In this paper 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 quite different from the Monte Carlo approach. In the Monte Carlo approach, the prediction uncertainty is obtained through a statistical post-processing of flow simulations, one for each of a large number of equiprobable realizations of the reservoir description.
We treat porosity and permeability as random space functions. 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 applicability of the moment equation approach relative to the Monte Carlo approach.