Weak formulation for a two-phase nonlinear flow in an undeformable porous medium

In this paper we present a mathematical model for the two-phase flow of a mono-component fluid in an undeformable porous medium. The main practical application is the problem of gas extraction in a geothermal reservoir for which the model can be used for predicting the extinction time of a specific phase in the reservoir. The system is modeled assuming that temperature is not evolving and that the driving mechanism in the case of co-existence of the two phases is capillarity. We also assume that the fluid can be found in liquid and gaseous phase and that there can be regions where this two phases co-exist. The various phases are separated by evolving boundaries (the mathematical formulation turns out to be a free boundary problem) which are determined imposing mass balance relations. We give an integral formulation for the so-called overall density, which is the sum of the densities of each phase weighted by saturation. Finally we present some numerical simulations to investigate the dependence of the solution on the physical parameters and on the boundary conditions involved in the system.