Copyright by Guozhong Gao 2005
Transcript of 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
SIMULATION OF BOREHOLE ELECTROMAGNETIC
MEASUREMENTS IN DIPPING AND ANISOTROPIC ROCK
FORMATIONS AND INVERSION OF ARRAY INDUCTION
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
SIMULATION OF BOREHOLE ELECTROMAGNETIC
MEASUREMENTS IN DIPPING AND ANISOTROPIC ROCK
FORMATIONS AND INVERSION OF ARRAY INDUCTION
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
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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
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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
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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
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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
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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
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. ............................................................................. 43
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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 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
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. ............................................................................. 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
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. .......... 160
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. .......... 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
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.................................................. 163
Figure 5.12: Comparison of the Hyy 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.................................................. 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
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. ................................................................... 197
xxviii
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. ................................................................... 198
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............................. 200
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............................. 201
xxix
Figure 6.6: Assessment of the accuracy of the integral equation
approximation of Hyy (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 6.7: 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)
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
xxx
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. .................................................................... 206
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. .................................................................... 207
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. .................................................................... 208
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................................ 209
xxxi
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................................ 210
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................................ 211
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................................ 212
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................................ 213
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
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. .................. 243
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. .................. 244
Figure 7.5: Scattered zzH 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 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. .................. 245
xxxiv
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. .................. 246
Figure 7.7: Scattered zzH 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 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. .................. 247
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. .................. 248
xxxv
Figure 7.9: Scattered zzH 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 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. .................. 249
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-
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
xxxvii
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 and the frequency is 220
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-
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 7.19: 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. ............................................ 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
contaminated with zero-mean, 2% random Gaussian additive
noise. The inversion was performed with data acquired at 25
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
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
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
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. 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
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. ......................................................... 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
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................................................................................... 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)
Substitution of equation (2.39) into equation (2.23) yields
( ) ( ) ( )
( )
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)
Substitution of equation (2.43) into equation (2.29) yields
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)
Substitution of equation (3.7) into equation (3.47) yields
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)
Substitution of equation (4.38) into equation (4.36) yields
( ) ( ) ( ) 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)
Substitution of equation (4.49) into equation (4.20) yields
( )( ) ( ) ( )zEzJzzgdzdi
zzJ
b ,,),;,(,
,0
ρρρρρωμρσρ
=′′⋅′′′′−Δ ∫ ∫
∞
∞−
∞, ( ) Rz ∈,ρ , (4.50)
where R is the spatial support of the inhomogeneous region.
Substitution of equation (4.30) into equation (4.50) yields
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 .
Substitution of equation (4.77) into equation (4.20) yields
( ) ( ) ( )( ) ( ) ( ) ( )
, , ,
, ; , , , , .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).
102
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
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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
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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.
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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 ,
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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.
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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
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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
nxxxxxxxx EEE σσσ === , (5A-4)
( ) ( ) ( ) ( ) ( ) ( )3131231231131131 ... ii nx
nxxxxxxxx EEE σσσ === , (5A-5)
( ) ( ) ( ) ( ) ( ) ( )1212212212112112 ... ii nx
nxxxxxxxx EEE σσσ === , (5A-6)
……
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
n jk n jkjk jk jk jk jkzx 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
n jk n jkjk jk jk jk jkzx 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
180
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
181
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 .
Substitution of equation (6.13) into equation (3.1) yields
( ) 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
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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
6.7.2 1D Anisotropic Rock Formation with Dip=60o
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
6.7.3 3D Anisotropic Rock Formation with Dip=0o
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.
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6.7.4 3D Anisotropic Rock Formation with Dip=60o
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.
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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.
232
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).
233
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
235
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
236
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
240
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
Figure 7.5: Scattered zzH 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 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.
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
Figure 7.7: Scattered zzH 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 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.
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
Figure 7.9: Scattered zzH 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 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.
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
and the frequency is 220 KHz.
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
and the frequency is 220 KHz.
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.
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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.
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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.
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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..
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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.
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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.
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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.
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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.
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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
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.
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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
Radial Distance (10-1m)
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.
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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
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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
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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
Radial Distance (10-1m)
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.
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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.
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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.
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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.
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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
329
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.,
330
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.
331
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.
332
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.