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: ε representing the distance between plates and aε$a_\varepsilon$ connected to the microstructure of the domain such that aε≪ε$a_\varepsilon \ll \varepsilon$. We consider the classical setting of perforated media, i.e. aε$a_\varepsilon$‐periodically distributed solid (not connected) obstacles of size aε$a_\varepsilon$. The goal of this paper is to introduce a version of the unfolding method, depending on both parameters ε and aε$a_\varepsilon$, and then to derive the corresponding 2D Darcy law.