Geoscience Reference
In-Depth Information
Fig. 8.10 Blow-up sketch of a water-air interface in a cylindrical
capillary tube of inside radius
R
; the radius of curvature of this
meniscus is
r
=
R
/
cos
α
, in which
α
is the wetting angle. The
pressure in the water at the meniscus in the center of the tube is
p
w
=
−γ
w
H
c
.
α
R
r
H
c
Example 8.1. Special case of a capillary tube
In the case of the capillary tube, in which water is in contact with air at atmospheric
pressure, as illustrated in Figure 8.10, the pressure increase
p
w
) in the tran-
sition from water to air equals the suction (i.e. negative pressure) in the water (
p
=
(
p
a
−
−
p
w
). If
the radius of the tube
R
is small enough so that the interface can be assumed to have the
curvature of a sphere, the radii of curvature are both equal to
R
is the
wetting angle of water with the glass. The wetting angle between water and quartz-like
materials and other soil minerals is usually (except in the presence of impurities) small;
the water pressure according to Equation (8.3) becomes then simply
/
cos
α
, where
α
p
w
=−
2
σ/
R
(8.5)
R
)at18
◦
C when
both
H
c
and
R
are in cm and the specific weight is constant. Equation (8.5) is valid for
the ideal case of a tube of circular cross section with a radius
R
. Figure 8.4 illustrates
how the water can be held in an analogous way in the pores of an irregular array of
particles; for such pores of irregular cross section, Equation (8.5) can be used to define
an effective radius: this is the radius
R
of a capillary tube of circular cross section, with
the same value of
and the capillary rise is
H
c
=−
p
w
/γ
w
(
=
H
) or roughly
H
c
=
(0
.
149
/
p
=−
p
w
across its water-air interface.
Pore size distribution
In numerous studies the effective radius of curvature
R
of the air-water interface in a
pore, as used in Equation (8.5), has also been taken as a measure of the size of that
pore; by analogy with pipes with a circular cross section,
R
is usually taken to be equal
to twice the hydraulic radius
R
h
, as defined in Equation (5.40), that is the ratio of the
cross-sectional area of the pore to its wetted perimeter. Equation (8.5) indicates that
the pressure drop across the air-water interface in any pore is inversely proportional
to the size of the pore. This means that with increasing negative pressures or suctions
−
=
γ
w
H
) increasingly smaller pores are being emptied. The soil water characteristic
relates the suction
H
with the water content
p
w
(
, that is the water still left in the soil. Hence,
if it is assumed that at a given suction
H
all pores above size
R
are empty, the soil water
characteristic with (8.5) is equivalent with a cumulative pore size distribution. In other
θ
