Stochastic Analysis of Immiscible Two-Phase Flow in Heterogeneous Media

We present a stochastic analysis of immiscible two-phase flow ~Buckley–Leverett displacement! in heterogeneous reservoirs. Since the detailed spatial variabilities of reservoir properties such as permeability and porosity cannot be described deterministically, both permeability and porosity, or either, are treated as random space functions. In turn, water saturation and oil production rate are also random variables. The main purpose of this study is to estimate the saturation field ~by its expected value! and the associated uncertainty ~by its standard deviation! for Buckley– Leverett displacement in such random media. The expected value and standard deviation may be used to construct confidence intervals for the saturation field in a reservoir and to evaluate the risk associated with a project caused by the incomplete knowledge of reservoir properties. Through transforming to coordinates attached to streamlines, the evaluation of saturation statistical moments is simplified to that of travel time and transverse displacement probability density functions. The latter do not vary with the twophase composition but entirely depend on the total velocity under some conditions. Through some oneand two-dimensional examples, we found that the well-known discontinuities existing in saturation profiles and oil production curves for homogeneous media disappear in the case of heterogeneous media. This is due to heterogeneity fingering, or the so-called ‘‘heterogeneity induced dispersion.’’ The effect of heterogeneity on oil production is the earlier water breakthrough and later oil arrival. The oil production is expected to take longer in heterogeneous media than in homogeneous media.