Effective governing equations for heterogenous porous media subject to inhomogeneous body forces

Abstract

We derive a new homogenized model for heterogeneous porous media driven by inhomogeneous body forces. We assume that the fine scale, characterizing the heterogeneities in the medium, is larger than the pore scale, but nonetheless much smaller than the size of the material (the coarse scale). We decouple spatial variations and assume periodicity on the fine scale. Fine scale variations are formally reflected in a locally unbounded source for the arising system of partial differential equations. We apply the asymptotic homogenization technique to obtain a well-defined coarse scale Darcy-type model. The resulting problem is driven by an effective source which comprises both the coarse scale divergence of the average body force, and additional contributions which are to be computed solving a well-defined diffusion-type cell problem which is driven solely by fine scale variations of the given force. The present model can be used to predict the effect of externally applied magnetic (or electric) fields on ferrofluids (or electrolytes) flowing in porous media. This work can, in perspective, pave the way for investigations of the effect of applied forces on complex and heterogeneous hierarchical materials, such as systems of fractures or cancerous biological tissues.

Keywords

• heterogeneous porous media
• asymptotic homogenization
• locally unbounded source
• ferrofluids
• hierarchical materials