Transport in Porous Media

, Volume 122, Issue 3, pp 527–546 | Cite as

Three-Dimensional Lattice Boltzmann Simulations of Single-Phase Permeability in Random Fractal Porous Media with Rough Pore–Solid Interface

  • Timothy A. Cousins
  • Behzad Ghanbarian
  • Hugh Daigle


Single-phase permeability k has intensively been investigated over the past several decades by means of experiments, theories and simulations. Although the effect of surface roughness on fluid flow and permeability in single pores and fractures as well as networks of fractures was studied previously, its influence on permeability in a random mass fractal porous medium constructed of pores of different sizes remained as an open question. In this study, we, therefore, address the effect of pore–solid interface roughness on single-phase flow in random fractal porous media. For this purpose, we apply a mass fractal model to construct porous media with a priori known mass fractal dimensions \(2.579 \le D_{\mathrm{m}} \le 2.893\) characterizing both solid matrix and pore space. The pore–solid interface of the media is accordingly roughened using the Weierstrass–Mandelbrot approach and two parameters, i.e., surface fractal dimension \(D_{\mathrm{s}}\) and root-mean-square (rms) roughness height. A single-relaxation-time lattice Boltzmann method is applied to simulate single-phase permeability in the corresponding porous media. Results indicate that permeability decreases sharply with increasing \(D_{\mathrm{s}}\) from 1 to 1.1 regardless of \(D_{\mathrm{m}}\) value, while k may slightly increase or decrease, depending on \(D_{\mathrm{m}}\), as \(D_{\mathrm{s}}\) increases from 1.1 to 1.6.


Mass fractal dimension Lacunarity Permeability Pore–solid interface Porosity Surface fractal dimension 



The authors are grateful to three anonymous reviewers for their fruitful comments. We also acknowledge Edmund Perfect (Department of Earth and Planetary, University of Tennessee) and Jung-Woo Kim (Radioactive Waste Disposal Research Division, Korea Atomic Energy Research Institute) for providing the Lacunarity MATLAB code used in this study.


