Uniqueness conditions in a hyperbolic model for oil recovery by steamdrive

In this paper we study a one-dimensional model for oil recovery by steamdrive. This model consists of two parts: a (global) interface model and a (local) steam condensation/capillary diffusion model. In the interface model a steam condensation front (SCF) is present as an internal boundary between the hot steam zone (containing water, oil and steam) and the cold liquid zone (containing only water and oil). Disregarding capillary pressure away from the SCF, a 2x2 hyperbolic system arises for the water and steam saturation. This system cannot be solved uniquely without additional conditions at the SCF. To find such conditions we make a blow up of the SCF and consider a parabolic transition model, including capillary diffusion. We study in detail the existence conditions for travelling wave solutions. These conditions translate into the missing matching conditions at the SCF in the hyperbolic limit and thus provide uniqueness. We show that different transition models yield different matching conditions, and thus different solutions of the interface model. We also give a relatively straightforward approximation and investigate its validity for certain ranges of model parameters.