Using Graph Neural Networks to Predict the Permeability of Porous Media
DOI:
https://doi.org/10.69631/ipj.v2i3nr47Keywords:
Porous media, Lattice Boltzmann method, Machine learning, Graph neural networksAbstract
The permeability of porous media is often calculated using correlations or computationally expensive simulations. Several methods have been developed which use neural networks to predict porous media properties, but little work has been done on the development of a model that can handle porous media at the representative elementary volume (REV) scale. This work describes the framework for developing a graph neural network (GNN) to predict the permeability of porous media based on representative pore networks extracted from the structures, rather than representative structure volumes. This allows for consistent input sizes for the neural network, irrespective of the average pore size, which is more difficult when the entire voxelized structure is the model input. A GNN was trained to predict the permeability of porous media based on lattice Boltzmann method (LBM) simulations of the flow through the structures. The GNN showed a good agreement with the LBM simulations over samples with permeabilities spanning several orders of magnitude. The GNN was able to outperform the Carman-Kozeny equation with a mean squared error (MSE) for the unseen testing dataset of 0.00190 and a mean absolute error (MAE) of 0.0302 compared to an MSE of 1.125 and an MAE of 0.783 for the Carman-Kozeny equation, when comparing against the LBM ground truth. The inference time of the GNN alone was several orders of magnitude faster than the LBM simulations, and nearly 10 times faster when including the pore network extraction time needed for the GNN. This work demonstrates the potential of using GNNs to predict the permeability of representative porous media, and the benefits of using model architectures that take pore networks as the inputs.
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1. Akiba, T., Sano, S., Yanase, T., Ohta, T., & Koyama, M. (2019). Optuna: A next-generation hyperparameter optimization framework. Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2623–2631. https://doi.org/10.1145/3292500.3330701
2. Alibrahim, H., & Ludwig, S. A. (2021). Hyperparameter optimization: Comparing genetic algorithm against grid search and bayesian optimization. 2021 IEEE Congress on Evolutionary Computation (CEC), 1551–1559. https://doi.org/10.1109/CEC45853.2021.9504761
3. Alqahtani, N., Alzubaidi, F., Armstrong, R. T., Swietojanski, P., & Mostaghimi, P. (2020). Machine learning for predicting properties of porous media from 2D X-ray images. Journal of Petroleum Science and Engineering, 184, 106514. https://doi.org/10.1016/j.petrol.2019.106514
4. Alzahrani, M. K., Shapoval, A., Chen, Z., & Rahman, S. S. (2023). Pore-GNN: A graph neural network-based framework for predicting flow properties of porous media from micro-CT images. Advances in Geo-Energy Research, 10(1), 39–55. https://doi.org/10.46690/ager.2023.10.05
5. Arlot, S., & Celisse, A. (2010). A survey of cross-validation procedures for model selection. Statistics Surveys, 4 (none). https://doi.org/10.1214/09-SS054
6. Bardenet, R., Brendel, M., Kégl, B., & Sebag, M. (2013). Collaborative hyperparameter tuning. In S. Dasgupta & D. McAllester (Eds.), Proceedings of the 30th International Conference on Machine Learning (Vol. 28, pp. 199–207). PMLR. https://proceedings.mlr.press/v28/bardenet13.html
7. Boccardo, G., Plato, L., Marchisio, D., Augier, F., Haroun, Y., et al. (2014, June). Pore-scale simulation of fluid flow in packed-bed reactors via rigid-body simulations and CFD. https://www.researchgate.net/publication/264897796_PORE-SCALE_SIMULATION_OF_FLUID_FLOW_IN_PACKED-BED_REACTORS_VIA_RIGID-BODY_SIMULATIONS_AND_CFD
