Adaptive POD model reduction for solute transport in heterogeneous porous media

We study the applicability of a model order reduction technique to the solution of transport of passive scalars in homogeneous and heterogeneous porous media. Transport dynamics are modeled through the advection-dispersion equation (ADE) and we employ Proper Orthogonal Decomposition (POD) as a strategy to reduce the computational burden associated with the numerical solution of the ADE. Our application of POD relies on solving the governing ADE for selected times, termed snapshots. The latter are then employed to achieve the desired model order reduction. We introduce a new technique, termed Snapshot Splitting Technique (SST), which allows enriching the dimension of the POD subspace and damping the temporal increase of the modeling error. Coupling SST with a modeling strategy based on alternating over diverse time scales the solution of the full numerical transport model to its reduced counterpart allows extending the benefit of POD over a prolonged temporal window so that the salient features of the process can be captured at a reduced computational cost. The selection of the time scales across which the solution of the full and reduced model are alternated is linked to the Péclet number (Pe), representing the interplay between advective and dispersive processes taking place in the system. Thus, the method is adaptive in space and time across the heterogenous structure of the domain through the combined use of POD and SST and by way of alternating the solution of the full and reduced models. We find that the width of the time scale within which the POD-based reduced model solution provides accurate results tends to increase with decreasing Pe. This suggests that the effects of local-scale dispersive processes facilitate the POD method to capture the salient features of the system dynamics embedded in the selected snapshots. Since the dimension of the reduced model is much lower than that of the full numerical model, the methodology we propose enables one to accurately simulate transport at a markedly reduced computational cost

POD-DEIM Global-Local Model Reduction for Multi-phase Flows in Heterogeneous Porous Media

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

Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media

We propose an online adaptive local-global POD-DEIM model reduction method for flows in heterogeneous porous media. The main idea of the proposed method is to use local online indicators to decide on the global update, which is performed via reduced cost local multiscale basis functions. This unique local-global 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 approach used for constructing a global reduced system is based on Proper Orthogonal Decomposition (POD) Galerkin projection. The nonlinearities are approximated by the Discrete Empirical Interpolation Method (DEIM). The online adaption is performed by incorporating new data, which become available at the online stage. Once the criterion for updates is satisfied, we adapt the reduced system online by changing the POD subspace and the DEIM approximation of the nonlinear functions. The main contribution of the paper is that the criterion for adaption and the construction of the global online modes are based on local error indicators and local multiscale basis function which can be cheaply computed. Since the adaption is performed infrequently, the new methodology does not add significant computational overhead associated with when and how to adapt the reduced basis. Our approach is particularly useful for situations where it is desired 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. Our method also offers an alternative of constructing a robust reduced system even if a potential initial poor choice of snapshots is used. Applications to single-phase and two-phase flow problems demonstrate the efficiency of our method