Preprint A573/2007
The Riemann solution for thermal flows with mass transfer between phases

Dan Marchesin | Lambert, Wanderson

**Keywords: **
Balance laws | conservation laws | Riemann problem | asymptotic expansion | porous medium | steamdrive | local thermodynamic equilibrium | geothermal energy | multiphase flow

We are interested in solving systems of balance equations under the approximation of local
thermodynamical equilibrium except at very localized locations. This equilibrium occurs for states
on a stratified variety called the 'thermodynamical equilibrium variety', which is obtained as the
domain of the zero-order approximation of an asymptotic expansion for these balance laws. Waves far
from thermodynamical equilibrium occur in thin regions of physical space and they form shocks
connecting sheets of the stratified variety. In this scenario, we develop the general theory for
fundamental solutions of a large class of systems of balance equations. We study all bifurcation
loci, such as coincidence and inflection locus and develop a systematic approach to solve problems
described by similar equations.
For concreteness, we exhibit the bifurcation theory for a representative system with four equations. This class of equations models thermal flow
with mass interchange between phases in porous media appearing in
oil recovery. 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 initially saturated with water, with
applications to geothermal energy recovery.