Computational Geosciences

, Volume 22, Issue 1, pp 309–327 | Cite as

Truncated conjugate gradient and improved LBFGS and TSVD for history matching

  • Flávio Dickstein
  • Paulo Goldfeld
  • Gustavo T. Pfeiffer
  • Renan V. Pinto
Original Paper


We propose a new algorithm for solving the history matching problem in reservoir simulation, truncated conjugate gradient (TCG), which involves a model reparameterization based on the factorization of the prior covariance matrix, C M = L L T . We also revisit the LBFGS algorithm, framing it into the same reparametrization, introducing M-LBFGS. We present numerical evidence that this reparameterization has an important regularizing impact on the solution process. We show how TCG and M-LBFGS, as well as TSVD, can be implemented without the need of actually computing the factor L. Our numerical experiments, including the PUNQ-S3 and the Brugge cases, indicate that TCG and M-LBFGS are effective schemes for history matching.


History matching Truncated conjugate gradient TSVD LBFGS PUNQ-S3 Brugge 


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The authors are grateful to Petrobras (Brazil) for the support provided in the development of this work. F. Dickstein also acknowledges the support of CNPq (Brazil). Finally, the authors thank ESSS for providing their software Kraken for pre- and post-processing of simulation data.


  1. 1.
    Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., Vallès, B.: The ensemble Kalman filter in reservoir engineering–a review. SPE J. 14(3), 393–412 (2009). CrossRefGoogle Scholar
  2. 2.
    Averick, B.M., Carter, R.G., Moré, J.J.: MINPACK-2 Project. Argonne National Laboratory and University of Minnesota (1993).
  3. 3.
    Bell, B.M., Cathey, F.W.: The iterated Kalman filter upyear as a Gauss-Newton method. IEEE Trans. Automat. Contr. 38(2), 294–297 (1993)CrossRefGoogle Scholar
  4. 4.
    Chen, Y., Oliver, D.S.: Levenberg–Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. Comput. Geosci. 17, 689–703 (2013). CrossRefGoogle Scholar
  5. 5.
    Dickstein, F., Goldfeld, P., Pfeiffer, G., Amorim, E., dos Santos, R., Gomez, S.: A study of the impact of 4D-seismic data on TSVD-based schemes for history matching. In: Latin American and Caribbean Petroleum Engineering Conference 2010, pp 643–653 (2010).
  6. 6.
    van den Doel, K., Ascher, U.: Adaptive and stochastic algorithms for EIT and DC resistivity problems with piecewise constant solutions and many measurements. SIAM J. Sci. Comput. 34(1), A185–A205 (2012). CrossRefGoogle Scholar
  7. 7.
    Dong, Y., Oliver, D.S.: Quantitative use of 4D seismic data for reservoir description. SPE J. 10(1), 91–99 (2005). CrossRefGoogle Scholar
  8. 8.
    Emerick, A.A., Moraes, R., Rodrigues, J.: History matching 4D seismic data with efficient gradient based methods. In: EUROPEC/EAGE Conference and Exhibition, 11–14 June, London (2007).
  9. 9.
    Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013). CrossRefGoogle Scholar
  10. 10.
    Emerick, A.A., Reynolds, A.C.: Investigation of the sampling performance of ensemble-based methods with a simple reservoir model. Comput. Geosci. 17(2), 325–350 (2013). CrossRefGoogle Scholar
  11. 11.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  12. 12.
    Floris, F.J.T., Bush, M.D., Cuypers, M., Roggero, F., Syversveen, A.R.: Methods for quantifying the uncertainty of production forecasts: a comparative study. Pet. Geosci. 7(S), S87–S96 (2001). CrossRefGoogle Scholar
  13. 13.
    Fossum, K., Mannseth, T., Oliver, D.S., Skaug, H.J.: Numerical comparison of ensemble Kalman filter and randomized maximum likelihood. In: ECMOR XIII - 13th European Conference on the Mathematics of oil Recovery (2012).
  14. 14.
