Pratap N. Sahay

On the Biot slow S-wave

It is accepted widely that the Biot theory predicts only one shear wave representing the in-phase/unison shear motions of the solid and fluid constituent phases (fast S-wave). The Biot theory also contains a shear mode wherein the two constituent phases essentially undergo out-of-phase shear motions (slow S-wave). From the outset of the development of the Biot framework, the existence of this mode has remained unnoticed because of an oversight in decoupling its system of two coupled equations governing shear processes. Moreover, in the absence of the fluid strain-rate term in the Biot constitutive relation, the velocity of this mode is zero. Once the Biot constitutive relation is corrected for the missing fluid strain-rate term (i.e., fluid viscosity), this mode turns out to be, in the inertial regime, a diffusive process akin to a viscous wave in a Newtonian fluid. In the viscous regime, it degenerates to a process governed by a diffusion equation with a damping term. Although this mode is damped so heavily that it dies off rapidly near its source, overlooking its existence ignores a mechanism to draw energy from seismic waves (fast P- and S-waves) via mode conversion at interfaces and at other material discontinuities and inhomogeneities. To illustrate the consequence of generating this mode at an interface, I examine the case of a horizontally polarized fast S-wave normal incident upon a planar air-water interface in a porous medium. Contrary to the classical Biot framework, which suggests that the incident wave should be transmitted practically unchanged through such an interface, the viscosity-corrected Biot framework predicts a strong, fast S-wave reflection because of the slow S-wave generated at the interface.

Fast compressional wave attenuation and dispersion due to conversion scattering into slow shear waves in randomly heterogeneous porous media.

Within the viscosity-extended Biot framework of wave propagation in porous media, the existence of a slow shear wave mode with non-vanishing velocity is predicted. It is a highly diffusive shear mode wherein the two constituent phases essentially undergo out-of-phase shear motions (slow shear wave). In order to elucidate the interaction of this wave mode with propagating wave fields in an inhomogeneous medium the process of conversion scattering from fast compressional waves into slow shear waves is analyzed using the method of statistical smoothing in randomly heterogeneous poroelastic media. The result is a complex wave number of a coherent plane compressional wave propagating in a dynamic-equivalent homogeneous medium. Analysis of the results shows that the conversion scattering process draws energy from the propagating wave and therefore leads to attenuation and phase velocity dispersion. Attenuation and dispersion characteristics are typical for a relaxation process, in this case shear stress relaxation. The mechanism of conversion scattering into the slow shear wave is associated with the development of viscous boundary layers in the transition from the viscosity-dominated to inertial regime in a macroscopically homogeneous poroelastic solid.

Generalized poroelasticity framework for micro‐inhomogeneous rocks

Poroelastic modelling of micro-inhomogeneous rocks is of interest for applications in rock physics and geomechanics. Laboratory measurements from both communities indicate that the Biot poroelasticity framework is not adequate. For the case of a macroscopically homogeneous and isotropic rock, we present the most general poroelasticity framework within the scope of equilibrium thermodynamics that is able to capture the effects of micro-inhomogeneities in a natural way. Within this generalized poroelasticity framework, the concept of micro-inhomogeneity is generically related to partial localization of the deformational potential energy either in the solid phase, including the interfacial region or in the fluid phase. The former case can occur in the presence of surface roughness or multi-mineralic frame and the latter case can be related to suspended particles residing in the fluid phase. A measure for micro-inhomogeneity is the coefficient that governs the effective pressure dependence of porosity changes as described by the porosity perturbation equation of this framework. It can be therefore equivalently interpreted as porosity effective pressure coefficient or as micro-inhomogeneity parameter. We show how this parameter and the other poroelastic constants embedded in this framework can be expressed in terms of experimentally accessible poroelastic constants.

Acoustic signatures of wave-induced vorticity diffusion in porous media

The importance of the viscous boundary layer flow for porous medium acoustics has been recognized by Biot who analyzed this effect for cylindrical tubes and subsequently extended his low-frequency theory to the full frequency range. More recently, the transition from the viscosityto the inertiadominated regime has been modeled using the so-called dynamic permeability model. In this paper we develop an alternative approach to model attenuation and dispersion associated with the transition from the viscosityto inertia-dominated regimes. Instead of analyzing the oscillatory Stokes flow in elastically-rigid tubes, we base our analysis on the viscosityextended Biot framework. This framework includes the fluid strain rate tensors in the poroelastic constitutive relations and predicts an additional shear wave mode corresponding to outof-phase shear motions of fluid and solid phases. As it has the nature of a diffusion process this shear wave is termed slow S-wave in analogy to its compressional wave counterpart, namely the slow P-wave We show that the conversion scattering process from a fast P-wave into the diffusive slow shear wave is a descriptor of the vorticity diffusion process occurring within the viscous boundary layer. To this end we make use of previously obtained results for conversion scattering from fast Pto slow S-waves and derive a dynamic-equivalent Pwavenumber from which we deduce attenuation and velocity dispersion. Comparison with the predictions of other models shows that the conversion scattering approach can model attenuation and dispersion associated with the transition from the viscosityto inertia-dominated regimes.

