Preprint A251/2003
Initial boundary value problems for a quasilinear parabolic system in three-phase capillary flow in porous media
We study two types of initial boundary value problems for a quasilinear parabolic system motivated by three-phase flow in porous medium in the presence of capillarity effects. The first type of problem prescribes a boundary condition of mixed type involving a combination of the value of the solution and its normal derivative at the boundary. The second type of problem prescribes Dirichlet boundary conditions and its solution is obtained as a limit case of the first type. The main assumption about the viscosity'' 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 porous medium. Namely, we derive an important practical consequence of the assumption that the diffusion term in the equation of one of the phases, say gas, depends only on the saturation of the corresponding 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. As an example, we include the analysis of a very special model of three-phase capillary flow where the capillarity matrix results to be degenerate, but we are still able to solve it, due to the particular form of the flux functions.