Biot-Rayleigh theory of wave propagation in double-porosity media

We derive the equations of motion of a double-porosity medium based on Biot's theory of poroelasticity and on a generalization of Rayleigh's theory of fluid collapse to the porous case. Spherical inclusions are imbedded in an unbounded host medium having different porosity, permeability, and compressibility. Wave propagation induces local fluid flow between the inclusions and the host medium because of their dissimilar compressibilities. Following Biot's approach, Lagrange's equations are obtained on the basis of the strain and kinetic energies. In particular, the kinetic energy and the dissipation function associated with the local fluid flow motion are described by a generalization of Rayleigh's theory of liquid collapse of a spherical cavity. We obtain explicit expressions of the six stiffnesses and five density coefficients involved in the equations of motion by performing gedanken experiments. A plane wave analysis yields four wave modes, namely, the fast P and S waves and two slow P waves. As an example, we consider a sandstone and compute the phase velocity and quality factor as a function of frequency, which illustrate the effects of the mesoscopic loss mechanism due to wave-induced fluid flow.