Rapid estimation of permeability from digital rock using 3D convolutional neural network


Permeability and its anisotropy are of central importance for groundwater and hydrocarbon migration. Existing fluid dynamics methods for computing permeability have common shortcomings, i.e., high computational complexity and long computational time, reducing the potential of these methods in practical applications. In view of this, a 3D CNN-based approach for rapidly estimating permeability in anisotropic rock is proposed. Using high-resolution X-ray microtomographic images of a sandstone sample, numerous samples of the size of 100-cube voxels were generated firstly by a series of image manipulation techniques. The shrinking and expanding algorithms are employed as the data augmentation methods to strengthen the role of porosity and specific surface area (SSA) since these two parameters are critical to estimate permeability. Afterwards, direct pore-scale modeling with Lattice-Boltzmann method (LBM) was utilized to compute the permeabilities in the direction of three coordinate axes and mean permeability as the ground truth. A dataset including 3158 samples for training and 57 samples for testing were obtained. Four 3D CNN models with the same network structure, corresponding to permeabilities in 3 directions and in average, were built and trained. Based on those trained models, the satisfactory predictions of the permeabilities in x-, y-, and z-axis directions and the mean permeability were achieved with R2 scores of 0.8972, 0.8821, 0.8201, and 0.9155, respectively. Furthermore, those proposed 3D CNN models achieved good generalization ability in predicting the permeability of other samples. The trained model takes only tens of milliseconds on average to predict the permeability of one sample in one axial direction, about 10,000 times faster than LBM. The promising performance clearly demonstrates the effectiveness of 3D CNN-based approach in rapidly estimating permeability in anisotropic rock. This new approach provides an alternative way to calculate permeability with low computing cost, and it has the potential to be extended to the estimation of relative permeability and other properties of rocks.

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  1. 1.

    Blunt, M.J., et al.: Pore-scale imaging and modelling. Adv. Water Resour. 51, 197–216 (2013)

    Article  Google Scholar 

  2. 2.

    Wildenschild, D., Sheppard, A.P.: X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv. Water Resour. 51, 217–246 (2013)

    Article  Google Scholar 

  3. 3.

    Andrä, H., et al.: Digital rock physics benchmarks—part II: computing effective properties. Comput. Geosci. 50, 33–43 (2013)

    Article  Google Scholar 

  4. 4.

    Bultreys, T., De Boever, W., Cnudde, V.: Imaging and image-based fluid transport modeling at the pore scale in geological materials: a practical introduction to the current state-of-the-art. Earth Sci. Rev. 155, 93–128 (2016)

    Article  Google Scholar 

  5. 5.

    Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181(3), 375–389 (1977)

    Article  Google Scholar 

  6. 6.

    Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30(1), 329–364 (1998)

    Article  Google Scholar 

  7. 7.

    Martys, N.S., Chen, H.: Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics. 53(1), 743–750 (1996)

    Google Scholar 

  8. 8.

    Kandhai, D., et al.: A comparison between lattice-Boltzmann and finite-element simulations of fluid flow in static mixer reactors. Int. J. Mod. Phys. C. 9(08), 1123–1128 (1998)

    Article  Google Scholar 

  9. 9.

    Meakin, P. and Tartakovsky, A. M.: Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media. Rev. Geophys.. 47(3) (2009)

  10. 10.

    Koroteev, D., et al.: Direct hydrodynamic simulation of multiphase flow in porous rock. Petrophysics. 55(04), 294–303 (2014)

    Google Scholar 

  11. 11.

    Yang, J., Crawshaw, J., Boek, E.S.: Quantitative determination of molecular propagator distributions for solute transport in homogeneous and heterogeneous porous media using lattice Boltzmann simulations. Water Resour. Res. 49(12), 8531–8538 (2013)

    Article  Google Scholar 

  12. 12.

    Yoon, H., Kang, Q., Valocchi, A.J.: Lattice Boltzmann-based approaches for pore-scale reactive transport. Rev. Mineral. Geochem. 80(1), 393–431 (2015)

    Article  Google Scholar 

  13. 13.

    Joekar-Niasar, V., et al.: Trapping and hysteresis in two-phase flow in porous media: a pore-network study. Water Resour. Res. 49(7), 4244–4256 (2013)

    Article  Google Scholar 

  14. 14.

    Blunt, M.J.: Flow in porous media — pore-network models and multiphase flow. Curr. Opin. Colloid Interface Sci. 6(3), 197–207 (2001)

    Article  Google Scholar 

  15. 15.

    Vogel, H.J., Roth, K.: Quantitative morphology and network representation of soil pore structure. Adv. Water Resour. 24(3), 233–242 (2001)

    Article  Google Scholar 

  16. 16.

    Sok, R.M., et al.: Direct and stochastic generation of network models from tomographic images; effect of topology on residual saturations. Transp. Porous Media. 46(2), 345–371 (2002)

    Article  Google Scholar 

  17. 17.