  1. Adler, P.M.: Fractal porous media III: transversal Stokes flow through random and Sierpinski carpets. Transp. Porous Media 3(2), 185–198 (1988)CrossRefGoogle Scholar
  2. Adler, P.M., Jacquin, C.G.: Fractal porous media I: longitudinal Stokes flow in random carpets. Transp. Porous Media 2(6), 553–569 (1987)CrossRefGoogle Scholar
  3. Allain, C., Cloitre, M.: Characterizing the lacunarity of random and deterministic fractal sets. Phys. Rev. A 44(6), 3552–3558 (1991)CrossRefGoogle Scholar
  4. Andrade Jr., J.S., Street, D.A., Shinohara, T., Shibusa, Y., Arai, Y.: Percolation disorder in viscous and nonviscous flow through porous media. Phys. Rev. E 51(6), 5725–5731 (1995)CrossRefGoogle Scholar
  5. Ausloos, M., Berman, D.H.: A multivariate Weierstrass–Mandelbrot function. Proc. R. Soc. Lond. A 400, 331–350 (1985)CrossRefGoogle Scholar
  6. Bahrami, M., Yovanovich, M.M., Culham, J.R.: Pressure drop of fully developed, laminar flow in rough microtubes. J. Fluids Eng. 128(3), 632–637 (2006)CrossRefGoogle Scholar
  7. Berkowitz, B., Hadad, A.: Fractal and multifractal measures of natural and synthetic fracture networks. J. Geophys. Res. B Solid Earth 102(B6), 12205–12218 (1997)CrossRefGoogle Scholar
  8. Boek, E.S., Venturoli, M.: Lattice-Boltzmann studies of fluid flow in porous media with realistic rock geometries. Comput. Math. Appl. 59(7), 2305–2314 (2010)CrossRefGoogle Scholar
  9. Brown, S.R.: Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. B Solid Earth 92(B2), 1337–1347 (1987)CrossRefGoogle Scholar
  10. Brown, S.R.: Simple mathematical model of a rough fracture. J. Geophys. Res. B Solid Earth 100(B4), 5941–5952 (1995)CrossRefGoogle Scholar
  11. Brown, S.R., Stockman, H.W., Reeves, S.J.: Applicability of the Reynolds equation for modeling fluid flow between rough surfaces. Geophys. Res. Lett. 22(18), 2537–2540 (1995)CrossRefGoogle Scholar
  12. Carr, J.R.: Statistical self-affinity, fractal dimension, and geologic interpretation. Eng. Geol. 48(3), 269–282 (1997)CrossRefGoogle Scholar
  13. Chen, Y., Cheng, P.: Fractal characterization of wall roughness on pressure drop in microchannels. Int. Commun. Heat Mass 30(1), 1–11 (2003)CrossRefGoogle Scholar
  14. Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30(1), 329–364 (1998)CrossRefGoogle Scholar
  15. Chen, Y., Zhang, C., Shi, M., Peterson, G.P.: Role of surface roughness characterized by fractal geometry on laminar flow in microchannels. Phys. Rev. E 80(2), 026301 (2009)CrossRefGoogle Scholar
  16. Chen, L., Kang, Q., Mu, Y., He, Y.L., Tao, W.Q.: A critical review of the pseudopotential multiphase lattice Boltzmann model: methods and applications. Int. J. Heat Mass Transf. 76, 210–236 (2014)CrossRefGoogle Scholar
  17. Cihan, A., Perfect, E., Tyner, J.S.: Water retention models for scale-variant and scale-invariant drainage of mass prefractal porous media. Vadose Zone J. 6(4), 786–792 (2007)CrossRefGoogle Scholar
  18. Cihan, A., Sukop, M.C., Tyner, J.S., Perfect, E., Huang, H.: Analytical predictions and lattice Boltzmann simulations of intrinsic permeability for mass fractal porous media. Vadose Zone J. 8(1), 187–196 (2009)CrossRefGoogle Scholar
  19. Cousins, T.A.: Effect of rough fractal pore-solid interface on single-phase permeability in random fractal porous media. M.Sc. Thesis, University of Texas at Austin (2016).
  20. Croce, G., D’Agaro, P.: Numerical simulation of roughness effect on microchannel heat transfer and pressure drop in laminar flow. J. Phys. D Appl. Phys. 38(10), 1518–1530 (2005)CrossRefGoogle Scholar
  21. Daigle, H., Reece, J.S.: Permeability of two-component granular materials. Transp. Porous Media 106(3), 523–544 (2015)CrossRefGoogle Scholar
