A multistage MCMC method with nonparametric error model for efficient uncertainty quantification in history matching

Uncertainty quantification is generally carried out in a Bayesian framework whereby multiple reservoir models can be evaluated by sampling from a posterior distribution that incorporates the observed production data and the prior parameter distribution. Rigorous sampling methods such as the Markov Chain Monte Carlo (MCMC) method provide accurate sampling but at a high cost because of their high rejection rates and the need to run flow simulation for every proposed candidate. We propose a methodology that combines coarse- and fine-scale information to improve the efficiency of MCMC methods without loss of its rigorousness in sampling. Our proposed method employs off-line computations for establishing a mathematical relation between coarse- and fine-scale error responses. This relation is modeled using non-parametric approaches that correlate the error responses via optimal transformations. The resulting statistical models are then used in efficient sampling within an MCMC framework. We propose a multi-stage MCMC where inexpensive coarse-scale simulations are performed to determine whether or not to run the fine-scale simulations. The latter is determined based on the non-parametric error model developed off-line. Our proposed method substantially improves the efficiency of existing multistage MCMC methods that only rely on coarse-scale or other approximate models. Most importantly, our proposed method does not depend on the proximity of the coarse and fine-scale models and thus, can employ much coarser and inexpensive models to guide the fine-scale simulations. We will present a synthetic and a field example involving three-phase flow to demonstrate that the proposed approach is both robust and computationally efficient. In these examples, water cut and GOR are integrated to generate multiple reservoir models and assess uncertainty in the production forecasts.