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Scaling Invariant Effects on the Permeability of Fractal Porous Media

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Abstract

Porous media are interconnected systems, in which the distribution of pore sizes might follow scaling invariant property and will affect the fluid flow through it significantly. Thus, except for a detailed understanding of the fundamental mechanism at pore scale, hard is it to determine the appropriate relationship between the permeability and the basic properties. In this study, in terms of the size distribution and spatial arrangement of the pores, we analytically derived a permeability model using series–parallel flow resistance mode firstly. And then, together with the scaling invariant characteristics of the porosity, specific area and hydraulic tortuosity, the analytical permeability model is reformulated into a fractal permeability–pore structure relationship. The results indicate that: (1) the square of the porosity (\(\varphi \)) is proportional to the permeability in a fractal porous media, not the cubic law described in Kozeny–Carman (KC) equation; (2) the hydraulic tortuosity is a power law model of the minimum particle size with the exponent \(\epsilon (D_{\mathrm{f}}-d)\), where \(D_{\mathrm{f}}\) and d are the pore size fractal dimension and Euclidean space dimension, respectively, while \(\epsilon \) is a parameter characterizing the spatial arrangement of pores; (3) the KC numerical prefactor is not a constant in fractal porous media. Its value, however, increases linearly with the size ratio of the minimum to the maximum pores but decreases exponentially with \(D_{\mathrm{f}}\). More importantly, it is found to be a parameter characterizing the difference of fluid flow in porous media from that in a straight tube described by the Poiseuille law. The performance of the new fractal permeability–pore model is verified by lattice Boltzmann simulations, and the numerical prefactor universality is examined as well.

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References

  • Adler, P.M., Thovert, J.F.: Fractal porous media. Transp. Porous Med. 13, 41–78 (1993)

    Article  Google Scholar 

  • Adler, P.M., Thovert, J.: Real porous media: local geometry and macroscopic properties. Appl. Mech. Rev. 51, 537–585 (1998)

    Article  Google Scholar 

  • Ahmadi, M.M., Mohammadi, S., Hayati, A.N.: Analytical derivation of tortuosity and permeability of monosized spheres: a volume averaging approach. Phys. Rev. E 83, 026312 (2011)

    Article  Google Scholar 

  • Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)

    Google Scholar 

  • Bear, J., Verruijt, A.: Modeling Groundwater Flow and Pollution. Springer, New York (1987)

    Book  Google Scholar 

  • Cai, J.C., Yu, B.M., Zou, M.Q., Mei, M.F.: Fractal analysis of surface roughness of particles in porous media. Chin. Phys. Lett. 27, 024705 (2010)

    Article  Google Scholar 

  • Carman, P.C.: Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15, 150–166 (1937)

    Google Scholar 

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

    Article  Google Scholar 

  • Carman, P.C.: Flow of Gases Through Porous Media. Butterworths Scientific Publications, London (1956)

    Google Scholar 

  • Chen, Q., Zhang, X.B., Zhang, J.F.: Improved treatments for general boundary conditions in the lattice Boltzmann method for convection–diffusion and heat transfer processes. Phys. Rev. E 88, 033304 (2013)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Collins, R.E.: Flow of Fluids Through Porous Materials. Reinhold Pub. Corp, New York (1961)

    Google Scholar 

  • Comiti, J., Renaud, M.: A new model for determining mean structure parameters of fixed beds from pressure drop measurements: application to beds packed with parallelepipedal particles. Chem. Eng. Sci. 44, 1539–1545 (1989)

    Article  Google Scholar 

  • Costa, A.: Permeability–porosity relationship: a reexamination of the Kozeny–Carman equation based on a fractal pore-space geometry assumption. Geophys. Res. Lett. 33, L2318 (2006)

    Article  Google Scholar 

  • Croce, G., D’Agaro, P., Nonino, C.: Three-dimensional roughness effect on microchannel heat transfer and pressure drop. Int. J. Heat Mass Transf. 50, 5249–5259 (2007)

    Article  Google Scholar 

  • Dathe, A., Thullner, M.: The relationship between fractal properties of solid matrix and pore space in porous media. Geoderma 129, 279–290 (2005)

