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

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Abstract

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.

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Abbreviations

A :

Cross-sectional area (cm2)

A p :

Total pore cross-sectional area (cm2)

C 1 :

Constant, as defined by Eq. 1

C 2 :

Water absorption coefficient

d :

Euclidean dimension

D :

Fractal dimension

D f :

Pore fractal dimension

D T :

Tortuosity fractal dimension

K :

Effective permeability (Darcy)

k :

Time exponent

L f :

Actual length that the flow travels (cm)

L s :

Straight-line length (cm)

M :

Imbibed weight in porous media (g)

m :

Imbibed weight in a single capillary (g)

N wt :

Imbibed volume in porous media

n :

Pores/capillaries number

P c :

Capillary pressure (Pa)

\({\overline r }\) :

Effective radius for pores (cm)

r eff :

Statistical effective radius (cm)

S wf :

Water saturation behind imbibition front (fraction)

S wi :

Initial water saturation (fraction)

t :

Imbibition time (s)

V b :

Bulk volume

V p :

Pore volume

δ :

Pore shape factor

σ :

Surface tension (N/m)

μ :

Viscosity (Pa s)

\({\varepsilon }\) :

Measuring unit

λ:

Pore diameter (cm)

\({\phi _2}\) :

Areal porosity

\({\phi _3}\) :

Volume porosity

Ω:

Integrating region ranging the minimum to the maximum capillaries

θ :

Contact angle

ρ :

Liquid density (g/cm3)

τ :

Tortuosity

av:

Average value

max:

Maximum value

min:

Minimum value

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Authors and Affiliations

  1. Institute of Geophysics and Geomatics, China University of Geosciences, 388 Lumo Road, Wuhan, 430074, Hubei, P. R. China

    Jianchao Cai

  2. School of Physics, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, 430074, Hubei, P. R. China

    Jianchao Cai & Boming Yu

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  1. Jianchao Cai
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Cai, J., Yu, B. A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media. Transp Porous Med 89, 251–263 (2011). https://doi.org/10.1007/s11242-011-9767-0

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