Wave propagation in anisotropic, saturated porous media: Plane-wave theory and numerical simulation

Abstract

Porous media are anisotropic due to bedding, compaction, and the presence of aligned microcracks and fractures. Here, it is assumed that the skeleton (and not the solid itself) is anisotropic. The rheological model also includes anisotropic tortuosity and permeability. The poroelastic equations are based on a transversely isotropic extension of Biot’s theory, and the problem is of plane strain type, i.e., two dimensional, describing qP−qS propagation. In the high-frequency case, the (two) viscodynamic operators are approximated by Zener relaxation functions that allow a closed differential formulation of Biot’s equation of motion. A plane-wave analysis derives expressions for the slowness, attenuation, and energy velocity vectors, and quality factor for homogeneous viscoelastic waves. The slow wave shows an anomalous polarization behavior. In particular, when the medium is strongly anisotropic the polarization is quasishear and the wave presents cuspidal triangles. Anisotropic tortuosity affects mainly the slow wavefront, and anisotropic permeability produces strong anisotropic attenuation of the three modes. The diffusive characteristics of the slow mode are predicted by the plane-wave analysis. As in the single-phase case, it is confirmed that the phase velocity is the projection of the energy velocity vector onto the propagation direction. Moreover, some fundamental energy relations, valid for a single-phase medium, are generalized to two-phase media. Transient propagation is solved with a direct grid method and a time-splitting integration algorithm, allowing the solution of the stiff part of the differential equations in closed analytical form. The snapshots show that the three waves are propagative when the fluid is ideal (zero viscosity). It is confirmed that, when the fluid is viscous, the slow wave becomes diffusive and appears as a static mode at the source location. The modeling confirms the triplication (cusps) of the slow wave and the polarization behavior predicted by the plane analysis.