Mathematical Models for Fluids with Pressure-Dependent Viscosity Flowing in Porous Media

In this paper we study three filtration problems through porous media, assuming that the viscosity of the fluid depends on pressure. After showing that in this case Darcy’s law is “formally” preserved (meaning that the formal relation remains unchanged except for viscosity that now depends on pressure), we focus on the following problems: Green–Ampt infiltration through a dry porous medium; the Dam problem; the Muskat problem. For each model (free boundary problems) we obtain explicit solutions that allow to quantify the detachment from the classical case, where with the word “classical” we mean that viscosity is taken constant.