Rx a y b z c z czz x a y b

z z

z z c e z z c ev y y dy dxy R R

α απ

+ +

− −

⎡ ⎤− − − +∂= − −⎢ ⎥∂ ⎣ ⎦

∫ ∫ ,

(3G-12)

1 21 2

0 0 05 51 2

( ) ( )1( ) ( )4

b z b zc c

c c

ik R ik Rx a y b z c z czz x a y b

z z

z z c e z z c ev x x dy dxx R R

α απ

+ +

− −

⎡ ⎤− − − +∂= − −⎢ ⎥∂ ⎣ ⎦

∫ ∫ ,

(3G-13)

and

1 21 2

0 05 51 2

1( )4

b z b zc c

c c

ik R ik Rx a y bz z

zz x a y bz z

e ev dy dxz R R

β βπ

+ +

− −

⎡ ⎤∂= −⎢ ⎥∂ ⎣ ⎦

∫ ∫ , (3G-14)

where

67

33 121

21 +−−= xbxbx RikRkα , (3G-15)

33 222

22 +−−= xbxbx RikRkα , (3G-16)

33 121

21 +−−= ybyby RikRkα , (3G-17)

33 22

22

2 +−−= ybyby RikRkα , (3G-18)

33 121

21 +−−= zbzbz RikRkα , (3G-19)

33 222

22 +−−= zbzbz RikRkα , (3G-20)

3 2 2 2 21 1 1 1(1 ( ) ) 3( ) ( 1)x b x b c x c b xik R k x x a R x x a ik Rβ = − + − − − − − − , (3G-21)

3 2 2 2 22 2 2 2(1 ( ) ) 3( ) ( 1)x b x b c x c b xik R k x x a R x x a ik Rβ = − + − + − − + − , (3G-22)

3 2 2 2 21 1 1 1(1 ( ) ) 3( ) ( 1)y b y b c y c b yik R k y y b R y y b ik Rβ = − + − − − − − − , (3G-23)

( )3 2 2 2 22 2 2 2(1 ( ) ) 3( ) 1y b y b c y c b yik R k y y b R y y b ik Rβ = − + − + − − + − , (3G-24)

( )3 2 2 2 21 1 1 1(1 ( ) ) 3( ) 1z b z b c z c b zik R k z z c R z z c ik Rβ = − + − − − − − − , (3G-25)

and

( )3 2 2 2 22 2 2 2(1 ( ) ) 3( ) 1z b z b c z c b zik R k z z c R z z c ik Rβ = − + − + − − + − . (3G-26)

68

Chapter 4: Numerical Simulation of EM measurements in Axisymmetric Media

This chapter describes numerical techniques developed to simulate EM borehole

measurements acquired in axisymmetric media. The logging environment is introduced

within the context of mud-filtrate invasion. Governing partial differential equations and

integral equations are derived for axisymmetric media and solved using various full-wave

techniques, such as the BiCGSTAB(L)-FFT, the BiCGSTAB(l)-FFHT, and finite

differences, as well as with various approximation strategies, such as a Preconditioned

Extended Born Approximation (PEBA), and a High-order Generalized Extended Born

Approximation (Ho-GEBA). These simulation techniques are validated by simulating the

response of borehole array induction tool in axisymmetric media.

4.1 INTRODUCTION

Axisymmetric inhomogeneous media are commonly encountered in borehole

geophysical applications. In such cases, a borehole with the surrounding fluid-invaded

rock formation is modeled as a radially and vertically inhomogeneous medium. The rock

formation is generally interpreted as a horizontally layered medium, which only exhibits

variations of material properties in the vertical direction. During and after drilling, the

near-wellbore rock formations are often altered by stress and stress releases, mud-filtrate

invasion, chemical reactions, and many other factors. Among these factors, mud-filtrate

invasion is responsible for the greatest changes of the electrical conductivity of the

formation layers in the horizontal direction. Mud-filtrate invasion is a phenomenon

whereby mud-filtrate invades the formation layers and displaces in-situ fluids. Variations

69

of electrical conductivity are mainly caused by the difference in salt concentration

between the mud-filtrate and the connate water and/or the displacement of hydrocarbon

by mud-filtrate. The process of mud-filtrate invasion is further complicated by capillary

pressure and gravity effects.

In well logging, the parameter of greatest interest for formation evaluation is tR ,

that is, the resistivity of a bed under consideration which has not been contaminated by

mud filtrate. However, logging tools measure the overall apparent resistivity, aR , and, in

order to accurately determine tR , perturbations caused by the electrical resistivity of

adjacent regions must be taken into account. As shown in Figure 4.1 (Schlumberger,

1987; Anderson, 2001), such adjacent regions include

(1) A borehole with diameter hd , filled with drilling mud of resistivity mR ,

Figure 4.1: Graphical illustration of the borehole logging environment.

70

(2) An invaded zone with resistivity xoR and diameter id , completely flushed by

mud filtrate,

(3) A transition zone with a diameter jd , partially flushed by mud filtrate, and

(4) Shoulder beds that are adjacent layers of differing resistivity, with resistivity

sR and various thicknesses.

For the case of a vertical well, the logging environment can be viewed as an

axisymmetric layered medium, i.e., the formation properties are invariant in the

azimuthal direction. Within each layer, the resistivity varies in the radial direction only.

Figure 4.2 shows a typical two-front invasion resistivity profile with an annulus. In

reality, the resistivity in the transition zone is not constant and different radial zones are

not separated by sharp boundaries. This is because capillary pressure effects and gravity

segregation tend to smooth the fluid displacement front. Single-front invasion profiles

which only include xoR and tR are frequently assumed in the logging industry to interpret

borehole induction measurements.

71

4.2 GOVERNING PARTIAL DIFFERENTIAL EQUATION FOR MODELING AXISYMMETRIC

MEDIA

This section is devoted to deriving the governing PDE (Partial Differential

Equation) for modeling borehole EM measurements acquired in axisymmetric media.

Although both solenoidal and toroidal sources are considered, the derivation will focus

on the solenoidal source.

In a cylindrical coordinate system ( )z,,φρ , an axisymmetric inhomogeneous

medium is invariant in the azimuthal φ direction; However, the EM fields excited by an

arbitrary source generally depend on ρ ,φ , and z. This type of EM simulation problem

with 2D inhomogeneities but 3D EM field variations is often referred to as a 2.5-

Figure 4.2: Illustration of a typical annulus invasion profile.

72

dimensional problem. However, when the excitation source exhibits azimuthal symmetry,

this 2.5 dimensional problem simplifies to a 2D one. A coaxial loop antenna (solenoidal

source) is one such symmetric source of excitation which generates a transverse electric

field, while a toroidal source is a symmetric excitation source which generates a

transverse magnetic field. Notice that when the tool is offset from the borehole axis, the

simulation problem remains 2.5 dimensional.

Assume a loop antenna with radius 0ρ that carries an electric current EI and is

located at z=z0, with time harmonic dependence tie ω− , where ω is angular frequency in

radian/s and t is time in seconds. The impressed current density can be expressed as

( )0 0ˆ( ) ( )s EI z zδ ρ ρ δ= − −J r φ , (4.1)

where ( )δ r is the Dirac-delta function, ˆ ˆzρ= +r ρ z , and ˆˆ zρ, φ, are the unit vectors of the

cylindrical coordinate system.

By denoting E and H as the electric and magnetic field vectors, respectively, and

by assuming non-magnetic material, Maxwell’s equations in the frequency domain can be

written as

( ) )(rHrE ωμi=×∇ , (4.2)

and

( ) ( ) ( )rJrErH(r s+=×∇ σ) . (4.3)

By taking the ×∇ operator on both sides of equation (4.2) together with

substitution from equation (4.3) yields the vector Helmholtz equation for E, given by

)()()()( rJrErrE sii ωμωμσ +=×∇×∇ . (4.4)

Rewriting the above equation gives

73

)()()( 2 rJrErE sik ωμ=−×∇×∇ , (4.5)

where

)(2 rωμσik = . (4.6)

In equation (4.6), k is the propagation constant, which is a function of position, and

)()()( 0 rrr ri εωεσσ −′= . (4.7)

In an axisymmetric inhomogeneous medium, a coaxial loop antenna will generate

an axisymmetric EM field, where the only non-zero component of the electric field is the

azimuthal φE component. In other words, a coaxial loop antenna generates a pure

transverse electric (TE) field in an axisymmetric medium. Thus, in a cylindrical

coordinate system, E×∇×∇ can be written as

22

2 2

1ˆ E EE

zφ φ

φρρ ρ ρ ρ

⎛ ⎞∂∂ ∂∇×∇× = −∇ = − − +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

E E φ . (4.8)

Substitution of equation (4.8) into equation (4.5), together with the use of

equation (4.1) yields

( ) ( )2

20 02 2

1 1Ek E i I z z

z φρ ωμ δ ρ ρ δρ ρ ρ ρ

⎛ ⎞∂ ∂ ∂+ − + = − − −⎜ ⎟∂ ∂ ∂⎝ ⎠

. (4.9)

Finally, we obtain the partial differential equation (PDE) for the electrical field in

an axisymmetric medium for a coaxial loop antenna, namely,

( ) ( )2

20 02

1Ek E i I z z

z φρ ωμ δ ρ ρ δρ ρ ρ

⎛ ⎞∂ ∂ ∂+ + = − − −⎜ ⎟∂ ∂ ∂⎝ ⎠

. (4.10)

Because μ is a constant, equation (4.10) can be rewritten as

( ) ( )20 0

1 1Ek E i I z z

z z φρμ μ ρ ωμ ρδ ρ ρ δρ ρμ ρ μ

⎛ ⎞∂ ∂ ∂ ∂+ + = − − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

. (4.11)

74

A toroidal source generates a transverse magnetic field and the corresponding

PDE can be written as

( ) ( )20 0

1 1mk H i I z z

z z φρε ε ρ ωε ρδ ρ ρ δρ ρε ρ ε

⎛ ⎞∂ ∂ ∂ ∂+ + = − − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

, (4.12)

where mI is given by equation (2.37) and ε is given by

0r iσε ε εω′

= + . (4.13)

Careful comparison of equation (4.11) and equation (4.12) indicates that these

two equations are completely dual. Thus, they can be written in a unified form as

( ) ( )20 0

1 1 k A i I z zz z φ ζρζ ζ ρ ωζ ρδ ρ ρ δ

ρ ρζ ρ ζ⎛ ⎞∂ ∂ ∂ ∂

+ + = − − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠, (4.14)

where, for the TE mode,ζ μ= , A Eφ φ= , and EI Iζ = ; for the TM mode,ζ ε= , A Hφ φ= ,

and mI Iζ = .

4.3 GOVERNING INTEGRAL EQUATION AND GREEN’S FUNCTIONS

The solution of equation (4.10) can also be obtained using integral equations. To

this end, first define a Green’s function ( ), ; ,g z zρ ρ′ ′ for an unbounded, homogeneous,

and isotropic background with constant complex conductivity bσ , in which the radiation

condition has been imposed at infinity. The governing equation is

( ) ( ) ( )2

22

1 , ; ,bk g z z z zz

ρ ρ ρ δ ρ ρ δρ ρ ρ

⎛ ⎞∂ ∂ ∂ ′ ′ ′ ′+ + = − − −⎜ ⎟∂ ∂ ∂⎝ ⎠, (4.15)

where the wave number bk of the background is given by

2b bk iωμσ= , (4.16)

75

and

0b b rbiσ σ ωε ε′= − . (4.17)

where bσ ′ is the background ohmic conductivity in S/m, and rbε is the background

dielectric constant.

The solution of equation (4.15) is given by (Torres-Verdín and Habashy, 2001),

( )0

, ; , cos2

bik Reg z z dR

πρρ ρ φ φπ′

′ ′ ′ ′= ∫ , (4.18)

where

( )2 2 2 2 cosR z z ρ ρ ρρ φ′ ′ ′ ′= − + + − . (4.19)

Using the definition of the Green’s function and the principle of linear

superposition, one derives the integral equation for the electric field as

( ) ( ) ( ) ( ) ( ), , , ; , , ,bE z E z i g z z z E z d dzφ φ φτρ ρ ωμ ρ ρ σ ρ ρ ρ′ ′ ′ ′ ′ ′ ′ ′= + Δ∫ , (4.20)

where ( ),bE zφ ρ is the electric field at ( ), zρ excited by the source in the background,

and is given by

( ) ( )0 0, , ; ,b EE z i I g z zφ ρ ωμ ρ ρ= . (4.21)

In equation (4.20), σΔ is the conductivity anomaly with respect to the

background, given by

( ) ( ), , bz zσ ρ σ ρ σΔ = − , (4.22)

and τ is the spatial support of non-zero conductivity variations with respect to the

assumed background.

Equation (4.20) is valid for observation points inside and outside of the

conductivity anomaly support. Using the method of moments (MoM) (Harrington, 1968),

76

the electric field inside the anomaly can be solved, and then the electric field at the

receiver locations can be obtained from the computed internal electric fields using

equation (4.20). Because in geophysical logging measurements often consist of magnetic

fields, we find that for small 0ρ latter can be written as

( ) 000 /,2),( ωμρρρ φ izEzH RRz = , (4.23)

where ( )0 , Rzρ is the location of the receiver.

We remark that no computationally efficient method exists to compute the

integral in equation (4.18). Thus, we proceed to derive alternative forms of ),;,( zzg ′′ρρ

using Fourier Transform (FT) and Hankel Transform (HT) techniques that are amenable

to efficient computation of the Green’s function.

Assuming that the Fourier and Hankel transforms of ),;,( zzg ′′ρρ exist and

noting that the derivatives with respect to ρ in equation (4.10) resemble a Bessel

equation of order 1, we make use of the Hankel transform of order 1. Now express

),;,( zzg ′′ρρ as

( ) ( ) ( ) zikzz

zekJkzkkGdkdkzzg ρρπ

ρρ ρρρρ 10,;,

21,;, ′′=′′ ∫ ∫

∞

∞−

∞, (4.24)

where ( )1J ⋅ is the Bessel function of the first kind of order 1.

The problem then consists of solving ),;,( zkkG z ′′ρρ in equation (4.24). Notice

that equation (4.24) combines an inverse Fourier transform and an inverse Hankel

transform.

Differentiation of equation (4.24) with respect to z twice yields

( ) ( ) ( ) zikzz

z zekJkzkkGdkkzzgz

ρρπ

ρρ ρρρ 10

2

2

2

,;,2

,;, ∫ ∫∞

∞−

∞′′−=′′

∂∂ . (4.25)

77

Similarly

( ) ( ) ( ) zikzz

zekJkzkkGdkdkg ρρπ

ρρρρ ρρρρ 1

3

0,;,

211 ′′−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

∫ ∫∞

∞−

∞. (4.26)

Due to the orthogonal property of Bessel and sinusoidal functions, the Delta

function can be expressed in terms of Fourier and Hankel transforms as

( )( ) ( ) ( ) zikzikz eekJkJkdkdkzz z ′−∞∞

∞−′

′=′−′− ∫∫ ρρ

πρρρδ ρρρρ 1102

. (4.27)

Substitution of equations (4.25), (4.26), and (4.27) into equation (4.10) yields

( ) ( )

( ) ( ) zikzikz

zikzbzz

zz

z

eekJkJkdkdk

ekkkkJkzkkGdkdk

′−∞

∞−

∞

∞

∞−

∞

′′

−=

−−′′

∫ ∫

∫ ∫

ρρπρ

ρρπ

ρρρρ

ρρρρρ

110

22210

2

)(,;,21

. (4.28)

Because equality (4.28) is valid for any ρ , ρ′ , z , and z′ , one can write

( ) ( )222

1,;,bz

zik

z kkkekJ

zkkGz

−+

′′=′′

′−

ρ

ρρ

ρρρ . (4.29)

The final expression for ),;,( zzg ′′ρρ is then given by,

( ) ( ) ( ) zik

bz

zik

zz

z

ekJkkkk

ekJdkdkzzg ρ

ρρπ

ρρ ρρρ

ρρ 1222

1

021,;,

−+

′′=′′

′−∞

∞−

∞

∫ ∫ . (4.30)

Another from of ),;,( zzg ′′ρρ can be obtained directly from the Hankel

transform. First assume that ),;,( zzg ′′ρρ can be expressed as

( ) ( )ρρρρ ρρρρ kJzzkGkdkzzg 120),;,(,;, ′′=′′ ∫

∞. (4.31)

By noting that ( )ρρδ ′− can be written in terms of a Hankel transform, and by

substituting equation (4.31) into equation (4.10), one obtains

( ) ( ) ( )ρρδρ ρρρ ′′′−−=′′⎥⎦

⎤⎢⎣

⎡−+

∂∂ kJzzzzkGkkz b 12

222

2

,;, . (4.32)

78

Now denote

222ργ kkb −= , (4.33)

whereupon equation (4.32) becomes

( ) ( ) ( )ρρδργ ρρ ′′′−−=′′⎥⎦

⎤⎢⎣

⎡+

∂∂ kJzzzzkGz 12

22

2

,;, . (4.34)

The solution of equation (4.34) is given by

( ) ( )γ

ρρργ

ρρ 2,;, 12

zziekJizzkG′−

′′=′′ . (4.35)

Finally, ),;,( zzg ′′ρρ takes on the form

( ) ( ) ( ) zziekJkJk

dkizzg ′−∞′

′=′′ ∫ γ

ρρρ

ρ ρργ

ρρρ 1102,;, . (4.36)

Equation (4.30) can be readily obtained from equation (4.36) through Fourier

transform. The Fourier transform of zie γ is

( ) zikzizi zeedzeF −∞

∞−∫=γγ

22

2γγ−

−=zk

i222

2

bz kkki−+

−=ρ

γ . (4.37)

Thus,

( )zzik

bzz

zzi zekkk

idke ′−∞

∞−

′−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+−= ∫ 222

2

ρ

γ γ . (4.38)


( ) ( ) ( ) zik

bz

zik

zz

z

ekJkkkk

ekJdkdkzzg ρ

ρρπ

ρρ ρρρ

ρρ 1222

1

021,;,

−+

′′=′′

′−∞

∞−

∞

∫ ∫ . (4.39)

79

4.4 FULL-WAVE MODELING TECHNIQUES

In this section, we introduce three full-wave numerical modeling techniques: the

BiCGSTAB(L)-FFT (Bi-Conjugate Gradient STABilized(L)-Fast Fourier Transform), the

BiCGSTAB(L)-FFHT (Bi-Conjugate Gradient STABilized(L)-Fast Fourier Hankel

Transform), and finite differences.

4.4.1 The BiCGSTAB(L)-FFT Technique

This technique considers the Green’s function in equation (4.18). Chapter 5

provides additional details of the same technique in the context of 3D modeling. The

BiCGSTAB(L)-FFT makes use of the spatial shift-invariant property of the Green’s

function. Such a property provides a way to solve large-scale EM scattering problems

with reduced memory storage, Green’s function evaluations, and CPU time to solve the

linear system. Chapter 5 shows that, in Cartesian system, the FFT can be used in all the

three directions. However, the Green’s function given by equation (4.18) is only shift-

invariant in the z-direction, whereupon the FFT can only be used in the z-direction. Such

a condition reduces the efficiency of the FFT technique, but still substantially remains

more efficient than the direct implementation of the integral equation.

4.4.1.1 Computation of the Integrals of the Green’s Function

We proceed to develop a method to efficiently evaluate the integrals of the

Green’s function shown in equation 4.18. From equation (4.18), the integral of

),;,( zzg ′′ρρ over a small volume V is given by

( )0

1, cos2

bik Rd z dz

I d z dz

eG z d dz dR

ρ ρ π

ρ ρρ φ ρ φ ρ

π′ ′+ +

′ ′− −′ ′ ′ ′ ′= ∫ ∫ ∫ . (4.40)

80

The expression R

e Rikb

can be written as the superposition of a DC part and an

auxiliary part as

Re

RRe RikRik bb 11 −

+= . (4.41)

Accordingly, GI can be rewritten as

( )0

0

1 1, cos2

1 1cos2

b

d z dz

I d z dz

ik Rd z dz

d z dz

G z d dz dRed dz d

R

π ρ ρ

ρ ρ

π ρ ρ

ρ ρ

ρ φ φ ρ ρπ

φ φ ρ ρπ

′ ′+ +

′ ′− −

′ ′+ +

′ ′− −

′ ′ ′ ′ ′=

−′ ′ ′ ′ ′+

∫ ∫ ∫

∫ ∫ ∫. (4.42)

In this last expression, the integral

1d z dz

d z dzdz d

Rρ ρ

ρ ρρ ρ

′ ′+ +

′ ′− −′ ′ ′∫ ∫ , (4.43)

can be solved analytically using the procedure described by Torres-Verdín and Habashy

(2001).

The integration with respect to ρ and z in the second term of the right-hand side

of equation (4.42) is estimated roughly through simple trapezoidal rule, then the

integration with respect to φ′ in equation (4.42) is completed using one dimensional

Gaussian quadrature.

4.4.1.2 Computation of Background Electric Fields

Background electric fields enter the linear system of equations as the right-hand

side vector. They can be computed exactly using equation (4.21) for a finite-size loop

antenna source. However, one still needs to evaluate the integral in equation (4.18)

numerically.

81

In geophysical induction logging, antennas are designed in such a way that their

EM fields are not far from those due to a point dipole source. Thus, when 0ρ ρ , R

e Rikb

can be expanded in the form

( )021 1 cos

b bik R ik r

be e ik r

R r rρρ φ⎡ ⎤′≈ + −⎢ ⎥⎣ ⎦

, (4.44)

where

( )2 2 20r z z ρ ρ′= − + + . (4.45)

By making use of the integration identity

∫ =π πφφ

2

0

2

2cos d , (4.46)

one obtains

( ) ( )20

0 3

1, ; , 14

bik rbg z z ik r e

rρρρ ρ ′ ≈ − . (4.47)

Substitution of equation (4.47) into equation (4.21) gives expression for the

background electric field, namely,

( ) ( )20

3

1,

4

bik rE b

b

i I ik r eE z

rφ

ωμ ρ ρρ

−= . (4.48)

4.4.1.3 Code Development

A code was developed using the BiCGSTAB(L)-FFT technique and the method

of moments. Compared to the direct implementation of the MoM, this implementation

reduces computer storage to 2

12

z

z

NN − times that of the MoM matrix, where Nz is the

number of spatial discretization cells in the z direction. Moreover, the same

82

implementation reduces the time required for the evaluations of the integrals of the

Green’s function by the same magnitude. For instance, if Nz=128, then the factor is

0.0156, which means that we only need to store and compute 1.56% of the original

matrix. This represents a significant savings in memory storage and CPU time. Using the

BiCGSTAB(L)-FFT, the computational cost of solving the complex linear system is

reduced to ( )zz NNNO 22 logρ , where Nρ is the number of the spatial discretization cells

in the ρ direction. When ρNN z >> , the corresponding reduction in computational time

is even more dramatic.

4.4.2 The BiCGSTAB(L)-FFHT Technique

The technique described in the previous section only accelerates the algorithm in

the z direction. When the number of the cells in the radial direction increases, the

algorithm becomes less and less efficient. However, the use of the Green’s function given

by equation (4.30) provides an efficient algorithm that applies to both the radial and

vertical direction.

Following Liu and Chew (1994), we introduce the concepts of induced currents,

given by

( ) ( ) ( )zEzzJ ,,, ρρσρ φΔ= . (4.49)


( )( ) ( ) ( )zEzJzzgdzdi

zzJ

b ,,),;,(,

,0

ρρρρρωμρσρ

=′′⋅′′′′−Δ ∫ ∫

∞

∞−

∞, ( ) Rz ∈,ρ , (4.50)

where R is the spatial support of the inhomogeneous region.


83

( )( )

( )( ) ( ) ( )

( )

112 2 20 0

,,

, 2

, .

z zik z ik zz

z b

b

k J kJ z i dk e dk dz d J z J k ez k k k

E z

ρ ρρ ρ

ρ

φ

ρρ ωμ ρ ρ ρ ρσ ρ π

ρ

∞ ∞ ∞ ∞ ′−

−∞ −∞

⎡ ⎤′ ′ ′ ′ ′ ′− ⎢ ⎥⎣ ⎦Δ + −

=

∫ ∫ ∫ ∫

(4.51)

Finally, we obtain

( )( ) ( )[ ] ( )zEzJFH

kkkiFH

zzJ

bbz

,,,

,222

1 ρρωμρσρ

ρ

=⎟⎟⎠

⎞⎜⎜⎝

⎛

−+−

Δ− , ( ) Rz ∈,ρ , (4.52)

where FH stands for Fourier and Hankel Transform, namely Fourier transform in the z

direction and Hankel transform in the ρ direction. We will make use of the FFT (Fast

Fourier Transform) and the FHT (Fast Hankel Transform) to solve equation (4.52). When

combined with the iterative algorithm BiCGSTAB(L), one can solve equation (4.52) with

a cost proportional to O(Nlog2N), and computer memory storage proportional to O(N),

where N is the number of discretization cells.

Once obtaining J within the inhomogeneity, the electric fields within the

inhomogeneity can be computed via equation (4.49). Subsequently the internal electric

fields can be propagated to receiver locations. Likewise, the magnetic fields at the

receivers can be computed via equation (4.23).

Supplement 4A gives a detailed description of the use of the FHT to solve

equation (4.52). Based on the theory developed in this section, a computer code was

developed to simulate the response of multi-frequency array induction tools in

axisymmetric rock formations.

84

4.4.3 Finite Differences

The finite-difference method (FDM) has been widely used to simulate EM

phenomena in the frequency and time domains (Wang and Hohman (1993); Druskin and

Knizhnerman (1994); Wang and Fang (2001); Weiss and Newman (2002); and

Davydycheva et al. (2003), to name a few). The staggered grid modeling approach was

proposed by Yee (1966) and has been applied to the simulation of EM fields in arbitrary

inhomogeneous isotropic media. This method yields a coercive approximation, that is,

every continuous Maxwell’s equation has its discrete counterpart satisfying conservation

laws such as Gauss and Stokes theorems. In this section, finite difference schemes are

developed to solve equations (4.9) and (4.14).

The TE equation (4.9) is rather easy to handle using finite differences given that

the magnetic permeability μ is assumed constant. No derivative of the material property

is involved in that equation. In this section, we focus our attention to solving equation

(4.9). For the general equation (4.14), the corresponding finite differencing procedure is

not trivial. Supplement 4B provides a detailed derivation of the finite-difference

procedure applied to equation (4.14). Notice that we choose to discretize equation (4.9)

instead of equation (4.10) in order to avoid the essential singularity that exists at 0ρ = .

85

Let us assume that the inhomogeneous space in zρ − plane is discretized into

Nρ cells in the ρ direction and zN cells in the z direction, as shown in Figure 4.3. The

grid nodes in the ρ and z directions are

, 1, , 1

, 1, , 1i

k z

i N

z k Nρρ = +

= +. (4.53)

Following Yee’s staggered grid discretization scheme, Eφ is sampled at half grid

numbers (i+1/2, k+1/2), Hρ is sampled at horizontal cell edges (i+1/2, k), and zH is

sampled at vertical cell edges (i, k+1/2).

We start by introducing the notation

1 3/ 2 1/ 2i i iρ ρ ρ+ + +Δ = − , (4.54)

1/ 2 1/ 2i i iρ ρ ρ+ −Δ = − , (4.55)

1i i iρ ρ ρ+Δ = − , (4.56)

E φ ρ

z

φ

Borehole Axis

Hρ

zH

Figure 4.3: Illustration of the finite-difference grid used to discretize the TE equation.

86

1 3/ 2 1/ 2i i iz z z+ + +Δ = − , (4.57)

1/ 2 1/ 2i i iz z z+ −Δ = − , (4.58)

and

1i i iz z z+Δ = − . (4.59)

Using a central finite-difference approximation, it follows that

( ) ( ) ( ) ( )2

2f a h f a h

f a O hh

+ − −′ = + . (4.60)

At the location ( )1/ 2, 1/ 2i k+ + one has

( ) ( )( ) ( ) ( )( )3/ 2, 1/ 2 1/ 2, 1/ 2 1/ 2, 1/ 2 1/ 2, 1/ 21

1/ 2 1

1 1 ,i k i k i k i ki i

i i i i

EE E E Eφφ φ φ φ

ρ ρρρ ρ ρ ρ ρ ρ ρ

+ + + + + + − ++

+ +

∂ ⎡ ⎤∂= − − −⎢ ⎥∂ ∂ Δ Δ Δ⎣ ⎦

(4.61)

and

( ) ( )( ) ( ) ( )( )2

1/ 2, 3/ 2 1/ 2, 1/ 2 1/ 2, 1/ 2 1/ 2, 1/ 22

1

1 1 1i k i k i k i k

k k k

EE E E E

z z z zφ

φ φ φ φ+ + + + + + + −

+

∂ ⎡ ⎤= − − −⎢ ⎥∂ Δ Δ Δ⎣ ⎦

.

(4.62)

Substitution of equations (4.61) and (4.62) into equation (4.9), and multiplication

by i kzρΔ Δ yields

( ) ( ) ( ) ( ) ( )

( ) ( )

1/ 2, 1/ 2 1/ 2, 1/ 2 1/ 2, 1/ 2 3/ 2, 1/ 2 1/ 2, 3/ 2, , , , ,

0 0

i k i k i k i k i ki k i k i k i k i k

E i k

A E B E C E D E E E

i I z z zφ φ φ φ φ

ωμ ρ δ ρ ρ δ

+ − − + + + + + + ++ + + +

= − Δ Δ − −,

(4.63)

where

,i

i kk

AzρΔ

=Δ

, (4.64)

87

,1/ 2

k ii k

i i

zB ρρ ρ+

Δ=

Δ, (4.65)

21, 2

1/ 2 1/ 21 1

1 1k i i i ki k i i k

i ii i k k

z zC k zz z

ρ ρ ρρ ρρ ρρ ρ

+

+ ++ +

⎛ ⎞ ⎛ ⎞Δ Δ Δ= − + −Δ + − + Δ Δ⎜ ⎟ ⎜ ⎟

Δ Δ Δ Δ⎝ ⎠ ⎝ ⎠, (4.66)

1,

1/ 2 1

k ii k

i i

zD ρρ ρ

+

+ +

Δ=

Δ, (4.67)

and

,1

ii k

k

Ezρ

+

Δ=Δ

. (4.68)

The coefficients A, B, C, D and E above are determined by the grid geometry,

except for C which is also determined by the conductivity distribution. As illustrated in

Figure 4.4, these 5 coefficients form a five-point stencil.

i+1/2, k+1/2 C

i+1/2, k+3/2 E

i+1/2, k-1/2A

i-1/2, k+1/2 B

i+3/2, k+1/2 D

i-1/2, k-1/2

i+3/2, k-1/2

i-1/2, k+3/2

i+3/2, k+3/2

Figure 4.4: Graphical description of the five-point stencil used in the finite-difference approximation of Maxwell’s equation in axisymmetric media.

88

The corresponding boundary conditions are given by:

(1) lim 0Eφρ→∞= ; (4.69)

(2) No mandrel:

0

1lim 0Eφρ ρ→= , (4.70)

(3) Metallic mandrel (radius is assumed to be a ):

1lim 0a

Eφρ ρ→= . (4.71)

Equations (4.63) through (4.71) give rise to a complex sparse matrix system. The

corresponding matrix contains five bands corresponding to the five coefficients. The

system is solved using BiCGSTAB(L). From a programming point of view, the following

steps are taken to solve the linear system: (1) To reduce memory storage, the matrix is

stored using row-based/column-based storage form; (2) the matrix needs to be computed

only once, since only C needs to be updated for different logging points; (3) the Delta

function is represented by a rectangular pulse function, whose amplitude is the inverse of

the area of the corresponding cell; (4) The radial grid is designed to have fine

discretization steps near the wellbore and coarse discretization steps away from the

wellbore.

In practice, to obtain accurate simulation results, a scattered field equation is

preferred over a total field equation. In so doing, an infinite uniform background medium

with conductivity bσ is chosen. The corresponding propagation constant bk is given by

equation (4.16) and the total electric field Eφ can be expressed as the superposition of the

scattered electric field sEφ and the background field bEφ , i.e.

89

sbE E Eφ φ φ= + . (4.72)

Substitution of equation (4.72) into equation (4.10) together with equation (4.15)

yields

( )2

2 2 22

1 sb bk E k k E

z φ φρρ ρ ρ

⎛ ⎞∂ ∂ ∂+ + = − −⎜ ⎟∂ ∂ ∂⎝ ⎠

. (4.73)

Comparison of equations (4.73) and (4.10) shows that the scattered field

formulation does not change the linear-system matrix. The only change takes place in the

right-hand side of the linear system, which can be computed from equation (4.21). The

solution of equation (4.73) instead of equation (4.10) using finite differences provides a

more accurate computation of the background fields.

Supplement 4B provides a detailed derivation of the finite-difference algorithm

needs to solve the general wave equation (4.14). An efficient computer code was

developed using the above-mentioned strategies for simulating the response of multi-

frequency array induction tools in axisymmetric rock formations. This code can handle

both solenoidal and toroidal sources. For all the numerical examples given in this chapter,

the default EM source is a solenoidal source.

4.4.4 Numerical Examples

Apparent resistivity/conductivity values are customarily used in well-logging

applications to describe induction sonde response, instead of EM fields. Supplement 4C

describes the transformation between EM fields and apparent conductivities, as well as

the related skin-effect corrections. Since EM fields and apparent conductivities are

equivalent, EM fields are used for the numerical examples considered in this dissertation.

90

4.4.4.1 Solenoidal Source

To validate the codes developed using the various full-wave techniques developed

in the previous sections, a modified Oklahoma model adapted from Torres-Verdín and

Habashy (2001) is considered to simulate the response of multi-frequency array induction

tools. The modified Oklahoma model is shown in Figure 4.5, and the corresponding

formation electrical and geometrical parameters are detailed in Table 4.1.

The borehole EM tool contains one transmitter and one receiver separated by a

distance of 0.5 m and operates at 10k Hz.

Figures 4.6 and 4.7 compare the real and imaginary parts of the magnetic

response obtained with the three full-wave simulation techniques described in this

chapter, respectively. These figures show that all three simulation algorithms provide

accurate simulation results.

91

Figure 4.5: A modified Oklahoma model. Left panel: Invasion radius versus depth. Right panel: Conductivity versus depth. In the figures, xoσ is the conductivity of the flushed zone, and tσ is the conductivity of the uninvaded formation. Electrical and geometrical parameters for this model are given in Table 4.1.

92

Layer No. Thickness

m Invasion Radius

Cm xoR mΩ⋅

tR mΩ⋅

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

5.18 2.44 1.22 0.91 2.13 1.22 1.83 0.91 1.52 2.13 5.48 2.44 2.13 3.05 1.22 1.52 0.91 1.22 1.22 1.22 1.52 0.91 0.61 0.61 0.61 3.66

64 0 0 0 64 0 38 64 89 64 114 0 0 64 114 38 0 0 0 0 38 0 89 0 38 0

5 2.5

0.333 1

2.5 1.428 0.667 0.133 0.01 0.333 0.333 0.025 0.667 0.4 2.5 0.4

0.667 0.133 0.1 0.4 5

0.667 2.5

0.133 3.333 1.428

0.1 2.5

0.333 1

0.06 1.428 0.01 0.133 0.4

0.333 0.007 0.025 0.667 0.01 0.06 0.01 0.667 0.133 0.1 0.4 0.1

0.667 0.056 0.133 0.056 1.428

Table 4.1: Description of the modified Oklahoma formation model illustrated in Figure

4.5.

93

Figure 4.6: Graphical comparison of the three full-wave simulation techniques applied to the modified Oklahoma model shown in Figure 4.5. The real part of the magnetic response is shown on the figure. The tool consists of one transmitter and one receiver, with a spacing of 0.5 m, and operates at 10 KHz. On the figure, “2DIE” designates the BiCGSTAB(L)-FFT; “FFHT” designates the BiCGSTAB(L)-FFHT; and “FD2D” designates the finite-difference code.

94

Figure 4.7: Graphical comparison of the three full-wave simulation techniques applied to the modified Oklahoma model shown in Figure 4.5. The imaginary part of the magnetic response is shown on the figure. The tool consists of one transmitter and one receiver, with a spacing of 0.5 m, and operates at 10 KHz. On the figure, “2DIE” designates the BiCGSTAB(L)-FFT; “FFHT” designates the BiCGSTAB(L)-FFHT; and “FD2D” designates the finite-difference code.

95

4.4.4.2 Toroidal Source

This section describes simulation results for the case of a toroidal EM source. The

tool operates at 25 KHz and consists of one transmitter and one receiver spaced at a

distance of 0.5 m. A generic toroidal coil is shown in Figure 2.1b. Following the notation

introduced in section 2.3.2, the radius of the toroidal coil in the ρ φ− plane, a is 0.03 m,

and the radius of the toroidal coil in the zρ − plane, tr , is 0.005 m.

The toroidal source option is only incorporated in the finite-difference code, and

the corresponding finite-difference algorithm is given in Supplement 4C. The code was

validated with the analytical solution given by equation (2.38) for an infinite

homogeneous rock formation. For inhomogeneous media, because there is no alternative

solution to compare to, in this section we only show simulation results for the three-layer

formation model shown in Figure 4.8.

