Averaged Momentum Equation for Flow Through a Nonhomogenenous Porous Structure

This paper addresses the derivation of the macroscopic momentum equation for flow through a nonhomogeneous porous matrix, with reference to dendritic structures characterized by evolving heterogeneities. A weighted averaging procedure, applied to the local Stokes' equations, shows that the heterogeneous form of the Darcy's law explicitly involves the porosity gradients. These extra terms have to be considered under particular conditions, depending on the rate of geometry variations. In these cases, the local closure problem becomes extremely complex and the full solution is still out of reach. Using a simplified two-phase system with continuous porosity variations, we numerically analyze the limits where the usual closure problem can be retained to estimate the permeability of the structure.