Environmental Engineering Reference
In-Depth Information
Table 5.6 Curve-Fitting Parameters for Seven Soils Using Simplified SWCC Equation
Curve-Fitting Parameters
ID
Soil
w
s
S
1
a
b
w
r
ψ
r
, kPa
Regina clay
b
10
33
1
1.057
0.261
1.07
×
9.838
0.131
2520
10
4
2
Jossigny silt
0.467
0.086
7.13
×
1.404
0.064
5810
10
9
3
Kaolin
0.830
0.145
1.92
×
2.527
0.049
6990
10
3
×
4
Beaver Creek sand
0.242
0.0046
3.70
6.186
0.020
4.43
10
4
5
Saskatchewan silty sand
0.244
0.015
5.30
×
4.052
0.098
18.7
10
5
6
Processed silt
0.250
0.036
3.01
×
2.824
0.114
124.
Indian Head till
c
10
8
7
0.382
0.084
1.47
×
2.627
0.078
1880
a
Residual soil suctions were calculated using the curve-fitting parameters
a
and
b
.
b
Several empirical data points were added along the desaturation part of the Regina clay for ease in curve fitting.
c
Data points for the SWCC at suctions higher than 1500 kPa were measured by Sillers (1997).
are independent and have physical soil propertymeaning (e.g.,
water content at 1 kPa soil suction, slope of the curve at soil
suctions less than the air-entry value, the air-entry value,
residual soil suction, maximum slope of the SWCC). The
curve-fitting parameters for the SWCC can also be estimated
from basic soil classification properties. The proposed SWCC
equations closely represent gravimetric water content-soil
suction conditions over the entire range of soil suction values.
The simplified SWCC equation also requires five curve-
fitting parameters, but only four parameters are truly inde-
pendent (i.e., since residual soil suction can be approximated
using the curve-fitting parameters
a
and
b
). Of the four curve-
fitting parameters, only two parameters have physical mean-
ing (i.e., the gravimetricwater content at 1 kPa soil suction and
the slope of the SWCC at soil suctions less than the air-entry
value). The advantage of this equation is its mathematical
simplicity while still retaining an accurate representation of
the entire SWCC. The simplified SWCC equation retains a
defined slope at soil suctions less than the air-entry value.
The equation is simple to use and can be readily implemented
in analytical solutions for geotechnical engineering problems.
consistent geometric relationship exists between the shape of
the SWCC and the equation parameters.
Wetting SWCCs that achieve a maximum degree of sat-
uration that is less than 100% can be represented by multi-
plying the proposed equations by the maximum achievable
degree of saturation.
5.7.1 Unimodal Equation with One Bending Point
A rotated and translated hyperbola was used to represent the
first type of SWCC curve: the two straight lines defined by
the coordinates (0, 1)
(ψ
aev
,
1
)
and
(
10
6
,
0
)
for the hyperbola
asymptotes. The degree of saturation, S, equation is written
as follows:
r
2
)
ln
(ψ/ψ
aev
)
tan
θ
g
(
1
+
S
=
(
1
−
r
2
tan
2
θ
g
)
r
2
ln
2
ψ
ψ
aev
tan
2
θ
g
a
2
(
1
r
2
tan
2
θ
g
)
1
+
−
−
+
r
2
tan
2
θ
g
tan
2
θ
g
)
(5.74)
1
−
(
1
+
where:
ψ
=
any soil suction value,
ψ
aev
=
air-entry value for the soil,
θ
g
=
-
λ
/2, the hyperbola rotation angle,
5.7 GITIRANA AND FREDLUND (2004) SWCC
r
=
tan (
λ
/2), the aperture angle tangent, and
Gitirana and Fredlund (2004) proposed an equation with
independent physical parameters for the SWCC. The equation
is a combination of two rotated and translated hyperboles.
The equation was intended for use with the degree-of-
saturation SWCCs.
Three types of equations were proposed to fit various shapes
of the SWCC: (1) a unimodal equationwith one bending point,
(2) a unimodal equation with two bending points, and (3) a
bimodal equation. The equations are based on the general
hyperbolic equation in the coordinate system, log
ψ
versus
degree of saturation
S
. The equation parameters are defined
as the coordinates where the hyperbola asymptotes meet. A
arctan [1
/
ln
(
10
6
/ψ
aev
)
], the desaturation slope.
The first derivative of Eq. 5.74 with respect to suction,
ψ
, is required to define the water storage function for a
transient seepage analysis and can be written as follows:
λ
=
tan
θ
g
(
1
r
2
)
+
dS
dψ
1
ψ
=
1
−
r
2
tan
2
θ
g
⎤
r
2
ln
(ψ/ψ
aev
)(
1
tan
2
θ
g
)/(
1
r
2
tan
2
θ
g
)
+
−
⎦
−
r
2
ln
2
(ψ/ψ
aev
)
+
a
2
(
1
−
r
2
tan
2
θ
g
/(
1
+
tan
2
θ
g
)
(5.75)
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