Figure 4.9 describes the simulation results obtained for the three-layer formation

model shown in Figure 4.8. In Figure 4.9, the left panel shows the real part of z

Figure 4.8: Graphical description of the three-layer rock formation model used to simulate the EM response of a toroidal source.

1 S/m

10 S/m

1 S/m

0.5 S/m 4 m

0.2 m

96

component of the electric field, zE , versus depth, while the right panel shows the

imaginary part of zE versus depth.

4.5 APPROXIMATE MODELING TECHNIQUES

Approximate strategies are important in solving large-scale EM scattering

problems in that they represent a compromise between accuracy and efficiency.

Moreover, approximate strategies are extremely useful for solving inverse problems. A

Figure 4.9: Simulation results obtained for the formation model given in Figure 4.8. The tool operates at 25 KHz and consists of one transmitter and one receiver spaced at a distance of 0.5 m. The radius of the toroidal coil in the ρ φ− plane is 0.03 m, and the radius of the toroidal coil in the zρ −plane is 0.005 m. The left panel shows the real part of zE , and the right panel shows the imaginary part of zE .

97

good example is the Born approximation (Born, 1933), which enforces a linear

relationship between the material property and the tool response. However, the

application of the Born approximation is limited to low frequencies and small

conductivity contrasts. The Extended Born Approximation (EBA) (Habashy et al, 1993;

Torres-Verdín and Habashy, 1994) has broader applications than the Born approximation.

However, when the source is close to the scatterer, the accuracy of the EBA could

degrade significantly.

This section introduces two approximate simulation strategies that are suitable

for geophysical induction logging. The first one is referred to as Preconditioned Extended

Born Approximation (PEBA) (Gao and Torres-Verdín, 2003), and the second one is

referred to as High-Order Generalized Extended Born Approximation (HO-GEBA) (Gao

and Torres-Verdín, 2005). The theory of the Born approximation, the EBA, and the Ho-

GEBA will be detailed in Chapters 6 and 7 in the context of solving 3D EM simulation

problems. In this chapter, we describe the implementation of these two approximations to

simulate the EM response of axisymmetric media. In Chapter 8, the PEBA provides an

efficient way to compute the Jacobian matrix for nonlinear least-squares inversion.

4.5.1 A Preconditioned Extended Born Approximation (PEBA)

To develop the PEBA, we begin with the EBA. The EBA was introduced by

Habashy (1993), and Torres-Verdín and Habashy (1994). This approximation makes use

of the singularity of the Green’s function when the integral equation (4.20) is specialized

for receiver locations within the scatterer. Because of this, equation (4.20) can be

rewritten as

98

( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( )

, ,

, ; , , ,

, ; , , , , .

bE z E z

i g z z z d dz E z

i g z z z E z E z d dz

φ φ

φτ

φ φτ

ρ ρ

ωμ ρ ρ σ ρ ρ ρ

ωμ ρ ρ σ ρ ρ ρ ρ

=

′ ′ ′ ′ ′ ′+ Δ

′ ′ ′ ′ ′ ′ ′ ′+ Δ −

∫∫

(4.74)

By neglecting the third term on the right-hand side of equation (4.74), one obtains

( ) ( ) ( )zEzzE b ,,, 1 ρρρ φφ−Λ≈ , (4.75)

where

( ) ( ) ( ), 1 , ; , ,z i g z z z d dzτ

ρ ωμ ρ ρ σ ρ ρ⎡ ⎤′ ′ ′ ′ ′ ′Λ = − Δ⎣ ⎦∫ . (4.76)

Without explicitly forming and inverting a large stiffness matrix, the computation

cost of the EBA is comparable to that of the standard first-order Born approximation.

However, one still has to evaluate the integrals of the Green’s function and to perform the

matrix-vector multiplication contained in equation (4.76). The latter operation entails a

computation cost proportional to ( )2NO , where N is the number of cells used in the

spatial discretization of the EM scatterers.

Numerical examples have shown that the EBA yields accurate results for a

number of practical applications of EM scattering (Habashy et al., 1993; Torres-Verdín

and Habashy, 1994). However, the accuracy of the EBA significantly degrades when the

inhomogeneity is large and/or is located close to both the source of EM excitation and the

receiver (Gao et al., 2003). Gao et al. (2002) showed that the EBA can be modified to

take into account the proximity of scatterers to the source of EM excitation. This work

showed that by using the background field as a preconditioner of the original linear

system of equations one could significantly improve the accuracy of the EBA.

To describe how the background electric field can be used as a preconditioner of

the EBA, we first define

99

( ) ( ) ( ), , ,b

E z F z E zφ φρ ρ ρ= , (4.77)

where F is an auxiliary function that relates Eφ and bEφ .


( ) ( ) ( )( ) ( ) ( ) ( )

, , ,

, ; , , , , .b b

b

F z E z E z

i g z z z F z E z d dzφ φ

φτ

ρ ρ ρ

ωμ ρ ρ σ ρ ρ ρ ρ

=

′ ′ ′ ′ ′ ′ ′ ′ ′ ′+ Δ∫ (4.78)

Thus,

( ) ( ) ( ) ( ) ( ), 1 , ; , , , ,F z i g z z z F z W z d dzτ

ρ ωμ ρ ρ σ ρ ρ ρ ρ′ ′ ′ ′ ′ ′ ′ ′ ′ ′= + Δ∫ , (4.79)

where

( ) ( )( )

,,

,b

b

E zW z

E zφ

φ

ρρ

ρ′ ′

′ ′ = . (4.80)

By imposing operating conditions on the scalar function F similar to those used in

the derivation of the EBA one obtains

( ) ( )11, ,F z zρ ρ−≈ Λ , (4.81)

where

( ) ( ) ( ) ( )1 , 1 , ; , , ,z i g z z z W z d dzτ

ρ ωμ ρ ρ σ ρ ρ ρ⎡ ⎤′ ′ ′ ′ ′ ′ ′ ′Λ = − Δ⎣ ⎦∫ . (4.82)

Substitution of this last expression into equation (4.77) yields the solution for the

internal electric field, Eφ , i.e.,

( ) ( ) ( )11, , ,

bE z z E zφ φρ ρ ρ−= Λ . (4.83)

Since 1Λ is a weighted version ofΛ , the solution given by equation (4.83) is here

termed “Preconditioned Extended Born Approximation” and is identified with the

acronym PEBA. The latter is essentially the direct result of preconditioning the stiffness

100

matrix of the MoM using a diagonal matrix whose elements are the corresponding

background electric fields (Gao and Torres-Verdín, 2002).

4.5.2 A High-Order Generalized Extended Born Approximation (Ho-GEBA)

A High-Order Generalized Extended Born Approximation (Ho-GEBA) is

developed for the numerical simulation of EM scattering due to rock formations that

exhibit axial symmetry around a wellbore. The resulting equations are solved via a

numerical procedure that is as efficient as the Extended Born Approximation (EBA).

With the acceleration of a fast Fourier transform, the operation count is proportional to

( )O CN , where N is the total number of spatial discretization cells, and C << N, is a

constant that depends on the number of discretization cells in the radial direction. The

Ho-GEBA remains accurate in the near-source scattering region and accounts for

multiple scattering in the presence of large conductivity contrasts and relatively large

frequencies.

4.5.2.1 Introduction

Accurate and rapid simulation of EM scattering phenomena in the vicinity of a

wellbore has been a subject of continuous research in the geophysical logging

community. Electrical conductivities estimated from borehole EM measurements are one

of the key parameters for estimating hydrocarbon saturation in porous and permeable

rock formations. To date, most of the approaches used in the industry for the

interpretation of EM well logs are based on a linear assumption between a perturbation in

the conductivity distribution and the ensuing perturbation in the tool response, which

leads to the first-order Born approximation (Born 1933). The geometrical theory,

101

developed first by Doll (1946), and subsequently improved by Zhang (1982), and Moran

(1982) remains one of the most important applications of the Born approximation in the

petroleum industry. Because of the assumption of linearity, the Born approximation can

be used for real-time interpretation of borehole EM induction data. However, extensive

numerical studies have shown that the Born approximation remains accurate only at low

frequencies and in the presence of small conductivity contrasts (Habashy et al, 1993).

To extend the validity of the Born approximation, Habashy et al. (1993) proposed

a non-linear approximation under the name of the Extended Born approximation (EBA)

which was extensively studied by Torres-Verdín and Habashy (1994). The EBA also has

been applied to the numerical simulation of axisymmetric well induction data (Torres-

Verdín and Habashy, 2001) and it does not assume linearity between the conductivity of

the rock formations and the tool response. However, the EBA remains as fast to evaluate

as the first-order Born approximation. Extensive numerical experiments have shown that

the EBA remains accurate for cases with much larger conductivity contrasts and higher

frequencies than the first-order Born approximation (Habashy et al., 1993; Torres-Verdín

and Habashy, 1994; Torres-Verdín and Habashy, 2001). However, due to the assumption

of the spatial smoothness of the internal electrical fields implicit in the EBA, the ensuing

accuracy is very sensitive to the location and proximity of the source with respect to the

scatterer (Torres-Verdín and Habashy, 2001; Gao et al., 2003; Gao and Torres-Verdín,

2004). Another deficiency of the EBA is that it only marginally includes multiple

scattering effects and this leads to low accuracy in some practical applications (Gao and

Torres-Verdín, 2004).

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To reduce the influence of the source and to account for additional multiple

scattering effects, Gao and Torres-Verdín (2004) proposed a High-order Generalized

Extended Born Approximation (Ho-GEBA) of EM scattering. Theoretical analysis and

numerical exercises on 3D scattering media have shown that the Ho-GEBA provides

much more accurate simulation results for a broader frequency range and for larger

conductivity contrasts than the first-order Born approximation and the EBA. Moreover,

the Ho-GEBA remains as efficient to compute as the EBA.

4.5.2.2 A Generalized Series Expansion of the Electric Field

For convenience, we define a linear integral operator as

[ ][ ] ( , )g i g dτ τωμ ′ ′⋅ = ⋅∫ r r r , (4.84)

where the subscript τ refers to the spatial support of the operator gτ .

Gao and Torres-Verdín (2004) derived a generalized series expansion for the

internal electric field from a new integral equation formulation. In the axisymmetric case,

the new integral equation can be written as

( )bE E g E Eφ φ τ φ φα α σ β= + Δ + , (4.85)

where

11

αγ

=+

, 1β α= − , (4.86)

and γ is given by

2 b

σγσΔ

=′

. (4.87)

103

Following Gao and Torres-Verdín (2004), using the method of successive

iterations one can derive a series for the internal electric fields as

( ) ( ) ( )0

nCB

nE Eφ φ

∞

=

=∑r r , (4.88)

where

( ) ( )( ) ( )1 1n n nCB CB CBE g E Eφ τ φ φα σ β− −= Δ + , n=2, 3, 4, … (4.89)

and

( ) ( )( ) ( )( )1 0 0CB CB b CBE g E E Eφ τ φ φ φα σ α= Δ + − , (4.90)

We call the series expansion given by equation (4.88) a Generalized Series (GS)

for the internal electric field, because the choice of ( )0CBEφ in equation (4.90) is arbitrary.

For example, the classical Born series, the modified Born series by Zhdanov and Fang

(1997), and the quasi-linear series by Zhdanov and Fang (1997), become special forms of

equation (4.88). One can also derive the extended Born series expansion, if one sets ( )0CBEφ

equal to the solution of the EBA (Gao and Torres-Verdín, 2004).

4.5.2.3 A Generalized Extended Born Approximation (GEBA)

Gao and Torres-Verdín (2004) derived the GEBA in terms of dyadic Green’s

functions and vectorial fields. In this section, we derive a GEBA for the scalar integral

equation. The derivation procedure is similar to that of Gao and Torres-Verdín (2004).

However, for completeness, here we give a detailed derivation procedure.

Let M be the total number of spatial discretization cells, and rewrite equation

(4.20) in component form as

( )m bmE E g Eφ φ τ φσ= + Δ ⋅ , m=1, 2, …, M. (4.91)

104

We decompose the domain τ into two sub-domains, sτ and sτ τ− , such that sτ is

a sub-domain that includes the m-th cell. Thus, equation (4.91) can be rewritten as

( ) ( )s sm bmE E g E g Eφ φ τ φ τ τ φσ σ−= + Δ ⋅ + Δ ⋅ . (4.92)

By moving the second term on the right-hand side of equation (4.92) to the left-

hand side of the same equation, one obtains

( ) ( )s sm bmE g E E g Eφ τ φ φ τ τ φσ σ−− Δ ⋅ = + Δ ⋅ . (4.93)

In connection to the last two equations, we advance the following Remark:

Remark 1: If there exists a sτ which satisfies the following two conditions:

1) Condition 1: Within sτ , the electric field E can be assumed spatially invariant,

and

2)Condition 2: Outside sτ the Green’s function decreases in amplitude

sufficiently fast to become negligible, then the second term on the right-hand side of

equation (4.93) can be neglected without affecting the accuracy of the result.

According to Remark 1, for such a sub-domain sτ , equation (4.93) can be

rewritten as

( )1s m bmg E Eτ φ φσ⎡ ⎤− Δ =⎣ ⎦ . (4.94)

Finally, equation (4.94) can be further rewritten as

m m bmE Eφ φ= Λ , (4.95)

where mΛ is a scattering function for the m-th cell, given by

105

( )( ) 11

sm gτ σ−

Λ = − Δ . (4.96)

Equation (4.95) is the fundamental equation of the GEBA. The closer the sub-

domain sτ is to satisfying Remark 1, the more accurate the solution of the internal electric

field obtained via equation (4.95) becomes. The choice of sτ possibly depends on the

source location(s), the frequency, and the conductivity contrast. Notice that the geometric

center of sτ is not necessarily the m-th cell. How to optimally determine sτ is not the

objective here. However, one can envision that the existence of such a sub-domain sτ

reduces a dense matrix problem to a banded one. Of course, by using the GEBA one does

not need to solve the banded system, which is another advantage of the approximation

described in this chapter.

As emphasized in section 4.5.1, following an extensive study on the properties of

the spectrum of the stiffness matrix, Gao and Torres-Verdín (2003) concluded that the 2D

Green’s function is not as diagonally dominant as the 3D Green’s tensor. They proposed

a weighted scattering function to make the stiffness matrix more diagonally dominant.

The weighted scattering function is readily derived as

( )1

1sm g Wτ σ

−⎡ ⎤Λ = − Δ⎣ ⎦ , (4.97)

where

( ) ( )( )

b

b m

EW

Eφ

φ

=r

rr

. (4.98)

Two special cases of the GEBA can be considered as follows:

Special Case 1: When s mτ τ→ , where mτ is the singular domain, which only

encloses the m-th cell. This condition does not change equation (4.95); however, it does

106

change the scattering function given by equation (4.95). With this change, the scattering

function becomes

( ) ( )11 1

m

sm gτ σ

−⎡ ⎤Λ = − Δ⎣ ⎦ . (4.99)

or

( ) ( )11 1

m

sm g Wτ σ

−⎡ ⎤Λ = − Δ⎣ ⎦ . (4.100)

This case is the simplest case of the GEBA since the computation of the scattering

function is trivial. However, the above expression may not provide sufficiently accurate

solutions since such the operating condition violates condition 2 in Remark 1, i.e. the

Green’s function may not fall off sufficiently fast to cause the second term on the right-

hand side of equation (4.93) to be negligible.

Special Case 2: When sτ τ→ , the scattering function becomes

( ) ( ) 12 1sm gτ σ

−Λ = − Δ⎡ ⎤⎣ ⎦ . (4.101)

or

( ) ( ) 12 1sm g Wτ σ

−Λ = − Δ⎡ ⎤⎣ ⎦ . (4.102)

Equation (4.101) is identical to that of the EBA (Habashy et al., 1993; Torres-

Verdín and Habashy, 1994; and Torres-Verdín and Habashy, 2001). Such a situation

represents the most complex case of the GEBA because the computation of the scattering

function given by equation (4.101) requires computational resources proportional

to ( )2O M . However, the computation of the scattering function can be accelerated with

the use of FFTs. We note that this treatment may not provide accurate simulations, since

it violates the condition 1 in Remark 1, i.e., the electric field, in general, may not be

107

spatially invariant in the whole domain. Numerical experiments, however, show that this

form of the scattering function is adequate for the 2D axisymmetric case.

4.5.2.4 A High-Order Generalized Extended Born Approximation (Ho-GEBA)

In the previous section, we make the assumption that a sub-domain sτ satisfies

Remark 1. Instead of finding an optimal sub-domain sτ , this section introduces an

alternative strategy. This strategy allows us to choose a sub-domain sτ , which satisfies

Condition 1 as closely as possible, and we approximate the electric field E on the right-

hand side of equation (4.93) in some fashion. We now show how to develop such a

strategy using the generalized series (GS) expansion of the internal electric field.

If a sub-domain sτ satisfies Condition 1 and that partially satisfies Condition 2,

equation (4.94) can be rewritten as

( ) ( ) ( )1s sm bmg E E g E g Eτ φ φ τ φ τ φσ σ σ⎡ ⎤− Δ = + Δ ⋅ − Δ ⋅⎣ ⎦ , m=1, 2, …, M. (4.103)

Notice that in this last equation the second term in equation (4.93) has been split into two

terms.

By substituting the GS of E (keeping the first N terms, for convenience) in

equation (4.84) into the right-hand side of equation (4.103), one can derive the equation

for the Ho-GEBA as follows:

( ) ( ) ( ) ( )1

( )

0( )

NNn

m CBm m CBmn

E E Eφ φ φ

−

=

′≈ + Λ ⋅∑r r r r , m=1, 2, …, M. (4.104)

where ( )NCBmEφ′ is given by

( ) ( )( ) ( )1 1N N NCBm CBm CBmE g E Eφ τ φ φσ γ− −′ = Δ ⋅ + ⋅ , N=2, 3, … (4.105)

and

108

( ) ( )( ) ( )1 0 0CBm CBm bm CBmE g E E Eφ τ φ φ φσ′ = Δ ⋅ + − . (4.106)

Equation (4.104) is the fundamental equation of the Ho-GEBA.

Two special cases can also be derived for the Ho-GEBA:

Special Case 1: Substitution of mΛ in equation (4.104) for 1smΛ yields

( ) ( ) ( ) ( )1

( ) 1

0( )

NNn s

m CBm m CBmn

E E Eφ φ φ

−

=

′≈ + Λ ⋅∑r r r r , m=1, 2, …, M. (4.107)

This form of the Ho-GEBA closely follows the assumptions made in the derivation of the

Ho-GEBA and consequently, becomes an accurate approximation to solve EM scattering

problems. Since the scattering function may be far from optimal, equation (4.107) may

converge slower than equation (4.104) with an optimal scattering function.

Special Case 2: One may posit that the substitution of mΛ in equation (4.104) for

2smΛ yields an approximation corresponding to the special case 2 of the GEBA. As a

matter of fact, we remark here that such a derivation is not possible because when sτ τ→ ,

the term involving sτ τ− in equation (4.104) automatically approaches zero, and only the

term bmEφ remains. However, Gao and Torres-Verdín (2004) showed that a similar

equation can be derived from the original equation from which the EBA is derived. The

final equation is given by

( ) ( ) ( ) ( )1

( ) 2

0( )

NNn s

m CBm m CBmn

E E Eφ φ φ

−

=

≈ + Λ ⋅∑r r r r , m=1, 2, …, M. (4.108)

Incidentally, simple substitution from mΛ to 2smΛ gives rise to equation (4.108).

Equation (4.104) is the basic equation for the Ho-GEBA in axisymmetric media.

The physical significance of the Ho-GEBA has been detailed by Gao and Torres-Verdín

109

(2004). Here, we remark that, because the Ho-GEBA includes source and multiple

scattering effects, it is in general more accurate than both the EBA and the first-order

Born approximation. In addition, the Ho-GEBA is computationally as efficient as the

EBA. Because the scattering terms in the GS expansion can be computed with the FFT,

the final computational cost is close to ( )2logO M M , where M is the total number of

spatial discretization cells.

We developed a computer program to incorporate all the options described in the

previous sections, including (a) the GS with different initial guesses, 9b) the GEBA with

different possible scattering functions, and (c) the Ho-GEBA with different initial guesses

and scattering functions. For the numerical examples shown in next section, we make use

of the scattering function given by equation (4.102), whereas the initial guess of the GS is

the solution of the GEBA.

4.5.2.5 Numerical Examples

Figure 4.10a shows the geometry of a generic three-layer axisymmetric

formation that includes a borehole and mud-filtrate invasion. In that figure, wr is the

radius of the wellbore, and xor is the radius of the invaded zone. The zone where xor r>

is the original (uninvaded) formation. Figure 4.10b shows the spatial distribution of

electrical resistivity corresponding to the geometrical model decribed in Figure 4.10a.

The figure has been simplified to show only a radial plane about the axis of symmetry.

On the figure, bR is the mud resistivity in the borehole, xoR is the resistivity of the

invaded zone, and tR is the resistivity of the original (uninvaded) formation.

110

xoR

xoR

xoR

tR

tR

tR

Borehole

bR

o

ρ

z

wr

xor

(a) (b)

Figure 4.10: (a) Diagram describing the geometry of a three-layer generic axisymmetric formation system that includes a borehole and mud-filtrate invasion. In the figure, wr is the radius of the wellbore, and xor is the radius of the invaded zone. The zone where xor r> corresponds to the original (uninvaded) formation. (b) Spatial distribution of formation resistivity corresponding to the geometry described in (a), where bR is the mud resistivity in the well, xoR is the resistivity of the invaded zone, and tR is the resistivity of the original formation.

111

Figure 4.11 shows the three-coil tool configuration assumed in the numerical

simulation exercises. The tool consists of one transmitter and two receivers, where the

spacing between the transmitter and the first receiver ( 1L ) is 0.6 m, and the spacing

between the transmitter and the second receiver ( 2L ) is 0.65 m. The assumed measured

signal ( zHΔ ) is the difference between the signal in Receiver 1 ( 1zH ), and Receiver 2

( 2zH ). The operating frequencies are 25 KHz and 100 KHz.

Figure 4.12 describes the Resistivity Model 1. It consists of a one-layer formation

embedded in a background medium that has the same resistivity as the mud in the

wellbore, equal to 1bR m= Ω⋅ , and the background dielectric constant is assumed to be 1.

The thickness of the layer is 3.2 m,

and 0.3xor m= , 0.1wr m= , 0.5xoR m= Ω⋅ , 0.2tR m= Ω⋅ . The maximum conductivity

contrast included in this synthetic model is equal to 1:5, which reflects the contrast

Figure 4.11: Three-coil tool configuration. The assumed borehole induction tool consists of one transmitter and two receivers, with the spacing between the transmitter and the first receiver ( 1L ) equal to 0.6 m, and the spacing between the transmitter and the second receiver ( 2L ) equal to 0.65 m. The measured signal ( zHΔ ) is the difference between the signal at Receiver 1 ( 1zH ), and the signal at Receiver 2 ( 2zH ). The operating frequencies are 25 KHz and 100 KHz.

0.6m

0.05m

Tx

Rx1 Rx2

zρ

112

between the borehole and the deep formation conductivity within the layer. Figures 4.13

and 4.14 describe the simulation results for Model 1 at 25 KHz and 100 KHz,

respectively. On the two figures, the left panel corresponds to the real part of zHΔ

(“REAL”), and the right panel corresponds to the imaginary part of zHΔ (“IMAG”). The

Ho-GEBA (up to the 3rd-order term, the HoGEBA-O2 and the HoGEBA-O3 on the

figure, the GEBA is equivalent to the HoGEBA-O1, and is also equivalent to the PEBA

for this case) results are plotted against the accurate solution from “2DIE” (the

BiCGSTAB(L)-FFT) and solutions from “Born” (the Born approximation) and “EBA”

(the EBA). By carefully studying Figures 4.4 and 4.5, one can conclude that the Ho-

GEBA (1st order to 3rd order) provides much more accurate results for this model and for

both frequencies, compared to the Born approximation and the EBA. The accuracy of the

Ho-GEBA increases with the order of the approximation, which can be seen if one

enlarges the curves shown in Figures 4.13 and 4.14.

113

Figure 4.12: Formation Model 1. The model consists of a one-layer formation embedded in a background medium with resistivity equal to that of the mud in the well, where 1bR m= Ω⋅ , and the background dielectric constant is 1. The thickness of the layer is 3.2 m, and 0.3xor m= , 0.1wr m= ,

0.5xoR m= Ω⋅ , 0.2tR m= Ω⋅ .

Borehole 0.5xoR m= Ω ⋅ 0.2tR m= Ω⋅

1bR m= Ω ⋅

1bR m= Ω ⋅

3.2 m

0.1 m

0.2 m

114

Figure 4.13: Numerical simulation results for Resistivity Model 1 at 25 KHz. The left panel shows the real part of zHΔ , and the right panel shows the imaginary part of zHΔ . The Ho-GEBA (up to the 3rd order) results are plotted against the accurate solution “2DIE”, and the solutions obtained with the Born approximation and the EBA. Note that the GEBA is equivalent to the PEBA for this case.

115

Figure 4.14: Numerical simulation results for Resistivity Model 1 at 100 KHz. The left panel shows the real part of zHΔ , and the right panel shows the imaginary part of zHΔ . The Ho-GEBA (up to the 3rd order) results are plotted against the accurate solution “2DIE”, and the solutions obtained with the Born approximation and the EBA. Note that the GEBA is equivalent to the PEBA for this case.

116

Figure 4.15 graphically describes the Resistivity Model 2. It consists of a one-

layer formation model embedded in a background medium that exhibits the same

resistivity as the mud in the well, equal to 1bR m= Ω⋅ , and the background dielectric

constant is assumed to be 1. The thickness of the layer is 3.2 m, with 0.3xor m= ,

0.1wr m= , 2xoR m= Ω⋅ , 10tR m= Ω⋅ . Compared to Model 1, Model 2 is a resistive

model. The maximum conductivity contrast included in this synthetic model is equal to

1:10, which reflects the contrast between the borehole and the deep formation

conductivity within the layer. Figures 4.16 and 4.17 describe the simulation results for

Model 2 at 25 KHz and 100 KHz, respectively. On the two figures, the left panel

corresponds to the real part of zHΔ , and the right panel corresponds to the imaginary part

of zHΔ . The Ho-GEBA results (up to the 3rd order, the HoGEBA-O2 and the HoGEBA-

O3 on the figures, the GEBA is equivalent to the HoGEBA-O1, and is also equivalent to

the PEBA) are plotted against the accurate solution from “2DIE” and solutions from

“Born” and “EBA”. From Figures 4.16 and 4.17, one can conclude that for this example

the Ho-GEBA provides more accurate results than either the Born approximation or the

EBA. The accuracy of the HO-EBA also increases with the order of approximation.

117

Figure 4.15: Formation Model 2. The model consists of a one-layer formation embedded in a background medium with electrical resistivity equal to that of the mud in the well, where 1bR m= Ω⋅ , and the background dielectric constant is 1. The thickness of the layer is 3.2 m, and 0.3xor m= , 0.1wr m= , 2xoR m= Ω⋅ ,

10tR m= Ω⋅ .

Borehole 2xoR m= Ω ⋅ 10tR m= Ω⋅

1bR m= Ω ⋅

1bR m= Ω ⋅

3.2 m

0.1 m

0.2 m

118

Figure 4.16: Numerical simulation results for Resistivity Model 2 at 25 KHz. The left panel shows the real part of zHΔ , and the right panel shows the imaginary part of zHΔ . Ho-GEBA (up to the 3rd order) results are plotted against the accurate solution “2DIE”, and the solutions obtained with the Born approximation and the EBA. Note that the GEBA is equivalent to the PEBA for this case.

119

Resistivity Model 3 is the modified Oklahoma model described in detail in Table

4.1. This model corresponds to a sequence of 26 layers, each of which is characterized by

a simple invasion front. The objective of this resistivity model is to test the accuracy of

the Ho-GEBA in a highly structured axisymmetric model exhibiting a variety of bed

thicknesses (2 ft-18 ft) and conductivity contrast (0.2 S/m – 143 S/m). Figure 4.18

summarizes the simulation results at 25 KHz, and Figure 4.19 summarizes the simulation

Figure 4.17: Numerical simulation results for resistivity model 2 at 100 KHz. The left panel shows the real part of zHΔ , while the right panel shows the imaginary part of zHΔ . The Ho-GEBA (up to the 3rd order) results are plotted against the accurate solution “2DIE”, and the solutions obtained with the Born approximation and the EBA. Note that GEBA is equivalent to PEBA for this case.

120

results at 100 KHz. Again, on each figure the left panel corresponds to the real part

of zHΔ , while the right panel corresponds to the imaginary part of zHΔ . From inspection

of Figures 4.18 and 4.19, one can conclude that the Ho-GEBA consistently provides

more accurate simulation results than either the Born approximation or the EBA, for both

the real and the imaginary part, and for both frequencies. The solutions obtained with the

Born approximation and the EBA are so inaccurate that they may not be used for

practical simulation purposes. The accuracy of the Ho-GEBA increases with the order of

the approximation.

To assess the effects of frequency on the accuracy of the Ho-GEBA, we consider

a frequency range of 100 Hz-2 MHz, which represents a typical frequency range in

induction logging. The formation model considered is Model 1, and the logging point

considered corresponds to a depth of -2 m, where relatively large oscillations in the

magnetic fields occur. We compare the accuracy of the Ho-GEBA, the Born

approximation, and the EBA by calculating the percentage error of the simulation both in

amplitude and phase. Results of this exercise are shown in Figure 4.20, where the

horizontal axis describes the frequency and the vertical axis describes the corresponding

percentage error in amplitude (left panel) and phase (right panel). Figure 4.20 indicates

that the EBA and the Born approximation have very similar performance for this

particular example. The two approximations remain accurate only below 10 KHz (at

10KHz, although both the EBA and the Born approximation have small errors in

amplitude, they tend to have large errors in phase). On the other hand, the Ho-GEBA

provides accurate simulations for the whole frequency range considered in Figure 4.20.

121

This exercise confirms that the Ho-GEBA possesses a much broader range of

applications than either the Born approximation or the EBA.

For all the numerical examples considered in this section, the computation of the

Ho-GEBA is as efficient as the Born approximation and the EBA. When FFTs are used in

the computer algorithm, the total computational cost is proportional to O(CN), where C is

a constant much smaller than N ( the number of the discretization cells).

Figure 4.18: Numerical simulation results for the modified Oklahoma model (described in Table 4.1) at 25 KHz. The left panel shows the real part of zHΔ , and the right panel shows the imaginary part of zHΔ . The Ho-GEBA (up to the 3rd

order) results are plotted against the accurate solution “2DIE”, and the solutions obtained with the Born approximation and the EBA. The GEBA is equivalent to the PEBA for this case.

122

Figure 4.19: Numerical simulation results for the modified Oklahoma formation model (described in Table 4.1) at 100 KHz. The left panel shows the real part of

zHΔ , and the right panel shows the imaginary part of zHΔ . The Ho-GEBA (up to the 3rd order) results are plotted against the accurate solution “2DIE”, and the solutions obtained with the Born approximation and the EBA. The GEBA is equivalent to the PEBA for this case.

123

Figure 4.20: Comparison of simulation results obtained with three approximations in the frequency range between 100 Hz and 2 MHz. The formation model considered is Model 1 and the logging point corresponds to a depth of -2 m. In the figure, the horizontal axis corresponds to frequency and the vertical axis describes the percentage errors in amplitude (left panel) and phase (right panel) of zHΔ .

124

4.6 CONCLUSIONS

This chapter developed efficient full-wave techniques and approximate techniques

to simulate the response of multi-frequency induction tools in axisymmetric media. The

full-wave techniques are the BiCGSTAB(L)-FFT, the BiCGSTAB(L)-FFHT, and the

finite differences, respectively. Several numerical exercises confirm that computer codes

based on these three techniques provide accurate simulation results for complex rock

formations that exhibit mud-filtrate invasion.

Approximate simulation techniques considered in this chapter include a

Preconditioned Extended Born Approximation (PEBA) and a High-Order Generalized

Extended Born Approximation (Ho-GEBA). Numerical exercises show that the PEBA

and the Ho-GEBA provide more accurate simulation results than the Born approximation

and the EBA but entails a similar computational efficiency.

125

Supplement 4A: Fast Hankel Transform (FHT)

The Hankel transform is defined as

( ) ( ) ( ) ρρρρ ρρ dkJfkF v∫∞

=0

, (4A-1)

and its inverse transform is given by

( ) ( ) ( ) ρρρρ ρρ dkkJkkFf v∫∞

=0

, (4A-2)

where ( )⋅νJ is the Bessel function of the first kind and v-th order. Notice that equations

(4A-1) and (4A-2) assume continuous functions.

Many methods have been proposed to compute the FHT (Fast Hankel Transform),

such as those based on linear filters (Anderson, 1979; Johansen and Sorensen, 1977), as

well as methods based on the FFT.

In this work, we choose to use the FFT method. Given that equations (4A-1) and

(4A-2) are completely dual, without loss of generality we choose to solve equation (4A-

2).

Let

xek −=ρ , and ye=ρ , (4A-3)

where ( )∞∞−∈ ,x , ( )∞∞−∈ ,y . Substitution of equation (4A-3) into equation (4A-2)

yields

( ) ( ) ( ) dxeeJeeFefe xyxyv

xxyy ∫∞

∞−

−−−−−= . (4A-4)

Equation (4A-4) can be written as a convolution operation, namely,

( ) vv HEdxxyHxEyG *)()( =−= ∫∞

∞−, (4A-5)

126

where

( )yy efeyG =)( , (4A-6)

( ) ( ) xx eeFxE −−−= , (4A-7)

and

( ) ( ) xvv exJxH = . (4A-8)

Using the convolution theorem, equation (4A-5) can be written as a simple

product in the Fourier domain,

( ) ( )sHsEsG vˆˆ)(ˆ ⋅= , (4A-9)

where G , E , and vH are the Fourier transforms of G, E, and Hv , respectively.

Equation (4A-9) can be efficiently evaluated using the FFT technique, namely

first applying the FFT on E and Hv, and then applying the inverse FFT on the product of

the corresponding FFTs. However, this procedure involves the approximation of the

continuous Fourier transform using the DFT, which suffers from numerical errors.

Here, we point out that, by making use of the following integral, vH can be

evaluated analytically to reduce the effects of truncation:

( )⎟⎠⎞

⎜⎝⎛ +−

Γ

⎟⎠⎞

⎜⎝⎛ ++

Γ= −−∞

∫2

12

1

2 1

0 uv

uv

adtatJt uuv

u a >0, ( ) 1Re −>+ vu , ( )21Re <u , (4A-10)

where

( ) ∫∞ −−=Γ0

1 dtetz tz , ( ) 0Re >z . (4A-11)

Thus,

127

( ) ( ) dyeeJesH syjyyv

π21

ˆ −∞

∞−∫= . (4A-12)

Substitution of the transformation yet = into equation (4A-10) yields

( ) ( ) ( )( )sj

sjdttJtsH sjsjv π

πππ

+Γ−Γ

== −∞ −∫ 112ˆ 2

10

2 , (4A-13)

with ( ) 12Re −>− sj π , and ( )212Re <− sj π .

Notice that we only make use of the first-order of Bessel function in this work.

Moreover, vH has the following property

1ˆ ≡vH , (4A-14)

which explains why linear-filters are also used to numerically implement the FHT.

Supplement 4B: Finite Differencing of the General Wave Equation (4.14)

Equation (4.14) describes the general equation for both TE and TM waves.

Section 4.4.3 derived the finite-difference equation for the TE wave equation. Since

equation (4.14) involves the derivatives of the material property ζ with respect to ρ and

z, the finite-difference formulation of equation (4.14) involves additional difficulties. The

finite-difference equation should satisfy the following conditions:

(1) No singularity at 0ρ = ,

(2) Boundary conditions are satisfied,

128

(3) If the rock formation simplifies to a homogeneous one, the solution of the

finite-difference equation should match the analytical solution exactly, i.e., whenζ μ= ,

it should reduces to that of the TE wave equation, and

(4) Direct evaluation of the derivatives of the material property should be avoided

to retain the accuracy for the case of large material contrasts.

Direct finite differencing of equation (4.14) does not satisfy the above conditions.

A new formulation is required to approach this problem. After careful considerations, we

arrived at the following PDE that is suitable for a finite-difference formulation:

( ) ( )20 02

1 1 1 1 .k A i I z zz z φ ζ

ζ ρ ζ ζ ωζ δ ρ ρ δρ ρ ζ ρ ρ ρ ζ ρ ρ ρ ζ

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ − − + + = − − −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

. (4B-1)

The corresponding finite-difference grid configuration is shown in Figure 4.3.

