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Capillary hyperdisperson of wetting liquids in fractal porous media

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Abstract

Recent displacement experiments show ‘anomalously’ rapid spreading of water during imbibition into a prewet porous medium. We explain this phenomenon, calledhyperdispersion, as viscous flow along fractal pore walls in thin films of thicknessh governed by disjoining forces and capillarity. At high capillary pressure, total wetting phase saturation is the sum of thin-film and pendular stucture inventories:S w =S tf +S ps . In many cases, disjoining pressure ∏ is inversely proportional to a powerm of film thicknessh, i.e. ∏ ∞h −m , so thatS tf ∞P c −1/m. The contribution of fractal pendular structures to wetting phase saturation often obeys a power lawS ps ∞P c (3−D), whereD is the Hausdorff or fractal dimension of pore wall roughness. Hence, if wetting phase inventory is primarily pendular structures, and if thin films control the hydraulic resistance of wetting phase, the capillary dispersion coefficient obeysD c ∞S v w , where v=[3−m(4−D) ]/m(3−D). The spreading ishyperdispersive, i.e.D c (S w ) rises as wetting phase saturation approaches zero, ifm>3/(4−D),hypodispersive, i.e.D c (S 2) falls as wetting phase saturation tends to zero, ifm<3/(4−D), anddiffusion-like ifm=3/(4−D). Asymptotic analysis of the ‘capillary diffusion’ equation is presented.

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Abbreviations

a :

Pre-exponential factor in power-law scaling ofD c withS w

b :

Exponent in power-law scaling ofD c withS w

B :

Ratio of gravitational forces to viscous forces

D :

Hausdorff or fractal dimension

D c :

Capillary dispersion coefficient (m2s−1)

f(r) :

Pore radius probability density function

F :

Fractional flow

G :

Mutual permeability (m3s Kg−1)

g :

Gravitational potential per unit mass (m s−)

h :

Film thickness (m)

k :

Absolute permeability (m2)

k i :

Effective permeability of phasei(m2)

m :

Power law exponent of ∏ (h) in the limit of smallh

P c :

Capillary pressure,P nw −P w (Pa)

P i :

Pressure in bulk phasei (Pa)

S :

Wetting phase saturation

t :

Time coordinate (s)

v :

Net flow velocity (m s−1)

x :

Spatial coordinate (m)

z :

Elevation (m)

γ :

Interfacial tension (N m−1)

δ:

Density contrast

λ :

Boltzmann transform

Μ i :

Viscosity of phasei (Kg m−1 s−1)

v :

Exponent in power-law scaling ofD c withS w

∏:

Disjoining pressure (Pa)

ρ i :

Density of phasei (Kg m−3)

σ 2 :

Variance of parent normal distribution

Φ :

Porosity

nw :

Nonwetting phase (oil, air)

ps :

Pendular structure

tf :

Thin film

w :

Wetting phase (water)

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Authors and Affiliations

  1. Department of Chemical Engineering and Materials Science, University of Minnesota, 55455, Minneapolis, MN, USA

    Pedro G. Toledo, H. Ted Davis & L. E. Scriven

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  1. Pedro G. Toledo
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  2. H. Ted Davis
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  3. L. E. Scriven
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Toledo, P.G., Ted Davis, H. & Scriven, L.E. Capillary hyperdisperson of wetting liquids in fractal porous media. Transp Porous Med 10, 81–94 (1993). https://doi.org/10.1007/BF00617512

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