The Riemann Problem for Multiphase Flows in Porous Media With Mass Transfer Between Phases

Abstract

We are interested in solving systems of conservation laws modeling multiphase fluid flows under the approximation of local thermodynamical equilibrium except at very localized places. This equilibrium occurs for states on sheets of a stratified variety called the "thermodynamical equilibrium variety," obtained from thermodynamical laws. Strong deviation from equilibrium occurs in shocks connecting adjacent sheets of this variety. We assume that fluids may expand and we model the physical problem by a system of equations where a velocity variable appears only in the flux terms, giving rise to a wave with "infinite" characteristic speed. We develop a general theory for fundamental solutions for this class of equations. We study all bifurcation loci, such as coincidence and inflection loci and develop a systematic approach to solve problems described by similar equations. For concreteness, we exhibit the bifurcation theory for a representative system with three equations. We find the complete solution of the Riemann problem for two-phase thermal flow in porous media with two chemical species; to simplify the physics, the liquid phase consists of a single chemical species. We give an example of steam and nitrogen injection into a porous medium, with applications to geothermal energy recovery.

The Riemann Solution for the Injection of Steam and Nitrogen in a Porous Medium

“…The other unknown is u. We define the steam and nitrogen gas compositions, ψ gw and ψ gn , as follows, see Bruining and Marchesin (2006); Lambert and Marchesin (2008):…”

“…The spg configuration consists of a two-parameter set of triplets ( (2006); Lambert and Marchesin (2008). Here, (V 1 , V 2 ) = (ψ gw , T ) are the primary variables, and s w = 0, ψ gn = 1 − ψ gw .…”

“…It is a general feature for systems of form (1) that u appears isolated in the eigenvalues and that u appears only in the last component of the eigenvectors, see Lambert and Marchesin (2008).…”

“…In Lambert and Marchesin (2009), we propose a general formalism for solving this class of equations; see also Lambert and Marchesin (2008) for further details.…”

“…We present an example of the Riemann solution for data in two different phase configurations and also analyze the bifurcations in these solutions. This model is an example of a general theory developed for compositional models with two chemical species and two phases that can be applied for different thermal flows; see Lambert and Marchesin (2009) for more mathematical details.…”