E-Book, Englisch, Band Volume 38, 690 Seiten
Reihe: Handbook of Geophysical Exploration: Seismic Exploration
Carcione Wave Fields in Real Media
3. Auflage 2014
ISBN: 978-0-08-100003-8
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark
Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media
E-Book, Englisch, Band Volume 38, 690 Seiten
Reihe: Handbook of Geophysical Exploration: Seismic Exploration
ISBN: 978-0-08-100003-8
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark
José M. Carcione received the degree 'Licenciado in Ciencias Físicas' from Buenos Aires University in 1978, the degree 'Dottore in Fisica' from Milan University in 1984 and the PhD in Geophysics from Tel-Aviv University in 1987. He was awarded the Alexander von Humboldt scholarship for a position at the Geophysical Institute of Hamburg University, where he stayed from 1987 to 1989. Dr. Carcione received the 2007 Anstey award at the EAGE in London and the 2017 EAGE Conrad Schlumberger award in Paris. He has authored several books and has published more than 360 peer-reviewed articles.
Autoren/Hrsg.
Weitere Infos & Material
1;Front Cover;1
2;Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media;4
2.1;Copyright;5
2.2;Contents;6
2.3;Dedication;13
2.4;Preface;14
2.5;About the Author;22
2.6;Basic Notation;24
2.7;Glossary of Main Symbols;26
2.8;Chapter 1: Anisotropic Elastic Media;28
2.8.1;1.1 Strain-Energy Density and Stress–Strain Relation;29
2.8.2;1.2 Dynamical Equations;32
2.8.2.1;1.2.1 Symmetries and Transformation Properties;34
2.8.2.1.1;Symmetry Plane of a Monoclinic Medium;36
2.8.2.1.2;Transformation of the Stiffness Matrix;38
2.8.3;1.3 Kelvin–Christoffel Equation, Phase Velocity and Slowness;39
2.8.3.1;1.3.1 Transversely Isotropic Media;41
2.8.3.2;1.3.2 Symmetry Planes of an Orthorhombic Medium;43
2.8.3.3;1.3.3 Orthogonality of Polarizations;45
2.8.4;1.4 Energy Balance and Energy Velocity;45
2.8.4.1;1.4.1 Group Velocity;48
2.8.4.2;1.4.2 Equivalence Between the Group and Energy Velocities;50
2.8.4.3;1.4.3 Envelope Velocity;51
2.8.4.4;1.4.4 Example: Transversely Isotropic Media;52
2.8.4.5;1.4.5 Elasticity Constants from Phase and Group Velocities;53
2.8.4.6;1.4.6 Relationship Between the Slowness and Wave Surfaces;56
2.8.4.6.1;SH-Wave Propagation;56
2.8.5;1.5 Finely Layered Media;57
2.8.5.1;1.5.1 The Schoenberg–Muir Averaging Theory;62
2.8.5.1.1;Examples;63
2.8.6;1.6 Anomalous Polarizations;66
2.8.6.1;1.6.1 Conditions for the Existence of Anomalous Polarization;67
2.8.6.2;1.6.2 Stability Constraints;69
2.8.6.3;1.6.3 Anomalous Polarization in Orthorhombic Media;71
2.8.6.4;1.6.4 Anomalous Polarization in Monoclinic Media;71
2.8.6.5;1.6.5 The Polarization;72
2.8.6.6;1.6.6 Example;73
2.8.7;1.7 The Best Isotropic Approximation;76
2.8.8;1.8 Analytical Solutions;79
2.8.8.1;1.8.1 2D Green Function;79
2.8.8.2;1.8.2 3D Green Function;80
2.8.9;1.9 Reflection and Transmission of Plane Waves;81
2.8.9.1;1.9.1 Cross-Plane Shear Waves;84
2.9;Chapter 2: Viscoelasticity and Wave Propagation;90
2.9.1;2.1 Energy Densities and Stress–Strain Relations;92
2.9.1.1;2.1.1 Fading Memory and Symmetries of the Relaxation Tensor;94
2.9.2;2.2 Stress–Strain Relation for 1D Viscoelastic Media;95
2.9.2.1;2.2.1 Complex Modulus and Storage and Loss Moduli;95
