Theoretical derivation of Darcy's law for fluid flow in thin porous media

In this paper we study stationary incompressible Newtonian fluid flow in a thin porous media. The media under consideration is a bounded perforated $3D$ domain confined between two parallel plates. The description of the domain includes two small parameters: $\epsilon$ representing the distance between pates and $a_{\epsilon }$ connected to the microstructure of the domain such that $a_{\epsilon }\ll \epsilon$. We consider the classical setting of perforated media, i.e. $a_{\epsilon }$-periodically distributed solid (not connected) obstacles of size $a_{\epsilon }$. The goal of this paper is to introduce a version of the unfolding method, depending on both parameters $\epsilon$ and $a_{\epsilon }$, and then to derive the corresponding $2D$ Darcy’s law.