    Article  Google Scholar 

  • Degruyter, W., Burgisser, A., Bachmann, O., Malaspinas, O.: Synchrotron X-ray microtomography and lattice Boltzmann simulations of gas flow through volcanic pumices. Geosphere 6, 470–481 (2010)

    Article  Google Scholar 

  • Duda, A., Koza, Z., Matyka, M.: Hydraulic tortuosity in arbitrary porous media flow. Phys. Rev. E 84, 036319 (2011)

    Article  Google Scholar 

  • Dullien, F.A.: Porous Media: Fluid Transport and Pore Structure. Academic Press, Salt Lake City (1991)

    Google Scholar 

  • Dünweg, B., Schiller, U.D., Ladd, A.J.C.: Statistical mechanics of the fluctuating lattice Boltzmann equation. Phys. Rev. E 76, 036704 (2007)

    Article  Google Scholar 

  • Feng, Y.F., Yu, B.M.: Fractal dimension for tortuous streamtubes in porous media. Fractals 15, 385–390 (2007)

    Article  Google Scholar 

  • Ghanbarian, B., Hunt, A.G., Ewing, R.P., Sahimi, M.: Tortuosity in porous media: a critical review. Soil Sci. Soc. Am. J. 77, 1461–1477 (2013)

    Article  Google Scholar 

  • Ghanbarian, B., Hunt, A.G., Sahimi, M., Ewing, R.P., Skinner, T.E.: Percolation theory generates a physically based description of tortuosity in saturated and unsaturated porous media. Soil Sci. Soc. Am. J. 77, 1920–1929 (2013)

    Article  Google Scholar 

  • Ghanbarian-Alavijeh, B., Hunt, A.G.: Comments on More general capillary pressure and relative permeability models from fractal geometry by Kewen Li. J. Contam. Hydrol. 140–141, 21–23 (2012)

    Article  Google Scholar 

  • Ghanbarian-Alavijeh, B., Hunt, A.: Unsaturated hydraulic conductivity in porous media: percolation theory. Geoderma 187–188, 77 (2012)

    Article  Google Scholar 

  • Guarracino, L.: Estimation of saturated hydraulic conductivity Ks from the van Genuchten shape parameter. Water Resour. Res. 43, W11502 (2007)

    Google Scholar 

  • Happel, J.: Viscous flow relative to arrays of cylinders. AiChE J. 5, 174–177 (1959)

    Article  Google Scholar 

  • Henderson, N., Brêttas, J.C., Sacco, W.F.: A three-parameter Kozeny–Carman generalized equation for fractal porous media. Chem. Eng. Sci. 65, 4432–4442 (2010)

    Article  Google Scholar 

  • Jin, Y., Song, H.B., Hu, B., Zhu, Y.B., Zheng, J.L.: Lattice Boltzmann simulation of fluid flow through coal reservoir’s fractal pore structure. Sci. China Earth Sci. 56, 1519–1530 (2013)

    Article  Google Scholar 

  • Jin, Y., Dong, J.B., Li, X., Wu, Y.: Kinematical measurement of hydraulic tortuosity of fluid flow in porous media. Int. J. Mod. Phys. C 26, 1550017 (2015)

    Article  Google Scholar 

  • Kandhai, D., Vidal, D.J.E., Hoekstra, A.G., Hoefsloot, H., Iedema, P., Sloot, P.M.A.: Lattice-Boltzmann and finite element simulations of fluid flow in a SMRX Static Mixer Reactor. Int. J. Numer. Methods Fluids 31, 1019–1033 (1999)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Kozeny, J.: Uber Kapillare Leitung Des Wassers in Boden. Stizungsber. Akad. Wiss. Wien 136, 271–306 (1927)

    Google Scholar 

  • Krohn, C.E., Thompson, A.H.: Fractal sandstone pores: automated measurements using scanning-electron-microscope images. Phys. Rev. B. 33, 6366 (1986)

    Article  Google Scholar 

  • Ladd, A.J.C.: Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271, 311–339 (1994)

