A Surrogate-based Adaptive Sampling Approach for History Matching and Uncertainty Quantification
A critical procedure in reservoir simulations is history matching (or data assimilation in a broader sense), which calibrates model parameters such that the simulation results are consistent with field measurements, and hence improves the credibility of the predictions given by the simulations. Often there exist non-unique combinations of parameter values that all yield the simulation results matching the measurements. For such ill-posed history matching problems, Bayesian theorem provides a theoretical foundation to represent different solutions and to quantify the uncertainty with the posterior PDF. Lacking an analytical solution in most situations, the posterior PDF may be characterized with a sample of realizations, each representing a possible scenario. A novel sampling algorithm is presented here for the Bayesian solutions to history matching problems. We aim to deal with two commonly encountered issues: 1) as a result of the nonlinear input-output relationship in a reservoir model, the posterior distribution could be in a complex form, such as multimodal, which violates the Gaussian assumption required by most of the commonly used data assimilation approaches; 2) a typical sampling method requires intensive model evaluations and hence may cause unaffordable computational cost. In the developed algorithm, we use a Gaussian mixture model as the proposal distribution in the sampling process, which is simple but also flexible to approximate non-Gaussian distributions and is particularly efficient when the posterior is multimodal. Also, a Gaussian process is utilized as a surrogate model to speed up the sampling process. Furthermore, an iterative scheme of adaptive surrogate refinement and re-sampling ensures sampling accuracy while keeping the computational cost at a minimum level. The developed approach is demonstrated with an illustrative example and shows its capability in handling the above-mentioned issues. Multimodal posterior of the history matching problem is captured and are used to give a reliable production prediction with uncertainty quantification. The new algorithm reveals a great improvement in terms of computational efficiency comparing previously studied approaches for the sample problem.