Summary
Accurate modeling of flow in oil/gas reservoirs requires a detailed description of reservoir properties such as permeability and porosity. However, such reservoirs are inherently heterogeneous and exhibit a high degree of spatial variability in medium properties. Significant spatial heterogeneity and a limited number of measurements lead to uncertainty in characterization of reservoir properties and thus to uncertainty in predicting flow in the reservoirs. As a result, the equations that govern flow in such reservoirs are treated as stochastic partial differential equations. The current industrial practice is to tackle the problem of uncertainty quantification by Monte Carlo simulations (MCS). This entails generating a large number of equally likely random realizations of the reservoir fields with parameter statistics derived from sampling, solving deterministic flow equations for each realization, and post-processing the results over all realizations to obtain sample moments of the solution. This approach has the advantages of applying to a broad range of both linear and nonlinear flow problems, but it has a number of potential drawbacks. To properly resolve high-frequency space-time fluctuations in random parameters, it is necessary to employ fine numerical grids in space-time. Therefore, the computational effort for each realization is usually large, especially for large-scale reservoirs. As a result, a detailed assessment of the uncertainty associated with flow-performance predictions is rarely performed.
In this work, we develop an accurate yet efficient approach for solving flow problems in heterogeneous reservoirs. We do so by obtaining higher-order solutions of the prediction and the associated uncertainty of reservoir flow quantities using the moment-equation approach based on Karhunen-Loéve decomposition (KLME). The KLME approach is developed on the basis of the Karhunen-Loéve (KL) decomposition, polynomial expansion, and perturbation methods. We conduct MCS and compare these results against different orders of approximations from the KLME method. The 3D computational examples demonstrate that this KLME method is computationally more efficient than both Monte Carlo simulations and the conventional moment-equation method. The KLME approach allows us to evaluate higher-order terms that are needed for highly heterogeneous reservoirs. In addition, like the Monte Carlo method, the KLME approach can be implemented with existing simulators in a straightforward manner, and they are inherently parallel. The efficiency of the KLME method makes it possible to simulate fluid flow in large-scale heterogeneous reservoirs.
Introduction
Owing to the heterogeneity of geological formations and the incomplete knowledge of medium properties, the medium properties may be treated as random functions, and the equations describing flow and transport in these formations become stochastic. Stochastic approaches to flow and transport in heterogeneous porous media have been extensively studied in the last 2 decades, and many stochastic models have been developed (Dagan 1989; Gelhar 1993; Zhang 2002). Two commonly used approaches for solving stochastic equations are MCS and the moment-equation method. A major disadvantage of the Monte Carlo method, among others, is the requirement for large computational efforts. An alternative to MCS is an approach based on moment equations, the essence of which is to derive a system of deterministic partial differential equations governing the statistical moments [usually the first two moments (i.e., mean and covariance)], and then solve them analytically or numerically.
THE MONTE CARLO METHOD APPLIED TO A PROBLEM IN $gamma$-RAY DIFFUSION