2.9.2.2;2.2.2 Energy and Significance of the Storage and Loss Moduli;97
2.9.2.3;2.2.3 Non-negative Work Requirements and Other Conditions;98
2.9.2.4;2.2.4 Consequences of Reality and Causality;99
2.9.2.5;2.2.5 Summary of the Main Properties;101
2.9.2.5.1;Relaxation Function;101
2.9.2.5.2;Complex Modulus;101
2.9.3;2.3 Wave Propagation in 1D Viscoelastic Media;101
2.9.3.1;2.3.1 Wave Propagation for Complex Frequencies;106
2.9.4;2.4 Mechanical Models and Wave Propagation;108
2.9.4.1;2.4.1 Maxwell Model;110
2.9.4.2;2.4.2 Kelvin–Voigt Model;112
2.9.4.3;2.4.3 Zener or Standard Linear Solid Model;116
2.9.4.4;2.4.4 Burgers Model;119
2.9.4.5;2.4.5 Generalized Zener Model;121
2.9.4.5.1;Nearly Constant Q;123
2.9.4.6;2.4.6 Nearly Constant-Q Model with a ContinuousSpectrum;124
2.9.5;2.5 Constant-Q Model and Wave Equation;126
2.9.5.1;2.5.1 Phase Velocity and Attenuation Factor;127
2.9.5.2;2.5.2 Wave Equation in Differential Form: Fractional Derivatives;128
2.9.5.2.1;Propagation in Pierre Shale;129
2.9.6;2.6 Equivalence Between Source and Initial Conditions;130
2.9.7;2.7 Hysteresis Cycles and Fatigue;132
2.9.8;2.8 Distributed-Order Fractional Time Derivatives;136
2.9.8.1;2.8.1 The n Case;137
2.9.8.2;2.8.2 The Generalized Dirac CombFunction;137
2.9.9;2.9 The Concept ofCentrovelocity;138
2.9.9.1;2.9.1 1D Green Function and Transient Solution;139
2.9.9.2;2.9.2 Numerical Evaluation of the Velocities;140
2.9.9.3;2.9.3 Example;142
2.9.10;2.10 Memory Variables and Equation of Motion;144
2.9.10.1;2.10.1 Maxwell Model;144
2.9.10.2;2.10.2 Kelvin–VoigtModel;146
2.9.10.3;2.10.3 Zener Model;147
2.9.10.4;2.10.4 Generalized Zener Model;148
2.10;Chapter 3: Isotropic Anelastic Media;150
2.10.1;3.1 Stress–Strain Relation;151
2.10.2;3.2 Equations of Motion and Dispersion Relations;152
2.10.3;3.3 Vector Plane Waves;155
2.10.3.1;3.3.1 Slowness, Phase Velocity and Attenuation Factor;155
2.10.3.2;3.3.2 Particle Motion of the P-Wave;157
2.10.3.3;3.3.3 Particle Motion of the S-Waves;159
2.10.3.4;3.3.4 Polarization and Orthogonality;161
2.10.4;3.4 Energy Balance, Velocity and Quality Factor;162
2.10.4.1;3.4.1 P-Wave;164
2.10.4.2;3.4.2 S-Waves;170
2.10.5;3.5 Boundary Conditions and Snell Law;171
2.10.6;3.6 The Correspondence Principle;172
2.10.7;3.7 Rayleigh Waves;173
2.10.7.1;3.7.1 Dispersion Relation;174
2.10.7.2;3.7.2 Displacement Field;175
2.10.7.3;3.7.3 Phase Velocity and Attenuation Factor;176
2.10.7.4;3.7.4 Special Viscoelastic Solids;177
2.10.7.4.1;Incompressible Solid;177
2.10.7.4.2;Poisson Solid;177
2.10.7.4.3;Hardtwig Solid;177
2.10.7.5;3.7.5 Two Rayleigh Waves;177
2.10.8;3.8 Reflection and Transmission of SH Waves;178
2.10.9;3.9 Memory Variables and Equation of Motion;182
2.10.10;3.10 Analytical Solutions;184
2.10.10.1;3.10.1 Viscoacoustic Media;184
2.10.10.2;3.10.2 Constant-Q Viscoacoustic Media;185
2.10.10.3;3.10.3 Viscoelastic Media;186
2.10.10.4;3.10.4 Pekeris Solution for Lamb Problem;188
2.10.11;3.11 Constant-Q P- and S-Waves;189
2.10.11.1;3.11.1 Time Fractional Derivatives;189
2.10.11.2;3.11.2 Spatial Fractional Derivatives;191
2.10.12;3.12 Wave Equations Based on the Burgers Model;191
2.10.12.1;3.12.1 Propagation of P–SV Waves;192
2.10.12.2;3.12.2 Propagation of SH Waves;193
2.10.13;3.13 The Elastodynamic of a Non-Ideal Interface;193
