Incompressible flows through a solid matrix with mass exchange: the double scale approach versus mixture theory

We consider a class of processes in which the flow of an incompressible fluid through a porous matrix is accompanied by a mass exchange between the constituents. This paper is in two parts. In the first part we use an upscaling procedure to derive a macroscopic law for the flow, starting from the analysis of the phenomena at the pore scale. To this end we utilize a simplified geometry of the solid matrix. The resulting governing equation is of Darcy type, as expected, but, owing to mass exchange, it contains a complicated nonlocal dependence on the hydraulic conductivity. In the second part we look directly for the formulation of a macroscopic model in the framework of mixture theory. We emphasize the basic role of energy dissipation: the two methods lead to the same conclusions, provided that the same dissipation rates are postulated.