Trajectory-based DEIM (TDEIM) model reduction applied to reservoir simulation
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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.
KeywordsModel reduction Trajectory piecewise linearization Discrete empirical interpolation method
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We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. We also appreciate Dr. Lou Durlofsky for his valuable ideas to improve this paper.
The authors received financial support from ExxonMobil Upstream Research Company, and permission to publish this paper.
- 1.Adams, R., Fournier, J.: Sobolev Spaces. Academic Press, Oxford (1975)Google Scholar
- 3.Aziz, K., Settari, A.: Petroleum reservoir simulation. Elsevier Applied Science Publishers, New York (1986)Google Scholar
- 10.Christie, M., Blunt, M., et al.: Tenth spe comparative solution project: a comparison of upscaling techniques. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2001)Google Scholar
- 11.van Doren, J.F., Markovinović, R., Jansen, J.D.: Reduced-order optimal control of water flooding using proper orthogonal decomposition. Comput. Geosci. 10(1), 137–158 (2006)Google Scholar
- 12.van Essen, G., Van den Hof, P., Jansen, J.D., et al.: A two-level strategy to realize life-cycle production optimization in an operational setting. SPE J. 18(06), 1–057 (2013)Google Scholar
- 13.Florez, H.: Domain decomposition methods for geomechanics. Ph.D. thesis The University of Texas at Austin (2012)Google Scholar
- 14.Florez, H., Argáez, M.: Applications and comparison of model-order reduction methods based on wavelets and POD. In: Fuzzy Information Processing Society (NAFIPS), 2016 Annual Conference of the North American, pp. 1–8. IEEE (2016)Google Scholar
- 16.Florez, H., Argaez, M.: A reduced order Gauss-Newton method for nonlinear problems based on compressed sensing for PDE applications, chap. 6, pp. 107–128. In Nonlinear Systems - Modeling, Estimation, and Stability. InTech Open. https://doi.org/10.5772/intechopen.74439. ISBN 978-1-78923-405-3 (2018)
- 18.Ghasemi, M., Ibrahim, A., Gildin, E., et al.: Reduced order modeling in reservoir simulation using the bilinear approximation techniques. In: SPE Latin America and Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers (2014)Google Scholar
- 19.Gildin, E., Ghasemi, M.: A new model reduction technique applied to reservoir simulation. In: ECMOR XIV-14Th European Conference on the Mathematics of Oil Recovery (2014)Google Scholar
- 20.Gildin, E., Ghasemi, M., Romanovskay, A., Efendiev, Y.: Nonlinear complexity reduction for fast simulation of flow in heterogeneous porous media. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2013)Google Scholar
- 24.Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Dimension Reduction of Large-Scale Systems, Pp. 261–306. Springer (2005)Google Scholar
- 26.Lie, K.A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source matlab implementation of consistent discretisations on complex grids, vol. 16. https://doi.org/10.1007/s10596-011-9244-4(2012)
- 27.Ljung, L.: System ddentification. In: Signal Analysis and Prediction, pp. 163–173. Springer (1998)Google Scholar