Nonlinear Expectation Inference for Efficient Uncertainty Quantification and History Matching of Transient Darcy Flows in Porous Media with Random Parameters Under Distribution Uncertainty

TL;DR

The paper addresses uncertainty quantification for transient Darcy flows under distribution (Knightian) uncertainty and introduces nonlinear expectation inference (NEI) to enable efficient history matching. NEI performs forward simulations on a fixed set of prior realizations and carries the inference to data space using upper and lower nonlinear expectations over subsets of priors, avoiding repeated forward runs typical of Bayesian methods. Across 2D and 3D test cases with Gaussian and non-Gaussian parameter fields, NEI achieves accurate history matching with substantially fewer forward simulations than traditional approaches like ESMDA, while providing robust posterior-subset predictions and uncertainty bounds. This approach offers a scalable, distribution-uncertainty-resilient alternative for reservoir forecasting, with public code and demonstrated applicability to complex geological settings.

Abstract

The uncertainty quantification of Darcy flows using history matching is important for the evaluation and prediction of subsurface reservoir performance. Conventional methods aim to obtain the maximum a posterior or maximum likelihood estimate (MLE) using gradient-based, heuristic or ensemble-based methods. These methods can be computationally expensive for high-dimensional problems since forward simulation needs to be run iteratively as physical parameters are updated. In the current study, we propose a nonlinear expectation inference (NEI) method for efficient history matching and uncertainty quantification accounting for distribution or Knightian uncertainty. Forward simulation runs are conducted on prior realisations once, and then a range of expectations are computed in the data space based on subsets of prior realisations with no repetitive forward runs required. In NEI, no prior probability distribution for data is assumed. Instead, the probability distribution is assumed to be uncertain with prior and posterior uncertainty quantified by nonlinear expectations. The inferred result of NEI is the posterior subsets on which the expected flow rates are consistent with observation. The accuracy and efficiency of the new method are validated using single- and two-phase Darcy flows in 2D and 3D heterogeneous reservoirs.

Figures (9)

• Figure 1: The ground truth for generating the observed flow rate data (a), and the pressure field at end of simulation using the true permeability field (b) for test case 1.
• Figure 2: The predicted flow rate at wells using posterior subsets by NEI (a) and the predicted flow rate at wells using posterior realisations by ESMDA (b) for test case 1.
• Figure 3: The ground truth for generating the observed flow rate data in test case 2.
• Figure 4: The pressure (b) and saturation (b) fields at end of simulation using the true permeability field for test case 2.
• Figure 5: The simulated flow rates at producers using prior realisations and posterior subsets by NEI for test case 2.