  22. Daigle, H., Ghanbarian, B., Henry, P., Conin, M.: Universal scaling of the formation factor in clays: example from the Nankai Trough. J. Geophys. Res. B Solid Earth 120(11), 7361–7375 (2015)CrossRefGoogle Scholar
  23. Dathe, A., Thullner, M.: The relationship between fractal properties of solid matrix and pore space in porous media. Geoderma 129(3), 279–290 (2005)CrossRefGoogle Scholar
  24. Deng, Z., Chen, Y., Shao, C.: Gas flow through rough microchannels in the transition flow regime. Phys. Rev. E 93(1), 013128 (2016)CrossRefGoogle Scholar
  25. Drazer, G., Koplik, J.: Permeability of self-affine rough fractures. Phys. Rev. E 62(6), 8076–8085 (2000)CrossRefGoogle Scholar
  26. Drazer, G., Koplik, J.: Transport in rough self-affine fractures. Phys. Rev. E 66(2), 026303 (2002)CrossRefGoogle Scholar
  27. Eker, E., Akin, S.: Lattice Boltzmann simulation of fluid flow in synthetic fractures. Transp. Porous Media 65(3), 363–384 (2006)CrossRefGoogle Scholar
  28. Feder, J.: Fractals (Physics of Solids and Liquids). Plennum, New York (1998)Google Scholar
  29. Ghanbarian, B., Hunt, A.G.: Universal scaling of gas diffusion in porous media. Water Resour. Res. 50(3), 2242–2256 (2014)CrossRefGoogle Scholar
  30. 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(6), 1920–1929 (2013)CrossRefGoogle Scholar
  31. Ghanbarian, B., Hunt, A.G., Ewing, R.P., Skinner, T.E.: Universal scaling of the formation factor in porous media derived by combining percolation and effective medium theories. Geophys. Res. Lett. 41(11), 3884–3890 (2014)CrossRefGoogle Scholar
  32. Ghanbarian, B., Hunt, A.G., Skinner, T.E., Ewing, R.P.: Saturation dependence of transport in porous media predicted by percolation and effective medium theories. Fractals 23(01), 1540004 (2015)CrossRefGoogle Scholar
  33. Ghanbarian, B., Hunt, A.G., Daigle, H.: Fluid flow in porous media with rough pore-solid interface. Water Resour. Res 52, 2045–2058 (2016)CrossRefGoogle Scholar
  34. Ghanbarian-Alavijeh, B., Millán, H., Huang, G.: A review of fractal, prefractal and pore-solid-fractal models for parameterizing the soil water retention curve. Can. J. Soil Sci. 91(1), 1–14 (2011)CrossRefGoogle Scholar
  35. Gostick, J.T., Weber, A.Z.: Resistor-network modeling of ionic conduction in polymer electrolytes. Electrochim. Acta 179, 137–145 (2015)CrossRefGoogle Scholar
  36. Hansen, J.P., Skjeltorp, A.T.: Fractal pore space and rock permeability implications. Phys. Rev. B 38(4), 2635 (1988)CrossRefGoogle Scholar
  37. He, X., Zou, Q., Luo, L.S., Dembo, M.: Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. J. Stat. Phys. 87(1–2), 115–136 (1997)CrossRefGoogle Scholar
  38. Hunt, A., Ewing, R., Ghanbarian, B.: Percolation theory for flow in porous media. Springer, Berlin (2014)CrossRefGoogle Scholar
  39. Jacquin, C.G., Adler, P.M.: Fractal porous media II: geometry of porous geological structures. Transp. Porous Media 2(6), 571–596 (1987)CrossRefGoogle Scholar
  40. Kadanoff, L.P.: On two levels. Phys. Today 39, 7–9 (1986)Google Scholar
  41. Kandlikar, S.G., Schmitt, D., Carrano, A.L., Taylor, J.B.: Characterization of surface roughness effects on pressure drop in single-phase flow in minichannels. Phys. Fluids 17(10), 100606 (2005)CrossRefGoogle Scholar
  42. Katz, A., Thompson, A.H.: Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54(12), 1325 (1985)CrossRefGoogle Scholar
  43. Keehm, Y., Mukerji, T., Nur, A.: Permeability prediction from thin sections: 3D reconstruction and lattice-Boltzmann flow simulation. Geophys. Res. Lett. 31(4), L04303 (2004)CrossRefGoogle Scholar
  44. Kim, J.W., Perfect, E., Choi, H.: Anomalous diffusion in two-dimensional Euclidean and prefractal geometrical models of heterogeneous porous media. Water Resour. Res. 43(1), W01405 (2007)Google Scholar
