Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media

Transcription

1 SECTION I. SEISMIC EXPLORATION Volume 38 WAVE FIELDS IN REAL MEDIA: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media (SECOND EDITION, REVISED AND EXTENDED) by Jose M. CARCIONE Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Borgo Grotta Gigante 42c, Sgonico, Trieste, Italy ELSEVIER Amsterdam - Boston - Heidelberg - London - New York - Oxford Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo

2 Contents Preface About the author Basic notation Glossary of main symbols xiii xix xx xxi Anisotropic elastic media Strain-energy density and stress-strain relation Dynamical equations Symmetries and transformation properties 6 Symmetry plane of a monoclinic medium 7 Transformation of the stiffness matrix Kelvin-Christoffel equation, phase velocity and slowness Transversely isotropic media Symmetry planes of an orthorhombic medium Orthogonality of polarizations Energy balance and energy velocity Group velocity Equivalence between the group and energy velocities Envelope velocity Example: Transversely isotropic media Elasticity constants from phase and group velocities Relationship between the slowness and wave surfaces 24 SH-wave propagation Finely layered media Anomalous polarizations Conditions for the existence of anomalous polarization Stability constraints Anomalous polarization in orthorhombic media Anomalous polarization in monoclinic media The polarization Example The best isotropic approximation Analytical solutions for transversely isotropic media...' D Green's function 40

3 vi CONTENTS D Green's function Reflection and transmission of plane waves Cross-plane shear waves 45 2 Viscoelasticity and wave propagation Energy densities and stress-strain relations Fading memory and symmetries of the relaxation tensor Stress-strain relation for 1-D viscoelastic media Complex modulus and storage and loss moduli Energy and significance of the storage and loss moduli Non-negative work requirements and other conditions Consequences of reality and causality Summary of the main properties 60 Relaxation function 60 Complex modulus Wave propagation concepts for 1-D viscoelastic media Wave propagation for complex frequencies Mechanical models and wave propagation Maxwell model Kelvin-Voigt model... ' Zener or standard linear solid model Burgers model Generalized Zener model 79 Nearly constant Q Nearly constant-q model with a continuous spectrum Constant-Q model and wave equation Phase velocity and attenuation factor Wave equation in differential form. Fractional derivatives 85 Propagation in Pierre shale The concept of centrovelocity ' D Green's function and transient solution Numerical evaluation of the velocities Example Memory variables and equation of motion Maxwell model Kelvin-Voigt model Zener model Generalized Zener model 95 3 Isotropic anelastic media Stress-strain relation Equations of motion and dispersion relations Vector plane waves Slowness, phase velocity and attenuation factor Particle motion of the P wave Particle motion of the S waves Polarization and orthogonality 106

4 CONTENTS vii 3.4 Energy balance, energy velocity and quality factor P wave S waves Boundary conditions and SnelPs law The correspondence principle Rayleigh waves Dispersion relation Displacement field Phase velocity and attenuation factor Special viscoelastic solids 120 Incompressible solid 120 Poisson solid 120 Hardtwig solid Two Rayleigh waves Reflection and transmission of cross-plane shear waves Memory variables and equation of motion Analytical solutions Viscoacoustic media L0.2 Constant-Q viscoacoustic media Viscoelastic media The elastodynamic of a non-ideal interface The interface model 130 Boundary conditions in differential form Reflection and transmission coefficients of SH waves 132 Energy loss Reflection and transmission coefficients of P-SV waves 133 Energy loss 135 Examples Anisotropic anelastic media Stress-strain relations Model 1: Effective anisotropy Model 2: Attenuation via eigenstrains Model 3: Attenuation via mean and deviatoric stresses Wave velocities, slowness and attenuation-vector Energy balance and fundamental relations Plane waves. Energy velocity and quality factor Polarizations The physics of wave propagation for viscoelastic SH waves Energy velocity Group velocity Envelope velocity Perpendicularity properties Numerical evaluation of the energy velocity Forbidden directions of propagation Memory variables and equation of motion in the time domain 162

