Abstract:

We apply a hybrid formulation combining the stochastic reduced basis methods with poly- nomial chaos expansions, which has been introduced recently by Nair [1] for solving the linearized stochastic partial di®erential equation governing single-phase °ow in porous media. We use a generalization of stochastic reduced basis projection schemes to non- Gaussian uncertainty models. The Karhunen-Loeve expansion is used to model the input log-permeability ¯eld; for non-Gaussian input we employ Polynomial Chaos expansion to model the nonlinearity in terms of Hermite polynomials. For the pressure equation, we em- ploy basis vectors spanning the preconditioned stochastic Krylov subspace which e±ciently reduces the dimensions of the solution space. Then the Galerkin projection scheme is used to estimate the coe±cients of the reduced basis approximations. We present a detailed comparison between high resolution Monte Carlo simulation and the Stochastic Reduced Basis Method (SRBM). We also study the di®erence between predictions obtained using SRBM with low-order Statistical Moment Equations (SME) and a Probabilistic Collocation Method (PCM). Natural formations with high permeability variability and large spatial cor- relation scales are of great interest. Consequently, we examine SRBM for systems with a variance of log-permeability ¾2 lnK from 0.1 to 3 and correlation scales (normalized by do- main length) of 0.05 to 0.5. In order to avoid issues related to statistical convergence and resolution level, we used 9000 highly detailed realizations of permeability for Monte Carlo Simulation (MCS). We show that SRBM gives reasonably close results with MCS using a small number of Krylov subspace basis at lower computational cost.

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