8. Bryntesson, L. M. (2002). Pore network modelling of the behaviour of a solute in chromatography media: Transient and steady-state diffusion properties. Journal of Chromatography A, 945(1–2), 103–115. https://doi.org/10.1016/S0021-9673(01)01485-6
9. Buyrukoğlu, S., & Akbaş, A. (2022). Machine learning based early prediction of type 2 diabetes: A new hybrid feature selection approach using correlation matrix with heatmap and SFS. Balkan Journal of Electrical and Computer Engineering, 10(2), 110–117. https://doi.org/10.17694/bajece.973129
10. Cai, C., Vlassis, N., Magee, L., Ma, R., Xiong, Z., et al. (2021). Equivariant geometric learning for digital rock physics: Estimating formation factor and effective permeability tensors from Morse graph. https://doi.org/10.48550/ARXIV.2104.05608
11. Cai, C., & Wang, Y. (2020). A note on over-smoothing for graph neural networks. arXiv. https://doi.org/10.48550/ARXIV.2006.13318
12. Carman, P. C. (1997). Fluid flow through granular beds. Chemical Engineering Research and Design, 75, S32–S48. https://doi.org/10.1016/S0263-8762(97)80003-2
13. Chen, Y., Liu, J., Zhang, X., Qi, X., & Jia, J. (2023). Largekernel3d: Scaling up kernels in 3D sparse CNNS. 2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 13488–13498. https://doi.org/10.1109/CVPR52729.2023.01296
14. Cohen, J. (1977). Differences between correlation coefficients. In Statistical Power Analysis for the Behavioral Sciences (pp. 109–143). Elsevier. https://doi.org/10.1016/B978-0-12-179060-8.50009-8
15. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, Clifford. (2001). Representations of Graphs. In Introduction to Algorithms (2nd ed., pp. 527–531). MIT Press. https://mitpress.mit.edu/9780262531962/introduction-to-algorithms/
16. Cummings, D., & Nassar, M. (2020). Structured citation trend prediction using graph neural networks. ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 3897–3901. https://doi.org/10.1109/ICASSP40776.2020.9054769
17. Dai, M., Demirel, M. F., Liang, Y., & Hu, J.-M. (2021). Graph neural networks for an accurate and interpretable prediction of the properties of polycrystalline materials. Npj Computational Materials, 7(1), 103. https://doi.org/10.1038/s41524-021-00574-w
18. Dong, G., & Liu, H. (Eds.). (2018). Feature engineering for machine learning and data analytics. CRC Press.
19. Duval, A. A., Schmidt, V., Hernández-García, A., Miret, S., Malliaros, F. D., et al. (2023). Faenet: Frame averaging equivariant GNN for materials modeling. In A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, & J. Scarlett (Eds.), Proceedings of the 40th International Conference on Machine Learning (Vol. 202, pp. 9013–9033). PMLR. https://proceedings.mlr.press/v202/duval23a.html
20. Eshghinejadfard, A., Daróczy, L., Janiga, G., & Thévenin, D. (2016). Calculation of the permeability in porous media using the lattice Boltzmann method. International Journal of Heat and Fluid Flow, 62, 93–103. https://doi.org/10.1016/j.ijheatfluidflow.2016.05.010
21. Fan, W., Ma, Y., Li, Q., He, Y., Zhao, E., et al. (2019). Graph neural networks for social recommendation. The World Wide Web Conference, 417–426. https://doi.org/10.1145/3308558.3313488
22. Feurer, M., & Hutter, F. (2019). Hyperparameter optimization. In F. Hutter, L. Kotthoff, & J. Vanschoren (Eds.), Automated Machine Learning: Methods, Systems, Challenges (pp. 3–33). Springer International Publishing. https://doi.org/10.1007/978-3-030-05318-5_1
23. Fey, M., & Lenssen, J. E. (2019). Fast graph representation learning with pytorch geometric. arXiv. https://doi.org/10.48550/ARXIV.1903.02428
24. Flam-Shepherd, D., Wu, T. C., Friederich, P., & Aspuru-Guzik, A. (2021). Neural message passing on high order paths. Machine Learning: Science and Technology, 2(4), 045009. https://doi.org/10.1088/2632-2153/abf5b8
