Fast simulation algorithms based on reduced-order modeling have been developed in order to facilitate large-scale and complex computationally intensive reservoir simulation and optimization. Methods like proper orthogonal decomposition (POD) and Dynamic Mode Decomposition (DMD) have been successfully used to efficiently capture and predict the behavior of reservoir fluid flow. Non-intrusive techniques (e.g., DMD), are especially attractive as it is a data-driven approach that do not require code modifications (equation free). In this paper, we will further enhance the application of the DMD, by investigating sparse approximations of the snapshots. This is particularly useful when there is a limited number of sparse measurements as in the case of reservoir simulation.

The approach taken here is the snapshot-based model reduction, whereby one computes a sequence of reservoir simulation solutions (e.g., pressures and water saturations in the case of two-phase flow model) forming a big data matrix – we call this the offline step - that is used to compute basis for representing the states of the system for different input parameters – the online step. The selection of these few basis is the core of the model reduction methods. DMD selects the basis and apply the reduction without knowledge of the inner works of the reservoir simulator, as opposed to the POD methods. Sparse DMD has been introduced recently to determine the subset of the DMD models that has the most profound influence on the quality of the approximation of the snapshot sequence. Two model reduction process are involved. One is offline process, which does not require running the simulator but rather predicting future behavior with linear combination of DMD modes. The other online process incorporates sparsity DMD modes in numerical simulator to release the burden of linear matrix solver.

We first show the methodology applied to a 3-D single phase flow problem. Here we show the DMD modes and its physical interpretations, and then move to two phase flow for 2-D heterogeneous reservoir using the SPE-10 benchmark. Both online and offline process will be used for evaluation. We observe that with a few DMD modes we can capture the behavior of the reservoir models. Sparse DMD leads to the optimal selection of the few DMD modes. We also assess the trade-offs between problem size and computational time for each reservoir model. The novelty of our method is the application of sparse DMD, which is a data-driven technique and the ability to select few optimal basis for the case of reservoir simulation.

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