  45. Kim, J.W., Sukop, M.C., Perfect, E., Pachepsky, Y.A., Choi, H.: Geometric and hydrodynamic characteristics of three-dimensional saturated prefractal porous media determined with lattice Boltzmann modeling. Transp. Porous Media 90(3), 831–846 (2011)CrossRefGoogle Scholar
  46. Kleinstreuer, C., Koo, J.: Computational analysis of wall roughness effects for liquid flow in micro-conduits. J. Fluids Eng. 126(1), 1–9 (2004)CrossRefGoogle Scholar
  47. Koponen, A., Kataja, M., Timonen, J.: Permeability and effective porosity of porous media. Phys. Rev. E 56(3), 3319–3325 (1997)CrossRefGoogle Scholar
  48. Koza, Z., Matyka, M., Khalili, A.: Finite-size anisotropy in statistically uniform porous media. Phys. Rev. E 79(6), 066306 (2009)CrossRefGoogle Scholar
  49. Krohn, C.E.: Fractal measurements of sandstones, shales, and carbonates. J. Geophys. Res. B Solid Earth 93(B4), 3297–3305 (1988)CrossRefGoogle Scholar
  50. Larson, R.G., Scriven, L.E., Davis, H.T.: Percolation theory of two phase flow in porous media. Chem. Eng. Sci. 36(1), 57–73 (1981)CrossRefGoogle Scholar
  51. Latt, J.: Palabos, parallel lattice Boltzmann solver. (2009)
  52. Lemaitre, R., Adler, P.M.: Fractal porous media IV: three-dimensional stokes flow through random media and regular fractals. Transp. Porous Media 5(4), 325–340 (1990)CrossRefGoogle Scholar
  53. Madadi, M., Sahimi, M.: Lattice Boltzmann simulation of fluid flow in fracture networks with rough, self-affine surfaces. Phys. Rev. E 67(2), 026309 (2003)CrossRefGoogle Scholar
  54. Madadi, M., VanSiclen, C.D., Sahimi, M.: Fluid flow and conduction in two-dimensional fractures with rough, self-affine surfaces: a comparative study. J. Geophys. Res. B Solid Earth 108(B8), ECV11 (2003)CrossRefGoogle Scholar
  55. Majumdar, A., Bhushan, B.: Fractal model of elastic-plastic contact between rough surfaces. J. Tribol. 113(1), 1–11 (1991)CrossRefGoogle Scholar
  56. Majumdar, A., Tien, C.L.: Fractal characterization and simulation of rough surfaces. Wear 136(2), 313–327 (1990)CrossRefGoogle Scholar
  57. Mandelbrot, B.B.: How long is the coast of Britain. Science 156(3775), 636–638 (1967)CrossRefGoogle Scholar
  58. Mandelbrot, B.B.: The Fractal Geometry of Nature. Macmillan, London (1983)Google Scholar
  59. Manwart, C., Aaltosalmi, U., Koponen, A., Hilfer, R., Timonen, J.: Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media. Phys. Rev. E 66(1), 016702 (2002)CrossRefGoogle Scholar
  60. Martys, N.S., Chen, H.: Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. Phys. Rev. E 53(1), 743–750 (1996)CrossRefGoogle Scholar
  61. Mavko, G., Nur, A.: The effect of a percolation threshold in the Kozeny–Carman relation. Geophysics 62(5), 1480–1482 (1997)CrossRefGoogle Scholar
  62. Mourzenko, V.V., Thovert, J.F., Adler, P.M.: Percolation and conductivity of self-affine fractures. Phys. Rev. E 59(4), 4265–4284 (1999)CrossRefGoogle Scholar
  63. Mourzenko, V.V., Thovert, J.F., Adler, P.M.: Permeability of self-affine fractures. Transp. Porous Media 45(1), 89–103 (2001)CrossRefGoogle Scholar
  64. Pan, C., Luo, L.S., Miller, C.T.: An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput. Fluids 35(8), 898–909 (2006)CrossRefGoogle Scholar
  65. Perrier, E., Rieu, M., Sposito, G., Marsily, G.: Models of the water retention curve for soils with a fractal pore size distribution. Water Resour. Res. 32(10), 3025–3031 (1996)CrossRefGoogle Scholar
  66. Power, W.L., Tullis, T.E., Weeks, J.D.: Roughness and wear during brittle faulting. J. Geophys. Res. B Solid Earth 93(B12), 15268–15278 (1988)CrossRefGoogle Scholar
  67. Radliński, A.P., Radlińska, E.Z., Agamalian, M., Wignall, G.D., Lindner, P., Randl, O.G.: Fractal geometry of rocks. Phys. Rev. Lett. 82(15), 3078–3081 (1999)CrossRefGoogle Scholar
