Wave Fields in Real Media (eBook)
538 Seiten
Elsevier Science (Verlag)
978-0-08-046890-7 (ISBN)
This book examines 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 analysis starts by introducing the relevant stress-strain relation. This relation and the equations of momentum conservation are combined to give the equation of motion. The differential formulation is written in terms of memory variables, and Biot's theory is used to describe wave propagation in porous media. For each rheology, a plane-wave analysis is performed in order to understand the physics of wave propagation. The book contains a review of the main direct numerical methods for solving the equation of motion in the time and space domains. The emphasis is on geophysical applications for seismic exploration, but researchers in the fields of earthquake seismology, rock acoustics, and material science - including many branches of acoustics of fluids and solids - may also find this text useful.
* Presents the fundamentals of wave propagation in anisotropic, anelastic and porus media
* Contains a new chapter on the analogy between acoustic and electromagnetic waves, incorporating the subject of electromagnetic waves
* Emphasizes geophysics, particularly, seismic exploration for hydrocarbon reservoirs, which is essential for exploration and production of oil
Wave Fields in Real Media examines 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 analysis starts by introducing the relevant stress-strain relation. This relation and the equations of momentum conservation are combined to give the equation of motion. The differential formulation is written in terms of memory variables, and Biot's theory is used to describe wave propagation in porous media. For each rheology, a plane-wave analysis is performed in order to understand the physics of wave propagation. The book contains a review of the main direct numerical methods for solving the equation of motion in the time and space domains. The emphasis is on geophysical applications for seismic exploration, but researchers in the fields of earthquake seismology, rock acoustics, and material science - including many branches of acoustics of fluids and solids - may also find this text useful. - Presents the fundamentals of wave propagation in anisotropic, anelastic and porus media- Contains a new chapter on the analogy between acoustic and electromagnetic waves, incorporating the subject of electromagnetic waves- Emphasizes geophysics, particularly, seismic exploration for hydrocarbon reservoirs, which is essential for exploration and production of oil
Front Cover 1
Wave Fields in Real Media 4
Copyright Page 5
Table of Contents 6
Preface 14
About the author 20
Basic notation 21
Glossary of main symbols 22
Chapter 1 Anisotropic elastic media 24
1.1 Strain-energy density and stress-strain relation 24
1.2 Dynamical equations 27
1.2.1 Symmetries and transformation properties 29
Symmetry plane of a monoclinic medium 30
Transformation of the stiffness matrix 32
1.3 Kelvin-Christoffel equation, phase velocity and slowness 33
1.3.1 Transversely isotropic media 34
1.3.2 Symmetry planes of an orthorhombic medium 36
1.3.3 Orthogonality of polarizations 37
1.4 Energy balance and energy velocity 38
1.4.1 Group velocity 40
1.4.2 Equivalence between the group and energy velocities 41
1.4.3 Envelope velocity 43
1.4.4 Example: Transversely isotropic media 43
1.4.5 Elasticity constants from phase and group velocities 45
1.4.6 Relationship between the slowness and wave surfaces 47
SH-wave propagation 47
1.5 Finely layered media 48
1.6 Anomalous polarizations 52
1.6.1 Conditions for the existence of anomalous polarization 52
1.6.2 Stability constraints 55
1.6.3 Anomalous polarization in orthorhombic media 56
1.6.4 Anomalous polarization in monoclinic media 56
1.6.5 The polarization 57
1.6.6 Example 58
1.7 The best isotropic approximation 61
1.8 Analytical solutions for transversely isotropic media 63
1.8.1 2-D Green's function 63
1.8.2 3-D Green's function 65
1.9 Reflection and transmission of plane waves 65
1.9.1 Cross-plane shear waves 68
Chapter 2 Viscoelasticity and wave propagation 74
2.1 Energy densities and stress-strain relations 75
2.1.1 Fading memory and symmetries of the relaxation tensor 77
