Convolutional surrogate for 3D discrete fracture–matrix tensor upscaling

Abstract

Modeling groundwater flow in three-dimensional fractured crystalline media requires capturing the spatial heterogeneity introduced by fractures. Direct numerical simulations using fine-scale discrete fracture–matrix (DFM) models are computationally demanding, particularly when repeated evaluations are needed. We aim to use a multilevel Monte Carlo (MLMC) method in the future to reduce computational cost while retaining accuracy. When transitioning between accuracy levels, numerical homogenization is used to upscale the impact of the hydraulic conductivity of sub-resolution fractures. To reduce the computational cost of conventional 3D numerical homogenization, we develop a surrogate model that predicts the equivalent hydraulic conductivity tensor, $K^{eq}$, from a voxelized 3D domain representing a tensor-valued random field of matrix and fracture hydraulic conductivities. Fracture properties, including 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 to capture both local spatial patterns and global interactions. Three surrogates are trained on data generated by discrete fracture–matrix (DFM) simulations, each corresponding to a different fracture-to-matrix conductivity ratio. Their performance is evaluated across varying fracture network parameters and correlation lengths of the matrix field. The trained surrogates achieve high prediction accuracy ($\text{NRMSE}<0.22$) in a wide range of test scenarios. To demonstrate practical applicability, we compare conductivities upscaled by numerical homogenization and by our surrogates in two macro-scale problems: computation of equivalent tensors of hydraulic conductivity and prediction of outflow from a constrained 3D area. In both cases, the surrogate-based approach preserves accuracy while substantially reducing computational cost. Surrogate-based upscaling achieves speedups exceeding $100\times$ when inference is performed on a GPU.