Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media

Front Cover
Wave Fields in Real Media
Copyright
Contents
About the author
Preface
Basic notation
Glossary of main symbols
1 Anisotropic elastic media
1.1 Strain-energy density and stress-strain relations
1.2 Dynamical equations
1.2.1 Symmetries and transformation properties
Symmetry plane of a monoclinic medium
Transformation of the stiffness matrix
1.3 Kelvin-Christoffel equation, phase velocity and slowness
1.3.1 Transversely isotropic media
1.3.2 Symmetry planes of an orthorhombic medium
1.3.3 Orthogonality of polarizations
1.4 Energy balance and energy velocity
1.4.1 Group velocity
1.4.2 Equivalence between the group and energy velocities
1.4.3 Envelope velocity
1.4.4 Example: transversely isotropic media
1.4.5 Elasticity constants from phase and group velocities
1.4.6 Relationship between the slowness and wave surfaces
SH-wave propagation
1.5 Finely layered media
1.5.1 The Schoenberg-Muir averaging theory
Examples
1.6 Anomalous polarizations
1.6.1 Conditions for the existence of anomalous polarization
1.6.2 Stability constraints
1.6.3 Anomalous polarization in orthorhombic media
1.6.4 Anomalous polarization in monoclinic media
1.6.5 The polarization
1.6.6 Example
1.7 The best isotropic approximation
1.8 Analytical solutions
1.8.1 2-D Green function
1.8.2 3-D Green function
1.9 Reflection and transmission of plane waves
1.9.1 Cross-plane shear waves
2 Viscoelasticity and wave propagation
2.1 Energy densities and stress-strain relations
2.1.1 Fading memory and symmetries of the relaxation tensor
2.2 Stress-strain relation for 1-D viscoelastic media
2.2.1 Complex modulus and storage and loss moduli
2.2.2 Energy and significance of the storage and loss moduli
2.2.3 Non-negative work requirements and other conditions
2.2.4 Kramers-Kronig relations
Stability and passivity
2.2.5 Summary of the main properties
Relaxation function
Complex modulus
2.3 Wave propagation in 1-D viscoelastic media
2.3.1 Energy balance and related quantities
2.3.2 Kramers-Kronig relations for velocity and attenuation
Numerical evaluation
2.3.3 Wave propagation for complex frequencies
2.4 Mechanical models and wave propagation
2.4.1 Maxwell model
2.4.2 Kelvin-Voigt model
2.4.3 Zener or standard-linear-solid model
The dispersion index
2.4.4 Burgers model
2.4.5 Generalized Zener model
Nearly constant Q
2.4.6 Nearly constant-Q model with a continuous spectrum
2.5 Constant-Q model and wave equation
2.5.1 Phase velocity and attenuation factor
2.5.2 Wave equation in differential form. Fractional derivatives
Propagation in Pierre shale
2.6 The Cole-Cole model
2.7 Equivalence between source and initial conditions
2.8 Hysteresis cycles and fatigue
2.9 Distributed-order fractional time derivatives
2.9.1 The βn case
2.9.2 The generalized Dirac comb function
2.10 The concept of centrovelocity
2.10.1 1-D Green function and transient solution
2.10.2 Numerical evaluation of the velocities
2.10.3 Example
2.11 Memory variables and equation of motion
2.11.1 Maxwell model
2.11.2 Kelvin-Voigt model
2.11.3 Zener model
2.11.4 Generalized Zener model
2.12 Instantaneous frequency and quality factor
2.12.1 The instantaneous frequency
2.12.2 The instantaneous quality factor
2.12.3 Examples
The Mandel signal
Propagation in a lossy homogeneous medium
3 Isotropic anelastic media
3.1 Stress-strain relations
3.2 Equations of motion and dispersion relations
3.3 Vector plane waves
3.3.1 Slowness, phase velocity, and attenuation factor
3.3.2 Particle motion of the P wave
3.3.3 Particle motion of the S waves
3.3.4 Polarization and orthogonality
3.4 Energy balance, velocity and quality factor
3.4.1 P wave
3.4.2 S waves
3.4.3 Expression relating different quality factors
