Development of Reduced-order Oil Reservoir Models using Localized DEIM

Abstract

For 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.

Efficient Adaptive Schemes to Use Reduced-Order Models for Fast Simulation of Water-Oil Displacement

“…In the next step, the models based on TPWL were used to perform the different simulation scenarios. Yoon et al (2014) applied discrete empirical interpolation method (DEIM) and localized discrete empirical interpolation method (LDEIM) to reach further reduction in calculations load and performance time of the reduced simulation. In DEIM method, only the blocks with the highest importance are used for the ROM.…”

Reduced-Order Modeling of Subsurface Multi-phase Flow Models Using Deep Residual Recurrent Neural Networks

“…In that work, machine learning technique was used to construct a number of POD-DEIM local reduced-order models and then a specific local reduced-order model was selected with respect to the current state of the dynamical system during the gradient-based optimization task. Similarly, Yoon et al (2014) used multiple local DEIM approximations in POD reduced model framework to reduce the computational costs of high-fidelity reservoir simulations.…”

Non-intrusive model reduction for a 3D unstructured mesh control volume finite element reservoir model and its application to fluvial channels

“…The hypersurfaces are established by POD and RBF interpolation, Smolyak sparse grid and Taylor series expansion method (Xiao et al, 2015a(Xiao et al, , 2015b. In addition, the hypersurface method based on NIROM has been successfully applied to fluid-structure interaction problems (Xiao et al, 2016(Xiao et al, , 2017a and free surface flow problems (Xiao et al, 2017b) Recently, reduced order methods (e.g., POD, POD/DEIM, trajectory piecewise linearisation and bilinear approximation techniques) have been applied to reservoir modelling (Heijn et al, 2004;Cardoso, 2009;Cardoso et al, 2009;Chaturantabut and Sorensen, 2011;Yang et al, 2015;Yoon et al, 2014;Ghasemi et al, 2014;Klie et al, 2013). Heijn et al (2004), Cardoso (2009) and Cardoso et al (2009) first developed POD ROMs for reservoir simulation.…”

“…Heijn et al (2004), Cardoso (2009) and Cardoso et al (2009) first developed POD ROMs for reservoir simulation. Chaturantabut and Sorensen (2011), Yang et al (2015) and Yoon et al (2014) further introduced DEIM into model reduction for nonlinear flows (Chaturantabut and Sorensen, 2011). Again, these reduced order methods are intrusive and equations/codes dependent.…”