Combining the Ensemble Kalman Filter with Markov Chain Monte Carlo for Improved History Matching and Uncertainty Characterization
Authors: Emerick Alexandre A. , Reynolds Albert C. 
Published: February 2011

Abstract
Although there is intense interest in employing the ensemble Kalman filter (EnKF) for assisted history matching, it is well known that the approximation of covariance matrices from a finite ensemble of states and a finite ensemble of predicted data vectors can lead to a large underestimation of uncertainty in the posterior probability density (pdf) for the state vector. Here, we demonstrate that, regardless of whether covariance localization is used, EnKF 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 much 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 the ensemble of states generated by assimilating data with EnKF are of relatively low probability. Under reasonable assumptions, a Markov chain Monte Carlo (MCMC) algorithm will theoretically generate a correct sampling of the posterior pdf conditional to dynamic data. However, standard implementations of MCMC are not computationally feasible as one must typically generate thousands of states in the chain to complete the transitionary burn-in period before one begins to generate states that represent samples from the target pdf, and each proposed state in the chain requires a run of the forward model (reservoir simulator) to evaluate the probability of accepting the proposal. In this work, the EnKF and MCMC methodologies are combined to obtain a relatively efficient algorithm for sampling the posterior pdf for the 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 evaluate the probability of accepting the state without running the simulator, which results in an efficient algorithm. Because the posterior covariance obtained after assimilating data with EnKF depends strongly on the initial ensemble, to obtain a good estimate of the posterior pdf, we generate several ensembles with EnKF and several Markov chains which are then combined and resampled with importance sampling. We demonstrate the potential utility of the proposed EnKF-MCMC algorithm on a small three-dimensional two-phase-flow reservoir example. The problem is sufficiently small that using only MCMC, we are able to generate a long-chain based on two million proposed states. The posterior pdf calculated from this long-chain is assumed to be true. The posterior pdf's for predicted cumulative oil and water production generated with the new EnKF-MCMC algorithm gives results that are in reasonable agreement with the posterior pdf's for the cumulative productions obtained from the long Markov chain, whereas the corresponding pdf's obtained using EnKF with and without covariance localization exhibit a significantly higher variance, i.e., overestimate uncertainty in the predicted cumulative oil and water production.