Reduced Degrees of Freedom Gaussian Mixture Model Fitting for Large Scale History Matching Problems

Gaussian-mixture-model (GMM) fitting has been proved a robust method to generate high quality, independent conditional samples of the posterior probability density function (PDF) by conditioning reservoir models to production data. However, the number of degrees-of-freedom (DOF) for all unknown GMM parameters may become huge for large-scale history-matching problems. A new formulation of GMM fitting with reduced number of DOF is proposed in this paper, to save memory-usage and reduce computational cost. Its performance is compared with other methods of GMM. The GMM fitting method can significantly improve the accuracy of the GMM approximation by adding more Gaussian components. In the full-rank GMM fitting formulation, both memory-usage and computational cost are proportional to the number of Gaussian components. In the reduced DOF GMM fitting formulation, the covariance matrix of the newly added Gaussian component is efficiently parameterized, using products of a low number of vectors and their transposes, whereas the other Gaussian components are simply modified by multipliers. Thus, memory usage and computational cost increase only modestly as the number of Gaussian components increases. Unknown GMM parameters, including the parameterized covariance matrix and mixture weighting factor for each Gaussian component, are first determined by minimizing the error that measures the distance between the GMM approximation and the actual posterior PDF. Then, performance of the new method is benchmarked against other methods using test problems with different numbers of uncertain parameters. The new method is found to perform more efficiently than the full-rank GMM fitting formulation, e.g., it further reduces the memory usage and computational cost by a factor of 5 to 10, while it achieves comparable accuracy. Although it is less efficient than the L-GMM approximation based on local linearization, it achieves much higher accuracy, e.g., it manages to further reduce the error by a factor of 20 to 600. Finally, the new method together with the parallelized acceptance-rejection (AR) algorithm is applied to a history matching problem. It is found to reduce the computational cost (i.e., the number of simulations required to generate an accepted conditional realization on average) by a factor of 200 when compared with the Markov chain Monte Carlo (MCMC) method, while the quality of accepted GMM samples is comparable to the MCMC samples. Uncertainty of reservoir model parameters and production forecasts can be properly quantified with accepted GMM samples by conditioning to production data.