Exuding porous media: deviations from Darcy's law

Abstract

This work addresses the question of a pertinent macroscale model describing creeping, incompressible and single-phase flow of a Newtonian fluid in an exuding, rigid and homogeneous porous medium. The macroscopic model is derived by upscaling the pore-scale Stokes equations considering a normal mass flux at the solid–fluid interface. The upscaled mass equation shows that the average velocity is non-solenoidal. In addition, the macroscopic momentum equation involves a Darcy term with the classical permeability tensor accounting for macroscopic drag and a correction velocity vector which is a signature of the local fluid displacements induced by the exuding phenomenon. This correction is the sum of a term accounting for the local exuding effect and a compensation term associated with the assumption of spatial periodicity. Both the first term and the permeability tensor are obtained from the solution of the same unique and intrinsic closure problem, which corresponds to that involved in the classical Darcy's law. The upscaled model is validated by comparisons with pore-scale numerical simulations in several illustrative examples. The different configurations evidence the richness of the problem, despite the apparent simplicity of its formulation. The results of this work motivate further investigation about the influence of internal flow sources in transport phenomena in porous media.