5 viii CONTENTS Strain memory variables Memory-variable equations SH equation of motion qp-qsv equation of motion Analytical solution for SH waves in monoclinic media The reciprocity principle Sources, receivers and reciprocity The reciprocity principle Reciprocity of particle velocity. Monopoles Reciprocity of strain Single couples 174 Single couples without moment 177 Single couples with moment Double couples 177 Double couple without moment. Dilatation 177 Double couple without moment and monopole force 178 Double couple without moment and single couple Reciprocity of stress Reflection and transmission of plane waves Reflection and transmission of SH waves Symmetry plane of a homogeneous monoclinic medium Complex stiffnesses of the incidence and transmission media Reflection and transmission coefficients Propagation, attenuation and energy directions Brewster and critical angles Phase velocities and attenuations Energy-flux balance Energy velocities and quality factors Reflection and transmission of qp-qsv waves Propagation characteristics Properties of the homogeneous wave Reflection and transmission coefficients Propagation, attenuation and energy directions Phase velocities and attenuations Energy-flow balance Umov-Poynting theorem, energy velocity and quality factor Reflection of seismic waves Incident inhomogeneous waves 224 Generation of inhomogeneous waves 225 Ocean bottom Reflection and transmission at fluid/solid interfaces Solid/fluid interface Fluid/solid interface The Rayleigh window Reflection and transmission coefficients of a set of layers 231

6 CONTENTS ix 7 Biot's theory for porous media Isotropic media. Strain energy and stress-strain relations Jacketed compressibility test Unjacketed compressibility test The concept of effective stress Effective stress in seismic exploration 242 Pore-volume balance 244 Acoustic properties Analysis in terms of compressibilities Anisotropic media. Strain energy and stress-strain relations Effective-stress law for anisotropic media Summary of equations 255 Pore pressure 256 Total stress 256 Effective stress 256 Skempton relation 256 Undrained-modulus matrix Brown and Korringa's equations 256, Transversely isotropic medium Kinetic energy Anisotropic media Dissipation potential Anisotropic media : Lagrange's equations and equation of motion The viscodynamic operator Fluid flow in a plane slit Anisotropic media Plane-wave analysis Compressional waves 271 Relation with Terzaghi's law 274 The diffusive slow mode The shear wave Strain energy for inhomogeneous porosity Complementary energy theorem Volume-averaging method Boundary conditions Interface between two porous media 284 Deresiewicz and Skalak's derivation 284 Gurevich and Schoenberg's derivation Interface between a porous medium and a viscoelastic medium Interface between a porous medium and a viscoacoustic medium Free surface of a porous medium The mesoscopic loss mechanism. White model Green's function for poro-viscoacoustic media Field equations The solution 296

7 CONTENTS 7.12 Green's function at a fluid/porous medium interface Poro-viscoelasticity Anisotropic poro-viscoelasticity Stress-strain relations Biot-Euler's equation Time-harmonic fields Inhomogeneous plane waves Homogeneous plane waves Wave propagation in femoral bone 316 The acoustic-electromagnetic analogy Maxwell's equations The acoustic-electromagnetic analogy Kinematics and energy considerations A viscoelastic form of the electromagnetic energy Umov-Poynting's theorem for harmonic fields Umov-Poynting's theorem for transient fields 333 The Debye-Zener analogy 337 The Cole-Cole model., The analogy for reflection and transmission Reflection and refraction coefficients 342 Propagation, attenuation and ray angles 343 Energy-flux balance Application of the analogy 344 Refraction index and Fresnel's formulae 344 Brewster (polarizing) angle 345 Critical angle. Total reflection 346 Reflectivity and transmissivity 349 Dual fields 349 Sound waves The analogy between TM and TE waves 351 Green's analogies Brief historical review D electromagnetic theory and the analogy The form of the tensor components Electromagnetic equations in differential form Plane-wave theory Slowness, phase velocity and attenuation Energy velocity and quality factor Analytical solution for anisotropic media The solution Finely layered media The time-average and CRIM equations The Kramers-Kronig dispersion relations The reciprocity principle Babinet's principle 375

8 CONTENTS xi 8.13 Alford rotation Poro-acoustic and electromagnetic diffusion Poro-acoustic equations Electromagnetic equations 380 The TM and TE equations 380 Phase velocity, attenuation factor and skin depth 381 Analytical solutions Electro-seismic wave theory Numerical methods Equation of motion Time integration Classical finite differences Splitting methods Predictor-corrector methods 390 The Runge-Kutta method Spectral methods Algorithms for finite-element methods Calculation of spatial derivatives Finite differences Pseudospectral methods The finite-element method Source implementation Boundary conditions Absorbing boundaries Model and modeling design - Seismic modeling Concluding remarks Appendix Electromagnetic-diffusion code Finite-differences code for the. SH-wave equation of motion Finite-differences code for the'sh-wave and Maxwell's equations Pseudospectral Fourier Method Pseudospectral Chebyshev Method 424 Examinations 427 Chronology of main discoveries 431 Leonardo's manuscripts 443 A list of scientists 447 Bibliography 457 Name index 491 Subject index 503