Spectral Green's Dyadic for Point Sources in Poroelastic Media

Abstract

A fairly detailed derivation of the spectral Green's dyadic for point sources in unbounded poroelastic media is presented. It is assumed that the motion of the poroelastic medium is governed by Biot's theory of poroelasticity; thus in a source-free unbounded space the wave field consists of two (fast and slow) longitudinal waves and a transverse wave. Considering a three-dimensional source-receiver system and then through a decomposition of the displacement and body force fields, the dilatational and rotational components of motion are separated. Separation yields two sets of systems of two partial differential equations representing scalar wave equations of poroelasticity, which unlike the case of elastic propagation are still coupled in terms of the motion of the pore fluid and that of the frame material, respectively. General solutions are derived from the fundamental eigenvalue problems of poroelasticity, which are associated with the systems of the homogeneous wave equations. Singular solutions for point sources are then obtained by superimposing the latter with particular solutions of the inhomogeneous wave equations. Consistent with previous studies, the spectral Green's dyadic shows that the body force singularity generates three distinct waves. These waves are radiating from the source with wave speeds, attenuations, and amplitudes, which depend on frequency and consequently on the level and type of dissipation in the two-phase medium.