Determination of the Effective Permeability of Doubly Porous Materials by a Two-Scale Homogenization Approach

The present work is dedicated to determining the effective permeability of doubly porous materials made of a solid phase comprising a network of interconnected pores at the nanoscale and a network of non-interconnected pores at the microscopic scale. The fluid flow at microscopic scale through the solid phase containing nanopores is described by the Darcy law, while fluid flow in the nano- and microscopic pores is governed by the Stokes equations. A two-scale homogenization approach is proposed to estimate the effective permeability of doubly porous materials in question. In the nanoscopic-to-microscopic upscaling, a micromechanical model based on the generalized self-consistent scheme (GSCS) is elaborated to estimate the microscopic permeability. In the microscopic-to-macroscopic upscaling, the equivalent inclusion method combined with the dilute, Mori–Tanaka, differential schemes is used to obtain different estimates of the macroscopic permeability. In the two-scale homogenization approach elaborated, the pore size and shape effects as well as the solid/fluid interface influence are taken into account. The results given by the proposed two-scale homogenization approach are discussed and compared with the relevant numerical results provided by the finite element method.