Solutions for Effective Shear Properties in a Three-Phase Poroelastic Sphere Model

An analytical solution is presented for the effective shear modulus in a heterogeneous fluid-filled poroelastic medium containing spherical inclusions. Our model consists of three phases: a spherical inclusion, a shell of matrix material with different mechanical and/or hydraulic properties and an outer region of effective homogeneous medium of infinite extent. The behavior of the inclusion and the matrix is described by Biot's equations, while the outer medium is regarded as an equivalent elastic or viscoelastic material with complex and frequency-dependent moduli to be determined. It is shown that for the double porosity structure (inclusions having a different solid frame than the matrix but the same pore fluid as the matrix), the mesoscopic-scale wave-induced fluid flow gives arise to a complex and frequency-dependent shear modulus. The mixed heterogeneity in the solid frame and pore fluid has important influences on the frequency-dependent shear wave attenuation.