Predicting Effective Diffusivity of Porous Media from Images by Deep Learning

Haiyi Wu; Wen-Zhen Fang; Qinjun Kang; Wen-Quan Tao; Rui Qiao

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Predicting Effective Diffusivity of Porous Media from Images by Deep Learning

Haiyi Wu, Wen-Zhen Fang, Qinjun Kang, Wen-Quan Tao, Rui Qiao

arXiv:1912.05532v1Dec 2, 2019physics.geo-pheess.IV

TL;DR

The paper asks whether effective diffusivity of diverse two-dimensional porous media can be predicted accurately without repeatedly performing expensive simulations. It trains CNNs on porous-structure images paired with LBM-computed diffusivities and finds good performance across broad porosity and topology ranges, while identifying reduced accuracy at very low diffusivity.

Problem
Accurately predicting transport properties for diverse porous media remains challenging because existing approaches can be computationally costly or insufficiently specific.
Method
The study trains CNN models using binary porous-media images and effective diffusivities computed with lattice Boltzmann simulations.
Results
Effective diffusivity spanning 0.1 ≲ De < 1 is predicted well for porous media with porosity 0.28–0.98 and complex topologies.
Takeaways & Limitations
Image-based deep learning can predict diffusivity of realistic, diverse porous media with good accuracy and substantially lower computational cost than direct calculations.
Takeaways & Limitations
The current model has limitations for porous media with very low effective diffusivity, motivating further approaches to address this regime.

Abstract

from arXiv · show

We report the application of machine learning methods for predicting the effective diffusivity (De) of two-dimensional porous media from images of their structures. Pore structures are built using reconstruction methods and represented as images, and their effective diffusivity is computed by lattice Boltzmann (LBM) simulations. The datasets thus generated are used to train convolutional neural network (CNN) models and evaluate their performance. The trained model predicts the effective diffusivity of porous structures with computational cost orders of magnitude lower than LBM simulations. The optimized model performs well on porous media with realistic topology, large variation of porosity (0.28-0.98), and effective diffusivity spanning more than one order of magnitude (), e.g., >95% of predicted De have truncated relative error of <10% when the true De is larger than 0.2. The CNN model provides better prediction than the empirical Bruggeman equation, especially for porous structure with small diffusivity. The relative error of CNN predictions, however, is rather high for structures with De < 0.1. To address this issue, the porosity of porous structures is encoded directly into the neural network but the performance is enhanced marginally. Further improvement, i.e., 70% of the CNN predictions for structures with true De < 0.1 have relative error <30%, is achieved by removing trapped regions and dead-end pathways using a simple algorithm. These results suggest that deep learning augmented by field knowledge can be a powerful technique for predicting the transport properties of porous media. Directions for future research of machine learning in porous media are discussed based on detailed analysis of the performance of CNN models in the present work.

I. Introduction

The paper addresses the challenge of predicting porous-media transport properties accurately and cheaply for diverse structures. It proposes image-based deep learning, particularly CNNs, to extract structural features and predict effective diffusivity efficiently.

Predicting transport properties for millions of porous structures can make direct numerical evaluation prohibitively expensive.
Empirical correlations and effective-medium theories are inexpensive but can be inaccurate for complex porous media because they use few structural parameters.
The central challenge is achieving low computational cost and high accuracy across diverse porous structures.
CNNs can extract geometrical features from binary images and predict transport properties without repeating expensive simulations for new structures.
This study develops a CNN framework for effective diffusivity across porosities of 0.28–0.98 and structures containing tortuous, trapped, and dead-end pathways.
Most porous media are predicted with less than 10% truncated relative error, but relative error increases when true diffusivity is especially low, De < 0.1.

II. Computational framework

The computational framework generates image–diffusivity data, constructs and trains CNN models, and evaluates their predictions on held-out porous structures.

The framework comprises dataset generation, CNN architecture design, and training and testing procedures.

A. Generation of datasets

The study generates two-dimensional binary porous-media images over broad porosity and topology ranges, computes their effective diffusivities with LBM, and partitions the resulting data for CNN development and evaluation.

