Homogenized models for a short-time filtration and for acoustic waves propagation in a porous media

Abstract

We consider a linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic and a characteristic time of processes is small enough. Such kind of models may describe, for example, hydraulic fracturing or acoustic or seismic waves propagation. As the results, we derive different types of homogenized equations involving non-isotropic Stokes system for fluid velocity coupled with acoustic equations for the solid component or different types of acoustic equations, depending on ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.