Here, Aφ is sampled at the center of the cell face, while its counterparts (the ρ and z

components of E or H) are sampled at the center of the cell edges. Thus, using central

finite differences, at location ( )1/ 2, 1/ 2i k+ + , one has

( ) ( )( ) ( )( )

( )( ) ( )( )

3/ 2, 1/ 2 1/ 2, 1/ 211, 1/ 21/ 2, 1/ 2

1

1/ 2 1/ 2, 1/ 2 1/ 2, 1/ 2, 1/ 2

i k i kii ki k

i

i i i k i kii k

i

A AA

A A

φ φ

φ

φ φ

ρζ ρζ ρ ζ

ρ ρ ζ ρ ρ ρ ρζ ρ

+ + + +++ ++ +

+

+ + + − ++

⎧ ⎫⎡ ⎤− −⎪ ⎪⎢ ⎥

Δ∂ ⎪ ⎪∂ ⎣ ⎦= ⎨ ⎬∂ ∂ Δ ⎡ ⎤⎪ ⎪−⎢ ⎥⎪ ⎪Δ⎣ ⎦⎩ ⎭

,(4B-2)

( )

( )( )

( )

( )

( )

3/ 2, 1/ 2 1/ 2, 1/ 21/ 2, 1/ 2

3/ 2, 1/ 2 1/ 2, 1/ 21/ 2 1

i k i ki k

i k i ki i i

A A Aφ φ φζ ζρ ρ ζ ζ ζρ ρ ρ

+ + − ++ +

+ + − ++ +

⎡ ⎤⎛ ⎞∂= −⎢ ⎥⎜ ⎟∂ Δ + Δ⎝ ⎠ ⎢ ⎥⎣ ⎦

, (4B-3)

( )( ) ( )3/ 2, 1/ 2 1/ 2, 1/ 2

1/ 2 1

1 1 i k i k

i i i

AA Aφφ φρ ρ ρ ρ ρ

+ + − +

+ +

∂ ⎡ ⎤= −⎣ ⎦∂ Δ + Δ, (4B-4)

and

129

( ) ( )( ) ( )( )

( )( ) ( )( )

1/ 2, 3/ 2 1/ 2, 1/ 21/ 2, 11/ 2, 1/ 2

1

1/ 2, 1/ 2 1/ 2, 1/ 21/ 2,

11

1

i k i ki ki k

k

i k i kki k

k

A Az

Az z z A A

z

φ φ

φ

φ φ

ζζζζ

ζ

+ + + ++ ++ +

+

+ + + −+

⎡ ⎤− −⎢ ⎥Δ∂ ∂ ⎢ ⎥=⎢ ⎥∂ ∂ Δ

−⎢ ⎥Δ⎢ ⎥⎣ ⎦

. (4B-5)

Substitution of equations (4B-2) through (4B-5) into equation (4B-1) yields

( ) ( ) ( ) ( ) ( )

( ) ( )

1/ 2, 1/ 2 1/ 2, 1/ 2 1/ 2, 1/ 2 3/ 2, 1/ 2 1/ 2, 3/ 2, , , , ,

0 0

i k i k i k i k i ki k i k i k i k i k

i k

A A B A C A D A E A

i I z z zφ φ φ φ φ

ζωζ ρ δ ρ ρ δ

+ − − + + + + + + ++ + + +

= − Δ Δ − −,

(4B-6)

where

( )

( )

1/ 2, 1/ 2

, 1/ 2,

i ki

i k i kk

Azρ ζ

ζ

+ +

+

Δ=Δ

, (4B-7)

( )

( ) ( )( )

( )

1/ 2, 1/ 2 1/ 2, 1/ 2

, , 1/ 2 1/ 2, 1/ 21/ 2 1/ 2 1

1i k i k

k i i ki k i k i k

i i i i i

z zB ρ ρζ ζρ ρ ζ ζρ ρ ρ

+ + + +

+ − ++ + +

⎛ ⎞Δ Δ Δ= + −⎜ ⎟⎜ ⎟Δ Δ + Δ ⎝ ⎠

, (4B-8)

( )

( )

( )

( )

( )

( )

( )

( )

1/ 2, 1/ 2 1/ 2, 1/ 21

, 1, 1/ 2 , 1/ 21/ 2 1

1/ 2, 1/ 2 1/ 2, 1/ 22

21/ 2, 1 1/ 2,1/ 21

i k i kk i i

i k i k i ki i i

i k i ki k

i i ki k i kik k

zC

z k zz z

ρ ρζ ζρ ρ ρζ ζ

ρζ ζρ ρρζ ζ

+ + + ++

+ + ++ +

+ + + +

+ + +++

⎛ ⎞Δ= − +⎜ ⎟⎜ ⎟Δ Δ⎝ ⎠

⎛ ⎞ Δ Δ−Δ + − + Δ Δ⎜ ⎟⎜ ⎟Δ Δ⎝ ⎠

, (4C-9)

( )

( ) ( )( )

( )

1/ 2, 1/ 2 1/ 2, 1/ 21

, 1, 1/ 2 3/ 2, 1/ 21/ 2 1 1/ 2 1

1i k i k

k i i ki k i k i k

i i i i i

z zD ρ ρζ ζρ ρ ζ ζρ ρ ρ

+ + + ++

+ + + ++ + + +

⎛ ⎞Δ Δ Δ= − −⎜ ⎟⎜ ⎟Δ Δ + Δ ⎝ ⎠

, (4B-10)

and

( )

( )

1/ 2, 1/ 2

, 1/ 2, 11

i ki

i k i kk

Ezρ ζ

ζ

+ +

+ ++

Δ=Δ

. (4B-11)

Coefficients A, B, C, D and E above represent the five-point stencil of the finite

difference equation. We point out that for homogeneous formation media, these

130

coefficients reduce to those of the TE equation given by equations (4.64) through (4.68).

Such a treatment provides a more accurate computation of the background fields. Also,

notice that if the material property ζ is not homogeneous, the scattered field equation

(4.73) does not hold for the general wave equation (4.14).

Supplement 4C: The Apparent Conductivity and Its Skin-Effect Correction

Following Moran and Kunz (1962), and assuming a time dependence of the

form i te ω− , for a two-coil induction sonde, the induced voltage at a receiver with R wire

turns is given by

( ) ( )2 ,R z RV i RH Lωμ πρ ρ= , (4C-1)

where Rρ is the radius of the receiver coil and L is the spacing between the transmitter

and receiver coils.

Using equation (4.48), for a uniform medium with a propagation constant k , the

induced voltage at the receiver coil is given by

( )2

2 1 ikLiV K ikL eLωμ

= − , (4C-2)

where K is the tool constant, given by

( ) ( )( )2 2 2

4T R ETRI

KL

ωμ πρ πρ

π= , (4C-3)

and where Tρ , T , and EI are the radius, number of turns, and the electric current

circulating in the transmitter coil, respectively.

Finally, the complex apparent conductivity is expressed as

/a R Xi V Kσ σ σ= + = − , (4C-4)

131

where Rσ and Xσ are referred to as “R-signal” and “X-signal,” respectively.

In equation (4C-4), Rσ is the targeted apparent conductivity. However, it is

subject to skin effects. The relation between the induction tool response and the

formation conductivity is in general nonlinear. This nonlinearity is caused by skin effects.

The skin effect is primarily due to the mutual interaction with one another of different

portions of the second current flow in the formation. It increases with increasing values

of conductivity, tool spacing, and frequency. To obtain the apparent conductivity devoid

of skin effect, and to compare it to the geometrical factor theory, we expand Rσ and

Xσ in powers of Lδ

as,

32 213 15R

L Lσ σδ δ

⎛ ⎞⎛ ⎞= − + −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠, (4C-5)

and

2

2 23x

LL

σ σωμ δ

⎛ ⎞= − + +⎜ ⎟⎝ ⎠

, (4C-6)

where σ is the conductivity given by the Doll geometrical factor theory (no skin effect),

and δ is the skin depth defined as

2 1 ik

δωμσ

+= = . (4C-7)

In equation (4C-6), the term 2

2Lωμ

− refers to the direct mutual coupling between

the transmitter and receiver coils, and it remains independent of conductivity.

The following methods have been proposed in the literature for the calculation of

the apparent conductivity devoid of skin effect, caσ :

132

1) According to Pai and Huang (1988), caσ is expressed as

32 213 15

Rca

R R

L L

σσ

δ δ

=⎛ ⎞

− + ⎜ ⎟⎝ ⎠

, (4C-8)

where Rδ is given by

2δωμσ

=RR

. (4C-9)

2) Careful comparison of equations (4C-5) and (4C-6) indicates that after the

direct coupling term in the X-signal is removed, the X-signal provides a first-

order approximation to the skin effect. Thus, for a low-conductivity formation

and at low frequencies, the skin-effect-corrected apparent conductivity can be

written as

2

2ca R x L

σ σ σωμ

= + + . (4C-10)

Notice that the above methods do not consider the dependence of apparent

conductivity on frequency and are subject to limitations. A better method was proposed

in the US Patent 5,666,057 (Beard and Zhou, 1997) to determine the skin-effect-corrected

conductivity from multi-frequency induction measurements. Accordingly, for multi-coil

sondes, the total tool response is the normalized sum of the individual two-coil responses,

weighted by the appropriate coil strengths and spacings, namely,

,

,

i j aij

i j ijta

i j

i j ij

T RL

T RL

σ

σ =∑

∑. (4C-11)

133

Chapter 5: A BiCGSTAB(L)-FFT Method for Three-Dimensional EM Modeling in Dipping and Anisotropic Media

In this chapter, we introduce the concept of electrical anisotropy, and develop an

efficient full-wave modeling technique that uses the FFT (Fast Fourier Transform)

technique and the BiCGSTAB(l) (Bi-Conjugate Gradient Stabilized(l)) algorithm for 3D

EM modeling in electrically anisotropic media (Fang et al., 2003). This technique

exploits the convolution property of the integral equation. Accordingly, the matrix-vector

multiplications that arise when solving a linear system using BiCGSTAB(l) is accelerated

with the use of FFTs. As a result, the method circumvents all the computational

difficulties of the MoM (see Chapter 3), e.g., matrix filling, memory storage, and linear-

system solving.

Numerical simulations of measurements performed with a tri-axial induction tool

in dipping and anisotropic rock formations are benchmarked against an accurate 3D

finite-difference code and a 1D code. These benchmark exercises show that the

BiCGSTAB(L)-FFT algorithm produces accurate and efficient simulations for a variety

of borehole and formation conditions.

5.1 INTRODUCTION

Formation conductivity/resistivity determination from wellbore measurements is

probably the oldest geophysical technique. However, as shown in Chapter 4, to date,

most of the simulation efforts have focused on axisymmetric distributions of electrical

conductivity. It was a long-standing notion that, particularly for induction instruments,

the calibrated instrument output known as apparent resistivity is close enough to the

134

virgin formation resistivity to be used in water saturation calculations under most

wellbore conditions without the need for corrections. The effects of resistivity anisotropy

do not manifest themselves when formations exhibit a zero relative deviation with respect

to the borehole axis (actually, conventional instruments can only detect the horizontal

resistivity in vertical wells, thereby resulting in the belief that reservoirs are

predominantly isotropic). Recently, however, at least two reasons have propelled us to

consider the effects of resistivity anisotropy. First, in vertical wells the main effect of

electrical anisotropy is the decrease of apparent resistivity in some pay zones, resulting in

the so-called low-resistivity pay sands. One important example is the thinly-bedded

laminar sand-shale sequences, in which electrical anisotropy is the effective macroscopic

result of a packet of thin layers below the vertical resolution of induction measurements

(Schoen et al., 1999). Second, in highly deviated wells, the spatial distribution of

electrical conductivity is no longer axial symmetric. A study by Klein, Martin, and Allen

(1997) revealed that, in horizontal wells, at least two separate orthogonal components of

electrical conductivity could influence induction measurements.

The presence of electrical anisotropy has been recognized as a potential source of

error in traditional induction log interpretation. A critical component to understanding

this problem is the ability to accurately predict the behavior of induced electromagnetic

fields in anisotropic media. The advent of a commercial EM multi-component borehole

logging tool with capabilities to measure electric anisotropy, has spearheaded efforts to

simulate numerically the corresponding measurements in complex 3D logging

environments (Wang and Fang, 2001; Avdeev et al., 2002; and Newman and Alumbaugh,

2002). The numerical simulation of formation electrical anisotropy effects is of

135

significance to the accurate petrophysical interpretation of induction logging tool

responses that to date remains an open challenge (Moran and Gianzero, 1979; Klein, et.

al., 1997; Gianzero, 1999; and Kriegshauser et al., 2000).

So far, both finite-difference (FD) and integral equation (IE) approaches have

been developed to simulate induction measurements acquired in general 3-D anisotropic

media. The FD approach is flexible in handling the complexity of formation models but

is time consuming when simulating the induction response of fine structures (Wang and

Fang, 2001, and Newman and Alumbaugh, 2002). The IE method is adequate for solving

small-scale EM problems. For large-scale EM problems, the IE approach is usually

considered as an improper method because of the expenses in solving the resulting large

and dense linear-system equation (see Chapter 3). However, recent developments show a

trend that the IE method may be applied to solve large-scale EM problems within a

reasonable time frame (Gan and Chew, 1994). Moreover, the IE approach has the

advantage to yield approximations that are exceedingly faster to compute than alternative

finite-difference approaches (e.g. Born, 1933; Habashy et al., 1993; Zhdanov and Fang,

1996; Fang and Wang, 2000; and Gao et al., 2003). In this chapter, we focus on new

developments of the IE approach and on their application to simulate the induction tool

response in the presence of anisotropic formations.

Integral equation techniques require the calculation of a dyadic Green’s function

for a chosen background model and entail the solution of a full and complex-valued

linear-system equation that results from the discretization of the anomalous scattering

medium (anomalous domain). The background model can be chosen arbitrarily as long as

the Green’s function remains amenable to efficient computations. In subsurface

136

geophysical applications, a layer background is usually assumed for the computation of

the Green’s functions (e.g. Hohmann, 1983; Wannamaker, et. al., 1983; Xiong, 1992; and

Avdeev et al., 2002). The integral equation approach becomes less efficient when the

number of cells used in the discretization of the anomalous domain is relatively large. A

large number of cells substantially increases both the computation time required to solve

the large linear-system equation and the memory required for storage. In a paper by

Xiong and Tripp (1995), a block system iteration method was used to solve the linear-

system equation. This method extends the capability of the IE method to deal with large-

scale problems. However, the problem size is still very limited. The overall

computational complexity was measured at ( )2O N , where N is the total number of

discretization cells, whereas the memory requirement was roughly proportional to

( )2O N . On the other hand, Avdeev et al. (2002) and Hursan and Zhdanov (2002)

reported a CG-FFT (Conjugate Gradient - Fast Fourier Transform) approach (Catedra et

al., 1995) to perform the computations. The CG-FFT type method achieves a

computation complexity of O(Nlog2N). However, in these methods, FFT techniques were

applied in the horizontal directions only because of the assumption of a layer background.

The overall computational complexity was of the order of x y z 2 x 2 yO(N N N log N log N ) ,

where xN , yN , and zN are the number of discretization cells in each coordinate direction,

respectively. The reported memory size was roughly ( )2x y zO N N N bytes. This approach is

efficient when zN is small, which is not the case in well-logging applications. The time

required for the calculation of the Green’s function also becomes critical when zN is

large.

137

In order to make efficient use of the properties of the CG-FFT approach, which

entails a computational complexity of the order of O(Nlog2N), the FFT technique needs

to be applied in all three coordinate directions. The similar approach has been used to

solve EM scattering problems in the presence of isotropic media by Gan and Chew

(1994), and Liu et al. (2001). However, to date there are no equivalent algorithms

reported in the open technical literature to solve a full 3-D anisotropy diffusion problem.

In this chapter, we introduce a CG-FFT-type method to simulate the response of an

induction borehole tool in the presence of dipping and anisotropic rock formations.

Accordingly, we assume, a uniform isotropic background formation and the anomalous

domain is uniformly discretized in each coordinate direction. Such a strategy requires the

explicit calculation and memory storage of only the first row of the associated electric

Green’s function matrix in each direction, thereby substantially improving the efficiency

of the algorithm.

5.2 ELECTRICAL ANISOTROPY

Electrical anisotropy is a material property that varies with direction. It is very

common in sedimentary strata. Detection of the electrical anisotropy of geologic

formations is a problem that has attracted the attention of geophysicists for nearly 70

years. Applications include ground water investigations (Christensen, 2000), hydrocarbon

exploration (Moran and Gianzero, 1979; Kriegshauser et al., 2000; and Anderson et al.,

2001), and regional-scale lithospheric mapping (Weidelt, 1999; and Everett and

Constable, 1999). Some materials, such as single crystal olivine, exhibit an inherent

electrical anisotropy (Constable et al., 1992). Other materials, such as clastic sedimentary

138

reservoir rocks, exhibit a macroscopic electrical anisotropy that is due to small-scale

petrophysical variations (Anderson et al., 1994).

The petrophysical origin of macroscopic electrical anisotropy in hydrocarbon

reservoirs can be classified into three categories. The first of these is anisotropy due to

variations in water saturation. For example, in cross-bedded sandstones, variations in

grain size and pore space geometry result in a graded water saturation profile across strata

within a stratigraphic set. The variable electrical conductivity contrast between the grains

and the pores space results in macroscopic electrical anisotropy (Klein et al., 1997) in

which the electrical conductivity in the direction perpendicular to the set is smaller in

amplitude than the electrical conductivity in the plane of the set. The second mechanism

arises from thin interbeds of sediments with different electrical properties. Klein et al.

(1997) showed that the high conductivity contrast between shales and sands results in a

pronounced anisotropy for shaly sand sequences. Finally, porosity variations (i.e. bimodal

porosity models) have recently been identified as a potential source of electrical

anisotropy in uniformly saturated water sands (Schon et al., 2000).

139

In sedimentary strata, one of the most common types of electrical anisotropy is

the so-called transversely isotropic (TI) anisotropy, in which the horizontal resistivity, hR ,

remains constant in the horizontal bedding plane, while the vertical resistivity, vR , normal

to the bedding plane is different from hR . Figure 5.1 shows a typical TI rock formation. It

has been shown that a TI anisotropic medium could be rescaled to an isotropic one with

an effective resistivity, R, given by

h vR R R= ⋅ . (5.1)

In such cases, the anisotropy coefficient λ is defined as

2 /v hR Rλ = , (5.2)

for both galvanic and induction tools. The apparent resistivity, aR , in a TI anisotropic

medium can be calculated with the expression (Moran and Gianzero, 1979)

hRvR

Figure 5.1: Example of a typical TI anisotropic rock formation.

140

2 2 2cos sinh

aRR λ

λ α α

⋅=

+, (5.3)

where α is the angle between the tool axis and the vertical direction.

When 0α = , a hR R= , and therefore vertical resistivity can not be detected by

conventional resistivity logging tools in vertical wells. This situation is commonly

referred to as the “paradox of electrical anisotropy”.

Accurate determination of the anisotropy coefficient is crucial for the

petrophysical evaluation of hydrocarbon reservoirs. For example, in a sand-shale

laminated formation, hR tends to reflect the resistivity of the shale, while vR tends to

reflect the resistivity of the sand. Since the resistivity of the sand is needed to evaluate

hydrocarbon saturation, knowledge of the anisotropy coefficient is necessary to estimate

vR from the measured hR . Because in such a situation hR is usually smaller than vR , the

corresponding measurement will indicate a low-resistivity pay sand.

Electrical anisotropy may be determined by joint inversion of induction and

laterolog measurements. This is possible because of the discrepancy between the two

measurements in shales and laminated sand-shale sequences (Hagiwara et al., 1999).

However, due to the use of focusing guard electrodes, laterolog measurements are more

sensitive to horizontal resistivity. Yang (2001) introduced the use of the lateral log

instead of the laterolog log to detect and assess electrical anisotropy. The basic idea is as

follows: suppose that the layer thickness is h , and that the corresponding layer thickness

that the lateral tool can sense is lath . If the resistivity that the lateral log can measure is

given by latR , then the following relation holds:

lath hλ= ⋅ , (5.4)

141

and

lat h v hR R R Rλ= ⋅ = ⋅ . (5.5)

Because induction tools preserve the true thickness and measure hR , the estimation of h

and hR are inverted from induction logs allows the subsequent estimation of λ from the

corresponding lateral measurements.

Although the above methods may be useful, the best way to detect and estimate

electrical anisotropy is to develop a new generation of induction tools that can measure

electrical anisotropy directly. To this end, the industry has seen the commercialization of

multi-component induction tools by Baker Atlas (3DEXTM, Kriegshauser, 2000) and

Schlumberger (Rosthal et al., 2003). These tools may exhibit some differences in tool

design and ensuing data interpretation. However, the basic tool configuration consists of

has three transmitting coils and three receiving coils aligned in three different directions,

respectively, which make it possible to measure all nine magnetic field components at

every tool location. Electrical anisotropy, the relative dip angle and the relative rotation

angle, can all be estimated from the measured nine magnetic field components. In this

dissertation, such a tool is referred to as “tri-axial induction tool.” Figure 5.2 illustrates

the components of a generic tri-axial induction tool. This tool measures the three-

dimensional tensor magnetic field response as

xx xy xz

yx yy yz

zx zy zz

H H HH H HH H H

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

H , (5.6)

where the entries in the first, second and third column are due to magnetic sources

oriented in the x, y, and z directions, respectively.

142

5.3 COORDINATE-SYSTEM TRANSFORMATION

If the angle between the formation bed strike and the well axis is not 90o, as in the

case of dipping rock formations or deviated wells, it is necessary to differentiate between

two different coordinate systems: the formation coordinate system and the instrument

Ry

Rz

Rx

xTx

Tz

Ty

z

y

Figure 5.2: Illustration of a generic tri-axial induction tool. The tool consists of 3 transmitters and 3 receivers oriented along the three coordinate axes. The transmitters could be deployed at the same point (collected transmitters). The same is true for the receiver (collected receivers).

143

coordinate system. Also, very often one needs to rotate a vector or a tensor from one

system to another.

Let ( zyx ′′′ ,, ) represent the instrument system, and ( ), ,x y z represent the

formation system. Assume that α is the relative dip angle, which is the angle between

the z′ axis and z axis when one rotates the z axis to z′ along the x z− plane, and β is

the relative rotation angle, which is the angle between x′ and x axis when one rotates

the x axis to x′ in the x y− plane. These two rotations can be expressed as a rotation

matrix T given by (Moran and Gianzero, 1979)

cos cos cos sin sinsin cos 0

sin cos sin sin cos

α β α β αβ β

α β α β α

−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

T . (5.7)

Notice that the property

1 T− =T T (5.8)

holds for the rotation matrix, where the superscript T denotes the transpose of a matrix.

Let A be a vector in the formation system, and A′ be a vector in the instrument

system. We can obtain A from A′ with the linear projection,

′=A TA . (5.9)

Assume that the conductivity tensor in the formation system is σ . For a TI

anisotropic formation, this tensor can be expressed as

0 00 00 0

h

h

v

σσ σ

σ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

. (5.10)

144

We proceed to derive the expressions of the conductivity tensor σ′ in the

instrument system fromσ . By denoting the current density by J when referred to

( , , )x y z and J′when referred to ( zyx ′′′ ,, ) together with the use of equation (5.9) yields

′=J TJ . (5.11)

The same relation applies for the electric field E. In the instrument system, one obtains

σ′′ ′=J E . (5.12)

Substitution of equation (5.12) into equation (5.11) together with equation (5.9) yields

1σ σ σ−′ ′′= = =J T E T T E E . (5.13)

Finally, one obtains

1σ σ −′= T T , (5.14)

or

1xx xy xz

yx yy yz

zx zy zz

σ σ σσ σ σ σ σ

σ σ σ

−

⎡ ⎤′ ⎢ ⎥= = ⎢ ⎥

⎢ ⎥⎣ ⎦

T T . (5.15)

We emphasize here that the conductivity tensor σ′ should be symmetric and non-

negative:

(1) Symmetry

The conductivity tensor is symmetric whenever the magnetic field does not play a

role in the conduction process. Therefore, the conductivity tensor remains symmetric in

the presence of a purely ohmic conduction system.

(2) Non-negativity

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The conductivity tensor σ′ remains positive semidefinite because the time-

averaged specific energy dissipation, * *1 12 2

E J E Eσ′⎛ ⎞ ⎛ ⎞⋅ = ⋅ ⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ is non-negative.

Explicit expressions of the conductivity tensor for some special cases are as

follows:

(1) TI anisotropic formation (uniaxial anisotropy)

βασσσσ 22 cossin)( hvhxx −+= ,

ββασσσ cossinsin)( 2hvxy −= ,

βαασσσ coscossin)( hvxz −= ,

βασσσσ 22 sinsin)( hvhyy −+= ,

βαασσσ sincossin)( hvyz −= ,

and

ασσσσ 2sin)( hvvzz −−= .

(2) Biaxially anisotropic formation

Biaxial anisotropy arises when the three principal tensor elements are different.

The conductivity tensor can then be written as

1

2

3

0 00 00 0

σσ σ

σ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

. (5.16)

After rotation, one obtains

βασσβσσσσ 2213

2212 cossin)(cos)( −+−+=xx ,

ββασσββσσσ cossinsin)(cossin)( 21321 −+−=xy ,

146

βαασσσ coscossin)( 13 −=xz ,

βασσβσσσσ 2213

2212 sinsin)(sin)( −+−+=yy ,

βαασσσ sincossin)( 13 −=yz ,

and

ασσσσ 2133 sin)( −−=zz .

In addition to conductivity anisotropy, the electromagnetic properties of

anisotropic media are characterized by two rank-two real and symmetric tensors, namely,

electrical permittivity and magnetic permeability. These tensors obey the same

transformation rules as the conductivity tensor.

5.4 AVERAGING OF THE CONDUCTIVITY TENSOR

In the numerical simulation of EM phenomena, the modeling domain is

discretized into many small domains, which are called “cells.” One usually assigns one

conductivity value for each cell. However, because the spatial distribution of conductivity

in the whole modeling domain is inhomogeneous and the cell size is finite, often

electrical conductivity at different locations of a cell are different, e.g., when crossing a

material boundary. Thus, the concept of “conductivity averaging” must be introduced to

calculate the conductivity value assigned to each cell. Conductivity averaging for the case

of anisotropic media is rather complicated, since several tensors need to be averaged to

produce the final result.

In this dissertation, we adopt the conductivity averaging method developed by

Wang and Fang (2001). This method is based on Kirchhoff’s theorem. Supplement 5A

gives a detailed derivation of this conductivity averaging method.

147

5.5 SOLUTION OF THE LINEAR SYSTEM OF EQUATIONS

As emphasized in Chapter 3, the MoM results in a complex linear system of

equations. The linear system can be solved by direct methods, such as LU decomposition.

However, LU decomposition entails a computation cost proportional to ( )3O N , where N

is the size of the matrix. Therefore, in general, direct matrix solution methods are only

efficient when solving small-scale problems.

As pointed out in Chapter 3, there are three main computational issues inherent to

the solution of large-scale EM simulation problems. The matrix-filling problem has been

approached with the techniques developed in Chapter 3. Memory storage requirements

are so large that it is not possible to store the whole matrix in memory, nor it is possible

to store it in hard disk. As a result, the linear system cannot be solved via direct methods.

In this chapter, we show how to utilize the space-shift invariant property of the dyadic

Green’s function to reduce computer storage requirements. In addition, the use of block

Toeplitz matrices resulting from the space-shift invariant property of the dyadic Green’s

function together with an iterative algorithm allow one to perform matrix-vector

multiplications by way of FFTs. The latter strategy reduces the computational cost to

approximately ( )2logO N N , where N is the number of spatial discretization cells.

In this dissertation, we make use of stabilized versions of the iterative algorithm

BiCG (Bi-Conjugate Gradient), also referred to as BiCGSTAB(L) (Bi-Conjugate

Gradient STABilized(L)) (Sleijpen and Fokkema,1993), where L identifies the various

levels of stabilization. Supplement 5B provides the pseudo code associated with the

BiCGSTAB(L) algorithm.

148

To illustrate the performance of the BiCGSTAB(L) over the BiCG, Figure 5.3 is a

comparison of the convergence behavior of the BiCG, the BiCGSTAB(1) and the

BiCGSTAB(2), when solving a 3D EM simulation problem by finite differences (Hou

and Torres-Verdín, 2004). The comparison clearly indicates that the BiCGSTAB(L) is

more efficient than the BiCG in terms of both convergence speed and stabilization.

5.6 THE BICGSTAAB(L)-FFT ALGORITHM

As emphasized in Chapter 3, the MoM transforms the electrical integral equation

into a complex matrix system, which can be symbolically written as

Figure 5.3: Comparison of the convergence behavior of the BiCG and the BiCGSTAB(L).

149

[ ][ ]{ }[ ] [ ]bG E EσΙ − Δ = , (5.17)

where matrix [ ]G is a full matrix with its entries being the integrals of the electrical

dyadic Green’s function; matrix [ ]σΔ is a diagonal matrix with each diagonal entry

being the tensor of the conductivity anomaly for the corresponding cell; [ ]E is the

unknown total electric field vector, and [ ]bE is the background field vector. Chapter 3

details the evaluation of the integrals of the electric dyadic Green’s function.

Careful examination of equation (2.58) indicates that the integrals of the dyadic

Green’s function only depend on the distance between two spatial locations. This feature

suggests that if a regular grid distribution is used, matrix [ ]G will exhibit the structure of

a block Toeplitz matrix. Toeplitz matrices exhibit several unique features that facilitate

both memory storage and the solution of the ensuing linear system of equations.

5.6.1 Toeplitz Matrices

A Toeplitz matrix is an n n× matrix ,k jT t= , where ,k j k jt t −= , i.e., a matrix of the

form

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−

−

−−−−

01

012

101

)1(210

tt

tttttt

tttt

T

n

n

………

. (5.18)

Examples of such matrices are covariance matrices of weakly stationary stochastic time

series and matrix representations of linear time-invariant discrete time filters. For 1D EM

problems, it can be shown that the MoM is associated with a matrix [G] that has the

150

properties of a scalar Toeplitz matrix similar to equation (5.18) whenever a uniform grid

is used in the discretization.

The Toeplitz matrix given by equation (5.18) possesses several important

features:

(1) Equal diagonal entries.

(2) The entries are fully described by its first row and first column.

In this sense, a Toeplitz matrix resembles a sparse matrix. Such a property

is very important for computer memory storage, since only the entries of the first

row and the first column need to be computed and stored.

(3) A Toeplitz matrix can be easily transformed into a circulant matrix with the

Toeplitz matrix embedded in the circulant matrix, namely,

** *T

C ⎡ ⎤= ⎢ ⎥⎣ ⎦

. (5.19)

Explicitly, the circulant matrix exhibits the form

0 1 2 1

1 0 1 2

2 1 0 3

1 2 3 0

n n

n

n n n

c c c cc c c c

C c c c c

c c c c

− −

−

− − −

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

………

…

. (5.20)

The first column of a circulant matrix is composed of the entries of the first

column Tncol ttt ][ 110 −=t and the reverse arrangement of the first row

Tnnrow ttt ][ 1)2()1( −−−−−=t of the associated Toeplitz matrix T, i.e.,

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

row

col

t0

tc , (5.21)

151

where zeros are added to make the dimension of c equal to an integer power of 2. The

circulant matrix can be constructed using vector c and the identity matrix, namely,

C R R R −= 2 n 1(c c c c) , (5.22)

where R=(e2 e3 … en e1) and ek is the k-th column of the identity matrix.

An important property of the circulant matrix is that the multiplication of a

circulant matrix times a vector can be performed using the FFT. In equation form,

(( ( ).*( ( )))C ifft fft fft= =y x c x , (5.23)

where c is the first column of the circulant matrix, .* means element-wise multiplication,

fft refers to the FFT, and ifft refers to the inverse FFT.

For the case of a matrix-vector multiplication between a Toeplitz matrix T and a

vector x, one first needs to transform the Toeplitz matrix T into a circulant matrix C

similar to equation (5.19). Then one needs to pad vector x with zeros to make its size

conformal to matrix C, i.e.

0⎛ ⎞

= ⎜ ⎟⎝ ⎠

xx . (5.24)

Finally,

** * 0 *T T

C ⎛ ⎞⎛ ⎞ ⎛ ⎞= = =⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠

x xy x . (5.24)

Therefore, the results of the multiplication between a Toeplitz matrix and a vector are the

first n entries of vector y .

Because the main computational cost associated with an iterative algorithm, i.e.

the BiCGSTAB(L), is the matrix-vector multiplication (direct evaluation entails a

computational cost proportional to ( )2O N ), using the FFT technique reduces the

152

computational cost to ( )2logO N N . In equation (5.17), the product of [ ] [ ][ ]{ }G EσΔ

can be performed via the above-mentioned technique, thereby reducing the computational

cost to ( )2logO N N . Moreover, since only the first row and the first column in matrix

[ ]G need to be computed and stored, the matrix-filling time and the computer memory

storage are both proportional to ( )O N .

5.6.2 Block Toeplitz Matrices

If the dimensionality of the simulation problem is either 2D or 3D, then the matrix

[ ]G is no longer a scalar Toeplitz matrix, but rather a block Toeplitz matrix. However,

the generalization of the linear-system algorithm from a scalar Toeplitz matrix to a block

Toeplitz matrix is straightforward. The only difference is that 2D FFTs and 3D FFTs are

needed for 2D and 3D simulation problems, respectively.

A block Toeplitz matrix is a Toeplitz matrix with Toeplitz blocks. For a 3D

problem, suppose that there are xn , yn , and zn cells in the x, y, and z directions,

respectively, and that the cells are numbered in the order of x, y, and z axis. It is easy to

show that each zn corresponds to a block Toeplitz matrix of size yn . Each block Toeplitz

matrix corresponds to a Toeplitz matrix of size xn , and each entry of the Toeplitz matrix

is a 3 by 3 matrix. The structure of matrix [ ]G is graphically described in Figure 5.4,

where the whole matrix is block Toeplitz, each zn corresponds to a block Toeplitz matrix

of size yn , each yn corresponds to a Toeplitz matrix, T, of size xn .

153

( ) ( )

( )

( )

( ) ( )

0 01 1

0 0

1 0 1 00 1

0 1

0

1 00

0 01 1

0 0

1 0 1 01 0

y y

y yz

y

y

y y

y yz

n n

n nn

n

n

n n

n nn

T T T T

T TT T T T

T T

TT T

T T T T

T TT T T T

− − − −

− −− −

− −

−

− − − −

− −−

⎡ ⎡ ⎤ ⎡ ⎤⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢⎢ ⎡ ⎤⎢ ⎢ ⎥⎢ ⎢ ⎥⎢ ⎢ ⎥⎢ ⎢ ⎥⎣ ⎦⎢⎢⎡ ⎤ ⎡ ⎤⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Thus, for a 3D EM simulation problem, one needs to compute the first row and

first column of each Toeplitz matrix for each direction. As a result, before being padded

with zeros, the vector c described above now has three dimensions, namely

( )2 1, 2 1, 2 1x y zn n n= − − −c c . (5.25)

Therefore, the total number of entries needed to compute the entries of this vector

are ( )( )( )2 1 2 1 2 1x y zn n n− − − . Vectorc can be padded with zeros in each direction

according to equation (5.21). In addition, vector x is now also a 3D vector, and hence

needs to be padded with zeros according to equation (5.24). Finally, for a 3D simulation

problem the matrix-vector multiplication can be performed as

3(( 3( ).*( 3( )))ifft fft fft=y c x , (5.26)

where 3fft refers to a 3D FFT, and 3ifft refers to an inverse 3D FFT.

Figure 5.4: Structure of a block Toeplitz matrix resulting from a 3D EM simulation problem. Each matrix T is a Toeplitz matrix of size xn , and each entry of the Toeplitz matrix is 3 by 3 matrix.

154

The final solution of the matrix-vector multiplication is obtained following the

algorithm described in section 5.6.1.

In this dissertation, the method discussed above is referred to as a

“BiCGSTAB(L)-FFT” method. In summary, the BiCGSTAB(L)-FFT method efficiently

addresses all three computational difficulties inherent to the MoM, namely

(1) Savings in Memory Storage

Only the first row and the first column of the [G] matrix in each direction

need to calculated and stored in memory. For example, the total number of

entries needed to store the whole matrix is ( )2

x y zn n n , while using the

BiCGSTAB(L)-FFT method, the number of entries needs to store in memory is

only ( )( )( )2 1 2 1 2 1x y zn n n− − − . The ratio is roughly ( )8 / x y zn n n , which

represents significant memory storage savings for large-scale EM simulation

problems.

(2) Substantial Savings in Matrix-Filling Time

Only the first row and the first column of the [G] matrix in each direction

need to be evaluated, which also suggests a matrix-filling time ratio of

( )8 / x y zn n n . Actually, only the entries of the first row need to be evaluated; the

entries of the first column can be obtained by simple manipulation of the indices,

thereby providing an additional reduction in matrix-filling time.

(3) Solution of the Linear System of Equations

The BiCGSTAB(L)-FFT algorithm reduces the computation cost of a

matrix-vector multiplication from 2N to O(Nlog2N).

155

5.7 NUMERICAL EXAMPLES

Figure 5.5 describes the rock formation model used in this chapter to test the

BiCGSTAB(L)-FFT algorithm. This rock formation model was adapted from an example

proposed by Wang and Fang (2001). It consists of 5 horizontal layers in which the top

and bottom layers are isotropic and exhibit a resistivity of 50 Ω⋅m. The third layer is a 50

Ω⋅m isotropic layer of thickness equal to 12.0 ft. Finally, the second and fourth layers are

electrically anisotropic with a horizontal resistivity of 3 Ω⋅m and a vertical resistivity of

15 Ω⋅m. These two layers are 2.0 ft and 10.0 ft thick, respectively. Mud-filtrate invasion

may also be present in these last two layers, with an invasion radius equal to 36.0 in, and

with the resistivity in the invaded zone equal to 3 Ω⋅m. The diameter of the borehole is

equal to 8.0 in. and its resistivity is equal to 1 Ω⋅m.

