Improved History Matching and Uncertainty Characterization

Abstract

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 141336, ’Combining the Ensemble Kalman Filter With Markov-Chain Monte Carlo for Improved History Matching and Uncertainty Characterization,’ by Alexandre A. Emerick, SPE, Petrobras, and Albert C. Reynolds, SPE, University of Tulsa, prepared for the 2011 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 21-23 February. The paper has not been peer reviewed. With the use of the ensemble Kalman filter (EnKF) for assisted history matching, 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 function (PDF) for the state vector. Regardless of whether covariance localization is used, EnKF can lead to an overestimation of the uncertainty in predictions of reservoir performance. Under reasonable assumptions, a Markov-chain Monte Carlo (MCMC) algorithm will, theoretically, generate a correct sampling of the posterior PDF conditional to dynamic data. Introduction In a reservoir-simulation study, history matching is essential to establish a measure of legitimacy of reservoir-performance predictions. Over the last decade, increased importance has been attached to quantifying uncertainty in reservoir-performance predictions and in reservoir description to manage risk. Because of this interest in characterizing uncertainty, it is common to generate multiple history-matched models. However, generating multiple history-matched models does not necessarily lead to a correct assessment of uncertainty. Uncertainty has no scientific meaning outside the realm of statistics and probability. The MCMC process provides a theoretically attractive method for sampling the posterior PDF for reservoir-model parameters. MCMC does not require knowledge of the normalizing constant in the target PDF to be sampled. A properly designed MCMC method will sample the target PDF correctly in the limit as the number of states in the chain approaches infinity. However, for high-dimension problems, MCMC typically requires a large number of iterations to sample the target PDF correctly. When MCMC is used to estimate the posterior PDF of reservoir-model parameters conditional to observed production data, calculation of the probability of accepting the transition from the current state to the proposed state requires a run of the reservoir simulator to evaluate the likelihood portion of the posterior PDF. Therefore, the direct application of MCMC can be prohibitively expensive computationally for realistic reservoir cases.