25. Ghanbarian, B., Hunt, A. G., Ewing, R. P., & Sahimi, M. (2013). Tortuosity in porous media: A critical review. Soil Science Society of America Journal, 77(5), 1461–1477. https://doi.org/10.2136/sssaj2012.0435
26. Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., & Dahl, G. E. (2017). Neural message passing for quantum chemistry. arXiv. https://doi.org/10.48550/ARXIV.1704.01212
27. Gong, L., & Cheng, Q. (2019). Exploiting edge features for graph neural networks. 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 9203–9211. https://doi.org/10.1109/CVPR.2019.00943
28. Google-research/tuning_playbook. (2025). [Computer software]. Google Research. https://github.com/google-research/tuning_playbook (Original work published 2023)
29. Gostick, J., Aghighi, M., Hinebaugh, J., Tranter, T., Hoeh, M. A., et al. (2016). OpenPNM: A pore network modeling package. Computing in Science & Engineering, 18(4), 60–74. https://doi.org/10.1109/MCSE.2016.49
30. Gostick, J., Khan, Z., Tranter, T., Kok, M., Agnaou, M., Sadeghi, M., & Jervis, R. (2019). Porespy: A python toolkit for quantitative analysis of porous media images. Journal of Open Source Software, 4(37), 1296. https://doi.org/10.21105/joss.01296
31. Gostick, J. T. (2017). Versatile and efficient pore network extraction method using marker-based watershed segmentation. Physical Review E, 96(2), 023307. https://doi.org/10.1103/PhysRevE.96.023307
32. Guibert, R., Nazarova, M., Horgue, P., Hamon, G., Creux, P., & Debenest, G. (2015). Computational permeability determination from pore-scale imaging: Sample size, mesh and method sensitivities. Transport in Porous Media, 107(3), 641–656. https://doi.org/10.1007/s11242-015-0458-0
33. Hall, M. A. (1999). Correlation-based feature selection for machine learning [The University of Waikato]. https://researchcommons.waikato.ac.nz/handle/10289/15043
34. Han, Y., & Cundall, P. A. (2011). Lattice Boltzmann modeling of pore‐scale fluid flow through idealized porous media. International Journal for Numerical Methods in Fluids, 67(11), 1720–1734. https://doi.org/10.1002/fld.2443
35. Hasanov, A. K., Dugan, B., Batzle, M. L., & Prasad, M. (2019). Hydraulic and poroelastic rock properties from oscillating pore pressure experiments. Journal of Geophysical Research: Solid Earth, 124(5), 4473–4491. https://doi.org/10.1029/2018JB017276
36. Haugen, H. J., & Bertoldi, S. (2017). Characterization of morphology—3D and porous structure. In Characterization of Polymeric Biomaterials (pp. 21–53). Elsevier. https://doi.org/10.1016/B978-0-08-100737-2.00002-9
37. Henderson, N., Brêttas, J. C., & Sacco, W. F. (2010). A three-parameter Kozeny–Carman generalized equation for fractal porous media. Chemical Engineering Science, 65(15), 4432–4442. https://doi.org/10.1016/j.ces.2010.04.006
38. Hewamalage, H., Ackermann, K., & Bergmeir, C. (2023). Forecast evaluation for data scientists: Common pitfalls and best practices. Data Mining and Knowledge Discovery, 37(2), 788–832. https://doi.org/10.1007/s10618-022-00894-5
39. Ioffe, S., & Szegedy, C. (2015). Batch normalization: Accelerating deep network training by reducing internal covariate shift. In F. Bach & D. Blei (Eds.), Proceedings of the 32nd International Conference on Machine Learning (Vol. 37, pp. 448–456). PMLR. https://proceedings.mlr.press/v37/ioffe15.html
40. Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv. https://doi.org/10.48550/ARXIV.1412.6980
41. Kipf, T. N., & Welling, M. (2016). Semi-supervised classification with graph convolutional networks. arXiv. https://doi.org/10.48550/ARXIV.1609.02907
42. Kočí, P., Isoz, M., Plachá, M., Arvajová, A., Václavík, M., et al. (2019). 3D reconstruction and pore-scale modeling of coated catalytic filters for automotive exhaust gas aftertreatment. Catalysis Today, 320, 165–174. https://doi.org/10.1016/j.cattod.2017.12.025
