Computational Geosciences

, Volume 16, Issue 2, pp 423–436 | Cite as

Population MCMC methods for history matching and uncertainty quantification

  • Linah Mohamed
  • Ben Calderhead
  • Maurizio Filippone
  • Mike Christie
  • Mark Girolami
Original Paper


This paper presents the application of a population Markov Chain Monte Carlo (MCMC) technique to generate history-matched models. The technique has been developed and successfully adopted in challenging domains such as computational biology but has not yet seen application in reservoir modelling. In population MCMC, multiple Markov chains are run on a set of response surfaces that form a bridge from the prior to posterior. These response surfaces are formed from the product of the prior with the likelihood raised to a varying power less than one. The chains exchange positions, with the probability of a swap being governed by a standard Metropolis accept/reject step, which allows for large steps to be taken with high probability. We show results of Population MCMC on the IC Fault Model—a simple three-parameter model that is known to have a highly irregular misfit surface and hence be difficult to match. Our results show that population MCMC is able to generate samples from the complex, multi-modal posterior probability distribution of the IC Fault model very effectively. By comparison, previous results from stochastic sampling algorithms often focus on only part of the region of high posterior probability depending on algorithm settings and starting points.


History matching Uncertainty quantification MCMC methods Population MCMC 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aanonsen, S., Naevdal, G., Oliver, D., Reynolds, A., Vallès, B.: The ensemble Kalman filter in reservoir engineering – a review. Soc. Pet. Eng. J. 14(3), 393–412 (2009) SPE 117274-PAGoogle Scholar
  2. 2.
    Busby, D.: Hierarchical adaptive experimental design for Gaussian process emulators. Reliab. Eng. Syst. Saf. 94(7), 1183–1193 (2009)CrossRefGoogle Scholar
  3. 3.
    Busby, D., Feraille, M.: Adaptive design of experiments for calibration of complex simulators—An application to uncertainty quantification of a mature oil field. J. Phys.: Conference Series 135(1), 012026 (2008)CrossRefGoogle Scholar
  4. 4.
    Busby, D., Farmer, C., Iske, A.: Uncertainty evaluation in reservoir forecasting by Bayes linear methodology. In: Iske, A., and Levesley, J. (eds.) Algorithms for Approximation. Proceedings of the 5th International Conference, Chester, pp. 187–196. Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Busby, D., Farmer, C., Iske, A.: Hierarchical nonlinear approximation for experimental design and statistical data fitting. SIAM J. Sci. Comput. 29(1), 49–69 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bush, M., Carter, J.: Applications of a modified genetic algorithm to parameter estimation in the petroleum industry. In: Dagli, D., Akay, E., Chen, C., Fernandez, B., and Ghosh, J. (eds.) Intelligent Engineering Systems through Artificial Neural Networks, pp. 6. ASME, New York (1996)Google Scholar
  7. 7.
    Calderhead, B., Girolami, M.: Estimating Bayes factors via thermodynamic integration and population MCMC. Comput. Stat. Data Anal. 53(12), 4028–4045 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Carter, J., and Ballester, P.: A real-parameter genetic algorithm for clustering identification in history matching, A012. In: Proceedings of the 9th European Conference on Mathematics of Oil Recovery, Cannes, France, 30 August–02 September (2004)Google Scholar
  9. 9.
    Carter, C., Ballester, P., Tavassoli, Z., King, P.: Our calibrated model has poor predictive value: An example from the petroleum industry. Reliab. Eng. Syst. Saf. 91, 1373–1381 (2006)CrossRefGoogle Scholar
  10. 10.
    Christie, M., Demyanov, V., Erbas, D.: Uncertainty quantification for porous media flows. J. Comput. Phys. 217(1), 143–158 (2006)MATHCrossRefGoogle Scholar
  11. 11.
    Demyanov, V., Pozdnoukhov, A., Christie, M., Kanevski, M.: Detection of optimal models in high dimensional parameter space with support sector machines. In: Proceedings of geoENV VII European Conference on Geostatistics for Environmental Applications (2008)Google Scholar
  12. 12.
    Duane, S., Kennedy, A., Pendleton, B., Roweth, D.: Hybrid Monte Carlo. Phys. Lett. B 195(2), 216–222 (1987)CrossRefGoogle Scholar
  13. 13.
    Erbas, D., Christie, M.: Effect of sampling strategies on prediction uncertainty estimation. In: Proceedings of the SPE Reservoir Simulation Symposium, SPE 106229, Houston, Texas, USA, 26–28 February (2007a)Google Scholar
  14. 14.
    Erbas, D., Christie, M.: How does sampling strategies affect uncertainty estimations? Oil Gas Sci. Technol. – Rev. IFP 62(2), 155–167 (2007b)CrossRefGoogle Scholar