2.10.13.1;3.13.1 The Interface Model;194
2.10.13.1.1;Boundary Conditions in Differential Form;195
2.10.13.2;3.13.2 Reflection and Transmission Coefficients of SH Waves;196
2.10.13.2.1;Energy Loss;196
2.10.13.3;3.13.3 Reflection and Transmission Coefficients of P–SV Waves;198
2.10.13.3.1;Energy Loss;199
2.10.13.3.2;Examples;200
2.11;Chapter 4: Anisotropic Anelastic Media;204
2.11.1;4.1 Stress–Strain Relations;206
2.11.1.1;4.1.1 Model 1: Effective Anisotropy;208
2.11.1.2;4.1.2 Model 2: Attenuation via Eigenstrains;209
2.11.1.3;4.1.3 Model 3: Attenuation via Mean and Deviatoric Stresses;211
2.11.2;4.2 Fracture-Induced Anisotropic Attenuation;212
2.11.2.1;4.2.1 The Equivalent Monoclinic Medium;214
2.11.2.2;4.2.2 The Orthorhombic Equivalent Medium;215
2.11.2.3;4.2.3 HTI Equivalent Media;217
2.11.3;4.3 Stiffness Tensorfrom OscillatoryExperiments;219
2.11.4;4.4 Wave Velocities, Slowness and Attenuation Vector;222
2.11.5;4.5 Energy Balance and Fundamental Relations;224
2.11.5.1;4.5.1 Plane Waves: Energy Velocity and Quality Factor;226
2.11.5.2;4.5.2 Polarizations;231
2.11.6;4.6 Propagation of SH Waves;232
2.11.6.1;4.6.1 Energy Velocity;232
2.11.6.2;4.6.2 Group Velocity;234
2.11.6.3;4.6.3 Envelope Velocity;235
2.11.6.4;4.6.4 Perpendicularity Properties;235
2.11.6.5;4.6.5 Numerical Evaluation of the Energy Velocity;237
2.11.6.6;4.6.6 Forbidden Directions of Propagation;239
2.11.7;4.7 Wave Propagation in Symmetry Planes;241
2.11.7.1;4.7.1 Properties of the Homogeneous Wave;243
2.11.7.2;4.7.2 Propagation, Attenuation and Energy Directions;243
2.11.7.3;4.7.3 Phase Velocities and Attenuations;244
2.11.7.4;4.7.4 Energy Balance, Velocity and Quality Factor;244
2.11.7.5;4.7.5 Explicit Equations in Symmetry Planes;245
2.11.8;4.8 Memory Variables and Equation of Motion;248
2.11.8.1;4.8.1 Strain Memory Variables;249
2.11.8.2;4.8.2 Memory-Variable Equations;251
2.11.8.3;4.8.3 SH Equation of Motion;253
2.11.8.4;4.8.4 qP–qSV Equation of Motion;253
2.11.9;4.9 Analytical Solution for SH Waves;255
2.12;Chapter 5: The Reciprocity Principle;258
2.12.1;5.1 Sources, Receivers and Reciprocity;259
2.12.2;5.2 The Reciprocity Principle;260
2.12.3;5.3 Reciprocity of Particle Velocity: Monopoles;261
2.12.4;5.4 Reciprocity of Strain;262
2.12.4.1;5.4.1 Single Couples;263
2.12.4.1.1;Single Couples Without Moment;264
2.12.4.1.2;Single Couples with Moment;264
2.12.4.2;5.4.2 Double Couples;265
2.12.4.2.1;Double Couple Without Moment: Dilatation;265
2.12.4.2.2;Double Couple Without Moment and Monopole Force;265
2.12.4.2.3;Double Couple Without Moment and SingleCouple;267
2.12.5;5.5 Reciprocity of Stress;267
2.12.6;5.6 Reciprocity Principle for Flexural Waves;269
2.12.6.1;5.6.1 Equation of Motion;270
2.12.6.2;5.6.2 Reciprocity of the Deflection;271
2.12.6.3;5.6.3 Reciprocity of the Bending Moment;272
2.13;Chapter 6: Reflection and Transmission of Plane Waves;274
2.13.1;6.1 Reflection and Transmission of SH Waves;275
2.13.1.1;6.1.1 Symmetry Plane of a Homogeneous Monoclinic Medium;276
2.13.1.2;6.1.2 Complex Stiffnesses;278
2.13.1.3;6.1.3 Reflection and TransmissionCoefficients;279
2.13.1.4;6.1.4 Propagation, Attenuation and Energy Directions;282
2.13.1.5;6.1.5 Brewster and Critical Angles;289
2.13.1.6;6.1.6 Phase Velocities and Attenuations;293
