Compositional flow in porous media: Riemann problem for three alkanes

Abstract

We consider the flow in a porous medium of three fluid compounds such as alkanes with different boiling points; the compounds partition into a liquid and a gaseous phase. Under some judiciously chosen physical assumptions, the flow is governed by a system of conservation laws; we derive the expression for the Rankine-Hugoniot locus, which involves a parameter dependent fifth degree polynomial in two variables. This expression allows us to establish in detail the bifurcation behavior of the locus Supplemented by the analysis of characteristic speeds and eigenvectors, the bifurcation analysis of the Rankine-Hugoniot locus is the enabling fulcrum for solving the Riemann problem for all data, which should be a prototype for general three component flow of two phases in porous media. Despite the existence of many similarities between this model and earlier models where proofs were not possible, here we managed to prove analytically many features. This system of conservation laws has three equations yet it leads to a characteristic polynomial of degree two; this peculiar feature has been unveiled recently, and it is typical of flow of fluids that change density upon changing phase.