Mathematical analysis for reservoir models

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.