Existence of transverse waves in anisotropic poroelastic media

Out of the four waves in an anisotropic poroelastic medium, two are termed as quasi-transverse waves. The prefix 'quasi' refers to their polarizations being nearly, but not exactly, perpendicular to direction of propagation. In this composite medium, unlike perfectly elastic medium, the propagation of a longitudinal wave along a phase direction may not be accompanied by transverse waves. The existence of a transverse wave in anisotropic poroelastic media is ensured by the two equations restricting the choice of elastic coefficients of porous aggregate as well as fluid–solid coupling. Necessary and sufficient conditions for the existence of transverse waves along the coordinate axes and in the coordinate planes for general anisotropy are discussed. The discussion is extended to the case of orthotropic materials and existence for few specific phase directions is also explored. The conditions for the transverse waves decided on the basis of their apparent polarizations, that is, particle motion being perpendicular to ray direction, are also discussed. For a particular numerical model, the existence of these apparent transverse waves is solved numerically for phase directions in coordinate planes. For general directions of phase propagation, the existence of these transverse waves is checked graphically for the chosen numerical model.     

Energy flux in viscoelastic anisotropic media

We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P , SV and SH waves.     

On the computation of shear waves by the ray method

Summary. The ray series solution of the elastodynamic equation of motion for shear waves propagating through a laterally inhomogeneous three-dimensional medium can be simplified by the use of a particular coordinate system that accompanies the wave front along the ray of investigation. The system is entirely determined by parameters that are obtainable from the ray. The transport equations for the principal shear wave components are then no longer coupled, but reduce to the same type of equation which determines the principal compressional wave component.     

Surface-wave propagation in a cracked poroelastic half-space lying under a uniform layer of fluid

The effect of cracks on the elastic properties of an isotropic elastic solid is studied when the cracks are saturated with a soft fluid. A polynomial equation in effective Poisson's ratio is obtained, whose coefficients are functions of Poisson's ratio of the uncracked solid, crack density and saturating fluid parameter. Elastic and dynamical constants used in Blot's theory of wave propagation in poroelastic solids are modified for the introduction of cracks. The effects of cracks on the velocities of three types of waves are observed numerically. The frequency equation is derived for the propagation of Rayleigh-type surface waves in a saturated poroelastic half-space lying under a uniform layer of liquid. Dispersion curves for a particular model of oceanic crust containing cracks are plotted. The effects of variations in crack density and saturation on the phase and group velocity are also analysed.     

Quasi-isotropic approximation of ray theory for anisotropic media

Wave propagation in weakly anisotropic inhomogeneous media is studied by the quasi-isotropic approximation of ray theory. The approach is based on the ray-tracing and dynamic ray-tracing differential equations for an isotropic background medium. In addition, it requires the integration of a system of two complex coupled differential equations along the isotropic ray.
The interference of the qS waves is described by traveltime and polarization corrections of interacting isotropic S waves. For qP waves the approach leads to a correction of the traveltime of the P wave in the isotropic background medium.
Seismograms and particle-motion diagrams obtained from numerical computations are presented for models with different strengths of anisotropy.
The equivalence of the quasi-isotropic approximation and the quasi-shear-wave coupling theory is demonstrated. The quasi-isotropic approximation allows for a consideration of the limit from weak anisotropy to isotropy, especially in the case of qS waves, where the usual ray theory for anisotropic media fails.     

Hydraulic crack propagation in a porous medium

We develop a model for the propagation of a fluid-filled crack in a porous medium. The problem is motivated by the mechanism whereby drainage networks may form in partially molten rock below the Earth's lithosphere. Other applications include the propagation of hydraulic fractures in jointed rocks and in oil drilling operations, and the formation of dessication cracks in soils. Motivated by the lithospheric problem, we study a situation in which gravity acts in the direction of crack propagation. The model couples the elastoliydrodynamic equations of crack propagation with a pore pressure field in the porous rock, which drives the fluid flow which supplies the crack. The effect of the pore flow is to include a diffusional term in the evolution equation for the crack width, thus allowing a crack initiated at the base of the lithosphere to propagate down into the asthenosphere. Asymptotic and numerical solutions are presented for this crack evolution. However, the predicted drainage of melt into this crack is tiny compared with the upward percolative melt migration, and the predicted width of cracks (millimetres) is much too small to allow propagation of melt into the lithosphere without freezing. As a mechanism to explain magma fracturing in the lithosphere, the process described here therefore requires further refinement.     