    Article  Google Scholar 

  • Liu, S., Masliyah, J.H.: Single fluid flow in porous media. Chem. Eng. Commun. 148, 653–732 (1996a)

    Article  Google Scholar 

  • Liu, S., Masliyah, J.H.: Steady developing laminar flow in helical pipes with finite pitch. Int. J. Comput. Fluid Dyn. 6, 209–224 (1996b)

    Article  Google Scholar 

  • Li, J., Yu, B.: Tortuosity of flow paths through a Sierpinski carpet. Chin. Phys. Lett. 28, 34701 (2011)

    Article  Google Scholar 

  • Mandelbrot, B.B.: The Fractal Geometry of Nature. Macmillan, New York (1983)

    Google Scholar 

  • Matyka, M., Khalili, A., Koza, Z.: Tortuosity–porosity relation in porous media flow. Phys. Rev. E 78, 026306 (2008)

    Article  Google Scholar 

  • Mauret, E., Renaud, M.: Transport phenomena in multi-particle systems. Limits of applicability of capillary model in high voidage beds-application to fixed beds of fibers and fluidized beds of spheres. Chem. Eng. Sci. 52, 1807–1817 (1997)

    Article  Google Scholar 

  • Mortensen, N.A., Okkels, F., Bruus, H.: Reexamination of Hagen–Poiseuille flow: shape dependence of the hydraulic resistance in microchannels. Phys. Rev. E 71, 057301 (2005)

    Article  Google Scholar 

  • Mota, M., Teixeira, J.A., Bowen, W.R., Yelshin, A.: Binary spherical particle mixed beds porosity and permeability relationship measurement. Trans. Fit. Soc. 1, 101–106 (2001)

    Google Scholar 

  • Nasta, P., Vrugt, J.A., Romano, N.: Prediction of the saturated hydraulic conductivity from Brooks and Corey’s water retention parameters. Water Resour. Res. 49, 2918–2925 (2013)

    Article  Google Scholar 

  • Nithiarasu, P., Ravindran, K.: A new semi-implicit time stepping procedure for buoyancy driven flow in a fluid saturated porous medium. Comput. Methods Appl. Mech. 165, 147–154 (1998)

    Article  Google Scholar 

  • Panda, M.N., Lake, L.W.: Estimation of single-phase permeability from parameters of particle-size distribution. AAPG Bull. 78, 1028–1039 (1994)

    Google Scholar 

  • Perrier, E., Bird, N., Rieu, M.: Generalizing the fractal model of soil structure: the pore solid fractal approach. Geoderma 88, 137–164 (1999)

    Article  Google Scholar 

  • Perrier, E.M.A., Bird, N.R.A.: Modelling soil fragmentation: the pore solid fractal approach. Soil Tillage Res. 64, 91–99 (2002)

    Article  Google Scholar 

  • Pitchumani, R., Ramakrishnan, B.: A fractal geometry model for evaluating permeabilities of porous preforms used in liquid composite molding. Int. J. Heat Mass Transf. 42, 2219–2232 (1999)

    Article  Google Scholar 

  • Qian, Y.H., D’Humières, D., Lallemand, P.: Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17, 479–488 (1992)

    Article  Google Scholar 

  • Rahli, O., Tadrist, L., Miscevic, M., Santini, R.: Fluid flow through randomly packed monodisperse fibers: the Kozeny–Carman parameter analysis. J. Fluids Eng. 119, 188–192 (1997)

    Article  Google Scholar 

  • Rawls, W.J., Brakensiek, D.L., Logsdon, S.D.: Predicting saturated hydraulic conductivity utilizing fractal principles. Soil Sci. Soc. Am. J. 57, 1193–1197 (1993)

    Article  Google Scholar 

  • Sahimi, M.: Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Mod. Phys. 65, 1393–1534 (1993)

    Article  Google Scholar 

  • Sangani, A.S., Acrivos, A.: Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiph. Flow 8, 193–206 (1982)

    Article  Google Scholar 

  • Smidt, J.M., Monro, D.M.: Fractal modeling applied to reservoir characterization and flow simulation. Fractals 6, 401–408 (1998)

