Environmental Engineering Reference
In-Depth Information
5.7.2 Unimodal Equation with Two Bending Points
Two rotated and translated hyperbola are needed to define
an entire unimodal SWCC with two bending points. The
three straight lines defined by the coordinates (0, 1)
(ψ
aev
,
1
)
(ψ
res
,S
r
)
and
(
10
6
,
0
)
are the asymptotes of the hyperbola.
These two hyperbola are merged through a third equation to
produce a continuous function with a smooth transition. The
proposed equation is as follows:
coordinates (0, 1)
(ψ
aev1
,
1
)(ψ
r
1
,S
r
1
)(ψ
aev2
,S
aev
)(ψ
r
2
,S
r
2
)
and 10
6
,
0
)
:
S
1
−
S
2
S
2
−
S
3
S
=
(ψ/
ψ
aev
ψ
res1
)
d
1
+
(ψ/
ψ
res1
ψ
aev2
)
d
2
1
+
1
+
S
3
−
S
4
+
(ψ/
ψ
aev2
ψ
res2
)
d
3
+
S
4
(5.78)
1
+
where:
S
1
−
S
2
S
=
(ψ/
ψ
aev
ψ
res
)
d
+
S
2
(5.76)
1
+
S
i
,θ
gi
,r
i
,
and
λ
i
are as defined in Eq. 5.76,
i
=
1, 2, 3, 4,
where
S
1
1,
S
2
S
r
1
,S
3
S
aev
,S
4
S
r
2
,S
5
=
=
=
=
=
0,
r
i
)
ln
(ψ/ψ
i
)
tan
2
θ
gi
tan
θ
gi
(
1
+
1
)
i
1
+
S
aev2
=
degree of saturation at the air-entry value for the
second part of the SWCC.
S
i
=
+
(
−
r
i
tan
2
θ
gi
r
i
tan
θ
gi
1
−
1
−
r
i
ln
ψ
ψ
1
=
ψ
aev1
,
a
2
(
1
r
i
tan
2
θ
gi
)
−
ψ
2
=
ψ
res1
,
S
i
×
+
+
ψ
i
ψ
3
(
1
+
tan
2
θ
gi
)
=
ψ
aev2
,
ψ
4
=
ψ
res2
,
and
ψ
5
10
6
, and
=
2 exp[1
/
ln
(ψ
j
+
1
/ψ
j
)
] are weight factors,
j
d
j
=
=
1,
1, 2;
θ
gi
=−
i
=
2, 3.
(λ
i
−
1
+
λ
i
)/
2, hyperbola rotation angles,
r
i
=
tan[
(λ
i
−
1
-
λ
i
)/
2], aperture angle tangents;
The derivative of Eq. 5.78 with respect to
ψ
can be obtained
in a manner similar to that shown above for Eq. 5.76.
(S
i
S
i
+
1
)/
[ln
(ψ
i
+
1
/ψ
i
)
]
λ
0
=
0 and
λ
i
=
arctan
{
−
}
are
the desaturation slopes;
S
1
=
1,
S
2
S
r
,
S
3
0, where
S
r
is the degree of satu-
ration at residual conditions,
=
=
5.7.4 Parametric Analysis of Gitirana and Fredlund
(2004) SWCC Equations
A parametric analysis shows the influence of the fitting
properties on the proposed equations. Figures 5.63-5.65
illustrate the independent influence of each of the equation
parameters. The SWCC curve parameters are shown to be
mathematically independent.
Figure 5.63 illustrates that larger values of
a
provide a
smoother curve. The air-entry value might appear to be
reduced as
a
increases but that is not the case. The apparent
reduction is simply a smoothing effect evenly distributed to
suction values lower and higher than both
ψ
aev
and
ψ
res
.
The curve limits may start deviating from
S
ψ
1
ψ
aev
,ψ
2
ψ
res
,ψ
3
10
6
,
=
=
=
d
=
2 exp[1
/ln(ψ
res
/ψ
aev
)
], a weight factor for
S
1
and
S
2
that produces a continuous and smooth curve, and,
ψ
res
=
residual suction
The first derivative of Eq. 5.76 with respect to
ψ
is
dS
dψ
dS
1
/dψ
−
dS
2
/dψ
=
(ψ/
ψ
aev
ψ
res
)
d
1
+
d
S
1
−
S
2
ψ
ψ
aev
ψ
res
d
ψ
dS
2
dψ
(5.77)
−
(ψ/
ψ
aev
ψ
res
)
d
]
2
+
[1
+
=
100% and
S
=
0% when values of
a
are greater than 0.2 (Fig. 5.63). It
is suggested that
a
values should lie within the range from
0 to 0.15.
where
tan
θ
gi
(
1
r
i
)
+
dS
i
dψ
=
1
ψ
1
)
i
r
i
tan
2
θ
gi
+
(
−
1
−
5.7.5 Fitting Gitirana and Fredlund (2004) SWCC
Equation to Experimental Data
Figure 5.66 shows the fitting of two data sets using the
Gitirana and Fredlund (2004) SWCC equation. The labora-
tory test data was from Patience Lake silt (Bruch 1993) and
Beaver Creek sand (Sillers, 1997). The best-fit parameters
are shown in Fig. 5.66.
Figure 5.67 illustrates how the Gitirana and Fredlund
(2004) SWCC equation can also be used to fit bimodal
SWCCs. Experimental data for two bimodal soils are used to
show the fitting capability of the proposed bimodal equation.
⎤
⎦
r
i
ln
(ψ/ψ
i
)(
1
tan
2
θ
gi
)/(
1
r
i
tan
2
θ
gi
)
+
−
×
r
i
ln
2
(ψ/ψ
i
)
r
i
tan
2
θ
gi
)/(
1
+
a
2
(
1
−
+
tan
2
θ
gi
)
i
=
1
,
2
5.7.3 Bimodal SWCC Equation
Four hyperbolae are needed to model a bimodal SWCC
delineated by the five asymptotes that are defined by
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