3.5 Boundary conditions and Snell law
3.6 The correspondence principle
3.7 Rayleigh waves
3.7.1 Dispersion relation
3.7.2 Displacement field
3.7.3 Phase velocity and attenuation factor
3.7.4 Special viscoelastic solids
Incompressible solid
Poisson solid
Hardtwig solid
3.7.5 Two Rayleigh waves
3.8 Reflection and transmission of SH waves
3.9 Memory variables and equation of motion
3.10 Analytical solutions
3.10.1 Viscoacoustic media
3.10.2 Constant-Q viscoacoustic media
3.10.3 Viscoelastic media
3.10.4 Pekeris solution for Lamb problem
3.11 Constant-Q P- and S-waves
3.11.1 Time fractional derivatives
3.11.2 Spatial fractional derivatives
3.12 Wave equations based on the Burgers model
3.12.1 Propagation of P-SV waves
3.12.2 Propagation of SH waves
3.13 P-SV wave equation based on the Cole-Cole model
3.14 The elastodynamic of a non-ideal interface
3.14.1 The interface model
Boundary conditions in differential form
3.14.2 Reflection and transmission coefficients of SH waves
Energy loss
3.14.3 Reflection and transmission coefficients of P-SV waves
Energy loss
Examples
3.15 SH-wave transfer function
4 Anisotropic anelastic media
4.1 Stress-strain relations
4.1.1 Model 1: effective anisotropy
4.1.2 Model 2: attenuation via eigenstrains
4.1.3 Model 3: attenuation via mean and deviatoric stresses
4.2 Fracture-induced anisotropic attenuation
4.2.1 The equivalent monoclinic medium
4.2.2 The orthorhombic equivalent medium
4.2.3 HTI equivalent media
4.3 Stiffness tensor from oscillatory experiments
4.4 Wave velocities, slowness and attenuation vector
4.5 Energy balance and fundamental relations
4.5.1 Plane waves. Energy velocity and quality factor
4.5.2 Polarizations
4.6 Propagation of SH waves
4.6.1 Energy velocity
4.6.2 Group velocity
4.6.3 Envelope velocity
4.6.4 Perpendicularity properties
4.6.5 Numerical evaluation of the energy velocity
4.6.6 Forbidden directions of propagation
4.7 Wave propagation in symmetry planes
4.7.1 Properties of the homogeneous wave
4.7.2 Propagation, attenuation and energy directions
4.7.3 Phase velocities and attenuations
4.7.4 Energy balance, velocity and quality factor
4.7.5 Explicit equations in symmetry planes
4.8 Summary of plane-wave equations
4.8.1 SH wave
Elastic anisotropic medium
Anelastic anisotropic medium. Homogeneous waves
Anelastic anisotropic medium. Inhomogeneous waves
4.8.2 qP-qS waves
Anelastic anisotropic medium. Homogeneous waves
Anelastic anisotropic medium. Inhomogeneous waves
4.9 Memory variables and equation of motion
4.9.1 Strain memory variables
4.9.2 Memory-variable equations
4.9.3 SH equation of motion
4.9.4 qP-qSV equation of motion
4.10 Fermat principle and its relation to Snell law
4.11 Analytical transient solution for SH waves
5 The reciprocity principle
5.1 Sources, receivers and reciprocity
5.2 The reciprocity principle
5.3 Reciprocity of particle velocity. Monopoles
5.4 Reciprocity of strain
5.4.1 Single couples
Single couples without moment
Single couples with moment
5.4.2 Double couples
Double couple without moment. Dilatation
Double couple without moment and monopole force
Double couple without moment and single couple
5.5 Reciprocity of stress
5.6 Reciprocity principle for flexural waves
5.6.1 Equation of motion
5.6.2 Reciprocity of the deflection
5.6.3 Reciprocity of the bending moment
6 Reflection and transmission coefficients
6.1 Reflection and transmission of SH waves
6.1.1 Symmetry plane of a homogeneous monoclinic medium
6.1.2 Complex stiffnesses
6.1.3 Reflection and transmission coefficients
6.1.4 Propagation, attenuation and energy directions
6.1.5 Brewster and critical angles
6.1.6 Phase velocities and attenuations
6.1.7 Energy-flux balance
6.1.8 Energy velocities and quality factors