2.2 Stress-strain relation for 1-D viscoelastic media 78
2.2.1 Complex modulus and storage and loss moduli 78
2.2.2 Energy and significance of the storage and loss moduli 80
2.2.3 Non-negative work requirements and other conditions 80
2.2.4 Consequences of reality and causality 81
2.2.5 Summary of the main properties 83
Relaxation function 83
Complex modulus 83
2.3 Wave propagation concepts for 1-D viscoelastic media 84
2.3.1 Wave propagation for complex frequencies 88
2.4 Mechanical models and wave propagation 91
2.4.1 Maxwell model 91
2.4.2 Kelvin-Voigt model 94
2.4.3 Zener or standard linear solid model 97
2.4.4 Burgers model 100
2.4.5 Generalized Zener model 102
Nearly constant Q 103
2.4.6 Nearly constant-Q model with a continuous spectrum 105
2.5 Constant-Q model and wave equation 106
2.5.1 Phase velocity and attenuation factor 107
2.5.2 Wave equation in differential form. Fractional derivatives. 108
Propagation in Pierre shale 109
2.6 The concept of centrovelocity 110
2.6.1 1-D Green's function and transient solution 111
2.6.2 Numerical evaluation of the velocities 112
2.6.3 Example 113
2.7 Memory variables and equation of motion 115
2.7.1 Maxwell model 115
2.7.2 Kelvin-Voigt model 117
2.7.3 Zener model 118
2.7.4 Generalized Zener model 118
Chapter 3 Isotropic anelastic media 120
3.1 Stress-strain relation 121
3.2 Equations of motion and dispersion relations 121
3.3 Vector plane waves 123
3.3.1 Slowness, phase velocity and attenuation factor 123
3.3.2 Particle motion of the P wave 125
3.3.3 Particle motion of the S waves 127
3.3.4 Polarization and orthogonality 129
3.4 Energy balance, energy velocity and quality factor 130
3.4.1 P wave 131
3.4.2 S waves 137
3.5 Boundary conditions and Snell's law 137
3.6 The correspondence principle 139
3.7 Rayleigh waves 139
3.7.1 Dispersion relation 140
3.7.2 Displacement field 141
3.7.3 Phase velocity and attenuation factor 142
3.7.4 Special viscoelastic solids 143
Incompressible solid 143
Poisson solid 143
Hardtwig solid 143
3.7.5 Two Rayleigh waves 143
3.8 Reflection and transmission of cross-plane shear waves 144
3.9 Memory variables and equation of motion 147
3.10 Analytical solutions 149
3.10.1 Viscoacoustic media 149
3.10.2 Constant-Q viscoacoustic media 150
3.10.3 Viscoelastic media 151
3.11 The elastodynamic of a non-ideal interface 152
3.11.1 The interface model 153
Boundary conditions in differential form 154
3.11.2 Reflection and transmission coefficients of SH waves 155
Energy loss 156
3.11.3 Reflection and transmission coefficients of P-SV waves 156
Energy loss 158
Examples 159
Chapter 4 Anisotropic anelastic media 162
4.1 Stress-strain relations 163
4.1.1 Model 1: Effective anisotropy 165
4.1.2 Model 2: Attenuation via eigenstrains 165
4.1.3 Model 3: Attenuation via mean and deviatoric stresses 167
4.2 Wave velocities, slowness and attenuation vector 168
4.3 Energy balance and fundamental relations 170
4.3.1 Plane waves. Energy velocity and quality factor 172
4.3.2 Polarizations 177
4.4 The physics of wave propagation for viscoelastic SH waves 178
4.4.1 Energy velocity 178
4.4.2 Group velocity 179
4.4.3 Envelope velocity 180
4.4.4 Perpendicularity properties 180
4.4.5 Numerical evaluation of the energy velocity 182
4.4.6 Forbidden directions of propagation 184
4.5 Memory variables and equation of motion in the time domain 185
4.5.1 Strain memory variables 186
4.5.2 Memory-variable equations 188
4.5.3 SH equation of motion 189
4.5.4 qP-qSV equation of motion 189
4.6 Analytical solution for SH waves in monoclinic media 191
Chapter 5 The reciprocity principle 194
5.1 Sources, receivers and reciprocity 195
5.2 The reciprocity principle 195
5.3 Reciprocity of particle velocity. Monopoles 196
5.4 Reciprocity of strain 197
5.4.1 Single couples 197
Single couples without moment 200
Single couples with moment 200
5.4.2 Double couples 200
Double couple without moment. Dilatation. 200
Double couple without moment and monopole force 201
Double couple without moment and single couple 201
5.5 Reciprocity of stress 202
Chapter 6 Reflection and transmission of plane waves 206
6.1 Reflection and transmission of SH waves 207
6.1.1 Symmetry plane of a homogeneous monoclinic medium 207
