Uncertainty Quantification for History Matching Problems With Multiple Best Matches Using a Distributed Gauss-Newton Method

It is crucially important for decision making and an extremely challenging task to properly quantify uncertainty of model parameters and production forecasts after conditioning to production data. A novel approach is proposed to generate approximate conditional realizations using the distributed Gauss-Newton (DGN) method together with a multiple local Gaussian approximation technique. Results are compared with those obtained from other approaches such as Randomized Maximum Likelihood (RML), Ensemble-Kalman-Filter (EnKF), and Markov-Chain-Monte-Carlo (MCMC) simulation. The DGN method is developed to find multiple local minima of the objective function in parallel, by collecting and sharing information from dispersed regions in parameter space dynamically. Around each local minimum, the estimated Hessian obtained from the Gauss-Newton approximation along with the prior inverse covariance matrix is used as a local approximation of the posterior inverse covariance matrix. The posterior joint PDF can then be approximated as a weighted linear superposition of multiple local Gaussian distributions, which can be sampled very efficiently without having to resort to expensive MCMC methods. The proposed approach is first validated using a nonlinear history matching toy problem with multiple modes. In terms of efficiency, the new approach can significantly reduce the computational cost and accelerate the uncertainty quantification process, when compared to the traditional RML method or traditional MCMC approaches. In terms of accuracy, uncertainty characteristics obtained from the proposed approach are comparable to those generated from the MCMC simulation, and they are much better than those obtained from EnKF or RML. The approach is then also applied to a real field history matching problem, where the dynamic system of multi-phase flow in the reservoir exhibits very strong nonlinear behavior, and the objective function has multiple local minima. Uncertainty ranges of production forecasts for the real field case are quantified by generating an ensemble of conditional realizations. The production forecasts for all conditional realizations are consistent with the production data observed after the history matching period, which further validates the applicability of the proposed method to real field problems. Its high efficiency makes the new approach practical for large-scale problems, for which methods based on Design of Experiment break down. Furthermore, as was argued by Vink, Gao, and Chen (2016), inevitable under-modeling reduces the urgency of using very accurate methods to quantify uncertainty, and using a multi-Gaussian approximation of the posterior should be more than sufficient. The proposed approach has been implemented in our next generation reservoir simulation and history matching system called PetroSigns.