Stochastic formulation for uncertainty analysis of two-phase flow in heterogeneous reservoirs

In this article we use a direct approach to quantify the uncertainty in Row performance predictions due to uncertainty in the reservoir description. W e solve moment equations derived from a stochastic mathematical statement o f immiscible nonlinear two-phase how in heterogeneous reservoirs. Our stoch astic approach is different from the Monte Carlo approach. In the Monte Car lo approach, the prediction uncertainty is obtained through a statistical p ostprocessing of flow simulations, one for each of a large number of equipr obable realizations of the reservoir description. We treat permeability as a random space function. In rum, saturation and fl ow velocity are random fields. We operate in a Lagrangian framework to deal with the transport problem. That is, we transform to a coordinate system a ttached to streamlines (time, travel time, and transverse displacements). W e retain the normal Eulerian (space and time) framework for the total veloc ity, which we take to be dominated by the heterogeneity of the reservoir. W e derive and solve expressions for the first (mean) and second (variance) m oments 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. Gen eral expressions for the moments of total velocity, travel time, transverse displacement, water saturation, production rate, and cumulative recovery a re presented and analyzed. A detailed comparison of the moment solution app roach with high-resolution Monte Carlo simulations for a variety of two-dim ensional problems is presented. We also discuss the advantages and limits o f the applicability of the moment equation approach relative to the Monte C arlo approach.