Quantifying Uncertainty for the PUNQ-S3 Problem in a Bayesian Setting With RML and EnKF

Summary The well known PUNQ-S3 reservoir model represents a synthetic problem which was formulated to test the ability of various methods and research groups to quantify the uncertainty in the prediction of cumulative oil production. Previous results reported on this project suggest that the randomized maximum likelihood (RML) method gives a biased characterization of the uncertainty. A major objective of this paper is to show that this is incorrect. With a correct implementation of the RML method within a Bayesian framework, we show that RML does an adequate job of sampling the a posteriori distribution for the PUNQ problem. In particular, the true predicted oil production lies within the band of predictions generated with the RML method and is not biased. We also apply the ensemble Kalman Filter (EnKF) method to the PUNQ data set, and show that this method also gives a reasonable quantification of the uncertainty in performance predictions with an uncertainty range similar to the one obtained with RML. Introduction We consider conditioning models to production data in a Bayesian framework and wish to generate a suite (ensemble) of models which represent a correct sampling of the conditional probability density function (pdf). By predicting future reservoir performance with each realization, we obtain a characterization of the uncertainty in predicted performance. Both the rejection algorithm and Markov chain Monte Carlo (MCMC) are theoretically sound sampling procedures, but they are too computationally inefficient for practical applications (Liu and Oliver 2003). Oliver et al. (1996) and Kitanidis (1986) independently proposed the randomized maximum likelihood (RML) method to generate an approximate sampling of the a posteriori pdf. Two different proofs (Oliver 1996; Reynolds et al. 1999) have been presented which show that the RML method samples the posterior probability density function (pdf) correctly if data are linearly related to the model; however, no rigorous theoretical foundation exists for the method when the relation between data and model is nonlinear, which is the case when the data represent production data. Computational results indicate that the RML method generates reasonable characterization of uncertainty for single-phase flow (Oliver et al. 1996; Reynolds et al. 1999; Liu and Oliver 2003). Our first objective is to show that, contrary to a previous claim (Floris 2001), RML gives a reasonable characterization of the uncertainty in predicted performance for the PUNQ-S3 problem; our second objective is to compare the quantification of uncertainty obtained with RML with the one obtained with the ensemble Kalman filter (EnKF). The PUNQ-S3 reservoir represents a synthetic model based on an actual reservoir (Floris et al. 2001; Barker et al. 2001). The problem was set up as a test case to allow various research groups to test their own methodology for the characterization of the uncertainty in reservoir performance predictions given some geologic information on the reservoir, hard data at well gridblocks and some scattered production data from the first 8 years of production. Then participants were asked to predict cumulative oil production for 16.5 years of total production and characterize the uncertainty in this prediction.