Combustion Fronts in Porous Media

Abstract

The equations governing combustion in situ in petroleum reservoirs are cast in a way that is appropriate to the study of the time asymptotic fundamental waves: rarefactions and shocks. The physical diffusion terms are maintained so that the internal structure of shocks can be analyzed. In this work, we determine the planar traveling wave solutions for the system of three evolution equations modeling combustion in two phase flow, obtained by neglecting heat losses and volume change effects. The chemical reaction rate is governed by Arrhenius' reaction rate law. Our analysis is intended to capture the main effect of combustion: the reduction of oil viscosity due to temperature increase and its practical result---the increase in oil recovery. It also is intended to resolve the difficulty introduced by the singularity in the Arrhenius reaction rate law. Our analysis shows that the combustion wave is represented by a heteroclinic connection in a nonhyperbolic system of three ordinary differential equations. It follows that numerical simulations need to resolve completely the small parabolic terms in the model to be reliable. The analysis also indicates that the phase diagram should be stable under perturbations of interest. Thus, the general structure of combustion waves in a more realistic model should be similar to the one we found here.