Nguetseng's Two-scale Convergence Method For Filtration and Seismic Acoustic Problems in Elastic Porous Media

Abstract

A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. 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. As the results, we derive Biot's equations of poroelasticity, equations of viscoelasticity, or decoupled system consisting of non-isotropic Lam\'{e}'s equations and Darcy's system of filtration, depending on ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.