A Stochastic Approach to the Two-Phase Displacement Problem in Heterogeneous Porous Media

The problem of two-phase immiscible flow in heterogeneous porous media in the case of a horizontal displacement of some fluid by another, which is of practical importance in industrial oil recovery, is considered. Assuming that (a) the saturation jump on the displacement front is constant, (b) the log-permeability of the medium obeys Gaussian statistics, and (c) the case when the front is stable, the displacement front position and the saturation distribution are described analytically in terms of generalized functions. Note that in our analysis we do not assume that the front shape fluctuations are small, and in this respect our results may be regarded as exact. The assumption that the log-permeability fluctuations are small was only used in deriving the linear relation between the log-permeability of a porous medium and the total flow velocity (Nœtinger et al. in Fluid Dyn 41(5):830–842, 2006). By means of ensemble averaging, the mean saturation and saturation variance are found in the vicinity of the front. These characteristics are related to the variance of front displacements, which, in turn, can be calculated analytically. Next, a method for reconstructing the full solution for the saturation (rarefaction wave) is proposed. Such a full solution satisfies the mass conservation requirement. Finally, the theoretical predictions are compared with the results of numerical simulations carried out within the framework of Monte-Carlo method.