Trajectory-Based DEIM TDEIM Model Reduction Applied to Reservoir Simulation

Abstract

Abstract Two well-known model reduction methods, namely the trajectory piecewise linearization (TPWL) approximation and the discrete empirical interpolation method (DEIM) are combined to utilize 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. Specifically, the nonlinear terms in the test simulation can be expressed as the sum of the nonlinear terms evaluated at the closest available training point from the high-fidelity training trajectory and a perturbed term defined as the difference between the the test and the training terms. We only interpolate this perturbed term in the reduced space of DEIM instead of the original nonlinear term, 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 two 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 one more derivative term. An example with a 3D reservoir model is then presented to show the capability of the extended TDEIM to further improve the accuracy of the reduced model.