Mathematical Analysis for Reservoir Models

In the rst part of this paper, the mathematical analysis is presented in detail for the single-phase, miscible displacement of one uid 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. Then in the second part of this paper, it is proven how the analysis can be extended to two-phase uid ow and transport equations in a porous medium. The ow equations are written in a fractional ow formulation so that a degenerate elliptic-parabolic partial diierential 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 uids. The analysis here does not utilize any regularized problem; a weak solution is obtained as a limit of solutions to discrete time problems.