    Gao, G., Reynolds, A.C.: An improved implementation of the LBFGS algorithm for automatic history matching. SPE J. 11(01), 5–17 (2006). CrossRefGoogle Scholar
  15. 15.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)Google Scholar
  16. 16.
    Hanke, M.: Regularizing properties of a truncated newton-CG algorithm for nonlinear inverse problems. Numer. Funct. Anal. Optim. 18(9–10), 971–993 (1997). CrossRefGoogle Scholar
  17. 17.
    Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia (1998).
  18. 18.
    Imperial-College London, webpage of Department of Earth Science and Engineering, Standard Models: (Accessed on 01/30/2017)
  19. 19.
    Liu, N., Oliver, D.S.: Evaluation of Monte Carlo methods for assessing uncertainty. SPE J. 8(2), 188–195 (2003). CrossRefGoogle Scholar
  20. 20.
    Makhlouf, E.M., Chen, W.H., Wasserman, M.L., Seinfeld, J.H.: A general history matching algorithm for three-phase, three-dimensional petroleum reservoirs. SPE Adv. Technol. Ser. 1(2), 83–92 (1993). CrossRefGoogle Scholar
  21. 21.
    Manly, B.F.J.: Multivariate Statistical Methods: a Primer, 4th edn. CRC Press, Boca Raton, FL (2017)Google Scholar
  22. 22.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006). Google Scholar
  23. 23.
    Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  24. 24.
    Peters, E., Arts, R.J., Brouwer, G.K., Geel, C.R., Cullick, S., Lorentzen, R., Chen, Y., Dunlop, K.N.B., Vossepoel, F., Xu, R., Sarma, P., Alhutali, A., Reynolds, A.: Results of the Brugge benchmark study for flooding optimization and history matching. SPE Reserv. Eval. Eng. 13(3), 391–405 (2010). CrossRefGoogle Scholar
  25. 25.
    Petrobras/CENPES: SIMPAR 2.0 Manual do Usuário (in Portuguese) (1995)Google Scholar
  26. 26.
    Reynolds, A.C., Zafari, M., Li, G.: Iterative forms of the ensemble Kalman filter. In: Proceedings of 10th European Conference on the Mathematics of Oil Recovery. Amsterdam (2006).
  27. 27.
    Rodrigues, J.R.P.: Calculating Derivatives for History Matching in Reservoir Simulators. SPE Reservoir Simulation Symposium, 31 January-2 Feburary. The Woodlands, Texas (2005). Google Scholar
  28. 28.
    Shirangi, M.G.: History matching production data and uncertainty assessment with an efficient TSVD parameterization algorithm. J. Petrol. Sci. Eng. 113, 54–71 (2014). CrossRefGoogle Scholar
  29. 29.
    Shirangi, M.G., Emerick, A.A.: An improved TSVD-based Levenberg–Marquardt algorithm for history matching and comparison with Gauss–Newton. J. Petrol. Sci. Eng. 143, 258–271 (2016). CrossRefGoogle Scholar
  30. 30.
    Tavakoli, R., Reynolds, A.C.: History matching with parameterization based on the SVD of a dimensionless sensitivity matrix. SPE J. 15(12), 495–508 (2010). CrossRefGoogle Scholar
  31. 31.
    Tavakoli, R., Reynolds, A.C.: Monte Carlo simulation of permeability fields and reservoir performance predictions with SVD parameterization in RML compared with enKF. Comput. Geosci. 15(1), 99–116 (2011). CrossRefGoogle Scholar
  32. 32.
    Yang, P.H., Watson, A.T.: Automatic history matching with variable-metric methods. SPE Reserv. Eng. 3(3), 995–1001 (1988). CrossRefGoogle Scholar
  33. 33.
    Zhang, F., Reynolds, A.C.: Optimization algorithms for automatic history matching of production data. In: ECMOR VIII - 8th European Conference on the Mathematics of oil Recovery (2002).
  34. 34.
    Zhao, H., Li, G., Reynolds, A., Yao, J.: Large-scale history matching with quadratic interpolation models. Comput. Geosci. 17(1), 117–138 (2013). CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.IM/UFRJRio de JaneiroBrazil
  2. 2.Institute of Industrial Sciencethe University of TokyoMeguroJapan
  3. 3.LabMAPetro/IM/UFRJRio de JaneiroBrazil

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