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.

Reduced-Order Modeling of Coupled Flow and Quasistatic Geomechanics

SPE Journal ◽  
2019 ◽  
Vol 25 (01) ◽  
pp. 326-346 ◽  
Author(s):  
Zhaoyang Larry Jin ◽  
Timur Garipov ◽  
Oleg Volkov ◽  
Louis J. Durlofsky
Keyword(s):  
Automatic Differentiation ◽  
Dimensional Subspace ◽  
General Purpose ◽  
Reduced Order Modeling ◽  
Global Maximum ◽  
Test Case ◽  
Coupled Flow ◽  
Piecewise Linearization ◽  
Reduced Order ◽  
Low Dimensional

Summary A reduced-order-modeling (ROM) framework is developed and applied to simulate coupled flow/quasistatic-geomechanics problems. The reduced-order model is constructed using POD-TPWL, in which proper orthogonal decomposition (POD), which enables representation of the solution unknowns in a low-dimensional subspace, is combined with trajectory-piecewise linearization (TPWL), where solutions with new sets of well controls are represented by means of linearization around previously simulated (training) solutions. The overdetermined system of equations is projected into the low-dimensional subspace using a least-squares Petrov-Galerkin (LSPG) procedure, which has been shown to maintain numerical stability in POD-TPWL models. The states and derivative matrices required by POD-TPWL, generated by an extended version of the Stanford University Automatic Differentiation General Purpose Research Simulator (AD-GPRS), are provided in an offline (preprocessing or training) step. Offline computational requirements correspond to the equivalent of five to eight full-order simulations, depending on the number of training runs used. Run-time (online) speedups of O(100) or more are typically achieved for new POD-TPWL test-case simulations. The POD-TPWL model is tested extensively for a 2D coupled problem involving oil/water flow and geomechanics. It is shown that POD-TPWL provides predictions of reasonable accuracy, relative to full-order simulations, for well-rate quantities, global pressure and saturation fields, global maximum- and minimum-principal-stress fields, and the Mohr-Coulomb rock-failure criterion, for the cases considered. A systematic study of POD-TPWL error is conducted using various training procedures for different levels of perturbation between test and training cases. The use of randomness in the well-bottomhole-pressure (BHP) profiles used in training is shown to be beneficial in terms of POD-TPWL solution accuracy. The procedure is also successfully applied to a prototype 3D example case.

scholarly journals GENETIC ALGORITHM-BASED CALIBRATION OF REDUCED ORDER GALERKIN MODELS

2011 ◽  
Vol 16 (1) ◽  
pp. 233-247 ◽  
Author(s):  
Witold Stankiewicz ◽  
Robert Roszaka ◽  
Marek Morzyńskia
Keyword(s):  
Genetic Algorithm ◽  
Bluff Body ◽  
Calibration Procedure ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Data Set ◽  
Reduced Order ◽  
Low Dimensional ◽  
Fluid Behaviour ◽  
Proper Orthogonal

Low-dimensional models, allowing quick prediction of fluid behaviour, are key enablers of closed-loop flow control. Reduction of the model's dimension and inconsistency of high-fidelity data set and the reduced-order formulation lead to the decrease of accuracy. The quality of Reduced-Order Models might be improved by a calibration procedure. It leads to global optimization problem which consist in minimizing objective function like the prediction error of the model. In this paper, Reduced-Order Models of an incompressible flow around a bluff body are constructed, basing on Galerkin Projection of governing equations onto a space spanned by the most dominant eigenmodes of the Proper Orthogonal Decomposition (POD). Calibration of such models is done by adding to Galerkin System some linear and quadratic terms, which coefficients are estimated using Genetic Algorithm.

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.

scholarly journals Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Imran Akhtar ◽  
Jeff Borggaard ◽  
Alexander Hay
Keyword(s):  
Finite Difference ◽  
Standard Method ◽  
Elliptic Cylinder ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Shape Sensitivity ◽  
Reduced Order ◽  
Flow Models ◽  
Low Dimensional ◽  
Parameter Values

Reduced-order models have a number of practical engineering applications for unsteady flows that require either low-dimensional approximations for analysis and control or repeated simulation over a range of parameter values. The standard method for building reduced-order models uses the proper orthogonal decomposition (POD) and Galerkin projection. However, this standard method may be inaccurate when used “off-design” (at parameter values not used to generate the POD). This phenomena is exaggerated when parameter values describe the shape of the flow domain since slight changes in shape can have a significant influence on the flow field. In this paper, we investigate the use of POD sensitivity vectors to improve the accuracy and dynamical system properties of the reduced-order models to problems with shape parameters. To carry out this study, we consider flows past an elliptic cylinder with varying thickness ratios. Shape sensitivities (derivatives of flow variables with respect to thickness ratio) computed by finite-difference approximations are used to compute the POD sensitivity vectors. Numerical studies test the accuracy of the new bases to represent flow solutions over a range of parameter values.

Error Estimates for Reduced Order POD Models of Navier-Stokes Equations

2008 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Error Estimates ◽  
Incompressible Flows ◽  
Orthogonal Decomposition ◽  
Galerkin Projection ◽  
Dimensional Model ◽  
Reduced Order ◽  
Infinite Dimensional ◽  
Low Dimensional ◽  
Proper Orthogonal

Reduced order modeling for the purpose of constructing a low dimensional model from high dimensional or infinite dimensional model has important applications in science and engineering such as fast model evaluations and optimization/control. A popular method for constructing reduced-order model is based on finding a suitable low dimensional basis by proper orthogonal decomposition (POD) and forming a model by Galerkin projection of the infinite dimensional model onto the basis. In this paper, we will discuss error estimates for Galerkin proper orthogonal decomposition method for an unsteady nonlinear coupled partial differential equations arising in viscous incompressible flows. A specific finite element in space and finite difference in time discretization scheme will be discussed.

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.

Data-Driven Reduced Order Modeling of Flows Around Two-Dimensional Bluff Bodies of Various Shapes

2019 ◽  
Author(s):  
Kazuto Hasegawa ◽  
Kai Fukami ◽  
Takaaki Murata ◽  
Koji Fukagata
Keyword(s):  
Number Fields ◽  
Flow Fields ◽  
Reduced Order Modeling ◽  
Reduced Order Model ◽  
Data Driven ◽  
Bluff Bodies ◽  
Two Dimensional ◽  
Order Model ◽  
Reduced Order ◽  
Low Dimensional

Abstract We propose a reduced order model for predicting unsteady flows using a data-driven method. As preliminary tests, we use two-dimensional unsteady flow around bluff bodies with different shapes as the training datasets obtained by direct numerical simulation (DNS). Our machine-learned architecture consists of two parts: Convolutional Neural Network-based AutoEncoder (CNN-AE) and Long Short Term Memory (LSTM), respectively. First, CNN-AE is used to map into a low-dimensional space from the flow field data. Then, LSTM is employed to predict the temporal evolution of the low-dimensional data generated by CNN-AE. Proposed machine-learned reduced order model is applied to two-dimensional circular cylinder flows at various Reynolds numbers and flows around bluff bodies of various shapes. The flow fields reconstructed by the machine-learned architecture show reasonable agreement with the reference DNS data. Furthermore, it can be seen that our machine-learned reduced order model can successfully map the high-dimensional flow data into low-dimensional field and predict the flow fields against unknown Reynolds number fields and shapes of bluff body. As concluding remarks, we discuss the extension study of machine-learned reduced order modeling for various applications in experimental and computational fluid dynamics.

Sign in / Sign up

Export Citation Format

Share Document