Simulation results are obtained for borehole deviations of 60o for two cases of

rock formation model: first, the rock formation is assumed to exhibit no invasion and no

10.0 ft

50Ω⋅m

Rh=3 Ω⋅m RV=15 Ω⋅m

Rxo= 3 Ω⋅m

50Ω⋅m

Rh=3 Ω⋅m RV=15 Ω⋅m

Rxo= 3 Ω⋅m

50 Ω⋅m

Borehole (D=8.0 in, R=1 Ω⋅m)

2.0 ft

12.0 ft

Figure 5.5: Graphical description of the generic 5-layer electrical conductivity model used in this chapter to test the BiCGSTAB(L)-FFT algorithm (not to scale).

156

borehole, i.e. to consist of a 1D stack of layers. The second model does assume a rock

formation with borehole and invasion, with the invasion and borehole parameters

described in the preceding paragraph. We compared simulation results with those

obtained using a 1D code (identified as “1D” in the corresponding figures) and the 3D

finite-difference simulation algorithm (identified as “3D FDM” in the figures)

developed by Wang and Fang (2001). The 3D FDM simulation results reported in this

chapter have been validated and benchmarked for accuracy by Wang and Fang (2001). In

the descriptions and figures below, the identifier “3D IE” is used to designate simulation

results obtained with the BiCGSTAB(L)-FFT algorithm.

Figure 5.6 graphically illustrates shows the borehole induction instrument

assumed in the numerical simulations. It consists of one transmitter and two receivers

moving in tandem along the borehole axis. The transmitters and receivers can be oriented

Tx

Rx 1

Rx 2

1.0 m

0.6 m

xy

z

Figure 5.6: Graphical description of the assumed double receiver, single transmitter instrument for borehole induction logging (not to scale). In general, the transmitter and receivers can be oriented in the x, y, or z directions.

157

in either the x, y, or z directions. The spacing between the transmitter and the first

receiver is 1.0 m (L1), whereas the spacing between the transmitter and the second

receiver is 1.60 m (L2). It is further assumed that the instrument measurement is a

combination of the response measured by the first magnetic receiver (H1) and the second

magnetic receiver (H2), given by the formula

132

31

2 HLLHH −= . (5.27)

The result given by equation (5.27) is called the compensated magnetic field.

Such a measurement is commonly used in array induction logging to remove the direct

coupling between transmitters and receivers thereby enhancing EM response from the

rock formation. Apparent conductivities can be calculated from the compensated

magnetic fields using the formulas (Yu et al., 2001):

22

1

2

8 Im( )

1xx xx

L HLL

πσ

μω

=⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

, (5.28)

22

1

2

8 Im( )

1yy yy

L HLL

πσ

μω

=⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

, (5.29)

and

22

1

2

8 Im( )

1zz zz

L HLL

πσ

μω

=⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

, (5.30)

where Im(H) represents the quadrature component of H, and H is computed using

equation (5.27).

158

Equations (5.28) through (5.30) indicate that apparent conductivities are

proportional to the quadrature component of the corresponding compensated magnetic

fields. In this dissertation, we choose to show magnetic fields instead of apparent

conductivities.

Moreover, the numerical simulations reported in this section consider only the

imaginary component of the variable H in equation (5.27). This choice is made because

of the availability of only the imaginary components of 1D and 3D FDM simulation

results. According to our observations, the real component of the same variable

approaches zero at low frequencies (of the order of 25 KHz) and becomes approximately

equal to its imaginary counterpart at high frequencies (of the order of 250 KHz).

A grid size of 128 x 64 x 128 was constructed to perform the calculations at 20

KHz. Accordingly, the discretization in each direction was made uniform and equal to

0.1m x 0.2m x 0.1m. The average CPU time required for the calculations was

approximately 40 minutes per tool location using a 900MHz Sun Workstation.

Simulation of 200 KHz model responses required a grid size of 64 x 64 x 128. The

discretization in all three directions was made uniform with step sizes equal to 0.1m.

These calculations required an average CPU time of approximately 20 minutes per tool

location using a 900MHz Sun Workstation.

5.7.1 1D Anisotropic Rock Formation

Figures 5.7 through 5.9 show simulation results (Hzz, Hxx, Hyy) obtained with the

BiCGSTAB(L)-FFT algorithm assuming a 1D anisotropy rock formation with a dip angle

of 60o. Simulation results for two frequencies (20 KHz and 220 KHz) are compared to

159

those obtained with the 1D code and the finite-difference code (3D FDM) developed by

Wang and Fang (2001). The comparison clearly indicates that the BiCGSTAB(L)-FFT

algorithm provides accurate simulation results.

Figure 5.7: Comparison of the Hzz field component simulated with the BiCGSTAB(L)-FFT algorithm and a 1D code assuming a 1D formation. The tool and the formation form an angle of 60o. Results for 20 KHz and 220 KHz are shown on this figure.

220 KHz

20 KHz

160

Figure 5.8: Comparison of the Hxx field component simulated with the BiCGSTAB(L)-FFT algorithm and a 1D code assuming a 1D formation. The tool and the formation form an angle of 60o. Results for 20 KHz and 220 KHz are shown on this figure.

20 KHz

220 KHz

161

Figure 5.9: Comparison of the Hyy field component simulated with the BiCGSTAB(L)-FFT algorithm and a 1D code assuming a 1D formation. The tool and the formation form an angle of 60o. Results for 20 KHz and 220 KHz are shown on this figure.

20 KHz

220 KHz

162

5.7.2 3D Anisotropic Rock Formation

Figures 5.10 through 5.12 show simulation results (Hzz, Hxx, Hyy) obtained with

the BiCGSTAB(L)-FFT algorithm assuming a 3D rock formation (borehole and invasion)

and a dip angle of 60o. Simulation results for two frequencies (20 KHz and 220 KHz) are

compared against those obtained with the 3D FDM code. By comparing Figure 5.10 to

Figure 5.7, one can visualize the effects of borehole and invasion for the two simulated

frequencies, both in the magnitude and shape of the tool’s response. The comparison

shows that the BiCGSTAB(L)-FFT algorithm yields accurate results in the presence of

reasonably complex 3D anisotropy models.

Figure 5.10: Comparison of the Hzz field component simulated with the BiCGSTAB(L)-FFT algorithm and a 3D FDM code assuming a 3D formationwith borehole and mud-filtrate invasion. The tool and the formation form an angle of 60o. Results for 20 KHz and 220 KHz are shown on this figure.

220 KHz

20 KHz

163

Figure 5.11: Comparison of the Hxx field component simulated with the BiCGSTAB(L)-FFT algorithm and a 3D FDM code assuming a 3D formationwith borehole and mud-filtrate invasion. The tool and the formation form an angle of 60o. Results for 20 KHz and 220 KHz are shown on this figure.

20 KHz

220 KHz

164

Figure 5.12: Comparison of the Hyy field component simulated with the BiCGSTAB(L)-FFT algorithm and a 3D FDM code assuming a 3D formationwith borehole and mud-filtrate invasion. The tool and the formation form an angle of 60o. Results for 20 KHz and 220 KHz are shown on this figure.

220 KHz

20 KHz

165

5.8 CONCLUSIONS

This chapter described the basic algorithm elements of EM modeling in

electrically anisotropic media, including the concept of electrical anisotropy, modern tri-

axial induction tools, conductivity tensor averaging, and coordinates transformation.

More importantly, this chapter developed an efficient BiCGSTAB(L)-FFT algorithm to

circumvent the computational difficulties associated with the MoM when solving large-

scale EM simulation problems, namely, matrix-filling time, memory storage, and linear-

system solving. A detailed description was given of the properties of Block Toeplitz

matrices, of the FFT implementation of Toeplitz matrix-vector multiplications, and of the

BiCGSTAB(L) algorithm. Numerical examples show that the BiCGSTAB(L)-FFT

algorithm provides accurate simulation results compared to 1D and 3D FDM codes for

complex 3D anisotropic rock formations on a SUN workstation.

166

Supplement 5A: Conductivity Tensor Averaging

Assume that one cell is divided into ninjnk sub-cells, where (i, j, k) identifies the

sub-cell (i=1, …,ni, j=1, …,nj, k=1, …,nk).

The conductivity tensor for sub-cell (i, j, k) can be expressed as

( )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

zzzyzx

yzyyyx

xzxyxxkji

σσσσσσσσσ

σ,,

, (5A-1)

and the average conductivity tensor can be written as

x

y z

Figure 5A-1: Graphical description of the spatial discretization of a cell into a collection of sub-cells.

167

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

zzzyzx

yzyyyx

xzxyxx

σσσσσσσσσ

σ . (5A-2)

We now proceed to derive the average conductivity tensor for the volume cell shown in

Figure 5A-1.

The application of a voltage V0 across the cell in the x-direction together with the

assumption that the electric field is uniform across each sub-cell yields

( ) ( ) ( ) ( ) ( ) ( )1111211211111111 ... ii nx

nxxxxxxxx EEE σσσ === , (5A-3)

( ) ( ) ( ) ( ) ( ) ( )2121221221121121 ... ii nx


( ) ( ) ( ) ( ) ( ) ( )3131231231131131 ... ii nx


( ) ( ) ( ) ( ) ( ) ( )1212212212112112 ... ii nx


……

In FORTRAN language format, the above operations can be performed with the

following lines of code:

Do j=1, …, nj

Do k=1, …, nk

( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 i in jk n jkjk jk jk jkxx x xx x xx xE E Eσ σ σ= = = . (5A-7)

End do

End do

In addition, one has

168

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) xnn

nnxxx E

xxxV

xxxxExExE

ii

ii

0112111110

11211111

1111211211111111

.........

=Δ++Δ+Δ

=Δ++Δ+Δ

Δ++Δ+Δ ,

(5A-8)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) xnn

nnxxx E

xxxV

xxxxExExE

ii

ii

0212211210

21221121

2121221221121121

.........

=Δ++Δ+Δ

=Δ++Δ+Δ

Δ++Δ+Δ ,

(5A-9)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) xnn

nnxxx E

xxxV

xxxxExExE

ii

ii

0312311310

31231131

3131231231131131

.........

=Δ++Δ+Δ

=Δ++Δ+Δ

Δ++Δ+Δ ,

(5A-10)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) xnn

nnxxx E

xxxV

xxxxExExE

ii

ii

0122121120

12212112

1212212212112112

.........

=Δ++Δ+Δ

=Δ++Δ+Δ

Δ++Δ+Δ ,

(5A-11)

…..

In FORTRAN language format, equations (5A-3) through (5A-11) can be

implemented with the following lines of code:

Do j = 1, …, nj

Do k = 1, …, nk

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 1 2 2

1 2

001 2

......

...

i i

i

i

n jk n jkjk jk jk jkx x x

n jkjk jk

xn jkjk jk

E x E x E xx x x

V Ex x x

Δ + Δ + + ΔΔ + Δ + + Δ

= =Δ + Δ + + Δ

. (5A-12)

End do

End do

From equation (5A-3), one obtains

169

( )( ) ( )

( )211

111111211

xx

xxxx

EE

σσ

= … ( )( ) ( )

( )11

11111111

i

in

xx

xxxnx

EE

σσ

= . (5A-13)

Substitution of equation (5A-13) into equation (5A-7) yields

( ) ( )( ) ( )

( )( )

( ) ( )

( )( )

( ) ( ) ( ) xn

nn

xx

xxx

xx

xxxx

Exxx

xE

xE

xE

i

i

i

011211111

1111

111111211

211

111111111111

=Δ++Δ+Δ

Δ++Δ+Δσ

σσ

σ

. (5A-14)

Equation (5A-14) can be further written as

( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )

11211 311 111 111 311111

11 1,11 11211 111 311

011 11211 311 111 211

i

i i i

i i

nxx xx xx xx xx

x n n nxx xx xx xx

xn nxx xx xx

xE

x xE

x x x

σ σ σ σ σ

σ σ σ σ

σ σ σ

−

⎡ ⎤Δ + +⎢ ⎥⎢ ⎥Δ + + Δ⎣ ⎦ =

Δ + Δ + + Δ. (5A-15)

Subsequently, equation (5A-15) can be rewritten as

( )( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

11 11211 311 111 211

111011211 311 111 111 311

11 1,11 11211 111 211

i i

i

i i i

n nxx xx xx

x xnxx xx xx xx xx

n n nxx xx xx xx

x x xE E

x

x x

σ σ σ

σ σ σ σ σ

σ σ σ σ −

Δ + Δ + + Δ=

⎡ ⎤Δ + +⎢ ⎥⎢ ⎥Δ + + Δ⎣ ⎦

. (5A-16)

The expressions for ( )ijkxE can be derived in analogous fashion.

Subsequently, the averaged x-directed current density can be expressed as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

111 111 111 111 121 121 121 121

111 111 121 121xx x xx x

xE y z E y zj

y z y zσ σΔ Δ + Δ Δ +

=Δ Δ + Δ Δ +

. (5A-17)

In FORTRAN language format, equation (5A-17) can be implemented with the following

lines of code:

170

Do j = 1, … , nj

Do k = 1, … , nk

( ) ( ) ( ) ( )

( ) ( )∑∑

ΔΔ

ΔΔ=

kj

jkjkkj

jkjkjkx

jkxx

x zy

zyEj

,

11,

1111σ. (5A-18)

End do

End do

Finally, the xx-component of the averaged conductivity can be calculated from

x

xxx E

j

0

=σ

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1

1 2 1 1 1 2

1 2 2 1 3

1

.i i

i i

jk jk

jk

n jk n jkjk jk jk jk jk jkxx xx xx

n jk n jkjk jk jk jk jkjk xx xx xx xx xx

y z

y z x x x

x x

σ σ σ

σ σ σ σ σ

=Δ Δ

Δ Δ Δ + Δ + + Δ⋅

Δ + Δ +

∑

∑

(5A-19)

Following the same procedure, one obtains

yyσ =

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1

1 2 1 1 1 2

1 2 2 1 3

1

,j j

j j

i k i k

ik

in k in ki k i k i k i k i k i kyy yy yy

in k in ki k i k i k i k i kik yy yy yy yy yy

x z

x z y y y

y y

σ σ σ

σ σ σ σ σ

Δ Δ

Δ Δ Δ + Δ + + Δ⋅

Δ + Δ +

∑

∑

(5A-20)

and

zzσ =

171

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1

1 2 1 1 1 2

1 2 2 1 3

1

.k k

k k

ij ij

ij

ijn ijnij ij ij ij ij ijzz zz zz

ijn ijnij ij ij ij ijij zz zz zz zz zz

x y

x y z z z

z z

σ σ σ

σ σ σ σ σ

Δ Δ

Δ Δ Δ + Δ + + Δ⋅

Δ + Δ +

∑

∑

(5A-21)

To compute the off-diagonal elements of the averaged conductivity tensor, a

voltage applied in the x-direction causes currents flowing in the y-direction. Graphically,

this situation can be illustrated with the diagram

It follows that

( ) ( ) ( )ijkx

ijkyx

ijky Ej σ= . (5A-22)

Therefore, the average y-directed current density can be approximated as

( ) ( ) ( ) ( )

( ) ( )

ijk ijk ijk ijkyx x

ijky ijk ijk

ijk

E x zj

x z

σ Δ Δ=

Δ Δ

∑

∑. (5A-23)

Finally,

x

yyx E

j

0

=σ

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

1 2 1 1 1

1 2 3 2 1 3

2 1 2 2 1

1 2 3 2 1

1

i i

i i

i i

i

n jk n jkjk jk jk jk jkyx xx xx

n jk n jkjk jk jk jk jk jkjk xx xx xx xx xx xx

n jk n jkjk jk jk jk jkyx xx xx

ijk ijk n jkjk jk jk jk jkxx xx xx xx x

ijk

x z x x

x x

x z x x

x z x x

σ σ σ

σ σ σ σ σ σ

σ σ σ

σ σ σ σ σ

Δ Δ Δ + + Δ

Δ + Δ +

Δ Δ Δ + + Δ= +

Δ Δ Δ + Δ

∑

∑ ( ) ( )3 in jkjkjk x xxσ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ ,

V0

x yj

172

(5A-24)

x

zzx E

j

0

=σ

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

1 2 1 1 1

1 2 3 2 1 3

2 1 2 2 1

1 2 3 2 1

1

i i

i i

i i

i

n jk n jkjk jk jk jk jkzx xx xx

n jk n jkjk jk jk jk jk jkjk xx xx xx xx xx xx


ijk ijk n jkjk jk jk jk jkxx xx xx xx x

ijk

x y x x

x x

x y x x

x y x x

σ σ σ

σ σ σ σ σ σ

σ σ σ

σ σ σ σ σ

Δ Δ Δ + + Δ

Δ + Δ +

Δ Δ Δ + + Δ= +

Δ Δ Δ + Δ

∑

∑ ( ) ( )3 in jkjkjk x xxσ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ ,

(5A-25)

y

zzy E

j

0

=σ

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

1 2 1 1 1

1 2 3 2 1 3

2 1 2 2 1

1 2 3 2 1

1

j j

j j

j j

j

in k in ki k i k i k i k i kzy yy yy

in k in ki k i k i k i k i k i kik yy yy yy yy yy yy

in k in ki k i k i k i k i kzy yy xx

ijk ijk in ki k i k i k i k i kyy yy yy yy y

ijk

x y y y

y y

x y y y

x y y y

σ σ σ

σ σ σ σ σ σ

σ σ σ

σ σ σ σ σ

Δ Δ Δ + + Δ

Δ + Δ +

Δ Δ Δ + + Δ= +

Δ Δ Δ + Δ

∑

∑ ( ) ( )3 jin ki kik y yyσ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ ,

(5A-26)

173

y

xxy E

j

0

=σ

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

1 2 1 1 1

1 2 3 2 1 3

2 1 2 2 1

1 2 3 2 1

1

j j

j j

j j

j

in k in ki k i k i k i k i kxy yy yy

in k in ki k i k i k i k i k i kik yy yy yy yy yy yy

in k in ki k i k i k i k i kxy yy xx

ijk ijk in ki k i k i k i k i kyy yy yy yy y

ijk

y z y y

y y

y z y y

y z y y

σ σ σ

σ σ σ σ σ σ

σ σ σ

σ σ σ σ σ

Δ Δ Δ + + Δ

Δ + Δ +

Δ Δ Δ + + Δ= +

Δ Δ Δ + Δ

∑

∑ ( ) ( )3 jin ki kik y yyσ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ ,

(5A-27)

z

xxz E

j

0

=σ

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

1 2 1 1 1

1 2 3 2 1 3

2 1 2 2 1

1 2 3 2 1

1

k k

k k

k k

k

ijn ijnij ij ij ij ijxz zz zz

ijn ijnij ij ij ij ij ijij zz zz zz zz zz zz

ijn ijnij ij ij ij ijxz zz zz

ijk ijk ijnij ij ij ij ijzz zz zz zz z

ijk

y z z z

z z

y z z z

y z z z

σ σ σ

σ σ σ σ σ σ

σ σ σ

σ σ σ σ σ

Δ Δ Δ + + Δ

Δ + Δ +

Δ Δ Δ + + Δ= +

Δ Δ Δ + Δ

∑

∑ ( ) ( )3 kijnijij z zzσ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ ,

(5A-28)

and

z

yyz E

j

0

=σ

174

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

1 2 1 1 1

1 2 3 2 1 3

2 1 2 2 1

1 2 3 2 1

1

k k

k k

k k

k

ijn ijnij ij ij ij ijyz zz zz

ijn ijnij ij ij ij ij ijij zz zz zz zz zz zz

ijn ijnij ij ij ij ijyz zz zz

ijk ijk ijnij ij ij ij ijzz zz zz zz z

ijk

x z z z

z z

x z z z

x z z z

σ σ σ

σ σ σ σ σ σ

σ σ σ

σ σ σ σ σ

Δ Δ Δ + + Δ

Δ + Δ +

Δ Δ Δ + + Δ= +

Δ Δ Δ + Δ

∑

∑ ( ) ( )3 kijnijij z zzσ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ .

(5A-29)

For uniformly spaced sub-cells, equations (5A-19) through (5A-21) and (5A-24) through

(5A-29) can be simplified as

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 2

2 1 3

i

i i

n jkjk jki xx xx xx

xx n jk n jkjk jk jkjkj k xx xx xx xx xx

nn n

σ σ σσσ σ σ σ σ

=+ +

∑ , (5A-30)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 2

2 1 3

j

j j

in ki k i kj yy yy yy

yy in k in ki k i k i kiki k yy yy yy yy yy

nn n

σ σ σσ

σ σ σ σ σ=

+ +∑ , (5A-31)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 2

2 1 3

k

k k

ijnij ijk zz zz zz

zz ijn ijnij ij ijiji j zz zz zz zz zz

nn n

σ σ σσσ σ σ σ σ

=+ +

∑ , (5A-32)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2

2 3 1 3

2 1

2 3 1 3

1

i

i i

i

i i

n jkjk jkyx xx xx

n jk n jkjk jk jk jkjk xx xx xx xx xx xx

n jkjk jkyx xx xx

yx n jk n jkjk jk jk jkjkj k xx xx xx xx xx xxn n

σ σ σ

σ σ σ σ σ σ

σ σ σσ

σ σ σ σ σ σ

⎛ ⎞⎜ ⎟

+ +⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟+ +⎜ ⎟⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑ , (5A-33)

175

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

1 2

2 3 1 3

2 1 2 2 1

2 3 1 3

1

i

i i

i i

i i

n jkjk jkzx xx xx

n jk n jkjk jk jk jkjk xx xx xx xx xx xx


zx n jk n jkjk jk jk jkjkj k xx xx xx xx xx xx

x y x x

n n

σ σ σσ σ σ σ σ σ

σ σ σσ

σ σ σ σ σ σ

⎛ ⎞⎜ ⎟

+ +⎜ ⎟⎜ ⎟

Δ Δ Δ + +Δ⎜ ⎟= +⎜ ⎟+ +⎜ ⎟

⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑ , (5A-34)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2

2 3 1 3

2 1

2 3 1 3

1

j

j j

j

j j

in ki k i kzy yy yy

in k in ki k i k i k i kik yy yy yy yy yy yy

in ki k i kzy yy xx

zy in k in ki k i k i k i kiki k yy yy yy yy yy yy

n n

σ σ σ

σ σ σ σ σ σ

σ σ σσ

σ σ σ σ σ σ

⎛ ⎞⎜ ⎟⎜ ⎟+ +⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟+ +⎜ ⎟⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑ , (5A-35)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2

2 3 1 3

2 1

2 3 1 3

1

j

j j

j

j j

in ki k i kxy yy yy

in k in ki k i k i k i kik yy yy yy yy yy yy

in ki k i kxy yy xx

xy in k in ki k i k i k i kiki k yy yy yy yy yy yy

n n

σ σ σ

σ σ σ σ σ σ

σ σ σσ

σ σ σ σ σ σ

⎛ ⎞⎜ ⎟⎜ ⎟+ +⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟+ +⎜ ⎟⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑ , (5A-36)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2

2 3 1 3

2 1

2 3 1 3

1

k

k k

k

k k

ijnij ijxz zz zz

ijn ijnij ij ij ijij zz zz zz zz zz zz

ijnij ijxz zz zz

xz ijn ijnij ij ij ijiji j zz zz zz zz zz zzn n

σ σ σσ σ σ σ σ σ

σ σ σσσ σ σ σ σ σ

⎛ ⎞⎜ ⎟

+ +⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟+ +⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑ , (5A-37)

and

176

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 2

2 3 1 3

2 1

2 3 1 3

1

k

k k

k

k k

ijnij ijyz zz zzijn ijnij ij ij ij

ij zz zz zz zz zz zz

ijnij ijyz zz zz

yz ijn ijnij ij ij ijiji j zz zz zz zz zz zzn n

σ σ σ

σ σ σ σ σ σ

σ σ σσ

σ σ σ σ σ σ

⎛ ⎞⎜ ⎟

+ +⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟+ +⎜ ⎟⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑ . (5A-38)

177

Supplement 5B: Pseudocode Describing the BiCGSTAB(l) Algorithm

Solving the linear system Ax=b

Begin

k=-l,

choose 0x , 0~r ,

compute 00 Axb −=r ,

take 1,0,1,,0 0001 =====− ωαρxxu .

repeat until 1+kr is small enough.

k=k+l,

put kk ru ru == − 010 ˆ,ˆ and kx x=0ˆ ,

00 ωρρ −= ,

For j=0,…,l-1 (BiCG part)

( ) 100

101 ,,~,ˆ ρρ

ρρ

αββρ ==== + jkj rr ,

For i=0,…,j

iii uru ˆˆˆ β−= ,

end

jj uAu ˆˆ 1 =+ ,

( )γρ

αγ 001 ,~,ˆ === ++ jkj aru ,

For i=0,…,j

1ˆˆˆ +−= iii uarr ,

178

end

0001 ˆˆˆ,ˆˆ uxxrAr jj α+==+ ,

end

For j=1,…,l (Mod. G-S)(MR part)

For i=1,…,j-1

( )iji

ij rr ˆ,ˆ1σ

τ = ,

iijjj rrr ˆˆˆ τ−= ,

end

( )jj

jjjj rrrr ˆ,ˆ1),ˆˆ( 0, σγσ =′= ,

end

lll γωγγ =′= , .

For j=l-1,…,1 ( γγ ′= −1T )

∑ +=−′=

l

ji ijijj 1γτγγ ,

end

For j=1,…,l-1 ( γγ TS=′′ )

∑ −

+= ++ +=′′1

1 11l

ji ijijj γτγγ ,

end

llll uuurrrrxx ˆˆˆ,ˆˆˆ,ˆˆˆ 00000100 γγγ −=′−=+= . (update)

For j=1,…,l-1

jjuuu ˆˆˆ 00 γ−= ,

179

jj rxx ˆˆˆ 00 γ ′′+= ,

jj rrr ˆˆˆ 00 γ ′−= .

end

put 0101 ˆ,ˆ ru klk == +−+ ru and 01 xk =+x

end

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Chapter 6: A Smooth Approximation Technique for Three-Dimensional EM modeling in Dipping and Anisotropic Media

Macroscopic electrical anisotropy of rock formations can substantially impact

estimates of fluid saturation performed with borehole electromagnetic (EM)

measurements. Accurate and expedient numerical simulation of the EM response of

electrically anisotropic and dipping rock formations remains an open challenge,

especially in the presence of borehole and invasion effects.

In the past, several scattering approximations have been developed to efficiently

simulate complex EM problems arising in the probing of subsurface rock formations.

These approximations include Born, Rytov, Extended Born (ExBorn), and Quasi-Linear

(QL), among others. However, so far none of these approximations has been adapted to

simulate scattering in the presence of anisotropic conductive media.

This chapter introduces a novel efficient 3D EM approximation based on a new

integral equation formulation (Gao, Torres-Verdín and Fang, 2004). The approximation

is developed with the main objective to simulate the multi-component borehole EM

response of electrically anisotropic rock formations. Firstly, the internal electrical field is

expressed as the product of spatially smooth and rough components. The rough

component is a scalar function of location, and is governed by the background electric

field. A vectorial function of location is used to describe the smooth component of the

internal electric field, here referred to as the polarization vector. Secondly, an integral

equation is constructed to describe the polarization vector. Because of the smooth nature

of the polarization vector, relatively few unknowns are needed to describe it, thereby

making its solution extremely efficient. One of the main features of the new

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approximation is that it properly accounts for the coupling of EM fields necessary to

simulate the response of electrically anisotropic rock formations.

Tests of accuracy and computer efficiency against 1D and 3D finite-difference

simulations of the EM response of tri-axial induction tools show that the new

approximation successfully competes with accurate finite-difference formulations, and

provides superior accuracy to that of standard approximations. Numerical simulations

involving more than 106 discretization cells require only several minutes per frequency

and instrument location for their simulation on a Silicon Graphics workstation furbished

with a 300 MHz, IP30 processor.

6.1 INTRODUCTION

Integral equations have been widely used to simulate EM scattering, including

applications in geophysical prospecting and antenna design. Hohmann (1971) first

discussed the application of integral equations for the simulation of 2D subsurface

geophysical problems. Since then, a number of applications and developments have been

reported that include 3D EM scattering in the presence of complex geometrical structures

(e.g. Hohmann, 1975 and 1983; Wannamaker, 1983; Xiong, 1992; Gao et al., 2002;

Hursan et al., 2002; and Fang et al., 2003, among others).

Simulation of EM scattering via integral equations includes two sequential steps.

First, the spatial distribution of electric fields within scatterers is computed with a

discretization scheme. Second, the internal scattering currents are “propagated” to

receiver locations. It is often necessary to discretize the scatterers into a large number of

cells depending on (a) frequency, (b) conductivity contrast, (c) size of the scatterers, and

(d) proximity of the source and/or the receiver to the scatterers. This discretization gives

182

rise to a full complex linear system of equations whose solution yields the spatial

distribution of internal electric fields. Requirements of computer memory storage

increase quadratically with an increase in the number of discretization cells. Moreover,

the need to solve a large, full, and complex linear system of equations places significant

constraints on the applicability of 3D integral equation methods.

There are several numerical strategies used to overcome the difficulties associated

with integral equation formulations of EM scattering. Fang et al. (2003) recently reported

one such strategy. Their simulation approach makes explicit use of the symmetry

properties of Toeplitz matrices. Fang et al.’s (2003) algorithm also applies a suitable

combination of BiCGSTAB(l) (Bi-Conjugate Gradient STABilized (l)) (Gerard and

Diederik, 1993) and the FFT to iteratively solve the linear system of equations. This

technique has been detailed in Chpater 5. The latter method is a natural extension of the

widely used CG-FFT (Conjugate Gradient-Fast Fourier Transform) strategy (Catedra et

al., 1995) to compute EM fields. Despite these significant improvements, integral

equation methods are still impractical for routine use in the interpretation of borehole EM

data. An alternative approach is to develop an approximate solution. Several

approximations to the integral equation formulation have been proposed in the past.

These include Born (1933), Extended Born (Habashy et al, 1993; and Torres-Verdín and

Habashy, 1994), and Quasi-Linear (Zhdanov and Fang, 1996; and Zhdanov, 2002).

However, none of the integral equation approximations published to date has been

formulated to approach the simulation of 3D EM scattering in the presence of electrically

anisotropic media. Developing such an approximation is the main thrust of this chapter.

183

This chapter describes a novel approximation technique, termed “Smooth

Approximation”. The approximation attempts to synthesize the spatial variability of the

secondary electric currents within a scatterer in two manners. First, a multiplicative term

is introduced to “capture” the spatial variability of the secondary electric currents due to

the close proximity of the EM source to the scatterer. A second multiplicative term is

used to synthesize spatial variations in the phase and polarization of the secondary

electric currents due to spatial variations in electrical conductivity, including those due to

electrical anisotropy. It is shown that for borehole logging applications the latter

multiplicative term is spatially smoother than the first term and hence can be described

with fewer discretization blocks. Moreover, the accuracy of the proposed approximation

depends on both the choice of the background model and the spatial distribution and

number of discretization blocks.

This chapter is organized as follows: First existing approximations are introduced

and analyzed. Then, the smooth approximation is introduced. Subsequently, technical

details are provided concerning the choice of the background conductivity value. A

section is also included to assess the influence of the spatial block discretization

constructed within EM scatterers. Simulation examples are used to compare the accuracy

of the new approximation against alternative integral equation approximations, i.e. Born,

and Extended Born. Finally, several examples are provided to illustrate the performance

of the new approximation in the presence of finite-size boreholes, mud-filtrate invasion,

and electrical anisotropy of rock formations. These examples assume EM sources and

receivers in the form of tri-axial multi-component borehole logging instruments.

184

6.2 APPROXIMATIONS TO EM SCATTERING

6.2.1 Born Approximation

The Born approximation was introduced by Born (1933) for solving optical

scattering problems. The physical justification of the Born approximation is that for

small and weak scatterers, the scattered electric field can be neglected such that the total

electric field can be approximated by the background electric field, namely,

b≈E E . (6.1)

As a result, a linear expression is obtained to describe the relationship between the

anomalous conductivity and the external EM scattered fields, which is particularly useful

for solving inverse problems. The Born approximation remains accurate only for small

conductivity contrasts, relatively small-size inhomogeneities, and low probing

frequencies (Habashy et al., 1993; Zhdanov and Fang, 1996; Fang and Wang, 2000; and

Gao et al., 2002).

6.2.2 Extended Born Approximation

An extended 3D EM Born approximation (EBA) was introduced by Habashy et

al. (1993), and Torres-Verdín and Habashy (1994). This approximation has been widely

used in the field of geophysical prospecting as it considerably extends the range of

accuracy of a standard Born approximation. The EBA can be viewed as the first-term

approximation of the Taylor series expansion of the electric field distribution within

scatterers. This is equivalent to assuming that the electric fields within scatterers are

locally smooth. In mathematical form,

( ) ( ) ( )b≈ Λ ⋅E r r E r , (6.2)

185

where ( )Λ r is a scattering tensor, given by

( ) ( ) ( )1

0 0 0,e

G dτ

σ−

⎛ ⎞Λ = Ι − ⋅Δ⎜ ⎟⎝ ⎠∫r r r r r . (6.3)

The physical significance of the scattering tensor ( )Λ r has been detailed by Torres-

Verdín and Habashy (1994). Clearly, the relation between E and σΔ given by equation

(6.2) is nonlinear.

Numerical examples, however, have shown that when the EM source is close to

the scatterers internal electric fields can vary in an abrupt manner, thereby rendering the

EBA inaccurate (Torres-Verdín and Habashy, 2001; and Gao et al. 2002). Efforts have

been made to improve the accuracy of the EBA. Torres-Verdín and Habashy (2001)

proposed a modified Extended Born approximation; Gao et al. (2002) constructed a set of

natural preconditioners of the MoM’s stiffness matrix and showed how different versions

of such preconditioners may yield, as special cases, solutions equivalent to Born and

Extended Born approximations (see Chapter 4). Recently, Liu and Zhang (2001)

successfully used the EBA as a preconditioner of the CG-FFT technique in an effort to

improve the efficiency of their solvers.

6.2.3 Quasi-Linear (QL) Approximation

The quasi-linear (QL) approximation was developed by Zhdanov and Fang

(1996). It relates the background and scattering fields within a given discretization cell by

an electrical reflectivity tensor, which is assumed a smooth function of the position. In

mathematical form,

( ) ( ) ( )bλ= Ι + ⋅E r E r , (6.4)

186

where the tensor λ is referred to as the electrical reflectivity tensor. A least-squares

minimization technique is used to solve for the electrical reflectivity tensor. The internal

scattered fields are computed using the reflectivity tensor and the background fields

(Zhdanov and Fang, 1996).

It has been shown that the QL approximation remains accurate and efficient for

3D modeling when the material property is isotropic (Zhdanov and Fang, 1996).

However, the scalar and diagonal formulation of the QL approximation cannot provide

accurate results for anisotropy modeling for cases in which the background/incident

electric fields exhibit null components. Specifically, suppose that the background

medium is homogeneous and unbounded, and that EM scattering is imposed with a z-

directed magnetic dipole. Then for each cell, the z-component of the background electric

field (Ebz) will remain null regardless of the specific properties of the spatial distribution

of electrical conductivity. As shown in equations (6.5) and (6.6) below, regardless of the

nature of the reflectivity tensor (scalar or diagonal), the z-component of the total electric

field (Ez) within each cell will remain null, i.e.,

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

bz

by

bx

z

y

x

EEE

EEE

λ , (6.5)

and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

bz

by

bx

z

y

x

EEE

EEE

3

2

1

000000

λλ

λ. (6.6)

Clearly, the above formulation is not capable of providing the proper EM field coupling

behavior expected in generally anisotropic media. Now let us further explore how this

187

zero-component affects the accuracy of the results. After the internal electric fields are

computed, the scattering currents can be computed from the electric fields and the

conductivity tensor via the relationship

EJ ⋅Δ= σs . (6.7)

The scattered magnetic field can then be obtained at each receiver location by applying

the magnetic Green’s tensor on the scattering currents, i.e.,

sJH ⋅=h

G . (6.8)

By expanding equation (6.8), one obtains

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

y

x

zzzyzx

yzyyyx

xzxyxx

hzz

hzy

hzx

hyz

hyy

hyx

hxz

hxy

hxx

z

y

x

EEE

GGGGGGGGG

HHH

σσσσσσσσσ

, (6.9)

or, in explicit form,

( )

( )

( )

hx xx xx x xy y xz z

hxy yx x yy y yz z

hxz zx x zy y zz z

H G E E E

G E E E

G E E E

σ σ σ

σ σ σ

σ σ σ

= + +

+ + +

+ + +

. (6.10)

( )

( )

( )

hy yx xx x xy y xz z

hyy yx x yy y yz z

hyz zx x zy y zz z

H G E E E

G E E E

G E E E

σ σ σ

σ σ σ

σ σ σ

= + +

+ + +

+ + +

. (6.11)

and

( )

( )

( )

hz zx xx x xy y xz z

hzy yx x yy y yz z

hzz zx x zy y zz z

H G E E E

G E E E

G E E E

σ σ σ

σ σ σ

σ σ σ

= + +

+ + +

+ + +

. (6.12)

Therefore, if Ez=0, then the contribution to the external magnetic field due to the

presence of xzσ , yzσ , and zzσ will not be accounted for by the above expressions. In a

188

similar fashion, if the scatterer is isotropic, then the contributions to the external magnetic

field due to the presence of hxzG , ,h

yzG and hzzG will remain unaccounted for by equations

(6.10) through (6.12).

