Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization

In this chapter of the special issue of the journal ”Transport in Porous Media”, on the topic ”Flow and transport above permeable domains”, we present modeling of flow and transport above permeable domains using the homogenization method. Our goal is to develop a heuristic approach which can be used by the engineering community for treating this type of problems and which has a solid mathematical background. The rigorous mathematical justification, of the presented results, is given in the corresponding articles of the authors. The plan is as follows: We start with the Introduction were we give an overview and comparison with interface conditions obtained using other approaches. In Section 2, we give a very short derivation of the Darcy law by homogenization, using the two-scale expansion in the typical pore size parameter ε. It gives us the definition of various auxiliary functions and typical effective properties as permeability. In Section 3, we introduce our approach to the effective interface laws on a simple 1D example. The approximation is obtained heuristically using the two steps strategy. For the 1D problem we calculate the approximation and the effective interface law explicitly and show that it is valid at order O(ε). Next, in Section 4 we give a derivation of the Beavers-Joseph-Saffman interface condition and of the pressure jump condition, using homogenization. We construct the corresponding boundary layer and present a heuristic calculation, leading to the interface law and being based on the rigorous mathematical result. In addition, we show the invariance ∗The research of A.M. was partially supported by the GDR MOMAS (Modélisation Mathématique et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN).