A filtration problem in a composite porous material with two free boundaries

In this paper the authors study a free boundary problem describing filtration of a liquid in a porous medium containing water absorbing granules (hydrophilic grains), with applications to ultra-napkins and diapers. The model is a saturation (mass balance) equation for liquid based on Darcy’s law, coupled to an ODE describing volume conservation. The saturation equation describes evolution of saturation S in porous medium of porosity " with hydraulic conductivity k(S, ") and pressure p. Saturation S is assumed to be a linear function of p. The total volume, which is a sum of porosity " and volume of grains V , does not change. The overall problem is an elliptic-parabolic Stefan problem in which two free boundaries separating three regions: dry, unsaturated and saturated, are present. The first boundary is the wetting front, the second is the saturation front. The authors prove existence and uniqueness of “almost classical” solutions.