Abstract
This paper presents a non-intrusive subdomain POD-TPWL (SD POD-TPWL) for reservoir history matching through integrating domain decomposition (DD), proper orthogonal decomposition (POD), radial basis function (RBF) interpolation, and the trajectory piecewise linearization (TPWL). It is an efficient approach for model reduction and linearization of general non-linear time-dependent dynamical systems without accessing to the legacy source code. In the subdomain POD-TPWL algorithm, firstly, a sequence of snapshots over the entire computational domain is saved and then partitioned into subdomains. From the local sequence of snapshots over each subdomain, a number of local basis vectors is formed using POD, and then the RBF interpolation is used to estimate the derivative matrices for each subdomain. Finally, those derivative matrices are substituted into a POD-TPWL algorithm to form a reduced-order linear model in each subdomain. This reduced-order linear model makes the implementation of the adjoint easy and results in an efficient adjoint-based parameter estimation procedure. Comparisons with the classic finite-difference-based history matching show that our proposed subdomain POD-TPWL approach is obtaining comparable results. The number of full-order model simulations required is roughly 2–3 times the number of uncertain parameters. Using different background parameter realizations, our approach efficiently generates an ensemble of calibrated models without additional full-order model simulations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- POD:
-
Proper orthogonal decomposition
- PCA:
-
Principal component analysis
- RBF:
-
Radial basis function
- TPWL:
-
Trajectory piecewise linearizaton
- DD:
-
Domain decomposition
- FOM:
-
Full-order model
References
Altaf, M.U., Heemink, A.W., Verlaan, M.: Inverse shallow-water flow modeling using model reduction. Int. J. Multiscale Comput. Eng. 7(6), 577–594 (2009)
Amsallem, D., Zahr, M.J., Farhat, C.: Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Methods Eng. 92(10), 891–916 (2012)
Baiges, J., Codina, R., Idelsohn, S.: A domain decomposition strategy for reduced order models. application to the incompressible navier–stokes equations. Comput. Methods Appl. Mech. Eng. 267, 23–42 (2013)
Bian, X., Li, Z., Karniadakis, G.E.: Multi-resolution flow simulations by smoothed particle hydrodynamics via domain decomposition. J. Comput. Phys. 297, 132–155 (2015)
Bishop, C.H., Frolov, S., Allen, D.R., Kuhl, D.D., Hoppel, K.: The local ensemble tangent linear model: an enabler for coupled model 4d-var. Q. J. Roy. Meteorol. Soc. 143(703), 1009–1020 (2017)
Bruyelle, J., Guérillot, D.: Neural networks and their derivatives for history matching and reservoir optimization problems. Comput. Geosci. 18(3-4), 549 (2014)
Cardoso, M., Durlofsky, L., Sarma, P.: Development and application of reduced-order modeling procedures for subsurface flow simulation. Int. J. Numer. Methods Eng. 77(9), 1322–1350 (2009)
Cardoso, M., Durlofsky, L.J.: Linearized reduced-order models for subsurface flow simulation. J. Comput. Phys. 229(3), 681–700 (2010)
Chaturantabut, S.: Temporal localized nonlinear model reduction with a priori error estimate. Appl. Numer. Math. 119, 225–238 (2017)
Chinchapatnam, P.P., Djidjeli, K., Nair, P.B.: Domain decomposition for time-dependent problems using radial based meshless methods. Numer. Methods Partial Differential Equations 23(1), 38–59 (2007)
Courant, R., Hilbert, D.: Methods of mathematical physics: Wiley interscience (1962)
Courtier, P., Thépaut, J.N., Hollingsworth, A.: A strategy for operational implementation of 4d-var, using an incremental approach. Q. J. Roy. Meteorol. Soc. 120(519), 1367–1387 (1994)
Evans, D.J.: Parallel sor iterative methods. Parallel Comput. 1(1), 3–18 (1984)
Frolov, S., Bishop, C.H.: Localized ensemble-based tangent linear models and their use in propagating hybrid error covariance models. Mon. Weather. Rev. 144(4), 1383–1405 (2016)
Fukunaga, K., Koontz, W.L.: Application of the Karhunen-Loeve expansion to feature selection and ordering. IEEE Trans. Comput. 100(4), 311–318 (1970)
Golub, G., Ortega, J.M.: Scientific computing: an introduction with parallel computing. Academic Press, Cambridge (1993)
He, J., Durlofsky, L.J.: Constraint reduction procedures for reduced-order subsurface flow models based on pod-tpwl. Int. J. Numer. Methods Eng. 103(1), 1–30 (2015)
He, J., Durlofsky, L.J., et al.: Reduced-order modeling for compositional simulation by use of trajectory piecewise linearization. SPE J. 19(05), 858–872 (2014)
