POD-DEIM Global-Local Model Reduction for Multi-phase Flows in Heterogeneous Porous Media
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Many applications such as production optimization and reservoir management are computationally demanding due to a large number of forward simulations. Typically, each forward simulation involves multiple scales and is computationally expensive. The main objective of this dissertation is to develop and apply both local and global model-order reduction techniques to facilitate subsurface flow modeling. We develop a POD-DEIM global model reduction method for multi-phase flow simulation. The approach entails the use of Proper Orthogonal Decomposition (POD)-Galerkin projection, and Discrete Empirical Interpolation Method (DEIM). POD technique constructs a small POD subspace spanned by a set of global basis that can approximate the solution space. The reduced system is set up by projecting the full-order system onto the POD subspace. Discrete Empirical Interpolation Method (DEIM) is used to reduce the nonlinear terms in the system. DEIM overcomes the shortcomings of POD in the case of nonlinear PDEs by retaining nonlinearities in a lower dimensional space. The POD-DEIM global reduction method enjoys the merit of significant complexity reduction. We also propose an online adaptive global-local POD-DEIM model reduction method. This unique global-local online combination allows (1) developing local indicators that are used for both local and global updates; (2) computing global online modes via local multiscale basis functions. The multiscale basis functions consist of offline and some online local basis functions. The main contribution of the method is that the criteria for adaptivity and the construction of the global online modes are based on local error indicators and local multiscale basis functions which can be cheaply computed. The approach is particularly useful for situations where one needs to solve the reduced system for inputs or controls that result in a solution outside the span of the snapshots generated in the offline stage. Another aspect of my dissertation is the development of a local model reduction method for multiscale problems. We use global coupling in the coarse grid level via the mortar framework to link the sub-grid variations of neighboring coarse regions. The mortar framework offers some advantages, such as the flexibility in the constructions of the coarse grid and sub-grid capturing tools. By following the framework of the Generalized Multiscale Finite Element Method (GMsFEM), we design an enriched multiscale mortar space. Using the proposed multiscale mortar space, we (1) construct a multiscale finite element method to solve the flow problem on a coarse grid; (2) design two-level preconditioners as exact solver for the flow problem.
Yang, Yanfang (2016). POD-DEIM Global-Local Model Reduction for Multi-phase Flows in Heterogeneous Porous Media. Doctoral dissertation, Texas A & M University. Available electronically from