Combining the micromechanical approach and boundary element method for estimating the effective permeability of 2D porous materials with arbitrarily shaped pores

The primary objective of this work is to determine the effective permeability of porous media consisting of an isotropic permeable solid matrix containing pores of arbitrary shapes. Fluid flow through the matrix phase is modeled by Darcy's law, while the flow inside the pores follows the Stokes equations. The interfaces between the matrix phase and inclusions are defined by the general form of the Beavers-Joseph-Saffman conditions. To achieve this objective, the Boundary Element Method (BEM) is first developed to solve the coupled Darcy and Stokes problem related to fluid flow through an infinite solid phase containing an arbitrarily shaped pore under a uniform prescribed pressure gradient at infinity. In contrast to the classical BEM where integration equations are often singular, our method, incorporating both finite difference and analytical integration schemes, overcomes this inconvenience. Additionally, compared to the commonly used numerical method based on the finite element method, our approach, which only requires discretization of the solid/fluid interface, significantly enhances computational speed and efficiency. Subsequently, each pore is substituted with an equivalent permeable inclusion, and its permeability is determined. Finally, employing classical micromechanical schemes, the macroscopic permeabilities of the porous material under consideration are estimated. These macroscopic permeability estimates are then compared with the relevant data available in the literature, as well as several numerical results provided by the finite element method.