    Article  Google Scholar 

  • Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford, New York (2001)

  • Sukop, M.C., Thorne, D.T.: Lattice Boltzmann Modeling : An Introduction for Geoscientisits and Engineers. Springer, New York (2007)

    Google Scholar 

  • Tarafdar, S., Franz, A., Schulzky, C., Hoffmann, K.H.: Modelling porous structures by repeated Sierpinski carpets. Phys. A 292, 1–8 (2001)

    Article  Google Scholar 

  • Thovert, J.F., Wary, F., Adler, P.M.: Thermal conductivity of random media and regular fractals. J. Appl. Phys. 68, 3872–3883 (1990)

    Article  Google Scholar 

  • Valdés-Parada, F.J., Porter, M.L., Wood, B.D.: The role of tortuosity in upscaling. Transp. Porous Med. 88, 1–30 (2011)

    Article  Google Scholar 

  • Vita, M.C., De Bartolo, S., Fallico, C., Veltri, M.: Usage of infinitesimals in the Mengers Sponge model of porosity. Appl. Math. Comput. 218, 8187–8195 (2012)

    Article  Google Scholar 

  • Wang, B.Y., Jin, Y., Chen, Q., Zheng, J.L., Zhu, Y.B., Zhang, X.B.: Derivation of permeability–pore relationship for fractal porous reservoirs using series-parallel flow resistance model and lattice Boltzmann method. Fractals 22, 1440005 (2014)

    Article  Google Scholar 

  • Wheatcraft, S.W., Tyler, S.W.: An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resour. Res. 24, 566–578 (1988)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Young, I.M., Crawford, J.W.: The fractal structure of soil aggregates: its measurement and interpretation. Eur. J. Soil. Sci. 42, 187–192 (1991)

    Article  Google Scholar 

  • Yu, B.M.: Comments on A fractal geometry model for evaluating permeabilities of porous preforms used in liquid composite molding. Int. J. Heat Mass Transf. 44, 2787–2789 (2001)

    Article  Google Scholar 

  • Yu, B.M., Li, J.H., Li, Z.H., Zou, M.Q.: Permeabilities of unsaturated fractal porous media. Int. J. Multiph. Flow 29, 1625–1642 (2003)

    Article  Google Scholar 

  • Yu, B., Zou, M., Feng, Y.: Permeability of fractal porous media by Monte Carlo simulations. Int. J. Heat Mass Transf. 48, 2787–2794 (2005)

    Article  Google Scholar 

  • Yu, B., Cai, J., Zou, M.: On the physical properties of apparent two-phase fractal porous media. Vadose Zone J. 8, 177–186 (2009)

    Article  Google Scholar 

  • Yu, B.M., Li, J.H.: Some fractal characters of porous media. Fractals 9, 365–372 (2001)

    Article  Google Scholar 

  • Zhou, H., Perfect, E., Li, B., Lu, Y.: Comments on On the physical properties of apparent two-phase fractal porous media. Vadose Zone J. 9, 192 (2010)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Resource and Environment, Henan Polytechnic University, Jiaozuo, 454003, China

    Y. Jin, Y. B. Zhu, X. Li, J. L. Zheng & J. B. Dong

  2. Collaborative Innovation Center of Coalbed Methane and Shale Gas for Central Plains Economic Region, Henan Province, Jiaozuo, 454003, China

    Y. Jin, Y. B. Zhu, X. Li, J. L. Zheng & J. B. Dong

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  1. Y. Jin
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  2. Y. B. Zhu
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  3. X. Li
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  4. J. L. Zheng
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Correspondence to Y. Jin.

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This work was supported by the National Natural Science Foundation of China (Grant Nos. 41102093, 41472128), and CBM Union Foundation of Shanxi Province of China (Grant No. 2012012002).

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Jin, Y., Zhu, Y.B., Li, X. et al. Scaling Invariant Effects on the Permeability of Fractal Porous Media. Transp Porous Med 109, 433–453 (2015). https://doi.org/10.1007/s11242-015-0527-4

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