    Kozeny, J.: Vber kapillare leitung des wassers im boden. Sitzungsber Akad. Wiss. Wien. 136(2a), 271–306 (1927)

  18. 18.

    Carman, P.C.: Permeability of saturated sands, soils and clays. J. Agric. Sci. 29(2), 262–273 (1939)

    Article  Google Scholar 

  19. 19.

    Xu, P., Yu, B.: Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry. Adv. Water Resour. 31(1), 74–81 (2008)

    Article  Google Scholar 

  20. 20.

    Ozgumus, T., Mobedi, M., Ozkol, U.: Determination of Kozeny constant based on porosity and pore to throat size ratio in porous medium with rectangular rods. Eng Appl Comput Fluid Mech. 8(2), 308–318 (2014)

    Google Scholar 

  21. 21.

    Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2(4), 303–314 (1989)

    Article  Google Scholar 

  22. 22.

    Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Netw. 4(2), 251–257 (1991)

    Article  Google Scholar 

  23. 23.

    Sonoda, S., Murata, N.: Neural network with unbounded activation functions is universal approximator. Appl. Comput. Harmon. Anal. 43(2), 233–268 (2017)

    Article  Google Scholar 

  24. 24.

    Shaham, U., Cloninger, A., Coifman, R.R.: Provable approximation properties for deep neural networks. Appl. Comput. Harmon. Anal. 44(3), 537–557 (2018)

    Article  Google Scholar 

  25. 25.

    Hong, J., et al.: Classification of cerebral microbleeds based on fully-optimized convolutional neural network. Multimed. Tools Appl., (2018)

  26. 26.

    Hong, J., et al.: Improvement of cerebral microbleeds detection based on discriminative feature learning. Fund Inform. 168(2–4), 231–248 (2019)

    Article  Google Scholar 

  27. 27.

    Wang, S.-H., Hong, J. and Yang, M.: Sensorineural hearing loss identification via nine-layer convolutional neural network with batch normalization and dropout. Multimed. Tools Appl. (2018)

  28. 28.

    Hong, J., et al.: Detecting cerebral microbleeds with transfer learning. Mach. Vis. Appl. (2019)

  29. 29.

    Wang, S.-H., et al.: Alcoholism identification via convolutional neural network based on parametric ReLU, dropout, and batch normalization. Neural Comput. & Applic. (2018)

  30. 30.

    Zhang, Y., et al.: Adaptive convolutional neural network and its application in face recognition. Neural. Process. Lett. 43(2), 389–399 (2016)

    Article  Google Scholar 

  31. 31.

    Wu, Y., et al.: Deep Convolutional Neural Network with Independent Softmax for Large Scale Face Recognition, in Proceedings of the 24th ACM International Conference on Multimedia, ACM: Amsterdam, The Netherlands. p. 1063–1067 (2016)

  32. 32.

    Guo, S., Chen, S., and Li, Y.: Face recognition based on convolutional neural network and support vector machine. in 2016 IEEE International Conference on Information and Automation (ICIA). (2016)

  33. 33.

    Cecen, A., et al.: Material structure-property linkages using three-dimensional convolutional neural networks. Acta Mater. 146, 76–84 (2018)

    Article  Google Scholar 

  34. 34.

    Yang, Z., et al.: Deep learning approaches for mining structure-property linkages in high contrast composites from simulation datasets. Comput. Mater. Sci. 151, 278–287 (2018)

    Article  Google Scholar 

  35. 35.

    Cang, R., et al.: Improving direct physical properties prediction of heterogeneous materials from imaging data via convolutional neural network and a morphology-aware generative model. Comput. Mater. Sci. 150, 212–221 (2017)

    Article  Google Scholar 

  36. 36.

    Mosser, L., Dubrule, O., Blunt, M.J.: Reconstruction of three-dimensional porous media using generative adversarial neural networks. Phys. Rev. E. 96(4), 043309 (2017)

    Article  Google Scholar 

  37. 37.

    Wu, J., Yin, X., Xiao, H.: Seeing permeability from images: fast prediction with convolutional neural networks. Sci. Bull. 63(18), 53–60 (2018)

    Article  Google Scholar 

  38. 38.

    Sudakov, O., Burnaev, E., Koroteev, D.: Driving digital rock towards machine learning: predicting permeability with gradient boosting and deep neural networks. Comput. Geosci. 127, 91–98 (2019)

    Article  Google Scholar 

  39. 39.

    Urban, G., et al.: Multi-modal brain tumor segmentation using deep convolutional neural networks. MICCAI BraTS (Brain Tumor Segmentation) Challenge. Proceedings, winning contribution: p. 31–35 (2014)

  40. 40.

