Abstract
A large set of 2D random arrays of circular cylinders is generated to perform a statistical study on rarefied gas flow through micro-porous media. The flow regimes in this work lie for Knudsen numbers (Kn) ranging from the continuum to the transition regimes. Arrays are built by randomly placing cylinders with constant diameter with a uniform distribution without overlapping, and are generated for three target porosities. Fluid flow is assumed to be incompressible and isothermal. A modified lattice Boltzmann model is adopted to account for discrete effects, with slip-velocity boundary conditions and a Kn-dependent multi-relaxation time collision operator. The apparent permeability is modeled with Darcy’s law with a Klinkenberg-type relationship and compared with existing correlations. Velocity fields highlight the increasing contribution of fluid flow through small pores with increasing Kn. Numerical results show that porous media randomness leads to an uncertainty on rarefied gas permeability calculation despite the same structural characteristics and may not strictly follow a specific correlation. The influence of a local collision operator based on a local Kn instead of a global one in the numerical model is also studied. Results show that the permeability in rarefied regimes undergoes significant deviation when applying the local collision operator compared to the global one. These differences could result from a more accurate capture of the pore-scale behavior with a local Kn. Thus, it emphasizes the sensitivity of the model and the apparent permeability calculation to the appropriate definition of Kn.
Similar content being viewed by others
References
Agrawal, A.: A comprehensive review on gas flow in microchannels. Int. J. Micro-Nano Scale Transp. 2, 1 (2012)
Ansumali, S., Karlin, I.V.: Kinetic boundary conditions in the lattice Boltzmann method. Phys. Rev. E 66(2), 026311 (2002)
Arlemark, E.J., Dadzie, S.K., Reese, J.M.: An extension to the Navier–Stokes equations to incorporate gas molecular collisions with boundaries. J. Heat Transf. 132(4), 041006 (2010)
Avramenko, A., Kovetska, Y., Shevchuk, I., Tyrinov, A., Shevchuk, V.: Heat transfer in porous microchannels with second-order slipping boundary conditions. Transp. Porous Media 129(3), 673–699 (2019)
Azhdari, A., Talebi, F., Valipour, M.S.: Investigation of pore-scale random porous media using lattice Boltzmann method. J. Heat Mass Transf. Res. 2(1), 1–12 (2015)
Beard, D., Weyl, P.: Influence of texture on porosity and permeability of unconsolidated sand. AAPG Bull. 57(2), 349–369 (1973)
Beskok, A., Karniadakis, G.E.: Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng. 3(1), 43–77 (1999)
Borner, A., Panerai, F., Mansour, N.N.: High temperature permeability of fibrous materials using direct simulation Monte Carlo. Int. J. Heat Mass Transf. 106, 1318–1326 (2017)
Bosl, W.J., Dvorkin, J., Nur, A.: A study of porosity and permeability using a lattice Boltzmann simulation. Geophys. Res. Lett. 25(9), 1475–1478 (1998)
Cancelliere, A., Chang, C., Foti, E., Rothman, D.H., Succi, S.: The permeability of a random medium: comparison of simulation with theory. Phys. Fluids A: Fluid Dyn. 2(12), 2085–2088 (1990)
Carman, P.C.: Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15, 150–166 (1937)
Carman, P.C.: Permeability of saturated sands, soils and clays. J. Agric. Sci. 29(2), 262–273 (1939)
Cercignani, C.: Mathematical Methods in Kinetic Theory. Springer, Boston (1969)
Chai, Z., Shi, B., Guo, Z., Lu, J.: Gas flow through square arrays of circular cylinders with Klinkenberg effect: a lattice Boltzmann study. Commun. Comput. Phys. 8(5), 1052 (2010a)
