Mechanics of heterogeneous porous media with several spatial scales

Abstract

By extending the theory of homogenization, we consider a heterogeneous porous medium whose material structure is characterized by multiple periodicity over several disparate length scales. Because for geological and engineering applications the driving force (e. g. the pressure gradient) and the global motion of primary interest is usually at a scale much larger than the largest of structural periodicity, we derive the phenomenological equations of motion by using the perturbation technique of multiple scales. All the effective coefficients are defined by boundary-value problems on unit cells in smaller scales and no constitutive assumptions are added aside from the basic equations governing the mechanics of the pore fluid and the solid matrix. In Part I the porous matrix is assumed to be rigid; the case of three scales is treated first. Symmetry and positiveness of the effective Darcy permeability tensor are proven. Extensions to four and more scales are then discussed. In Part II we allow the solid matrix to be deformable and deduce the equations of consolidation of a two-phase medium. The coupling of fluid flow and the quasi-static elastic deformation of the matrix is considered when there are three disparate length scales. Effective coefficients of various kinds are deduced in terms of cell problems on the scale of the pores. Several general symmetry relations as well as specific properties of a medium composed of layers of porous matrix are discussed.