43. Kozeny, j. (1927) Ueber Kapillare Leitung des Wassers im Boden. Sitzungsber akad. Wiss. , Wien, 136(2a), pp 271-306. - References—Scientific research publishing. (n.d.). Retrieved August 16, 2025, from https://www.scirp.org/reference/referencespapers?referenceid=1306904
44. Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., & Viggen, E. M. (2017a). Boundary Conditions for Fluid-Structure Interaction. In The Lattice Boltzmann Method: Principles and Practice (pp. 433–491). Springer International Publishing. https://doi.org/10.1007/978-3-319-44649-3
45. Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., & Viggen, E. M. (2017b). MRT and TRTcollision operators. In T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, & E. M. Viggen (Eds.), The Lattice Boltzmann Method: Principles and Practice (pp. 407–431). Springer International Publishing. https://doi.org/10.1007/978-3-319-44649-3_10
46. Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., & Viggen, E. M. (2017c). Non-dimensionalisation and choice of simulation parameters. In T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, & E. M. Viggen (Eds.), The Lattice Boltzmann Method: Principles and Practice (pp. 265–294). Springer International Publishing. https://doi.org/10.1007/978-3-319-44649-3_7
47. Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., & Viggen, E. M. (2017d). Numerical methods for fluids. In T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, & E. M. Viggen, The Lattice Boltzmann Method (pp. 31–58). Springer International Publishing. https://doi.org/10.1007/978-3-319-44649-3_2
48. Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., & Viggen, E. M. (2017e). The lattice boltzmann equation. In T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, & E. M. Viggen, The Lattice Boltzmann Method (pp. 61–104). Springer International Publishing. https://doi.org/10.1007/978-3-319-44649-3_3
49. Lallemand, P., & Luo, L.-S. (2003). Lattice Boltzmann method for moving boundaries. Journal of Computational Physics, 184(2), 406–421. https://doi.org/10.1016/S0021-9991(02)00022-0
50. Latt, J., Malaspinas, O., Kontaxakis, D., Parmigiani, A., Lagrava, D., et al. (2021). Palabos: Parallel lattice boltzmann solver. Computers & Mathematics with Applications, 81, 334–350. https://doi.org/10.1016/j.camwa.2020.03.022
51. Li, Q., Luo, K. H., Kang, Q. J., He, Y. L., Chen, Q., & Liu, Q. (2016). Lattice Boltzmann methods for multiphase flow and phase-change heat transfer. Progress in Energy and Combustion Science, 52, 62–105. https://doi.org/10.1016/j.pecs.2015.10.001
52. Ly, H. B., Monchiet, V., & Grande, D. (2016). Computation of permeability with Fast Fourier Transform from 3-D digital images of porous microstructures. International Journal of Numerical Methods for Heat & Fluid Flow, 26(5), 1328–1345. https://doi.org/10.1108/HFF-12-2014-0369
53. Malekzadeh, M., Hajibabaee, P., Heidari, M., Zad, S., Uzuner, O., & Jones, J. H. (2021). Review of graph neural network in text classification. 2021 IEEE 12th Annual Ubiquitous Computing, Electronics & Mobile Communication Conference (UEMCON), 0084–0091. https://doi.org/10.1109/UEMCON53757.2021.9666633
54. Marcato, A., Boccardo, G., & Marchisio, D. (2021). A computational workflow to study particle transport and filtration in porous media: Coupling CFD and deep learning. Chemical Engineering Journal, 417, 128936. https://doi.org/10.1016/j.cej.2021.128936
55. Marcato, A., Estrada Santos, J., Boccardo, G., Viswanathan, H., Marchisio, D., & Prodanović, M. (2022). Prediction of local concentration fields in porous media with chemical reaction using a multi scale convolutional neural network. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.4167602