Chai, Z., Shi, B., Lu, J., Guo, Z.: Non-darcy flow in disordered porous media: a lattice Boltzmann study. Comput. Fluids 39(10), 2069–2077 (2010b)
Chai, Z., Lu, J., Shi, B., Guo, Z.: Gas slippage effect on the permeability of circular cylinders in a square array. Int. J. Heat Mass Transfer 54(13–14), 3009–3014 (2011)
Chen, L., Kang, Q., Dai, Z., Viswanathan, H.S., Tao, W.: Permeability prediction of shale matrix reconstructed using the elementary building block model. Fuel 160, 346–356 (2015a)
Chen, L., Kang, Q., Pawar, R., He, Y.L., Tao, W.Q.: Pore-scale prediction of transport properties in reconstructed nanostructures of organic matter in shales. Fuel 158, 650–658 (2015b)
Chen, L., Zhang, L., Kang, Q., Viswanathan, H.S., Yao, J., Tao, W.: Nanoscale simulation of shale transport properties using the lattice Boltzmann method: permeability and diffusivity. Sci. Rep. 5(1), 1–8 (2015c)
Chen, L., Zhang, R., Min, T., Kang, Q., Tao, W.: Pore-scale study of effects of macroscopic pores and their distributions on reactive transport in hierarchical porous media. Chem. Eng. J. 349, 428–437 (2018)
Childs, E.C., Collis-George, N.: The permeability of porous materials. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 201(1066), 392–405 (1950)
Coskun, S.B., Wardlaw, N.C.: Estimation of permeability from image analysis of reservoir sandstones. J. Pet. Sci. Eng. 10(1), 1–16 (1993)
Costa, V., Oliveira, L., Baliga, B., Sousa, A.: Simulation of coupled flows in adjacent porous and open domains using a control-volume finite-element method. Numer. Heat Transf. Part A: Appl. 45(7), 675–697 (2004)
Dongari, N., Zhang, Y., Reese, J.M.: Modeling of Knudsen layer effects in micro/nanoscale gas flows. J. Fluids Eng. 133(7), 071101 (2011)
Endruweit, A., Long, A.C.: Influence of stochastic variations in the fibre spacing on the permeability of bi-directional textile fabrics. Compos. Part A: Appl. Sci. Manuf. 37(5), 679–694 (2006)
Endruweit, A., Long, A.C., Robitaille, F., Rudd, C.D.: Influence of stochastic fibre angle variations on the permeability of bi-directional textile fabrics. Compos. Part A: Appl. Sci. Manuf. 37(1), 122–132 (2006)
Fraser, H.: Experimental study of the porosity and permeability of clastic sediments. J. Geol. 43(8, Part 1), 910–1010 (1935)
Fryer, G.: A theory of gas flow through capillary tubes. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 293(1434), 329–341 (1966)
Guo, Z., Shu, C.: Lattice Boltzmann method and its applications in engineering, vol. 3. World Scientific, Singapore (2013)
Guo, Z., Zhao, T.: Lattice Boltzmann model for incompressible flows through porous media. Phys. Rev. E 66(3), 036304 (2002)
Guo, Z., Zheng, C.: Analysis of lattice Boltzmann equation for microscale gas flows: relaxation times, boundary conditions and the Knudsen layer. Int. J. Comput. Fluid Dyn. 22(7), 465–473 (2008)
Guo, Z., Shi, B., Zhao, T., Zheng, C.: Discrete effects on boundary conditions for the lattice Boltzmann equation in simulating microscale gas flows. Phys. Rev. E 76(5), 056704 (2007)
Guo, Z., Zheng, C., Shi, B.: Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow. Phys. Rev. E 77(3), 036707 (2008)
Ho, M., Leclaire, S., Reggio, M., Trépanier, J.Y.: Investigation of advection-diffusion problems and simulations using the lattice Boltzmann method and the arrayfire library for HPC on GPU. Oral presentation at CFDSC 2019, London, CANADA (2019a)
Ho, M., Pérez, J.G., Reggio, M., Trépanier, J.Y.: Permeability calculation of rarefied gas flows through 2D porous structures using the lattice Boltzmann method. Phys. Chem. Earth, Parts A/B/C 113, 43–49 (2019b)
