Darcy’s Law for Flow in a Periodic Thin Porous Medium Confined Between Two Parallel Plates
John Fabricius, J. Gunnar I. Hellström, T. Staffan Lundström, Elena Miroshnikova, Peter Wall · Transport in Porous Media · 2016
We study stationary incompressible fluid flow in a thin periodic porous medium. The medium under consideration is a bounded perforated 3D-domain confined between two parallel plates. The distance between the plates is $$\delta $$ , and the perforation consists of $$\varepsilon $$ -periodically distributed solid cylinders which connect the plates in perpendicular direction. Both parameters $$\varepsilon $$ , $$\delta $$ are assumed to be small in comparison with the planar dimensions of the plates. By constructing asymptotic expansions, three cases are analysed: (1) $$\varepsilon \ll \delta $$ , (2) $$\delta /\varepsilon \sim \text {constant}$$ and (3) $$\varepsilon \gg \delta $$ . For each case, a permeability tensor is obtained by solving local problems. In the intermediate case, the cell problems are 3D, whereas they are 2D in the other cases, which is a considerable simplification. The dimensional reduction can be used for a wide range of $$\varepsilon $$ and $$\delta $$ with maintained accuracy. This is illustrated by some numerical examples.