Trajectory-based DEIM (TDEIM) model reduction applied to reservoir simulation

Abstract

Two well-known model reduction methods, namely the trajectory piecewise linearization (TPWL) approximation and the discrete empirical interpolation method (DEIM), are combined to take advantage of their benefits and avoid their shortcomings to generate reduced-order models for reservoir simulation. To this end, we use the trajectory-based DEIM (TDEIM) to approximate the nonlinear terms in the simulation. Explicitly, we express such quantities in the test simulation as the sum of their equivalents evaluated at the closest available training point from the high-fidelity training trajectory and a perturbed contribution defined as the difference between the test and the training runs. We only interpolate this difference term in the reduced space of DEIM instead of the original one, resulting in computational savings and improvement in accuracy. TDEIM is further combined with the proper orthogonal decomposition (POD) method to provide an efficient POD-TDEIM framework. We test our new methodology on three examples, involving two-phase (water-oil) heterogeneous reservoir models. First, the performance of POD-TDEIM is compared with POD-TPWL and POD-DEIM on a 2D reservoir model. For the same set of high-fidelity training runs, POD-TDEIM outperforms the other two methods. We further propose an extended TDEIM in which the nonlinear term is expanded along the training trajectory to include an additional higher order derivative. An example of a 3D reservoir model is then presented to show the capability of the extended TDEIM to improve the accuracy of the reduced model further when handling significant discrepancies between the training and test boundary controls. We also present another 3D example to demonstrate the superior correctness and numerical stability of using Petrov-Galerkin projection with TDEIM over the Galerkin projection.