An adaptive metropolis-hasting sampling algorithm for reservoir uncertainty quantification in Bayesian inference

Uncertainty quantification plays a crucial role in providing high quality and robust decisions for reservoir management. A set of diverse fitting models that represent the correct sampling of the posterior allow us to estimate posterior distribution, thus, to quantify uncertainty in performance prediction. We propose a novel uncertainty quantification method based on Metropolis-Hasting sampling with an adaptive multivariate proposal distribution. The proposed method involves inference from an ensemble of simulation models obtained by direct search algorithms in the history matching phase to approximate the posterior probability of the uncertainty parameters in model space and, consequently, the predictive parameter of the reservoir simulation model. The method is not sensitive to the initial covariance matrix and either of the ensemble's covariance or the identity matrix can be used as the initial covariance matrix, which is going to be updated during the burn-in period, whenever a certain number of accepted samples become available. When compared to NAB, another ensemble-based uncertainty quantification method, our MCMC based method with adaptive covariance matrix, was shown to be competitive to NAB. It provided results similar to the probability distribution function of multivariate Gaussian function and slightly closer to the truth case than NAB in the PUNQ-S3 model application, although it was slightly outperformed by NAB in the IC-Fault model application, with regard to the closeness of forecasted CDF to the database result. The proposed method performs well in problems with a misfit landscape and probability density that can be estimated with a Gaussian model (e.g. multivariate Gaussian function and PUNQ-S3 model), while it may struggle in problems with sharp minima and those than cannot be effectively estimated with Gaussian models.