Enhanced linearized reduced-order models for subsurface flow simulation.

*(English)*Zbl 1337.76064Summary: Trajectory piecewise linearization (TPWL) represents a promising approach for constructing reduced-order models. Using TPWL, new solutions are represented in terms of expansions around previously simulated (and saved) solutions. High degrees of efficiency are achieved when the representation is projected into a low-dimensional space using a basis constructed by proper orthogonal decomposition of snapshots generated in a training run. In recent work, a TPWL procedure applicable for two-phase subsurface flow problems was presented. The method was shown to perform well for many cases, such as those with no density differences between phases, though accuracy and robustness were found to degrade in other cases. In this work, these limitations are shown to be related to model accuracy at key locations and model stability. Enhancements addressing both of these issues are introduced. A new TPWL procedure, referred to as local resolution TPWL, enables key grid blocks (such as those containing injection or production wells) to be represented at full resolution; i.e., these blocks are not projected into the low-dimensional space. This leads to high accuracy at selected locations, and will be shown to improve the accuracy of important simulation quantities such as injection and production rates. Next, two techniques for enhancing the stability of the TPWL model are presented. The first approach involves a basis optimization procedure in which the number of columns in the basis matrix is determined to minimize the spectral radius of an appropriately defined amplification matrix. The second procedure incorporates a basis matrix constructed using snapshots from a simulation with equal phase densities. Both approaches are compatible with the local resolution procedure. Results for a series of test cases demonstrate the accuracy and stability provided by the new treatments. Finally, the TPWL model is used as a surrogate in a direct search optimization algorithm, and comparison with results using the full-order model demonstrates the efficacy of the enhanced TPWL procedures for this application.

##### MSC:

76S05 | Flows in porous media; filtration; seepage |

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76T99 | Multiphase and multicomponent flows |

##### Keywords:

trajectory piecewise linearization; TPWL; reservoir simulation; subsurface flow; two-phase flow; optimization; model order reduction; reduced-order model; surrogate model; proper orthogonal decomposition
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\textit{J. He} et al., J. Comput. Phys. 230, No. 23, 8313--8341 (2011; Zbl 1337.76064)

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##### References:

[1] | Vermeulen, P.T.M.; Heemink, A.W.; Stroet, C.B.M.T., Reduced models for linear groundwater flow models using empirical orthogonal functions, Advances in water resources, 27, 57-69, (2004) |

[2] | van Doren, J.F.M.; Markovinović, R.; Jansen, J.D., Reduced-order optimal control of water flooding using proper orthogonal decomposition, Computational geosciences, 10, 137-158, (2006) · Zbl 1161.86304 |

[3] | Heijn, T.; Markovinović, R.; Jansen, J.D., Generation of low-order reservoir models using system-theoretical concepts, SPE journal, 9, 2, 202-218, (2004) |

[4] | Cardoso, M.A.; Durlofsky, L.J.; Sarma, P., Development and application of reduced-order modeling procedures for subsurface flow simulation, International journal for numerical methods in engineering, 77, 9, 1322-1350, (2009) · Zbl 1156.76420 |

[5] | M.J. Rewienski, A Trajectory Piecewise-Linear Approach to Model Order Reduction of Nonlinear Dynamical Systems, Ph.D. thesis, Massachusetts Institute of Technology, 2003. |

[6] | Rewienski, M.; White, J., A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices, IEEE transactions on computer-aided design of integrated circuits and systems, 22, 2, 155-170, (2003) |

[7] | D. Gratton, K. Willcox, Reduced-order, trajectory piecewise-linear models for nonlinear computational fluid dynamics, in: 34th AIAA Fluid Dynamics Conference and Exhibit, Portland, Oregon, USA, 2004. |

[8] | Yang, Y.J.; Shen, K.Y., Nonlinear heat-transfer macromodeling for MEMS thermal devices, Journal of micromechanics and microengineering, 15, 2, 408-418, (2005) |

[9] | Vasilyev, D.; Rewienski, M.; White, J., Macromodel generation for biomems components using a stabilized balanced truncation plus trajectory piecewise-linear approach, IEEE transactions on computer-aided design of integrated circuits and systems, 25, 2, 285-293, (2006) |

[10] | Bond, B.N.; Daniel, L., A piecewise-linear moment-matching approach to parameterized model-order reduction for highly nonlinear systems, IEEE transactions on computer-aided design of integrated circuits and systems, 26, 12, 2116-2129, (2007) |

[11] | Cardoso, M.A.; Durlofsky, L.J., Linearized reduced-order models for subsurface flow simulation, Journal of computational physics, 229, 3, 681-700, (2010) · Zbl 1253.76122 |

[12] | P. Astrid, A. Verhoeven, Application of least squares MPE technique in the reduced order modeling of electrical circuits, in: 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, 2006. |

[13] | B.N. Bond, L. Daniel, Stabilizing schemes for piecewise-linear reduced order models via projection and weighting functions, in: Proceedings of the IEEE/ACM International Conference on Computer-Aided Design, San Jose, California, 2007, pp. 860-867. |

[14] | B.N. Bond, L. Daniel, Guaranteed stable projection-based model reduction for indefinite and unstable linear systems, in: Proceedings of the IEEE/ACM International Conference on Computer-Aided Design, San Jose, California, 2008, pp. 728-735. |

[15] | Bond, B.N.; Daniel, L., Stable reduced models for nonlinear descriptor systems through piecewise-linear approximation and projection, IEEE transactions on computer-aided design of integrated circuits and systems, 28, 10, 1467-1480, (2009) |

[16] | Bui-Thanh, T.; Damodaran, M.; Willcox, K., Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition, AIAA journal, 42, 8, 1505-1516, (2004) |

