Online adaptive POd-DEIM model reduction for fast simulation of flows in heterogeneous media

In the paper, we propose an online adaptive POD-DEIM model reduction method for fast multiscale reservoir simulations in highly heterogeneous porous media. The approach uses Proper Orthogonal Decomposition (POD) Galerkin projection to construct a global reduced system. The nonlinear terms are approximated by the Discrete Empirical Interpolation Method (DEIM). To adapt at the online stage the states (velocity, pressure and saturation) of the system, we incorporate new data, as it becomes available. Once the criterion for updates is satisfied, we adapt the reduced system online by updating the POD subspace and the DEIM approximation of the nonlinear functions. These global online basis function updates improve the accuracy of snapshot approximation. Since the adaption is performed infrequently, the new methodology does not add a significant computational overhead due to the adaptation of the reduced bases. Our 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. Our method is able to construct a robust reduced system even if an initial poor choice of snapshots is used. We demonstrate with a numerical experiment to demonstrate the efficiency of our method.