Conservation Laws of Mixed Type Describing Three-Phase Flow in Porous Media

Abstract

In this paper we examine the mathematical structure of a model for three-phase, incompressible flow in a porous medium. We show that, in the absence of diffusive forces, the system of conservation laws describing the flow is not necessarily hyperbolic. We present an example in which there is an elliptic region in saturation space for reasonable relative permeability data. A linearized analysis shows that in nonhyperbolic regions solutions grow exponentially. However, the nonhyperbolic region, if present, will be of limited extent which inherently limits the exponential growth. To examine these nonlinear effects we resort to fine grid numerical experiments with a suitably dissipative numerical method. These experiments indicate that the solutions of Riemann problems remain well behaved in spite of the presence of a linearly unstable elliptic region in saturation space. In particular, when initial states are outside the elliptic region the Riemann problem solution appears to stay outside the region. Furthermore, stable shocks are formed connecting states inside the nonhyperbolic regions with states outside.