Non-intrusive subdomain POD-TPWL for reservoir history matching

  • Cong XiaoEmail author
  • Olwijn Leeuwenburgh
  • Hai Xiang Lin
  • Arnold Heemink
Open Access
Original Paper


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.


Data assimilation Reduced-order modeling Model linearization Domain decomposition 



Proper orthogonal decomposition


Principal component analysis


Radial basis function


Trajectory piecewise linearizaton


Domain decomposition


Full-order model



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 information

This study received research funds from China Scholarship Council (CSC) and Delft University of Technology.

Supplementary material

10596_2018_9803_MOESM1_ESM.pdf (694 kb)
(PDF 693 KB)


  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Bruyelle, J., Guérillot, D.: Neural networks and their derivatives for history matching and reservoir optimization problems. Comput. Geosci. 18(3-4), 549 (2014)CrossRefGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    Cardoso, M., Durlofsky, L.J.: Linearized reduced-order models for subsurface flow simulation. J. Comput. Phys. 229(3), 681–700 (2010)CrossRefGoogle Scholar
  9. 9.
    Chaturantabut, S.: Temporal localized nonlinear model reduction with a priori error estimate. Appl. Numer. Math. 119, 225–238 (2017)CrossRefGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Courant, R., Hilbert, D.: Methods of mathematical physics: Wiley interscience (1962)Google Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Evans, D.J.: Parallel sor iterative methods. Parallel Comput. 1(1), 3–18 (1984)CrossRefGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    Fukunaga, K., Koontz, W.L.: Application of the Karhunen-Loeve expansion to feature selection and ordering. IEEE Trans. Comput. 100(4), 311–318 (1970)CrossRefGoogle Scholar
  16. 16.
    Golub, G., Ortega, J.M.: Scientific computing: an introduction with parallel computing. Academic Press, Cambridge (1993)Google Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    He, J., Sætrom, J., Durlofsky, L.J.: Enhanced linearized reduced-order models for subsurface flow simulation. J. Comput. Phys. 230(23), 8313–8341 (2011)CrossRefGoogle Scholar
  20. 20.
    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)CrossRefGoogle Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    Jansen, J.: Adjoint-based optimization of multi-phase flow through porous media-a review. Comput. Fluids 46(1), 40–51 (2011)CrossRefGoogle Scholar
  23. 23.
    Jones, D.R.: A taxonomy of global optimization methods based on response surfaces. J. Glob. Optim. 21(4), 345–383 (2001). CrossRefGoogle Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    Klie, H., et al.: Unlocking fast reservoir predictions via nonintrusive reduced-order models. In: SPE reservoir simulation symposium. Society of Petroleum Engineers (2013)Google Scholar
  26. 26.
    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)CrossRefGoogle Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    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)CrossRefGoogle Scholar
  29. 29.
    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)CrossRefGoogle Scholar
  30. 30.
    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)CrossRefGoogle Scholar
  31. 31.
    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). CrossRefGoogle Scholar
  32. 32.
    Nocedal, J., Wright, S.J.: Numerical optimization: Springer (1999)Google Scholar
  33. 33.
    Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  34. 34.
    Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Company, Amsterdam (1977)Google Scholar
  35. 35.
    Przemieniecki, J.S.: Matrix structural analysis of substructures. AIAA J. 1(1), 138–147 (1963)CrossRefGoogle Scholar
  36. 36.
    Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)Google Scholar
  37. 37.
    Tan, X., Gildin, E., Florez, H., Trehan, S., Yang, Y., Hoda, N.: Trajectory-based DEIM (TDEIM) model reduction applied to reservoir simulation. Comput. Geosci. (2018)
  38. 38.
    Tarantola, A.: Inverse problem theory: Methods for data fitting and model parameter estimation. DBLP (2005)Google Scholar
  39. 39.
    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)CrossRefGoogle Scholar
  40. 40.
    Vermeulen, P., Heemink, A.: Model-reduced variational data assimilation. Mon. Weather. Rev. 134(10), 2888–2899 (2006)CrossRefGoogle Scholar
  41. 41.
    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)CrossRefGoogle Scholar
  42. 42.
    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)Google Scholar
  43. 43.
    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)Google Scholar

Copyright information

© The Author(s) 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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.

Authors and Affiliations

  • Cong Xiao
    • 1
    Email author
  • Olwijn Leeuwenburgh
    • 2
    • 3
  • Hai Xiang Lin
    • 1
  • Arnold Heemink
    • 1
  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  2. 2.Civil Engineering and GeosciencesDelft University of TechnologyDelftThe Netherlands
  3. 3.TNOUtrechtThe Netherlands

Personalised recommendations