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

TL;DR

The paper investigates predicting permeability tensors of 2D porous media directly from binary images using three neural architectures—ResNet, ViT, and ConvNeXt. It leverages a large synthetic dataset generated via Gaussian-smoothed random fields and ground-truth permeability from Lattice-Boltzmann simulations, with extensive augmentation to enhance generalization. ConvNeXt-Small delivers the highest predictive accuracy (test R^2 ≈ 0.9946), while ViTs require more data and longer training to approach CNN-like performance; the study also highlights the benefits of dihedral-group data augmentation and analyzes convergence behavior. The results demonstrate that image-based DL offers a fast, accurate alternative to flow simulations for estimating permeability tensors and point toward extensions to larger domains and 3D media.

Abstract

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.

Figures (9)

• Figure 1: A synthetic porous medium with periodic geometry. The colouring indicates relative flow speed. The shaded area contains parts of periodic images of the domain.
• Figure 2: Distribution of permeability component values over porosity for all $24,000$ samples. The permeability tensor values are $K_{ij}$ for $i,j=x$ or $y$. $KC$ is the Kozeny-Carman fit to the data.
• Figure 3: The 8 elements of the dihedral group $D_4$ acting on the synthetic porous medium.
• Figure 4: Plot of validation $1-R^2$ score for ConvNeXt and ViT in the tiny and small configuration over different sizes of the dataset.
• Figure 5: Best (lowest) $1-R^2$ value obtained for each training run over the total number of epochs the model was trained for.