Inhomogeneous plane waves in attenuative anisotropic porous media

TL;DR

This paper develops a comprehensive theoretical framework for analyzing inhomogeneous plane wave propagation in attenuative anisotropic porous media, incorporating velocity and attenuation anisotropy, and providing explicit expressions for wave characteristics and energy dissipation.

Contribution

It introduces a novel fractional differential equation and an alternative complex slowness vector formulation for inhomogeneous waves in anisotropic poro-viscoelastic media, extending classical Biot theory.

Findings

Derived explicit expressions for phase velocities and complex slownesses of various wave modes.

Presented new energy balance equations and formulas for energy velocities.

Analyzed dissipation factors with two different energy-based measures.

Abstract

We investigate the propagation of inhomogeneous plane waves in poro-viscoelastic media, explicitly incorporating both velocity and attenuation anisotropy. Starting from classical Biot theory, we present a fractional differential equation describing wave propagation in attenuative anisotropic porous media that accommodates arbitrary anisotropy in both velocity and attenuation. Then, instead of relying on the traditional complex wave vector approach, we derive new Christoffel and energy balance equations for general inhomogeneous waves by employing an alternative formulation based on the complex slowness vector. The phase velocities and complex slownesses of inhomogeneous fast and slow quasi-compressional (qP1 and qP2) and quasi-shear (qS1 and qS2) waves are determined by solving an eighth-degree algebraic equation. By invoking the derived energy balance equation along with the computed complex slowness, we present explicit and concise expressions for energy velocities. Additionally, we analyze dissipation factors defined by two alternative measures: the ratio of average dissipated energy density to either average strain energy density or average stored energy density. We clarify and discuss the implications of these definitional differences in the context of general poro-viscoelastic anisotropic media. Finally, our expressions are degenerated to give their counterparts of the homogeneous waves as a special case, and the reduced forms are identical to those presented by the existing poro-viscoelastic theory. Several examples are provided to illustrate the propagation characteristics of inhomogeneous plane waves in unbounded attenuative vertical transversely isotropic porous media.