Pressure Preconditioning Using Proper Orthogonal Decomposition a Thesis Submitted to the Department of Energy Resources Engineering of Stanford University in Partial Fulfillment of the Requirements for the Degree of Master of Science

A linear-solver preconditioner for the pressure systems in reservoir simulation based on Proper Orthogonal Decomposition (POD) is developed and investigated. The POD approach entails the use of a lower-dimensional space to approximate the original high-dimensional space. Essentially, POD can be considered as a Singular Value Decomposition (SVD) approach. The POD approximation error can be estimated by examining the singular values. The POD-based Reduced-Order Modeling (ROM) strategy projects the full linear system (Ax = b) into a reduced form (Arz = br). The resulting ROM can be used to explore the parameter space of interest at a much lower computational cost compared with the full model. The utility of the ROM model depends on its accuracy in representing the solutions of the full-order model. It is shown that among the different schemes, Galerkin projection gives the best accuracy. In order to quantify the quality of a POD basis for a specific linear system, algebraic indicators are proposed. Various issues are discussed related to designing an adaptive pressure preconditioning strategy, in which the simulator can adjust the POD space adaptively when the approximation becomes less accurate. A spectral-analysis approach for assessing POD as a pressure preconditioner is presented. A two-dimensional example is used to demonstrate the high and low frequencies that comprise the solution space, of which the low-frequency errors are more difficult to resolve. Numerical experiments for oil/water reservoir iii models using the IMPES scheme show that the POD preconditioner can resolve certain eigenvectors, though the errors associated with significant parts of the spectrum are left untouched. The POD method is effective when the right-hand-side (RHS) encountered in the simulation contains only a few components with frequencies that the POD basis has no effect on. Unlike other preconditioners that solve the problem equally well for any RHS, the POD preconditioner is RHS dependent. Hence, the POD preconditioner does not appear to be an ideal choice as a generic pressure preconditioner.