6.1.9 The physical solution in ray-tracing algorithms
6.2 Reflection and transmission of qP-qSV waves
6.2.1 Phase velocities and attenuations
6.2.2 Energy-flow balance
6.2.3 Reflection of seismic waves
6.2.4 Incident inhomogeneous waves
Generation of inhomogeneous waves
Ocean bottom
6.3 Interfaces separating a solid and a fluid
6.3.1 Solid/fluid interface
6.3.2 Fluid/solid interface
6.3.3 The Rayleigh window
6.4 Scattering coefficients of a set of layers
7 Biot theory for porous media
7.1 Isotropic media. Stress-strain relations
7.1.1 Jacketed compressibility test
7.1.2 Unjacketed compressibility test
7.2 Gassmann equation and effective stress
7.2.1 Effective stress in seismic exploration
Pore-volume balance
Acoustic properties
7.2.2 Analysis in terms of compressibilities
7.3 Pore-pressure build-up in source rocks
7.4 The asperity-deformation model
7.5 Anisotropic media. Stress-strain relations
7.5.1 Effective-stress law for anisotropic media
7.5.2 Summary of equations
Pore pressure
Total stress
Effective stress
Skempton relation
Undrained-modulus matrix
7.5.3 Brown and Korringa equations
Transversely isotropic medium
7.6 Kinetic energy
7.6.1 Anisotropic media
7.7 Dissipation potential
7.7.1 Anisotropic media
7.8 Lagrange equations and equation of motion
7.8.1 The viscodynamic operator
7.8.2 Fluid flow in a plane slit
7.8.3 Anisotropic media
7.9 Plane-wave analysis
7.9.1 Compressional waves
Relation with Terzaghi law and the second P wave
The diffusive slow mode
7.9.2 The shear wave
7.10 Strain energy for inhomogeneous porosity
7.10.1 Complementary energy theorem
7.10.2 Volume-averaging method
7.11 Boundary conditions
7.11.1 Interface between two porous media
Deresiewicz and Skalak's derivation
Gurevich and Schoenberg's derivation
7.11.2 Interface between a porous medium and a viscoelastic medium
7.11.3 Interface between a porous medium and a viscoacoustic medium
7.11.4 Free surface of a porous medium
7.12 Reflection and transmission coefficients
7.13 Extension of Biot theory to a composite frame
7.13.1 Strain-energy density
7.13.2 Example: frozen porous media
Stress-strain relations
Conservation of momentum
Phase velocity and attenuation factor
7.14 Extension of Biot theory to partial saturation
7.14.1 Conservation of momentum
7.14.2 Stress-strain relations
7.14.3 Phase velocity and attenuation factor
7.15 Squirt-flow dissipation
7.16 The mesoscopic-loss mechanism. White model
7.17 The Biot-Gardner effect
7.17.1 Radial effect
7.17.2 Axial effect
7.18 Green function for poro-viscoacoustic media
7.18.1 Field equations
7.18.2 The solution
7.19 Green function for a fluid/solid interface
7.20 Poro-viscoelasticity
7.21 Bounds and averages on velocity and quality factor
7.21.1 Stiffness bounds
7.21.2 Stiffness averages
7.22 Anelasticity models based on ellipsoidal pores
7.23 Effect of melt on wave propagation
7.23.1 Arrhenius viscosity related to grain-boundary relaxation
7.23.2 Stress-strain relations and wave properties
Walsh model
CPA model
7.24 Anisotropy and poro-viscoelasticity
7.24.1 Stress-strain relations
7.24.2 Biot-Euler equation
7.24.3 Time-harmonic fields
7.24.4 Inhomogeneous plane waves
7.24.5 Homogeneous plane waves
7.24.6 Wave propagation in femoral bone
7.25 Gassmann equation for a solid pore infill
7.26 Mesoscopic loss in layered and fractured media
7.26.1 Effective fractured medium
7.27 Fluid-pressure diffusion
7.27.1 Solution to the linear diffusion equation
7.27.2 Solutions to the non-linear diffusion equation
Non-linear pressure equation
Solution for radial horizontal flow and constant mass injection
Solution in spherical coordinates and constant pressure injection
Triggering front
7.27.3 Biot classical equation with fractional derivatives