  68. Rieu, M., Perrier, E.: Fractal models of fragmented and aggregated soils. In: Baveye, P., Parlange, J.-Y., Stewart, B.A. (eds.) Advances in Soil Science. Fractals in Soil Science, pp. 169–202. CRC Press, Boca Raton (1998)Google Scholar
  69. Rieu, M., Sposito, G.: Fractal fragmentation, soil porosity, and soil water properties: I. Theory. Soil Sci. Soc. Am. J. 55(5), 1231–1238 (1991)CrossRefGoogle Scholar
  70. Sahimi, M.: Fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media. Transp. Porous Media 13(1), 3–40 (1993)CrossRefGoogle Scholar
  71. Sahimi, M.: Applications of Percolation Theory. CRC Press, Boca Raton (1994)Google Scholar
  72. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. Wiley, Hoboken (2011)CrossRefGoogle Scholar
  73. Sheikh, B., Pak, A.: Numerical investigation of the effects of porosity and tortuosity on soil permeability using coupled three-dimensional discrete-element method and lattice Boltzmann method. Phys. Rev. E 91(5), 053301 (2015)CrossRefGoogle Scholar
  74. Stauffer, D., Aharony, A.: Introduction to Percolation Theory. CRC Press, London (1994)Google Scholar
  75. Succi, S., Foti, E., Higuera, F.: Three-dimensional flows in complex geometries with the lattice Boltzmann method. Europhys. Lett. 10(5), 433 (1989)CrossRefGoogle Scholar
  76. Sukop, M.C., van Dijk, G.J., Perfect, E., van Loon, W.K.: Percolation thresholds in 2-dimensional prefractal models of porous media. Transp. Porous Media 48(2), 187–208 (2002)CrossRefGoogle Scholar
  77. Taylor, J.B., Carrano, A.L., Kandlikar, S.G.: Characterization of the effect of surface roughness and texture on fluid flow—past, present, and future. Int. J. Therm. Sci. 45(10), 962–968 (2006)CrossRefGoogle Scholar
  78. Thompson, M.E., Brown, S.R.: The effect of anisotropic surface roughness on flow and transport in fractures. J. Geophys. Res. B Solid Earth 96(B13), 21923–21932 (1991)CrossRefGoogle Scholar
  79. Tsang, Y.W., Tsang, C.F.: Channel model of flow through fractured media. Water Resour. Res. 23(3), 467–479 (1987)CrossRefGoogle Scholar
  80. Tyler, S.W., Wheatcraft, S.W.: Fractal processes in soil water retention. Water Resour. Res. 26(5), 1047–1054 (1990)CrossRefGoogle Scholar
  81. van der Marck, S.C.: Network approach to void percolation in a pack of unequal spheres. Phys. Rev. Lett. 77(9), 1785 (1996)CrossRefGoogle Scholar
  82. 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 (2016)CrossRefGoogle Scholar
  83. Xie, S., Cheng, Q., Ling, Q., Li, B., Bao, Z., Fan, P.: Fractal and multifractal analysis of carbonate pore-scale digital images of petroleum reservoirs. Mar. Pet. Geol. 27(2), 476–485 (2010)CrossRefGoogle Scholar
  84. Yang, S., Yu, B., Zou, M., Liang, M.: A fractal analysis of laminar flow resistance in roughened microchannels. Int. J. Heat Mass Transf. 77, 208–217 (2014)CrossRefGoogle Scholar
  85. Yang, S., Liang, M., Yu, B., Zou, M.: Permeability model for fractal porous media with rough surfaces. Microfluid. Nanofluid. 18(5–6), 1085–1093 (2015)CrossRefGoogle Scholar
  86. Zhang, X., Knackstedt, M.A., Sahimi, M.: Fluid flow across mass fractals and self-affine surfaces. Phys. A Stat. Mech. Appl. 233(3), 835–847 (1996)CrossRefGoogle Scholar
  87. Zimmerman, R.W., Kumar, S., Bodvarsson, G.S.: Lubrication theory analysis of the permeability of rough-walled fractures. Int. J. Rock Mech. Min. Sci. 28(4), 325–31 (1991)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2018

Authors and Affiliations

  • Timothy A. Cousins
    • 1
  • Behzad Ghanbarian
    • 2
    • 3
  • Hugh Daigle
    • 1
  1. 1.Hildebrand Department of Petroleum and Geosystems EngineeringUniversity of Texas at AustinAustinUSA
  2. 2.Bureau of Economic Geology, Jackson School of GeosciencesUniversity of Texas at AustinAustinUSA
  3. 3.Department of GeologyKansas State UniversityManhattanUSA

Personalised recommendations