Environmental Engineering Reference
In-Depth Information
f(r)
A
, where
A
is a constant. Equation 5.27 can
then be written as
=
integration form can be used as a general form to approxi-
mate the SWCC:
∞
ψ
max
AC
1
ψ
AC
h
2
1
ψ
max
B
ψ
θ(ψ)
=
θ
s
f(h)
dh
(5.31)
θ(ψ)
=
dh
=
−
=
−
D
ψ
ψ
(5.28)
where:
f(h)
where:
=
pore-size distribution as a function of
soil
suction.
B
=
AC
=
constant and
D
=
AC
/ψ
max
=
constant.
Equation 5.31 will generally produce a nonsymmetrical
S-shaped curve. Several special cases follow.
Let us consider the case of a normal distribution curve
where the
f(h)
function is assumed to take the form of a
normal distribution:
2. Case where pore-size function varies inversely as
r
2
.
For the case where
f(r)
A/r
2
, the relationship
between volumetric water content and suction can be
written as
=
1
√
2
πσ
e
−
(h
−
μ)
2
/
2
σ
2
f(h)
=
(5.32)
ψ
max
Ah
2
C
2
C
h
2
dh
θ(ψ)
=
=
B
−
Dψ
(5.29)
where:
μ
ψ
=
mean value of the distribution of
f(h)
,
e
=
irrational constant equal to 2.71828, and
where:
σ
=
standard deviation of the distribution of
f(h)
.
B
=
Aψ
max
/C
=
constant and
The SWCC defined by Eq. 5.31 can be expressed as
follows:
D
=
A/C
=
constant.
∞
∞
θ
s
2
2
√
π
Equation 5.29 represents a linear variation in the dis-
tribution of pore sizes. In other words, there is a linear
relationship between volumetric water content and soil
suction.
3. Case where pore-size function varies inversely as
r
(m
+
1
)
. For the case where
f(r)
e
−
y
2
θ(ψ)
=
θ
s
f(h)dh
=
μ)/
√
2
σ
ψ
(ψ
−
erfc
ψ
θ
s
2
μ
√
2
σ
−
dy
=
(5.33)
A/r
(m
+
1
)
, with
m
as an integer, the relationship between volumetric
water content and soil suction can be written as
=
where:
∞
2
√
π
e
−
y
2
/dy
erfc
(x)
=
=
1
−
erf
(x)
=
1
−
ψ
max
Ah
m
+
1
C
m
+
1
C
h
2
dh
x
Dψ
m
θ(ψ)
=
=
B
−
(5.30)
∞
2
√
π
e
−
y
2
/dy
and
ψ
0
where:
erfc
(x)
=
complement of the error function, erf
(x)
.
A(ψ
max
)
m
/(mC
m
)
B
=
=
constant and
Equation 5.33 takes the form of a symmetrical S-shaped
curve. The SWCC of the soil will be close to a symmetrical
S-shaped curve if the pore-size distribution of a soil can be
approximated as a normal distribution. Equation 5.33 can
then be used as a model to describe the SWCC relationship.
The two fitting parameters (i.e., the mean value
μ
and the
standard deviation
σ
) in Eq. 5.33 are related to the air-entry
value of the soil and the slope at the inflection point on
the SWCC. If the slope at the inflection point is
s
and the
air-entry value is
ψ
aev
, then the standard deviation
σ
can be
written as
A/(mC
m
)
D
=
=
constant.
The power law relationship (i.e., Eq. 5.19) proposed by
Brooks and Corey (1964) is a special case of Eq. 5.30. In
other words, the Brooks and Corey (1964) power law rela-
tionship is valid when the pore-size distribution is close to
the distribution function
f(r)
A/r
m
+
1
.
The volumetric water content
θ
over the entire soil suc-
tion range from 0 to 10
6
kPa can be referenced to zero water
content; otherwise, the normalized water content becomes
negative at water contents less than the residual value
θ
r
.In
this case, water content is written in a dimensionless form
(i.e.,
d
=
=
θ
s
√
2
πs
σ
=
(5.34)
θ/θ
s
). Equation 5.27 suggests that the following
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