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

Abstract

We solve the model for the flow of nitrogen, vapor, and water in a porous medium, neglecting compressibility, heat conductivity, and capillary effects. Our choice of injection conditions is determined by the application to clean up polluted sites. We study all mathematical structures, such as rarefaction, shock waves, and their bifurcations; we also develop a systematic method to find fundamental solutions for thermal compositional flows in porous media. In addition, we unexpectedly find a rarefaction evaporation wave which has not been previously reported in any other study.

A unsplitting finite volume method for models with stiff relaxation source terms

“…The quantity is the time scale related to the mass phase transfer per unit volume that are activated in non-equilibrium regions. The mass transfer are very fast and reversible [18,19,20]. The generalized eigenvalue system for (8)…”

“…In this problem, we analyse two different methodologies to model thermal flow in porous media, which in turn are used to enhanced oil recovery [18] and in situ combustion [19]. Under the thermodynamic equilibrium such system is given by [18,19],…”

“…The mathematical formalism for (14) was developed in [18,19]. However, for a general approach of (15) much more deeper studies are required.…”

“…The full model with all parameters is found in [19]. However, we can also propose a model far from equilibrium.…”

“…The two positive constants g and α denote the gravity constant and the friction coefficient, respectively; the quantity is relate to a compressible Euler flow [21,23]. The system (19) is closed by the classical definition of the total energy E = e + u 2 /2, where e is the internal energy, and by a pressure law p = p(ρ, e), which in turn satisfies classical gas dynamic assumptions [23]. We consider a compressible flow with = 1/2 and initial condition (ρ, u, p) = (1, 0, 1) for 1/2 ≤ x ≤ 2/3 and (ρ, u, p) = (1, 0, 2) otherwise.…”