Modelling of Flow Through Porous Media Over the Complete Flow Regime

  • Ashes Banerjee
  • Srinivas PasupuletiEmail author
  • Mritunjay Kumar Singh
  • Sekhar Chandra Dutta
  • G. N. Pradeep Kumar


A mathematical model is developed based on the empirical power law equation for post-laminar flow through porous media. Hydraulic conductivity and the critical Reynolds number are used as boundary conditions. The developed model can predict hydraulic gradients for specific velocities, irrespective of the media sizes or porosities, over the complete flow transition. Therefore, the model can be very useful to recognise the specific flow regime or to predict the velocity and hydraulic gradient for a given flow regime. A parametric study is carried out concerning the behaviour of binomial (Forchheimer) and power law (Izbash and Wilkins) coefficients subjected to different media sizes, porosities and flow regimes. The observed behaviour of Forchheimer and Izbash coefficients with different media sizes and porosities are similar to the experimental results reported in the literature. However, the values of these coefficients differ when subjected to different flow regimes for any specific packing. The ratios of non-Darcy and Darcy coefficients of the Forchheimer equation suggest an increasing influence of inertia towards the turbulent regime. The maximum and minimum values of β are found to be 1.38 and 0.69 for laminar and turbulent regime, respectively. However, these values are found to be unaffected by the media size and porosity variation. The value of Wilkins coefficient w in the laminar regime is found to be 4841.72 for all media sizes and porosities. However, the coefficient represents a decreasing variation trend towards the turbulent regime which is also dependent of the media size variation.


Flow modelling Porous media Post-laminar flow Binomial equation Power law equation 



The authors would like to thank the authorities of IIT (ISM) Dhanbad for their support and in utilising the facilities in various departments. The authors would also acknowledge the funding received from IIT (ISM), Dhanbad under MRP and FRS projects for fabrication of the permeameters, procurement of fluid, media and other accessories utilised for experimentation and investigation.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they do not have any conflict of interest.


