Computational Geosciences

, Volume 22, Issue 1, pp 329–346 | Cite as

A new insight into onset of inertial flow in porous media using network modeling with converging/diverging pores

  • Maziar Veyskarami
  • Amir Hossein Hassani
  • Mohammad Hossein Ghazanfari
Original Paper


The network modeling approach is applied to provide a new insight into the onset of non-Darcy flow through porous media. The analytical solutions of one-dimensional Navier-Stokes equation in sinusoidal and conical converging/diverging throats are used to calculate the pressure drop/flow rate responses in the capillaries of the network. The analysis of flow in a single pore revealed that there are two different regions for the flow coefficient ratio as a function of the aspect ratio. It is found that the critical Reynolds number strongly depends on the pore geometrical properties including throat length, average aspect ratio, and average coordination number of the porous media, and an estimation of such properties is required to achieve more reliable predictions. New criteria for the onset of non-Darcy flow are also proposed to overcome the lack of geometrical data. Although the average aspect ratio is the main parameter which controls the inertia effects, the effect of tortuosity on the onset of non-Darcy flow increases when the coordination number of media decreases. In addition, the higher non-Darcy coefficient does not essentially accelerate the onset of inertial flow. The results of this work can help to better understand how the onset of inertial flow may be controlled/changed by the pore architecture of porous media.


Onset of inertial flow Pore network modeling Sinusoidal/conical throats Critical Reynolds number Forchheimer number Non-darcy flow criterion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank the Salehi’s graphical group and Amir Hossein Mousavi for their help in providing the graphical figures.