Ho, M., Leclaire, S., Trépanier, J.Y., Reggio, M., Martin, A.: Permeability calculation of a fibrous thermal insulator using the lattice Boltzmann method (2020)
Javadpour, F., et al.: Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Pet. Technol. 48(08), 16–21 (2009)
Jeong, N., Choi, D.H., Lin, C.L.: Prediction of Darcy–Forchheimer drag for micro-porous structures of complex geometry using the lattice Boltzmann method. J. Micromech. Microeng. 16(10), 2240 (2006)
Klinkenberg, L., et al.: The permeability of porous media to liquids and gases. In: Drilling and production practice, American Petroleum Institute (1941)
Landry, C.J., Prodanović, M., Eichhubl, P.: Direct simulation of supercritical gas flow in complex nanoporous media and prediction of apparent permeability. Int. J. Coal Geol. 159, 120–134 (2016)
Lange, K.J., Sui, P.C., Djilali, N.: Pore scale simulation of transport and electrochemical reactions in reconstructed PEMFC catalyst layers. J. Electrochem. Soc. 157(10), B1434–B1442 (2010)
Latt, J., Chopard, B.: Lattice Boltzmann method with regularized pre-collision distribution functions. Math. Comput. Simul. 72(2–6), 165–168 (2006)
Latt, J., Chopard, B., Malaspinas, O., Deville, M., Michler, A.: Straight velocity boundaries in the lattice Boltzmann method. Phys. Rev. E 77(5), 056703 (2008)
Li, J., Brown, D., Calo, V., Efendiev, Y., Iliev, O., et al.: Multiscale lattice Boltzmann method for flow simulations in highly heterogenous porous media. In: SPE Reservoir Characterization and Simulation Conference and Exhibition, Society of Petroleum Engineers (2013a)
Li, Q., He, Y., Tang, G., Tao, W.: Lattice Boltzmann modeling of microchannel flows in the transition flow regime. Microfluid. Nanofluidics 10(3), 607–618 (2011)
Li, X., Cai, J., Xin, F., Huai, X., Guo, J.: Lattice Boltzmann simulation of endothermal catalytic reaction in catalyst porous media. Appl. Therm. Eng. 50(1), 1194–1200 (2013b)
Li, X., Chen, J., Xu, M., Huai, X., Xin, F., Cai, J.: Lattice Boltzmann simulation of catalytic reaction in porous media with buoyancy. Appl. Therm. Eng. 70(1), 586–592 (2014)
Liao, Q., Yang, Y.X., Zhu, X., Chen, R., Fu, Q.: Pore-scale lattice Boltzmann simulation of flow and mass transfer in bioreactor with an immobilized granule for biohydrogen production. Sci. Bull. 62(1), 22–30 (2017)
Liu, L., Yao, J., Zhang, L., An, S., Zhao, J., Sun, H.: Rev-scale simulation of micro-fractured unconventional gas reservoir. J. Nat. Gas Sci. Eng. 48, 100–110 (2017)
Liu, X., Guo, Z.: A lattice Boltzmann study of gas flows in a long micro-channel. Comput. Math. Appl. 65(2), 186–193 (2013)
Liu, Z., Wu, H.: Pore-scale study on flow and heat transfer in 3D reconstructed porous media using micro-tomography images. Appl. Therm. Eng. 100, 602–610 (2016)
Ma, Q., Chen, Z.: Lattice Boltzmann simulation of multicomponent noncontinuum diffusion in fractal porous structures. Phys. Rev. E 92(1), 013025 (2015)
Martys, N.S., Chen, H.: Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. Phys. Rev. E 53(1), 743 (1996)
Michalis, V.K., Kalarakis, A.N., Skouras, E.D., Burganos, V.N.: Rarefaction effects on gas viscosity in the Knudsen transition regime. Microfluid. Nanofluidics 9(4–5), 847–853 (2010)
Mohammadmoradi, P., Kantzas, A.: Pore-scale permeability calculation using CFD and DSMC techniques. J. Pet. Sci. Eng. 146, 515–525 (2016)
Montessori, A., Prestininzi, P., La Rocca, M., Succi, S.: Lattice Boltzmann approach for complex nonequilibrium flows. Phys. Rev. E 92(4), 043308 (2015)
Mosavat, N., Hasanidarabadi, B., Pourafshary, P.: Gaseous slip flow simulation in a micro/nano pore-throat structure using the lattice Boltzmann model. J. Pet. Sci. Eng. 177, 93–103 (2019)
