An Efficient Stochastic Decomposition Approach for Large-Scale Subsurface Flow Problems

Abstract

Subsurface formations are of large scales and are inherently heterogeneous at a multiplicity of scales. Significant spatial heterogeneity and a limited number of measurements lead to uncertainty in characterization of formation properties and thus, to uncertainty in predicting flow in the formations. Such uncertainties add another dimension in probability space to the already large-scale subsurface problems. In this work, we develop an accurate yet efficient approach for solving flow problems in large-scale heterogeneous formations. 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-Loeve decomposition (KLME). In the KLME approach, the log permeability (lnK) field is first expanded into a multiscale series in terms of orthogonal standard Gaussian random variables with their coefficients obtained from the eigen-decomposition of the lnK covariance. Next, the pressure and velocity fields are all decomposed with perturbation expansions in which each individual term is further expanded into a polynomial series of orthogonal Gaussian random products. The coefficients associated with these series are deterministic and solved recursively from low to high expansion orders. The pressure and velocity moments (such as the means, covariances, and higher moments) can then be calculated from these coefficients using simple algebraic operations. There are two attractive computational features in this new approach. First, all equations for the deterministic coefficients share exactly the same structure as the original equation, which greatly simplifies its implementation as the existing simulators/solvers can be utilized as well as significantly reduces the computation effort as the coefficient matrix remains unchanged and only the right-hand-side vector needs to be updated across different orders. Second, at each expansion order, the equations are independent of each other, which allows for performing massively parallel computation. The new approach is validated and its efficiency and accuracy is demonstrated with traditional Monte Carlo simulations in large-scale three-dimensional subsurface problems.