Environmental Engineering Reference
In-Depth Information
1.0
0.8
(
i
,
θ
i
)
0.6
0.4
0.2
0
0
5
10
15
20
25
Soil suction, kPa
Figure 8.5
Typical SWCC for calculating water permeability function (after Marshall, 1958;
Kunze et al., 1968).
proposed for the calculation of the permeability function
k
w
(θ)
using Eq. 8.8. The main difference between the vari-
ous procedures lies in the interpretation of the pore interac-
tion term (i.e.,
θ
s
/
N
in Eq. 8.8) (Green and Corey, 1971a).
where:
[i.e.,
T
s
ρ
w
g/
2
v
w
θ
s
/N
2
A
d
=
adjustment
factor
kPa
2
)
].
s
−
1
(
m
·
·
The matching factor
k
s
/k
sc
based on the saturated coef-
ficient of permeability is used to provide a smooth transition
between the measured saturated coefficient of permeabil-
ity and the remainder of the permeability function. The
above computational procedure for obtaining the perme-
ability function has been found to be most successful for
sandy soils having a relatively narrow pore-size distribution
(Nielsen et al., 1972). A comparison between the permeabil-
ity function
k
w
(θ)
computed from Eq. 8.8 and experimental
data is shown in Fig. 8.6 for a fine sand. The SWCC for
the sand and the comparison of its permeability function are
shown in Figs. 8.6a and 8.6b, respectively. If the information
shown in Figs. 8.6a and 8.6b is cross plotted, a more con-
ventional form for the permeability function is obtained (i.e.,
logarithm of permeability versus logarithm of soil suction),
as shown in Fig. 8.7.
The coefficient of permeability
k
w
at a specific volumetric
water content
θ
is computed directly from Eq. 8.8. The shape
of the permeability function is determined by the terms
inside the summation sign portion of the equation which
are obtained from the SWCC. However, the magnitude of
the permeability function needs to be adjusted to coincide
with the measured saturated coefficient of permeability
k
s
when soil suction is zero. The permeability function can be
computed directly from the SWCC if the saturated coeffi-
cient of permeability is measured because all of the terms in
front of the summation sign in Eq. 8.8 can be considered as
an adjustment factor. The permeability function
k
w
(θ)
can
therefore be written as follows:
The terms in the “adjustment factor” in Eq. 8.8 result from
attempting to calculate the saturated coefficient of perme-
ability from Poiseuille's theory. In geotechnical engineering
practice the saturated coefficient of permeability of the soil is
usually measured in the laboratory. Then the above integra-
tion procedure is used to simply calculate the shape of the
unsaturated permeability function. The unsaturated coeffi-
cient of permeability is therefore equal to the saturated coef-
ficient of permeability multiplied by the summation terms
shown in Eq. 8.9.
The data for a SWCC are all that is required for the
computation of the shape of the permeability function [i.e.,
k
w
(θ)
]. The following example is used to illustrate the tech-
nique by which the coefficient of permeability
k
w
(θ)
can be
computed as a function of volumetric water content.
Let us consider the SWCC shown in Fig. 8.8. The compu-
tation of
k
w
(θ)
is illustrated for the drying curve. The drying
curve is first divided into
m
equal intervals of volumetric
water content, as shown in Fig. 8.9. In this case, the drying
curve has maximum and minimum volumetric water con-
tents (percentage form) of 38.8 and 10.9%, respectively. The
drying curve is divided into 20 intervals with 20 midpoints.
The first volumetric water content corresponds to saturated
conditions. Each volumetric water content midpoint,
θ
w
i
,
corresponds to a particular matric suction,
u
a
−
u
w
i
.The
midpoints are numbered starting from point 1 (i.e.,
i
=
1)
to point 20 (i.e.,
i
m
). The permeability function
k
w
(θ)
is computed in accordance with the following equation:
=
(
2
j
j
m
2
i)
u
a
−
u
w
−
2
k
s
k
sc
k
w
(θ)
i
=
A
d
+
1
−
(
2
j
j
(8.10)
m
2
i)
u
a
−
u
w
−
2
j
=
i
k
w
(θ)
i
=
k
s
+
1
−
i
=
1
,
2
,...,m
(8.9)
j
=
i
Search WWH ::
Custom Search