He, J., Sætrom, J., Durlofsky, L.J.: Enhanced linearized reduced-order models for subsurface flow simulation. J. Comput. Phys. 230(23), 8313–8341 (2011)
He, J., Sarma, P., Durlofsky, L.J.: Reduced-order flow modeling and geological parameterization for ensemble-based data assimilation. Comput. Geosci. 55, 54–69 (2013)
Heijn, T., Markovinovic, R., Jansen, J., et al.: Generation of low-order reservoir models using system-theoretical concepts. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2003)
Jansen, J.: Adjoint-based optimization of multi-phase flow through porous media-a review. Comput. Fluids 46(1), 40–51 (2011)
Jones, D.R.: A taxonomy of global optimization methods based on response surfaces. J. Glob. Optim. 21(4), 345–383 (2001). https://doi.org/10.1023/A:1012771025575
Kaleta, M.P., Hanea, R.G., Heemink, A.W., Jansen, J.D.: Model-reduced gradient-based history matching. Comput. Geosci. 15(1), 135–153 (2011)
Klie, H., et al.: Unlocking fast reservoir predictions via nonintrusive reduced-order models. In: SPE reservoir simulation symposium. Society of Petroleum Engineers (2013)
Lie, K.A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source matlab implementation of consistent discretisations on complex grids. Comput. Geosci. 16(2), 297–322 (2012)
Liu, C., Xiao, Q., Wang, B.: An ensemble-based four-dimensional variational data assimilation scheme. part i: Technical formulation and preliminary test. Mon. Weather. Rev. 136(9), 3363–3373 (2008)
Liu, C., Xiao, Q., Wang, B.: An ensemble-based four-dimensional variational data assimilation scheme. part ii: Observing system simulation experiments with advanced research wrf (arw). Mon. Weather. Rev. 137(5), 1687–1704 (2009)
Lucia, D.J., King, P.I., Beran, P.S.: Reduced order modeling of a two-dimensional flow with moving shocks. Comput. Fluids 32(7), 917–938 (2003)
Markovinović, R., Jansen, J.: Accelerating iterative solution methods using reduced-order models as solution predictors. Int. J. Numer. Methods Eng. 68(5), 525–541 (2006)
Matthews, J.D., Carter, J.N., Stephen, K.D., Zimmerman, R.W., Skorstad, A., Manzocchi, T., Howell, J.A.: Assessing the effect of geological uncertainty on recovery estimates in shallow-marine reservoirs: the application of reservoir engineering to the saigup project. Pet. Geosci. 14(1), 35–44 (2008). https://doi.org/10.1144/1354-079307-791. http://pg.lyellcollection.org/content/14/1/35
Nocedal, J., Wright, S.J.: Numerical optimization: Springer (1999)
Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press, Cambridge (2008)
Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Company, Amsterdam (1977)
Przemieniecki, J.S.: Matrix structural analysis of substructures. AIAA J. 1(1), 138–147 (1963)
Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)
Tan, X., Gildin, E., Florez, H., Trehan, S., Yang, Y., Hoda, N.: Trajectory-based DEIM (TDEIM) model reduction applied to reservoir simulation. Comput. Geosci. https://doi.org/10.1007/s10596-018-9782-0 (2018)
Tarantola, A.: Inverse problem theory: Methods for data fitting and model parameter estimation. DBLP (2005)
Trehan, S., Durlofsky, L.J.: Trajectory piecewise quadratic reduced-order model for subsurface flow, with application to pde- constrained optimization. J. Comput. Phys. 326, 446–473 (2016)
Vermeulen, P., Heemink, A.: Model-reduced variational data assimilation. Mon. Weather. Rev. 134(10), 2888–2899 (2006)
Wu, Y., Wang, H., Zhang, B., Du, K.L.: Using radial basis function networks for function approximation and classification. ISRN Applied Mathematics 2012, 34 (2012)
Xiao, D., Fang, F., Pain, C., Navon, I., Muggeridge, A.: Non-intrusive reduced order modelling of waterflooding in geologically heterogeneous reservoirs. In: ECMOR XV-15th European conference on the mathematics of oil recovery (2016)
Xiao, D., Fang, F., Pain, C., Navon, I., Salinas, P., Muggeridge, A.: Non-intrusive model reduction for a 3d unstructured mesh control volume finite element reservoir model and its application to fluvial channels. Computers & Geosciences (2016)
Acknowledgements
We thank the research funds by China Scholarship Council (CSC) and Delft University of Technology. We are also very grateful to the editor and reviewers for their reviews and insightful comments.
Funding
This study received research funds from China Scholarship Council (CSC) and Delft University of Technology.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Xiao, C., Leeuwenburgh, O., Lin, H.X. et al. Non-intrusive subdomain POD-TPWL for reservoir history matching. Comput Geosci 23, 537–565 (2019). https://doi.org/10.1007/s10596-018-9803-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-018-9803-z