2.13.1.7;6.1.7 Energy-Flux Balance;295
2.13.1.8;6.1.8 Energy Velocities and Quality Factors;297
2.13.2;6.2 Reflection and Transmission of qP–qSV Waves;299
2.13.2.1;6.2.1 Phase Velocities and Attenuations;301
2.13.2.2;6.2.2 Energy-Flow Balance;302
2.13.2.3;6.2.3 Reflection of Seismic Waves;304
2.13.2.4;6.2.4 Incident Inhomogeneous Waves;312
2.13.2.4.1;Generation of Inhomogeneous Waves;315
2.13.2.4.2;Ocean Bottom;316
2.13.3;6.3 Interfaces Separating a Solid and a Fluid;318
2.13.3.1;6.3.1 Solid/Fluid Interface;318
2.13.3.2;6.3.2 Fluid/Solid Interface;319
2.13.3.3;6.3.3 The Rayleigh Window;320
2.13.4;6.4 Scattering Coefficients of a Set of Layers;322
2.14;Chapter 7: Biot Theory for Porous Media;326
2.14.1;7.1 Isotropic Media – Stress–Strain Relations;329
2.14.1.1;7.1.1 Jacketed Compressibility Test;330
2.14.1.2;7.1.2 Unjacketed Compressibility Test;331
2.14.2;7.2 The Concept of Effective Stress;332
2.14.2.1;7.2.1 Effective Stress in Seismic Exploration;335
2.14.2.1.1;Pore-Volume Balance;337
2.14.2.1.2;Acoustic Properties;339
2.14.2.2;7.2.2 Analysis in Terms of Compressibilities;340
2.14.3;7.3 Pore-Pressure Buildup in Source Rocks;344
2.14.4;7.4 The Asperity-Deformation Model;346
2.14.5;7.5 Anisotropic Media – Stress–Strain Relations;350
2.14.5.1;7.5.1 Effective-Stress Law for Anisotropic Media;355
2.14.5.2;7.5.2 Summary of Equations;357
2.14.5.3;7.5.3 Brown and Korringa Equations;358
2.14.5.3.1;Transversely Isotropic Medium;358
2.14.6;7.6 Kinetic Energy;359
2.14.6.1;7.6.1 Anisotropic Media;362
2.14.7;7.7 Dissipation Potential;364
2.14.7.1;7.7.1 Anisotropic Media;365
2.14.8;7.8 Lagrange Equations and Equation of Motion;366
2.14.8.1;7.8.1 The Viscodynamic Operator;367
2.14.8.2;7.8.2 Fluid Flow in aPlane Slit;368
2.14.8.3;7.8.3 Anisotropic Media;374
2.14.9;7.9 Plane-Wave Analysis;375
2.14.9.1;7.9.1 Compressional Waves;375
2.14.9.1.1;Relation With Terzaghi Law and the Second P-Wave;379
2.14.9.1.2;The Diffusive Slow Mode;380
2.14.9.2;7.9.2 The Shear Wave;381
2.14.10;7.10 Strain Energy for Inhomogeneous Porosity;383
2.14.10.1;7.10.1 Complementary Energy Theorem;383
2.14.10.2;7.10.2 Volume-Averaging Method;385
2.14.11;7.11 Boundary Conditions;389
2.14.11.1;7.11.1 Interface Between Two Porous Media;389
2.14.11.1.1;Deresiewicz and Skalak's Derivation;389
2.14.11.1.2;Gurevich and Schoenberg's Derivation;392
2.14.11.2;7.11.2 Interface Between a Porous Medium and a ViscoelasticMedium;394
2.14.11.3;7.11.3 Interface Between a Porous Medium and a Viscoacoustic Medium;394
2.14.11.4;7.11.4 Free Surface of a Porous Medium;395
2.14.12;7.12 Squirt-Flow Dissipation;395
2.14.13;7.13 The Mesoscopic Loss Mechanism – White Model;398
2.14.14;7.14 Mesoscopic Loss in Layered and Fractured Media;405
2.14.14.1;7.14.1 Effective Fractured Medium;407
2.14.15;7.15 Green Function for Poro-Viscoacoustic Media;409
2.14.15.1;7.15.1 Field Equations;409
2.14.15.2;7.15.2 The Solution;411
2.14.16;7.16 Green Function for a Fluid/Solid Interface;414
2.14.17;7.17 Poro-Viscoelasticity;419
2.14.18;7.18 Fluid-Pressure Diffusion in Anisotropic Media;423
2.14.18.1;7.18.1 Biot Classical Equation and Fractional-Derivative Version;423
2.14.18.2;7.18.2 Frequency-Wavenumber Domain Analysis;425
2.14.19;7.19 Anisotropy and Poro-Viscoelasticity;430
