Wave propagation in a strongly heterogeneous elastic porous medium: Homogenization of Biot medium with double porosities

Abstract

Abstract We study wave propagation in an elastic porous medium saturated with a compressible Newtonian fluid. The porous network is interconnected whereby the pores are characterized by two very different characteristic sizes. At the mesoscopic scale, the medium is described using the Biot model, characterized by a high contrast in the hydraulic permeability and anisotropic elasticity, whereas the contrast in the Biot coupling coefficient is only moderate. Fluid motion is governed by the Darcy flow model extended by inertia terms and by the mass conservation equation. The homogenization method based on the asymptotic analysis is used to obtain a macroscopic model. To respect the high contrast in the material properties, they are scaled by the small parameter, which is involved in the asymptotic analysis and characterized by the size of the heterogeneities. Using the estimates of wavelengths in the double-porosity networks, it is shown that the macroscopic descriptions depend on the contrast in the static permeability associated with pores and micropores and on the frequency. Moreover, the microflow in the double porosity is responsible for fading memory effects via the macroscopic poroviscoelastic constitutive law.