A COMPARATIVE STUDY ON UNCERTAINTY QUANTIFICATION FOR FLOW IN RANDOMLY HETEROGENEOUS MEDIA USING MONTE CARLO SIMULATIONS AND CONVENTIONAL AND KL-BASED MOMENT-EQUATION APPROACHES ∗ ZHIMING LU † AND DONGXIAO ZHANG †‡ SIAM J. SCI. COMPUT. c  2004 Society for Industrial and Applied Mathematics Vol. 26, No. 2, pp. 558–577 Abstract. Geological formations are ubiquitously heterogeneous, and the equations that govern flow and transport in such formations can be treated as stochastic partial differential equations. The Monte Carlo method is a straightforward approach for simulating flow in heterogeneous porous media; an alternative based on the moment-equation approach has been developed in the last two decades to reduce the high computational expense required by the Monte Carlo method. However, the computational cost of the moment-equation approach is still high. For example, to solve head covariance up to first order in terms of σ 2 Y , the variance of log hydraulic conductivity Y = ln Ks, it is required to solve sets of linear algebraic equations with N unknowns for 2N times (N being the number of grid nodes). The cost is even higher if higher-order approximations are needed. Zhang and Lu [J. Comput. Phys., 194 (2004), pp. 773–794] developed a new approach to evaluate high-order moments (fourth order for mean head in terms of σ Y , and third order for head variances in terms of σ 2 Y ) of flow quantities based on the combination of Karhunen–Lo` eve decomposition and perturbation methods. In this study, we systematically investigate the computational efficiency and solution accuracy of three approaches: Monte Carlo simulations, the conventional moment-equation (CME) approach, and the moment-equation approach based on Karhunen–Lo` eve decomposition (KLME). It is evident that the computational cost for the KLME approach is significantly lower than those required by the Monte Carlo and CME approaches. More importantly, while the computational costs (in terms of the number of times for solving linear algebraic equations with N unknowns) for the CME approach depend on the number of grid nodes, the cost for the KLME approach is independent of the number of grid nodes. This makes it possible to apply the KLME method to solve more realistic large-scale flow problems. Key words. Monte Carlo simulations, moment-equation approach, Karhunen–Lo` eve decompo- sition, heterogeneity, flow and transport, porous media AMS subject classifications. 65C05, 65C30, 49N27, 76S05, 78M05, 82C31 DOI. 10.1137/S1064827503426826 1. Introduction. Owing to heterogeneity of geological formations and incom- plete knowledge of medium properties, the medium properties are treated as random space 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 past two decades, and many stochastic models have been developed [7, 9, 4, 35]. Monte Carlo simulation is a conceptually straightforward method for solving these stochastic partial differential equations. It entails generating a large number of equally likely random realizations of the parameter fields, solving deterministic flow and trans- ∗ Received by the editors April 30, 2003; accepted for publication (in revised form) September 30, 2003; published electronically December 22, 2004. This work was supported by DOE/NGOTP under contract AC1005000. This work was performed by an employee of the U.S. Government or under U.S. Government contract. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights. http://www.siam.org/journals/sisc/26-2/42682.html † Hydrology, Geochemistry, and Geology Group (EES-6), MS T003, Los Alamos National Labo- ratory, Los Alamos, NM 87545 (zhiming@lanl.gov, donzhang@lanl.gov). ‡ Current address: Mewbourne School of Petroleum and Geological Engineering, University of Oklahoma, 100 East Boyd, SEC T301, Norman, OK 73019. 558