6.1.2 Complex stiffnesses of the incidence and transmission media 209
6.1.3 Reflection and transmission coefficients 210
6.1.4 Propagation, attenuation and energy directions 213
6.1.5 Brewster and critical angles 218
6.1.6 Phase velocities and attenuations 222
6.1.7 Energy-flux balance 224
6.1.8 Energy velocities and quality factors 226
6.2 Reflection and transmission of qP-qSV waves 228
6.2.1 Propagation characteristics 228
6.2.2 Properties of the homogeneous wave 230
6.2.3 Reflection and transmission coefficients 231
6.2.4 Propagation, attenuation and energy directions 232
6.2.5 Phase velocities and attenuations 233
6.2.6 Energy-flow balance 233
6.2.7 Umov-Poynting theorem, energy velocity and quality factor 235
6.2.8 Reflection of seismic waves 236
6.2.9 Incident inhomogeneous waves 247
Generation of inhomogeneous waves 248
Ocean bottom 249
6.3 Reflection and transmission at fluid/solid interfaces 251
6.3.1 Solid/fluid interface 251
6.3.2 Fluid/solid interface 252
6.3.3 The Rayleigh window 253
6.4 Reflection and transmission coefficients of a set of layers 254
Chapter 7 Biot's theory for porous media 258
7.1 Isotropic media. Strain energy and stress-strain relations 260
7.1.1 Jacketed compressibility test 260
7.1.2 Unjacketed compressibility test 261
7.2 The concept of effective stress 263
7.2.1 Effective stress in seismic exploration 265
Pore-volume balance 267
Acoustic properties 269
7.2.2 Analysis in terms of compressibilities 269
7.3 Anisotropic media. Strain energy and stress-strain relations 273
7.3.1 Effective-stress law for anisotropic media 277
7.3.2 Summary of equations 278
Pore pressure 279
Total stress 279
Effective stress 279
Skempton relation 279
Undrained-modulus matrix 279
7.3.3 Brown and Korringa's equations 279
Transversely isotropic medium 280
7.4 Kinetic energy 280
7.4.1 Anisotropic media 283
7.5 Dissipation potential 285
7.5.1 Anisotropic media 286
7.6 Lagrange's equations and equation of motion 286
7.6.1 The viscodynamic operator 288
7.6.2 Fluid flow in a plane slit 288
7.6.3 Anisotropic media 293
7.7 Plane-wave analysis 294
7.7.1 Compressional waves 294
Relation with Terzaghi's law 297
The diffusive slow mode 299
7.7.2 The shear wave 299
7.8 Strain energy for inhomogeneous porosity 301
7.8.1 Complementary energy theorem 302
7.8.2 Volume-averaging method 303
7.9 Boundary conditions 307
7.9.1 Interface between two porous media 307
Deresiewicz and Skalak's derivation 307
Gurevich and Schoenberg's derivation 309
7.9.2 Interface between a porous medium and a viscoelastic medium 311
7.9.3 Interface between a porous medium and a viscoacoustic medium 312
7.9.4 Free surface of a porous medium 312
7.10 The mesoscopic loss mechanism. White model 312
7.11 Green's function for poro-viscoacoustic media 318
7.11.1 Field equations 318
7.11.2 The solution 319
7.12 Green's function at a fluid/porous medium interface 322
7.13 Poro-viscoelasticity 326
7.14 Anisotropic poro-viscoelasticity 330
7.14.1 Stress-strain relations 331
7.14.2 Biot-Euler's equation 332
7.14.3 Time-harmonic fields 332
7.14.4 Inhomogeneous plane waves 335
7.14.5 Homogeneous plane waves 337
7.14.6 Wave propagation in femoral bone 339
Chapter 8 The acoustic-electromagnetic analogy 344
8.1 Maxwell's equations 346
8.2 The acoustic-electromagnetic analogy 347
8.2.1 Kinematics and energy considerations 352
8.3 A viscoelastic form of the electromagnetic energy 354
8.3.1 Umov-Poynting's theorem for harmonic fields 355
8.3.2 Umov-Poynting's theorem for transient fields 356
The Debye-Zener analogy 360
The Cole-Cole model 364
8.4 The analogy for reflection and transmission 365
8.4.1 Reflection and refraction coefficients 365
Propagation, attenuation and ray angles 366
Energy-flux balance 366
8.4.2 Application of the analogy 367
Refraction index and Fresnel's formulae 367
Brewster (polarizing) angle 368
Critical angle. Total reflection 369
Reflectivity and transmissivity 372
Dual fields 372
Sound waves 373
8.4.3 The analogy between TM and TE waves 374
Green's analogies 375
8.4.4 Brief historical review 378
8.5 3-D electromagnetic theory and the analogy 379
8.5.1 The form of the tensor components 380
8.5.2 Electromagnetic equations in differential form 381
8.6 Plane-wave theory 382
8.6.1 Slowness, phase velocity and attenuation 384
