Deep learning surrogate for predicting hydraulic conductivity tensors from stochastic discrete fracture-matrix models

TL;DR

The paper addresses the computational bottleneck in predicting upscaled hydraulic properties for 2D stochastic discrete fracture–matrix models by replacing numerical homogenization with a CNN–FNN-based surrogate. The authors train separate surrogates on data with different fracture-to-matrix conductivity ratios and demonstrate that the surrogate accurately predicts the equivalent hydraulic conductivity tensor $\boldsymbol{K}^{eq}$, achieving high $R^2$ (often >0.95) for moderate ratios and enabling large speedups (up to about $28\times$) when applied across many homogenization blocks. They prove the surrogate's utility by comparing upscaled predictions in two macroscale problems (Aquifer and Anisotropy) against traditional homogenization, finding near-identical $\boldsymbol{K}^{eq}$ and favorable effects on the quantities of interest. The approach supports efficient MLMC implementations and offers a path toward 3D extensions, with code and datasets enabling reproducibility.

Abstract

Simulating 2D flow in fractured crystalline rock requires 2D stochastic discrete-fracture matrix (DFM) models. To obtain the simulation statistics of interest at an affordable computational cost, we aim to use the multilevel Monte Carlo method. To use this multiscale approach, one needs to upscale the hydraulic conductivity of the fractures by numerical homogenization. In this work, we substitute numerical homogenization with a surrogate model to speed up the computations. In particular, we resort to a deep convolutional neural network (CNN) connected to a deep feed-forward neural network. The equivalent hydraulic conductivity tensor $K_{eq}$ is predicted based on an input spatial random field (SRF) of hydraulic conductivity tensors, cross-section, and hydraulic conductivity of fractures. Three independent surrogates with the same architecture are trained using data from DFM models with three different ratios of hydraulic conductivities of fracture and bulk $K_f/K_b$. As the $K_f/K_b$ ratio increases, the multivariate $K_{eq}$ distribution becomes more complex, and thus, the prediction accuracy of the trained surrogates deteriorates. Regardless of $K_f/K_b$, however, an improvement in the prediction accuracy of the trained surrogates is noted as the considered fracture density of the modeling setup decreases. We also investigate prediction accuracy on input SRFs of different correlation lengths. Upscaling by numerical homogenization and by surrogate modeling is compared on two practical problems: upscaling of the hydraulic conductivity tensor and groundwater flow through a given surface. We obtained equally accurate results for the equivalent hydraulic tensor calculation of upscaled DFM models regardless of the upscaling method. For the groundwater flow problem, the accuracy of quantity of interest imitates the accuracy of $K_{eq}$ predictions.

Figures (10)

• Figure 1: Left: The fine-scale DFM model ($h<5$), grey fractures, and permeability tensor field are homogenized using overlapping square blocks of size $15$ depicted in the corner. Right: The coarse model ($H<10$) with homogenized permeability tensor component $K_{xx}$. Notice the reduced spread of the upscaled field.
• Figure 2: Distributions of $\boldsymbol{K}^{eq}$ components for datasets of different $K_f/K_m$, Dataset $\mathcal{A}$ for $K_f/ K_m = 1.0e3$, Dataset $\mathcal{B}$ for $K_f/ K_m = 1.0e5$, and Dataset $\mathcal{C}$ for $K_f/ K_m = 1.0e7$.
• Figure 3: Accuracy of the surrogates on test datasets
• Figure 4: Impact of fracture density $\rho^{\prime}_{2D}$ on surrogate's accuracy
• Figure 5: Impact of matrix correlation length $\lambda$ on surrogate's accuracy