Acta Mechanica

, Volume 230, Issue 10, pp 3703–3721 | Cite as

Two-scale analysis of the permeability of 3D periodic granular and fibrous media

  • M. K. BourbatacheEmail author
  • M. Hellou
  • F. Lominé
Original Paper


In this paper, a numerical study of slow flow through a filter viewed as a porous medium made of arrays of cubic solid particles or solid fibers of square cross section is considered. A double-scale asymptotic method is used to determine a system of equations that are then solved numerically to calculate the permeability. Simulations are made at the REV scale, and macroscopic properties are deduced. At the microscale, three arrangements (simple cubic, body-centered cubic and face-centered cubic) are analyzed. A parametric study is carried out, for both granular and fibrous cases, showing the porosity evolution with the size ratio between the solid particles and the periodic cell. At the macroscopic scale, the interest of this analysis is to compute the Darcy’s permeability of such arrays as a function of the porosity and the packing characteristics. Results are given over the full porosity range for SC, BCC and FCC arrays. On the other side, the microscopic analysis shows the influence of particle or fiber arrangement and size on the fluid velocity and the pressure field inside the porous structure.



  1. 1.
    Auriault, J.L.: Heterogeneous medium. is an equivalent macroscopic description possible? Int. J. Eng. Sci. 29(7), 785–795 (1991)zbMATHGoogle Scholar
  2. 2.
    Auriault, J.L., Boutin, C., Geindreau, C.: Homogenization of Coupled Phenomena in Heterogenous Media. Wiley, London (2009)Google Scholar
  3. 3.
    Babu, B.Z., Pillai, K.M.: Experimental investigation of the effect of fiber-mat architecture on the unsaturated flow in liquid composite molding. J. Compos. Mater. 38(3–4), 57–99 (2004)Google Scholar
  4. 4.
    Bourbatache, K., Millet, O., Ait-Mokhtar, A., Amiri, O.: Modeling the chlorides transport in cementitious materials by periodic homogenization. Transp. Porous Med. 94(1), 437–459 (2012)MathSciNetGoogle Scholar
  5. 5.
    Bourbatache, K., Millet, O., Ait-Mokhtar, A., Amiri, O.: Chloride transfer in cement-based materials. Part 2. Experimental study and numerical simulations. Int. J. Numer. Anal. Methods Geomech. 37, 1628–1641 (2013)Google Scholar
  6. 6.
    Boutin, C.: Study of permeability by periodic and self-consistent homogenisation. Eur. J. Mech. A Solids 19(4), 603–632 (2000)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brown, R.C.: A many-fibre model of airflow through a fibrous filter. J. Aerosol Sci. 15(5), 583–593 (1984)Google Scholar
  8. 8.
    Carman, P.C.: Fluid flow through granular beds. Chem. Eng. Res. Des. 75, S32–S48 (1997)Google Scholar
  9. 9.
    Cheung, C.S., Cao, Y.H., Yan, Z.D.: Numerical model for particle deposition and loading in electret filter with rectangular split-type fibers. Comput. Mech. 35(6), 449–458 (2005)zbMATHGoogle Scholar
  10. 10.
    Cho, H., Jeong, N., Sung, H.J.: Permeability of microscale fibrous porous media using the lattice boltzmann method. Int. J. Heat Fluid Flow 44(3–4), 435–443 (2013)Google Scholar
  11. 11.
    Costa, A.: Permeability-porosity relationship: a reexamination of the kozeny-carman equation based on a fractal pore-space geometry assumption. Geophys. Res. Lett. 33, 1–5 (2006)Google Scholar
  12. 12.
    Crevacore, E., Tosco, T., Sethi, R., Boccardo, G., Marchisio, D.L.: Recirculation zones induce non-fickian transport in three-dimensional periodic porous media. Phys. Rev. E 94(5), 1–12 (2016)Google Scholar
  13. 13.
    Davis, A.M.J., O’Neil, M.E., Dorrepaal, J.M., Ranger, K.B.: Separation from the surface of two equal spheres in Stokes flow. J. Fluid Mech. 77, 625–644 (1976)zbMATHGoogle Scholar
  14. 14.