  1. Antohe, B., Lage, J., Price, D., Weber, R.: Experimental determination of permeability and inertia coefficients of mechanically compressed aluminum porous matrices. J. Fluids Eng. 119, 404–412 (1997)CrossRefGoogle Scholar
  2. Banerjee, A., Pasupuleti, S.: Effect of convergent boundaries on post laminar flow through porous media. Powder Technol. 342, 288–300 (2019)CrossRefGoogle Scholar
  3. Banerjee, A., Pasupuleti, S., Singh, M.K., Kumar, G.: An investigation of parallel post-laminar flow through coarse granular porous media with the Wilkins equation. Energies 11, 320 (2018a)CrossRefGoogle Scholar
  4. Banerjee, A., Pasupuleti, S., Singh, M.K., Kumar, G.N.P.: A study on the Wilkins and Forchheimer equations used in coarse granular media flow. Acta Geophys. 66, 81–91 (2018b)CrossRefGoogle Scholar
  5. Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)Google Scholar
  6. Boomsma, K., Poulikakos, D.: The effects of compression and pore size variations on the liquid flow characteristics in metal foams. J. Fluids Eng. 124, 263–272 (2002)CrossRefGoogle Scholar
  7. Bordier, C., Zimmer, D.: Drainage equations and non-Darcian modelling in coarse porous media or geosynthetic materials. J. Hydrol. 228, 174–187 (2000)CrossRefGoogle Scholar
  8. Bu, S., Yang, J., Dong, Q., Wang, Q.: Experimental study of transition flow in packed beds of spheres with different particle sizes based on electrochemical microelectrodes measurement. Appl. Therm. Eng. 73, 1525–1532 (2014)CrossRefGoogle Scholar
  9. Bu, S., Yang, J., Dong, Q., Wang, Q.: Experimental study of flow transitions in structured packed beds of spheres with electrochemical technique. Exp. Therm. Fluid Sci. 60, 106–114 (2015)CrossRefGoogle Scholar
  10. Chen, C., Wan, J., Zhan, H.: Theoretical and experimental studies of coupled seepage-pipe flow to a horizontal well. J. Hydrol. 281, 159–171 (2003)CrossRefGoogle Scholar
  11. Chen, Y.-F., Liu, M.-M., Hu, S.-H., Zhou, C.-B.: Non-Darcy’s law-based analytical models for data interpretation of high-pressure packer tests in fractured rocks. Eng. Geol. 199, 91–106 (2015a)CrossRefGoogle Scholar
  12. Chen, Y.-F., Zhou, J.-Q., Hu, S.-H., Hu, R., Zhou, C.-B.: Evaluation of Forchheimer equation coefficients for non-Darcy flow in deformable rough-walled fractures. J. Hydrol. 529, 993–1006 (2015b)CrossRefGoogle Scholar
  13. Cheng, N.-S., Hao, Z., Tan, S.K.: Comparison of quadratic and power law for nonlinear flow through porous media. Exp. Therm. Fluid Sci. 32, 1538–1547 (2008)CrossRefGoogle Scholar
  14. Dan, H.C., He, L.H., Xu, B.: Experimental investigation on non-Darcian flow in unbound graded aggregate material of highway pavement. Transp. Porous Med. 112, 189–206 (2016)CrossRefGoogle Scholar
  15. Dudgeon, C.R.: An experimental study of the flow of water through coarse granular media. La Houille Blanche. 7, 785–801 (1966)CrossRefGoogle Scholar
  16. Dukhan, N., Ali, M.: Strong wall and transverse size effects on pressure drop of flow through open-cell metal foam. Int. J. Therm. Sci. 57, 85–91 (2012)CrossRefGoogle Scholar
  17. Dukhan, N., Bağcı, Ö., Özdemir, M.: Experimental flow in various porous media and reconciliation of Forchheimer and Ergun relations. Exp. Therm. Fluid Sci. 57, 425–433 (2014)CrossRefGoogle Scholar
  18. Dybbs, A., Edwards, R.: A new look at porous media luid mechanics—Darcy to turbulent. In: Fundamentals of transport phenomena in porous media, vol. 82, pp. 199–256. Springer, Dordrecht (1984)Google Scholar
  19. Fand, R., Kim, B., Lam, A., Phan, R.: Resistance to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres. J. Fluids Eng. 109, 268–274 (1987)CrossRefGoogle Scholar
  20. Garga, V.K., Hansen, D., Townsend, R.D.: Considerations on the design of flow through rockfill drains. In: 14th Annual British Columbia Mine Reclamation Symposium, Cranbrook, BC (1990)Google Scholar
  21. Giroud, J.P., Kavazanjian Jr., E.: Degree of turbulence of flow in geosynthetic and granular drains. J. Geotech. Geoenvironmental Eng. 140, 06014001 (2014)CrossRefGoogle Scholar
  22. Hassanizadeh, S.M., Gray, W.G.: High velocity flow in porous media. Transp. Porous Med. 2, 521–531 (1987)CrossRefGoogle Scholar
  23. Hellström, G., Lundström, S.: Flow through porous media at moderate Reynolds number. In: Proceedings, 4th International Scientific Colloquium Modelling for Material Processing, pp. 129–134 (2006)Google Scholar
  24. Horton, N., Pokrajac, D.: Onset of turbulence in a regular porous medium: an experimental study. Phys. Fluids 21, 045104 (2009)CrossRefGoogle Scholar
  25. Huang, K., Wan, J., Chen, C., He, L., Mei, W., Zhang, M.: Experimental investigation on water flow in cubic arrays of spheres. J. Hydrol. 492, 61–68 (2013)CrossRefGoogle Scholar
  26. Jolls, K., Hanratty, T.: Transition to turbulence for flow through a dumped bed of spheres. Chem. Eng. Sci. 21, 1185–1190 (1966)CrossRefGoogle Scholar
  27. Kovacs, G.: Seepage through saturated and unsaturated layers. Hydrol. Sci. J. 16, 27–40 (1971)Google Scholar
  28. Kumar, G.N.P., Venkataraman, P.: Non-Darcy converging flow through coarse granular media. J. Inst. Eng. India Civ. Eng. Div. 76, 6–11 (1995)Google Scholar
  29. Kundu, P., Kumar, V., Mishra, I.M.: Experimental and numerical investigation of fluid flow hydrodynamics in porous media: characterization of pre-Darcy, Darcy and non-Darcy flow regimes. Powder Technol. 303, 278–291 (2016)CrossRefGoogle Scholar
