Seismic wave equation formulated by generalized viscoelasticity in fluid-saturated porous media

Abstract

Biot’s theory of poroelasticity describes seismic waves propagating through fluid-saturated porous media, so-called two-phase media. The classic Biot’s theory of poroelasticity considers the wave dissipation mechanism as being the friction of relative motion between the fluid in the pores and the solid rock skeleton. However, within the seismic frequency band, the friction has a major influence only on the slow P-wave and an insignificant influence on the fast P-wave. To represent the intrinsic viscoelasticity of the solid skeleton, we incorporate a generalized viscoelastic wave equation into Biot’s theory for the fluid-saturated porous media. The generalized equation that unifies the pure elastic and viscoelastic cases is constituted by a single viscoelastic parameter, presented as the fractional order of the wavefield derivative in the compact form of the wave equation. The generalized equation that includes the viscoelasticity appropriately describes the dissipation characteristics of the fast P-wave. Plane-wave analysis and numerical solutions of our wave equation reveal that (1) the viscoelasticity in the solid skeleton causes the energy attenuation on the fast P-wave and the slow P-wave at the same order of magnitude and (2) the generalized viscoelastic wave equation effectively describes the dissipation effect of the waves propagating through the fluid-saturated porous media.