This chapter unveils a new approximation that circumvents the construction

problems associated with the QL approximation (scalar or diagonal) in the presence of

electric anisotropy. Numerical examples drawn from borehole multi-component

induction logging are used to evaluate the efficiency and accuracy of the new

approximation.

6.3 A SMOOTH EM APPROXIMATION (SA)

The EM scattering approximation reported in this chapter is based on a new

formulation of the integral equation (3.1). In so doing, the total electric field vector

within each discretization cell is expressed as the product of a scalar function of the

background field (spatially rough component) and a polarization vector (spatially smooth

component), namely,

( ) ( ) ( )rrdrE be= , (6.13)

where

( ) ( ) ( ) α)( * rErEr bbbe ⋅= , (6.14)

and

( ) ( )rrd⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

z

y

x

ddd

. (6.15)

189

The scalar component of the product in equation (6.13) is used to synthesize the

relative spatial changes in magnitude of the electric field, whereas the vector component

in the same equation is used to synthesize relative spatial changes in the polarization,

phase, and, to a less extent, of the amplitude of the electric field. In equation (6.14),

[ ]0,1α ∈ is a parameter that controls the spatial fluctuations of be and the spatial

smoothness of d. When ,0=α the scalar function be becomes spatially constant and

equal to one. In turn, this choice causes the vector function d to be identical to E, thereby

obtaining the original integral equation. For the numerical examples described in this

chapter we adopt the choice 2/1=α .


( ) 0 0 0( ) ( ) ( ) ( , ) ( ) ( )e

b b be G e dτ

σ= + ⋅Δ ⋅∫ 0 0r d r E r r r r r d r r , (6.16)

or, alternatively,

( ) 00 0

( )( ) ( ) ( , ) ( )( )

eb

bb

eG deτ

σ⎛ ⎞

= + ⋅Δ ⋅⎜ ⎟⎝ ⎠

∫ 0 0rd r d r r r r d r rr

, (6.17)

where

( ) ( ) ( )rerErd bbb /= . (6.18)

The new approximation stems directly from this last integral equation. In the

above expression, be embodies relative changes in the magnitude of the internal electric

field due to the proximity of the EM source. The larger the distance from the EM source

to the scatterer, the less significant the spatial changes of be within the scatterer. In the far

field, one would expect be to be spatially constant within the scatterer.

190

The spatial smoothness criterion necessary to accurately describe vector d

depends, to some extent, on the proximity of the EM receiver to the scatterer. Equation

(3.1) shows that the simulation of EM scattering at the receiver location is performed by

“propagating” the internal scattering electrical currents to the EM receiver location. In

this case, the “propagator” is given by the electric Green’s tensor,

),( 0rrR

eG ,

where Rr is the EM receiver’s location and 0r is a point within the scatterer. The effect

of the “propagator” can also be thought of as an operation wherein the scattering current,

)()( 00 rEr ⋅Δσ ,

is spatially low-pass filtered (i.e. it is smoothed in space) to provide the value of the

electric field at the EM receiver’s location. For a constant frequency, such a smoothing

operation becomes more pronounced as the receiver recedes away from the scatterer. In

the case of a fixed transmitter-receiver configuration, such as in borehole induction

logging, the “propagator” itself provides a precise measure of the degree of smoothness

necessary to compute accurate solutions of the scattered EM field at the receiver location.

In other words, even though scattering currents may exhibit large spatial variations within

the scatterer, these spatial variations are effectively smoothed when “propagated” to the

receiver location. Because of this important remark, it is only necessary to calculate

scattering currents with accuracies consistent with those of the spatial smoothing

properties of the “propagator.”

The criterion adopted in this chapter to control the degree of spatial smoothness of

the internal electric field consists of discretizing the scatterer into a collection of blocks,

each block consisting of several cells. This procedure assumes that within each block the

191

d vector is constant, whereas the scalar function be is assumed variable within a block but

constant within a cell. Because of the choice of a uniform spatial discretization grid, all

cells exhibit the same shape and size. The spatial distribution and size of blocks,

however, can be chosen in a more flexible manner. It is only required that blocks be built

to conform to cell boundaries. Finally, the d vector associated with a given block is

solved via equation (6.17). Such a procedure gives rise to an over-determined

(rectangular) complex linear system of equations for the unknown vector d within all of

the discretization blocks. The rectangular, over-determined nature of the linear system of

equations is due to the fact that the number of blocks is, by construction, smaller than the

number of cells. Following a procedure described in the Supplement 6A, the rectangular

linear system is reduced to a 3Nx3N linear system of equations where N is the number of

blocks. This reduction of the size of the linear system substantially decreases memory

storage and CPU time requirements.

Additional savings in computer storage and CPU execution time are achieved

with the use of uniform spatial discretization scheme and a Toeplitz matrix formulation.

When using uniform discretization grids, a Toeplitz matrix is constructed for each

discretization block. Matrix vector multiplications are further accelerated using the FFT

(see Chapter 5).

We remark that the spatial discretization of blocks and cells adopted in this

chapter is of a Cartesian type. Moreover, in an effort to properly model the borehole, the

Cartesian block discretization is chosen with orthogonal axes conformal to the axis of the

borehole (and hence conformal to the axis of the logging instrument). Cell locations and

distances between a given cell and the axis of the borehole are measured perpendicular to

192

the borehole axis. For the case of a non-conformal distribution of conductivity such as,

for instance, dipping anisotropic beds, a conductivity averaging technique is used to

assign a tensorial electrical conductivity to a specific cell. The material averaging

technique given in Supplement 5A is used to assign an electrical conductivity tensor to a

particular cell.

6.4 ON THE CHOICE OF THE BACKGROUND CONDUCTIVITY

As emphasized above, the integral equation approximation introduced in this

chapter makes use of a Green’s tensor defined over a homogeneous and isotropic

unbounded medium. The choice of the simplest possible Green’s tensor is made to limit

the complexity of the numerical computations associated with the integral equation

solution. Moreover, the new approximation involves two conformal spatial discretization

volumes. The first one is constructed using fine cells to describe the spatial variability of

the scalar term be . In turn, the specific value of be assigned to a given cell depends on the

assumed background model. This suggests the possibility of selecting the value of

background conductivity to provide the largest possible accuracy within the practical

limits of the approximation.

A criterion to choose the background conductivity is to make a compromise

between the contribution of small and large conductivity values in the rock formation

model, or else to minimize the difference between the minimum and maximum formation

conductivity values using some weighted metric. Extensive numerical experiments

suggest that the geometrical average of the minimum and maximum formation

conductivity values provides adequate results for the examples considered in this chapter.

This geometrical average is given by

193

maxmin σσσ ⋅=b . (6.19)

The variables minσ and maxσ in equation (6.19) are the minimum and maximum,

respectively, of all the conductivity values considered in the numerical simulation.

Equation (6.19) is also suggested by studies in the theory of effective media

involving the electrical conductivity of two-dimensional composites, a subject originally

considered by A.M. Dykhne (1970). Using Dykhne’s theory, it can be shown that a

symmetric mixture of two components exhibits an effective conductivity given by the

geometrical average of the conductivities of the constituent materials. Alternative

procedures could exist to choose an optimal background conductivity. These could

include weighted averages of the conductivity distribution, where the weights would be

determined by (a) proximity to the source(s), (b) proximity to the receiver(s), and (c)

block volume. Yet another variation of equation (6.19) could be constructed with

averages of electrical conductivity or resistivity taken along orthogonal or else arbitrary

directions. The latter possibility is enticing but we choose not explore it in the present

dissertation.

Quite obviously, the choice of background conductivity other than that of the

borehole conductivity causes the borehole itself to become part of the anomalous

conductivity region. Because of this, memory and CPU requirements increase when

computing the internal electric field. Despite such difficulties, numerical experiments

show that the choice of background conductivity different from that of the borehole does

not substantially compromise the efficiency of the simulation algorithm. The small

sacrifice in computer efficiency is drastically outweighed by the gain in numerical

accuracy. Moreover, the implementation of the integral equation algorithm described in

194

this chapter makes use of 3D FFTs that require of a uniform and spatially continuous

discretization grid for their implementation. The discretization does include the borehole

region, and therefore the algorithm does not explicitly enforce a choice of background

conductivity equal to the borehole conductivity.

6.5 SENSITIVITY TO THE CHOICE OF SPATIAL DISCRETIZATION

As emphasized above, there are two levels of spatial discretization involved in the

computation of the integral equation approximation described in this chapter. A fine cell

structure is first constructed to describe the spatial variations within the scatterer of the

scalar factor be contained in equations (6.16) or (6.17). The relative spatial variations of

this factor are primarily controlled by the proximity of the EM source to the scatterers.

On the other hand, a relatively larger conformal block structure is constructed to describe

the spatial variations of vector d. A given block in the discretization scheme of vector d

is composed of several cells used for the spatial discretization of the scalar factor be . The

specific choice of block and cell structure may have a significant influence on the

performance of the approximation.

The strategy chosen in this chapter to construct block structures is one in which

small blocks are placed in close proximity to the borehole, the transmitter(s), and/or the

receiver(s). Small discretization blocks are required near receivers because an accurate

representation of the polarization vector is needed in those blocks to properly account for

the relative large influence of the dyadic Green’s tensor when propagating the internal

electric field to receiver locations. Larger blocks are used to discretize the remaining

spatial regions in the scattering rock formations.

195

The formation model used in this chapter is described in Figure 5.5, and the tool

configuration is illustrated in Figure 5.6.

Notice that in the descriptions and figures included in this chapter, the identifier

“3DIE Appr.” is used to designate simulation results obtained with the smooth

approximation. It is also assumed that the instrument measurement is a combination of

the response measured by the first magnetic receiver (H1) and the second magnetic

receiver (H2), given by equation (5.27).

In this section, attention is focused to a model with borehole and invasion, and the

assumption is made of a borehole dip angle of 60o; the operating frequency is 220KHz.

Figure 6.1 describes the simulated Hzz field component, i.e. the vertical magnetic field

component due to a vertical magnetic source. This figure describes simulation results

obtained using 8, 216, 1000, and 2400 discretization blocks, together with the

corresponding results obtained with a 3D finite-difference code. In all of the above cases

the number of discretization cells is 640,000. Figures 6.2 and 6.3 show the Hxx and Hyy

components simulated for the same formation model, respectively. These figures suggest

that 1000 discretization blocks already provide an accuracy similar to that of the 3D

finite-difference code. Usage of 2400 blocks only provides a minor improvement over

that of 1000 blocks. The same figures indicate that usage of only 8 blocks already

produces the basic behavioral features of the magnetic field components Hxx, Hyy, and

Hzz, thereby lending credence to the validity of the new approximation.

The above exercises are not intended to serve as guide for choosing the number of

discretization blocks necessary to accurately simulate the EM response of a specific rock

formation model. However, they do confirm that only a few discretization blocks are

196

needed to reach an acceptable degree of accuracy. We remark that the minimum number

of discretization blocks that can be used with the algorithm described in this chapter is

eight. This restriction comes from the fact that by construction the discretization blocks

are laid out symmetrically in all three directions with respect to the borehole axis.

Therefore, the minimal structure that can be used for block discretization is the one with

one block per octant.

Figure 6.1: Assessment of the accuracy of the integral equation approximation of Hzz (imaginary part) for a given number of spatial discretization blocks. The formation dips at an angle of 60o and is modeled in the presence of both a borehole and mud-filtrate invasion. Simulation results are shown for a probing frequency of 220 KHz.

197

Figure 6.2: Assessment of the accuracy of the integral equation approximation of Hxx (imaginary part) for a given number of spatial discretization blocks. The formation dips at an angle of 60o and is modeled in the presence of both a borehole and mud-filtrate invasion. Simulation results are shown for a probing frequency of 220 KHz.

198

Figure 6.3: Assessment of the accuracy of the integral equation approximation of Hyy (imaginary part) for a given number of spatial discretization blocks. The formation dips at an angle of 60o and is modeled in the presence of both a borehole and mud-filtrate invasion. Simulation results are shown for a probing frequency of 220 KHz.

199

6.6 ASSESSMENT OF ACCURACY WITH RESPECT TO ALTERNATIVE APPROXIMATIONS

The objective of this section is to assess the accuracy and efficiency of the new

integral equation approximation in comparison with Born and extended Born

approximations.

Although similar comparisons of the above approximations have been reported by

a number of authors, including Habashy et al. (1993), and Zhdanov and Fang (1996),

none of the previous comparisons were performed in the context of electrically

anisotropic media. It can be readily shown that the Born approximation cannot reproduce

the coupling of EM fields in the presence of electrically anisotropic media. On the other

hand, the Extended Born approximation does account for some of the coupling of EM

fields but its accuracy is compromised when the source is close to the scatterer (Torres-

Verdín and Habashy, 2001; and Gao et al., 2003). Finally, it has been shown that the

scalar and diagonal quasi-linear approximations of Zhdanov and Fang (1996) cannot

account for the coupling of EM fields in the presence of electrically anisotropic media

because of the existence of null components in the background electric field (Gao et al.,

2003).

The rock formation model considered for this study is the same one used in the

previous section. Simulation results are identified as follows: the label “Born” is used to

designate simulations obtained with the first-order Born approximation, and the label

“ExBorn” is used to designate results obtained with the Extended Born approximation.

Again, 2400 blocks are used for the computation of the new approximation. Figures 6.4

through 6.6 show the comparisons between simulation results for the three

approximations for the Hzz, Hxx, and Hyy field components, respectively. Simulation

200

results summarized in these figures indicate a superior performance of the new

approximation with respect to the Born and extended Born approximations.

Figure 6.4: Assessment of the accuracy of the integral equation approximation of Hzz (imaginary part) with respect to alternative approximation strategies (Born and Extended Born). The formation dips at an angle of 60o and is modeled in the presence of both a borehole and mud-filtrate invasion. Simulation results are shown for a probing frequency of 220 KHz.

201

Figure 6.5: Assessment of the accuracy of the integral equation approximation of Hxx (imaginary part) with respect to alternative approximation strategies (Born and Extended Born). The formation dips at an angle of 60o and is modeled in the presence of both a borehole and mud-filtrate invasion. Simulation results are shown for a probing frequency of 220 KHz.

202

Figure 6.6: Assessment of the accuracy of the integral equation approximation of Hyy (imaginary part) with respect to alternative approximations (Born and Extended Born). The formation dips at an angle of 60o and is modeled in the presence of both a borehole and mud-filtrate invasion. Simulation results are shown for a probing frequency of 220 KHz.

203

6.7 NUMERICAL EXAMPLES

Additional rock formation models and probing frequencies have been considered

to further assess the accuracy and efficiency of the new approximation. These include: (a)

a 1D formation that exhibits no borehole and no invasion, with a well dipping at an angle

of 0o and 60o (the source direction dips at an angle of 0o and 60o with respect to the

formation’s horizontal layering plane), (b) a 3D formation with borehole and invasion,

with a well dipping at an angle of 0o and 60o (the borehole axis dips at angles of 0o and

60o with respect to the formation’s horizontal layering plane). The probing frequencies

considered in the simulations are 20 KHz and 220 KHz. All of the examples considered

in this section assume a multi-component induction instrument for borehole logging.

Although the main purpose of this chapter is to assess the accuracy and efficiency of the

new approximation in the simulation of borehole EM logging measurements, some

petrophysical comments are provided when interpreting the simulation examples. The

intent is to also assess the physical validity of the approximation.

The spatial discretization grid constructed for the simulations reported in this

chapter consists of 80 cells in the x direction, 80 cells in the y direction, and 100 cells in

the z-direction. Cell sizes are kept uniform and equal to 0.1 m. In total, 2400 blocks are

used for the discretization of the models considered in this chapter.

6.7.1 1D Anisotropic Rock Formation with Dip=0o

Figures 6.7 through 6.8 show simulation results (Hzz, Hxx, and Hyy) obtained with

the SA assuming a 1D rock formation and a borehole dip angle of 0o. Simulation results

for two frequencies (20 KHz and 220 KHz) are compared to those obtained with the 1D

code. The comparisons confirm the improved accuracy of the SA. It is remarked that Hxx

204

and Hyy remain identical when simulated along vertical wells and hence only one figure is

shown here. Likewise, it is found that Hzz remains identical to the Hzz field component

simulated for the case of an isotropic formation of conductivity equal to that of the

horizontal conductivity.

Figure 6.7: Comparison of the Hzz field component (imaginary part) simulated with the SA and a 1D code. In both cases, the simulations were performed assuming a 1D formation that exhibits electrical anisotropy. The induction logging tool is assumed to be oriented perpendicular to the formation. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

205

Figure 6.8: Comparison of the Hxx field component (imaginary part) simulated with the SA and a 1D code. In both cases, the simulations were performed assuming a 1D formation that exhibits electrical anisotropy. The induction logging tool is assumed to be oriented perpendicular to the formation. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

206


Figures 6.9 through 6.11 show simulation results (Hzz, Hxx, and Hyy) obtained with

the SA assuming a 1D rock formation and a borehole dipping at an angle of 60o.

Simulation results for two frequencies (20 KHz and 220 KHz) are compared to those

obtained with the 1D code. Again, the comparisons confirm the improved accuracy of the

SA. Figures 6.7 and 6.9 indicate a substantial sensitivity of Hzz to the presence of a dip

angle. Similarly, a comparison of Figures 6.8 and 6.10 indicates a substantial sensitivity

of Hxx to the presence of a dip angle. Yet greater effects due to the presence of a dip angle

can be observed by comparing the simulated Hyy components.

Figure 6.9: Comparison of the Hzz field component (imaginary part) simulated with the SA and a 1D code. In both cases, the simulations were performed assuming a 1D formation that exhibits electrical anisotropy and a borehole dipping at an angle of 60o. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

207

Figure 6.10: Comparison of the Hxx field component (imaginary part) simulated with the SA and a 1D code. In both cases, the simulations were performed assuming a 1D formation that exhibits electrical anisotropy and a borehole dipping at an angle of 60o. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

208

Figure 6.11: Comparison of the Hyy field component (imaginary part) simulated with the SA and a 1D code. In both cases, the simulations were performed assuming a 1D formation that exhibits electrical anisotropy and a borehole dipping at an angle of 60o. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

209


Figures 6.12 and 6.13 show simulation results (Hzz, Hxx, and Hyy) obtained with

the SA assuming a 3D rock formation. Simulation results for two frequencies (20 KHz

and 220 KHz) are compared against those obtained with the 3D FDM code. The

comparisons confirm the improved accuracy of the SA for a 3D rock formation that

includes borehole, invasion, and dip angle. The influence of a borehole and invasion can

be clearly observed on the behavior of the simulated magnetic field components Hzz and

Hxx.

Figure 6.12: Comparison of the Hzz field component (imaginary part) simulated with the SA and a 3D-FDM code assuming a 3D formation that includes both a borehole and invasion. The borehole dips at an angle of 0o. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

210

Figure 6.13: Comparison of the Hxx field component (imaginary part) simulated with the SA and a 3D-FDM code assuming a 3D formation that includes both a borehole and invasion. The borehole dips at an angle of 0o. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

211


Figures 6.14 through 6.16 show simulation results (Hzz, Hxx, and Hyy) obtained

with the SA assuming a 3D rock formation (including both a borehole and invasion) and

a dip angle of 60o. Simulation results for two frequencies (20 KHz and 220 KHz) are

compared against those obtained with the 3D FDM code. The comparisons confirm the

improved accuracy of the SA for a 3D rock formation that includes borehole, invasion,

and a 60o dip angle. A comparison of Figures 6.9 and 6.14 provides evidence of the

sensitivity of the simulated magnetic field components Hzz and Hxx to the presence of

both a borehole and invasion. By contrast, Hxx shows almost no sensitivity to the

presence of a borehole and/or invasion.

Figure 6.14: Comparison of the Hzz field component (imaginary part) simulated with the SA and a 3D-FDM code assuming a 3D formation that includes both a borehole and invasion. The borehole dips at an angle of 60o. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

212

Figure 6.15: Comparison of the Hxx field component (imaginary part) simulated with the SA and a 3D-FDM code assuming a 3D formation that includes both a borehole and invasion. The borehole dips at an angle of 60o. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

213

Figure 6.16: Comparison of the Hyy field component (imaginary part) simulated with the SA and a 3D-FDM code assuming a 3D formation that includes both a borehole and invasion. The borehole dips at an angle of 60o. Simulation results are shown for probing frequencies of 20 KHz and 220 KHz.

214

The above simulation exercises consistently show that the newly developed

approximation yields accurate results in the presence of complex 3D anisotropy models

for the two probing frequencies considered in this chapter (20 KHz and 220 KHz).

Simulation of EM fields for one single borehole profile location required approximately

3-4 minutes on a SGI OCTANE workstation (furbished with a 300 MHz IP30 processor).

By contrast, depending on the size of the spatial discretization grid, it takes anywhere

from 20 minutes to 1 hour of CPU time to simulate one EM borehole profiling location

with a full-wave integral equation code using BiCGSTAB(L)-FFT technique (see Chapter

5).

6.8 CONCLUSIONS

This chapter described a smooth EM scattering approximation introduced to

substantially reduce computation times in the simulation of borehole induction responses

of 3D anisotropic rock formations. The approximation makes use of a simple scalar-

vectorial product to synthesize the spatial smoothness properties of EM scattering

currents. Additional computer efficiency for the approximation is achieved with the use

of uniform discretization grids. Numerical simulations and comparisons against 1D and

3D finite-difference codes indicate that the new approximation remains accurate within

the frequency range of borehole induction instruments. Numerical experiments and

benchmark comparisons also indicate that the new approximate remains accurate in the

presence of a borehole, mud-filtrate invasion, dipping, and electrically anisotropic rock

formations.

It was shown that the accuracy of the SA depends on the choice of both the

background conductivity and the spatial block structure used for discretization. A

215

criterion was described in this chapter to select a background conductivity. Likewise, it

was shown that only a relatively small number of spatial discretization blocks are needed

to obtain accurate simulations of borehole EM measurements.

216

Supplement 6A: Algorithmic Implementation of the Smooth EM Approximation

We first divide the scattering domain into N blocks, with nV being the spatial

region occupied by the n-th block, and

( ) ( )1

N

n nn

P=

=∑d r d r , (6A-1)

where

( )10

nn

VP

elsewhere∈⎧

= ⎨⎩

rr . (6A-2)

Substitution of equation (6A-1) into equation (6.16) yields

( ) ( ) ( ) ( ) ( ) ( )0 0 0 01

,n

N e

b b n bVn

e G e dσ=

− ⋅ Δ =∑∫r d r r r r r r d E r . (6A-3)

Because the conductivity tensor is constant within a given block one can rewrite

equation (6A-3) as

( ) ( ) ( ) ( ) ( )0 0 01

,n

N e

nb b n bVn

e G e d σ=

− ⋅ Δ =∑∫r d r r r r r d E r . (6A-4)

We now divide block nV into nP cells and proceed to match the incident fields at

each cell location, mr . Equation (6A-4) becomes

( ) ( ) ( ) ( )0 01 1

,n

pn

PN ep

nb m m m b n b mVn p

e G d e σ= =

⎡ ⎤− Δ =⎢ ⎥

⎣ ⎦∑ ∑∫r d r r r r d E r . (6A-5)

For each cell, we define

( ), 0 0pn

xx xy xze

m yx yy yzV

zx zy zz

G G GG G d G G G

G G G

⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥⎣ ⎦

∫ r r r , (6A-6)

217

and

pbB Ge= . (6A-7)

Equation (6A-4) then becomes

)()()(11

mbnn

P

p

pn

N

nmmb

n

Be rEdrdr =Δ⎥⎦

⎤⎢⎣

⎡− ∑∑

==

σ . (6A-8)

Using matrix notation, equation (6A-7) can be written as

1313333333 )( ××××× =− MNNNNMNM RdSCA , (6A-9)

where M is the number of cells and

11

22

1 1N N

NN

AA

AA

A− −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

. (6A-10)

In equation (6A-10), each submatrix Aii , i=1…N is associated with the

background field within a given block, and has dimensions 33 ×nP . For example,

11

11

11

11

11

11

11

0 0

0 0

0 0

0 0

0 0

0 0

p

p

p

pb

pb

pb

pb

pb

pb

e

e

eA

e

e

e

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

, (6A-11)

( )1 , 1 1, , , , ,T

x y z Nx Ny Nzd d d d d d d= , (6A-12)

and

218

11 12 1 1 1

21 22 2 1 2

11 12 1 1 1

1 2 1

N N

N N

N N N N N N

N N NN NN

C C C CC C C C

CC C C CC C C C

−

−

− − − − −

−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

. (6A-13)

Each submatrix [ ] ∑==×

jP

p

pijij BCC

33 represents the contribution from the j-th

block on the i-th cell.

Also,

11

22

1 1N N

NN

SS

SS

S− −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, (6A-14)

where each submatrix iiS (i=1 … N) contains the average conductivity tensor for each

block. The size of this submatrix is 3x3, namely, iiiS σΔ= . The conductivity averaging

technique used to assemble the entries iiS is the one described by Supplement 5A.

Finally,

( )1 1 1, , , , , ,T

b x b y b z bMx bMy bMzR E E E E E E= . (6A-15)

To solve the over-determined complex linear system of equations represented by

equation (6A-9), we pre-multiply both sides of equation (6A-9) by matrix A* to obtain

RAdCSAAA *)**( =− , (6A-16)

where matrix A* is the transpose conjugate of matrix A. Because the matrices AA* , CA* ,

and RA* are all independent of conductivity, they can be stored in hard-disk memory

prior to performing the computations. Specifically, when the conductivity distribution

219

changes with a change of location of the induction-logging instrument, it is only

necessary to construct a new conductivity matrix. The remaining matrices included in

equation (6A-16) will not change with a change in instrument location.

It is pointed out that equation (6A-16) is different from the least-squares solution

of the over-determined complex linear system of equations described by equation (6A-9).

The way to obtain a least-squares solution of the over-determined linear system (6A-9) is

to pre-multiply both sides of the linear system by the matrix (A-CS)*. However, we

remark that such an operation may involve substantial computer resources. The rationale

for using equation (6A-16) instead of the standard least-squares solution is as follows.

From inspection of equations (6A-6), (6A-7) and (6A-13) one can conclude that, in

general, the entries of matrix A are much larger than those of matrix CS. One can easily

show that the entries of matrix C involve the entries of matrix A times values derived

from the Green’s tensor that, in turn, are normally much smaller than 1. In view of the

above, equation (6A-16) remains an accurate and expedient alternative to the least-

squares solution of equation (6A-9). Extensive numerical experiments have confirmed

the practical validity of equation (6A-16).

220

Chapter 7: A High-Order Generalized Extended Born Approximation for Three-Dimensional EM Modeling in Dipping and Anisotropic Media

Large computer resources are often needed to solve large-scale EM problems in

inhomogeneous and anisotropic media. This chapter introduces a generalized extended

Born approximation (GEBA) and its high-order variants (Ho-GEBA) to efficiently and

accurately simulate EM scattering problems. We make use of a generalized series

expansion of the internal electric field to construct high-order terms of the generalized

extended Born approximation (Ho-GEBA). A salient feature of the Ho-GEBA is its

enhanced accuracy over the Born approximation and the EBA, even when only the first-

order term of the series expansion is considered in the approximation. This behavior is

not conditioned by either the source location or the spatial distribution of the internal

electric field. A unique feature of the Ho-GEBA is that it can be used to simulate the EM

response of electrically anisotropic media. Such a feature is not possible with

approximations of the internal electric field that are based on the behavior of the

background electric field. Three-dimensional numerical examples are used to benchmark

the efficiency and accuracy of the Ho-GEBA. We also provide comparisons with the

first-order Born approximation and the EBA. These simulation examples are performed

assuming Vertical Magnetic Dipole (VMD), and Transverse Magnetic Dipole (TMD)

sources operating in the induction frequency range.

221

7.1 INTRODUCTION

In Chapter 5, we developed a BiCGSTAB(L)-FFT algorithm to efficiently solve

large-scale EM simulation problems using the integral equation approach. An alternative

approach to expedite the solution of EM scattering problems is to develop approximate

solutions. Approximation strategies are frequently used to solve EM scattering problems.

They represent a good compromise between computer efficiency and accuracy when

solving large-scale inverse scattering problems. Several approximations of the integral

equation formulation have been proposed and used in the past. These include the Born

approximation (1933), the EBA (Habashy et al., 1993; and Torres-Verdín and Habashy,

1994), and the Quasi-Linear approximation (Zhdanov and Fang, 1996). In addition, the

SA (Gao et al., 2003 a; Gao et al., 2003b; Gao et al., 2004) was developed to efficiently

simulate the EM response of electrically anisotropic media based on the theory of field

decomposition (see Chapter 6). The Born approximation is restricted to low frequencies

and low-conductivity contrasts (Habashy et al., 1993). On the other hand, the EBA

significantly improves the accuracy of the Born approximation because of the inclusion

of multiple scattering effects (Habashy et al., 1993). It has been found, however, that the

accuracy of the EBA deteriorates when the scatterer is close to the source region, or else

when the electric field exhibits significant spatial variations within the scatterer (Torres-

Verdín and Habashy, 1994; Gao et al., 2003). These two situations frequently arise in

applications of geophysical borehole induction logging, wherein the accuracy of the EBA

is sometimes inferior to that of the first-order Born approximation. Gao and Torres-

Verdín (2003) have made considerable progress in making use of the background electric

fields and the spatial distribution of conductivity to construct a preconditioning matrix

222

that accounts for the proximity of the source to the scatterers. This method has been

successfully used to solve 2.5 dimensional problems in cylindrical coordinate systems.

However, the method does not perform well when solving 3D EM scattering problems

(Gao and Torres-Verdín, 2003). Moreover, both the Born approximation and the EBA do

not effectively account for EM coupling due to electrically anisotropic media. The latter

situation has been discussed in great detail in several of our publications (Gao et al.,

2003a; Gao et al., 2003b; and Gao et al., 2004).

To properly account for the effects of source proximity, multiple scattering, and

EM coupling in the presence of electrically anisotropic media, in this chapter we develop

a Generalized Extended Born Approximations (GEBA) and its high-orders variants. We

show that the EBA is a special case of GEBA. Subsequently, a High-Order Generalized

Extended Born Approximations (Ho-GEBA) is proposed to further improve the accuracy

of the GEBA without sacrifice of computer efficiency. This is achieved by making use of

a generalized series (GS) expansion of the electric field. In the formulation of the Ho-

GEBA, the GEBA acts as the residual term of the GS. Theoretical analysis and numerical

experiments consistently confirm the high accuracy of the Ho-GEBA irrespective of the

source position or the spatial distribution of the internal electric field. Numerical

examples in the induction frequency range are included to quantify the accuracy and

efficiency of the Ho-GEBA for the cases of Vertical Magnetic Dipole (VMD) and

Transverse Magnetic Dipole (TMD) excitation.

The chapter is organized as follows: We first introduce the theory of the GS, the

GEBA and the Ho-GEBA. Subsequently, numerical examples are included to validate the

theory. We focus our attention to the physical significance of the GEBA and the HO-

223

GEBA and on their numerical validation by comparing it to both the first-order Born

approximation and the EBA.

7.2 A GENERALIZED SERIES (GS) EXPANSION OF THE ELECTRIC FIELD

The theory of the integral equation has been described in Chapter 3. In this

section, we develop a Generalized Series (GS) expansion for the internal electric field.

For convenience, we rewrite equation (3.1) using operator notation as

( ) ( )τ τσ= + = + Δ ⋅ = +E E E E E E Jb s b bG G , (7.1)

where

( ) ( ) ( )σ= Δ ⋅J r r E r , (7.2)

Gτ is a linear integral operator defined by

( ) 0( , )( )e

G G dτ τ⋅ = ⋅∫ 0r r r , (7.3)

and the subscript τ designates the spatial support of the operator.

In theory, equation (7.1) can be solved via the method of successive iterations

(Von Neumann series), namely

( ) ( )( )1N Nb Gτ σ −= + Δ ⋅E E E , N=1, 2, 3, … (7.4)

From the Banach theorem (Aubin, 1979), it is well known that the Von Neumann series

converges if the operator Gτ is a contraction operator, that is if

( ) ( )( ) ( ) ( )( )1 2 1 2τ σ κ σ⎡ ⎤Δ ⋅ − ≤ Δ ⋅ −⎢ ⎥⎣ ⎦

E E E EG , (7.5)

224

where is the 2 norm, 1κ < , and ( )1E , and ( )2E are any two different solutions. In

other words, to guarantee the Von Neumann series to converge, the norm of the operator

Gτ must be less than one, namely

1Gτ < . (7.6)

If one takes the background electric fields as the initial solution of equation (7.4),

one can derive the classical Born series expansion (Born, 1933) for E as

( ) ( ) ( )0

nB

n

∞

=

=∑E r E r , (7.7)

where

( ) ( )( )1n nB BGτ σ −= Δ ⋅E E , n=1, 2, 3, … (7.8)

and

( )0B b=E E . (7.9)

Each iteration of the Born Series in equation (7.7) involves only one matrix-vector

multiplication. However, usually the norm of operator Gτ is greater than 1, whereupon

the Born series expansion of equation (7.7) does not always converge, e.g., in the case of

highly conductive media. This situation greatly limits the range of applicability of the

Born series expansion for EM modeling.

Using an energy inequality, Zhdanov and Fang (1997) constructed a globally

convergent modified Born series expansion. Accordingly, a linear transformation was

used to transform the operator Gτ into a new operator cGτ . The 2 norm of cGτ is always

less than or equal to one, namely

1cGτ ≤ . (7.10)

225

and cGτ can be applied to any vector-valued function [see equation (7A-9) in Supplement

7A].

Starting with the same energy inequality used by Zhdanov and Fang (1997), in

Supplement 7A we derive a new formulation of the integral equation as

( )2τα α σ β β′= ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅E E E Eb ba a G , (7.11)

where the tensors a , α and β are given by equations (7A-13), (7A-18) and (7A-19),

respectively. The electric field E is computed via equation (7A-16) after E is solved

from equation (7.11). A proof that equation (7.11) is a contractive integral equation is

given in Supplement 7A.

Based on the new integral equation (7.11), and following the same procedure as

the derivation of the classical Born series expansion, a new series approximation can be

derived for the electric field. We start by assuming that the initial guess of E in equation

(7.11) is ( )0CBE , namely,

( ) ( )0 0CB=E E . (7.12)

Notice that ( )0CBE is unknown and that the subscript “CB” here has no specific meaning. In

Supplement 7B, we derive a series expansion for the electric field as

( ) ( ) ( )0

nCB

n

∞

=

=∑E r E r , (7.13)

where

( ) ( )( ) ( )1 1n n nCB CB CBGτα σ β− −= ⋅ Δ ⋅ + ⋅E E E , n=2, 3, 4, … (7.14)

and

226

( ) ( )( ) ( )( )1 0 0CB CB b CBGτα σ α= ⋅ Δ ⋅ + ⋅ −E E E E . (7.15)

We refer to the series given by equation (7.13) as a Generalized Series (GS) for the

electric field, given that any alternative series expansion can be derived from it. For

example, the classical Born series expansion, the modified Born series expansion of

Zhdanov and Fang (1997), and the quasi-linear series expansion of Zhdanov and Fang

(1997) are all special variants of equation (7.13). Table 7.1 summarizes the relationship

between the GS and other existing series expansions of the electric field. A salient feature

of GS is that it converges for arbitrary lossy media. The latter property is addressed in

detail in Appendices 7A and 7B.

Existing or Possible Series Expansions Relation to the Generalized Series (GS) Classical Born (Born 1933) ( )0

CB b=E E , α = Ι , 0β =

Modified Born (Zhdanov and Fang, 1997) ( )0CB b=E E

Quasi-linear (Zhdanov and Fang, 1997) ( )0CB bλ= ⋅E E , where λ is the electrical

reflectivity tensor in the quasi-linear approximation.

Extended Born ( )0CB b= Λ ⋅E E , where Λ is the scattering

tensor in the Extended Born Approximation.

Table 7.1: Relationship between the GS and other series expansions of the internal electric field reported in the open technical literature.

Cui et al. (2004) advanced an approximation to EM scattering similar to the

extended Born series; however, their approximation does not guarantee the convergence

of the high-order terms of the series because the formulation does not enforce a

contractive operator. Figure 7B-2 (Supplement 7B) compares the convergence of the

EBA series for the rock formation model shown in Figure 7B-1 both with and without

contraction. The left-hand panel of Figure 7B-2 shows the convergence behavior of the

227

EBA series without contraction (N.C.), while the right-hand panel shows the convergence

behavior of the same series with contraction (W.C.). This graphical comparison clearly

shows that, without contraction, high-order terms of the EBA series tend to diverge. We

remark here that the low-order terms (i.e., the 2nd order) may accidentally produce better

results for some cases (see, for example, Cui et al., 2004). However, the overall behavior

of the series is divergent. Figure 7B-2 (right panel) also indicates that the use of a better

starting point does not guarantee a faster convergence of the series [see the curve denoted

by EBA series (W.C.)]. Cui et al. (2004) also introduced the use of a backconditioner to

improve the accuracy of the approximation. A similar backconditioner strategy was

advanced and tested by Gao and Torres-Verdín (2003) in the inversion of array induction

data.

