yong Li

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

to ed, rical ent s of ns the ent Summary In this article we use a direct approach to quantify the uncerta in flow performance predictions due to uncertainty in the reserv description. We solve moment equations derived from a stocha mathematical statement of immiscible nonlinear two-phase fl in heterogeneous reservoirs. Our stochastic approach is diffe from the Monte Carlo approach. In the Monte Carlo approach, prediction uncertainty is obtained through a statistical po processing of flow simulations, one for each of a large numbe equiprobable realizations of the reservoir description. We treat permeability as a random space function. In turn, s ration and flow velocity are random fields. We operate in a L grangian framework to deal with the transport problem. That 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. 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 comp flow geometry. Closed outer boundaries and converging/diverg flows due to the presence of sources/sinks require special m ematical and numerical treatments. General expressions for moments of total velocity, travel time, transverse displacem water saturation, production rate, and cumulative recovery presented and analyzed. A detailed comparison of the mom solution approach with high-resolution Monte Carlo simulatio for a variety of two-dimensional problems is presented. We a discuss the advantages and limits of the applicability of the m ment equation approach relative to the Monte Carlo approach

Perturbation-based moment equation approach for flow in heterogeneous porous media: applicability range and analysis of high-order terms

We present detailed comparisons between high-resolution Monte Carlo simulation (MCS) and low-order numerical solutions of stochastic moment equations (SMEs) for the first and second statistical moments of pressure. The objective is to quantify the difference between the predictions obtained from MCS and SME. Natural formations with high permeability variability and large spatial correlation scales are of special interest for underground resources (e.g. oil and water). Consequently, we focus on such formations. We investigated fields with variance of log-permeability, σY2, from 0.1 to 3.0 and correlation scales (normalized by domain length) of 0.05 to 0.5. In order to avoid issues related to statistical convergence and resolution level, we used 9000 highly resolved realizations of permeability for MCS. We derive exact discrete forms of the statistical moment equations. Formulations based on equations written explicitly in terms of permeability (K-based) and log-transformed permeability (Y-based) are considered. The discrete forms are applicable to systems of arbitrary variance and correlation scales. However, equations governing a particular statistical moment depend on higher moments. Thus, while the moment equations are exact, they are not closed. In particular, the discrete form of the second moment of pressure includes two triplet terms that involve log-permeability (or permeability) and pressure. We combined MCS computations with full discrete SME equations to quantify the importance of the various terms that make up the moment equations. We show that second-moment solutions obtained using a low-order Y-based SME formulation are significantly better than those from K-based formulations, especially when σY2 > 1. As a result, Y-based formulations are preferred. The two triplet terms are complex functions of the variance level and correlation length. The importance (contribution) of these triplet terms increases dramatically as σY2 increases above one. We also show that one of the triplet terms is much more important than the other. When comparing K-based MCS with Y-based SME, model differences must be taken into consideration. These differences (model errors) are due to embedded assumptions and differences in implementing the discrete forms of the equations.