8.6.2 Energy velocity and quality factor 386
8.7 Analytical solution for anisotropic media 389
8.7.1 The solution 391
8.8 Finely layered media 392
8.9 The time-average and CRIM equations 395
8.10 The Kramers-Kronig dispersion relations 396
8.11 The reciprocity principle 397
8.12 Babinet's principle 398
8.13 Alford rotation 399
8.14 Poro-acoustic and electromagnetic diffusion 401
8.14.1 Poro-acoustic equations 401
8.14.2 Electromagnetic equations 403
The TM and TE equations 403
Phase velocity, attenuation factor and skin depth 404
Analytical solutions 404
8.15 Electro-seismic wave theory 405
Chapter 9 Numerical methods 408
9.1 Equation of motion 408
9.2 Time integration 409
9.2.1 Classical finite differences 411
9.2.2 Splitting methods 412
9.2.3 Predictor-corrector methods 413
The Runge-Kutta method 413
9.2.4 Spectral methods 413
9.2.5 Algorithms for finite-element methods 415
9.3 Calculation of spatial derivatives 415
9.3.1 Finite differences 415
9.3.2 Pseudospectral methods 417
9.3.3 The finite-element method 419
9.4 Source implementation 420
9.5 Boundary conditions 421
9.6 Absorbing boundaries 423
9.7 Model and modeling design – Seismic modeling 424
9.8 Concluding remarks 427
9.9 Appendix 428
9.9.1 Electromagnetic-diffusion code 428
9.9.2 Finite-differences code for the SH-wave equation of motion 432
9.9.3 Finite-differences code for the SH-wave and Maxwell's equations 438
9.9.4 Pseudospectral Fourier Method 445
9.9.5 Pseudospectral Chebyshev Method 447
Examinations 450
Chronology of main discoveries 454
Leonardo's manuscripts 466
A list of scientists 470
Bibliography 480
Name index 514
Subject index 526
Preface
(Second Edition, Revised and Extended)
This book presents the fundamentals of wave propagation in anisotropic, anelastic and porous media. I have incorporated in this second edition a chapter about the analogy between acoustic waves (in the general sense) and electromagnetic waves. The emphasis is on geophysical applications for seismic 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, non-destructive 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 porosity, anelasticity1 and anisotropy 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 modeling and inversion for the interpretation of the seismic 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, etc.
Wave simulation is a theoretical field of research that began nearly three 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.
This book examines 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 (stress-strain relation) appropriate for the particular rheology2. 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’s 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 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 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 modeling 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. We also consider finely layered and 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: Anelasticity and wave propagation. Attenuation is introduced in the form of Boltzmann’s 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 π/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’s 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.
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. 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 equal to the magnitude of the phase velocity.
Chapter 5: The reciprocity principle. Reciprocity is usually applied to concentrated point forces and point receivers. However, reciprocity has a much wider application potential; in many cases, it is not used at its full potential, either because a variety of source and receiver types are...
| Erscheint lt. Verlag | 24.1.2007 |
|---|---|
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Informatik |
| Naturwissenschaften ► Geowissenschaften ► Geografie / Kartografie | |
| Naturwissenschaften ► Geowissenschaften ► Geologie | |
| Naturwissenschaften ► Geowissenschaften ► Geophysik | |
| Naturwissenschaften ► Physik / Astronomie ► Mechanik | |
| Technik | |
| ISBN-10 | 0-08-046890-X / 008046890X |
| ISBN-13 | 978-0-08-046890-7 / 9780080468907 |
| Haben Sie eine Frage zum Produkt? |
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