    Turaga, S.C., Murray, J.F., Jain, V., Roth, F., Helmstaedter, M., Briggman, K., Denk, W., Seung, H.S.: Convolutional networks can learn to generate affinity graphs for image segmentation. Neural Comput. 22(2), 511–538 (2010)

    Article  Google Scholar 

  41. 41.

    Dou, Q., et al.: Automatic detection of cerebral microbleeds from MR images via 3D convolutional neural networks. IEEE Trans. Med. Imaging. 35(5), 1182–1195 (2016)

    Article  Google Scholar 

  42. 42.

    Zewei, D., et al.: Investigation of different skeleton features for CNN-based 3D action recognition. in 2017 IEEE International Conference on Multimedia & Expo Workshops (ICMEW). (2017)

  43. 43.

    Kim, J., et al.: Learning spectro-temporal features with 3D CNNs for speech emotion recognition. in 2017 Seventh International Conference on Affective Computing and Intelligent Interaction (ACII). IEEE (2017)

  44. 44.

    Wu, M., Xiao, F., Johnson-Paben, R.M., Retterer, S.T., Yin, X., Neeves, K.B.: Single-and two-phase flow in microfluidic porous media analogs based on Voronoi tessellation. Lab Chip. 12(2), 253–261 (2012)

    Article  Google Scholar 

  45. 45.

    Newman, M.S., Yin, X.: Lattice Boltzmann simulation of non-Darcy flow in stochastically generated 2D porous media geometries. SPE J. 18(01), 12–26 (2013)

    Article  Google Scholar 

  46. 46.

    Yong, Y., et al.: Direct simulation of the influence of the pore structure on the diffusion process in porous media. Comput. Math. Appl. 67(2), 412–423 (2014)

    Article  Google Scholar 

  47. 47.

    Stauffer, D., Aharony, A.: Introduction to percolation theory. Taylor & Francis, London (1992)

    Google Scholar 

  48. 48.

    Liu, J., Pereira, G.G., Regenauer-Lieb, K.: From characterisation of pore-structures to simulations of pore-scale fluid flow and the upscaling of permeability using microtomography: a case study of heterogeneous carbonates. J. Geochem. Explor. 144, 84–96 (2014)

    Article  Google Scholar 

  49. 49.

    Liu, J., Regenauer-Lieb, K.: Application of percolation theory to microtomography of structured media: percolation threshold, critical exponents, and upscaling. Phys. Rev. E. 83(1), 016106 (2011)

    Article  Google Scholar 

  50. 50.

    Ma, X., Haimson, B.C.: Failure characteristics of two porous sandstones subjected to true triaxial stresses. J. Geophys. Res. Solid Earth. 121(9), 6477–6498 (2016)

    Article  Google Scholar 

  51. 51.

    Terada, K., et al.: Simulation of the multi-scale convergence in computational homogenization approaches. Int. J. Solids Struct. 37(16), 2285–2311 (2000)

    Article  Google Scholar 

  52. 52.

    Pelissou, C., et al.: Determination of the size of the representative volume element for random quasi-brittle composites. Int. J. Solids Struct. 46(14), 2842–2855 (2009)

    Article  Google Scholar 

  53. 53.

    Liu, J., et al.: Improved Estimates of Percolation and Anisotropic Permeability from 3-D X-Ray Microtomography Using Stochastic Analyses and Visualization. Geochem. Geophys. Geosyst.. 10(5) (2009)

    Google Scholar 

  54. 54.

    Krizhevsky, A., Sutskever, I., Hinton, G.E.: ImageNet classification with deep convolutional neural networks. Commun. ACM. 60(6), 84–90 (2017)

    Article  Google Scholar 

  55. 55.

    Wang, S., et al.: Cerebral micro-bleed detection based on the convolution neural network with rank based average pooling. IEEE Access. 5, 16576–16583 (2017)

    Article  Google Scholar 

  56. 56.

    Wang, S.-H., Lv, Y.D., Sui, Y., Liu, S., Wang, S.J., Zhang, Y.D.: Alcoholism detection by data augmentation and convolutional neural network with stochastic pooling. J. Med. Syst. 42(1), 2 (2017)

    Article  Google Scholar 

  57. 57.

    Liu, J., et al.: Applications of microtomography to multiscale system dynamics: visualisation, characterisation and high performance computation. In: Yuen, D.A., et al. (eds.) GPU Solutions to Multi-scale Problems in Science and Engineering, pp. 653–674. Springer Berlin Heidelberg, Berlin (2013)

    Google Scholar 

  58. 58.

    Keehm, Y., Mukerji, T., and Nur, A.: Permeability prediction from thin sections: 3D reconstruction and Lattice-Boltzmann flow simulation. Geophys. Res. Lett.. 31(4) (2004)

  59. 59.