Mostaghimi, P., Liu, M., Arns, C.H.: Numerical simulation of reactive transport on micro-CT images. Math. Geosci. 48(8), 963–983 (2016)
Nabovati, A., Sousa, A.: Fluid flow simulation in random porous media at pore level using lattice Boltzmann method. In: New Trends in Fluid Mechanics Research, pp. 518–521. , Springer (2007)
Nabovati, A., Llewellin, E.W., Sousa, A.C.: A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method. Compos. Part A: Appl. Sci. Manuf. 40(6–7), 860–869 (2009)
Ostoja-Starzewski, M.: Material spatial randomness: from statistical to representative volume element. Probab. Eng. Mech. 21(2), 112–132 (2006)
Pan, Y., Kong, S.C.: Simulation of biomass particle evolution under pyrolysis conditions using lattice Boltzmann method. Combust. Flame 178, 21–34 (2017)
Panerai, F., White, J.D., Cochell, T.J., Schroeder, O.M., Mansour, N.N., Wright, M.J., Martin, A.: Experimental measurements of the permeability of fibrous carbon at high-temperature. Int. J. Heat Mass Transf. 101, 267–273 (2016)
Poovathingal, S., Stern, E.C., Nompelis, I., Schwartzentruber, T.E., Candler, G.V.: Nonequilibrium flow through porous thermal protection materials, part II: oxidation and pyrolysis. J. Comput. Phys. 380, 427–441 (2019)
Ren, J., Zheng, Q., Guo, P., Peng, S., Wang, Z., Du, J.: Pore-scale lattice Boltzmann simulation of two-component shale gas flow. J. Nat. Gas Sci. Eng. 61, 46–70 (2019)
Shou, D., Fan, J., Ding, F.: Hydraulic permeability of fibrous porous media. Int. J. Heat Mass Transf. 54(17–18), 4009–4018 (2011)
Stern, E., Nompelis, I., Schwartzentruber, T.E., Candler, G.V.: Microscale simulations of porous TPS materials: application to permeability. In: 11th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, p. 2247 (2014)
Succi, S., Foti, E., Higuera, F.: Three-dimensional flows in complex geometries with the lattice Boltzmann method. EPL (Europhys. Lett.) 10(5), 433 (1989)
Suga, K., Takenaka, S., Ito, T., Kaneda, M., Kinjo, T., Hyodo, S.: Lattice Boltzmann flow simulation in micro-nano transitional porous media. In: 2010 14th International Heat Transfer Conference, American Society of Mechanical Engineers Digital Collection, pp. 321–329 (2010)
Sullivan, S., Sani, F., Johns, M., Gladden, L.: Simulation of packed bed reactors using lattice Boltzmann methods. Chem. Eng. Sci. 60(12), 3405–3418 (2005)
Tucny, J.M., Vidal, D., Leclaire, S., Bertrand, F.: Comparison of existing and extended boundary conditions for the simulation of rarefied gas flows using the lattice Boltzmann method. Int. J. Mod. Phys. C 31(5), 2050070 (2016)
Verhaeghe, F., Luo, L.S., Blanpain, B.: Lattice Boltzmann modeling of microchannel flow in slip flow regime. J. Comput. Phys. 228(1), 147–157 (2009)
Vidal, D., Ridgway, C., Pianet, G., Schoelkopf, J., Roy, R., Bertrand, F.: Effect of particle size distribution and packing compression on fluid permeability as predicted by lattice-Boltzmann simulations. Comput. Chem. Eng. 33(1), 256–266 (2009)
Wang, J., Chen, L., Kang, Q., Rahman, S.S.: The lattice Boltzmann method for isothermal micro-gaseous flow and its application in shale gas flow: a review. Int. J. Heat Mass Transf. 95, 94–108 (2016a)
Wang, J., Kang, Q., Wang, Y., Pawar, R., Rahman, S.S.: Simulation of gas flow in micro-porous media with the regularized lattice Boltzmann method. Fuel 205, 232–246 (2017)
Wang, Z., Jin, X., Wang, X., Sun, L., Wang, M.: Pore-scale geometry effects on gas permeability in shale. J. Nat. Gas Sci. Eng. 34, 948–957 (2016b)
Wu, K., Li, X., Wang, C., Chen, Z., Yu, W.: A model for gas transport in microfractures of shale and tight gas reservoirs. AIChE J. 61(6), 2079–2088 (2015)