56. McKay, M. D., Beckman, R. J., & Conover, W. J. (1979). Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(2), 239–245. https://doi.org/10.1080/00401706.1979.10489755
57. Mean squared error. (2008). In The Concise Encyclopedia of Statistics (pp. 337–339). Springer New York. https://doi.org/10.1007/978-0-387-32833-1_251
58. Meinicke, S., Dubil, K., Wetzel, T., & Dietrich, B. (2020). Characterization of heat transfer in consolidated, highly porous media using a hybrid-scale CFD approach. International Journal of Heat and Mass Transfer, 149, 119201. https://doi.org/10.1016/j.ijheatmasstransfer.2019.119201
59. Meyers, J. J., & Liapis, A. I. (1999). Network modeling of the convective flow and diffusion of molecules adsorbing in monoliths and in porous particles packed in a chromatographic column. Journal of Chromatography A, 852(1), 3–23. https://doi.org/10.1016/S0021-9673(99)00443-4
60. Mostaghimi, P., Percival, J. R., Pavlidis, D., Ferrier, R. J., Gomes, J. L. M. A., et al. (2015). Anisotropic mesh adaptivity and control volume finite element methods for numerical simulation of multiphase flow in porous media. Mathematical Geosciences, 47(4), 417–440. https://doi.org/10.1007/s11004-014-9579-1
61. Nishiyama, N., & Yokoyama, T. (2017). Permeability of porous media: Role of the critical pore size. Journal of Geophysical Research: Solid Earth, 122(9), 6955–6971. https://doi.org/10.1002/2016JB013793
62. Niu, Z., Pinfield, V. J., Wu, B., Wang, H., Jiao, K., et al. (2021). Towards the digitalisation of porous energy materials: Evolution of digital approaches for microstructural design. Energy & Environmental Science, 14(5), 2549–2576. https://doi.org/10.1039/D1EE00398D
63. Novák, V., Kočí, P., Štěpánek, F., & Marek, M. (2011). Integrated multiscale methodology for virtual prototyping of porous catalysts. Industrial & Engineering Chemistry Research, 50(23), 12904–12914. https://doi.org/10.1021/ie2003347
64. Nowak, A., Villar, S., Bandeira, A. S., & Bruna, J. (2017). Revised note on learning algorithms for quadratic assignment with graph neural networks. arXiv. https://doi.org/10.48550/ARXIV.1706.07450
65. Ortega, A. (2022). Introduction to graph signal processing (1st ed.). Cambridge University Press. https://doi.org/10.1017/9781108552349
66. Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., et al. (2019). Pytorch: An imperative style, high-performance deep learning library. arXiv. https://doi.org/10.48550/ARXIV.1912.01703
67. Petrasch, J., Meier, F., Friess, H., & Steinfeld, A. (2008). Tomography based determination of permeability, Dupuit–Forchheimer coefficient, and interfacial heat transfer coefficient in reticulate porous ceramics. International Journal of Heat and Fluid Flow, 29(1), 315–326. https://doi.org/10.1016/j.ijheatfluidflow.2007.09.001
68. Rabbani, A., Babaei, M., Shams, R., Wang, Y. D., & Chung, T. (2020). DeePore: A deep learning workflow for rapid and comprehensive characterization of porous materials. Advances in Water Resources, 146, 103787. https://doi.org/10.1016/j.advwatres.2020.103787
69. Richter, M. L., Byttner, W., Krumnack, U., Wiedenroth, A., Schallner, L., & Shenk, J. (2021). (Input) Size Matters for CNN Classifiers. In I. Farkaš, P. Masulli, S. Otte, & S. Wermter (Eds.), Artificial neural networks and machine learning – icann 2021: 30th International Conference on Artificial Neural Networks, Bratislava, Slovakia, September 14–17, 2021, Proceedings, Part I (Vol. 12891, pp. 133–144). Springer International Publishing. https://doi.org/10.1007/978-3-030-86362-3
70. Santos, J. E., Yin, Y., Jo, H., Pan, W., Kang, Q., et al. (2021). Computationally efficient multiscale neural networks applied to fluid flow in complex 3d porous media. Transport in Porous Media, 140(1), 241–272. https://doi.org/10.1007/s11242-021-01617-y
