Convolutional Surrogate for 3D Discrete Fracture-Matrix Tensor Upscaling

Abstract

Modeling groundwater flow in three-dimensional fractured crystalline media requires accounting for strong spatial heterogeneity induced by fractures. Fine-scale discrete fracture-matrix (DFM) simulations can capture this complexity but are computationally expensive, especially when repeated evaluations are needed. To address this, we aim to employ a multilevel Monte Carlo (MLMC) framework in which numerical homogenization is used to upscale sub-resolution fracture effects when transitioning between accuracy levels. To reduce the cost of conventional 3D numerical homogenization, we develop a surrogate model that predicts the equivalent hydraulic conductivity tensor Keq from a voxelized 3D domain representing tensor-valued random fields of matrix and fracture conductivities. Fracture size, orientation, and aperture are sampled from distributions informed by natural observations. The surrogate architecture combines a 3D convolutional neural network with feed-forward layers, enabling it to capture both local spatial features and global interactions. Three surrogates are trained on data generated by DFM simulations, each corresponding to a different fracture-to-matrix conductivity contrast. Performance is evaluated across a wide range of fracture network parameters and matrix-field correlation lengths. The trained models achieve high accuracy, with normalized root-mean-square errors below 0.22 across most test cases. Practical applicability is demonstrated by comparing numerically homogenized conductivities with surrogate predictions in two macro-scale problems: computing equivalent conductivity tensors and predicting outflow from a constrained 3D domain. In both cases, surrogate-based upscaling preserves accuracy while substantially reducing computational cost, achieving speedups exceeding 100x when inference is performed on a GPU.

Figures (9)

• Figure 1: Comparison of the fine DFM model fracture network $\mathcal{F}_{h, L}$, with $h < 5$ (left), and the corresponding coarse DFM model fracture network $\mathcal{F}_{H, L}$, with $H < 10$ (right). The illustration also shows the homogenization blocks of size $l = 15$ and their overlap $l/2$.
• Figure 2: Left: The fine DFM model ($h<5$). Orange fractures ($\mathcal{F}_{h, H}$) and the hydraulic conductivity tensor field ($\boldsymbol{K}_h$) are homogenized using overlapping square blocks of size $l=15$. Right: Corresponding coarse DFM model ($H<10$) with homogenized hydraulic conductivity tensor component $K_{xx}$. Notice the reduced range in the upscaled field. Black fractures ($\mathcal{F}_{H, L}$) are not affected by homogenization. Only the $K_{xx}$ component of the hydraulic conductivity tensor is shown.
• Figure 3: An illustration of an input hydraulic conductivity tensors ($K_{xx}$ component) on a mesh and its voxelized representation.
• Figure 4: 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 $K_f/ K_m = 1.0e7$. Diagonal components are shown on a $\log_{10}$ scale.
• Figure 5: The prediction accuracy of the trained surrogates for the components of $\boldsymbol{K}^{eq}$ evaluated on test datasets. Diagonal components are shown on a $\log_{10}$ scale.