Development of Reduced-order Oil Reservoir Models using Localized DEIM

Abstract

AbstractFor applications of numerical reservoir simulators such as production optimization or uncertainty assessment, hundreds or thousands of reservoir simulation runs must be performed. Such huge computational cost is a major problem in petroleum engineering, and reduced-order model (ROM) based on proper orthogonal decomposition (POD) has been intensively studied in recent decade to overcome it. While POD-based methods have been shown to be efficient compared to full order model (FOM), it is not as efficient when it is applied to a typical large dimensional nonlinear reservoir system. One of the reasons for the inefficiency is that in a typical nonlinear reservoir system ROM based only on POD is still dependent on the dimension of FOM. This is due to the fact that to compute the reduced nonlinear term in mass balance equation of a reservoir system, one must first reconstruct the full-order state solution such as pressure and saturation, evaluate the full-order nonlinear term before projecting it onto a reduced subspace. Therefore, we developed a reduced-order reservoir model based on a discrete empirical interpolation method (DEIM) to approximate nonlinear phase potential terms so that the repeated online evaluations of the ROM in Newton iteration are independent of full-order dimension. The independence comes from the fact that DEIM just needs to evaluate the nonlinear term only at interpolation indices that represent grid blocks that are important in terms of preserving the continuity properties of the mass balance equation. A case study was carried out to investigate the performance of DEIM compared POD. Although the testing schedule of well controls is far apart from the training schedule of well controls, close matches are obtained. Thus, the ROM using DEIM is expected to enable the practical application of reservoir simulator, such as production optimization in which many simulation runs must be performed, in a reasonable time frame by significantly relieving the required numerical effort.