Hyper-Reduced-Order Models for Subsurface Flow Simulation

SPE Journal ◽  
2016 ◽  
Vol 21 (06) ◽  
pp. 2128-2140 ◽  
Author(s):  
Seonkyoo Yoon ◽  
Zeid M. Alghareeb ◽  
John R. Williams
Keyword(s):  
Subsurface Flow ◽  
Flow Simulation ◽  
Reduced Order Modeling ◽  
Production Optimization ◽  
Galerkin Projection ◽  
Flow Modeling ◽  
State Reduction ◽  
Constraint Reduction ◽  
Reduced Order ◽  
Modeling Procedure

Summary Subsurface flow modeling is an indispensable task for reservoir management, but the associated computational cost is burdensome because of model complexity and the fact that many simulation runs are required for its applications such as production optimization, uncertainty quantification, and history matching. To relieve the computational burden in reservoir flow modeling, a reduced-order modeling procedure based on hyper-reduction is presented. The procedure consists of three components: state reduction, constraint reduction, and nonlinearity treatment. State reduction based on proper orthogonal decomposition (POD) is considered, and the impact of state reduction, with different strategies for collecting snapshots, on accuracy and predictability is investigated. Petrov-Galerkin projection is used for constraint reduction, and a hyper-reduction that couples the Petrov-Galerkin projection and a “gappy” reconstruction is applied for the nonlinearity treatment. The hyper-reduction method is a Gauss-Newton framework with approximated tensors (GNAT), and the main contribution of this study is the presentation of a procedure for applying the method to subsurface flow simulation. A fully implicit oil/water two-phase subsurface flow model in 3D space is considered, and the application of the proposed hyper-reduced-order modeling procedure achieves a runtime speedup of more than 300 relative to the full-order method, which cannot be achieved when only constraint reduction is adopted.

scholarly journals Evolve Then Filter Regularization for Stochastic Reduced Order Modeling

2018 ◽  
Author(s):  
Xuping Xie ◽  
Feng Bao ◽  
Clayton G. Webster
Keyword(s):  
Stochastic Systems ◽  
Spatial Filter ◽  
Reduced Order Modeling ◽  
Reduced Order Model ◽  
Galerkin Projection ◽  
Order Model ◽  
Reduced Order ◽  
Stochastic Burgers Equation ◽  
Numerical Stabilization ◽  
Convection Dominated

In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems. The standard Galerkin projection reduced order model (G-ROM) yield numerical oscillations in a convection-dominated regime. The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model (ROM). Our numerical results based on a stochastic Burgers equation with linear multiplicative noise. It shows that the EF-ROM is significantly better results than G-ROM.

Reduced-Order Modeling for Compositional Simulation by Use of Trajectory Piecewise Linearization

SPE Journal ◽  
2014 ◽  
Vol 19 (05) ◽  
pp. 858-872 ◽  
Author(s):  
Jincong He ◽  
Louis J. Durlofsky
Keyword(s):  
Oil And Gas ◽  
Dimensional Subspace ◽  
Reduced Order Modeling ◽  
Galerkin Projection ◽  
Compositional Simulation ◽  
Piecewise Linearization ◽  
Reduced Order ◽  
Computational Optimization ◽  
Trajectory Piecewise Linearization ◽  
Low Dimensional

Summary Compositional simulation can be very demanding computationally as a result of the potentially large number of system unknowns and the intrinsic nonlinearity of typical problems. In this work, we develop a reduced-order modeling procedure for compositional simulation. The technique combines trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) to provide a highly efficient surrogate model. The compositional POD-TPWL method expresses new solutions in terms of linearizations around states generated (and saved) during previously simulated “training” runs. High-dimensional states are projected (optimally) into a low-dimensional subspace by use of POD. The compositional POD-TPWL model is based on a molar formulation that uses pressure and overall component mole fractions as the primary unknowns. Several new POD-TPWL treatments, including the use of a Petrov-Galerkin projection to reduce the number of equations (rather than the Galerkin projection, which was applied previously), and a new procedure for determining which saved state to use for linearization are incorporated into the method. Results are presented for heterogeneous 3D reservoir models containing oil and gas phases with up to six hydrocarbon components. Reasonably close agreement between full-order reference solutions and compositional POD-TPWL simulations is demonstrated for the cases considered. Construction of the POD-TPWL model requires preprocessing overhead computations equivalent to approximately three or four full-order runs. Runtime speedups by use of POD-TPWL are, however, very significant—up to a factor of 800 for the cases considered. The POD-TPWL model is thus well suited for use in computational optimization, in which many simulations must be performed, and we present an example demonstrating its application for such a problem.

Use of Reduced-Order Modeling Procedures for Production Optimization

SPE Journal ◽  
2009 ◽  
Vol 15 (02) ◽  
pp. 426-435 ◽  
Author(s):  
M.A.. A. cardoso ◽  
L.J.. J. Durlofsky
Keyword(s):  
Optimization Problems ◽  
Dimensional Space ◽  
Reduced Order Modeling ◽  
Production Optimization ◽  
High Fidelity ◽  
Control Variables ◽  
Reduced Order ◽  
First Case ◽  
High Fidelity Simulations

Summary 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.

Proper Orthogonal Decomposition Based Reduced Order Modeling of the Very High Temperature Reactor Lower Plenum Hydrodynamics

2014 ◽  
Author(s):  
Gregory A. Banyay ◽  
Mohammad Ahmadpoor ◽  
John C. Brigham
Keyword(s):  
Coherent Structures ◽  
Reduced Order Modeling ◽  
Orthogonal Decomposition ◽  
Galerkin Projection ◽  
Optimal Basis ◽  
Reduced Order ◽  
Very High Temperature Reactor ◽  
High Temperature Reactor ◽  
Proper Orthogonal ◽  
Very High

The feasibility of reduced order modeling for turbulent flows using Proper Orthogonal Decomposition (POD) based Surrogate modeling and Galerkin Projection is demonstrated for use in the hydrodynamic modeling of the Very High Temperature Reactor (VHTR) lower plenum. The lower plenum of the Helium-cooled VHTR consists of vertical cylinder arrays subjected to turbulent jetting and cross-flow. Unsteady Reynolds-Averaged Navier-Stokes (RANS) Computational Fluid Dynamics (CFD) simulations are used to acquire an ensemble of possible solution fields for flow around a circular cylinder in an open domain. Numerical results are validated to prior published literature. From the resultant data ensemble are extracted the coherent structures to create an optimal basis. POD is used to extract the coherent structures as this technique has been demonstrated to provide a basis of a chosen dimension such that the average L2-error is minimized for the best approximation of the basis to the data ensemble. The resultant optimal basis is used to construct accurate reduced order models. The computational effectiveness and insights revealed by this reduced order modeling approach are discussed for both the Surrogate modeling approach and Galerkin Projection.

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