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.
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P. Carman, ‘‘Permeability of saturated sands, soils and clays,’’ J. Agricult. Sci. 29, 262–273 (1939).
J. Kozeny, ‘‘Uber Kapillare Leitung des Wassers im Boden,’’ Ber. Wien Akad. 136A, 271–306 (1927).
P. Mostaghimi, M. J. Blunt, and B. Bijeljic, ‘‘Computations of absolute permeability on Micro-CT images,’’ Math. Geosci. 45, 103–125 (2013).
B. R. Gebart, ‘‘Permeability of unidirectional reinforcements for RTM,’’ J. Compos. Mater. 26, 1100–1133 (1992).
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).
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).
A. Koponen, M. Kataja, and J. Timonen, ‘‘Permeability and effective porosity of porous media,’’ Phys. Rev. E 56, 3319–3325 (1997).
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).
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).
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).
A. Koponen, M. Kataja, and J. Timonen, ‘‘Tortuous flow in porous media,’’ Phys. Rev. E 54, 406–410 (1996).
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).
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).
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).
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).
W. Sobieski, ‘‘Numerical investigations of tortuosity in randomly generated pore structures,’’ Math. Comput. Simul. 166, 1–20 (2019).
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).
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).
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).
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).
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).
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).
S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond (Oxford Univ. Press, UK, 2001).
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).
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).
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).
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).
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).
Q. Zou and X. He, ‘‘On pressure and velocity boundary conditions for the lattice Boltzmann BGK model,’’ Phys. Fluids 9, 1591–1598 (1997).
This study was supported by the Russian Science Foundation, project no. 19-77-00019.
(Submitted by A. M. Elizarov)
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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). https://doi.org/10.1134/S1995080221120404
- absolute permeability
- porous media
- Kozeny–Carman equation
- lattice Boltzmann equations