Single-scattering approximations for coefficients in Biot’s equations of poroelasticity

Three single-scattering approximations for coefficients in Biot’s equations of poroelasticity are considered: the average T-matrix approximation (ATA), the coherent potential approximation (CPA), and the differential effective medium (DEM). The scattering coefficients used here are exact results obtained previously for scattering from a spherical inclusion of one Biot material imbedded in another otherwise homogeneous Biot material. The CPA has been shown previously to guarantee that, if the coefficients for the scattering materials satisfy Gassmann’s equation, then the effective coefficients for the composite medium satisfy Brown and Korringa’s generalization of Gassmann’s equation. A collection of similar results is obtained here showing that the coefficients derived from ATA, CPA, or DEM all satisfy the required conditions for consistency. It is also shown that Gassmann’s equation will result from any of these single-scattering approximations if the collection of scatterers includes only spheres of fluid and of a single type of elastic solid.