7.3 THE EXTENDED BORN APPROXIMATION (EBA)

Based on equation (3.1), the EBA for EM scattering was developed that captures

some of the multiple scattering effects, and that is more accurate than the first-order Born

approximation for some practical EM scattering problems (Habashy et al., 1993; and

Torres-Verdín and Habashy, 1994). However, it has also been shown that if the source is

very close to the scatterer or if the electric field varies significantly within the scatterer,

such as commonly encountered in borehole induction logging, the accuracy of the EBA

seriously deteriorates (Gao et al., 2003a; Gao et al., 2003b).

To derive the EBA, one first rewrites equation (3.1) as

( )

( ) ( )

0 0

0 0 0

( ) ( ) ( , ) ( )

( , ) ( ) ( )

e

b

e

G d

G d

τ

τ

σ

σ

= + ⋅Δ ⋅

+ ⋅Δ ⋅ −

∫

∫

0

0

E r E r r r r E r r

r r r E r E r r. (7.16)

228

Habashy et al. (1993), and Torres-Verdín and Habashy (1994), omitted the third term on

the right-hand side of equation (7.16) by arguing that the contribution from this term is

marginal compared to the second term because of the singular behavior of the dyadic

Green’s function. Thus, by omitting the third term in equation (7.16) one obtains

( )0 0( ) ( ) ( , ) ( )e

b G dτ

σ≈ + ⋅Δ ⋅∫ 0E r E r r r r E r r . (7.17)

It immediately follows that

( ) ( ) ( )b≈ Λ ⋅E r r E r , (7.18)

where ( )Λ r is a scattering tensor, given by

( ) ( ) ( )1

0 0 0,e

G dτ

σ−

⎛ ⎞Λ = Ι − ⋅Δ⎜ ⎟⎝ ⎠∫r r r r r . (7.19)

The physical significance of the scattering tensor ( )Λ r has been detailed by Torres-

Verdín and Habashy (1994).

7.4 A GENERALIZED EXTENDED BORN APPROXIMATION (GEBA)

In the derivation of the EBA it is not clear whether the omission of the second

term on the right-hand side of equation (7.16) affects the final solution. Here, we derive a

generalized extended Born approximation (GEBA) based on a more mathematically and

physically consistent analysis.

Let M be the total number of spatial discretization cells, and rewrite equation

(7.1) into component form as

( )m bm Gτ σ= + Δ ⋅E E E , m=1, 2,…, M. (7.20)

229

We proceed to decompose the domain τ into two sub-domains, sτ and sτ τ− , in which

sτ is a sub-domain which encloses the m-th cell. Thus, equation (7.20) can be rewritten

as

( ) ( )s sm bm G Gτ τ τσ σ−= + Δ ⋅ + Δ ⋅E E E E . (7.21)

By transferring the second term on the right-hand side of equation (7.21) to the left-hand

side one obtains

( ) ( )s sm bmG Gτ τ τσ σ−− Δ ⋅ = + Δ ⋅E E E E . (7.22)

The following Remark is introduced to define the properties of the above

operator:

Remark 1: If there exists a spatial sub-domain sτ that satisfies the following two

conditions:

(1) Condition 1: Within sτ , the electric field E can be treated as spatially

invariant, and

(2) Condition 2: Outside sτ the Green’s dyadic function decreases in amplitude

sufficiently fast to have a negligible effect, then the second term on the right-

hand side of equation (7.22) can be neglected without affecting the accuracy

of the calculation of the internal electric field.

According to Remark 1, for such a sub-domain sτ , equation (7.20) can be

rewritten as

230

( )( )s m bmGτ σΙ − Δ =E E . (7.23)

or, equivalently,

mm bm= Λ ⋅E E , (7.24)

where mΛ is a scattering tensor for the m-th cell, and is given by

( )( ) 1

sm Gτ σ

−

Λ = Ι − Δ . (7.25)

Equation (7.24) is the fundamental equation for the GEBA. The more the sub-domain sτ

satisfies Remark 1, the more accurate the solution from equation (7.24) becomes. The

choice of sτ depends primarily on the source location(s), the frequency, the conductivity

contrast. Notice that the center of sτ is not necessarily the m-th cell. How to optimally

determine sτ goes beyond the scope of this work. However, one can envision that the

existence of such a sub-domain sτ reduces a dense matrix problem to a banded one.

Two special cases can be derived for the GEBA:

Special Case 1: When s mτ τ→ , where mτ is the singular domain, which only

encloses the m-th cell. This treatment does not modify equation (7.24); however, it does

modify the scattering tensor given by equation (7.25). The corresponding scattering

tensor can be written as

( ) ( )( ) 11

m

sm Gτ σ

−

Λ = Ι − Δ . (7.26)

This is the simplest case of the GEBA because the computation of the scattering tensor is

trivial. However, the above expression may not be sufficiently accurate since it violates

231

Condition 2 of Remark 1, i.e. the Green’s dyad may not decrease sufficiently fast to cause

the second term on the right-hand side of equation (7.22) to be negligible.

Special Case 2: When sτ τ→ , the scattering tensor becomes

( ) ( )( ) 12sm Gτ σ

−

Λ = Ι − Δ . (7.27)

The latter result is identical to that of the EBA (Habashy et al., 1993, and Torres-Verdín

and Habashy, 1994). This is the most complex case for the GEBA, since the computation

of the scattering tensor given by equation (7.27) requires numerical resources

proportional to ( )2O M . Also, this treatment may not provide accurate simulations, as it

violates Condition 1 of Remark 1, i.e., the electric field, in general may not be spatially

invariant in the whole scattering domain.

7.5 A HIGH-ORDER GENERALIZED EXTENDED BORN APPROXIMATION (HO-GEBA)

In the previous section, we assumed a sub-domain sτ that satisfied Remark 1.

However, we note that the two conditions in Remark 1 are not mutually complementary.

Thus the existence of sτ is a trade-off between meeting Condition 1 and Condition 2. In

this section, we introduce an alternative strategy that does not need the choice of an

optimal sub-domain. In such a strategy, one chooses a sub-domain sτ that satisfies

Condition 1 of Remark 1 as closely as possible; subsequently, one approximates the

electric field E on the right-hand side of equation (7.20) in some fashion. We now

develop such a strategy using the generalized series expansion (GS) of the internal

electric field.

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For a sub-domain sτ that satisfies Condition 1 and only satisfies Condition 2 in

some fashion, equation (7.22) can be rewritten as

( )( ) ( ) ( )s sm bmG G Gτ τ τσ σ σΙ − Δ = + Δ ⋅ − Δ ⋅E E E E , m=1, 2, …, M. (7.28)

Notice that the second term in equation (7.22) has been split into two terms in equation

(7.28). Then, by substituting the GS of E (keeping the first N terms, for convenience) in

equation (7.13) into the right-hand side of equation (7.28), one derives the equation for

the HO-GEBA as follows:

( ) ( ) ( ) ( )1

( )

0

( )N

Nnmm CBm CBm

n

−

=

′≈ + Λ ⋅∑E r E r r E r , m=1, 2, …, M. (7.29)

where ( )NCBm′E is given by equation (7C-10) and (7C-11). Supplement 7C gives a detailed

mathematical derivation of equation (7.29). We remark that equation (7.29) is the

fundamental equation of the HO-GEBA.

Two special cases can also be considered for the Ho-GEBA:

Special Case 1: Substitution of mΛ in equation (7.29) for 1s

mΛ yields

( ) ( ) ( ) ( )1 1

( )

0

( )N s

Nnmm CBm CBm

n

−

=

′≈ + Λ ⋅∑E r E r r E r , m=1, 2, …, M. (7.30)

This form of the Ho-GEBA closely follows the assumptions made in the derivation of the

Ho-GEBA. Therefore, equation (7.30) is a good approximation to solve EM scattering

problems. We remark here that, although an optimal scattering tensor is not needed for

the solution of equation (7.29), the choice of an optimal scattering tensor would improve

the rate of convergence of equation (7.29).

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Special Case 2: One may posit that by replacing mΛ in equation (7.29) for 2s

mΛ ,

an approximation ensues corresponding to Special Case 2 of the GEBA. As a matter of

fact, we remark here that one cannot directly derive such an approximation from equation

(7.28) because when sτ τ→ , the term involving sτ τ− in equation (7.28) automatically

approaches zero, and only the term bmE remains. In such a case the GS can be used

nowhere. However, a similar equation can be derived from the original equation that

gives rise to the EBA. Supplement 7D contains a detailed mathematical derivation for

this special case. The final equation is given by

( ) ( ) ( ) ( )1 2

( )

0

( )N s

Nnmm CBm CBm

n

−

=

′≈ + Λ ⋅∑E r E r r E r , m=1, 2, …, M. (7.31)

Incidently, by making a simple substitution from mΛ to 2s

mΛ , one can obtain exactly the

same form given by equation (7.31). In some sense, this exercise sheds light to the

difference between the derivation mechanisms behind the Ho-GEBA and the EBA.

7.6 THE PHYSICAL SIGNIFICANCE OF THE HO-GEBA

From the previous discussion, it follows that the Ho-GEBA is a combination of

the GS and the GEBA, in which the GEBA acts as the residual term of the GS. However,

numerical exercises indicate that the GEBA term can dramatically increase the speed of

convergence of the GS, thereby rendering the HO-GEBA extremely efficient to

accurately solve EM scattering problems. We remark that the GEBA with an optimal sub-

domain sτ can provide accurate solutions of EM scattering. However, as has been

pointed out by Gao et al. (2003a), Gao et al. (2003b), and Gao et al. (2004), because of

null components in the background field vector bE , the GEBA may not properly

234

reproduce cross-coupling EM terms in the presence of electrically anisotropic media.

This problem can be circumvented with the Ho-GEBA.

The physical significance of the GEBA over the EBA has been made clear in the

above derivation. We now explain how the Ho-GEBA improves the solution term by

term. To do so, we first expand equation (42) explicitly as follows:

1st order N=1 ( ) ( ) ( ) ( ) ( ) ( )0 1mm CBm CBm′≈ + Λ ⋅E r E r r E r , (7.42)

2nd order N=2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 2mm CBm CBm CBm′= + + Λ ⋅E r E r E r r E r , (7.43)

and

3rd order N=3 ( ) ( ) ( ) ( ) ( ) ( )2

3

0

nmm CBm CBm

n=

′= + Λ ⋅∑E r E r r E r . (7.44)

From equation (7.32), one can observe that the first-order GEBA (N=1) tends to keep the

zero-th order scattering term intact, and hence accounts for multiple-scattering terms via

the interaction between the scattering tensor and the first-scattering term. Since the zero-

th order scattering term is closely related to the source, one would expect it to reflect

some of the source effects. Because of this, it is expected that the first-order GEBA

would be more accurate than the Born approximation, the EBA, and the GEBA. Actually,

from the mathematical derivation of the GEBA and the HO-GEBA, one can expect the

first-order of the GEBA to provide accurate simulation results, including the case of

electrically anisotropic media.

The computation cost of low orders of Ho-GEBA is similar to that of the Born

approximation. However, because the FFT can be used to compute the GS terms, the

final computational cost is proportional to ( )2logO N N , where N is the total number of

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spatial discretization cells (Fang et al., 2003). For the EBA, the scattering tensor can also

be computed using FFTs.

7.7 NUMERICAL VALIDATION

To validate the Ho-GEBA theory, we focus on its special case 1, i.e., equation

(7.30). One can envision that the accuracy of the simulations could improve with a better

choice of scattering tensor. In this chapter, two kinds of formation models are considered

to validate the Ho-GEBA. The first kind of formation model is general 3D scatterer

models, which exhibit no electrical anisotropy, while the second kind of formation model

considered is dipping and anisotropic rock formations.

7.7.1 3D Scatterers

In this section, we consider examples of both conductive and resistive scattering

in the induction frequency range. Specifically, the frequencies used are 10 KHz, and 200

KHz. For all the numerical examples shown in this section, we only compute the results

up to the 3rd order of the Ho-GEBA. In addition to Vertical Magnetic Dipole (VMD)

sources, we investigate applications of the Ho-GEBA to the case of Transverse Magnetic

Dipole (TMD) sources due to the increasing relevance of transverse sources in

geophysical borehole induction logging (Gao et al., 2003a; Gao et al., 2003b; Gao et al.,

2004). We adopt the following notation to describe the simulation results: xxH refers to

the scattered magnetic field in the x-direction due to an x-directed source, and zzH refers

to the scattered magnetic field in the z-direction due to a z-directed source. Also, on the

figures, the label “Exact” designates the solution obtained with a full-wave 3D IE code,

“Born” designates the solution obtained with the Born approximation, “EBA” designates

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the solution obtained with the EBA, and “HOGEBA-n” (n=1, 2, 3) designates solutions

obtained with the n-th order terms of the Ho-GEBA. In addition, “REAL” designates the

in-phase component, and “IMAG” designates the quadrature component.

Figure 7.1 graphically describes the scattering models used in this paper. The

background Ohmic resistivity is 10 mΩ⋅ , and the background dielectric constant is 1.

One x-directed magnetic dipole source and one z-directed magnetic dipole source with a

magnetic moment of 1 2A m⋅ are assumed located at the origin, with 20 receivers

deployed along the z-axis uniformly separated at 0.2-meter intervals. No receiver is

assumed at the origin. A cubic scatterer with a side length of 2 m is centered about the x-

axis, and is symmetric about the y- and z-axes. Depending on the resistivity and the

distance between the scatterer and the source (located at the origin), the following four

models are considered in the simulations: Model 1: R=1 mΩ⋅ , L=4.0 m; Model 2:

R=1 mΩ⋅ , L=0.1 m; Model 3: R=100 mΩ⋅ , L=4.0 m; Model 4: R=100 mΩ⋅ , L=0.1 m.

Figure 7.2 shows the scattered xxH component as a function of receiver location

for two different frequencies: 10 KHz and 200 KHz. The assumed scattering model is

Model 1, and the left panel shows the results for 10 KHz, whereas the right panel shows

the results for 200 KHz. For each panel, the top figure describes the in-phase (real)

component of xxH , and the bottom figure describes the quadrature (imaginary)

component of xxH . Clearly, the accuracy of the Ho-GEBA is superior to either the EBA

or the first-order Born approximation at both frequencies. Notice that for this particular

case, the first-order Born approximation is more accurate than the EBA, and that the EBA

entails large errors in both the in-phase and quadrature components of xxH .

237

Figure 7.3 shows the scattered zzH component as a function of receiver location for two

different frequencies: 10 KHz and 200 KHz. The assumed scattering model is Model 1.

Figure 7.1: Graphical description of the scattering models considered in this section. The background ohmic resistivity is 10 mΩ⋅ and the background dielectric constant is 1. One x-directed and one z-directed magnetic dipole sources with a magnetic moment of 1 2A m⋅ are assumed located at the origin, and 20 receivers are deployed along the z-axis with a uniform separation of 0.2 meters. No receiver is at the origin. A cubic scatterer with a side length of 2 m is centered about the x-axis, and is symmetrical about the y and z axes. Depending on the resistivity, R, of the scatterer and the distance, L, between the source and the scatterer, a total of four scattering models are used in the numerical experiments: Model 1: R=1 mΩ⋅ , L=4.0 m; Model 2: R=1 mΩ⋅ , L=0.1 m; Model 3: R=100 mΩ⋅ , L=4.0 m; Model 4: R=100 mΩ⋅ , L=0.1 m.

Rx11

Rx1

Rx9

Rx10

Tx …

……

…

Rx12

Rx20

10bR m= Ω⋅ 2m

2m L

x

z

2m

238

The left-hand panel shows simulation results for 10 KHz, whereas the right-hand panel

shows simulation results for 200 KHz. For each panel, the top figure describes the in-

phase (real) component of zzH , and the bottom figure describes the quadrature

(imaginary) component of zzH . Again, the Ho-GEBA yields more accurate results than

either the EBA or the first-order Born approximation at both frequencies. Also, for this

case the EBA is more accurate than the first-order Born approximation. The EBA entails

errors in both the in-phase or quadrature components for the two frequencies, while the

first-order Born approximation entails large errors in both the in-phase and quadrature

components of zzH .

Next, we move the scatterer closer to the source until the distance between the

scatterer and the source is 0.1 m (such a distance is a common borehole radius in

geophysical logging applications). This is scattering Model 2. The remaining model

parameters are kept the same as those described for scattering Model 1. Figure 7.4 shows

the scattered xxH component as a function of receiver location for two different

frequencies: 10 KHz and 200 KHz. The left-hand panel shows simulation results for 10

KHz, whereas the right-hand panel shows simulation results for 200 KHz. For each panel,

the top figure describes the in-phase (real) component of xxH , and the bottom figure

describes the quadrature (imaginary) component of xxH . Clearly, the Ho-GEBA yields

more accurate results than either the EBA or the first-order Born approximation at both

frequencies. Notice that for this particular case, the first-order Born approximation is

more accurate than the EBA, especially for the quadrature component. The EBA exhibits

large errors in both the in-phase and quadrature components of xxH . For this particular

239

scattering model, and by comparison of Figures 7.2 and 7.4, it is found that the EBA

yields inaccurate results for xxH regardless of both the frequency of operation and the

distance between the source and the scatterer. On the other hand, the Ho-GEBA yields

accurate simulation results.

Figure 7.5 shows the scattered zzH component as a function of receiver location

for two different frequencies: 10 KHz and 200 KHz. The assumed scattering model is

Model 2. The left-hand panel shows simulation results for 10 KHz, whereas the right-

hand panel shows simulation results for 200 KHz. For each panel, the top figure describes

the in-phase (real) component of zzH , whereas the bottom figure describes the quadrature

(imaginary) component of zzH . We observe that the Ho-GEBA (especially the 2nd and

3rd order) yields much more accurate simulations than the EBA and the Born

approximations at both frequencies. Notice that for the in-phase component, the EBA

entails exceedingly large errors, which confirms our earlier statement that the accuracy of

the EBA considerably degrades when the scatterer is close to the source region.

By replacing the block resistivities included in Model 1 and Model 2 from 1

mΩ⋅ to 100 mΩ⋅ , we generate two resistive scattering models: Model 3 and Model 4.

Figure 7.6 shows the scattered xxH component as a function of receiver location at two

different frequencies: 10 KHz and 200 KHz. The assumed scattering model is Model 3.

The left-hand panel shows simulation results for 10 KHz, and the right-hand panel shows

simulation results for 200 KHz. For each panel, the top figure describes the in-phase

(real) component of xxH , whereas the bottom figure describes the quadrature (imaginary)

component of xxH . Clearly, the Ho-GEBA yields more accurate results than the EBA

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and Born approximation at both frequencies. Notice that for this case (as can be also

observed in Figure 7.2), the Born approximation yields more accurate results than the

EBA. The EBA entails large errors in both the in-phase and quadrature components.

Figure 7.7 shows the scattered zzH component as a function of receiver location,

at two different frequencies: 10 KHz and 200 KHz. The assumed scattering model is

Model 3. The left-hand panel shows simulation results for 10 KHz, and the right-hand

panel shows simulation results for 200 KHz. For each panel, the top figure describes the

in-phase (real) component of zzH , whereas the bottom figure describes the quadrature

(imaginary) component of zzH . In similar fashion to Figure 7.3, the Ho-GEBA entails

accurate results compared to the EBA and considerably more accurate results than the

Born approximation at both frequencies.

We proceed to displace Model 3 closer to the source, thereby constructing Model

4. Figure 7.8 shows the scattered xxH component as a function of receiver location at

two different frequencies: 10 KHz and 200 KHz. The left-hand panel shows simulation

results for 10 KHz, and the right-hand panel shows simulation results for 200 KHz. For

each panel, the top figure describes the in-phase (real) component of xxH , whereas the

bottom figure describes the quadrature (imaginary) component of xxH . Clearly, the Ho-

GEBA yields more accurate results than either the EBA or the Born approximation at

both frequencies. Notice that for this case, in similar fashion to Figure 7.6, the Born

approximation is more accurate than the EBA. The EBA exhibits errors in both the in-

phase and quadrature components of xxH (note that the EBA yields an in-phase

component with the wrong sign).

241

Figure 7.9 shows the scattered zzH component as a function of receiver location

at 10 KHz and 200 KHz. The assumed scattering model is Model 4. The left-hand panel

shows simulation results for 10 KHz, and the right-hand panel shows simulation results

for 200 KHz. For each panel, the top figure describes the in-phase (real) component of

zzH , whereas the bottom figure describes the quadrature (imaginary) component of zzH .

In similar fashion to Figure 7.7, the Ho-GEBA is more accurate than either the EBA or

the Born approximation at both frequencies. For this case, the EBA entails more accurate

quadrature components than the Born approximation. However, in similarity with Figure

7.8, the EBA yields an in-phase component with the wrong sign.

To further assess the accuracy of the Ho-GEBA with respect to frequency, we

now consider a fixed receiver located at -0.1 m. The assumed scattering model is Model

2. This model represents a typical conductive medium and exhibits substantial near-

source scattering effects. The frequency range considered for the simulations is between

10 KHz and 2 MHz, which is typical of borehole geophysical induction logging. Figures

7.10 and 7.11 graphically compare the scattered magnetic field components xxH and

zzH , respectively, simulated with the Ho-GEBA up to the 5th order together with the full-

wave solution, the EBA, and the Born approximation. This graphical comparison clearly

indicates that the Ho-GEBA yields consistent and accurate results that are superior to the

EBA and the Born approximation over the entire frequency range.

242

Figure 7.2: Scattered xxH component for Model 1. The left- and right-hand panels show simulation results for 10 KHz and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of xxH , and the bottom figure describes the quadrature (imaginary) component of xxH . Simulation solutions from the Born approximation, the EBA, and the Ho-GEBA (up to the 3rd order) are plotted against the exact solution.

243

Figure 7.3: Scattered zzH component for Model 1. The left- and right-hand panels show simulation results for 10 KHz and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of zzH , and the bottom figure describes the quadrature (imaginary) component of zzH . Simulation solutions from the Born approximation, the EBA, and the Ho-GEBA (up to the 3rd order) are plotted against the exact solution.

244

Figure 7.4: Scattered xxH component for Model 2. The left- and right-hand panels show simulation results for 10 KHz and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of xxH , and the bottom figure describes the quadrature (imaginary) component of

xxH . Simulation solutions from the Born approximation, the EBA, and the Ho-GEBA (up to the 3rd order) are plotted against the exact solution.

245


246

Figure 7.6: Scattered xxH component for Model 3. The left- and right-hand panels show simulation results for 10 KHz and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of xxH , and the bottom figure describes the quadrature (imaginary) component of

xxH . Simulation solutions from the Born approximation, the EBA, and the Ho-GEBA (up to the 3rd order) are plotted against the exact solution.

247


248

Figure 7.8: Scattered xxH component for Model 4. The left- and right-hand panels show simulation results for 10 KHz and 200 KHz, respectively. For each panel, the top figure describes the in-phase (real) component of xxH , and the bottom figure describes the quadrature (imaginary) component of xxH . Simulation solutions from the Born approximation, the EBA, and the Ho-GEBA (up to the 3rd order) are plotted against the exact solution.

249


250

Figure 7.10: Comparison of the EBA, the Born and the EBA over the frequency range of 10 KHz-2 MHz. The model considered is Model 2, and the signal is for the receiver at -0.1 m. The left figure describes the in-phase (real) component of xxH , and the right figure describes the quadrature (imaginary) component of xxH . Simulation solutions from the Born approximation, the EBA, and the Ho-GEBA (up to the 5rd order) are plotted against the exact solution.

251

Figure 7.11: Comparison of the EBA, the Born and the EBA over the frequencyrange of 10 KHz-2 MHz. The model considered is Model 2, and the signal is for the receiver at -0.1 m. The left figure describes the in-phase (real) component of zzH , and the right figure describes the quadrature (imaginary) component of zzH . Simulation solutions from the Born approximation, the EBA, and the Ho-GEBA (up to the 5rd order) are plotted against the exact solution.

252

Figure 7.12: Graphical comparison of the convergence rate of the Ho-GEBA and the GS. Model 1 is the assumed scattering and the numerical simulations correspond to the zzH component. The left-hand panel shows convergence results for 10 KHz, and the right-hand panel for 200 KHz.

253

We also want to emphasize that the Ho-GEBA can dramatically improve the

convergence rate of the GS. This is best explained with a simulation exercise. Figure

7.12 graphically compares the convergence of the GS and the Ho-GEBA for the

scattering Model 1 ( zzH ). The left-hand panel in that figure corresponds to 10 KHz, and

Figure 7.13: Graphical corroboration of some technical issues associated with the special case 2 of the Ho-GEBA. Model 2 is the assumed scattering model and the numerical simulations correspond to the zzH component. The nomenclature HoGEBAS2-n (n=1, 2, 3) identifies simulation results associated with the special case 2 of the Ho-GEBA. The left- and right-hand panels describe the real and imaginary parts of zzH , respectively.

254

the right-hand panel to 200 KHz. Clearly, for this simulation exercise the rate of

convergence of the Ho-GEBA is superior to that of the GS.

Another technical issue that needs consideration is the special case 2 of the Ho-

GEBA. At the outset, we emphasized that this special case may not applicable for some

cases of EM scattering. To clarify this point, we make use of another simulation exercise.

Figure 7.13 describes simulation results (in-phase components of zzH ) obtained for

Model 2. In that figure, the curves labeled HoGEBAS2-n, n=1, 2, 3, describe simulation

results obtained for the special case 2 of the Ho-GEBA. These results clearly indicate that

the special case 2 of the Ho-GEBA is not applicable for the problem at hand. One may

conclude that the special case 2 of the Ho-GEBA only applies to simulation cases where

the EBA remains accurate.

7.7.2 Dipping and Anisotropic Rock Formations

As mentioned above, a unique feature of the Ho-GEBA is that it is suitable for

simulating EM measurements in the presence of electrically anisotropic media. In this

section, we use the same formation models and tool configuration described in Chapter 5

and Chapter 6 to test the Ho-GEBA in the presence of electrically anisotropic media.

7.7.2.1 1D Anisotropic Rock Formation, Dip Angle= 60

The formation model is assumed to be a 1D anisotropic rock formation without

borehole and mud-filtrate invasion. The dip angle is 60 and the frequency is 220 KHz.

Figures 7.14 through 7.16 show the simulation results for the magnetic field

components xxH , yyH and zzH , respectively. Simulations obtained with the Ho-GEBA

(up to the 4th order) are plotted against those obtained with a 1D code, the Born

255

approximation and the EBA. The comparison clearly indicates that the Ho-GEBA

provides more accurate simulation results than the Born approximation and the EBA.

Moreover, the 4th order of the Ho-GEBA already provides accurate simulation results.

Figure 7.14: Comparison of the xxH field component simulated with the Ho-GEBA, the Born approximation, the EBA and an analytical 1D code assuming a 1D anisotropic rock formation. The tool and the formation form an angle of 60o

and the frequency is 220 KHz.

256

Figure 7.15: Comparison of the yyH field component simulated with the Ho-GEBA, the Born approximation, the EBA and an analytical 1D code assuming a 1D anisotropic rock formation. The tool and the formation form an angle of 60o


257

Figure 7.16: Comparison of the zzH field component simulated with the Ho-GEBA, the Born approximation, the EBA and an analytical 1D code assuming a 1D anisotropic rock formation. The tool and the formation form an angle of 60o


258

7.7.2.2 3D Anisotropic Rock Formation, Dip Angle= 60

The assumed formation model is a 3D anisotropic rock formation with borehole

and mud-filtrate invasion. The dip angle is 60 and the frequency is 220 KHz.

Figures 7.17 through 7.19 show the simulation results for the magnetic field

components xxH , yyH and zzH , respectively. Simulation results obtained with the Ho-

GEBA (up to the 4th order) are plotted against those obtained with a 3D FDM code, the

Born approximation and the EBA. The comparison clearly indicates that the Ho-GEBA

provides more accurate simulation results than the Born approximation and the EBA.

Moreover, the 4th order of the Ho-GEBA already provides accurate simulation results.

259

Figure 7.17: Comparison of the xxH field component simulated with the Ho-GEBA, the Born approximation, the EBA and a 3D FDM code assuming a 3D anisotropic rock formation with borehole and mud-filtrate invasion. The tool and the formation form an angle of 60o and the frequency is 220 KHz.

260

Figure 7.18: Comparison of the yyH field component simulated with the Ho-GEBA, the Born approximation, the EBA and a 3D FDM code assuming a 3D anisotropic rock formation with borehole and mud-filtrate invasion. The tool and the formation form an angle of 60o and the frequency is 220 KHz.

261

Figure 7.19: Comparison of the zzH field component simulated with the Ho-GEBA, the Born approximation, the EBA and a 3D FDM code assuming a 3D anisotropic rock formation with borehole and mud-filtrate invasion. The tool and the formation form an angle of 60o and the frequency is 220 KHz.

262

7.8 CONCLUSIONS

The following conclusions stem from the simulation exercises described above:

(1) In general, the Ho-GEBA is more accurate than the EBA regardless of the

distance between the source and scatterer, and the operating frequency. For some cases

where the source is far from the scatterer, the EBA also provides relatively accurate

results. However when the source is moved closer to the scatterer, the EBA yields in-

phase components with very large errors, and sometimes even with the wrong sign.

(2) For some of the examples described in this chapter, the Born approximation

outperforms the EBA, whereas in others the EBA outperforms the Born approximation.

However, in general the Born approximation and the EBA do not provide similarly

accurate results, except in some limiting situations, i.e., when the scattering tensor

approaches the unity tensor. The physical interpretation for this remark is that, while the

scattering tensor remains source independent, the EBA emphasizes the zero-th order

scattering term (the Born approximation) to account for some of the multiple scattering

via the scattering tensor. Clearly, when the Born approximation provides accurate results

the distorted Born approximation does not.

As a general conclusion, the GS is a generalized series expansion of the internal

electric field, whereas the GEBA is a generalized extended Born approximation, which is

based on more solid mathematical and physical assumptions than the EBA. Moreover, the

EBA is only a special case of the Ho-GEBA. The Ho-GEBA is a combination of the

GEBA and the GS. In general, the GEBA will converge substantially faster than the GS.

We validated the Ho-GEBA using simple 3D scatterers. Numerical experiments in the

induction frequency range show that the Ho-GEBA in general yields more accurate

263

simulation results than both the Born approximation and the EBA. The total

computational cost of the Ho-GEBA is proportional to ( )2logO M M , where M is the

number of spatial discretization cells. A unique feature of the Ho-GEBA is that it can be

used to simulate EM scattering due to electrically anisotropic media. This feature is not

possible with either the Born approximation or the EBA.

Supplement 7A: Derivation of the New Integral Equation

Singer (1995), Pankratov (1995), and Zhdanov and Fang (1997) derived an

energy inequality for the anomalous EM field. Such an energy inequality can be

generalized to the case wherein an electrical conductivity anomaly is embedded in an

infinite uniform conductive background (Singer, 1995).

Assume an electrical conductivity anomaly with a closed boundary Σ embedded

in an infinite uniform conductive background of conductivity equal to bσ ′ . Following

Zhdanov and Fang (1997), the per-period average of energy flow, Q, of anomalous EM

field through ∑ can be expressed as

( )*1Re2 s sQ dv ds

τ ∑= ∇ ⋅ = × ⋅∫ ∫P E H n , (7A-1)

where τ is the spatial support of the conductivity anomaly, *12 s s= ×P E H is the Poynting

vector, n is the outgoing unit vector normal to the surface ∑ , sE and sH are the

anomalous electric and magnetic fields, respectively, * denotes complex conjugate, and

264

( )Re ⋅ symbolizes the real part of the corresponding quantity. According to the Poynting

theorem and Maxwell’s equations (Harrington, 1961), Q can be rewritten as

( )2 *1 Re2 b s sQ dv

τσ ′= − + ⋅∫ E E J , (7A-2)

where J is the anomalous electric current vector.

It has been shown that the energy flow, Q, of the anomalous field must be

nonnegative (Pankratov, 1995). Thus, the following equation holds

( )2 *1 Re 02 b s s dv

τσ ′ + ⋅ ≤∫ E E J . (7A-3)

The integrand in equation (7A-3) can be rewritten as

( )2 2

2 *b s s b s

b b

Re2 4

σ σσ σ

′ ′+ ⋅ = + −′ ′

JJE E J E . (7A-4)

Substitution of equation (7A-4) into equation (7A-3) yields the energy inequality

2 2

2 4b sb b

dv dvτ τσ

σ σ′ + ≤

′ ′∫∫∫ ∫∫∫JJE . (7A-5)

Equation (7A-5) holds in the sense of the physics of the interaction between the EM

fields and the medium. Such a condition represents a physical constraint for our

derivations below.

Because bσ ′ is always positive, equation (7A-5) is equivalent to

1/ 21/ 22 22

22 2b s

b b

dv dvτ τ

σσ σ

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟′ + ≤ ⎜ ⎟⎜ ⎟′ ⎜ ⎟′⎝ ⎠ ⎝ ⎠∫∫∫ ∫∫∫

JJE . (7A-6)

Next, we note that

265

2 2b s b sb b

σ σσ σ

⎛ ⎞′ ′+ = +⎜ ⎟′ ′⎝ ⎠

J JE E , (7A-7)

and make use of equation (7.1) to obtain

2

22 2

2

b sb

b bb b

c

b

G

G

τ

τ

σσ

σ σσ σ

σ

⎛ ⎞′ +⎜ ⎟′⎝ ⎠

⎛ ⎞′ ′= +⎜ ⎟⎜ ⎟′ ′⎝ ⎠⎛ ⎞

= ⎜ ⎟⎜ ⎟′⎝ ⎠

JE

J J

J

, (7A-8)

where cGτ is an operator that can be applied to any vector-valued function and is given by

( ) ( )2cb bG Gτ τσ σ′ ′= +x x x . (7A-9)

From the physical constraint given by equation (7A-6), one can derive the following

inequality for the operator cGτ

( ) 1τ ≤ ⋅x xcG , (7A-10)

where ⋅ denotes the 2 -norm in a Hilbert space, and is defined as

( )1/ 22dv

τ⋅ = ⋅∫∫∫ . (7A-11)

By making use of equation (7.2), Zhdanov and Fang (1997) transformed equation (7A-8)

into

( )cs b s ba b G bτ

⎡ ⎤+ = +⎢ ⎥⎣ ⎦E E E E , (7A-12)

where

22

b

b

a σ σσ

′ + Δ=

′, and

2 b

b σσ

Δ=

′. (7A-13)

266

Equation (7A-12) can be treated as an integral equation with respect to the

product saE , i.e.,

( )s sa C a=E E , (7A-14)

where C is a new operator that remains contractive for any type of lossy background

medium (Zhdanov and Fang, 1997).

By making use of equation (7A-12) and equation (7.1), and after some

manipulations, one obtains

( )1

τ τσ σ−⎛ ⎞′ ′= + ⋅ = +⎜ ⎟

⎝ ⎠E E E E Ec c

b b b bG b a O , (7A-15)

where

1a−

=E E . (7A-16)

Following Zhdanov and Fang (1997), it can be shown that for any lossy

background medium ( b 0σ ′ > ), the following relation holds

11b a

−

⋅ < . (7A-17)

According to the Cauchy-Schwartz inequality, equations (7A-10) and (7A-17)

guarantee that the operator cOτ be contractive, namely,

( )1

1c cO b a Gτ τ

−

≤ ⋅ < ⋅x x x . (7A-18)

Equation (7A-15) eventually leads to the new integral equation

( )2τα α σ β β′= ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅E E E Eb ba a G , (7A-19)

where

267

( ) 1

2 2b bα σ σ σ−

′ ′= Ι + Δ , (7A-20)

and

β α= Ι − . (7A-21)

Notice that the contraction of the new integral equation (7A-19) is assured by equation

(7A-18). Finally, E is given by equation (7A-16).

Supplement 7B: Derivation of the Generalized Series (GS) Expansion for the Internal Electric Field

Assume that the initial guess of E in equation (7.11) is given by ( )0CBE , namely,

( ) ( )0 0CB=E E . (7B-1)

We remark that ( )0CBE is unknown and that the subscript “CB” here has no specific

meaning. Substitution of equation (B-1) into equation (23) together with equation (7A-

16) yields

( ) ( )( ) ( )

( ) ( )( ) ( )( )

1 0 0

0 0 0

b CB CB

CB CB b CB

G

G

τ

τ

α α σ β

α σ α

= ⋅ + ⋅ Δ ⋅ + ⋅

= + ⋅ Δ ⋅ + ⋅ −

E E E E

E E E E . (7B-2)

Notice that equation (7A-13) has been used to derive equation (7B-2).

Now define

( ) ( )( ) ( )( )1 0 0CB CB b CBGτα σ α= ⋅ Δ ⋅ + ⋅ −E E E E . (7B-3)

Equation (7B-2) can then be rewritten as

( ) ( ) ( )1 0 1CB CB= +E E E . (7B-4)

268

Substitution of equation (7B-4) into equation (7.11) together with equation (7A-16)

yields

( ) ( ) ( ) ( )2 0 1 2CB CB CB= + +E E E E , (7B-5)

where

( ) ( )( ) ( )2 1 1CB CB CBGτα σ β= ⋅ Δ ⋅ + ⋅E E E . (7B-6)

By repeating the same procedure, one derives the following series expansion

( ) ( ) ( )0

nCB

n

∞

=

=∑E r E r , (7B-7)

where

( ) ( )( ) ( )1 1n n nCB CB CBGτα σ β− −= ⋅ Δ ⋅ + ⋅E E E , n=2, 3, 4, … (7B-8)

and ( )1CBE is given by equation (7B-3).

