Abstract:

Reservoir simulation of realistic systems can be computationally demanding because of the large number of system unknowns and the intrinsic nonlinearity of typical problems. Compositional simulation, in which multiple components and complex phase behavior are present, can be particularly challenging. The high computational cost of reservoir simulation represents a substantial issue for applications such as production optimization and history matching, in which hundreds or thousands of simulation runs must be performed. Reduced-order modeling represents a promising approach for accelerating the simulations required for these important applications. In this work, we focus on the development and application of a reduced-order modeling technique called POD-TPWL, which combines trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) to provide highly efficient surrogate models. The POD-TPWL method expresses new solutions in terms of linearizations around states generated (and saved) during previously simulated “training” runs. Highdimensional states (e.g., pressure and saturation in every grid block in an oil-water problem) are projected optimally into a low-dimensional subspace using POD.

We first consider the application of POD-TPWL for data assimilation (or history matching) in oil-water systems. The POD-TPWL model developed for this application represents simulation results for new geological realizations in terms of a linearization around training cases. Geological models are expressed in reduced terms using a Karhunen-Loeve expansion of the log-transmissibility field. Thus, both the reservoir states (represented using POD) 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 relatively 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, EnKF results using 50 high-fidelity simulations along with 150 POD-TPWL simulations are shown to be much better than those using only 50 high-fidelity simulations (for which ensemble collapse is observed) and are, in fact, generally comparable to the results achieved using 200 high-fidelity simulations.

We next develop a POD-TPWL methodology for oil-gas compositional systems. This model is based on the molar formulation in Stanford’s General Purpose Research Simulator with Automatic Differentiation, AD-GPRS, which uses pressure and overall component mole fractions as the primary unknowns. Several new features, including the application of a Petrov-Galerkin projection to reduce the number of equations (rather than the Galerkin projection, which was used previously), and a new procedure for determining which saved state to use for linearization, are incorporated into the method. Results are presented for heterogeneous three-dimensional reservoir models with up to six hydrocarbon components. Reasonably close agreement between full-order reference solutions and compositional POD-TPWL simulations is demonstrated for the cases considered. Construction of the POD-TPWL model requires preprocessing overhead computations equivalent to about three to four fullorder runs. Runtime speedups using POD-TPWL are, however, very significant – about a factor of 500-800 for the cases considered. The POD-TPWL model is thus well suited for use in computational optimization, in which many simulations must be performed, and we present examples demonstrating its application for such problems.

Finally, we investigate the accuracy and stability of different constraint reduction treatments for POD-TPWL models. Following an error analysis of the general PODTPWL representation, two projection methods, namely Galerkin projection and Petrov- Galerkin projection, are derived by minimizing the constraint reduction error under different norms. These projection methods are assessed computationally for oil-water and compositional systems. For oil-water systems, Galerkin projection combined with a stabilization procedure is generally more accurate than Petrov-Galerkin projection, though even with this stabilization Galerkin projection is not guaranteed to be stable at all time steps. For compositional systems, the POD-TPWL model with Galerkin projection exhibits poor stability, while Petrov-Galerkin provides a consistently stable and robust POD-TPWL model. A hybrid procedure for oil-water systems, which applies different projections at different time steps to achieve both accuracy and stability, is presented. Two other constraint reduction methods, referred to as inverse projection and weighted inverse projection, are also formulated and tested. These approaches are computationally more expensive but do offer some theoretical advantages, and may be useful in realistic problems following further development.

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Copyright 2013, Jincong He: Please note that the reports and theses are copyright to their original authors. Authors have given written permission for their work to be made available here. Readers who download reports from this site should honor the copyright of the original authors and may not copy or distribute the work further without the permission of the author, Jincong He.