2.14.19.1;7.19.1 Stress–Strain Relations;431
2.14.19.2;7.19.2 Biot–Euler Equation;432
2.14.19.3;7.19.3 Time-HarmonicFields;432
2.14.19.4;7.19.4 Inhomogeneous Plane Waves;435
2.14.19.5;7.19.5 Homogeneous Plane Waves;438
2.14.19.6;7.19.6 Wave Propagation in Femoral Bone;440
2.14.20;7.20 Gassmann Equation for a Solid Pore Infill;444
2.15;Chapter 8: The Acoustic–Electromagnetic Analogy;448
2.15.1;8.1 Maxwell Equations;452
2.15.2;8.2 The Acoustic–Electromagnetic Analogy;453
2.15.2.1;8.2.1 Kinematics and Energy Considerations;458
2.15.3;8.3 A Viscoelastic Form of the Electromagnetic Energy;460
2.15.3.1;8.3.1 Umov–Poynting Theorem for Harmonic Fields;461
2.15.3.2;8.3.2 Umov–Poynting Theorem for Transient Fields;463
2.15.3.2.1;The Debye–Zener Analogy;466
2.15.3.2.2;The Cole–Cole Model;471
2.15.4;8.4 The Analogy for Reflection and Transmission;472
2.15.4.1;8.4.1 Reflection and Refraction Coefficients;473
2.15.4.1.1;Propagation, Attenuation and Ray Angles;473
2.15.4.1.2;Energy-Flux Balance;474
2.15.4.2;8.4.2 Application of the Analogy;474
2.15.4.2.1;Refraction Index and Fresnel Formulae;474
2.15.4.2.2;Brewster (Polarizing) Angle;475
2.15.4.2.3;Critical Angle: Total Reflection;477
2.15.4.2.4;Reflectivity and Transmissivity;478
2.15.4.2.5;Dual Fields;480
2.15.4.2.6;Sound Waves;481
2.15.4.3;8.4.3 The Analogy Between TM and TE Waves;483
2.15.4.3.1;Green Analogies;483
2.15.4.4;8.4.4 Brief HistoricalReview;487
2.15.5;8.5 The Single-Layer Problem;488
2.15.5.1;8.5.1 TM–SH–TE Analogy;492
2.15.5.2;8.5.2 Analogy with Quantum Mechanics: Tunnel Effect;492
2.15.6;8.6 3D Electromagnetic Theory and the Analogy;493
2.15.6.1;8.6.1 The Form of the Tensor Components;494
2.15.6.2;8.6.2 Electromagnetic Equations in Differential Form;495
2.15.7;8.7 Plane-Wave Theory;497
2.15.7.1;8.7.1 Slowness, Phase Velocity andAttenuation;500
2.15.7.2;8.7.2 Energy Velocity and Quality Factor;502
2.15.8;8.8 Electromagnetic Diffusion in Anisotropic Media;506
2.15.8.1;8.8.1 Differential Equations;506
2.15.8.2;8.8.2 Dispersion Relation;507
2.15.8.3;8.8.3 Slowness, Kinematic Velocities, Attenuation and Skin Depth;508
2.15.8.4;8.8.4 Umov–Poynting Theorem and Energy Velocity;509
2.15.8.5;8.8.5 Fundamental Relations;510
2.15.9;8.9 Analytical Solution for Anisotropic Media;511
2.15.9.1;8.9.1 The Solution;513
2.15.10;8.10 Elastic Medium with Fresnel Wave Surface;514
2.15.10.1;8.10.1 Fresnel Wave Surface;514
2.15.10.2;8.10.2 Equivalent Elastic Medium;516
2.15.11;8.11 Finely Layered Media;517
2.15.12;8.12 The Time-Average and CRIM Equations;520
2.15.13;8.13 The Kramers–Kronig Dispersion Relations;521
2.15.14;8.14 The ReciprocityPrinciple;523
2.15.15;8.15 Babinet Principle;524
2.15.16;8.16 Alford Rotation;525
2.15.17;8.17 Cross-Property Relations;527
2.15.18;8.18 Poro-Acoustic and Electromagnetic Diffusion;529
2.15.18.1;8.18.1 Poro-Acoustic Equations;530
2.15.18.2;8.18.2 Electromagnetic Equations;531
2.15.18.2.1;The TM and TEEquations;531
2.15.18.2.2;Phase Velocity, Attenuation Factor and Skin Depth;532
2.15.18.2.3;Analytical Solutions;533
2.15.19;8.19 Electro-Seismic Wave Theory;534
2.16;Chapter 9: Numerical Methods;536
2.16.1;9.1 Equation of Motion;537
2.16.2;9.2 Time Integration;538
2.16.2.1;9.2.1 Classical Finite Differences;540