In Supplement 7A, we demonstrated that the integral equation from which the

series expansion (7B-7) was derived is a contractive integral equation. This indicates that

the series expansion given by equation (7B-7) is always convergent. To confirm this, we

consider a numerical example for which the classical Born series is divergent. Figure 7B-

1 graphically describes the formation model, consisting of a conductive cube with a side

length of 2 m and conductivity equal to 10 S/m, embedded in a background medium of

conductivity equal to 1 S/m. The transmitter and the receiver are assumed to be vertical

magnetic dipoles operating at 20 KHz. The distance between the transmitter and the cube

is 0.1 m, and the spacing between the transmitter and receiver is 0.5 m. Measurements

consist of the scattered magnetic field at the receiver. Figure 7B-2 graphically compares

the convergence of the GS (right panel) against the convergence of the classical Born

269

series (left panel). On these figures, the horizontal axis describes the iteration number,

while the vertical axis describes the amplitude of the scattered magnetic field. This

exercise clearly indicates that the classical Born series expansion does not converge,

while the GS converges to the exact solution in a few iterations.

10 S/m Tx

Rx

2 m

2 m

2 m

z

x

0.1 m

0.5 m1 S/m

Figure 7B-1: Rock formation model used to numerically test the convergence

properties of the GS. A conductive cube with a side length of 2 m and a

conductivity of 10 S/m is embedded in a background medium of conductivity

equal to 1 S/m. The transmitter and the receiver are assumed to be vertical

magnetic dipoles operating at 20 KHz. The distance between the transmitter and

the cube is 0.1 m, and the spacing between the transmitter and receiver is 0.5 m.

270

Figure 7B-2: Graphical comparison of the convergence behavior of the classical Born

series expansion, the GS (starting from the background field), the EBA series

expansion [no contraction (N.C.)], and the EBA series expansion [with contraction

(W.C.)] for the rock formation model given in figure B-1. The left figure describes the

convergence behavior of both the classical Born series expansion and the EBA series

expansion (N.C.), while the right figure describes the convergence behavior of the GS

and EBA series expansion (W.C.). The solution line was calculated using a full-wave

3D integral-equation code (Fang, Gao, and Torres-Verdín, 2003)

271

Supplement 7C: Derivation of the Fundamental Equation of the Ho-

GEBA

Substitution of equation (7.13) into the right-hand side of equation (7.28) gives

( )( ) ( ) ( )1 1

0 0s s

N Nn n

m bm CB CBn n

G G Gτ τ τσ σ σ− −

= =

⎛ ⎞ ⎛ ⎞Ι − Δ = + Δ ⋅ − Δ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∑E E E E . (7C-1)

Since E is assumed spatially invariant within sub-domain sτ , one can rewrite equation

(7C-1) as

( )( ) ( )( ) ( ) ( )1 1

0 0s s

N Nn n

m bm CB CBmn n

G G Gτ τ τσ σ σ− −

= =

Ι − Δ = + Δ ⋅ − Δ ⋅∑ ∑E E E E . (7C-2)

From equations (7B-3) and (7B-8) one obtains

( )( ) ( ) ( )( ) ( ) ( )( )( ) ( )( )

( ) ( )

1 1 11 0 2 1

01

1

1

0

Nn

CB CBm CBm bm CBm CBmn

N NCBm CBm

Nn N

CBm bm CBmn

Gτ σ α α α

α

− − −

=

−−

−

=

⎡ ⎤Δ ⋅ = ⋅ + − + ⋅ −⎢ ⎥⎣ ⎦

+ + ⋅ −

′= − +

∑

∑

E E E E E E

E E

E E E

, (7C-3)

where

( ) ( )( ) ( )1 1N N NCBm CBm CBmGτ σ γ− −′ = Δ ⋅ + ⋅E E E , N=2, 3, … (7C-4)

( ) ( )( ) ( )1 0 0CBm CBm bm CBmGτ σ′ = Δ ⋅ + −E E E E , (7C-5)

and

2 b

σγσΔ

=′

. (7C-6)

Substitution of equation (7C-3) into equation (7C-2) yields

272

( )( ) ( )( ) ( ) ( )1

0s s

Nn N

m CBm CBmn

G Gτ τσ σ−

=

′Ι − Δ = Ι − Δ ⋅ +∑E E E . (7C-7)

Finally,

( ) ( )1

0

Nn N

mm CBm CBmn

−

=

′= + Λ ⋅∑E E E , (7C-8)

where mΛ is given by equation (7.25).

Supplement 7D: Derivation of Special Case No.2 of the Ho-GEBA

First, the generalized series expansion of ( )0E r can be written as

( ) ( ) ( )0 00

nCB

n

∞

=

=∑E r E r . (7D-1)

Subtraction of equation (7.13) from equation (7D-1) yields

( ) ( ) ( ) ( ) ( ) ( )( )0 00

n nCB CB

n

∞

=

− = −∑E r E r E r E r . (7D-2)

For convenience, we keep the first N+1 terms in equation (7D-2), and substitute

the ensuing expression into equation (7.16), to obtain

( )

( ) ( ) ( )( )0 0

0 0 00

( ) ( ) ( , ) ( )

( , ) ( ) ( )

e

b

Nen n

CB CBn

G d

G d

τ

τ

σ

σ=

= + ⋅Δ ⋅

⎛ ⎞+ ⋅Δ ⋅ −⎜ ⎟⎝ ⎠

∫

∑∫

0

0

E r E r r r r r E r

r r r E r E r r. (7D-3)

By expanding ( ) ( )NCBE r with a Taylor series about 0r one obtains

( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0N N N

CB CB CB= + − ∇ +E r E r r r E r (7D-4)

Further, by retaining only the first term on the right-hand side of equation (7D-4) one can

write

273

( ) ( ) ( ) ( )0N N

CB CB≈E r E r . (7D-5)

Using this last expression and rearranging the terms in equation (7D-3), one readily

obtains

( ) ( )

( ) ( )

1

0 00

1

0 0 00

( ) ( ) ( , ) ( ) ( )

( , ) ( )

Nen

b CBn

N en

CBn

G d

G d

τ

τ

σ

σ

−

=

−

=

⎛ ⎞≈ + ⋅Δ ⋅ −⎜ ⎟⎝ ⎠

+ ⋅Δ ⋅

∑∫

∑∫

0

0

E r E r r r r r E r E r

r r r E r r

. (7D-6)

Substitution of equation (7C-3) into equation (7D-6), together with some simple

mathematical manipulations yields

( ) ( ) ( ) ( )1

( )

0( )

NNn

CB CBn

−

=

′≈ + Λ ⋅∑E r E r r E r , (7D-7)

where ( )NCB′E is given by equation (7C-4), and (7C-5).

274

Chapter 8: Inversion of Multi-frequency Array Induction Measurements

Array induction tools play a crucial role in the petrophysical assessment of

hydrocarbon bearing rocks. Current procedures used for on-site processing and

interpretation of array induction data are based on a sequence of corrections and

approximations intended to expedite the on-site estimation of apparent resistivities. The

desired commercial product is a set of resistivity curves that exhibits (a) optimal vertical

resolution, (b) minimal shoulder-bed effect, and (c) selective deepening of the zone of

response away from the borehole wall. Rigorous inversion procedures, however, are

needed to properly account for shoulder-bed and invasion effects. A number of inversion

strategies have been advanced thus far, but the challenge is still open to develop

expedient, efficient, and robust algorithms that could possibly be run on-site with a

minimum number of simplifying assumptions.

In this chapter, we develop efficient inversion algorithms for the inversion of

array induction data. We differentiate two types of inverse problems depending on the

assumption of the formation model. One problem is Resistivity Imaging (RIM), which is

based on an assumption that the spatial distribution of resistivity is continuous; the other

one is Resistivity Inversion (RIN), which is based on an assumption that the spatial

distribution of resistivity exhibits a blocky structure. The underlying theory of nonlinear

inversion based on regularized Gauss-Newton iterations is introduced in this chapter. The

RIM is performed with an inner-loop and outer-loop optimization technique (Gao and

Torres-Verdín, 2003), while the RIN is performed only by a one-loop optimization. One

important feature of the algorithms developed in this chapter is that, the simulated

275

measurements and the Jacobian (or sensitivity) matrix are computed simultaneously with

only one forward simulation. This feature makes the inversion algorithm extremely

efficient. Inversion of multi-frequency array induction data based on multi-front mud-

filtrate invasion rock formation models show that robust and efficient inversions can be

obtained using the algorithms developed in this chapter. Moreover, inversion results

show that the RIM is more suitable for qualitatively estimating the resistivity profile in

the radial direction, while the RIN is superior to the RIM for the quantitative evaluation

of in situ hydrocarbon saturations.

8.1 INTRODUCTION

Borehole induction tools are routinely used to estimate electrical conductivity of

rock formations in the virgin-zone. Subsequently, electrical conductivity is used to

estimate in-situ hydrocarbon saturation, and hence to assess the economic value of

commercial reservoirs. Most of the commercial software used for on-site interpretation of

borehole induction measurements is based on the estimation of apparent resistivity. The

calculation of apparent resistivity curves is performed with several simplifying

assumptions to the influence of borehole effects, shoulder beds, and mud-filtrate

invasion. Estimation of true resistivity values can only be performed with rigorous

inversion procedures that can account for all of the existing environmental conditions in

an accurate manner.

Traditional methods used for the inversion of induction logs assume a linear

relationship between induction tool measurements and formation conductivity. The

formation is often assumed to be a layered model in which electrical conductivity varies

only with depth. Spurious inversion artifacts may occur if this assumption is not met.

276

Moreover, the linearity between the induction tool response and the formation

conductivity is only strictly valid for low values of conductivity and frequency. In cases

of deep conductive invasion, the inverted one-dimensional (1D) conductivity profile can

deviate considerably from the true formation resistivity profile. Increased demand has

been placed on the direct estimation of two-dimensional (2D) models of electrical

conductivity as a function of radial distance, ρ , and depth, z. The direct estimation of 2D

models of electrical conductivity implicitly does away with the need to perform borehole,

shoulder, and invasion corrections, and hence provides more accurate values of virgin-

zone conductivity.

Estimation of 2D spatial distributions of electrical conductivity can be performed

with a nonlinear inversion algorithm. In this approach, a method is required to

numerically simulate borehole induction data for an arbitrary 2D distribution of

formation conductivity. The inverse problem is initialized with a coarse electrical

conductivity model. Subsequently, a solution to the inverse problem is obtained by

iteratively varying the conductivity model until the numerically simulated borehole

induction data reproduces the measurements. Examples can be found in Lin et al. (1984)

(a least-squares inversion), Freedman et al. (1991) (a maximum entropy inversion), Chew

et al. (1994) (an inversion approach based on the distorted Born iterative method),

Torres-Verdín et al. (1994) (an inversion approach based on a nonlinear scattering

approximation), San Martin et al. (2001) (an inversion method based on neural

networks), and Gao (2002) (an inversion strategy that makes use of quasi-Newton

updates).

277

One common inversion procedure consists of minimizing a quadratic cost

function that emphasizes the sum of the squared differences between the measured and

the simulated data. Because in most cases the relationship between the property

distribution function and the simulated data is nonlinear, the minimization is performed

with a nonlinear search technique. This nonlinear search technique is constructed by way

of a suitable number of sequential linear steps that eventually trace the road toward an

extremum of the cost function. Each linear step requires computing a Jacobian/sensitivity

matrix, which describes the first-order variations of the simulated data with respect to a

variation in the model parameters. The main difficulties associated with iterative

nonlinear search techniques are: (a) computing the Jacobian matrix at each linear step, (b)

solving the forward problem accurately and efficiently, and (c) addressing the inherent

non-uniqueness and ill-posed nature of the inversion.

The following three approaches have been put forth to circumvent the above

drawbacks of nonlinear iterative minimization: (1) approximate the forward problem, (2)

approximate the Jacobian matrix, and (3) resort to alternative minimization approaches

that could be less taxing in the search for the extremum of the cost function. One such

alternative minimization approach, termed “inner-outer loop optimization”, is introduced

in the first part of this chapter. The outer loop is constructed with an exact forward solver

of the original numerical simulation problem. Concomitantly, an inner loop is constructed

using a fast and efficient approximate solver of the original numerical simulation

problem. Nonlinear optimization is used within the inner loop to find a local stationary

point of the least-squares cost function. Upon convergence of the inner loop, an outer

loop takes the outcome of the inner loop to check the corresponding data misfit. If the

278

computed data misfit is deemed acceptable, then the inversion stops, and the outcome

from the inner loop is taken as the final estimation of electrical conductivity. Otherwise,

the computed data misfit is used to construct a new data vector that is input to a new

inner-loop minimization. Computations within the inner loop are extremely fast and

efficient because of the use of the approximate forward solver PEBA. The simulated

measurements and the Jacobian matrix are computed simultaneously with only one

forward simulation. Accurate solvers and approximate solvers have been developed in

Chapter 4.

To perform the inversion, one begins by assuming a rock formation model with

prescribed structure. One objective of this work is to study the effects of different

formation models on the inversion of induction data. Two types of formation models are

studied in this chapter. The first formation model consists of a spatially continuous

resistivity distribution. The inversion procedure based on this type of formation model is

termed “Resistivity Imaging (RIM).” The second formation model assumes that the

resistivity distribution is a blocky structure, such as commonly assumed with multi-front

mud-filtrate invasion models in well logging interpretation (see Chapter 4). The inversion

procedure based on this type of formation model is termed “Resistivity Inversion (RIN).”

8.2 TWO-DIMENSIONAL RESISTIVITY IMAGING BASED ON AN INNER-LOOP AND OUTER-

LOOP OPTIMIZATION TECHNIQUE

This section describes a novel inversion algorithm that efficiently combines

approximate and accurate numerical simulations of borehole induction measurements.

Inversion is approached as the minimization of the weighted least-squares prediction

error using an inner-outer loop approach. Within the inner-loop, a fast approximation to

279

the forward problem is used to perform an efficient minimization. This yields an

approximate solution to the unknown model parameters, i.e. radial profiles of electrical

conductivity for each layer. Upon completion of the inner-loop minimization, an outer

loop performs an accurate simulation of the measurements corresponding to the

conductivity model rendered by the inner-loop. If the fit to the data is not satisfactory,

then the data misfit is used to construct a new inner-loop minimization. The procedure

repeats itself until the data misfit is brought down to an acceptable value. Substantial

savings in computer time can be achieved if the approximate forward solver is

constructed in an efficient manner.

A novel approximation of EM scattering is used in this section to construct the

fast and accurate solver required by the inner-loop minimization. Several synthetic

examples are described of the application of the inner-outer loop minimization approach

in the presence of additive random measurement noise. An inversion methodology is also

advanced in which 2D distributions of electrical conductivity are estimated using a serial

sequence of 1D, 1.5D, and 2D inverse problems. Robust estimation of electrical

resistivity of layered formations with invasion can be performed in considerably less

computer time than standard nonlinear inversion strategies, thereby providing an

opportunity for real-time, on-site estimation of 2D distributions of electrical conductivity.

8.2.1 Nonlinear Optimization

The quadratic (least-squares) cost function used in this section is written as

( ) ( ) 222 })({)(2 mmWmrdWx ⋅+−⋅= mdC λχ , (8.1)

where:

m is the vector of unknown parameters,

280

)(mr is the residual vector constructed from the difference between the

measured data, d, and the simulated data, ( )mf , i.e.

( ) dmfmr −= )( , (8.2)

2χ is a prescribed value of quadratic data misfit to be enforced during

inversion, and

λ is a scalar Lagrange multiplier, or regularization parameter, that assigns a

relative value of importance to the two additive terms included in the cost

function.

In equation (8.1), ( )dWd and ( )mWm are matrix operators in data and model

space, respectively. Matrix ( )dWd is used to weigh a particular measurement with respect

to a previously estimated signal-to-noise ratio. The same matrix is commonly taken as the

square root of the inverse of the data covariance matrix. Another option is to construct

( )dWd in the form of a diagonal matrix with entries equal to the inverse of the

measurements. This is equivalent to redefining the residual vector in equation (8.2) as

( ) ( )( ) iiii ddfr /−= mm , (8.3)

where i=1, 2, …, dN , and dN is the dimension of the data vector (i.e. the number of

data).

Matrix ( )mWm in equation (8.1) is a function of the parameter vector, m, and is

used to stabilize the inversion and to guarantee a unique solution. There are several

options considered in this chapter. The simplest choice for ( )mWm is the identity matrix.

For this particular choice, equation (8.1) will enforce the minimization of the 2 norm of

the vector of model parameters, i.e.

281

( )2 , min= =m m m . (8.4)

Another choice considered for ( )mmW is the discretized version of the gradient

operator,∇ , namely,

min2 =∇m . (8.5)

The latter option naturally biases the minimization toward a solution with minimal spatial

roughness, and is often referred to as Occam’s razor (Constable et al., 1987, and de

Groot-Hedlin, 2000). On occasion, however, this choice can result in spurious

oscillations when m is discontinuous (Portniaguine and Zhdanov, 1999). In the discrete

case, ∇ can be written in matrix notation as

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

=∇

110

11011

0

. (8.6)

A variant of equation (8.5) consists of minimizing the difference between the unknown

model and a priori reference model, refm , in the 2 norm sense, i.e.

min2=− refmm . (8.7)

This last criterion is adopted in this chapter to obtain solutions to the inverse problem

with increasing degrees of spatial complexity. Specifically, an inversion is first

performed to estimate an optimal homogeneous background model. Such an optimal

background model is subsequently treated as the input reference model in the inversion of

an optimal 1D formation model. Finally, the optimal 1D formation model is input as the

reference model for the inversion of a 2D distribution of electrical conductivity.

282

A strategy commonly used to minimize the cost function in equation (8.1) is

based on Gauss-Newton linear recursions, given by

( ) ( ) ( ) ( )1; ; ; ;T T Tk k m m k k kλ +′ ′ ′ ′⎡ ⎤+ =⎣ ⎦J m d J m d W W m J m d r m d , (8.8)

where

( ) ( ) ( )kdk mJdWdmJ =′ ; , (8.9)

and

( ) ( ) ( ) ( )kdkkk mrdWmdmJdmr −′=′ ;; . (8.10)

In the above equations, J is the Jacobian of the residual vector, r. By making use of the

definition of r in equation (8.2), one obtains

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∂∂∂∂∂∂

∂∂∂∂∂∂∂∂∂∂∂∂

=

NMMM

N

N

mfmfmf

mfmfmfmfmfmf

///

//////

21

22212

12111

mJ , (8.11)

where M and N are the number of measurements and the number of model parameters,

respectively.

The above linear system of equations can be written as the least-squares solution

of the over-determined linear system of equations

⎥⎦

⎤⎢⎣

⎡ ′=⎥

⎦

⎤⎢⎣

⎡ ′

0r

mW

J

mλ. (8.12)

In equation (8.1), the regularization parameter, λ, can change from iteration to

iteration depending on the criterion enforced for the rate of decrease of data misfit with

respect to iteration number. It can also be chosen through a line search to minimize the

first term in the cost function given by equation (8.21) (Constable et al., 1987). However,

a basic criterion for choosing λ is that it should lie in the interval (Anderson, 2001)

283

m mmax(small ) min(large )μ λ μ≤ ≤ , (8.13)

where mμ are the eigenvalues of the matrix JJ ′′T . The second part of the inequality in

equation (8.13) guarantees that the spectral content of the inverse operator remains

unaltered, while the first part of the inequality regularizes the inversion by suppressing

the null space of the inverse operator.

In the inversion examples presented in this section, equations (8.3) and (8.5) are

implemented as the default options.

8.2.2 An Inner-loop and Outer-loop Optimization Technique

In Torres-Verdín et al. (1999), a dual-grid inversion technique was developed to

invert subsurface DC resistivity data. A subset of the original finite-difference grid was

used to construct the auxiliary forward solver. The same objective can be achieved using

a stand-alone approximation of the original forward problem. This technique is here

termed “inner-outer loop” inversion. It consists of (a) an exact forward solver in the

outer-loop used to evaluate the data misfit, and (b) a fast, approximate forward solver

used within the inner-loop to perform the nonlinear optimization. The exact forward

solvers and approximate forward solvers have been detailed in Chapter 4.

This chapter uses an adaptation of the inversion strategy proposed by Torres-

Verdín et al. (1999) in which an approximate forward solver is constructed with Born,

EBA, and PEBA approximations. The computational overhead associated with the

calculation of the Jacobian matrix is considerably reduced within the inner loop because

of the approximate nature of the forward solver. Actually, the Jacobian matrix is only the

by-product of the corresponding forward run, which means that one forward run can

produce both the simulation of the measurements and the Jacobian matrix. After

284

convergence is achieved within the inner-loop, the estimated electrical conductivity

model is transferred to the outer loop. At this point, the outer loop performs an exact

numerical simulation of the measurements and quantifies the corresponding data misfit. If

the computed data misfit is within acceptable bounds, then the inversion stops, and the

current inverted conductivity model is accepted as the final solution. Otherwise, the

computed data misfit is transferred back to the inner-loop optimization. This process

repeats itself until the prescribed stopping criterion is met.

Following the notation of Torres-Verdín et al. (1999), the approximate forward

operator can be written as

)ˆ(ˆ mhd = , (8.14)

whereas the approximate inverse operator can be expressed as

)ˆ(ˆ dhm 1−= . (8.15)

This last equation is constructed in strict analogy with the original forward operator,

namely,

)(1 dfm −= . (8.16)

In equations (8.14) and (8.15), d is a modified measurement vector, m is a model vector

derived from the inverse operator 1h − , f is the exact forward operator, and m is the model

parameter vector.

Using the notation introduced in the previous paragraph, the inner-outer loop

optimization technique can be graphically described with the flowchart shown in Figure

8.1.

On convergence,

mmm ==+ kk ˆˆ 1 , (8.17)

285

where m is the solution from the operator 1−f .

8.2.3 Computation of the Jacobian Matrix Based on the PEBA

It was mentioned in the previous section that, for the PEBA, one forward run

yields both the simulated measurements and the Jacobian matrix. This section describes

how this is done mathematically.

By taking the derivative with respect to lσΔ (the l-th parameter) in equation

(4.23) and by making use of the chain rule, one obtains

( ) ( )0

2 , ; , ,l

z z

l l

r r

EH HE

g z z E z d dz

φ

φ

φτ

σ σ

ρ ρ ρ ρρ

∂∂ ∂=

∂Δ ∂ ∂Δ

′ ′ ′ ′ ′ ′= ∫

( ) ( )10

,2 , ; , .i

N

i r ri l

E zg z z d dzφ

τ

ρσ ρ ρ ρ

ρ σ=

′ ′∂′ ′ ′ ′+ Δ

∂Δ∑ ∫ (8.18)

δ<−dmf )(

m

Inner Loop h(m)

Outer Loop f(m)

Nonlinear Minimization

Initial Guess

YES

NO

dmfmhd +−= )()(ˆ

StopBegin

Figure 8.1: Flowchart of the inner-loop and outer-loop optimization algorithm.

286

Therefore, the problem of computing the entries of the Jacobian (sensitivity)

matrix centers about the computation of ( )

l

zEσρφ

Δ∂

′′∂ ,, which in turn depends on the

specific approximation used to compute the electric field internal to EM scatterers. For

the case of the PEBA, one approaches this problem by first taking the derivative with

respect to lσΔ in equation (4.75), and then by making use of equation (4.76), to obtain

( )( )

( ) ( ), ,1 ,

,l l

E z d zE z

z dφ

φ

ρ ρρ

σ ρ σ∂ Λ

= −∂Δ Λ Δ

, (8.19)

where

( ) ( ) ( ),, ; , ,

lb

l

d zi g z z E z d dz

d φτ

ρωμ ρ ρ ρ ρ

σΛ

′ ′ ′ ′ ′ ′= −Δ ∫ , (8.20)

and Λ is given by equation (4.76).

It is important to point out that the inversion parameters can be the conductivity

itself or the logarithm of the conductivity. Assume that σ denotes the conductivity, then

its logarithm q is given by

lnq σ= . (8.21)

Next, assume that d denotes one component of the data; the derivative of d with respect

to q is then given by

d dq q

σσ

∂ ∂ ∂=

∂ ∂ ∂. (8.22)

By noting that qσ σ∂=

∂, one finally obtains

d dq

σσ

∂ ∂=

∂ ∂. (8.23)

287

Equation (8.23) shows that the Jacobian matrix with respect to the logarithm of the

conductivity is simply the product of the Jacobian matrix with respect to the conductivity

itself and the corresponding conductivity values.

8.2.4 Resistivity Imaging Examples

Induction tool configurations assumed in this section are shown in Figure 8.2. The

first tool, identified as Tool No. 1, consists of one transmitter and three receivers. This

configuration is equivalent to three single-transmitter single-receiver arrays with

distances between transmitter and receiver equal to 15 inches, 27 inches, and 72 inches,

respectively. It is also assumed that this tool can operate at 25 KHz, 50 KHz, and 100

KHz.

The second tool, identified as Tool No. 2, is composed of three arrays, with each

array consisting of one transmitter and two receivers. In this configuration, the transmitter

is common to all the arrays. For each array, the two receivers are connected in phase

opposition to reduce transmitter-receiver coupling; the spacing between them is set to 2

inches. Thus, the measurement performed by each of the three arrays is defined as the

difference between the measurements performed by the two receivers. The separations

between transmitter and receiver are equal to 15 inches, 27 inches, and 72 inches,

respectively. In this particular case, transmitter-receiver separation is measured as the

distance between the transmitter and the midpoint between the two receivers. Tool No. 2

is assumed to operate at 25 KHz, 50 KHz, and 100 KHz.

For the two tool configurations described above, measurements consist of

complex-valued vertical magnetic fields and are assembled into entries of the data vector,

d. Real and imaginary parts of the measured vertical magnetic field are assembled

288

separately into d. Unknown parameters consist of formation conductivity values,

assembled into entries of the parameter vector, m. A filter is used to enforce the positivity

of the conductivity values before the conductivity profile is used in the next iteration. As

described in previous sections, electrical conductivity values contained in vector m can

also be described in terms of a logarithmic transformation. This transformation is widely

used as it explicitly enforces positivity of the estimated values of electrical conductivity.

However, occasionally the same transformation will exacerbate the nonlinear relationship

between measurements and model parameters, hence causing convergence difficulties to

the inversion as well as artificial local minima in the cost function.

For the inversion exercises described in this section, the outer loop is constructed

using an accurate integral equation solver. On the other hand, the inner loop of the

inversion algorithm has been designed with three possible options, namely, the Born

approximation, the EBA, and the PEBA. For the sake of conciseness, however, the

inversion exercises reported in this chapter are performed exclusively with the use of the

the PEBA. Also, inversion exercises for each model example are performed assuming

1D, 1.5D, or 2D spatial distributions of electrical conductivity. Notice that because of the

existence of the borehole the terminology “1D” here is different from a 1D layered

formation. A 1.5D inversion is initialized with a previously estimated 1D distribution of

electrical conductivity. Likewise, a 2D inversion is initialized with a previously estimated

1.5D distribution of electrical conductivity. Here, the nomenclature 1.5D refers to a 1D

formation with fixed zones of radial invasion. The radial location of the invasion zones

remains fixed, but there is no limit to the number of invasion zones. Clearly, as the

number of radial zones increases, the conductivity model effectively becomes a 2D

289

spatial distribution of electrical conductivity. The idea behind the use of a 1.5D

conductivity distribution is to improve the sensitivity of the data to the true formation

conductivity values far from the borehole wall. The inversion algorithm will estimate

conductivity values within each invasion zone as well as within the virgin zone. On the

other hand, the term 2D refers to an inversion performed to estimate a fully discrete 2D

spatial distribution of electrical conductivity.

In all of the inversion exercises reported in this section, a smoothing criterion is

enforced in the estimation of spatial distributions of conductivity. Likewise, data

weighting is performed with a diagonal matrix dW constructed with the inverse of data

values. All of the exercises reported in this chapter were performed using multi-

frequency data simulated at 25 KHz, 50 KHz, and 100 KHz.

8.2.4.1 One-Dimensional Formation Model

The first formation model considered in this section is the “chirp”-like layer

sequence shown in Figure 8.3. Synthetic induction log data were generated with a depth

sampling rate of 0.15 m, and consisted of 120 profiling points for each array. Borehole

conductivity and borehole radius are kept constant at 0.5 S/m and 0.1 m, respectively.

Two sets of synthetic data were generated for this formation model, which correspond to

Tool No. 1 and Tool No. 2, respectively. Inversions are performed in a sequential manner

to estimate 1D, 1.5D and 2D conductivity distributions for this formation model.

For the 1D inversion, the formation is discretized into 35 thin layers. This

inversion is initialized with a homogeneous formation model. Figure 8.4 shows the

estimated 1D profiles of electrical conductivity as a function of iteration. For comparison,

the original model is plotted on the same figure. Data used as input to the inversion were

290

simulated assuming noise-free measurements acquired with Tool No. 2. Figure 8.5 shows

the misfit of the imaginary part of the magnetic field data acquired with array no. 2 at 25

KHz (for illustration purposes, data misfit results are only shown for a particular array at

a particular frequency). The inverted profile of electrical conductivity already resembles

that of the original profile at the 4th iteration of the outer loop.

Figure 8.6 shows the schedule of convergence for both outer and inner loops.

Presence of “jumps” in the schedule of convergence within the inner-loop is a common

phenomenon for the inversion algorithm used in this chapter. They are most likely due to

the approximate nature of the inner loop minimization.

The addition of 2% zero-mean, Gaussian random noise to the input data causes

only slight changes to the inverted conductivity profile. This profile is shown in Figure

8.7.

1.5D and 2D distributions of electrical conductivity were also estimated from

induction data simulated for the 1D chirp-like formation model. The objective of this

inversion exercise was to assess the resolution of array induction data in the vertical and

radial directions. Figure 8.8 shows the 1.5D conductivity model estimated from borehole

induction data contaminated with 2% zero-mean, additive Gaussian random noise. The

inversion assumed 8 fixed invasion fronts, located at radial distances of 0.2, 0.3, 0.4, 0.5,

0.6, 0.7, 0.8, and 0.9 meters, respectively. The estimated vertical boundaries of the layers

are in good agreement with the original boundaries. Figure 8.8 also indicates that radial

resolution decreases from front to front, especially within thin layers.

Figure 8.9 shows the 2D distribution of electrical conductivity estimated from

induction data simulated for the 1D chirp-like formation model. The formation was

291

discretized into 2100 small cells. This number of cells is only slightly larger than the

number of data input to the inversion. The inverted distribution of electrical conductivity

exhibits decreasing resolution in the radial direction.

The previous inversion exercises were performed using data simulated for Tool

No. 2. Figures 8.10, 8.11, and 8.12 show the 1D, 1.5D, and 2D distributions of electrical

conductivity, respectively, inverted from induction data simulated for Tool No. 1 for the

same chirp-like formation model (Figure 8.3). The data are also contaminated with 2%

zero-mean, additive random Gaussian noise. Inverted conductivity distributions are very

similar to those obtained assuming data acquired with Tool No. 2.

8.2.4.2 Two-Dimensional Formation Model

A more detailed and spatially complex formation model is graphically described

in Figure 8.13. This model exhibits invasion and includes relatively large conductivity

contrasts. Induction logs were simulated at a depth sampling rate of 0.15 m, with a total

of 140 profiling locations for each array. Borehole conductivity and borehole radius are

kept constant at the values 0.5 S/m and 0.1 m, respectively. Two sets of synthetic data are

generated, which correspond to Tool No. 1 and Tool No.2, respectively. 1D, 1.5D and 2D

conductivity profiles are sequentially inverted for this formation model. A smoothing

criterion is used in the inversions together with a data-weighting matrix equal to a

diagonal matrix with entries equal to the reciprocal of the measurements.

Figure 8.14 shows the 1D profiles of electrical conductivity inverted from

synthetic data acquired with Tool No. 1 and Tool No. 2, respectively. In both cases, data

were contaminated with additive, zero-mean 2% random Gaussian noise. The formation

292

was discretized into 42 thin layers. There is great similarity between the two inverted

profiles of electrical resistivity. However, because induction data are naturally more

sensitive to the near-borehole region, inverted conductivity values are much closer to

those of the invaded zone. For instance, the conductivity value estimated for the first

invaded formation is around 0.4 S/m, which is closer to the invaded zone value (0.6 S/m)

than to the corresponding value for the virgin zone. The influence of near-borehole

conductivity values is much more critical in the case of larger conductivity contrasts, e.g.

within the 2nd, 3rd, and 4th invaded layers. This behavior poses significant problems to the

estimation of 1D conductivity values in the presence of invasion (see, for instance, Gao,

2003, and San Martin, et al., 2001). Even with sophisticated corrections, these effects

may still linger in the estimated values of electrical conductivity.

Figures 8.15 and 8.16 show the 1.5D and 2D spatial distributions of electrical

conductivity, respectively, inverted from data simulated for the model described in Figure

13. In the two cases, induction data input to the inversion were simulated numerically

assuming acquisition with Tool No. 1, and subsequently contaminated with 2% zero-

mean additive Gaussian random noise. For the estimation of a 1.5D spatial distribution of

conductivity, the inversion was performed using 12 fixed radial regions of invasion,

located at radial distances of 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, and 1.3 m,

respectively. On the other hand, the estimation of a 2D distribution of electrical

conductivity was performed enforcing a discretization with 2520 cells. In the two cases,

the estimated values of electrical conductivity in the invasion zone are in good agreement

with the original values described in Figure 8.13. Inverted vertical boundaries are also in

good agreement with the location of vertical boundaries in the original model. However,

293

the inverted 1.5D spatial distribution shows an anomalous radial transition zone before

reaching the virgin zone. The inverted 2D distribution of electrical conductivity exhibits a

tendency toward diminishing spatial resolution away from the borehole wall.

T

A1

A2

A3

Tool 1

T

A1

A2

A3

Tool 2

Figure 8.2: Schematic of the two array induction tools assumed in this paper. Both Tool No. 1 and Tool No. 2 consist of 3 arrays; the difference being that each array consists of one transmitter and one receiver for Tool No. 1, and of one transmitter and two receivers for Tool No. 2. Separations betweentransmitter and arrays of receivers are 15 inches, 27 inches, and 72 inches, respectively. Both tools operate at 25 KHz, 50 KHz, and 100 KHz.

294

0 1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1C

ondu

ctiv

ity (S

/m)

Depth(m)

Figure 8.3: One-dimensional chirp-like formation model used in the inversion. The widths of the 4 resistive beds are 0.3, 0.6, 1.2, and 2.4 meters, respectively.

295

0 1 2 3 4 5 6 7 8 9 100

1

2 OriginalInverted

0 1 2 3 4 5 6 7 8 9 100

1

2

0 1 2 3 4 5 6 7 8 9 100

1

2

Con

duct

ivity

(S/m

)

0 1 2 3 4 5 6 7 8 9 100

1

2

0 1 2 3 4 5 6 7 8 9 100

1

2

depth (m)

Iteration No. 1

Iteration No. 2

Iteration No. 3

Iteration No. 4

Iteration No. 5

Figure 8.4: Vertical profiles of electrical conductivity inverted as a function of the outer-loop iteration number. Inversion results for iterations 1, 2, 3, 4, and 5 are shown on the figure. Each outer-loop iteration consists of 4 inner-loop iterations. The inversions were performed using noise-free data “acquired” with Tool 2 at 25KHz, 50 KHz, and 100 KHz.

296

0246810121416180

5x 10

-4 Misfit with iterations

246810121416180

5x 10-4

0246810121416180

5x 10-4

Imag

(Hz)

(A/m

)

0246810121416180

5x 10-4

Originalinverted

0246810121416180

5x 10-4

Depth (m)

Initial Guess

iteration No. 1

Iteration No. 2

Iteration No. 3

Iteration No. 4

Figure 8.5: Data misfit for array-2 of Tool No. 2 at 25 KHz (imaginary part) as a function of the number of outer-loop iterations. Data misfit results for iterations 1, 2, 3, 4, and 5 are shown on the figure. Each outer-loop iteration consists of 4 inner-loop iterations. The inversions were performed using noise-free data “acquired” with Tool No. 2 at 25 KHz, 50 KHz, and 100 KHz.

297

0 2 4 60

0.05

0.1

0.15

0.2

0.25

Iteration No.

Dat

a M

isfit

Outer Loop

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Iteration No.

Dat

a M

isfit

Inner Loop

Figure 8.6: Plots of data misfit as a function of iteration number for the inversion results described in Figure 4. The left panel shows values of data misfit with respect to outer-loop iteration number. Data misfit values as a function of inner-loop iteration number are shown in the right panel.

298

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

depth (m)

Con

duct

ivity

(S/m

)

originalinverted

Figure 8.7: Vertical profile of electrical conductivity inverted from Tool No. 2 array-induction data simulated for the 1D chirp-like model and contaminated with zero-mean, 2% random Gaussian additive noise. The inversion was performed with data acquired at 25 KHz, 50 KHz, and 100 KHz.

