Machine-learning-based modeling of coarse-scale error, with application to uncertainty quantification

Abstract

The use of upscaled models is attractive in many-query applications that require a large number of simulation runs, such as uncertainty quantification and optimization. Highly coarsened models often display error in output quantities of interest, e.g., phase production and injection rates, so the direct use of these results for quantitative evaluations and decision making may not be appropriate. In this work, we introduce a machine-learning-based post-processing framework for modeling the error in coarse-model results in the context of uncertainty quantification. Coarse-scale models are constructed using an accurate global single-phase transmissibility upscaling procedure. The framework entails the use of high-dimensional regression (random forest in this work) to model error based on a number of error indicators or features. Many of these features are derived from approximations of the subgrid effects neglected in the coarse-scale saturation equation. These features are identified through volume averaging, and they are generated by solving a fine-scale saturation equation with a constant-in-time velocity field. Our approach eliminates the need for the user to hand-design a small number of informative (relevant) features. The training step requires the simulation of some number of fine and coarse models (in this work we perform either 10 or 30 training simulations), followed by construction of a regression model for each well. Classification is also applied for production wells. The methodology then provides a correction at each time step, and for each well, in the phase production and injection rates. Results are presented for two- and three-dimensional oil–water systems. The corrected coarse-scale solutions show significantly better accuracy than the uncorrected solutions, both in terms of realization-by-realization predictions for oil and water production rates, and for statistical quantities important for uncertainty quantification, such as P10, P50, and P90 predictions.

Appendix: Random-forest regression

Random forest is a decision-tree-based supervised statistical technique used here to construct both the classification and regression models from the EMML training dataset. We provide a succinct description here – see [5] for more details. A single decision tree can be used to recursively segment the domain of the EMML training dataset in the feature space, as shown in Fig. 22. Segmentation is achieved by minimizing the following expression in a top-down greedy approach:

$$ \sum\limits_{n,k: \boldsymbol{f}^{n}\in R_{1}(j,s)} (\delta^{n} - \hat{ \delta}_{R_{1}})^{2} + \sum\limits_{n,k: \boldsymbol{f}^{n} \in R_{2}(j,s)} (\delta^{n} - \hat{ \delta}_{R_{2}})^{2}. $$

(28)

Here, R₁(j, s) = {f | f_j < s} and R₂(j, s) = {f | f_j ≥ s} denote the feature-space regions as shown in Fig. 22, where j ∈{1,…,N_f} is the feature index and $s\in \mathbb {R}^{}$ is the cut-point that segments the domain, and $\hat { \delta }_{R_{k}} \in \mathbb {R}^{},\ k = 1,2$ denotes the mean value of the error for all of the training samples in R_k. The recursive segmentation enables the nonlinear interactions between the features to be captured. Methods of this type are known as tree-based because the recursive segmentation can be interpreted as a decision tree.

Fig. 22

Schematic of segmentation of the feature space using a decision tree. Figure modified from [24]

Single decision trees, however, tend to overfit the data. As a result the classification or regression model may suffer from low bias and high variance. To avoid this problem, in the random forest procedure we construct an ensemble of decision trees to improve prediction accuracy. This is accomplished by first selecting a random subsample of the EMML training dataset from the overall set. The subsamples are drawn with replacement (bootstrapping) such that the size of the new set is the same as that of the original set. Then, when constructing each decision tree, at any node of the tree, random forest selects a random subset of features from all possible features, and then segments the data based on this subset. This entails solving Eq. 28 using the subset of features. As a result of these treatments, random forest leads to reduction in the variance without a corresponding increase in the bias. The random forest implementation used in this work is provided in [6].