  1. 1.
    Forchheimer: Hydrolik. Teubner, Leipzing and Berlin (1914)Google Scholar
  2. 2.
    Ergun, S.: Fluid flow through packed column. Chem. Eng. Prog. 48(2), 89–94 (1952)Google Scholar
  3. 3.
    Huang, H., Ayoub, J.A.: Applicability of the Forchheimer equation for non-Darcy flow in porous media. SPE J. 13(01), 112–122 (2008)CrossRefGoogle Scholar
  4. 4.
    Chaudhary, K., Cardenas, M.B., Deng, W., Bennett, P.C.: The role of eddies inside pores in the transition from Darcy to Forchheimer flowsGeophysical Research Letters 38(24) (2011)Google Scholar
  5. 5.
    Ruth, D., Ma, H.: On the derivation of the Forchheimer equation by means of the average theorem. Transp. Porous Medias 7(3), 225–264 (1992)Google Scholar
  6. 6.
    Whitaker, S.: The Forchheimer equation: a theoretical development. Transp. Porous Media 25, 27–61 (1996)CrossRefGoogle Scholar
  7. 7.
    Barr, D.w.: Turbulent flow through porous media. Ground Water 39(5), 646–650 (2000)CrossRefGoogle Scholar
  8. 8.
    Forchheimer: Wasserbewegung durch Boden. Z. Ver. Dtsch. Ing. 45(5), 1781–1788 (1901)Google Scholar
  9. 9.
    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(3), 268–274 (1987)CrossRefGoogle Scholar
  10. 10.
    Chaudhary, K., Cardenas, M.B., Deng, W., Bennett, P.C.: Pore geometry effects on intrapore viscous to inertial flows and on effective hydraulic parameters. Water Resour. Res. 49(2), 1149–1162 (2013)CrossRefGoogle Scholar
  11. 11.
    Mei, C., Auriault, J.-L.: The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647–663 (1991)CrossRefGoogle Scholar
  12. 12.
    Fourar, M., Radilla, G., Lenormand, R., Moyne, C.: On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Adv. Water Resour. 27(6), 669–677 (2004)CrossRefGoogle Scholar
  13. 13.
    Balhoff, M., Mixkelić, A., Wheeler, M.F.: Polynomial filtration laws for low Reynolds number flows through porous media. Transp. Porous Medias 81(1), 35–60 (2010)CrossRefGoogle Scholar
  14. 14.
    Dullien, F.A.: Porous media: fluid transport and pore structure. Academic (2012)Google Scholar
  15. 15.
    Chen, Z., Lyons, S.L., Qin, G.: Derivation of the Forchheimer law via homogenization. Transp. Porous Medias 44(2), 325–335 (2001)CrossRefGoogle Scholar
  16. 16.
    Hassanizadeh, S.M., Gray, W.G.: High velocity flow in porous media. Transp. Porous Medias 2, 521–531 (1987)Google Scholar
  17. 17.
    Balhoff, M.T., Wheeler, M.F.: A predictive pore-scale model for non-Darcy flow in porous media. SPE J. 14, 579–587 (2009)CrossRefGoogle Scholar
  18. 18.
    Ewing, R.E., Lazarov, R.D., Lyons, S.L., Papavassiliou, D.V., Papavassiliou, J., Qin, G.: Numerical well model for non-Darcy flow through isotropic porous media. Computat. Geosci. 3, 185–204 (1999)CrossRefGoogle Scholar
  19. 19.
    Chilton, T.H., Colburn, A.P.: Pressure drop in packed tubes. Ind Engngc. Chem. 23(8), 913–919 (1931)CrossRefGoogle Scholar
  20. 20.
    Tek, M.R.: Development of a generalized Darcy equation. Trans. AIME 210, 376–377 (1957)Google Scholar
  21. 21.
    Wright, D.E.: Non-linear flow through granular media. J. Hyd. Div. Trans. ASCE 94, 851 (1968)Google Scholar
  22. 22.
    deVries, J.: Prediction of non-Darcy flow in porous media. J. lrrig. Drain. Div. ASCE IR2 (1979)Google Scholar
  23. 23.
    Green, L.J., Duwez, P.: Fluid flow through porous metals. J. Appl. Mech 18, 39–45 (1951)Google Scholar
  24. 24.
    Leonormand, R., Touboul, E., Zarcone, C.: Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189, 165–187 (1988)CrossRefGoogle Scholar
  25. 25.
    Dillard, L.A., Blunt, M.J.: Development of a pore network simulation model to study nonaqueous phase liquid dissolution. Water Resour. Res. 36(2), 439–454 (2000)CrossRefGoogle Scholar
  26. 26.
    Lopez, X., Valvatne, P.H., Blunt, M.J.: Predictive network modeling of single-phase non-Newtonian flow in porous media. J. Colloid. Interf. Sci. 264(1), 256–265 (2003)CrossRefGoogle Scholar
  27. 27.
    Balhoff, M.T., Thompson, K.E.: Modeling the steady flow of yield-stress fluids in packed beds. AIChE J. 50(12), 3034–3048 (2004)CrossRefGoogle Scholar
  28. 28.
    Chaouche, M., Rakotomalala, N., Salin, D., Xu, B., Yorstos, Y.C.: Capillary effects in drainage in heterogeneous porous media. Chem. Engng. Sci. 49, 2447–2466 (1994)CrossRefGoogle Scholar
  29. 29.
    Sahimi, M.: Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Mod. Phys. 65, 1393–1534 (1993)CrossRefGoogle Scholar
  30. 30.
    Thauvin, F., Mohanty, K.K.: Network modeling of non-Darcy flow through porous media. Transp. Porous Media 19, 19–37 (1998)CrossRefGoogle Scholar
  31. 31.
    Wang, X., Thauvin, F., Mohanty, K.K.: Non-Darcy flow through anisotropic porous media. Chem. Eng. Sci. 54, 1859–1869 (1999)CrossRefGoogle Scholar
  32. 32.
    Piri, M., Blunt, M.J.: Three-dimensional mixed-wet random pore-scale network modeling of two- and three-phase flow in porous media. I. Model description. Phys. Rev. E: Stat. Phys., Plasmas, Fluids 71, 026301 (2005)CrossRefGoogle Scholar
  33. 33.
    Stark, K.P.: Fundamentals of transport phenomena in porous media, vol. 2. Elsevier, Amsterdam (1972)Google Scholar
  34. 34.
    Du Plessis, J.P., Masliyah, J.H.: Mathematical modeling of flow through consolidated isotropic porous media. Transp. Porous Media 3, 145–161 (1988)CrossRefGoogle Scholar