Nature of waves in deformable porous media

The deformable porous media possess four distinct seismic wave processes, namely, fast- and slow-waves each associated with compression and shear deformations. The fast compressional (shear) wave is essentially motion of a porous medium as a whole such that constituent phases undergo volume (shape) change in the same manner. The linear momentum fluxes are associated with these motions and they describe transport of translational kinetic energy. The slow compressional (shear) wave is basically motions of constituent phases undergoing volume (shape) in equal but opposite manner such that the medium as a whole is at rest. This new mode of deformation amounts to existence of an intrinsic angular momentum (spin). The slow waves are basically spin fluxes and they describe transport of rotational kinetic energy.

Eigenmodes for an elastic halfspace surrounding a semi-rigid cylinder

Abstract The elastodynamics of a halfspace with a cylindrical hole belongs to a class of problems which involve solving the elastic wave equation in a geometry that is bounded by nonparallel surfaces. Integral transform techniques are not very suitable for the solution of such problems because the transform approach yields solutions in terms of wavefronts (or rays). For a nonparallel geometry one may need a large collection of such terms and this may be an inconvenient description. So, we have taken an approach to construct the eigenmodes (normal modes) for this geometry for a given set of boundary conditions. A set of linearly independent solutions, without any reference to sources or receivers, is developed and completeness of this set is verified explicitly. Using this complete set of eigenmodes the displacement field for any source problem in this geometry can be expressed following standard mathematical procedures.

Viscosity scaling of wave attenuation mechanisms in porous rocks: Theory and numerical simulations

Different wave-induced attenuation mechanisms in fluidsaturated porous rocks scale differently with the fluid shear viscosity though the only source of dissipation is a relative fluid-solid displacement followed by a viscous flow. To provide physical insight into this characteristic viscosity scaling behavior we analyze the underlying equilibration processes related to the mechanism of wave-induced flow at mesoscopic heterogeneities and the Biot global flow mechanism. We show that the the former mechanism is related to fluid pressure diffusion, while the latter is related to vorticity diffusion within the viscous boundary layer. The difference in the diffusing physical quantity, that is fluid pressure and vorticity, explains the characteristic viscosity scaling. We further show that the process of vorticity diffusion can be described by the so-called slow shear wave. This allow us to connect the process of vorticity diffusion to propagating waves. Finite-difference simulations of wave propagation in an idealized, fluid-saturated porous medium in form of a single channel illustrate the conversion process of a propagating wave into the slow shear wave.

Attenuation of elastic waves due to wave-induced vorticity diffusion in porous media

A theory for attenuation of elastic waves due to wave-induced vorticity diffusion in the presence of randomly correlated pore-scale heterogeneities in porous media is developed. It is shown that the vorticity field is associated with a viscous wave in the pore space, the so-called slow shear wave. The latter is linked to the porous medium acoustics through incorporation of the fluid strain rate tensor of a Newtonian fluid in the poroelastic constitutive relations. The method of statistical smoothing in random media is used to derive dynamic-equivalent elastic wave numbers accounting for the conversion scattering process into the slow shear wave. The result is a model for wave attenuation and dispersion associated with the transition from viscosity- to inertia-dominated flow regime in porous media. It is also shown that the momentum flux transfer from the slow compressional into the slow shear wave is a proxy for the dynamic permeability in porous media. A dynamic permeability model is constructed that con...

Natural Field Variables In Dynamic Poroelasticity

The natural fields in porous media are center of mass and internal motions. They are respectively the fields associated with total momentum flux and relative acceleration of the solid and fluid constituents. The pairs 0) and (0, are the natural modes of vibrations of porous media. The first mode represents motion of porous medium as a whole, i.e., there is no relative motion of the constituents, while in the second mode one constituent moves with respect to another in such a manner that the medium is at rest. In terms of these field variables we have a theory amicable to exploration seismics.