    Davit, Y., Bell, C.G., Byrne, H.M., Chapman, L.A.C., Kimpton, L.S., Lang, G.E., Leonard, K.H.L., Oliver, J.M., Pearson, N.C., Shipley, R.J., Waters, S.L., Whiteley, J.P., Wood, B.D., Quintard, M.: Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare? Adv. Water Resour. 62(Part B), 178–206 (2013)Google Scholar
  15. 15.
    Dhaniyala, S.: An asymmetrical, three-dimensional model for fibrous filters. Aerosol Sci. Technol. 30(4), 333–348 (1999)Google Scholar
  16. 16.
    Drummond, J.E., Tahir, M.I.: Laminar viscous flow through regular arrays of parallel solid cylinders. Int. J. Multiphase flow 10(5), 515–540 (1984)zbMATHGoogle Scholar
  17. 17.
    Fan, J., Lominé, F., Hellou, M.: A numerical analysis of pressure drop and particle capture efficiency by rectangular fibers using lb-de methods. Acta Mech. 229, 1–18 (2018)MathSciNetGoogle Scholar
  18. 18.
    Fardi, B., Liu, B.Y.H.: Flow field and pressure drop of filters with rectangular fibers. Aerosol Sci. Technol. 17(1), 36–44 (1992)Google Scholar
  19. 19.
    Firdaouss, M., Duplessis, J.P.: On the prediction of darcy permeability in nonisotropic periodic two-dimensional porous media. J. Porous Media 7(2) (2004)zbMATHGoogle Scholar
  20. 20.
    Happel, J.: Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. AIChE J. 4(2), 197–201 (1958)MathSciNetGoogle Scholar
  21. 21.
    Happel, J.: Viscous flow relative to arrays of cylinders. Am. Inst. Chem. Engng. J. 5, 174–177 (1959)Google Scholar
  22. 22.
    Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Springer, Netherlands (1983)zbMATHGoogle Scholar
  23. 23.
    Hasimoto, H.: On the periodic fundamental solution of the stokes equation and their application to viscous how past a cubic array of spheres. J. Fluid Mech. 5, 317–328 (1959)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hellou, M., Martinez, J., El Yazidi, M.: Stokes flow through microstructural model of fibrous media. Mech. Res. Commun. 61(3–4), 97–103 (2004)zbMATHGoogle Scholar
  25. 25.
    Higdon, J.J.L., Ford, G.D.: Permeability of three-dimensional models of fibrous porous media. J. Fluid Mech 308, 341–361 (1996)zbMATHGoogle Scholar
  26. 26.
    Hommel, J., Coltman, E., Class, H.: Porosity-permeability relations for evolving pore-space: a review with a focuson (bio-)geochemicallyaltered porous media. Transp. Porous Media 124, 1–41 (2018)MathSciNetGoogle Scholar
  27. 27.
    Hornung, U.: Homogenization and porous Media. IAM, (1997)Google Scholar
  28. 28.
    Huang, H., Wang, K., Zhao, H.: Numerical study of pressure drop and diffusional collection efficiency of several typical non circular fibers in filtration. Powder Technol. 292, 232–241 (2016)Google Scholar
  29. 29.
    Jackson, G.W., James, D.F.: The permeability of fibrous porous media. Can. J. Chem. Eng. 64, 364–374 (1986)Google Scholar
  30. 30.
    Kadaksham, A., Pillapakkam, S.B., Singh, P.: Permeability of periodic arrays of spheres. Mech. Res. Commun. 32, 659–665 (2005)zbMATHGoogle Scholar
  31. 31.
    Kozeny, J.: Uber kapillare Leitung des Wassers im Boden-Aufstieg, Versickerung und Anwendung auf die Bewasserung, Sitzungsberichte der Akademie der Wissenschaften Wien. Mathematisch Naturwissenschaftliche Abteilung 136, 271–306 (1927)Google Scholar
  32. 32.
    Kuwabara, S.: The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small reynolds number. J. Phys. Soc. Jpn. 14(4), 527–532 (1959)MathSciNetGoogle Scholar
  33. 33.
    Niya, S.M.R., Selvadurai, A.P.S.: The estimation of permeability of a porous medium with a generalized pore structure by geometry identification. Phys. Fluids. 29, 037101 (2017)Google Scholar
  34. 34.
    Ouyang, M., Liu, B.Y.H.: Analytical solution of flow field and pressure drop for filters with rectangular fibers. J. Aerosol Sci. 29(1), 187–196 (1998)Google Scholar
  35. 35.
    Pierce, R.S., Falzon, B.G.: Simulating resin infusion through textile reinforcement materials for the manufacture of complex composite structures. Engineering 3, 596–607 (2017)Google Scholar
  36. 36.
