Skip to main content

Evaluation of Absolute Permeability in Heterogeneous and Anisotropic Porous Media Using the Lattice Boltzmann Simulations


This paper presents a systematical study of the effect of porosity, pore-level heterogeneity and anisotropy on the absolute permeability of digital images of porous media. The main goal is to develop an analytical formula that estimates permeability as a function of these three parameters at once. Permeability is assessed based on numerical simulations using the lattice Boltzmann equations. Digital models of porous media are generated by a combined method consisting of Monte-Carlo and quartet structure generation set (QSGS) algorithms. Increase in heterogeneity negatively affects permeability. With an increase in porosity, the effect of heterogeneity on flow properties decreases. There was a linear decrease in permeability during the transition between favorable and unfavorable anisotropy. The influence of anisotropy is most pronounced in samples with high porosity and monotonically reduces with decreasing porosity. Heterogeneity negatively influences on the sensitivity of flow properties to changes in anisotropy and independent on porosity.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. P. Carman, ‘‘Permeability of saturated sands, soils and clays,’’ J. Agricult. Sci. 29, 262–273 (1939).

    Article  Google Scholar 

  2. J. Kozeny, ‘‘Uber Kapillare Leitung des Wassers im Boden,’’ Ber. Wien Akad. 136A, 271–306 (1927).

    Google Scholar 

  3. P. Mostaghimi, M. J. Blunt, and B. Bijeljic, ‘‘Computations of absolute permeability on Micro-CT images,’’ Math. Geosci. 45, 103–125 (2013).

    Article  MathSciNet  Google Scholar 

  4. B. R. Gebart, ‘‘Permeability of unidirectional reinforcements for RTM,’’ J. Compos. Mater. 26, 1100–1133 (1992).

    Article  Google Scholar 

  5. A. Eshghinejadfard, L. Daróczy, G. Janiga, and D. Thévenin, ‘‘Calculation of the permeability in porous media using the lattice Boltzmann method,’’ Int. J. Heat Fluid Flow 62, 93–103 (2016).

    Article  Google Scholar 

  6. A. Ebrahimi Khabbazi, J. S. Ellis, and A. Bazylak, ‘‘Developing a new form of the Kozeny–Carman parameter for structured porous media through lattice-Boltzmann modeling,’’ Comput. Fluid 75 (20), 35–41 (2013).

    Article  Google Scholar 

  7. A. Koponen, M. Kataja, and J. Timonen, ‘‘Permeability and effective porosity of porous media,’’ Phys. Rev. E 56, 3319–3325 (1997).

    Article  Google Scholar 

  8. H. Rumpf and A. R. Gupte, ‘‘Influence of porosity and particle size distribution in resistance of porous flow,’’ Chem. Ing. Tech. 43, 33–34 (1971).

    Article  Google Scholar 

  9. A. Nabovati, E. W. Llewellin, and A. C. M. Soussa, ‘‘A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method,’’ Composites, Part A 40, 860–869 (2009).

    Article  Google Scholar 

  10. A. Nabovati, E. W. Llewellin, and A. C. M. Soussa, ‘‘Fluid flow simulation in random porous media at pore level using lattice Boltzmann method,’’ J. Eng. Sci. Technol. 2, 226–237 (2007).

    Google Scholar 

  11. A. Koponen, M. Kataja, and J. Timonen, ‘‘Tortuous flow in porous media,’’ Phys. Rev. E 54, 406–410 (1996).

    Article  Google Scholar 

  12. Sh. Zhang, H. Yan, J. Teng, and D. Sheng, ‘‘A mathematical model of tortuosity in soil considering particle arrangement,’’ Vadose Zone J. 19, e24 (2020).

  13. T. Li, Min Li, X. Jing, W. Xiao, and Q. Cui, ‘‘Influence mechanism of pore-scale anisotropy and pore distribution heterogeneity on permeability of porous media,’’ Pet. Explor. Developm. 46, 594–604 (2019).

    Article  Google Scholar 

  14. Z. Wang, X. Jin, X. Wang, L. Sun, and M. Wang, ‘‘Pore-scale geometry effects on gas permeability in shale,’’ J. Nat. Gas Sci. Eng. 34, 948–957 (2016).

    Article  Google Scholar 

  15. L. Germanou, M. T. Ho, Y. Zhang, and L. Wu, ‘‘Intrinsic and apparent gas permeability of heterogeneous and anisotropic ultra-tight porous media,’’ J. Nat. Gas Sci. Eng. 60, 271–283 (2018).

