The 3D flow of a liquid through a porous medium with absorbing and swelling granules

Abstract

We study the flow of an incompressible fluid through a porous medium with hydrophile granules. The system is schematized as a periodic array of cubic cells, each containing one spherical swelling granule. The physical situation is such that the size of the granules is of the same order of the size of the cells and much larger than the microscopic constituents of the porous matrix. The porosity at each point of a cell is defined according to the size of the granules located at the cell vertices. The swelling of each granule is governed by a kinetic law involving the average moisture content of the medium over the granule surface. The notion of weak solution is introduced and we prove the existence of such solution using backward time differences. The discretized problem is studied in detail and appropriate a priori estimates are obtained. Passing to the limit requires a precise analysis of the convergence in the geometry evolving with the solution.