    Rocha, R.P.A., Cruz, M.E.: Calculation of the permeability and apparent permeability of three-dimensional porous media. Transp. Porous Med. 83(26), 349–373 (2010)MathSciNetGoogle Scholar
  37. 37.
    Rodriguez, E., Giacomelli, F., Vazquez, A.: Permeability-porosity relationship in rtm for different fiberglass and natural reinforcements. J. Compos. Mater. 38(3), 259–268 (2004)Google Scholar
  38. 38.
    Sanchez P.E.: Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics, vol. 129, Berlin (1980)Google Scholar
  39. 39.
    Sangani, A.S., Acrivos, A.: Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiph. Flow 8(3), 193–206 (1982)zbMATHGoogle Scholar
  40. 40.
    Sangani, A.S., Acrivos, A.: Slow flow through a periodic array of spheres. Int. J. Multiph. Flow 8(4), 343–360 (1982)zbMATHGoogle Scholar
  41. 41.
    Stylianopoulos, T., Yeckel, A., Derby, J.J., Luo, X.J., Shephard, M.S., Sander, E.A., Barocas, V.H.: Permeability calculations in three-dimensional isotropic and oriented fiber networks. Phys. Fluids 20, 123601 (2008)zbMATHGoogle Scholar
  42. 42.
    Sun, Z., Tang, X., Cheng, G.: Numerical simulation for tortuosity of porous media. Microporous Mesoporous Mater. 173, 37–42 (2013)Google Scholar
  43. 43.
    Tahir, M.A., Vahedi Tafreshi, H.: Influence of fiber orientation on the transverse permeability of fibrous media. Phys. Fluids 21(8), 083604 (2009)zbMATHGoogle Scholar
  44. 44.
    Tamayol, A., Bahrami, M.: Analytical determination of viscous permeability of fibrous porous media. Int. J. Heat Mass Transf. 52, 2407–2414 (2009)zbMATHGoogle Scholar
  45. 45.
    Tamayol, A., Bahrami, M.: Transverse permeability of fibrous porous media. Phys. Rev. E 83, 1–10 (2011)zbMATHGoogle Scholar
  46. 46.
    Tamayol, A., Wong, K.W., Bahrami, M.: Effects of microstructure on flow properties of fibrous porous media at moderate reynolds number. Phys. Rev. E 85, 026318 (2012)Google Scholar
  47. 47.
    Taneda, S.: Visualization of separating stokes flows. J. Phys. Soc. Jpn. 46(6), 1935–1942 (1979)Google Scholar
  48. 48.
    Tomadakis, M.M., Robertson, T.J.: Viscous permeability of random fiber structures: comparison of electrical and diffusional estimates with experimental and analytical results. J. Compos. Mater. 39(2), 163–187 (2005)Google Scholar
  49. 49.
    Valdés-Parada, F.J., Porter, M.L., Wood, B.D.: The role of tortuosity in upscaling. Transp. Porous Media 88(1), 1–30 (2011)MathSciNetGoogle Scholar
  50. 50.
    Verleye, B., Lomov, S.V., Long, A., Verpoest, I., Roose, D.: Permeability prediction for the meso-macro coupling in the simulation of the impregnation stage of resin transfer moulding. Compos. Part A Appl. Sci. Manuf. 41(1), 29–35 (2010)Google Scholar
  51. 51.
    Wang, C.Y.: Flow through a finned channel filled with a porous medium. Chem. Eng. Sci. 65, 1826–1831 (2010)Google Scholar
  52. 52.
    Wang, C.Y.: Stokes flow through an array of rectangular fibers. Int. J. Multiph. Flow 22(1), 185–194 (1996)zbMATHGoogle Scholar
  53. 53.
    Wang, K., Zhao, H.: The influence of fiber geometry and orientation angle on filtration performance. Aerosol Sci. Technol. 49(2), 75–85 (2015)MathSciNetGoogle Scholar
  54. 54.
    Yazdchi, K., Srivastava, S., Luding, S.: Microstructural effects on the permeability of periodic fibrous porous media. Int. J. Multiph. flow 37, 956–966 (2011)Google Scholar
  55. 55.
    Zhong, W.H., Currie, I.G., James, D.F.: Creeping flow through a model fibrous porous medium. Exp. Fluids 40, 119–126 (2006)Google Scholar
  56. 56.
    Zhu, C., Lin, C.-H., Cheung, C.S.: Inertial impaction-dominated fibrous filtration with rectangular or cylindrical fibers. Powder Technol. 112(1), 149–162 (2000)Google Scholar
  57. 57.
    Zick, A.A., Homsy, G.M.: Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115(26), 13 (1982)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Génie Civil et Génie MécaniqueINSA de RennesRennesFrance

Personalised recommendations