Transport in Porous Media

, Volume 89, Issue 1, pp 63–73 | Cite as

Capillary Pressure Curve for Liquid Menisci in a Cubic Assembly of Spherical Particles Below Irreducible Saturation

  • Jacob Bear
  • Boris Rubinstein
  • Leonid Fel


The capillary pressure–saturation relationship, P c(S w), is an essential element in modeling two-phase flow in porous media (PM). In most practical cases of interest, this relationship, for a given PM, is obtained experimentally, due to the irregular shape of the void space. We present the P c(S w) curve obtained by basic considerations, albeit for a particular class of regular PM. We analyze the characteristics of the various segments of the capillary pressure curve. The main features are the behavior of the P c(S w) curve as the wetting-fluid saturation approaches zero, and as this saturation is increased beyond a certain critical value. We show that under certain conditions (contact angle, distance between spheres, and saturation), the value of the capillary pressure may change sign.


Porous media Two-phase flow Wetting fluid Capillary pressure curve 

List of Symbols


Distance between spheres


Number of spheres per unit cell


Mean curvature of meniscus


Capillary pressure

Pw, Pn

Pressures in wetting and non-wetting fluids


Radius of sphere

Sw, Sn

Saturations of wetting and non-wetting fluids

Swr, Snr

Irreducible w-fluid and residual n-fluid saturations


Volume of pendular ring


Surface tension of the w–n interface


Contact angle




Filling angle


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bear J., Cheng A.H.-D: Modeling Groundwater Flow and Contaminant Transport. Springer, Berlin (2010)CrossRefGoogle Scholar
  2. Brooks R.J., Corey A.T.: Hydraulic properties of porous media. Hydrology Paper 3. Colorado State University, Fort Collins (1964)Google Scholar
  3. Collins R.E.: Flow of Fluids Through Porous Materials. Reinhold Publishing Corp, New York (1961)Google Scholar
  4. Gwirtzman H., Roberts P.V.: Pore scale spatial analysis of two immiscible fluids. Water Resour. Res. 27(6), 1165–1176 (1991)CrossRefGoogle Scholar
  5. Melrose J.C.: Model calculations for capillary condensation. AIChE J. 12(5), 986–994 (1966)CrossRefGoogle Scholar
  6. Nitao J.J., Bear J.: Potentials and their role in transport in porous media. Water Resour. Res. 32(2), 225–250 (1996)CrossRefGoogle Scholar
  7. Or D., Tuller M.: Liquid retention and interfacial area in variably saturated porous media: upscaling from single-pore to sample-scale model. Water Resour. Res. 35(12), 3591–3600 (1999)CrossRefGoogle Scholar
  8. Orr F.M., Scriven L.E., Rivas A.P.: Pendular rings between solids: meniscus properties and capillary force. J. Fluid Mech. 67(4), 723–742 (1975)CrossRefGoogle Scholar
  9. Patzek T.: Verification of a complete pore network simulator of drainage and imbibition. Soc. Petrol. Eng. 71310, 144–156 (2001)Google Scholar
  10. Plateau, J.: The figures of equilibrium of a liquid mass, pp. 338–369. The Annual Report of the Smithsonian Institution, Washington, DC (1864)Google Scholar
  11. Ramirez-Flores, J.C., Bachmann, J., Marmur, A.: Direct determination of contact angles of model soils in comparison with wettability characterization by capillary rise. J. Hydrol. (2010)Google Scholar
  12. Rose W.: Volume and surface areas of pendular rings. J. Appl. Phys. 29(4), 687–691 (1958)CrossRefGoogle Scholar
  13. Rossi C., Nimmo J.R.: Modeling of water retention from saturation to oven dryness. Water Resour. Res. 30(3), 701–780 (1994)CrossRefGoogle Scholar
  14. van Genuchten M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. 44, 892–898 (1980)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Civil EngineeringTechnionHaifaIsrael
  2. 2.Stowers Institute for Medical ResearchKansas CityUSA

Personalised recommendations