    Wu, K., et al.: 3D stochastic modelling of heterogeneous porous media – applications to reservoir rocks. Transp. Porous Media. 65(3), 443–467 (2006)

    Article  Google Scholar 

  60. 60.

    Manwart, C., et al.: Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media. Phys. Rev. E. 66(1), 016702 (2002)

    Article  Google Scholar 

  61. 61.

    Fredrich, J. T., DiGiovanni, A. A., and Noble, D. R.: Predicting macroscopic transport properties using microscopic image data. J. Geophys. Res. Solid Earth. 111(B3) (2006)

    Article  Google Scholar 

  62. 62.

    Khan, F., et al.: 3D simulation of the permeability tensor in a soil aggregate on basis of nanotomographic imaging and LBE solver. J. Soils Sediments. 12(1), 86–96 (2012)

    Article  Google Scholar 

  63. 63.

    Shah, S.M., et al.: Micro-computed tomography pore-scale study of flow in porous media: effect of voxel resolution. Adv. Water Resour. 95, 276–287 (2016)

    Article  Google Scholar 

  64. 64.

    Yoon, H., Dewers, T.A.: Nanopore structures, statistically representative elementary volumes, and transport properties of chalk. Geophys. Res. Lett. 40(16), 4294–4298 (2013)

    Article  Google Scholar 

  65. 65.

    Talon, L., et al.: Assessment of the two relaxation time Lattice-Boltzmann scheme to simulate Stokes flow in porous media. Water Resour. Res.. 48(4) (2012)

  66. 66.

    Qian, Y.H., Orszag, S.A.: Lattice BGK models for the Navier-Stokes equation: nonlinear deviation in compressible regimes. Europhys Lett. 21(3), 255–259 (1993)

    Article  Google Scholar 

  67. 67.

    Tan, J., Sinno, T.R., Diamond, S.L.: A parallel fluid–solid coupling model using LAMMPS and Palabos based on the immersed boundary method. J. Comput. Sci. 25, 89–100 (2018)

    Article  Google Scholar 

  68. 68.

    Szegedy, C., et al.: Going deeper with convolutions. in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). (2015)

  69. 69.

    He, K., et al.: Deep residual learning for image recognition. in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). (2016)

  70. 70.

    Lecun, Y., et al.: Gradient-based learning applied to document recognition. Proc. IEEE. 86(11), 2278–2324 (1998)

    Article  Google Scholar 

  71. 71.

    Dou, Q., Chen, H., Yu, L., Qin, J., Heng, P.A.: Multilevel contextual 3-D CNNs for false positive reduction in pulmonary nodule detection. IEEE Trans. Biomed. Eng. 64(7), 1558–1567 (2017)

    Article  Google Scholar 

  72. 72.

    Nair, V. and Hinton, G. E.: Rectified linear units improve restricted boltzmann machines, in Proceedings of the 27th International Conference on International Conference on Machine Learning, Omnipress: Haifa, Israel. p. 807-814 (2010)

  73. 73.

    Hong, J. and Liu, J.: Cerebral microbleeds detection via convolutional neural network with and without batch normalization, in Frontiers in Intelligent Computing: Theory and Applications, Springer. p. 152–162 (2020)

    Google Scholar 

  74. 74.

    Boureau, Y.-L., Ponce, J., and LeCun, Y.: A theoretical analysis of feature pooling in visual recognition. in Proceedings of the 27th international conference on machine learning (ICML-10). (2010)

  75. 75.

    Zeiler, M.D. and Fergus, R.: Stochastic pooling for regularization of deep convolutional neural networks. arXiv preprint arXiv:1301.3557. (2013)

  76. 76.

    Clavaud, J.-B., et al.: Permeability anisotropy and its relations with porous medium structure. J. Geophys. Res. Solid Earth. 113(B1) (2008)

  77. 77.

    Ketkar, N.: Stochastic gradient descent, in Deep Learning with Python: A Hands-on Introduction, Apress: Berkeley, CA. p. 113-132 (2017)

  78. 78.

    Che, Y., et al.: Petascale scramjet combustion simulation on the Tianhe-2 heterogeneous supercomputer. Parallel Comput. 77, 101–117 (2018)

    Article  Google Scholar 

  79. 79.

    Karpatne, A., et al.: Physics-guided neural networks (pgnn): an application in lake temperature modeling. arXiv preprint arXiv:1710.11431. (2017)

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Thanks to Dr. Ma Xiaodong for providing microCT images of sandstone samples.


This paper is supported by the National Natural Science Foundation of China (41574087).

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Hong, J., Liu, J. Rapid estimation of permeability from digital rock using 3D convolutional neural network. Comput Geosci (2020). https://doi.org/10.1007/s10596-020-09941-w

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  • Anisotropic rock
  • Digital rock physics
  • Permeability estimation
  • 3D convolutional neural network
  • Shrinking and expanding