Source-linked AI summary
Predicting Porosity, Permeability, and Tortuosity of Porous Media from Images by Deep Learning
Krzysztof M. Graczyk, Maciej Matyka
TL;DR
The paper addresses the costly computation of porous-media transport properties from obstacle configurations. It uses CNNs to predict porosity, permeability, and tortuosity from two-dimensional images, achieving good accuracy and reproducing the tortuosity–porosity relation qualitatively.
Problem
Computing porous-media transport properties is time- and resource-consuming, motivating a faster approach to connect obstacle configurations with porosity, permeability, and tortuosity.
Method
CNNs are trained on binary images of two-dimensional porous systems to predict porosity, permeability, and tortuosity, using flow-derived quantities as targets.
Results
For a large fraction of samples, prediction errors are below 6%, and the models qualitatively reproduce the empirical tortuosity–porosity relation.
Takeaways & Limitations
CNNs can predict the three fundamental porous-media quantities from images and reconstruct the previously observed tortuosity–porosity dependence.
Abstract
from arXiv · showhide
Convolutional neural networks (CNN) are utilized to encode the relation between initial configurations of obstacles and three fundamental quantities in porous media: porosity ($\varphi$), permeability $k$, and tortuosity ($T$). The two-dimensional systems with obstacles are considered. The fluid flow through a porous medium is simulated with the lattice Boltzmann method. It is demonstrated that the CNNs are able to predict the porosity, permeability, and tortuosity with good accuracy. With the usage of the CNN models, the relation between $T$ and $\varphi$ has been reproduced and compared with the empirical estimate. The analysis has been performed for the systems with $\varphi \in (0.37,0.99)$ which covers five orders of magnitude span for permeability $k \in (0.78, 2.1\times 10^5)$ and tortuosity $T \in (1.03,2.74)$.
Introduction
Porosity, permeability, and tortuosity describe transport through porous media but are costly to compute from complex pore structures. The paper uses CNNs to predict these quantities from obstacle images and examines their relation to tortuosity and porosity.
- Macroscopic transport properties depend on the microscopic structure and connectivity of the pore space.
- Porosity measures void fraction, permeability reflects fluid-transport ability, and tortuosity characterizes particle paths through porous media.
- Computing flow in realistic porous geometries is time- and resource-consuming, motivating a CNN-based shortcut.
- The CNN takes binary obstacle images as input and predicts the vector (ϕ,k,T), with flow-simulation outputs supplying permeability and tortuosity.
- The best model keeps prediction errors below 6% and reproduces the tortuosity–porosity dependence in qualitative agreement with empirical fits.
1 Lattice Boltzmann method
The lattice Boltzmann method generates pore-scale flow fields and the labels used for learning porous-medium properties. Simulations resolve velocity in the pore space under specified obstacle and boundary conditions.
- The dataset pairs binary obstacle images with porosity, permeability, and tortuosity labels calculated from LBM flow simulations.
- LBM transports a density distribution function according to a discrete Boltzmann equation, from which the velocity field is obtained.
- Obstacles are completely solid and pores completely permeable, so the velocity field is solved only in pore space.
- The simulations start from zero velocity and apply an external gravity force to drive flow toward steady state.
- Velocity-magnitude visualizations use brighter colors for larger pore-scale velocities and highlight the resulting flow paths.
- Porous samples depict black obstacles and interconnected light-gray fluid regions; inaccessible gaps are filled before solver and network input.
2 Deep learning approach
The approach uses CNNs to map binary obstacle images to porosity, permeability, and tortuosity, with architectures and preprocessing designed for porous-media data. Two analyses vary input size and network architecture, and training uses normalized labels, reweighting, and validation-based model selection.
- CNN approach: CNNs encode obstacle configurations into porosity, permeability, and tortuosity predictions.The networks analyze images of porous structures and output the three target quantities.
- Data generation: 100,000 binary 800×400 images were generated, with 200-pixel margins removed so CNN inputs were effectively 400×400.The margins reduce boundary effects in simulations but contain no information used by the network.
- Data preparation: Samples were reweighted using permeability–tortuosity bins to improve coverage of the label space during training.Each bin received a weight based on the total sample count divided by its population, and training sampled from the resulting distribution.
- Network architectures: The netA architecture has six convolutional blocks and two fully connected layers, while netB has seven convolutional blocks and two fully connected layers.Pooling appears in the convolutional blocks, and batch normalization is used in selected blocks of both networks.
- Data preparation: Labels were transformed toward zero, and permeability was represented using log(k) before rescaling.The final settings used ϕs = 1.0, ϕ = 0.5, Ts = 2.8, T = 0.5, log(ks) = 12.3, and log(k) = 0.5.
- Training and evaluation: Analysis A uses resized inputs, whereas analysis B uses original 400×400 inputs, trading faster training against retaining more image information.The data were split into 85% training and 15% validation sets; models were optimized with MSE and mini-batch SGD.
3 Results and summary
CNN models predicted porosity, permeability, and tortuosity from two-dimensional obstacle images, with generally good agreement against solver values. They also reproduced the empirical tortuosity–porosity relationship, while performance was weaker for low permeability and high tortuosity.
- Model settings: The final CNN settings used SGD with momentum 0.9 and a learning rate reduced by 10% every 50 training epochs.The study selected hyperparameters after testing various configurations.
- Model architecture: The two architectures increased filter counts with depth while reducing input size; netB additionally used large K = 10 kernels for long-distance correlations.Both architectures produced a final CNN-layer vector of length 400.
- Evaluation: Predictions were compared with lattice-Boltzmann solver values for porosity, permeability, and tortuosity using training and validation data.The figures also included histograms of the prediction-to-true-value ratio, summarized by mean and variance.
- Prediction quality: The networks showed good agreement overall, but low permeability and tortuosity T > 1.75 were difficult to predict accurately.Re-weighting partially addressed these difficulties, and batch normalization provided another improvement.
- Prediction quality: For analysis A, a large fraction of samples had relative prediction differences below 6%, with porosity and tortuosity differences below 1%.NetB matched netA for porosity and tortuosity but produced rather uncertain permeability predictions.
- Tortuosity–porosity relation: The reconstructed T(ϕ) relation agreed qualitatively with the empirical fit, although finite-size effects underestimated tortuosity near the percolation threshold.The study used 1,300 unlabelled samples for this reconstruction.
- Summary: Overall, CNNs predicted porosity, permeability, and tortuosity from images of two-dimensional porous systems and reconstructed the empirical tortuosity–porosity dependence.The study compared resized-input and original-size-input networks.
Author contributions statement
K.G. and M.M. jointly conceived, conducted, wrote, and reviewed the project, while K.G. led the deep-learning analysis and M.M. led fluid-flow simulations.
- K.G. designed and performed the deep-learning analysis, while M.M. designed and performed the fluid-flow simulations.
- K.G. and M.M. conceived and conducted the project and jointly wrote and reviewed the manuscript.