2.16.2.2;9.2.2 Splitting Methods;542
2.16.2.3;9.2.3 Predictor–Corrector Methods;542
2.16.2.3.1;The Runge–Kutta Method;542
2.16.2.3.2;Fractional Calculus;543
2.16.2.4;9.2.4 Spectral Methods;543
2.16.2.5;9.2.5 Algorithms forFinite-ElementMethods;545
2.16.3;9.3 Calculation of Spatial Derivatives;546
2.16.3.1;9.3.1 Finite Differences;546
2.16.3.2;9.3.2 PseudospectralMethods;548
2.16.3.3;9.3.3 The Finite-Element Method;550
2.16.4;9.4 Source Implementation;552
2.16.5;9.5 Boundary Conditions;553
2.16.6;9.6 Absorbing Boundaries;554
2.16.7;9.7 Model and Modelling Design. Seismic Modelling;556
2.16.8;9.8 Concluding Remarks;559
2.16.9;9.9 Appendix;560
2.16.9.1;9.9.1 The FractionalDerivative;560
2.16.9.1.1;Grünwald–Letnikov and Central-Difference Approximations;561
2.16.9.2;9.9.2 Electromagnetic-Diffusion Code;562
2.16.9.3;9.9.3 Finite-differences code for SH waves;568
2.16.9.4;9.9.4 Finite-Difference Code for SH (TM)Waves;575
2.16.9.5;9.9.5 Pseudospectral Fourier Method;584
2.16.9.5.1;Calculation of Fractional Derivatives;587
2.16.9.6;9.9.6 Pseudospectral Chebyshev Method;587
2.16.9.7;9.9.7 Pseudospectral Sine/cosine Method;590
2.16.9.8;9.9.8 Earthquake Sources. The Moment–Tensor;592
2.16.9.9;9.9.9 3D Anisotropic Media. Free-Surface Boundary Treatment;595
2.16.9.10;9.9.10 Modelling in Cylindrical Coordinates;598
2.16.9.10.1;Equations for Axis-Symmetric Single-Phase Media;598
2.16.9.10.2;Equations for Poroelastic Media;599
2.17;Examinations;602
2.18;Chronology of MainDiscoveries;606
2.19;Leonardo's Manuscripts;618
2.20;A List of Scientists;624
2.21;Bibliography;634
2.22;Name Index;664
2.23;Subject Index;678
Preface
This book presents the fundamentals of wave propagation in anisotropic, anelastic and porous media, including electromagnetic waves. This new edition incorporates research work performed during the last seven years on several relevant topics, which have been distributed in the various chapters. The emphasis is on geophysical applications for hydrocarbon exploration, but researchers in the fields of earthquake seismology, rock acoustics and material science – including many branches of acoustics of fluids and solids (acoustics of materials, nondestructive testing, etc.) – may also find this text useful. This book can be considered, in part, a monograph, since much of the material represents my own original work on wave propagation in anisotropic, viscoelastic media. Although it is biased to my scientific interests and applications, I have, nevertheless, sought to retain the generality of the subject matter, in the hope that the book will be of interest and use to a wide readership.
The concepts of anisotropy, anelasticity1 and poroelasticity in physical media have gained much attention in recent years. The applications of these studies cover a variety of fields, including physics and geophysics, engineering and soil mechanics, underwater acoustics, etc. In particular, in the exploration of oil and gas reservoirs, it is important to predict the rock porosity, the presence of fluids (type and saturation), the preferential directions of fluid flow (anisotropy), the presence of abnormal pore-pressures (overpressure), etc. These microstructural properties and in situ rock conditions can be obtained, in principle, from seismic