Mathematical Analysis for Reservoir Models

This is the authors' abstract. We don't add key points for this paper.

In the first part of this paper, the mathematical analysis is presented in detail for the single-phase, miscible displacement of one fluid by another in a porous medium. It is shown that initial boundary value problems with various boundary conditions for this miscible displacement possess a weak solution under physically reasonable hypotheses on the data. In the second part of this paper, it is proven how the analysis can be extended to two-phase fluid flow and transport equations in a porous medium. The flow equations are written in a fractional flow formulation so that a degenerate elliptic-parabolic partial differential system is produced for a global pressure and a saturation. This degenerate system is coupled to a parabolic transport equation which models the concentration of one of the fluids. The analysis here does not utilize any regularized problem; a weak solution is obtained as a limit of solutions to discrete time problems.