# Convolutional Surrogate for 3D Discrete Fracture-Matrix Tensor Upscaling Source: [Convolutional Surrogate for 3D Discrete Fracture-Matrix Tensor Upscaling]() Collection index: [llms.txt]() Data updated: 2026-04-07T05:00:04.607Z Authors: Martin \\v{S}petl\\'ik, Jan B\\v{r}ezina Published: 2026-01-19T23:53:58.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [3D Discrete Fracture-Matrix Tensor Upscaling on DFM (15x15x15, test)](): 3D Discrete Fracture-Matrix Tensor Upscaling; 3D Discrete Fracture-Matrix (DFM) models domain size 15x15x15 250 samples (test); NRMSE (kxx): 0.01222. - [Outflow Prediction on 3D Discrete Fracture-Matrix (DFM) Domain (60, 60, 60) (250 samples)](): Constraint problem outflow prediction; 3D Discrete Fracture-Matrix (DFM) Domain (60, 60, 60) (250 samples); NRMSE: 0.00381. - [Constraint problem outflow prediction on Constraint problem domain size 15x15x15](): Constraint problem outflow prediction; Constraint problem domain size 15x15x15; NRMSE: 0.01323. - [Equivalent Hydraulic Conductivity Tensor Upscaling on 3D DFM (60x60x60) (test)](): Equivalent hydraulic conductivity tensor upscaling; 3D Discrete Fracture-Matrix (DFM) domain size ΩUP=(60, 60, 60) (test); NRMSE (kxx): 0.00022. ## Original sources and implementation [Original paper]() [PDF]()