Existence of weak solutions for nonisothermal immiscible compressible two-phase flow in porous media

Abstract

We introduce a model of the time evolution of a flow of immiscible compressible fluids in porous media, taking into account the thermal effects. The problem leads to a coupled system of three nonlinear equations, two of which are degenerate. The time derivative has a new degeneracy in addition to the usual one in two-phase flows because of compressibility. We introduce a suitable weak formulation of the problem based on the total energy conservation principle. A new existence result of weak solutions of the more general model is obtained based on assumptions that are physically relevant to the problem data. The result is obtained in several steps involving an appropriate regularization and a time discretization. First we prove the existence of a weak solution for the non-degenerate problem based on obtaining a priori estimates, discrete maximum principle, and using the Leray–Schauder fixed point theorem. Finally, by using uniform estimates, our compactness result, and a suitable limit passages, we can establish the existence of a weak solution to the degenerate problem. This result is a further progress compared to the result obtained in [Math. Methods Appl. Sci. 40 (2017), no. 18, 7510–7539.], which deals with an incompressible model.