Darcy's law survival from no-slip to perfect-slip flow in porous media

A macroscopic model for perfect-slip flow in porous media is derived in this work, starting from the pore-scale flow problem and making use of an upscaling technique based on the adjoint method and Green's formula. It is shown that the averaged momentum equation has a Darcy form in which the permeability tensor, K-ps, is obtained from an associated adjoint (closure) problem that is to be solved on a (periodic) unit cell representative of the structure. Similarly to the classical permeability tensor, K, in the no-slip regime, K-ps is intrinsic to the porous medium under consideration and is shown to be symmetric and positive. Integral relationships between Kps, the partial-slip flow permeability tensor, K-s, and K are derived. Numerical simulations carried out on two-dimensional model porous structures, together with an approximate analytical solution and an empirical correlation for a particular configuration, confirm the validity of the macroscopic model and the relationship between K-ps and K.