Dataset generation: Two-dimensional porous structures are generated as binary images in which each pixel represents pore space or solid phase.
Dataset generation: The dataset contains 1,960 samples spanning porosity ε = 0.28–0.98, with 28 samples generated at each porosity.
Dataset generation: The generated samples include tortuous pathways, trapped regions, and dead-end pores, particularly at porosity below 0.5.
Effective diffusivity calculation: Effective diffusivity is computed by solving diffusion with the two-dimensional D2Q9 lattice Boltzmann model under imposed concentration and periodic boundary conditions.
Effective diffusivity calculation: The dimensionless effective diffusivity De spans approximately 10^-10 to 1.0.
Model evaluation: The data are randomly divided into training, validation, and testing sets containing 60%, 11.4%, and 28.6% of the samples, respectively.

B. Convolutional neural network for predicting effective diffusivity

The paper develops a CNN that predicts effective diffusivity from 2D porous-media images. Its architecture extracts image features through convolutional and pooling layers, then combines them with optional porosity information before producing the diffusivity prediction.

The CNN predicts porous-media effective diffusivity from 2D binary structure images.
The model uses convolutional and pooling layers to extract feature maps from the input image before fully connected layers make the final prediction.The architecture is adapted from AlexNet-like CNNs and contains pairs of convolutional and pooling layers followed by fully connected layers.
Convolutional filters learn spatial features through ReLU-activated convolutions, while filter weights and biases are learned during CNN training.
The illustrated CNN uses two convolutional layers with feature-map depths of 32 and 64, followed by max pooling with a 4 × 4 filter and stride 4.
Using field knowledge to augment CNN model: Porosity can be added to the flattened feature representation before the first fully connected layer as field-informed input.The final model uses a fixed porosity weight of 10/4096 after testing several weights with similar results.
Using field knowledge to augment CNN model: A preprocessing option removes trapped pore spaces through connected-component analysis before images enter the CNN.The framework also generates ground-truth diffusivities with LBM simulations and uses training, validation, and testing datasets.

C. Parameterization of convolutional neural networks

CNN hyperparameters are selected through cross-validation where feasible, while learnable weights and biases are optimized on the training data by minimizing mean squared error.

The study varies convolutional-layer count and filter size during cross-validation to reduce overfitting.Many other hyperparameters are selected empirically because exhaustive cross-validation would be prohibitively expensive.
The CNN learns convolutional and fully connected weights and biases from the training dataset.
The loss function is the average mean square error over the entire training dataset.
Weights are initialized from a truncated normal distribution, biases from 0.1, and training uses Adam with learning rate γ = 10^-4 for 1600 epochs.
The trained model saves the optimized parameters and restores them to evaluate the testing dataset without retraining.

III. Results and Discussions

The results section evaluates CNN hyperparameters, compares the regular CNN with an empirical model, and examines field-knowledge-informed CNNs across porous structures.

The study first examines key hyperparameters through cross-validation, then compares regular CNN performance with a classical empirical model.
The study also evaluates a field-knowledge-informed CNN over a wide range of porous structures.

A. Cross-validation for hyperparameters selection

Cross-validation shows that CNN performance depends on convolutional-layer count and filter size. The selected model uses two convolutional layers with 5 × 5 filters because it balances error and computational cost.

For a fixed filter size, mean square error is smaller for CNNs with fewer convolutional layers.
For a fixed number of convolutional layers, 3 × 3 filters produce the highest mean square error, likely because they capture fewer moderate-to-large-scale features.
CNNs with 5 × 5 and 7 × 7 filters have comparable performance, while 7 × 7 filters require higher computational cost.
The adopted architecture has two convolutional layers with 5 × 5 filters.

B. Performance of the regular CNN model

The regular CNN predicts effective diffusivity across diverse porous structures with strong absolute-error performance, but relative errors increase as true diffusivity decreases. Compared with the Bruggeman equation, the CNN performs better for structures with complex, low-diffusivity transport pathways.

