Initial Boundary Value Problems for a Quasi-linear Parabolic System in Three-Phase Capillary Flow in Porous Media

Abstract

We study two types of initial boundary value problems for a quasi-linear parabolic system motivated by three-phase flows in porous media in the presence of capillarity effects. The first type of problem prescribes a mixed boundary condition, involving a combination of the value of the solution and its normal derivative at the boundary. The second type prescribes the value of the solution at the boundary, which is the so-called Dirichlet boundary condition. We prove the existence and uniqueness of smooth solution for the first type of initial boundary value problem, and we obtain the existence of a solution for the second one as a limit case of the first type. The main assumption about the diffusion matrix of the system is that it is triangular with strictly positive diagonal elements. Another interesting feature is concerned specifically with the application to three-phase capillary flow in a porous medium. Namely, we derive an important practical consequence of the assumption that the capillarity matrix is upper triangular, if we further impose that the second diagonal element depends only on the second variable, i.e., the second phase. We show that this mathematical assumption in turn provides an efficient method for the definition of the capillary pressures in the interior of the triangle of saturations through the solution of a well-posed boundary value problem for a linear hyperbolic system. Finally, as an example, we include the analysis of a special case of three-phase capillary flow model where the capillarity matrix is degenerate, but we are still able to solve it due to the particular form of the flux functions.