and electromagnetic properties, such as travel times, amplitude information and wave polarization. These measurable quantities are affected by the presence of anisotropy and attenuation mechanisms. For instance, shales are naturally bedded and possess intrinsic anisotropy at the microscopic level. Similarly, compaction and the presence of microcracks and fractures make the skeleton of porous rocks anisotropic. The presence of fluids implies relaxation phenomena, which causes wave dissipation. The use of modelling and inversion for the interpretation of the wave response of reservoir rocks requires an understanding of the relationship between the seismic and electromagnetic properties and the rock characteristics, such as permeability, porosity, tortuosity, fluid viscosity, stiffness, dielectric permittivity, electrical conductivity, etc.
Wave simulation is a theoretical field of research that began nearly four decades ago, in close relationship with the development of computer technology and numerical algorithms for solving differential and integral equations of several variables. In the field of research known as computational physics, algorithms for solving problems using computers are important tools that provide insight into wave propagation for a variety of applications.
In this book, I examine the differences between an ideal and a real description of wave propagation, where ideal means an elastic (lossless), isotropic and single-phase medium, and real means an anelastic, anisotropic and multi-phase medium. The first realization is, of course, a particular case of the second, but it must be noted that, in general, the real description is not a simple and straightforward extension of the ideal description.
The analysis starts by introducing the constitutive equation (strain–stress relation) appropriate for the particular rheology.2 This relation and the equations of conservation of linear momentum are combined to give the equation of motion, a second-order or a first-order matrix differential equation in time, depending on the formulation of the field variables. The differential formulation for lossy media is written in terms of memory (hidden) variables or alternatively, fractional derivatives. Biot theory is essential to describe wave propagation in multi-phase (porous) media from the seismic to the ultrasonic frequency range, representative of field and laboratory experiments, respectively. The acoustic–electromagnetic analogy reveals that the different physical phenomena have the same mathematical formulation. For each constitutive equation, a plane-wave analysis is performed in order to understand the physics of the wave propagation (i.e., calculation of phase, group and energy velocities, and quality and attenuation factors). For some cases, it is possible to obtain an analytical solution for transient wave fields in the space-frequency domain, which is then transformed to the time domain by a numerical Fourier transform. The book concludes with a review of the so-called direct numerical methods for solving the equations of motion in the time-space domain. The plane-wave theory and the analytical solutions serve to test the performance (accuracy and limitations) of the modelling codes.
