SPE Journal
Volume 16, Number 1, March 2011, pp. 172-182

SPE-123611-PA

Quantifying Monte Carlo Uncertainty in the Ensemble Kalman Filter

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DOI  More information 10.2118/123611-PA http://dx.doi.org/10.2118/123611-PA

Citation

  • Thulin, K., Nævdal, G., Skaug, H.J., and Aanonsen, S.I. 2011. Quantifying Monte Carlo Uncertainty in the Ensemble Kalman Filter. SPE J.  16 (1): 172-182. SPE-123611-PA. doi: 10.2118/123611-PA.

Discipline Categories

  • 6.5.6 Dynamic Model Update Algorithms
  • 6.5.8 History Matching
  • 6.5.5 Evaluation of Uncertainties
  • 6.8 Fundamental Research in Reservoir Description and Dynamics

Keywords

  • ensemble Kalman filter, uncertainty quantification, data assimilation, posterior distribution, ensemble size

Summary

The ensemble Kalman filter (EnKF) is currently considered one of the most promising methods for conditioning reservoir-simulation models to production data. The EnKF is a sequential Monte Carlo method based on a low-rank approximation of the system covariance matrix. The posterior probability distribution of model variables may be estimated from the updated ensemble, but, because of the low-rank covariance approximation, the updated ensemble members become correlated samples from the posterior distribution. We suggest using multiple EnKF runs, each with a smaller ensemble size, to obtain truly independent samples from the posterior distribution. This allows a pointwise confidence interval to be constructed for the posterior cumulative distribution function (CDF). We investigate the methodology for finding an optimal combination of ensemble batch size n  and number of EnKF runs m while keeping the total number of ensemble members n×m constant. The optimal combination of n and m is found through minimizing the integrated mean-square error (MSE) for the CDFs. We illustrate the approach on two models, first a small linear model and then a synthetic 2D model inspired by petroleum applications. In the latter case, we choose to define an EnKF run with 10,000 ensemble members as having zero Monte Carlo error. The proposed methodology should be applicable also to larger, more-realistic models.

© 2010. Society of Petroleum Engineers

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History

  • Original manuscript received: 17 December 2008
  • Revised manuscript received: 4 May 2010
  • Manuscript approved: 26 May 2010
  • Published online: 27 October 2010
  • Version of record: 15 March 2011