71. Schulz, R., Ray, N., Zech, S., Rupp, A., & Knabner, P. (2019). Beyond Kozeny–Carman: Predicting the permeability in porous media. Transport in Porous Media, 130(2), 487–512. https://doi.org/10.1007/s11242-019-01321-y
72. Shorten, C., & Khoshgoftaar, T. M. (2019). A survey on image data augmentation for deep learning. Journal of Big Data, 6(1), 60. https://doi.org/10.1186/s40537-019-0197-0
73. Singh, A., Regenauer‐Lieb, K., Walsh, S. D. C., Armstrong, R. T., Van Griethuysen, J. J. M., & Mostaghimi, P. (2020). On representative elementary volumes of grayscale micro‐CT images of porous media. Geophysical Research Letters, 47(15), e2020GL088594. https://doi.org/10.1029/2020GL088594
74. Tarokh, A., Mohamad, A. A., & Jiang, L. (2013). Simulation of conjugate heat transfer using the lattice boltzmann method. Numerical Heat Transfer, Part A: Applications, 63(3), 159–178. https://doi.org/10.1080/10407782.2012.725009
75. Wagner, A., Eggenweiler, E., Weinhardt, F., Trivedi, Z., Krach, D., et al. (2021). Permeability estimation of regular porous structures: A benchmark for comparison of methods. Transport in Porous Media, 138(1), 1–23. https://doi.org/10.1007/s11242-021-01586-2
76. Wang, H., Yin, Y., Hui, X. Y., Bai, J. Q., & Qu, Z. G. (2020). Prediction of effective diffusivity of porous media using deep learning method based on sample structure information self-amplification. Energy and AI, 2, 100035. https://doi.org/10.1016/j.egyai.2020.100035
77. Wang, M., & Pan, N. (2007). Numerical analyses of effective dielectric constant of multiphase microporous media. Journal of Applied Physics, 101(11), 114102. https://doi.org/10.1063/1.2743738
78. Wang, M., Wang, J., Pan, N., Chen, S., & He, J. (2007). Three-dimensional effect on the effective thermal conductivity of porous media. Journal of Physics D: Applied Physics, 40(1), 260–265. https://doi.org/10.1088/0022-3727/40/1/024
79. Wu, H., Fang, W.-Z., Kang, Q., Tao, W.-Q., & Qiao, R. (2019). Predicting effective diffusivity of porous media from images by deep learning. Scientific Reports, 9(1), 20387. https://doi.org/10.1038/s41598-019-56309-x
80. Wu, Z., Pan, S., Chen, F., Long, G., Zhang, C., & Yu, P. S. (2021). A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems, 32(1), 4–24. https://doi.org/10.1109/TNNLS.2020.2978386
81. Xiong, Q., Baychev, T. G., & Jivkov, A. P. (2016). Review of pore network modelling of porous media: Experimental characterisations, network constructions and applications to reactive transport. Journal of Contaminant Hydrology, 192, 101–117. https://doi.org/10.1016/j.jconhyd.2016.07.002
82. Yagis, E., Atnafu, S. W., García Seco De Herrera, A., Marzi, C., Scheda, R., et al. (2021). Effect of data leakage in brain MRI classification using 2D convolutional neural networks. Scientific Reports, 11(1), 22544. https://doi.org/10.1038/s41598-021-01681-w
83. Yasuda, T., Ookawara, S., Yoshikawa, S., & Matsumoto, H. (2021). Machine learning and data-driven characterization framework for porous materials: Permeability prediction and channeling defect detection. Chemical Engineering Journal, 420, 130069. https://doi.org/10.1016/j.cej.2021.130069
84. Ye, L., Lv, W., Zhang, K. H. L., Wang, X., Yan, P., et al. (2015). A new insight into the oxygen diffusion in porous cathodes of lithium-air batteries. Energy, 83, 669–673. https://doi.org/10.1016/j.energy.2015.02.072
85. Zhang, Y., Yang, Z., Wang, F., & Zhang, X. (2021). Comparison of soil tortuosity calculated by different methods. Geoderma, 402, 115358. https://doi.org/10.1016/j.geoderma.2021.115358
86. Zhao, Q., Han, X., Guo, R., & Chen, C. (2025). A computationally efficient hybrid neural network architecture for porous media: Integrating convolutional and graph neural networks for improved property predictions (No. arXiv:2311.06418). arXiv. https://doi.org/10.48550/arXiv.2311.06418
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