Transport in Porous Media

, Volume 128, Issue 1, pp 303–318 | Cite as

A Two-Parametric Model for Gas Flow in Low-Permeable Porous Materials

  • Peter PauliniEmail author


Capillary flow at low pressure gradients is determined by the pore structure of the matrix and the viscosity of the fluid. Traditionally, the conductivity of porous materials is characterized by a single parameter—the coefficient of permeability—which includes all geometric and structural resistances of the capillary structure. The fluid is described by its dynamic viscosity, which is independent of pressure in liquids. Gases, on the other hand, are highly compressible which means that their coefficient of permeability is becoming pressure dependent. The Klinkenberg effect takes account of this behaviour by introducing the notion of intrinsic permeability; nevertheless, it remains applicable only to a limited pressure range. At low- and high-pressure gradients, the Klinkenberg linearization deviates considerably from the real flow behaviour. This paper demonstrates that microporous capillary resistance to gas flow cannot be adequately described by a single material parameter. On the basis of ideal gas law conditions, the mass flow is derived by a two-parametric power law. Pore pressure decline is characterized by a permeability exponent n and a reference velocity v1, replacing both dynamic viscosity of the fluid and permeability. The actual pressure distribution over the flow path is expressed by means of an equation of motion which leads to a logarithmic linearization of the material parameters. Velocity and acceleration vectors are derived and discussed in relation to the actual pressure distribution. The two parameters n and v1 remain constant throughout and independent of pore pressure and are thus used to describe the flow rates over the entire pressure range. This model is then validated by means of flow analyses based on data given in the literature and compared to the classical theory of permeability.


Permeability Microstructure Pressure distribution Capillary gas flow Klinkenberg effect 



Generous research and experimental conditions provided by the Austrian Science Act UOG 1975 are deeply appreciated. Special thanks to Prof. Peter Wagner for proofreading and valuable comments. Thanks are also expressed to Prof. Wu for giving permission to use his accurate test data.

Supplementary material

11242_2019_1245_MOESM1_ESM.xlsx (41 kb)
Supplementary material 1 (XLSX 41 kb)


  1. Al-Hussainy, R., Ramey, H.J., Crawford, P.B.: The flow of real gases through porous media. J. Petrol. Technol. 18(5), 624–636 (1966)CrossRefGoogle Scholar
  2. Baer, J.: Dynamics of Fluids in Porous Media. Dover Publ, New York (1972)Google Scholar
  3. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn. Wiley, New York (2002)Google Scholar
  4. Carman, P.C.: Flow of Gases Through Porous Media. Butterworth, London (1956)Google Scholar
  5. Carman, P.C.: Fluid flow through a granular bed. Trans. Inst. Chem. Eng Lond. 15, 150–156 (1937)Google Scholar
  6. Civan, F.: Effective correlation of apparent gas permeability in tight porous media. Transp. Porous Med. 82, 375–384 (2010)CrossRefGoogle Scholar
  7. Cornell, D., Katz, D.L.: The flow of gases through consolidated porous media. Ind. Eng. Chem. 45, 2145–2152 (1953)CrossRefGoogle Scholar
  8. Cui, X., Bustin, A.M.M., Bustin, R.M.: Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids 9, 208–223 (2009)CrossRefGoogle Scholar
  9. de Waele, A.: Die Aenderung der Viskosität mit der Schergeschwindigkeit disperser Systeme. Kolloid-Zeitschrift 36(6), 332–333 (1925)CrossRefGoogle Scholar
  10. Fetkovich, M.J., Fetkovich, E.J., Fetkovich, M.D.: Useful concepts for decline-curve forecasting, reserves estimation and analysis. SPE Reserv. Eng. 11, 13–22 (1996)CrossRefGoogle Scholar
  11. Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Levant Books, Kolkata (2005)Google Scholar
  12. Heller, R., Vermylen, J., Zoback, M.: Experimental investigation of matrix permeability of gas shales. AAPG Bull. 98(5), 975–995 (2014)CrossRefGoogle Scholar
  13. Hershey, D.: Transport Analysis. Plenum Press, New York (1973)CrossRefGoogle Scholar
  14. Javadpour, F.: Nanopores and apparent permeability of gas flow in mudrocks (Shales and Siltstone). JCPT 48(8), 16–21 (2009)CrossRefGoogle Scholar
  15. Klinkenberg, L.J.: The permeability of porous media to liquid and gases, Drilling and production practice. Production Practice, American Petroleum Institut, pp. 200–213 (1941)Google Scholar
  16. Kozeny, J.: Über kapillare Leitung des Wassers im Boden. Sitzungsber. Akad. Wiss. Wien 136, 271–306 (1927)Google Scholar
  17. Paulini P., Nasution F.: Air permeability of near surface concrete. In: Toutlemonde, F. (Ed) Concrete under Severe Conditions, Environment and Loading, Proc. 5th Int. Conf. Concrete under Severe Conditions Environment and Loading, LCPC, Paris, pp. 241–248 (2007)Google Scholar
  18. RILEM TC 116-PCD: Permeability of concrete as a criterion of its durability. Mater. Struct. 32, 174–179 (1999)CrossRefGoogle Scholar
  19. Scheidegger, A.E.: The Physics of Flow Through Porous Media, 3rd edn. University of Toronto Press, Toronto (1974)Google Scholar
  20. Tanikawa, W., Shimamoto, T.: Klinkenberg effect for gas permeability and its comparison to water permeability for porous sedimentary rocks. Hydrol. Earth Syst. Sci. Discuss. 3, 1315–1338 (2006)CrossRefGoogle Scholar
  21. Wang, G., Ren, T., Wang, K., Zhou, A.: Improved apparent permeability models of gas flow in coal with Klinkenberg effect. Fuel 128, 53–61 (2014)CrossRefGoogle Scholar
  22. Villani, C., Loser, R., West, M.J., Di Bella, C., Lura, P., Weiss, W.J.: An inter lab comparison of gas transport testing procedures: oxygen permeability and oxygen diffusivity. Cem. Concr. Compos. 53, 357–366 (2014)CrossRefGoogle Scholar
  23. Wu, Y.S., Pruess, K., Persoff, P.: Gas flow in porous media with Klinkenberg effects. Transp. Porous Med. 32, 117–137 (1998)CrossRefGoogle Scholar
  24. Zhu, G.Y., Liu, L., Yang, Z.M., Liu, X.G., Guo, Y.G., Cui, Y.T.: Experimental and mathematical model of gas flow in low permeability porous media. In: Proc. Fifth Int. Conf. Fluid Mechanics, Aug. 15–19, Shanghai, China, Tsinghua University Press & Springer, pp. 534–537 (2007)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Material TechnologyUniversity InnsbruckInnsbruckAustria

Personalised recommendations