Wyckoff, R., Botset, H., Muskat, M., Reed, D.: The measurement of the permeability of porous media for homogeneous fluids. Rev. Sci. Instrum. 4(7), 394–405 (1933)
Xu, Q., Long, W., Jiang, H., Zan, C., Huang, J., Chen, X., Shi, L.: Pore-scale modelling of the coupled thermal and reactive flow at the combustion front during crude oil in-situ combustion. Chem. Eng. J. 350, 776–790 (2018)
Xuan, Y., Zhao, K., Li, Q.: Investigation on mass diffusion process in porous media based on lattice Boltzmann method. Heat Mass Transf. 46(10), 1039–1051 (2010)
Yalamanchili, P., Arshad, U., Mohammed, Z., Garigipati, P., Entschev, P., Kloppenborg, B., Malcolm, J., Melonakos, J.: ArrayFire: a high performance software library for parallel computing with an easy-to-use API. https://github.com/arrayfire/arrayfire (2015)
Yang, G., Weigand, B.: Investigation of the Klinkenberg effect in a micro/nanoporous medium by direct simulation Monte Carlo method. Phys. Rev. Fluids 3(4), 044201 (2018)
Yang, P., Wen, Z., Dou, R., Liu, X.: Effect of random structure on permeability and heat transfer characteristics for flow in 2D porous medium based on MRT lattice Boltzmann method. Phys. Lett. A 380(37), 2902–2911 (2016)
Yang, P., Wen, Z., Dou, R., Liu, X.: Permeability in multi-sized structures of random packed porous media using three-dimensional lattice Boltzmann method. Int. J. Heat Mass Transf. 106, 1368–1375 (2017)
Yin, X., Chen, W., To, A., McVeigh, C., Liu, W.K.: Statistical volume element method for predicting microstructure-constitutive property relations. Comput. Methods Appl. Mech. Eng. 197(43–44), 3516–3529 (2008)
Young, J., Todd, B.: Modelling of multi-component gas flows in capillaries and porous solids. Int. J. Heat Mass Transf. 48(25–26), 5338–5353 (2005)
Zeng, Y., Ning, Z., Wang, Q., Sun, H., Huang, L., Ye, H.: Gas transport in self-affine rough microchannels of shale gas reservoir. J. Pet. Sci. Eng. 167, 716–728 (2018)
Zhang, D., Zhang, R., Chen, S., Soll, W.E.: Pore scale study of flow in porous media: scale dependency, rev, and statistical rev. Geophys. Res. Lett. 27(8), 1195–1198 (2000)
Zhang, T., Sun, S., Song, H.: Flow mechanism and simulation approaches for shale gas reservoirs: a review. Transp. Porous Media 126(3), 655–681 (2019)
Zhang, X., Xiao, L., Shan, X., Guo, L.: Lattice Boltzmann simulation of shale gas transport in organic nano-pores. Sci. Rep. 4, 4843 (2014)
Zhao, J., Yao, J., Li, A., Zhang, M., Zhang, L., Yang, Y., Sun, H.: Simulation of microscale gas flow in heterogeneous porous media based on the lattice Boltzmann method. J. Appl. Phys. 120(8), 084306 (2016a)
Zhao, J., Yao, J., Zhang, M., Zhang, L., Yang, Y., Sun, H., An, S., Li, A.: Study of gas flow characteristics in tight porous media with a microscale lattice Boltzmann model. Sci. Rep. 6, 32393 (2016b)
Zhao, T., Zhao, H., Li, X., Ning, Z., Wang, Q., Zhao, W., Zhang, J.: Pore scale characteristics of gas flow in shale matrix determined by the regularized lattice Boltzmann method. Chem. Eng. Sci. 187, 245–255 (2018)
Zhou, L., Qu, Z., Ding, T., Miao, J.: Lattice Boltzmann simulation of the gas-solid adsorption process in reconstructed random porous media. Phys. Rev. E 93(4), 043101 (2016)
Acknowledgements
Financial support from the Simulation-based Engineering Science (Génie Par la Simulation) program funded through the CREATE program from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Ho, M., Leclaire, S., Reggio, M. et al. Stochastic Effects of 2D Random Arrays of Cylinders on Rarefied Gas Permeability Using the Lattice Boltzmann Method. Transp Porous Med 136, 607–637 (2021). https://doi.org/10.1007/s11242-020-01532-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-020-01532-8