Robust Uncertainty Quantification through Integration of Distributed Gauss-Newton Optimization with 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 workflow by seamless integration of a distributed Gauss-Newton (DGN) optimization method with 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 located with the local-search DGN optimization method. A GMM is constructed to approximate the posterior probability density function, by fitting 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 (i.i.d.) samples that can be directly used for uncertainty quantification of model parameters and production forecasts. The proposed method is first validated with 1-D nonlinear synthetic problems having multiple MAP points. The AR-GMM samples are better than the original GMM samples. Then, it is tested with a synthetic history-matching problem using the SPE-1 reservoir model with 8 uncertain parameters. The proposed method generates conditional samples that are better than or equivalent to those generated by other methods, e.g., 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 5 to 100). Finally, it is applied to a real field reservoir model with synthetic data, having 235 uncertain parameters. A GMM with 27 Gaussian components is constructed to approximate the actual posterior PDF. 105 AR-GMM samples are accepted from the 1000 original GMM samples, and are used to quantify 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 workflow.