Wave structure for a nonstrictly hyperbolic system of three conservation laws

Abstract

This work presents the wave structure and the Riemann solution for a system of three conservation laws. The system is hyperbolic; however, in state space there are two surfaces and a curve on which distinct pairs of eigenvalues coincide. Even though one of the eigenvalues is linearly degenerate, there is no global ordering of the characteristic speeds as in the Euler equations for gas dynamics. The lack of global ordering causes complicated structures to exist in the wave curves and consequently in the construction of the Riemann solution. The system arises in polymer flooding of three-phase petroleum reservoirs, where a polymer in small concentration is added to the water phase to increase its viscosity and improve oil displacement efficiency. The Riemann solution consists of sequences of two kinds of waves: saturation waves, corresponding to slow, fast, and transitional waves of the subsystem of two equations for a fixed concentration, and concentration waves, corresponding to contact discontinuities. There are at most three constant states appearing in the Riemann solution. The maximum of three is attained when a transitional saturation wave occurs. A surprising feature is that a saturation wave may occur within a concentration wave; such a saturation wave is not adjacent to any constant states.