Filtration Law in Porous Media with Poor Separation of Scales

Abstract

We investigate the validity of Darcy’s law when the separation of scales is poor. We use the method of multi-scale asymptotic expansions which gives the macroscopic behaviour from the pore scale description. The first order approximation is the Darcy’s law. When the separation of scales is poor, eventual correctors to Darcy’s law are obtained by investigating the following orders of approximation, thus enabling us to study its robustness. We investigate the two first correctors. Thus, the accuracy of the macroscopic flow law is improved from ${\cal O}$ (ε) to ${\cal O}$ (ε³), where ε is the separation of scale parameter. The second corrector shows a Brinkman’s term. For macroscopically homogeneous porous media, the first corrector cancels out, that points out the robustness of Darcy’s law in this case.