Asymptotic eigenfrequencies of the Earth's normal modes

We derive asymptotic formulae for the toroidal and spheroidal eigenfrequencies of a SNREI earth model with two discontinuities, by considering the constructive interference of propagating SH and P-SV body waves. For a model with a smooth solid inner core, fluid outer core and mantle, there are four SH and 10 P-SV ray parameters regimes, each of which must be examined separately. The asymptotic eigenfrequency equations in each of these regimes depend only on the intercept times of the propagating wave types and the reflection and transmission coefficients of the waves at the free surface and the two discontinuities. If the classical geometrical plane-wave reflection and transmission coefficients are used, the final eigenfrequency equations are all real. In general, the asymptotic eigenfrequencies agree extremely well with the exact numerical eigenfrequencies; to illustrate this, we present comparisons for a crustless version of earth model 1066A.     

Guided waves in three-dimensional structures

A new formulation for the propagation of surface waves in three-dimensionally varying media is developed in terms of modal interactions. A variety of assumptions can be made about the nature of the modal field: a single set of reference modes, a set of local modes for the structure beneath a point, or a set of local modes for a laterally varying reference structure. Each modal contribution is represented locally as a spectrum of plane waves propagating in different directions in the horizontal plane. The influence of 3-D structure is included by allowing coupling between different modal branches and propagation directions. For anisotropic models, with allowance for attenuation, the treatment leads to a set of coupled 2-D partial differential equations for the weight functions for different modal orders.
The representation of the guided wavefield requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as different polarization states of the modal spectrum. The coupling equations describe the interaction between the different polarizations induced by the presence of the 3-D structure.
The level of lateral variation within the 3-D model is not required to be small. Horizontal refraction or reflection of the surface wavefield can be included by allowing for transfer between modes travelling in different directions. Approximate forms of the coupled equation system can be employed when the level of heterogeneity is small, for example the coupling between the fundamental mode and higher modes can often be neglected, or forward propagation can be emphasized by restricting the interaction to a limited band of plane waves covering the expected direction of propagation.     

Computation of qP-wave rays, traveltimes and slowness vectors in orthorhombic media

The eikonal equation is the equation of the phase slowness surface for isotropic and anisotropic media. In general anisotropic media, there is no simple explicit expression for the phase slowness surface. An approximate expression of the eikonal equation may be obtained in weakly anisotropic media. In orthorhombic media, the approximate eikonal equation of the qP wave is the sum of an ellipsoidal form and a more complicated term. The ellipsoidal form corresponds to what we call ellipsoidal anisotropy. Ray equations written in the Hamiltonian formulation are characteristics of the eikonal equation. Ray perturbation theory may be used to compute changes in ray paths and physical attributes (traveltime, polarization, amplitude) due to changes in the medium with respect to a reference medium. Examples obtained in homogeneous orthorhombic media show that a reference medium with ellipsoidal anisotropy is a better choice to develop the perturbation approach than an isotropic reference medium. Models with strong anisotropy can be considered. The comparison with results obtained by an exact ray program shows a relative traveltime error of less than 0.5 per cent for a model with relatively strong anisotropy. We propose a finite element approach in which the medium is divided into a set of elements with polynomial elastic parameter distributions. Inside each element, using a perturbation approach, analytical expressions for rays and traveltimes are obtained Ray tracing reduces to connecting these analytical solutions at the vertices of the cells.     

Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modelling

The perfectly matched layer (PML) absorbing boundary condition is incorporated into an irregular-grid elastic-wave modelling scheme, thus resulting in an irregular-grid PML method. We develop the irregular-grid PML method using the local coordinate system based PML splitting equations and integral formulation of the PML equations. The irregular-grid PML method is implemented under a discretization of triangular grid cells, which has the ability to absorb incident waves in arbitrary directions. This allows the PML absorbing layer to be imposed along arbitrary geometrical boundaries. As a result, the computational domain can be constructed with smaller nodes, for instance, to represent the 2-D half-space by a semi-circle rather than a rectangle. By using a smooth artificial boundary, the irregular-grid PML method can also avoid the special treatments to the corners, which lead to complex computer implementations in the conventional PML method. We implement the irregular-grid PML method in both 2-D elastic isotropic and anisotropic media. The numerical simulations of a VTI lamb's problem, wave propagation in an isotropic elastic medium with curved surface and in a TTI medium demonstrate the good behaviour of the irregular-grid PML method.     

