Transport in Porous Media

, Volume 89, Issue 2, pp 251–263 | Cite as

A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media

  • Jianchao Cai
  • Boming YuEmail author


In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the \({\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}}\) law is obtained (here D T is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (D T = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.


Capillary imbibition Fractal Porous media Time exponent 

List of symbols


Cross-sectional area (cm2)


Total pore cross-sectional area (cm2)


Constant, as defined by Eq. 1


Water absorption coefficient


Euclidean dimension


Fractal dimension


Pore fractal dimension


Tortuosity fractal dimension


Effective permeability (Darcy)


Time exponent


Actual length that the flow travels (cm)


Straight-line length (cm)


Imbibed weight in porous media (g)


Imbibed weight in a single capillary (g)


Imbibed volume in porous media


Pores/capillaries number


Capillary pressure (Pa)

\({\overline r }\)

Effective radius for pores (cm)


Statistical effective radius (cm)


Water saturation behind imbibition front (fraction)


Initial water saturation (fraction)


Imbibition time (s)


Bulk volume


Pore volume

Greek letters


Pore shape factor


Surface tension (N/m)


Viscosity (Pa s)

\({\varepsilon }\)

Measuring unit


Pore diameter (cm)

\({\phi _2}\)

Areal porosity

\({\phi _3}\)

Volume porosity


Integrating region ranging the minimum to the maximum capillaries


Contact angle


Liquid density (g/cm3)





Average value


Maximum value


Minimum value


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institute of Geophysics and GeomaticsChina University of GeosciencesWuhanP. R. China
  2. 2.School of PhysicsHuazhong University of Science and TechnologyWuhanP. R. China

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