  15. 15.
    Evensen, G.: Data assimilation, The Ensemble Kalman Filter, 2nd edn. Springer, New York (2009)Google Scholar
  16. 16.
    Filippone, M., Calderhead, B., Girolami, M., Mohamed, L., Christie M.: Inference for Gaussian process emulation of oil reservoir simulation codes. In: Proceedings of the ISBA 2010 World Meeting/9th Valencia Meeting, Benidorm, Spain, 3–8 June (2010)Google Scholar
  17. 17.
    Geyer, C.: Markov chain Monte Carlo maximum likelihood. In: Keramidas, M. (ed.) Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, pp. 156–163. Interface Foundation, Fairfax Station (1991)Google Scholar
  18. 18.
    Gilks, W., Richardson, S., Spiegelhalter, D.: Markov Chain Monte Carlo in Practice. Chapman & Hall, Boca Raton (1996)MATHGoogle Scholar
  19. 19.
    Girolami, M., Calderhead, B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods (with discussion). J. R. Stat. Soc., Ser. B 73(2), 1–37 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hajizadeh, Y., Christie, M., Demyanov, V.: History matching with differential evolution approach: A look at new search strategies. In: Proceedings of the 72nd EAGE Conference and Exhibition/SPE EUROPEC 2010, SPE 130253, Barcelona, Spain, 14–17 June (2010)Google Scholar
  21. 21.
    Holloman, C., Lee, H., Higdon, D.: Multiresolution genetic algorithms and Markov chain Monte Carlo. J. Comput. Graph. Stat. 15(4), 861–879 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    IC Fault Model: London: Department of Earth Science and Engineering, Imperial College London., accessed July (2010)
  23. 23.
    Jaynes, E.: Probability Theory: The Logic of Science, vol. 1. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  24. 24.
    Kim, W., Park, J., Lee, K.: Stereo matching using population–based MCMC. Int. J. Comput. Vis. 83(2), 195–209 (2009)CrossRefGoogle Scholar
  25. 25.
    Liang, F., Wong, W.: Evolutionary Monte Carlo: applications to Cp model sampling and change point problem. Stat Sin 10, 317–342 (2000)MATHGoogle Scholar
  26. 26.
    Liu, N., Oliver, D.: Evaluation of Monte Carlo methods for assessing uncertainty. Soc. Pet. Eng. J. 8(2), 188–195 (2003) SPE 84936–PAGoogle Scholar
  27. 27.
    Ma, X., Al-Harbi, M., Datta-Gupta, A., Efendiev, Y.: An efficient two-stage sampling method for uncertainty quantification in history matching geological models. Soc. Pet. Eng. J. 13(1), 77–87 (2008) SPE 102476-PAGoogle Scholar
  28. 28.
    Mohamed, L., Christie, M., Demyanov, V.: Comparison of stochastic sampling algorithms for uncertainty quantification. Soc. Pet. Eng. J. 15(1), 31–38 (2010a) SPE 119139–PAGoogle Scholar
  29. 29.
    Mohamed, L., Christie, M., Demyanov, V.: Reservoir model history matching with particle swarms. In: Proceedings of the Oil and Gas India Conference and Exhibition (OGIC), SPE 129152, Mumbai, India, 20–22 January (2010b)Google Scholar
  30. 30.
    Mohamed, L., Christie, M., Demyanov, V., Robert, E., Kachuma, D.: Application of particle swarms for history matching in the Brugge reservoir. In: Proceedings of the SPE Annual Technical Conference and Exhibition (ATCE), SPE 135264, Florence, Italy (2010c)Google Scholar
  31. 31.
    Petrovska, I., Carter, J.: Estimation of distribution algorithms for history matching. In: Proceedings of the 10th European Conference on the Mathematics of Oil Recovery, Amsterdam, The Netherlands, 4–7 September (2006)Google Scholar
  32. 32.
    Sambridge, M.: Geophysical inversion with a neighbourhood algorithm – II. Appraising the ensemble. Geophys. J. Int. 138(3), 727–746 (1999)CrossRefGoogle Scholar
  33. 33.
    Specht, D.: A general regression neural network. IEEE Trans. Neural Netw. 2(6), 568–576 (1991)CrossRefGoogle Scholar
  34. 34.
    Tavassoli, Z., Carter, C., King, P.: Errors in history matching. Soc. Pet. Eng. J. 9(3), 352–361 (2004) SPE 86883–PAGoogle Scholar
  35. 35.
    Vrugt, J., Braak, C., Diks, C., Robinson, B., Hyman, J., Higdon, D.: Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomised subspace sampling. Int. J. Nonlinear Sci. Numer. Simul. 10(3), 273–290 (2009)CrossRefGoogle Scholar
  36. 36.
    Williams, G., Mansfield, M., MacDonald, D., Bush, M.: Top–down reservoir modelling. In: Proceedings of the SPE Annual Technical Conference and Exhibition, SPE 89974, Houston, Texas, 26–29 September (2004)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Linah Mohamed
    • 1
  • Ben Calderhead
    • 2
  • Maurizio Filippone
    • 2
  • Mike Christie
    • 1
  • Mark Girolami
    • 2
  1. 1.Institute of Petroleum EngineeringHeriot-Watt UniversityEdinburghUK
  2. 2.Department of Statistical ScienceUniversity College LondonLondonUK

Personalised recommendations