When formulating history matching within the Bayesian framework, we may quantify the uncertainty of model parameters and production forecasts using conditional realizations sampled from the posterior probability density function (PDF). It is quite challenging to sample such a posterior PDF. Some methods e.g., Markov chain Monte Carlo (MCMC), are very expensive (e.g., MCMC) while others are cheaper but may generate biased samples. In this paper, we propose an unconstrained Gaussian Mixture Model (GMM) fitting method to approximate the posterior PDF and investigate new strategies to further enhance its performance.

To reduce the CPU time of handling bound constraints, we reformulate the GMM fitting formulation such that an unconstrained optimization algorithm can be applied to find the optimal solution of unknown GMM parameters. To obtain a sufficiently accurate GMM approximation with the lowest number of Gaussian components, we generate random initial guesses, remove components with very small or very large mixture weights after each GMM fitting iteration and prevent their reappearance using a dedicated filter. To prevent overfitting, we only add a new Gaussian component if the quality of the GMM approximation on a (large) set of blind-test data sufficiently improves.

The unconstrained GMM fitting method with the new strategies proposed in this paper is validated using nonlinear toy problems and then applied to a synthetic history matching example. It can construct a GMM approximation of the posterior PDF that is comparable to the MCMC method, and it is significantly more efficient than the constrained GMM fitting formulation, e.g., reducing the CPU time by a factor of 800 to 7300 for problems we tested, which makes it quite attractive for large scale history matching problems.

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