299

Figure 8.8: One-and-half (1.5D) electrical conductivity model inverted from array induction data simulated for the 1D chirp-like formation model withinvasion. Data input to the inversion were simulated numerically for Tool No. 2 and were subsequently contaminated with zero-mean, 2% random Gaussian additive noise. Eight fixed piston-like invasion fronts were assumed in the inversion, with radii of invasion equal to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 meters, respectively. The inversion was performed using data acquired at 25 KHz, 50 KHz, and 100 KHz.

300

Figure 8.9: Two-dimensional distribution of electrical conductivity inverted from array induction data simulated for the 1D chirp-like formation model with invasion. Data input to the inversion were simulated numerically for Tool No. 2 and were subsequently contaminated with zero-mean, 2% random Gaussian additive noise. The inversion was performed with data acquired at 25 KHz, 50 KHz, and 100 KHz.

301

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

depth (m)

Con

duct

ivity

(S/m

)

OriginalInverted

Figure 8.10: Vertical profile of electrical conductivity inverted from Tool No. 1 array-induction data simulated for the 1D chirp-like model and contaminated with zero-mean, 2% random Gaussian additive noise. The inversion was performed with data acquired at 25 KHz, 50 KHz, and 100 KHz..

302

Figure 8.11: Electrical 1.5D conductivity model inverted from array induction data simulated for the 1D chirp-like formation model with invasion. Data input to the inversion were simulated numerically for Tool No. 1, and were subsequently contaminated with zero-mean, 2% random Gaussian additive noise. Eight fixed piston-like invasion fronts were assumed in the inversion, with radii of invasion equal to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 meters, respectively. The inversion was performed with data acquired at 25 KHz, 50 KHz, and 100 KHz.

303

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 5 10 15 20

0

10

20

30

40

50

60

70

80

90

100

Radial Distance (10-1 m)

Dep

th (1

0-1 m

)

Figure 8.12: Two-dimensional distribution of electrical conductivity inverted from Tool No. 1 array induction data simulated for the 1D chirp-like formation model with invasion. Data input to the inversion were simulated numerically and were subsequently contaminated with zero-mean, 2% random Gaussian additive noise. Inverted 2D conductivity image for the chirp-like 1-D formation model. 2% Gaussian random noise is added. The inversion was performed with data acquired at 25 KHz, 50 KHz, and 100 KHz.

304

1 S/m

0.6S/m 0.1S/m

1 S/m

0.05S/m 0.2 S/m1 S/m

2 S/m 0.1S/m

0.05S/m 0.6S/m

1 S/m

Bor

ehol

e

Figure 8.13: Graphical description of the 2D formation model constructed to test the inversion algorithm. From top to bottom, the thickness of the 8 layers is 2.1, 2.1, 1.2, 1.8, 0.9, 1.5, 1.8, and 1.2 meters, respectively. Invasion radii for the 4 invaded layers are 0.6, 0.9, 0.6, and 0.9 meters, respectively, from top to bottom.

305

0 2 4 6 8 10 120

0.5

1

1.5co

nduc

tivity

(S/m

)

0 2 4 6 8 10 120

0.5

1

1.5

cond

uctiv

ity(S

/m)

Depth(10-1m)

Tool2

Tool1

Figure 8.14: One-dimensional profile of electrical conductivity inverted from array induction data simulated for the 2D formation model shown in Figure 8.13. Data input to the inversion were contaminated with zero-mean, 2% random Gaussian additive noise. The upper panel shows the conductivity profile inverted from data “acquired” with Tool No. 2, and the lower panel shows the conductivity profile inverted from data “acquired” with Tool No. 1. The inversion was performed with data acquired at 25 KHz, 50 KHz, and 100 KHz.

306

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 2 4 6 8 10 12 14 16 18 20

0

20

40

60

80

100

120

Radial Distance (10-1m)

Dep

th (1

0-1m

)

Figure 8.15: Electrical 1.5D conductivity model inverted from array induction data simulated for the 2D formation model with invasion. Data input to the inversion were simulated numerically for Tool No. 1, and were subsequently contaminated with zero-mean, 2% random Gaussian additive noise. Twelve fixed piston-like invasion fronts were assumed in the simulations, with radii of invasion equal to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, and 1.3 meters, respectively. The inversion was performed with data acquired at 25 KHz, 50 KHz, and 100 KHz.

307

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 2 4 6 8 10 12 14 16 18 20

0

20

40

60

80

100

120


Dep

th (1

0-1m

)

Figure 8.16: Two-dimensional distribution of electrical conductivity inverted from array induction data simulated for the 2D formation model shown in Figure 13. Data input to the inversion were simulated numerically for Tool No. 1 and were subsequently contaminated with zero-mean, 2% random Gaussian additive noise. The inversion was performed with data acquired at 25 KHz, 50 KHz, and 100 KHz.

308

8.3 TWO DIMENSIONAL RESISTIVITY INVERSION FOR CONDUCTIVITY MODELS WITH

MULTI-FRONT MUD-FILTRATE INVASION

Mud-filtrate invasion frequently alters the electrical conductivity of the zones

around the borehole whereby induction log measurements depart from the original

formation conductivity. Estimating true resistivity values of the original (uninvaded) rock

formation is necessary for accurate reservoir evaluation. On the other hand, mud-filtrate

invasion also can be viewed as a down-hole flow experiment, which could be used to

estimate multi-phase flow parameters, such as relative permeability, filtrate loss, initial

water saturation, etc. from multi-channel resistivity measurements (Ramakrishnan and

Wilkinson, 1999). Thus, information about the invasion front is also important for

formation evaluation.

The previous section described a procedure to estimate radial resistivity profile

using an imaging method based on an inner-outer loop optimization technique (Gao and

Torres-Verdín, 2003). For comparison, Figure 8.17 replots Figures 8.13 and 8.15 side by

side, in which the left panel is the original 2D formation model and the right panel is the

inverted image. The inverted resistivity image clearly shows that the overall formation

structure is properly estimated. However, due to the rapidly reduced sensitivity of the

measurements in the radial direction, the conductivity values inverted in the original zone

are not accurate enough for the reliable estimation of fluid saturation. Imaging does offer

a qualitative way to assess whether a rock formation is invaded by mud-filtrate invasion.

To estimate the conductivity of the original formation more accurately, in this

section we assume a blocky resistivity structure, i.e., a multi-front mud-filtrate invasion

model. Inversion parameters now include both the electrical parameters (conductivity and

309

dielectric constant) and the geometrical parameters, such as invasion fronts and layer

boundaries.

8.3.1 Constrained Nonlinear Least-Squares Inversion

The theory of nonlinear optimization has been described in sections 8.2 and 8.3.

Notice that the smoothing option is not used here because of the assumption of a blocky

model structure. The inversion algorithm is based on a Gauss-Newton procedure,

stabilized with subspace minimization and a truncated QR method for those cases where

the sensitivity matrix is rank-deficient. This treatment does away with the need to choose

a specific regularization parameter. The detailed algorithm can be found in Lindstrom

and Wedin (1984).

The computation of the Jacobian matrix is described in section 8.3.3, and the

inner-loop and outer-loop optimization is not used here since the PEBA already provides

accurate simulations of the measurements.

8.3.2 The Computer Code

The code developed in this section is particularly designed for blocky formation

models, such as those of multi-front mud-filtrate invasion. The number of radial fronts

can be specified by the user. Electrical parameters (conductivity and dielectric constant)

for each block and the radial locations of invasion fronts are treated as unknown

parameters. Layer boundaries are assumed known from other information; otherwise,

they can be inverted via an inversion procedure that assumes a borehole but no mud-

filtrate invasion. Likewise, the algorithm can enforce several levels of inversion

sequentially, and take the inversion results from the previous level as the initial guess for

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the current level. This option is of great importance when parameters such as layer

boundary and/or invasion front are estimated by the inversion. In total, there are four

levels of inversion, which are described as follows:

LEVEL 1: Inversion for conductivity models with borehole and no mud-filtrate

invasion. Layer boundaries are fixed.

LEVEL 2: Inversion for conductivity models with borehole and no mud-filtrate

invasion. Layer boundaries are entered to the inversion as unknown

parameters.

LEVEL 3: Inversion for conductivity models with borehole and mud-filtrate

invasion. Invasion radii are fixed.

LEVEL 4: Inversion for conductivity models with borehole and mud-filtrate

invasion. Invasion radii are entered to the inversion as unknown

parameters.

8.3.3 Inversion Examples

The code developed in this section supports two kinds of induction tool

configurations (Figure 8.2). Moreover, the tool is frequency-selective, which means that

some particular array only works at some particular frequency. The specific configuration

of induction tool used in subsequent inversion examples is shown in Figure 8.18. This

tool consists of four arrays with spacings of 15, 27, 54 and 72 inches, respectively. The

spacing is measured as the distance between the transmitter and the midpoint between the

two receivers. The radius of the coils is assumed to be 0.03 m.

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8.3.3.1 Inversion of a 2D Layered Formation with Borehole and No Invasion

Figure 8.19 shows a 1D layered formation model with borehole and no invasion

(note Figure 8.19 is the same figure as Figure 8.3, but oriented in a different direction).

We use this model to test the inversion algorithm for estimating the electrical

conductivity of each layer as well as the location of layer boundaries. At first, we fix the

layer boundaries and perform the LEVEL 1 inversion. We test the robustness of the

inversion using different levels of noise. Figure 8.20 shows inversion results and the

relative errors in the inverted conductivities for 0, 1, 2, 5, 10, and 20 percent zero-mean

Gaussian random noise added to the data, respectively. Results from this exercise indicate

that the inversion algorithm provides accurate results with up to 10% noise added to the

input data (except the 0.3 m thick layer). Acceptable results are also obtained with 20%

noise added to the data (10 % error for most cases). The biggest error is associated with

the 0.3 m thick layer, which is about the limit of the vertical resolution for most borehole

induction tools.

Next, we invert conductivity values and layer boundaries simultaneously, which is

the LEVEL 2 inversion. The code does this in two steps. Step 1: By fixing the layer

boundaries, we estimate the best conductivity of each layer to match the data in a certain

number of iterations. Step 2: By taking the inversion results from Step 1 as the initial

guess, we perform the simultaneous inversion for the conductivity and layer boundaries.

Figure 8.21 shows inversion results for 0, 1, 2, 5, 10, and 20 percent zero-mean Gaussian

random noise added to the data, respectively. Results indicate that the inversion provides

accurate results irrespective of the noise level in the data (except the 0.3 m thick layer),

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both for conductivities and layer-boundary locations. When 20% noise is added to the

data, the inverted boundary for the 0.3 m thick layer is not correct.

Figure 8.22 shows the RMS (Root Mean Square) error misfit versus iteration

number for different levels of noise added to the data. The left panel of the figure shows

the results from the LEVEL 1 inversion, while the right panel of the figure shows the

results from the LEVEL 2 inversion. Because the initial layer boundaries are slightly

offset from the true layer boundaries, the LEVEL 1 inversion does not converge to an

error misfit close to the noise level. For the LEVEL 2 inversion, layer boundaries and

conductivities are accurately inverted, and the misfit errors are very close to the

corresponding noise levels.

8.3.3.2 Inversion of a 2D Formation that Includes Borehole and Mud-filtrate

Invasion

The formation model including both borehole and mud-filtrate invasion is shown

in Figure 8.13. The borehole conductivity is assumed to be 0.5 S/m. For the inversions

reported in this section, we assume that layer boundaries are known from a priori

information. We test the inversion algorithm for estimating the conductivity value in each

block as well as the invasion fronts. Again, to test the validity and robustness of the

inversion, we first fix the invasion fronts and add different levels of noise to the data to

perform the LEVEL 3 inversion. Table 8.1 summarizes the inversion results obtained

for 0, 1, 2, 5, 10, and 20 percent zero-mean Gaussian random noise added to the data,

respectively. Results indicate that the inversion yields accurate results for up to 10%

noise in the input data. Reasonable results are obtained with up to 20% noise added to the

data.

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Next, we invert the electrical conductivity of each block and the invasion fronts

simultaneously (the LEVEL 4 inversion). In a similar fashion to the LEVEL 2

inversion, this is performed in two steps: (1) By fixing the invasion fronts, we estimate

the conductivities that best match the data in a certain number of iteration; (2) By taking

the inversion results of Step 1 as the initial guess, we invert the conductivities and

invasion fronts simultaneously. Table 8.2 shows inversion results for 0, 1, 2, 5, 10, and

20 percent zero-mean Gaussian random noise added to the data, respectively. This

exercise shows that the inversion provides relatively accurate estimates of radial invasion

fronts with up to 10% noise added to the input data. However, invasion fronts are more

difficult to estimate than layer boundaries. This is explained because of the rapidly-

reduced spatial sensitivity of the data in the radial direction.

Figure 8.23 shows the RMS (Root Mean Square) error misfit versus iteration

number for different levels of noise added to the data. The left panel of the figure shows

results from the LEVEL 3 inversion, while the right panel of the figure shows results

from the LEVEL 4 inversion. Because the initial invasion fronts are slightly offset from

the true invasion fronts, the LEVEL 3 inversion does not converge to an error misfit

close to the noise level. On the other hand, the LEVEL 4 inversion allows the

simultaneous inversion of invasion fronts and conductivity values, and the final error

misfit is very close to the corresponding noise level.

8.4 CONCLUSIONS

This chapter reviewed the theory of nonlinear inversion and developed novel

algorithms for the inversion of multi-frequency array induction data. Two types of

inversion were considered in this chapter, namely, Resistivity Imaging (RIM) and

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Resistivity Inversion (RIN). The RIM assumes a continuous spatial distribution of

resistivity, whereas the RIN is based the assumption of a blocky resistivity distribution.

An inner-loop and outer-loop optimization technique was developed for the RIM.

Numerical examples show that this technique can be used for the efficient and stable

inversion of 1D, 1.5D, and 2D spatial distributions of electrical conductivity in the

presence of noisy data. The inversion algorithm also lends itself to the joint inversion of

multi-frequency data in a way that permits selective and progressive deepening of the

zone of response away from the borehole wall. It was shown in this chapter that 1.5 and

2D distributions of electrical conductivity could be estimated using a sequence of

inversions with increasing degrees of spatial complexity. This approach naturally biases

the inversion toward models that exhibit a radial structure typical of mud-filtrate

invasion.

Based on formation models constructed with blocky structures, a RIN algorithm

was developed for inverting the electrical conductivities of each block and the

corresponding invasion fronts. The code is efficient in that the most time-consuming part,

the computation of the Jacobian matrix, is approached simultaneously with the simulation

of the measurements and hence requires only one forward simulation. A routine was also

developed to estimate the electrical conductivity of each layer and the layer boundaries.

Numerical examples show that accurate estimation of unknown parameters can be

performed even in the presence of a very high percentage of noise in the data (e.g. 20%).

Because of the rapidly reduced sensitivity of the measurements to radial variations of

electrical conductivity, radial invasion fronts are more difficult to estimate than layer

315

boundaries. However, reliable estimations can be achieved with relatively high values of

noise-to-signal ratios.

Comparison of inversion results obtained with the RIM and the RIN indicates that

the RIM is more suitable for the qualitative detection of mud-filtrate invasion, whereas

the RIN is superior to the RIM for the quantitative estimation of electrical conductivity in

the uninvaded region.

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0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 2 4 6 8 10 12 14 16 18 20

0

20

40

60

80

100

120


Dep

th (1

0-1m

)

1 S/m

0.6S/m 0.1S/m

1 S/m

0.05S/m 0.2 S/m

1 S/m 2 S/m 0.1S/m

0.05S/m 0.6S/m

1 S/m

Bo

reh

ole

Figure 8.17: Left Panel: the original 2D conductivity profile. Right Panel: the inverted 2D conductivity image. Data were generated as a subset of induction logging tool measurements acquired at 25 KHz, 50 KHz and 100 KHz; 2% additive Gaussian noise was added to the data before the inversion. From top to bottom, the thickness of the 8 layers is 2.1, 2.1, 1.2, 1.8, 0.9, 1.5, 1.8, and 1.2 meters, respectively. Invasion radii for the 4 invaded layers are 0.6, 0.9, 0.6, and 0.9 meters, respectively, from top to bottom. (From Gao and Torres-Verdín, 2003).

T

A1

A3

A4

A2

Figure 8.18: Array induction instrument assumed by the numerical examples considered in section 8.4. The instrument is a subset of the Array Induction Tool. Sounding frequencies are 25 KHz, 50 KHz, and 100 KHz.

317

Figure 8.19: A 1D formation model with borehole and without invasion. The borehole radius is 0.1 m, and the conductivity of the mud is 0.5 S/m. The shoulders are assumed to have a conductivity of 0.5 S/m.

318

Figure 8.20: Inversion results and relative error of the inverted conductivities for a 2D formation model with borehole and without invasion. The corresponding layer boundaries are assumed known and fixed. The borehole radius is 0.1 m, and the conductivity of the mud is 0.5 S/m. The shoulder is assumed to have a conductivity of 0.5 S/m. The initial guess for the conductivity of each layer is 0.2 S/m.

319

Figure 8.21: Inversion results and relative error of the inverted conductivities for a 2D formation model with borehole and without invasion. Both layer boundaries and conductivities are inverted simultaneously. The borehole radius is 0.1 m, and the conductivity of the mud is 0.5 S/m. The shoulder is assumed to have a conductivity of 0.5 S/m. Initial boundaries and conductivities areshown on the left figure.

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Figure 8.22: The RMS misfit error versus iteration number for different levels of noise added to the data. The formation model is shown in Figure 8.19. The left part of the figure shows the LEVEL 1 inversion results, while the right part of the figure shows the LEVEL 2 inversion results.

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Figure 8.23: The RMS misfit error versus iteration number for different levels of noise added to the data. The formation model is shown in Figure 8.13. The left part of the figure shows the LEVEL 3 inversion results, while the right part of the figure shows the LEVEL 4 inversion results.

322

Inverted Conductivity (S/m) Para. No.

Initial Guess (S/m)

True Value (S/m) 0%

Noise 1%

Noise 2%

Noise 5% Noise 10% Noise

20% Noise

1 0.2 1 0.999 0.998 0.996 0.990 0.972 0.903

2 0.2 1 0.996 0.991 0.985 0.967 0.926 0.792

3 0.2 0.6 0.600 0.599 0.599 0.596 0.589 0.568

4 0.2 0.1 0.100 0.102 0.104 0.108 0.112 0.098

5 0.2 1 1.000 1.002 1.003 1.006 1.004 0.955

6 0.2 1 1.000 0.997 0.994 0.981 0.948 0.801

7 0.2 5e-2 5.000e-2 4.989e-2 4.971e-2 4.883e-2 4.592e-2 3.012e-2

8 0.2 0.2 0.200 0.201 0.202 0.206 0.216 0.256

9 0.2 1 1.000 1.001 1.002 1.001 0.990 0.941

10 0.2 1 1.000 1.001 1.002 1.003 1.000 0.945

11 0.2 2 2.000 2.005 2.009 2.019 2.019 1.942

12 0.2 0.1 0.100 0.096 0.093 0.081 0.061 0.024

13 0.2 5e-2 5.000e-2 5.034e-2 5.064e-2 5.122e-2 5.110e-2 4.559e-2

14 0.2 0.6 0.600 0.600 0.601 0.602 0.600 0.593

15 0.2 1 1.000 1.005 1.010 1.022 1.033 1.016

16 0.2 1 1.000 0.992 0.984 0.958 0.905 0.769

Table 8.1: Summary of inversion results for the 2D formation model with borehole and invasion for different values of noise level added to the data. One fixed invasion front is assumed for each layer. Odd numbering is used for the invaded zone, while even numbering is used for the uninvaded zone within the same layer.

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Inverted Conductivity (S/m) Cond No.

Initial Guess (S/m)

True Value (S/m) 0%

Noise 1%

Noise 2%

Noise 5%

Noise 10% Noise

20% Noise

1 0.2 1 1.003 1.019 1.016 0.998 0.974 0.912

2 0.2 1 0.999 0.994 0.991 0.973 0.929 0.804

3 0.2 0.6 0.601 0.597 0.597 0.590 0.582 0.546

4 0.2 0.1 0.099 0.093 0.098 0.085 0.092 0.047

5 0.2 1 0.999 1.000 1.000 0.998 0.984 0.922

6 0.2 1 1.004 1.000 1.000 0.992 0.978 0.864

7 0.2 5e-2 4.937e-2 4.952e-2 4.909e-2 4.704e-2 4.099e-2 1.943e-2

8 0.2 0.2 0.180 0.191 0.175 0.166 0.143 0.145

9 0.2 1 1.000 1.003 1.003 1.007 0.959 0.951

10 0.2 1 1.002 1.002 1.004 1.006 1.002 0.940

11 0.2 2 2.004 2.006 2.016 2.026 2.035 1.98

12 0.2 0.1 0.122 0.103 0.134 0.120 0.121 0.202

13 0.2 5e-2 4.973e-2 5.032e-2 5.049e-2 5.153e-2 5.109e-2 4.180e-2

14 0.2 0.6 0.574 0.599 0.556 0.582 0.542 0.401

15 0.2 1 0.995 1.005 1.006 1.014 1.017 0.953

16 0.2 1 1.001 0.994 0.971 0.908 0.793 0.664

Front No.

Initial (m)

Original (m) Inverted Invasion Fronts (m)

1 0.5 0.6 0.602 0.613 0.612 0.641 0.642 0.734

2 0.5 0.9 0.834 0.868 0.803 0.741 0.615 0.535

3 0.5 0.6 0.600 0.601 0.594 0.585 0.582 0.544

4 0.5 0.9 0.885 0.905 0.875 0.891 0.862 0.721

Table 8.2: Summary of the inversion results for the 2D formation model with borehole and invasion for different noise levels added to the data. One invasion front is assumed for each layer, and the odd numbering is used for the invaded zone, while the even numbering is used for the original zone of the same layer. The invasion fronts and the conductivity ofeach block are inverted simultaneously.

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Chapter 9: Summary, Conclusions and Recommendations

9.1 SUMMARY

The first objective of this dissertation was to develop efficient numerical

algorithms to simulate the response of borehole EM logging instruments, including array-

induction and tri-axial induction tools, in the presence of axisymmetric media and 3D

dipping and anisotropic rock formations. Our focus was placed on the integral equation

method, although the finite-difference method was also used for EM modeling in

axisymmetric media. In addition, full-wave and approximate techniques were developed

for EM modeling in both axisymmetric media and 3D dipping and anisotropic rock

formations.

For axisymmetric media, full-wave techniques developed in this dissertation

include the BiCGSTAB(L)-FFT, the BiCGSTAB(L)-FFHT method based on an integral

equation formulation, and a finite-difference method based on the PDE formulation of

Maxwell’s equations. Solenoidal and toroidal sources were considered and integrated into

one formulation. Numerical exercises showed that these three simulation techniques

provide accurate and efficient simulations in the presence of complex rock formation

models. We also developed two approximate techniques for EM modeling in

axisymmetric media, the PEBA and the Ho-GEBA. Both techniques are almost matrix

free, a feature that makes it possible to substantially expedite the numerical simulation of

EM measurements. The approximate techniques provide more accurate simulations than

the Born approximation and the EBA.

The development of EM numerical simulation techniques in the presence of 3D

dipping and anisotropic rock formations was the main thrust of this dissertation. Full-

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wave and approximate modeling techniques were developed for the simulation of EM

measurements acquired in this type of rock formations. The full-wave modeling

technique is based on the implementation of the MoM. However, for large-scale EM

problems, the naïve implementation of the MoM involves three insurmountable

difficulties: matrix filling time, memory storage, and matrix-system solving. To reduce

matrix-filling time, we developed analytical techniques to integrate the spatial dyadic

Green’s functions. On the other hand, to reduce memory storage requirements, and to

solve the complex linear system of equations in an efficient manner, we developed the

BiCGSTAB(L)-FFT technique using the concept of block Toeplitz matrices resulting

from the space-shift invariant property of the dyadic Green’s functions. The

BiCGSTAB(L)-FFT technique also reduced matrix-filling time, for only a few entries of

the MoM stiffness matrix need to be evaluated to perform the simulation. These features

rendered the BiCGSTAB(L)-FFT technique adequate to solve large-scale EM simulation

problems on a standard computer workstation.

We also developed approximate simulation techniques for efficient EM modeling

in the presence of 3D dipping and anisotropic rock formations: the SA, and the Ho-

GEBA. Numerical exercises showed that the SA and the Ho-GEBA provide substantially

more accurate EM simulations in electrically anisotropic media that the Born

approximation and the EBA at approximately the same computer efficiency. Table 9.1

compares the computer efficiency for these three 3D modeling techniques. To perform

the comparison, we ran the three codes on a PC that included a 3.2 GHz Pentium 4 Intel

CPU processor. The grid consisted of 64,000 nodes for all three cases. Notice that the

computer efficiency of the SA depends primarily on the number of blocks, and for the

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comparison given in Table 9.1, the number of blocks is 2400. We also remark that, in

theory, the BiCGSTAB(L)-FFT algorithm and the Ho-GEBA have no limitations in the

range of operation. However, the SA may be subject to frequency limitations because the

oscillatory nature of EM fields increases with increasing frequency.

Algorithm CPU time BiCGSTAB(L)-FFT 12 minutes/tool location

SA 30 seconds/tool location Ho-GEBA 13 seconds/ tool location/ order

The second objective of this dissertation was to develop advanced algorithms for

the inversion of multi-frequency array induction measurements. Two types of inversion

were considered in this dissertation: “Resistivity Imaging (RIM),” which is based on the

assumption of a spatially continuous resistivity distribution, and “Resistivity Inversion

(RIN),” which assumes a blocky spatial distribution of electrical resistivity. An inner-

loop, outer-loop optimization technique was developed to perform the inversion. The

basic inversion algorithm is based on nonlinear least-squares minimization with

regularized Gauss-Newton iterations. The Jacobian matrix required by the minimization

is computed via the PEBA. The PEBA makes it possible to compute the simulated

measurements and the Jacobion matrix simultaneously with only one forward run. We

demonstrated the applicability and robustness of the inversion algorithms on synthetic

multi-frequency array induction measurements corrupted with various amounts of

additive random noise.

Table 9.1: Comparison of the computer efficiency of the BiCGSTAB(L)-FFT, the SA and the Ho-GEBA. The number of nodes is equal to be 64,000 in all three cases, and the computer platform is a PC that includes a 3.2 GHz Pentium 4 Intel processor. The number of blocks for the SA is 2400.

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9.2 CONCLUSIONS

The following conclusions stem from this dissertation:

(1) The work in this dissertation shows that the integral equation method is not

only ideal for solving small-scale EM problems, but also useful for solving large-scale

EM problems, such as those arising in well logging applications involving dipping beds

and electrical anisotropy.

(2) Analytical techniques can be derived to expedite the evaluation of the integrals

of the 3D and 2D spatial dyadic Green’s functions. These techniques circumvent one of

the computational difficulties inherent to the MoM: matrix-filling CPU time.

(3) The FFT technique can be used to expedite the solution of EM problems. A

combination of the BiCGSTAB(L) algorithm and the FFT can reduce a large-scale EM

problem to a nearly matrix-free one, thereby reducing the total computational cost to

approximately ( )2O N log N , where N is the total number of spatial discretization cells.

This circumvents the following two computational difficulties inherent to the MoM:

computer memory storage and matrix-system solving. It also partially helps to reduce

matrix-filling time.

(4) Approximate modeling techniques are useful for both forward modeling and

inversion. They represent a compromise between efficiency and accuracy. This

dissertation developed several new approximate modeling techniques both for EM

modeling in axisymmetric media and for simulation in the presence of 3D dipping and

anisotropic media. Theoretical analyses and numerical exercises showed that developing

approximate techniques for EM modeling in electrically anisotropic media is possible and

necessary. One advantage of approximate modeling techniques is that they are useful for

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developing fast inversion algorithms. For example, the PEBA developed in this

dissertation allows one to compute both the simulated measurements and the Jacobian

matrix with only one forward simulation.

(5) The total electric field vector can be decomposed into the product of a smooth

component (vector) and a rough component (scalar). In addition, the rough component

can be expressed as a function of the background field. The SA was developed based on

this concept.

(6) The GS, the GEBA, and the Ho-GEBA can be readily adapted for EM

modeling both in axisymmetric media and in 3D dipping and anisotropic rock formations.

They not only provide much more accurate solutions than the Born approximation and

the EBA, but are also suitable for EM modeling in the presence of electrically anisotropic

media.

(7) The inner-loop and outer-loop minimization technique considered in this

dissertation is efficient for fast inversion, provided that both full-wave modeling

techniques and approximate modeling techniques are available.

(8) Inversion of multi-frequency array induction data suggests that the RIN is

superior to the RIM for the quantitative evaluation of in-situ hydrocarbon saturation.

9.3 RECOMMENDATIONS FOR FUTURE WORK

This dissertation focused on solving large-scale EM problems using the integral

equation method. When solving the integral equation via the MoM, we chose to use the

pulse function as basis function. The pulse function is simple but does not preserve some

of the properties of EM fields. For example, on the edge of the cell, the normal

component of the electric current should be continuous, but the pulse function does not

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explicitly enforce such a condition. Thus, use of pulse basis functions may affect the

accuracy of the simulations; otherwise more discretization cells are needed to reduce this

effect. Future work is envisioned to use alternative basis functions, such as the rooftop

functions (Glisson and Wilton, 1980) defined on rectangular sub-domains, and the RWG

basis functions (Rao, Wilton and Glisson, 1982) defined on triangular sub-domains. The

rooftop functions are well suited for modeling that includes geometries that conform to

Cartesian coordinates, while the RWG functions are capable of modeling flat-face

approximations of arbitrary geometries. The advantage of these two basis functions is

that they are defined on two neighboring sub-domains and the unknown quantity is

associated with the common edge between the two sub-domains (Gurel et al., 1997). On

this common edge, the normal component of the current is continuous and has a constant

value, while on the other edges the current does not have a normal component,

whereupon no line charges exist at the boundaries of the basis functions. Notice that since

these last two basis functions are face-based, special treatments may be needed to

represent a general 3D function in terms of such basis functions.

We developed full-wave and approximate modeling techniques to simulate the

response of tri-axial induction tools in the presence of 3D dipping and anisotropic rock

formations. However, this dissertation did not consider the study of multi-component

induction measurements in a borehole environment (Wang et al., 2001). Future work can

focus on this subject. The topic can cover near-zone effects on coplanar and coaxial

measurements, such as those to the borehole (size, mud type), mud-filtrate invasion,

shoulder beds, and tool eccentricity. In addition, cross-bedding anisotropy (Wang and

Georgi, 2004) and fractured media are important topics of consideration for formation

evaluation.

Another possible extension of this dissertation is the inversion of multi-

component induction data. One-dimensional inversion (Wang et al., 2003; Lu and

Alumbaugh, 2001) and two-dimensional inversion (Zhang, 2001; Kriegshauser et al.,

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2001 ) have been investigated by various authors. However, to date the inversion of 3D

multi-component induction data remains an open challenge.

Due to lack of field data, we only tested our inversion algorithms on synthetic

array induction measurements. Testing on field data is needed to ensure the applicability

and robustness of the inversion algorithms.

Finally, we remark that most of the algorithms developed in this dissertation

would be readily adapted to parallel computer environments using state-of-the-art parallel

algorithms and parallel computing platforms.

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Appendix: Selected Publications Completed During the Course of the Ph.D. Research

[1] Fang, S., Gao, G., and Torres-Verdín, C., 2003, Efficient 3-D electromagnetic modeling in the presence of anisotropic conductive media using integral equations: Proceedings of the Third International Three-Dimensional Electromagnetics (3DEM-3) Symposium, in J. Macnae and G. Liu, Australian Society of Exploration Geophysicist.

[2] Gao, G., Fang, S. and Torres-Verdín, C., 2003, A new approximation for 3D electromagnetic scattering in the presence of anisotropic conductive media: Proceedings of the Third International Three-Dimensional Electromagnetics (3DEM-3) Symposium, in J. Macnae and G. Liu, Australian Society of Exploration Geophysicist.

[3] Gao, G., Torres-Verdín, C., and Fang, S., 2004, Fast 3D modeling of borehole induction data in dipping and anisotropic formations using a novel approximation technique: Petrophysics, Vol. 45, 335-349.

[4] Gao, G., Torres-Verdín, C., and Habashy, T. M., 2005, Analytical techniques to evaluate the integrals of 3D and 2D spatial dyadic Green’s functions: Progress in Electromagnetics Research, PIER 52, 47-80.

[5] Gao, G., Torres-Verdín, C., and Fang, S., 2003, Fast 3D modeling of borehole induction data in dipping and anisotropic formations using a novel approximation technique: Transactions of the 44th SPWLA Annual Logging Symposium, Chapter VV.

[6] Gao, G., and Torres-Verdín, C., 2003, Fast inversion of borehole induction data using an inner-outer loop optimization technique: Transactions of the 44th SPWLA Annual Logging Symposium, Paper TT.

[7] Gao, G., and Torres-Verdín, C., 2004, A high-order generalized extended Born approximation to simulate electromagnetic geophysical measurements in inhomogeneous and anisotropic media: SEG Expanded Abstracts, Denver, 628-631.

[8] Gao, G., and Torres-Verdín, C., 2005, A high-order generalized extended born approximations for electromagnetic scattering: Submitted to IEEE Trans. Antennas Propagat., in review.

[9] Gao, G., and Torres-Verdín, C., 2005, Efficient Numerical Simulation of Axisymmetric Electromagnetic Induction Data using a High-Order Generalized Extended Born Approximation: Submitted to IEEE Geoscience and Remote Sensing, in review.

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Nomenclature

Symbols

σ ′ = Ohmic conductivity, /S m .

0ε = Electrical permittivity of free space, F/m.

rε = Dielectric constant, dimensionless.

0μ = Magnetic permeability of free space, H/m.

rμ = Relative permeability, dimensionless. f = Frequency, Hz. ω = Angular frequency ( 2 fπ= ), radians/s. i = 1− . t = Time, s.

tie ω− = Time convention. σ = 0riσ ωε ε′ − , complex conductivity, S/m.

aσ = Apparent conductivity (complex), S/m.

caσ = Skin-effect-corrected apparent conductivity (real), S/m.

Rσ = R-Signal, S/m.

Xσ = X-Signal, S/m.

hσ = Horizontal conductivity, S/m.

vσ = Vertical conductivity, S/m.

ε = 0r iσε εω′

+ , complex electrical permittivity, F/m.

r = ( ), ,x y z , Cartesian coordinates, equal to zyx ˆˆˆ zyx ++ .

= Denotes a 3x3 tensor.

xxH = Magnetic field generated in x-direction by an x-directed source (the second x represents the source direction), A/m.

E = Electric field intensity, V/m H = Magnetic field intensity, A/m.

EI = Electric current, Ampere.

mI = Magnetic current, Volt. φ , = Porosity, dimensionless.

sφ = Sand porosity, dimensionless.

wR = Formation water resistivity, mΩ⋅ .

hR = Horizontal resistivity, mΩ⋅ .

vR = Vertical resistivity, mΩ⋅ .

sR = Sand resistivity, or shoulder bed resistivity, mΩ⋅ .

333

aR = Apparent resistivity, mΩ⋅ .

tR = True resistivity, mΩ⋅ .

mR = Mud resistivity, mΩ⋅ .

xoR = Flushed zone resistivity, mΩ⋅ .

shV = Volume of shale, dimensionless. λ = Anisotropy coefficient, dimensionless. d = Electrical polarization vector. M = Magnetic current density, 2/V m .

EJ = Electric current density, 2/A m . ρ = Electric charge density, 3/C m .

bk = Propagation constant, 1/m.

3C = 3D geometric factor, 3m .

2C = 2D geometric factor, 2m . Acronyms 1D = One Dimensional. 1.5D = One and Half Dimensional. 2D = Two Dimensional. 3D = Three Dimensional. BiCGSTAB(L) = Bi-Conjugate Gradient STABilized(L). DFT = Discrete Fourier Transform. EBA = Extended Born Approximation. EM = Electromagnetic. FDM = Finite Difference Method. FFT = Fast Fourier Transform. FHT = Fast Hankel Transform. GEBA = Generalized Extended Born Approximation. GS = Generalized Series. Ho-GEBA = High-order Generalized Extended Born Approximation. IE = Integral Equation. LWD = Logging While Drilling. MoM = Method of Moments. PDE = Partial Differential Equation. PEBA = Preconditioned Extended Born Approximation. QL = Quasi-Linear approximation. RMS = Root Mean Square. RIM = Resistivity Imaging. RIN = Resistivity Inversion. SA = Smooth Approximation. TI = Transversely Isotropic. TE = Transverse Electric field. TM = Transverse Magnetic field.

334

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Vita

Guozhong Gao was born in the village of Xi Zhai Zi, He Fang Xiang, Hui Min

County, Shan Dong Province, China, on May 23rd, 1974, the son of Hongsheng Gao and

Jieying Fu. He received his B.S. degree from Southwest Petroleum Institute (China) in

1996 and his M.S. degree from the University of Petroleum, Beijing in 2000,

respectively, all in Applied Geophysics. He worked with Baker Atlas during the summers

of 2001 and 2002 and with Schlumberger during the summers of 2003 and 2004. Since

the fall of 2000, he has been pursuing a Ph.D. degree in the Department of Petroleum and

Geosystems Engineering of the University of Texas at Austin.

Permanent address:

Xi Zhai Zi Cun,

He Fang Xiang,

Hui Min County,

Shan Dong Province, 251702

China

This dissertation was typed by the author.

Copyright by Guozhong Gao 2005

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