Conditional Stochastic Moment Equations for Uncertainty Analysis of Flow in Heterogeneous Reservoirs

Abstract

SummaryWe assume that permeability is a random space function defined by its mean and covariance. The stochastic nature of the permeability description leads to uncertainty in flow-related quantities such as pressure, saturation, and production rate. We extended our statistical moment equation (SME) approach to accommodate conditioning. The conditional statistical moment equations (CSME) framework is a direct approach for quantifying the uncertainty in flow performance caused by uncertainty in the reservoir description. It is quite different from Monte Carlo Simulation (MCS). In MCS, the performance 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 developed a CSME computational tool for flow in heterogeneous domains, which we use here to analyze the behavior of the second statistical moment of pressure and velocity in the presence of permeability measurements. We study the effect of both the number and spatial arrangement of available measurements on the computed variances of pressure and velocity. This numerical CSME tool allows us to quantify the value of existing and future information, and that helps in the evaluation of existing projects and in steering future developments. We present several examples that demonstrate how to choose the best sampling locations to obtain maximum reduction in prediction uncertainty. We compare our results with high-resolution MCS.