~1.5×108 FLOPs enables CNN prediction, compared with roughly one hour required by LBM simulations.
8.64 × 10−4 MSE is achieved across the entire testing dataset despite diverse pore shapes and topologies.
The Bruggeman equation performs comparably above 𝐷𝑒~0.6 but overpredicts low diffusivity and performs worse than the CNN.
Over 95% of predictions in each diffusivity group have absolute error smaller than 0.1.
~95% of predictions for 0.2 < 𝐷𝑒< 0.6 have truncated relative error below 10%, whereas ~60% for 𝐷𝑒< 0.1 meet that threshold.
41.7% mean truncated relative error from a truncated-relative-error loss improves on the regular CNN’s 68.6%, but favors samples with 𝐷𝑒<0.2 over those with 𝐷𝑒>0.8.

C.1. Porosity-informed CNN model

The porosity-informed CNN adds overall porosity as an input, but this produces only marginal improvement because porosity does not capture complex transport features such as dead-end pathways and trapped space.

Model design: The model introduces overall porosity as an additional input to the CNN's first fully connected layer.This design supplies a global physical parameter alongside image-derived features.
Performance: Both MSE and truncated relative error decrease by about 12% relative to the regular CNN.The improvement is reported for the testing dataset.
Performance: The porosity-informed model does not greatly alleviate errors for structures with D_e≲0.1.Low-diffusivity structures remain a difficult prediction regime.
Interpretation: Porosity is an unreliable indicator of D_e at low diffusivity because dead-end pathways and trapped space also hinder transport.These structural features extend beyond the overall amount of pore space.
Interpretation: Adding porosity may be redundant because convolutional feature maps can already encode it, while local filters may miss large-scale structure.The paper links this redundancy and limited global-feature extraction to the marginal performance gain.

C.2. The CNN model using preprocessed porous structure as input

Preprocessing removes trapped and dead-end pore regions before CNN inference, substantially improving predictions, especially for samples with very small effective diffusivity.

Preprocessing: The preprocessing retains components connected across the inlet and outlet and removes trapped pores and dead-end pores.The remaining connected pore space becomes the effective pore space used to reconstruct the input image.
Results: The preprocessed images produce CNN predictions in good agreement with LBM predictions.The model is trained using the preprocessed images and their corresponding D_e values.
Results: The MSE falls to 6.92×10^-4, 20% below the regular CNN model.This comparison is reported for the CNN trained on preprocessed structures.
Results: Over 70% of samples with D_e<0.1 have truncated relative error within 20%.The preprocessing particularly improves predictions in the very-low-diffusivity regime.
Limitations: About 10% of predictions for D_e<0.1 still have truncated relative error above 200%.These failures occur when true diffusivity is extremely small, around 10^-4 or below.
Interpretation: Accurate prediction of the hardest samples requires capturing both global tortuous pathways and local critical throats.CNN convolutional layers may struggle because they primarily retrieve relatively local characteristics.

IV. Conclusions

The study develops CNN models for effective diffusivity prediction from porous-media images and finds good accuracy at much lower computational cost, while identifying low-diffusivity and three-dimensional settings as important boundaries.

Contribution: The work develops CNN models for predicting the effective diffusivity of 2D porous media from binary images.The study evaluates model performance using MSE, mean truncated relative error, and error distributions across diffusivity ranges.
Results: Six orders of magnitude lower computational cost is reported relative to direct pore-scale calculations.This cost comparison accompanies the image-based prediction approach.
Results: Effective diffusivity is predicted well for realistic porous media spanning 0.1≲D_e<1 and porosity 0.28-0.98.The evaluated structures include complex and diverse topologies.
Model refinement: Porosity incorporation yields marginal improvement, whereas removing trapped pore space and dead-end pathways improves CNN predictions.The preprocessing-based improvement is presented as a field-knowledge augmentation.
Limitations: The present model performs poorly for porous media with very low effective diffusivity.The paper identifies multiscale feature extraction and advanced geometrical encoding as possible directions for improvement.
Future directions: Accurate low-diffusivity prediction requires effective extraction and weighting of geometric features at both global and local scales.The relevant structures may contain a few narrow throats along one or a few tortuous pathways.