Homogenization of fluid saturated double porosity media with a new type of the contrast in the Biot mesoscopic model

Abstract

This paper presents a mathematical model of a double poro-elastic medium derived by the homogenization of a two-component periodically heterogeneous Biot continuum characterized by strong heterogeneities in the permeability and poroelastic coefficients of mesoscopic structure. A new scaling of the mesoscopic material parameters with respect to the small parameter which is involved in the asymptotic analysis is introduced to capture this high contrast quality of the mesoscopic model. It is shown that such a scaling ansatz of the mesoscopic material parameters can be justified using different micromodels associated with the two mesoscopic components. While the “matrix” is a little permeable stiff phase, the “conductive channels” are made of a very soft fibrous structure equivalently represented by an network of helical springs. The unfolding method of the homogenization is employed to derive the macroscopic model involving the frequency-dependent effective parameters. Semipermeable interfaces between hard dual porosity and soft primary porosity are considered. The wave dispersion of two shear wave modes S1, S2, and two pressure wave modes P 1, P 2, is illustrated in an numerical example and validated, in a part, using the reference dispersion analysis based on the Bloch wave decomposition.