I n s t r u m e n t s

a n d

M e t h o d s

Pore-Space

CharacterizationofWetSnowinthePendularRegime

JimFrankenfield

Cyberspace Snow and Avalanche Center (CSAC) snow@csac.org or snowman@aip.org

ABSTRACT

The geometry of wetting fluid pendular rings in a porous
media can be used to characterize the media in this satura-
tion regime.A number of expressions forthe volume and
surface area of pendularrings in an ideal soil have been
publishedbutarenotinagreement.Acorrectsetof
equations has been derived for an idealsoil of spheres and
alsoforasinteredmediarepresentedbyallowingthe
spherestooverlap.Thesevolumeandarea expressions
allow for thepossibility of a non-zero contactanglebetween
the wetting fluid and the solid.
The transition between the pendular and funicular re-
gimes occurs when either the pendular rings merge or the
capillary pressure (Pc) across the wetting/non-wetting in-
terface becomes zero. Use of the Laplace Equation and the
ring geometry leads toan implicit equationforthe P
The roots ofequation are inagreement^C=0
point.thisgeneral
withmeasur edvaluesofthetransition
saturation.CharacteristiccurvesofPvs.Saturation can
be generated as well.The basis forthis^C
is the thermody-
namic relation P=dE/dV. Energy can be calculated by us-
ing the interfacial area expressions and the relatedsurface
energies.Young's equation and symmetry can be used to
reducethe necessary surface energies to only the wetting/
non-wettingsurface tension.Aspreadsheet can thenbe
used to generate the characteristic curves.

VOLUMES AND AREAS OF PENDULAR RINGS

Volume and area expressions forpendular rings around a
point contact between two spherical particles (Fig 1) have
been previously published. However, volume expressions
given by Rose [1],Gardner and Gardner [2],Dallavalle [3]

and Gvirtzman and Roberts [4] are found to differ.
The correct expression forthe volume of such a ring with
an arbitrary contact angle between the wetting fluid and
the solid particle has been derived by Frankenfield [5] and

[!]
V_p= 2 p R ^3[!][!]j )^2[!][!]
[!]^{(1 -cos}_[!]^{[!] 1 +cotw [!] sinj+cotw [!]}
[!]^[!][!][!]1

is
When the contact angle is zero this reduces to the result
given by Rose [1].The correct area forthe wetting fluid -
nonwettingfluidinterfaceofsucharingwasgivenby
Gvirtzman and Roberts [4] and is

A_wn=4p R ²

cosj)cot
w+sin
j_]-(
1-cosj)
}

The areaof the interface between thewetting fluid and the

A_ws=2pR²(1-cosj)

solid is
A somewhat more realistic approximation tothe particle
interface in wet snow is obtained by allowing it to flatten.
Geometrically,this amounts to letting the two spheres in
the ideal soil case overlap. (Figure 2) This is still a rough
approximation. The nature of this interface is discussed in
more detail by Colbeck [6]. For this geometry, Frankenfield
[5] has shown that the correctvolume and areaexpressions

Fig 2 - Pendular ringabout a flattened contact between two ice parti-
cles

129