This numerical flux replaces the Godunov numerical flux in the algorithm, thereby eliminating the need for a global Riemann problem solution. A generalization of the Engquist-Osher flux for systems is then used to compute a numerical flux based on this approximation. The method constructs an approximation of the Riemann problem from local wave information. This extension allow the method to be applied to problems that are not strictly hyperbolic and exhibit local linear degeneracies in the wave fields. We describe an extension of higher order Godunov methods to general systems of hyperbolic conservation laws. Efficient algorithms for computing the exact solution are also given, making the analytical developments presented here readily applicable to interpretation of lab displacement experiments, and implementation of streamline =, number = 1, We demonstrate that assuming the saturation paths of the solution are straightlines may result in inaccurate predictions for some realistic systems. We present the complete analytical solution to the Riemann problem (constant initial and injected states) in detail, and describe the characteristic waves that may arise, concluding that only nine combinations of rarefactions, shocks and rarefaction-shocks are possible. These properties, which are natural extensions of the two-phase flow model, ensure that the solution is physically sensible. Under these conditions, the mathematical problem reduces to a 2 x 2 system of conservation laws whose essential features are: (1) the system is strictly hyperbolic (2) both characteristic fields are nongenuinely nonlinear, with single, connected inflection loci. The model uses the common multiphase flow extension of Darcy's equation, and does not include gravity and capillarity effects. In this paper we study one-dimensional three-phase flow through porous media of immiscible, incompressible fluids.

Analytic rarefaction