Reduced-order flow modeling and geological parameterization for ensemble-based data assimilation

Reduced-order modeling represents an attractive approach for accelerating computationally expensive reservoir simulation applications. In this paper, we introduce and apply such a methodology for data assimilation problems. The technique applied to provide flow simulation results, trajectory piecewise linearization (TPWL), has been used previously for production optimization problems, where it has provided large computational speedups. The TPWL model developed here represents simulation results for new geological realizations in terms of a linearization around previously simulated (training) cases. The high-dimensional representation of the states is projected into a low-dimensional subspace using proper orthogonal decomposition. The geological models are also represented in reduced terms using a Karhunen-Loeve expansion of the log-transmissibility field. Thus, both the reservoir states and geological parameters are described very concisely. The reduced-order representation of flow and geology is appropriate for use with ensemble-based data assimilation procedures, and here it is incorporated into an ensemble Kalman filter (EnKF) framework to enrich the ensemble at a low cost. The method is able to reconstruct full-order states, which are required by EnKF, whenever necessary. The combined technique enables EnKF to be applied using many fewer high-fidelity reservoir simulations than would otherwise be required to avoid ensemble collapse. For two- and three-dimensional example cases, it is demonstrated that EnKF results using 50 high-fidelity simulations along with 150 TPWL simulations are much better than those using only 50 high-fidelity simulations (for which ensemble collapse is observed) and are, in fact, comparable to the results achieved using 200 high-fidelity simulations. (C) 2012 Elsevier Ltd. All rights reserved.