Neural Networks for Predicting Permeability Tensors of 2D Porous Media: Comparison of Convolution- and Transformer-based Architectures

Permeability is a central concept in the macroscopic description of flow through porous media, with applications spanning from oil recovery to hydrology. Traditional methods for determining the permeability tensor involving flow simulations or experiments can be time consuming and resource-intensive, while analytical methods, e.g., based on the Kozeny-Carman equation, may be too simplistic for accurate prediction based on pore-scale features. In this work, we explore deep learning as a more efficient alternative for predicting the permeability tensor based on two-dimensional binary images of porous media, segmented into solid ($1$) and void ($0$) regions. We generate a dataset of 24,000 synthetic random periodic porous media samples with specified porosity and characteristic length scale. Using Lattice-Boltzmann simulations, we compute the permeability tensor for flow through these samples with values spanning three orders of magnitude. We evaluate three families of image-based deep learning models: ResNet (ResNet-$50$ and ResNet-$101$), Vision Transformers (ViT-T$16$ and ViT-S$16$) and ConvNeXt (Tiny and Small). To improve model generalisation, we employ techniques such as weight decay, learning rate scheduling, and data augmentation. The effect of data augmentation and dataset size on model performance is studied, and we find that they generally increase the accuracy of permeability predictions. We also show that ConvNeXt and ResNet converge faster than ViT and degrade in performance if trained for too long. ConvNeXt-Small achieved the highest $R^2$ score of $0.99460$ on $4,000$ unseen test samples. These findings underscore the potential to use image-based neural networks to predict permeability tensors accurately.

💡 Research Summary

This paper investigates the use of image‑based deep neural networks to predict the permeability tensor of two‑dimensional porous media directly from binary microstructure images. The authors generate a synthetic dataset of 24,000 periodic porous samples, each represented as a 2‑D binary image where solid pixels are 1 and void pixels are 0. For every sample, a high‑fidelity Lattice‑Boltzmann simulation is performed to compute the full 2 × 2 permeability tensor, yielding values that span three orders of magnitude (≈10⁻⁶ – 10⁻³). The dataset is split into training, validation, and a held‑out test set of 4,000 images.

Three families of modern computer‑vision architectures are evaluated: (i) Residual Convolutional Networks (ResNet‑50 and ResNet‑101), (ii) Vision Transformers (ViT‑T16 and ViT‑S16), and (iii) ConvNeXt (Tiny and Small). All models receive the same 224 × 224 input size and are adapted to output three continuous values corresponding to the independent components of the symmetric permeability tensor. Training employs the AdamW optimizer with a cosine learning‑rate schedule, an initial learning rate of 1e‑3, and weight decay of 1e‑4. To improve generalisation, the authors apply extensive data augmentation (random rotations, flips, scaling, and Gaussian noise) and investigate the impact of dataset size (6 k, 12 k, 24 k samples).

Results show that ConvNeXt‑Small achieves the highest predictive performance, reaching an R² of 0.99460 on the unseen test set. ResNet‑101 follows closely with R² ≈ 0.993, while ConvNeXt‑Tiny and ResNet‑50 remain above 0.990. Vision Transformers lag behind: ViT‑S16 converges more slowly, requires many more epochs (≈120) and suffers from over‑fitting if trained too long, leading to a drop in validation R². In contrast, ConvNeXt and ResNet models converge rapidly (≈50–70 epochs) and begin to degrade only after excessive training. Data augmentation consistently improves accuracy by 0.003–0.007 in R², and increasing the training set from 6 k to 24 k samples yields comparable gains, confirming that both diversity and quantity of data are beneficial.

The study demonstrates that deep learning can replace costly Lattice‑Boltzmann simulations for permeability prediction, delivering orders‑of‑magnitude faster inference while maintaining near‑perfect correlation with ground‑truth values. The authors discuss practical implications for real‑time flow modelling, reservoir optimisation, and rapid screening of porous designs. They also outline future work, including extension to three‑dimensional microstructures, incorporation of non‑periodic boundary conditions, multi‑phase flow scenarios, and physics‑informed loss functions to further embed Darcy‑scale constraints into the learning process. Overall, the paper provides a comprehensive benchmark and clear evidence that modern convolutional architectures, particularly ConvNeXt, are currently the most effective tools for image‑to‑property mapping in porous media.