Use of Reduced-Order Modeling Procedures for Production Optimization

Abstract

The determination of optimal well settings is very demanding computationally because the simulation model must be run many times during the course of the optimization. For this reason, reduced-order modeling procedures, which are a family of techniques that enable highly efficient simulations, may be very useful for optimization problems. In this paper, we describe a recently developed reduced-order modeling (ROM) technique that has been used in other application areas, the trajectory piecewise linearization (TPWL) procedure, and incorporate it in production-optimization computations. The TPWL methodology represents solutions encountered during the optimization runs in terms of Taylor-series expansions around previously simulated states. This requires a small number of preprocessing (training) simulations using the full (high-fidelity) model, during which pressure and saturation states and Jacobian matrices are saved. These states and matrices are then projected into a low-dimensional space using proper orthogonal decomposition (POD). Simulations in this reduced space can be performed very efficiently; in this work, we observe runtime speedups of a factor of 450. Overall speedups are, however, less because of the preprocessing overhead.We assess the TPWL representation for simulations of waterflood in a heterogeneous 3D model containing more than 20,000 gridblocks and six wells. The high degree of accuracy of the TPWL model is first demonstrated for several testing simulations in which producer-and injector-well settings differ from those used in the training runs. The TPWL representations are then used in optimizations involving the determination of optimal bottomhole pressures (BHPs) for a reservoir model with four production wells and two injection wells. A gradient-based algorithm is applied for the optimizations. In the first case, the BHPs of the producers and injectors are optimized at six different times (36 control variables) and in the second case at 15 different times (90 control variables). Results for optimized net present value (NPV) using TPWL are shown to be in consistently close agreement with those computed using high-fidelity simulations. Most significantly, when the optimal well settings obtained using the TPWL procedure are applied in high-fidelity models, the resulting NPVs are within approximately 0.5% of the values determined using the high-fidelity simulations. Our overall conclusion is that the TPWL representation may be quite useful in production-optimization problems.

Complexity Reduction of Multi-Phase Flows in Heterogeneous Porous Media

“…We observe error values reaching a maximum value of 10%. One can reduce this error, as discussed in Cardoso and Durlofsky (2010), by excluding the gridblocks containing wells from reduced subspace and solve the pressure and saturation equations in original space. This shows the robustness of the projection-based approach, combining multiscale FEM, POD, and DEIM, to derive a reliable reduced-order model while moderately varying forcing inputs (injection rates).…”

“…In the case of nonintrusive methods, data-driven model reduction has been the choice in material-balance-type modeling such as the capacitance/resistance models (Yousef et al 2006) and flownetwork models (Lerlertpakdee et al 2014), and in the use of artificial intelligence, such as neural networks and fuzzy-logic techniques (Mohaghegh et al 2012). For the intrusive schemes, reduced-order modeling by projection has been used in the systems/controls-like framework, such as the balanced truncation (Heijn et al 2004), proper orthogonal decompositions (PODs) (Volkwein and Hinze 2005), and the trajectory-piecewise linear (TPWL) techniques (Cardoso and Durlofsky 2010), bilinear Krylov subspace methods , and quadratic bilinear model order reduction .…”

Complexity Reduction of Multiphase Flows in Heterogeneous Porous Media

“…This method was coupled with the TPWL model reduction in Cardoso and Durlofsky (2010). In this approach, one performs several linearizations of the nonlinear systems about several known states so that one can use POD or other linear model-reduction techniques to derive good approximations.…”

Fast Multiscale Reservoir Simulations using POD-DEIM Model Reduction

“…In recent years, many reduced-order modeling techniques have been adapted to speed up reservoir simulation and production optimization. Reduced-order modeling by projection has been used in the systems/controlslike framework, such as the balanced truncation (Heijn et al, 2004), proper orthogonal decompositions (POD) (Doren et al, 2004), the trajectory-piecewise linear (TPWL) techniques Cardoso and Durlofsky (2010), empirical interpolation methods Efendiev et al (2013); Ghommem et al (2013), bilinear Krylov subspace methods ) and quadratic bilinear model order reduction . In this work we focus on the POD based model order reduction technique, which performs a Galerkin projection on the space identified by important modes.…”

“…There are different techniques to alleviate this problem. One approach is to linearize these nonlinear functions around several known states and use these piecewise linear solutions, see (Cardoso and Durlofsky, 2010). Here, we use DEIM, where one construct another subspace for reducing the nonlinear function evaluations (Chaturantabut and Sorensen, 2010).…”