  35. 35.
    Ma, H., Ruth, D.W.: The microscopic analysis of high Forchheimer number flow in porous media. Transp. Porous Media 13, 139–160 (1993)CrossRefGoogle Scholar
  36. 36.
    Zeng, Z., Grigg, R.: A criterion for non-Darcy flow in porous media. Transp. Porous Media 63, 57–69 (2006)CrossRefGoogle Scholar
  37. 37.
    Martins, A.A., Laranjeira, P.E., Lopes, J.C.B., Dias, M.M.: Network modeling of flow in a packed bed. AIChE J. 53(1), 91–107 (2007)CrossRefGoogle Scholar
  38. 38.
    Fatt, M.: The network model of porous media. I. Capillary pressure characteristics. Pet. Trans. 207, 142–164 (1956)Google Scholar
  39. 39.
    Thompson, K.E., Fogler, H.S.: Modelling flow in disordered packed beds from pore-scale fluid mechanics. AIChE J. 43, 1377–1389 (1997)CrossRefGoogle Scholar
  40. 40.
    Krohn, C.E., Thompson, A.H.: Fractal sandstones pores: automated measurements using scanning-electron microscope images. Phys. Rev. B Condens. Matter. 33, 6366–6374 (1986)CrossRefGoogle Scholar
  41. 41.
    Caruso, L., Simmons, G., Wilkens, R.: The physical properties of a set of sandstone—part 1. The samples. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 22, 381–392 (1985)CrossRefGoogle Scholar
  42. 42.
    Lao, H.-W., Neeman, H.J., Papavassilou, D.V.: A pore network model for the calculation of non-Darcy flow coefficients in fluid flow through porous media. Chem. Eng. Comm. 191(10), 1285–1322 (2004)CrossRefGoogle Scholar
  43. 43.
    Mohanty, K.K., Salter, S.J.: Multiphase flow in porous media: II. pore-level modeling. Paper presented at the Annual Fall Technical Conference of the SPE-AIME, New OrleansGoogle Scholar
  44. 44.
    Veyskarami, M., Hassani, A.H., Mohammad Hossein Ghazanfari, M.H.: Modeling of non-Darcy flow through anisotropic porous media: role of pore space profiles. Chem. Eng. Sci. 151, 93–104 (2016)CrossRefGoogle Scholar
  45. 45.
    Arns, J.-Y., Robins, V., Sheppard, A.P., Sok, R.M., Pinczewski, W.V., Knackstedt, M.A.: Effect of network topology on relative permeability. Transp. Porous Media 55(1), 21–46 (2004)CrossRefGoogle Scholar
  46. 46.
    Raoof, A., Hassanizadeh, S.M.: A new method for generating pore-network models of porous media. Transp. Porous Media 81(3), 391–407 (2010)CrossRefGoogle Scholar
  47. 47.
    Ioannidis, M., Chatzis, I.: On the geometry and topology of 3D stochastic porous media. J. Colloid Interface Sci. 229(2), 323–334 (2000)CrossRefGoogle Scholar
  48. 48.
    Sok, R.M., Knackstedt, M.A., Sheppard, A.P., Pinczewski, W.V., Lindquist, W., Venkatarangan, A., Paterson, L.: Direct and stochastic generation of network models from tomographic images; effect of topology on residual saturations. Transp. Porous Media 46(2-3), 345–371 (2002)CrossRefGoogle Scholar
  49. 49.
    Vasilyev, L., Raoof, A., Nordbotten, J.M.: Effect of mean network coordination number on dispersivity characteristics. Transport Porous Media (2012)Google Scholar
  50. 50.
    Ioannidis, M., Chatzis, I.: Network modelling of pore structure and transport properties of porous media. Chem. Eng. Sci. 54, 1859–1869 (1993)Google Scholar
  51. 51.
    Diaz, C.E., Chatzis, I., Dullien, F.A.L.: Simulation of capillary pressure curves using bond correlated site percolation on a simple cubic network. Transp. Porous Med 2, 215–240 (1987)CrossRefGoogle Scholar
  52. 52.
    Sochi, T.: Newtonian flow in converging-diverging capillaries. arXiv:1108.0163v2 (2012)
  53. 53.
    Formaggia, L., Lamponi, D., Quarteroni, A.: One-dimensional models for blood flow in arteries. J. Eng. Math. 47(3/4), 251–276 (2003)CrossRefGoogle Scholar
  54. 54.
    Sochi, T.: Newtonian flow in converging-diverging capillaries. Int. J. Model. Simul. Sci. Comput. 4(03), 1350011 (2013)CrossRefGoogle Scholar
  55. 55.
    Cengel, Y., Cimbala, J.: Fluid mechanics; fundamentals and application. McGraw-Hill, New York (2006)Google Scholar
  56. 56.
    Jones, S.: Using the inertial coefficient, b, to characterize heterogeneity in reservoir rock. Society of Petroleum Engineers, SPE Annual Technical Conference and Exhibition (1987)Google Scholar
  57. 57.
    Janicek, J.D., Katz, D.L.V.: Applications of unsteady state gas flow calculations (1955)Google Scholar
  58. 58.
    Coles, M., Hartman, K.: Non-Darcy measurements in dry core and the effect of immobile liquid. In: SPE Gas Technology Symposium. Society of Petroleum Engineers (1998)Google Scholar
  59. 59.
    Geertsma, J.: Estimating the coefficient of inertial resistance in fluid flow through porous media. Soc. Pet. Eng. J. 14(05), 445–450 (1974)CrossRefGoogle Scholar
  60. 60.
    Firoozabadi, A., Katz, D.L.: An analysis of high-velocity gas flow through porous media. J. Pet. Technol. 31(02), 211–216 (1979)CrossRefGoogle Scholar
  61. 61.
    Kataja, M., Rybin, A., Timonen, J.: Permeability of highly compressible porous medium. J. Appl. Phys. 72, 1271 (1992)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Maziar Veyskarami
    • 1
  • Amir Hossein Hassani
    • 1
  • Mohammad Hossein Ghazanfari
    • 1
  1. 1.Chemical and Petroleum Engineering DepartmentSharif University of TechnologyTehranIran

Personalised recommendations