Generating an estimate of uncertainty in production forecasts has become almost standard in the oil industry but is often done with procedures that yield at best a highly approximate uncertainty quantification. Formally, the uncertainty quantification of a production forecast can be achieved by generating a correct characterization of the posterior probability density function (pdf) of reservoir model parameters conditional to dynamic data and sampling this pdf correctly. While Markov chain Monte Carlo (MCMC) provides a theoretically rigorous method for sampling any target pdf that is known up to a normalizing constant, in reservoir engineering applications, researchers have found that it may require extraordinarily long chains containing millions to hundreds of million of states to obtain a a correct characterization of the target pdf. When the target pdf has a single mode or has multiple modes concentrated in a small region, it is possible to implement a proposal distribution based essentially on random walk so that the resulting Markov chain Monte Carlo algorithm based on the Metropolis-Hastings acceptance probability can yield a good characterization of the posterior pdf. However, such a method may still require the generation of millions of states in the chain in order to obtain a proper sampling of the posterior pdf. For a high-dimensional multimodal pdf with modes separated by large regions of low or zero probability, characterizing the pdf with MCMC based on random walk is not feasible. While methods such as population MCMC exist for characterizing a multimodal pdf, their computational cost generally makes the application of these algorithm far too costly for field application. In this paper, we design a new proposal distribution based on a Gaussian mixture pdf for use in MCMC where the posterior pdf can be a multimodal pdf, possibly with the modes spread far apart. Simply put, the method generates modes using a gradient-based optimization method and constructs a Gaussian mixture model to use as the basic proposal distribution. Tests on three simple problems are presented to establish the validity of the method. The performance of the new MCMC algorithm is compared with random walk MCMC and is also compared with population MCMC for a target pdf which is multimodal.

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