Scattering of surface waves modelled by the integral equation method

The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at   R = 0  , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.     

Gaussian beams for surface waves in laterally slowly-varying media

Summary. Asymptotic ray theory is applied to surface waves in a medium where the lateral variations of structure are very smooth. Using ray-centred coordinates, parabolic equations are obtained for lateral variations while vertical structural variations at a given point are specified by eigenfunctions of normal mode theory as for the laterally homogeneous case. Final results on wavefields close to a ray can be expressed by formulations similar to those for elastic body waves in 2-D laterally heterogeneous media, except that the vertical dependence is described by eigenfunctions of 'local' Love or Rayleigh waves. The transport equation is written in terms of geometrical-ray spreading, group velocity and an energy integral. For the horizontal components there are both principal and additional components to describe the curvature of rays along the surface, as in the case of elastic body waves. The vertical component is decoupled from the horizontal components. With complex parameters the solutions for the dynamic ray tracing system correspond to Gaussian beams: the amplitude distribution is bell-shaped along the direction perpendicular to the ray and the solution is regular everywhere, even at caustics. Most of the characteristics of Gaussian beams for 2-D elastic body waves are also applicable to the surface wave case. At each frequency the solution may be regarded as a set of eigenfunctions propagating over a 2-D surface according to the phase velocity mapping.     

Reflection coefficients for weak anisotropic media

The interaction of plane elastic waves with a plane boundary between two anisotropic elastic half-spaces is investigated. The anisotropy dealt with in this study is of a general type. Explicit expressions for energy-related reflection and transmission coefficients are derived. They represent an approximation which is valid for a small deviation of the elastic parameters from isotropy.
Classical perturbation theory is applied on a 6times6 non-symmetric real eigenvalue problem to calculate first-order corrections for the polarization and stress of the plane waves. The explicit solution of the isotropic problem is used as a reference case. Degenerate perturbation theory is used to consider the splitting of the isotropic S -wave into two anisotropic qS-waves. The boundary conditions for two half-spaces in welded contact lead to a 6times6 system of linear equations. A correction to the isotropic solution is calculated by linearization. The resultant coefficients are functions of horizontal slowness, Lamé parameters and densities of the reference media, and of the perturbation of the elasticity tensors from isotropy.     

Wave propagation in fluid-saturated media: waveform and spectral analysis

A new numerical approach to the solution of waves propagating in a fluid-saturated medium, using Biot's theory as a foundation, has important implications for oil reservoir management and earthquake prediction. A numerical scheme is developed using an exponential transformation that explicitly treats the petrophysical and fluid properties of the medium within the framework of a generalized model. The scheme accounts for wave dissipation and velocity modifications. The numerical solution is used to perform numerical experiments to study the dynamic behaviour of waves in a fluid-saturated medium at well-logging frequencies (15 kHz). The results from the numerical experiments indicate that the degree of saturation by a high-viscosity fluid (HVF) such as oil, the temperature and the porosity of a medium strongly influence the spectral power distribution, frequency content and the velocity of waves propagating through the medium. An increase in HVF saturation causes enhanced attenuation of the low-frequency components, and increases the seismic velocity. An increase in porosity, however, enriches the low-frequency components and decreases the seismic velocity. A spectral quantification procedure is suggested and used to obtain information about the petrophysical and fluid properties of the medium from the spectral characteristics of the transmitted waveform. The procedure involves segmentation of the energy or power distribution of the transmitted waveforms into specified energy bands. The energy or power in these bands is then estimated. The extracted quantification variables are found to have strong correlations with the degree of HVF saturation, and the temperature and the porosity of the medium.     

Real and complex spectra – a generalization of WKBJ seismograms

Real plane-waves constitute the building blocks for recently developed spectral techniques in synthetic seismology. While providing numerical convenience, real slowness-spectra model certain wave phenomena in a distributed 'unnatural' way, whereas complex spectra model these phenomena in a compact, more 'natural' way. The theory of complex spectra, called by us the 'Spectral Theory of Transients' (STT) and developed elsewhere, is summarized here and contrasted with the real-spectrum approach. Relying strongly on the theory of analytic functions, STT permits the transient responses to be classified and evaluated according to the singularities they introduce in the complex slowness plane. The method is illustrated for a number of 2-D SH -wave model propagation environments, including interface reflection, head waves, multiple encounters with caustics due to concave boundaries or ducting medium inhomogeneities, and diffraction by structures with edges.     

