Macroscopic permeability of doubly porous solids with spheroidal macropores: closed-form approximate solutions of the longitudinal permeability

Abstract

The presence of macropores and fractures significantly affects the effective transport properties of porous solids such as concrete and rocks. The dimensions of the fractures are generally large behind that of the initial porosity, so that the problem contains two porosities. The influence of the macroporosity is studied in the homogenization framework by solving at the intermediate scale, that of the macropores, a coupled Darcy/Stokes problem with the Beavers–Joseph–Saffman (BJS) interface condition. We derive analytic expressions of the macroscopic permeability in the case of an isotropic permeable matrix containing spheroidal-shaped macropores. To this aim, we consider a representative volume element (RVE) on which uniform boundary conditions are considered for the velocity and pressure fields. The local problem is written as minimum principles; kinematic and static approaches are developed to derive rigorous bounds for the macroscopic permeability. Closed-form expressions of the longitudinal permeability (along the revolution axe of the spheroid) are determined by considering a simplified RVE constituted of two confocal spheroids. They depend on the volume fraction and the eccentricity of the spheroidal macropores, the scale factor between the two porosities and the slip coefficient of the BJS model. Illustrations show the influence of these parameters.