Stochastic Formulation for Uncertainty Assessment of Two-Phase Flow in Heterogeneous Reservoirs

Summary In this article we use a direct approach to quantify the uncertainty in flow performance predictions due to uncertainty in the reservoir description. We solve moment equations derived from a stochastic mathematical statement of immiscible nonlinear two-phase flow in heterogeneous reservoirs. Our stochastic approach is different from the Monte Carlo approach. In the Monte Carlo approach, the prediction uncertainty is obtained through a statistical postprocessing of flow simulations, one for each of a large number of equiprobable realizations of the reservoir description. We treat permeability as a random space function. In turn, saturation and flow velocity are random fields. We operate in a Lagrangian framework to deal with the transport problem. That is, we transform to a coordinate system attached to streamlines ~time, travel time, and transverse displacements!. We retain the normal Eulerian ~space and time! framework for the total velocity, which we take to be dominated by the heterogeneity of the reservoir. We derive and solve expressions for the first ~mean! and second ~variance! moments of the quantities of interest. We demonstrate the applicability of our approach to complex flow geometry. Closed outer boundaries and converging/diverging flows due to the presence of sources/sinks require special mathematical and numerical treatments. General expressions for the moments of total velocity, travel time, transverse displacement, water saturation, production rate, and cumulative recovery are presented and analyzed. A detailed comparison of the moment solution approach with high-resolution Monte Carlo simulations for a variety of two-dimensional problems is presented. We also discuss the advantages and limits of the applicability of the moment equation approach relative to the Monte Carlo approach.