A brief description of the main concepts discussed in this book follows.
Chapter 1: Anisotropic Elastic Media. In anisotropic lossless media, the directions of the wavevector and Umov–Poynting vector (ray or energy-flow vector) do not coincide. This implies that the phase and energy velocities differ. However, some ideal properties prevail: there is no dissipation, the group-velocity vector is equal to the energy-velocity vector, the wavevector is normal to the wave-front surface, the energy-velocity vector is normal to the slowness surface, plane waves are linearly polarized and the polarization of the different wave modes are mutually orthogonal. Methods used to calculate these quantities and provide the equation of motion for inhomogeneous media are shown. I also consider the seismic properties of finely stratified media composed of anisotropic layers, anomalously polarized media and the best isotropic approximation of anisotropic media. Finally, the analysis of a reflection–transmission problem and analytical solutions along the symmetry axis of a transversely isotropic medium are discussed.
Chapter 2: Viscoelasticity and Wave Propagation. Attenuation is introduced in the form of Boltzmann superposition law, which implies a convolutional relation between the stress and strain tensors through the relaxation and creep matrices. The analysis is restricted to the one-dimensional case, where some of the consequences of anelasticity become evident. Although phase and energy velocities are the same, the group velocity loses its physical meaning. The concept of centrovelocity for non-harmonic waves is discussed. The uncertainty in defining the strain and rate of dissipated-energy densities is overcome by introducing relaxation functions based on mechanical models. The concepts of memory variable and fractional derivative are introduced to avoid time convolutions and obtain a time-domain differential formulation of the equation of motion.
Chapter 3: Isotropic Anelastic Media. The space dimension reveals other properties of anelastic (viscoelastic) wave fields. There is a distinct difference between the inhomogeneous waves of lossless media (interface waves) and those of viscoelastic media (body waves). In the former case, the direction of attenuation is normal to the direction of propagation, whereas for inhomogeneous viscoelastic waves, that angle must be less than p/2. Furthermore, for viscoelastic inhomogeneous waves, the energy does not propagate in the direction of the slowness vector and the particle motion is elliptical in general. The phase velocity is less than that of the corresponding homogeneous wave (for which planes of constant phase coincide with planes of constant amplitude); critical angles do not exist in general, and, unlike the case of lossless media, the phase velocity and the attenuation factor of the transmitted waves depend on the angle of incidence. There is one more degree of freedom, since the attenuation vector is playing a role at the same level as the wavenumber vector. Snell law, for instance, implies continuity of the tangential components of both vectors at the interface of discontinuity. For homogeneous plane waves, the energy-velocity vector is equal to the phase-velocity vector. The last part of the chapter analyzes the viscoelastic wave equation expressed in terms of fractional time derivatives, and provides expressions of the reflection and transmission coefficients corresponding to a partially welded interface.
Chapter 4: Anisotropic Anelastic Media. In isotropic media there are two well-defined relaxation functions, describing purely dilatational and shear deformations of the medium. The problem in anisotropic media is to obtain the time dependence of the relaxation components with a relatively reduced number of parameters. Fine layering has an “exact” description in the long-wavelength limit. The concept of eigenstrain allows us to reduce the number of relaxation functions to six; an alternative is to use four or two relaxation functions when the anisotropy is relatively weak. Fracture-induced anisotropic attenuation is studied, and harmonic quasi-static numerical experiments are designed to obtain the stiffness components of anisotropic anelastic media. The analysis of SH waves suffices to show that in anisotropic viscoelastic media, unlike the lossless case: the group-velocity vector is not equal to the energy-velocity vector, the wavevector is not normal to the energy-velocity surface, the energy-velocity vector is not normal to the slowness surface, etc. However, an energy analysis shows that some basic fundamental relations still hold: for instance, the projection of the energy velocity onto the propagation direction is...
