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Robust Uncertainty Quantification Through Integration of Distributed-Gauss-Newton Optimization With a Gaussian Mixture Model and Parallelized Sampling Algorithms


 Uncertainty quantification of production forecasts is crucially important for business planning of hydrocarbon–field developments. This is still a very challenging task, especially when subsurface uncertainties must be conditioned to production data. Many different approaches have been proposed, each with their strengths and weaknesses. In this work, we develop a robust uncertainty–quantification work flow by seamless integration of a distributed–Gauss–Newton (GN) (DGN) optimization method with a Gaussian mixture model (GMM) and parallelized sampling algorithms. Results are compared with those obtained from other approaches.
 Multiple local maximum–a–posteriori (MAP) estimates are determined with the local–search DGN optimization method. A GMM is constructed to approximate the posterior probability–density function (PDF) by reusing simulation results generated during the DGN minimization process. The traditional acceptance/rejection (AR) algorithm is parallelized and applied to improve the quality of GMM samples by rejecting unqualified samples. AR–GMM samples are independent, identically distributed samples that can be directly used for uncertainty quantification of model parameters and production forecasts.
 The proposed method is first validated with 1D nonlinear synthetic problems with multiple MAP points. The AR–GMM samples are better than the original GMM samples. The method is then tested with a synthetic history–matching problem using the SPE01 reservoir model (Odeh 1981; Islam and Sepehrnoori 2013) with eight uncertain parameters. The proposed method generates conditional samples that are better than or equivalent to those generated by other methods, such as Markov–chain Monte Carlo (MCMC) and global–search DGN combined with the randomized–maximum–likelihood (RML) approach, but have a much lower computational cost (by a factor of five to 100). Finally, it is applied to a real–field reservoir model with synthetic data, with 235 uncertain parameters. A GMM with 27 Gaussian components is constructed to approximate the actual posterior PDF. There are 105 AR–GMM samples accepted from the 1,000 original GMM samples, and they are used to quantify the uncertainty of production forecasts. The proposed method is further validated by the fact that production forecasts for all AR–GMM samples are quite consistent with the production data observed after the history–matching period.
 The newly proposed approach for history matching and uncertainty quantification is quite efficient and robust. The DGN optimization method can efficiently identify multiple local MAP points in parallel. The GMM yields proposal candidates with sufficiently high acceptance ratios for the AR algorithm. Parallelization makes the AR algorithm much more efficient, which further enhances the efficiency of the integrated work flow.