Modelling Complex Flows in Porous Media by Means of Upscaling Procedures

Abstract

We review a series of problems arising in the field of flows through porous media and that are highly nontrivial either because of the presence of mass exchange between the fluid and the porous matrix (or other concurrent phenomena of physical or chemical nature), or because of a particularly complex structure of the medium. In all these cases there is a small parameter $\varepsilon $, representing the ratio between the microscopic and the macroscopic space scale. Our attention is focussed on a modelling technique (upscaling) which start from the governing equations written at the pore scale, introduces an expansion in power series of $\varepsilon $of all the relevant quantities and eventually leads to the formulation of the macroscopic governing equations at the various orders in $\varepsilon $ by a matching procedure, followed by suitable averaging. Two problems will be analyzed with some detail: soil erosion and the dynamics of water ultrafiltration devices. Moreover other problems will be occasionally discussed and open questions will be proposed.