Synthetic logs of multipole sources in boreholes based on the Kelvin–Voigt stress–strain relation

We design a numerical algorithm for wave simulation in a borehole due to multipole sources. The stress–strain relation of the formation is based on the Kelvin–Voigt mechanical model to describe the attenuation. The modelling, which requires two anelastic parameters and twice the spatial derivatives of the lossless case, simulates 3-D waves in an axisymmetric medium by using the Fourier and Chebyshev methods to compute the spatial derivatives along the vertical and horizontal directions, respectively. Instabilities of the Chebyshev differential operator due to the implementation of the fluid–solid boundary conditions are solved with a characteristic approach, where the characteristic variables are evaluated at the source central frequency. The algorithm uses two meshes to model the fluid and the solid. The presence of the logging tool is modelled by imposing rigid boundary conditions at the inner surface of the fluid mesh. Examples illustrating the propagation of waves are presented, namely, by using monopoles, dipoles and a quadrupoles as sources in hard and soft formations. Moreover, the presence of casing and layers is considered. The modelling correctly simulates the features—traveltime and attenuation—of the wave modes observed in sonic logs, namely, the P and S body waves, the Stoneley wave, and the dispersive S waves in the case of multipole sources.     

Guided wave propagation in laterally varying media - I. Theoretical development

Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed to describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution of the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH -waves, P - SV -waves and full anisotropy.
The reflected and transmitted wavefields for a region of heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields to those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able to derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg - and Sn -waves in a multilayered model for which there are 19 coupled modes.
The method is applicable to three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse to the propagation path, leads to a net rate of increase of the transverse components of the seismogram at the expense of the other components.     

Time-averaged and time-dependent energy-related quantities of harmonic waves in inhomogeneous viscoelastic anisotropic media

The energy–flux vector and other energy-related quantities play an important role in various wave propagation problems. In acoustics and seismology, the main attention has been devoted to the time-averaged energy flux of time-harmonic wavefields propagating in non-dissipative, isotropic and anisotropic media. In this paper, we investigate the energy–flux vector and other energy-related quantities of wavefields propagating in inhomogeneous anisotropic viscoelastic media. These quantities satisfy energy-balance equations, which have, as we show, formally different forms for real-valued wavefields with arbitrary time dependence and for time-harmonic wavefields. In case of time-harmonic wavefields, we study both time-averaged and time-dependent constituents of the energy-related quantities. We show that the energy-balance equations for time-harmonic wavefields can be obtained in two different ways. First, using real-valued wavefields satisfying the real-valued equation of motion and stress–strain relation. Second, using complex-valued wavefields satisfying the complex-valued equation of motion and stress–strain relation. The former approach yields simple results only for particularly simple viscoelastic models, such as the Kelvin–Voigt model. The latter approach is considerably more general and can be applied to viscoelastic models of unrestricted anisotropy and viscoelasticity. Both approaches, when applied to the Kelvin–Voigt viscoelastic model, yield the same expressions for the time-averaged and time-dependent constituents of all energy-related quantities and the same energy-balance equations. This indicates that the approach based on complex-valued representation of the wavefield may be used for time harmonic waves quite universally. This study also shows importance of joint consideration of time-averaged and time-dependent constituents of the energy-related quantities in some applications.     

Quadrangle-grid velocity–stress finite difference method for poroelastic wave equations

A quadrangle-grid velocity–stress finite difference method, based on a first-order hyperbolic system that is equivalent to Biot's equations, is developed for the simulation of wave propagation in 2-D heterogeneous porous media. In this method the velocity components of the solid material and of the pore fluid relative to that of the solid, and the stress components of three solid stresses and one fluid pressure are defined at different nodes for a staggered non-rectangular grid. The scheme uses non-orthogonal grids, allowing surface topography and curved interfaces to be easily modelled in the numerical simulation of seismic responses of poroelastic reservoirs. The free-surface conditions of complex geometry are achieved by using integral equilibrium equations on the surface, and the source implementations are simple. The algorithm is an extension of the quadrangle-grid finite difference method used for elastic wave equations.