Environmental Engineering Reference
In-Depth Information
Therefore,
the relative permeability
k
r
(ψ)
at suction
the measured permeability data. Results show that the pre-
dicted permeability function closely matches the measured
permeability data.
Permeability function calculations for Columbia sandy
silt (Brooks and Corey, 1964) are presented in Figs. 8.15
and 8.16. The calculated relative coefficient-of-permeability
function closely matches
ψ
is
θ
e
y
i
−
N
θ
e
y
i
θ(ψ)
k
r
(ψ)
≈
e
y
i
i
=
j
θ
e
y
i
−
N
θ
s
θ
e
y
i
(8.27)
the experimentally measured
e
y
i
values.
A best-fit SWCC for the experimental data for Super-
stition sand (Richards, 1952) is shown in Fig. 8.17. The
relative coefficients of permeability computed from the mea-
sured SWCC are shown in Fig. 8.18. The calculated rela-
tive coefficient-of-permeability values deviate slightly from
the measured coefficients of permeability as soil suction
increases. Residual conditions are reached at soil suctions
around 10 kPa.
The differences between the calculated and measured
coefficients of permeability may, in part, be due to inade-
quacies in the fit of the experimental SWCC data, as shown
in Fig. 8.17. To test this hypothesis, a second and more
accurate fit was performed on the experimental permeability
data. The improved best-fit curve to the experimental
soil-water characteristic for the Superstition sand is shown
in Fig. 8.19. The predicted permeability function based
on the improved best-fit curve provides a more accurate
fit of the experimental permeability data, as shown in
Fig. 8.20. The results indicate that it is important to obtain
an accurate fit of the experimental soil-water characteristic
data in order to accurately compute the coefficient-of-
permeability function.
The SWCC and permeability function results for Yolo
light clay (Moore, 1939) are presented in Figs. 8.21
and 8.22, respectively. The predicted permeability function
is close to the measured data in the low-soil-suction
range. The unusual shape of the measured permeability
function data suggests that there may have been errors in
the measured permeability data. The clay material may
have experienced some volume change as soil suction was
changed. If this is the case, then the degree-of-saturation
SWCC would have provided a better indication of the actual
air-entry value for Yolo light clay. The estimation of the
coefficient-of-permeability function for clayey soils is gener-
ally less accurate than permeability function estimations for
sandy soils.
SWCC and permeability data on Guelph silt (Elrick
and Bowman, 1964) are shown in Figs. 8.23 and 8.24.
This example illustrates the effect of hysteresis on the
SWCCs and the permeability function. The results indicate
that, although there is considerable hysteresis observed
in the volumetric water content SWCC, hysteresis is less
pronounced in the permeability function
k (θ)
.
i
=
1
The lower limit of the integration given by Eq. 8.21 cor-
responds to the saturated water content
θ
s
. Although the
air-entry value is used as the lower limit of the integration,
any other value between 0 and
ψ
aev
can be used. In other
words, the air-entry value does not have to be known pre-
cisely. However, the value must be positive to perform the
integration on a logarithm scale.
8.2.6 Numerical Results and Comparisons with
Experimental Data
The numerical integration of Eq. 8.21 can be accomplished
as follows. First, it is necessary to determine the four fit-
ting parameters for the SWCC (i.e.,
a
f
,n
f
,m
f
, and
ψ
r
),
as shown in Eq. 8.19. These parameters can be manually
determined or computed using a nonlinear least-squares rou-
tine. Neither of the residual values
θ
r
and
ψ
r
are explicitly
required as part of the curve-fitting procedure. The perme-
ability function can be computed once the SWCC fitting
parameters are known. Equation 8.21 is used in conjunction
with the saturated coefficient of permeability.
Following are some comparisons between measured and
predicted coefficient of permeability curves for several soils.
The soil properties and values of the fitting parameters for
Eq. 8.19 for each soil are listed in Table 8.1.
Figure 8.13 shows a curve that is best-fit to the experi-
mental data for Touchet silt (GE3) from Brooks and Corey
(1964). The predicted coefficient-of-permeability function
based on the best-fit curve in Fig. 8.14 is compared with
Table 8.1 Soil Properties and Fitting parameters for
Fredlund and Xing (1994) Equation on Several Soils
10
−
6
k
s
×
Soil Type
θ
s
(m/s)
a
f
n
f
m
f
ψ
r
Touchet silt
(GE3)
0.430
—
8.34
9.90 0.44
30.0
Columbia
sandy silt
0.458
—
6.01 11.86 0.36
30.0
Yolo light clay 0.375
0.123
2.70
2.05 0.36 100.0
Guelph silt
0.520
3.917
5.61
2.24 0.40 300.0
drying
0.430
—
3.12
4.86 0.23 100.0
wetting
Superstition
sand
8.2.7 Comparison of Data Fit and Theory
The examples of experimental soil-water characteristic data
and measured permeability data presented in the previous
— 18.3
2.77 11.20 0.45 300.0
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