Stochastic Modeling of Immiscible Flow with Moment Equations

We present numerical and analytical results for a stochastic representation of immiscible flow in a heterogeneous reservoir. Using a perturbation method, we derive equations of statistical moments to second order in the standard deviation of log permeability. We allow these moments to depend on location, hence the common assumption of statistical homogeneity (stationarity) does not apply. This method is developed as an approach either to scale-up, or to uncertainty propagation, and is an alternative to ensemble simulation and post-processing (Monte Carlo). The second-order theory gives a correction to first-order, macrodispersion models. Finally, we discuss the applicability of a Gaussian joint distribution assumption, with which the method becomes third-order in the standard deviation of log permeability.