Riemann problems requiring a viscous profile entropy condition

The Riemann problem is solved for 2 × 2 systems of non-strictly hyperbolic conservation laws abstracted from a three-phase Buckley-Leverett model for oil reservoir flow. The example presented here completes a program for the solution of Riemann problems with quadratic flux functions by allowing the region of parameters most relevant for oil reservoirs. In this example there are shocks which have viscous profiles but do not satisfy the Lax conditions and shocks which satisfy the Lax entropy conditions but fail to have viscous profiles. Therefore these two fundamental notions of entropy are properly distinct. Combining results from numerical and theoretical examinations, we show that the Lax entropy condition is incomplete in this example. The Riemann problem in general fails to have a solution in the class of Lax shocks; the solution of the Riemann problem, however, exists and is in the class of shocks with viscous profiles. Global analysis of dynamical systems defined by traveling wave solutions to the associated viscosity equation is an essential tool for our study. The shapes of the Hugoniot loci were obtained numerically. Numerical methods were also used to distinguish between and show the actual occurrence of all distinct phase configurations which the mathematical theory allowed.