  30. Lacey, R.: The characteristic flow equation: a tool for engineers and scientists. Geotext. Geomembr. 44, 534–548 (2016)CrossRefGoogle Scholar
  31. Lage, J., Antohe, B., Nield, D.: Two types of nonlinear pressure-drop versus flow-rate relation observed for saturated porous media. J. Fluids Eng. 119, 700–706 (1997)CrossRefGoogle Scholar
  32. Larsson, I.A.S., Lundström, T.S., Lycksam, H.: Tomographic PIV of flow through ordered thin porous media. Exp. Fluids 59, 96 (2018)CrossRefGoogle Scholar
  33. Lasseux, D., Valdés-Parada, F.J.: On the developments of Darcy’s law to include inertial and slip effects. Comptes Rendus Mécanique 345, 660–669 (2017)CrossRefGoogle Scholar
  34. Latifi, M., Midoux, N., Storck, A., Gence, J.: The use of micro-electrodes in the study of the flow regimes in a packed bed reactor with single phase liquid flow. Chem. Eng. Sci. 44, 2501–2508 (1989)CrossRefGoogle Scholar
  35. Macini, P., Mesini, E., Viola, R.: Laboratory measurements of non-Darcy flow coefficients in natural and artificial unconsolidated porous media. J. Pet. Sci. Eng. 77, 365–374 (2011)CrossRefGoogle Scholar
  36. Mathias S.A., Todman, L.C.: Step-drawdown tests and the Forchheimer equation. Water Resour. Res. 46, W07514 (2010)Google Scholar
  37. Moutsopoulos, K.N., Papaspyros, I.N., Tsihrintzis, V.A.: Experimental investigation of inertial flow processes in porous media. J. Hydrol. 374, 242–254 (2009)CrossRefGoogle Scholar
  38. Munson, B.R., Young, D.F., Okiishi, T.H., Huebsch, W.W.: Fundamentals of Fluid Mechanics, p. 69. Wiley, Hoboken (2006)Google Scholar
  39. Nezhad, M.M., Rezania, M., Baioni, E.: Transport in porous media with nonlinear flow condition. Transp. Porous Med. 126, 5–22 (2019)CrossRefGoogle Scholar
  40. Ovalle-Villamil, W., Sasanakul, I.: Investigation of non-Darcy low for fine grained materials. Geotech. Geol. Eng. 37, 413–429 (2019)Google Scholar
  41. Qian, J., Zhan, H., Zhao, W., Sun, F.: Experimental study of turbulent unconfined groundwater flow in a single fracture. J. Hydrol. 311, 134–142 (2005)CrossRefGoogle Scholar
  42. Reddy, N.B., Rao, P.R.: Effect of convergence on nonlinear flow in porous media. J. Hydraul. Eng. 132, 420–427 (2006)CrossRefGoogle Scholar
  43. Salahi, M.-B., Sedghi-Asl, M., Parvizi, M.: Nonlinear flow through a packed-column experiment. J. Hydrol. Eng. 20, 04015003 (2015)CrossRefGoogle Scholar
  44. Sedghi-Asl, M., Rahimi, H., Farhoudi, J., Hoorfar, A., Hartmann, S.: One-dimensional fully developed turbulent flow through coarse porous medium. J. Hydrol. Eng. 19, 1491–1496 (2013)CrossRefGoogle Scholar
  45. Sedghi-Asl, M., Rahimi, H., Salehi, R.: Non-Darcy flow of water through a packed column test. Transp. Porous Med. 101, 215–227 (2014)CrossRefGoogle Scholar
  46. Seguin, D., Montillet, A., Comiti, J.: Experimental characterisation of flow regimes in various porous media—I: limit of laminar flow regime. Chem. Eng. Sci. 53, 3751–3761 (1998a)CrossRefGoogle Scholar
  47. Seguin, D., Montillet, A., Comiti, J., Huet, F.: Experimental characterization of flow regimes in various porous media—II: transition to turbulent regime. Chem. Eng. Sci. 53, 3897–3909 (1998b)CrossRefGoogle Scholar
  48. Sidiropoulou, M.G., Moutsopoulos, K.N., Tsihrintzis, V.A.: Determination of Forchheimer equation coefficients a and b. Hydrol. Process. 21, 534–554 (2007)CrossRefGoogle Scholar
  49. Skjetne, E., Auriault, J.-L.: High-velocity laminar and turbulent flow in porous media. Transp. Porous Med. 36, 131–147 (1999)CrossRefGoogle Scholar
  50. Thiruvengadam, M., Kumar, G.N.P.: Validity of Forchheimer equation in radial flow through coarse granular media. J. Eng. Mech. 123, 696–704 (1997)CrossRefGoogle Scholar
  51. Trussell, R.R., Chang, M.: Review of flow through porous media as applied to head loss in water filters. J. Environ. Eng. 125, 998–1006 (1999)CrossRefGoogle Scholar
  52. van Lopik, J.H., Snoeijers, R., van Dooren, T.C., Raoof, A., Schotting, R.J.: The effect of grain size distribution on nonlinear flow behavior in sandy porous media. Transp. Porous Med. 120, 37–66 (2017)CrossRefGoogle Scholar
  53. Venkataraman, P., Rao, P.R.M.: Darcian, transitional, and turbulent flow through porous media. J. Hydraul. Eng. 124, 840–846 (1998)CrossRefGoogle Scholar
  54. Venkataraman, P., Rao, P.R.M.: Validation of Forchheimer’s law for flow through porous media with converging boundaries. J. Hydraul. Eng. 126, 63–71 (2000)CrossRefGoogle Scholar
  55. Wen, Z., Huang, G., Zhan, H.: Non-Darcian flow in a single confined vertical fracture toward a well. J. Hydrol. 330, 698–708 (2006)CrossRefGoogle Scholar
  56. Wilkins, J.K.: Flow of water through rockfill and its application to the design of dams. N. Z. Eng. 10, 382–387 (1955)Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  • Ashes Banerjee
    • 1
  • Srinivas Pasupuleti
    • 1
    Email author
  • Mritunjay Kumar Singh
    • 2
  • Sekhar Chandra Dutta
    • 1
  • G. N. Pradeep Kumar
    • 3
  1. 1.Department of Civil EngineeringIndian Institute of Technology (Indian School of Mines)DhanbadIndia
  2. 2.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia
  3. 3.Department of Civil Engineering, SVU College of EngineeringSri Venkateswara UniversityTirupatiIndia

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