Online 1. Reducedorder modeling for oilwater and compositional systems, with application to data assimilation and production optimization [electronic resource] [2013]
 He, Jincong.
 2013.
 Description
 Book — 1 online resource.
 Summary

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. Reducedorder 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 reducedorder modeling technique called PODTPWL, which combines trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) to provide highly efficient surrogate models. The PODTPWL 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 oilwater problem) are projected optimally into a lowdimensional subspace using POD. We first consider the application of PODTPWL for data assimilation (or history matching) in oilwater systems. The PODTPWL 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 KarhunenLoeve expansion of the logtransmissibility field. Thus, both the reservoir states (represented using POD) and geological parameters are described very concisely. The reducedorder representation of flow and geology is appropriate for use with ensemblebased 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 fullorder states, which are required by EnKF, whenever necessary. The combined technique enables EnKF to be applied using many fewer highfidelity reservoir simulations than would otherwise be required to avoid ensemble collapse. For two and threedimensional example cases, EnKF results using 50 highfidelity simulations along with 150 PODTPWL simulations are shown to be much better than those using only 50 highfidelity simulations (for which ensemble collapse is observed) and are, in fact, generally comparable to the results achieved using 200 highfidelity simulations. We next develop a PODTPWL methodology for oilgas compositional systems. This model is based on the molar formulation in Stanford's General Purpose Research Simulator with Automatic Differentiation, ADGPRS, which uses pressure and overall component mole fractions as the primary unknowns. Several new features, including the application of a PetrovGalerkin 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 threedimensional reservoir models with up to six hydrocarbon components. Reasonably close agreement between fullorder reference solutions and compositional PODTPWL simulations is demonstrated for the cases considered. Construction of the PODTPWL model requires preprocessing overhead computations equivalent to about three to four fullorder runs. Runtime speedups using PODTPWL are, however, very significant  about a factor of 500800 for the cases considered. The PODTPWL 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 PODTPWL models. Following an error analysis of the general PODTPWL representation, two projection methods, namely Galerkin projection and PetrovGalerkin projection, are derived by minimizing the constraint reduction error under different norms. These projection methods are assessed computationally for oilwater and compositional systems. For oilwater systems, Galerkin projection combined with a stabilization procedure is generally more accurate than PetrovGalerkin projection, though even with this stabilization Galerkin projection is not guaranteed to be stable at all time steps. For compositional systems, the PODTPWL model with Galerkin projection exhibits poor stability, while PetrovGalerkin provides a consistently stable and robust PODTPWL model. A hybrid procedure for oilwater 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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