[17] | Bui-Thanh, T.; Willcox, K.; Ghattas, O.; van Bloemen Waanders, B., Goal-oriented, model-constrained optimization for reduction of large-scale systems, Journal of computational physics, 224, 2, 880-896, (2007) · Zbl 1123.65081 |

[18] | Gerritsen, M.; Lambers, J., Integration of local-global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations, Computational geosciences, 12, 193-218, (2008) · Zbl 1159.76362 |

[19] | Efendiev, Y.R.; Durlofsky, L.J., A generalized convection-diffusion model for subgrid transport in porous media, Multiscale modeling and simulation, 1, 504-526, (2003) · Zbl 1191.76098 |

[20] | Chen, Y.; Li, Y., Local-global two-phase upscaling of flow and transport in heterogeneous formations, Multiscale modeling and simulation, 8, 125-153, (2009) · Zbl 1404.76252 |

[21] | Hou, T.Y.; Wu, X.H., A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of computational physics, 134, 169-189, (1997) · Zbl 0880.73065 |

[22] | Jenny, P.; Lee, S.H.; Tchelepi, H.A., Adaptive multiscale finite-volume method for multiphase flow and transport in porous media, Multiscale modeling and simulation, 3, 50-64, (2004) · Zbl 1160.76372 |

[23] | Efendiev, Y.; Ginting, V.; Hou, T.; Ewing, R., Accurate multiscale finite element methods for two-phase flow simulations, Journal of computational physics, 220, 1, 155-174, (2006) · Zbl 1158.76349 |

[24] | Tartakovsky, A.M.; Tartakovsky, D.M.; Scheibe, T.D.; Meakin, P., Hybrid simulations of reaction-diffusion systems in porous media, SIAM journal on scientific computing, 30, 2799-2816, (2007) · Zbl 1175.76141 |

[25] | I. Battiato, D.M. Tartakovsky, A.M. Tartakovsky, T.D. Scheibe, Hybrid models of reactive transport in porous and fractured media, Advances in Water Resources, doi:10.1016/j.advwatres.2011.01.012, in press. · Zbl 1327.76130 |

[26] | Aziz, K.; Settari, A., Fundamentals of reservoir simulation, (1986), Elsevier Applied Science Publishers |

[27] | Adamjan, V.; Arov, D.; Krein, M., Analytic properties of Schmidt pairs for a Hankel operator and the generalized schur – takagi problem, Math. USSR sbornik, 15, 31-73, (1971) · Zbl 0248.47019 |

[28] | M. Bettayeb, L. Silverman, M. Safonov, Optimal approximation of continuous-time system, in: Proceedings of the IEEE Conference on Decision and Control, vol. 1, Albuquerque, NM, 1980. |

[29] | Moore, B., Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE transactions on automatic control, 26, 17-31, (1981) · Zbl 0464.93022 |

[30] | Feldmann, P.; Freund, R.W., Efficient linear circuit analysis by Padé approximation via the Lanczos process, IEEE transactions on computer-aided design of integrated circuits and systems, 14, 639-649, (1995) |

[31] | Lumley, J.L., Atmospheric turbulence and radio wave propagation, Journal of computational chemistry, 23, 13, 1236-1243, (1967) |

[32] | Pearson, K., On lines and planes of closest fit to points in space, Philosophical magazine, 2, 559-572, (1901) · JFM 32.0246.07 |

[33] | Cardoso, M.A.; Durlofsky, L.J., Use of reduced-order modeling procedures for production optimization, SPE journal, 15, 426-435, (2010) · Zbl 1253.76122 |

[34] | Berkooz, G.; Titi, E.Z., Galerkin projections and the proper orthogonal decomposition for equivariant equations, Physics letter A, 174, 1-2, 94-102, (1993) |

[35] | S.A. Castro, A Probabilistic Approach to Jointly Integrate 3D/4D Seismic Production Data and Geological Information for Building Reservoir Models, Ph.D. thesis, Stanford University, 2007. |

[36] | H. Cao, Development of Techniques for General Purpose Simulators, Ph.D. thesis, Stanford University, 2002. |

[37] | Y. Jiang, A Flexible Computational Framework for Efficient Integrated Simulation of Advanced Wells and Unstructured Reservoir Models, Ph.D. thesis, Stanford University, 2007. |

[38] | Mardia, K.V.; Kent, J.T.; Bibby, J.M., Multivariate analysis, (1979), Academic Press London · Zbl 0432.62029 |

[39] | Hastie, T.; Tibshirani, R.; Freidman, J., The elements of statistical learning; data mining, inference, and prediction, (2009), Springer New York |

[40] | Gelfand, I.M., Zur theorie der charaktere der abelschen topologischen gruppen, Recueil mathematique (matematicheskiisbornik) N.S., 9, 1, 49-50, (1941) · Zbl 0024.32301 |

[41] | Vempala, S.S., The random projection method, (2004), American Mathematical Society · Zbl 1058.68063 |

[42] | Christie, M.A.; Blunt, M.J., Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE reservoir evaluation & engineering, 4, 4, 308-317, (2001) |

[43] | Kolda, T.G.; Lewis, R.M.; Torczon, V., Optimization by direct search: new perspectives on some classical and modern methods, SIAM review, 45, 3, 385-482, (2003) · Zbl 1059.90146 |

[44] | Echeverría Ciaurri, D.; Isebor, O.J.; Durlofsky, L.J., Application of derivative-free methodologies to generally constrained oil production optimisation problems, International journal of mathematical modelling and numerical optimisation, 2, 134-161, (2011) |

[45] | J. He, Enhanced Linearized Reduced-order Models for Subsurface Flow Simulation, Master’s thesis, Stanford University, 2010. |

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