Shear properties of heterogeneous fluid-filled porous media with spherical inclusions

An exact analytical solution is presented for the effective dynamic shear modulus in a heterogeneous fluid-filled poroelastic medium containing spherical inclusions. The complex and frequency-dependent properties of the derived shear modulus are solely caused by the physical mechanism of mesoscopic-scale wave-induced fluid flow whose scale is assumed to be smaller than wavelength but larger than the size of pores. 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. This three-phase model represents a self-consistent model or an approximate model of composite having periodically distributed inclusions. The behaviors of both the inclusion and the matrix are described by Biot's equations (1941) with standard conditions of Deresiewicz and Skalak (1963) at the inclusion-matrix interface. The effective medium is regarded as an equivalent elastic or viscoelastic material with complex and frequency-dependent moduli to be determined. The derived effective shear modulus is used to quantify the shear-wave attenuation and velocity dispersion. For the problem of fluid patchy saturation (inclusions with the same solid frame as the matrix but with a different pore fluid from that in the matrix), the gas pocket does not affect the shear attenuation and dispersion characteristic of the water-filled matrix medium at all. For the problem of double porosity structure (inclusions having a different solid frame than the matrix but the same pore fluid as the matrix), with the increase of frequency the heterogeneous medium transitions from a low-frequency state having drained inclusions and drained matrix with no pore pressure difference to a higher-frequency state having undrained inclusions and undrained matrix with no fluid communication at the inclusion's surface. The relaxation frequency at which the maximum value of inverse quality factor occurs moves to frequencies by two orders of magnitude lower if the size of a unit cell increases by one order of magnitude. Stiff inclusions imbedded in a relatively soft matrix can cause significant and observable attenuation at seismic frequency bands, but softer inclusions imbedded in a relatively stiff matrix cause very weak attenuation. The mixed heterogeneity in both the solid frame and pore fluid also has important influences on the frequency-dependent shear wave attenuation.