SPE Journal
Volume 17, Number 2, June 2012, pp. 418-440

Summary

It is well known that when applied to reservoir history-matching problems, the ensemble Kalman filter (EnKF) can lead to a large underestimation of uncertainty in the posterior probability-density function (PDF) for reservoir-model parameters. Here, we demonstrate that, regardless of whether covariance localization is used, EnKF also can lead to an overestimation of the uncertainty in future predictions of reservoir performance. This overestimation occurs because, even though the data matches obtained with EnKF tend to appear reasonable, these matches are significantly worse than those that can be obtained when history matching dynamic data with a gradient-based method. The relatively poor data match obtained with EnKF means that the ensemble of states generated by assimilating data with EnKF are of relatively low probability (low value of posterior PDF). Under reasonable assumptions, a Markov-chain Monte Carlo (MCMC) algorithm will theoretically generate an accurate sampling of the posterior PDF conditional to dynamic data. However, standard implementations of MCMC are not computationally feasible because each proposed state in the chain requires a run of the forward model (reservoir simulator). In this work, the EnKF and MCMC methodologies are combined to obtain a relatively efficient algorithm for sampling the posterior PDF for reservoir-model parameters. In this algorithm, a symmetric square root of the state vector's posterior covariance matrix is calculated from an ensemble of state vectors obtained from EnKF. It is shown that this square root can be used both to propose new states in the Markov chain and to evaluate the probability of accepting the state without running the simulator, which results in an efficient algorithm. In order to improve sampling, we also propose to generate multiple ensembles with EnKF and multiple Markov chains, which are then combined and resampled based on the normalized objective function. We tested the proposed EnKF-MCMC algorithm on a small 3D two-phase-flow reservoir example. The problem is sufficiently small that by using MCMC only, we are able to generate a long chain based on 2 million proposed states. The posterior PDF calculated from this long chain is assumed to be accurate. The posterior PDFs for predicted cumulative oil and water production generated with the new EnKF-MCMC algorithm provide results that are in reasonable agreement with the posterior PDFs for the cumulative oil and water production obtained from the long Markov chain, whereas the corresponding PDFs obtained using EnKF with and without covariance localization exhibit a significantly higher variance (i.e., overestimate uncertainty in the predicted cumulative reservoir production).

© 2012. Society of Petroleum Engineers

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History

• Original manuscript received: 10 December 2010
• Meeting paper published: 22 February 2011
• Revised manuscript received: 12 August 2011
• Manuscript approved: 9 September 2011
• Published online: 17 April 2012
• Version of record: 11 June 2012