    Article  Google Scholar 

  16. W. Sobieski, ‘‘Numerical investigations of tortuosity in randomly generated pore structures,’’ Math. Comput. Simul. 166, 1–20 (2019).

    Article  MathSciNet  Google Scholar 

  17. P. A. Slotte, C. F. Berg, and H. H. Khanamiri, ‘‘Predicting resistivity and permeability of porous media using Minkowski functionals,’’ Transp. Porous Media 131, 705–722 (2020).

    Article  MathSciNet  Google Scholar 

  18. S. M. Shah, F. Gray, J. P. Crawshaw, and E. S. Boek, ‘‘Micro-computed tomography pore-scale study of flow in porous media: Effect of Voxel resolution,’’ Adv. Water Resour. 95, 276–287 (2015).

    Article  Google Scholar 

  19. P. Yang, Z. Wena, R. Dou, and X. Liu, ‘‘Permeability in multi-sized structures of random packed porous media using three-dimensional lattice Boltzmann method,’’ Int. J. Heat Mass Transfer 106, 1368–1375 (2017).

    Article  Google Scholar 

  20. M. Wang, J. Wang, N. Pan, and Sh. Chen, ‘‘Mesoscopic predictions of the effective thermal conductivity for microscale random porous media,’’ Phys. Rev. E 75, 036702 (2007).

  21. T. R. Zakirov and M. G. Khramchenkov, ‘‘Prediction of permeability and tortuosity in heterogeneous porous media using a disorder parameter,’’ Chem. Eng. Sci. 227, 115893 (2020).

  22. H. Laubie, S. Monfared, F. Radjai, R. Pellenq, and F.-J. Ulm, ‘‘Disorder-induced stiffness degradation of highly disordered porous materials,’’ J. Mech. Phys. Solids 106, 207–228 (2017).

    Article  MathSciNet  Google Scholar 

  23. S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond (Oxford Univ. Press, UK, 2001).

    MATH  Google Scholar 

  24. T. R. Zakirov and A. A. Galeev, ‘‘Absolute permeability calculations in micro-computed tomography models of sandstones by Navier–Stokes and lattice Boltzmann equations,’’ Int. J. Heat Mass Transfer 129, 415–426 (2019).

    Article  Google Scholar 

  25. M. J. Blunt, B. Bijeljic, H. Dong, O. Gharbi, S. Iglauer, P. Mostaghimi, A. Paluszny, and C. Pentland, ‘‘Pore-scale imaging and modeling,’’ Adv. Water Resour. 51, 197–216 (2013).

    Article  Google Scholar 

  26. T. R. Zakirov, A. A. Galeev, E. O. Statsenko, and L. I. Khaidarova, ‘‘Calculation of filtration characteristics of porous media by their digitized images,’’ J. Eng. Phys. Thermophys. 91, 1069–1078 (2018).

    Article  Google Scholar 

  27. C. Pan, L. S. Luo, and C. T. Miller, ‘‘An evaluation of lattice Boltzmann schemes for porous medium flow simulation,’’ Comput. Fluids 35, 898–909 (2006).

    Article  Google Scholar 

  28. E. Aslan, I. Taymaz, and A. C. Benim, ‘‘Investigation of the lattice Boltzmann SRT and MRT stability for lid driven cavity flow,’’ Int. J. Mater. Mech. Manuf. 2, 317–324 (2014).

    Google Scholar 

  29. Q. Zou and X. He, ‘‘On pressure and velocity boundary conditions for the lattice Boltzmann BGK model,’’ Phys. Fluids 9, 1591–1598 (1997).

    Article  MathSciNet  Google Scholar 

Download references


This study was supported by the Russian Science Foundation, project no. 19-77-00019.

Author information

Authors and Affiliations


Corresponding authors

Correspondence to T. R. Zakirov, A. N. Kolchugin, A. A. Galeev or M. G. Khramchenkov.

Additional information

(Submitted by A. M. Elizarov)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zakirov, T.R., Kolchugin, A.N., Galeev, A.A. et al. Evaluation of Absolute Permeability in Heterogeneous and Anisotropic Porous Media Using the Lattice Boltzmann Simulations. Lobachevskii J Math 42, 3048–3059 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • absolute permeability
  • heterogeneity
  • anisotropy
  • porous media
  • tortuosity
  • Kozeny–Carman equation
  • lattice Boltzmann equations