Frequency-wavenumber domain analysis
7.28 Thermo-poroelasticity
7.28.1 Equations of motion
7.28.2 Plane-wave analysis
7.28.3 Example
7.28.4 Kjartansson model
8 The acoustic-electromagnetic analogy
8.1 Maxwell equations
8.2 The acoustic-electromagnetic analogy
8.2.1 Kinematics and energy considerations
8.3 A viscoelastic form of the electromagnetic energy
8.3.1 Umov-Poynting theorem for harmonic fields
8.3.2 Umov-Poynting theorem for transient fields
The Debye-Zener analogy
The Cole-Cole model
8.4 The analogy for reflection and transmission
8.4.1 Reflection and refraction coefficients
Propagation, attenuation and ray angles
Energy-flux balance
8.4.2 Application of the analogy
Refraction index and Fresnel formulae
Brewster (polarizing) angle
Critical angle. Total reflection
Reflectivity and transmissivity
Dual fields
Sound waves
8.4.3 The analogy between TM and TE waves
Green analogies
8.4.4 Brief historical review
8.5 The layer problem
8.5.1 TM-SH-TE analogy
8.5.2 Analogy with quantum mechanics. The tunnel effect
8.5.3 SH-TM analogy for two layers. Magnetotellurics
Surface impedance and apparent viscosity
Surface impedance and apparent resistivity
Example
8.6 3-D electromagnetic theory and the analogy
8.6.1 The form of the tensor components
8.6.2 Electromagnetic equations in differential form
8.7 Plane-wave theory
8.7.1 Slowness, phase velocity and attenuation
8.7.2 Energy velocity and quality factor
8.8 Electromagnetic diffusion in anisotropic media
8.8.1 Differential equations
8.8.2 Dispersion relation
8.8.3 Slowness, kinematic velocities, attenuation and skin depth
8.8.4 Umov-Poynting theorem and energy velocity
8.8.5 Fundamental relations
8.9 Analytical solution for anisotropic media
8.9.1 The solution
8.10 Elastic medium with Fresnel wave surface
8.10.1 Fresnel wave surface
8.10.2 Equivalent elastic medium
8.11 Finely layered media
8.12 The time-average and CRIM equations
8.13 Kramers-Kronig relations
8.14 The reciprocity principle
8.15 Babinet principle
8.16 Alford rotation
8.17 Cross-property relations
8.18 Poroacoustic and electromagnetic diffusion
8.18.1 Poroacoustic equations
8.18.2 Electromagnetic equations
The TM and TE equations
Phase velocity, attenuation factor and skin depth
Analytical solutions
8.19 Electro-seismic wave theory
8.20 Gravitational waves
9 Numerical methods
9.1 Equation of motion
9.2 Time integration
9.2.1 Classical finite differences
9.2.2 Splitting methods
9.2.3 Predictor-corrector methods
The Runge-Kutta method
9.2.4 Fractional calculus
9.2.5 Spectral methods
9.2.6 Algorithms for finite-element methods
9.3 Calculation of spatial derivatives
9.3.1 Finite differences
9.3.2 Pseudospectral methods
9.3.3 The finite-element method
9.4 Source implementation
9.5 Boundary conditions
9.6 Absorbing boundaries
9.7 Model and modeling design. Seismic modeling
9.8 Concluding remarks
9.9 Appendix
9.9.1 Fractional calculus
Grünwald-Letnikov and central-difference approximations
Euler scheme
CL method
GMMP method
Adams-Bashforth-Moulton scheme
Discretization of the 2-D P-SV Cole-Cole wave equation
9.9.2 Electromagnetic-diffusion code
9.9.3 Finite-differences code for SH waves
9.9.4 Finite-difference code for SH (TM) waves
9.9.5 Pseudospectral Fourier method
Calculation of fractional derivatives
9.9.6 Pseudospectral Chebyshev method
9.9.7 Pseudospectral sine/cosine method
9.9.8 Earthquake sources. The moment-tensor
9.9.9 3-D anisotropic media. Free-surface boundary treatment
9.9.10 Modeling in cylindrical coordinates
Equations for axis-symmetric single-phase media
Equations for poroelastic media
Examinations
Chronology of main discoveries
Leonardo's manuscripts
A list of scientists
Bibliography
Name index
Subject index
Back Cover