Environmental Engineering Reference
In-Depth Information
Equation 8.19 fits experimental data over the entire suc-
tion range from 0 to 10
6
kPa (Fredlund and Xing, 1994). It is
convenient to perform the integration along the soil suction
axis when calculating the coefficient of permeability using
Eqs. 8.18 and 8.19. Equation 8.18 can also be transformed
into the following form:
where:
upper limit of integration [i.e., ln
10
6
],
b
=
y
=
dummy variable of integration representing the log-
arithm of suction,
θ
=
derivative of the SWCC equation, and
e
y
=
natural number raised to the dummy variable power.
θ
(y)
dy
ψ
r
ψ
r
θ(y)
−
θ(ψ)
θ(y)
−
θ
s
8.2.5 Numerical Integration for Relative Coefficient of
Permeability
The permeability function is expressed in a form where inte-
gration can be performed on a logarithm scale (Eq. 8.21).
The two integrals in Eq. 8.21 can be evaluated using the fol-
lowing numerical integration method. Let
a
and
b
denote the
lower and upper limits of the integration. Then
a
θ
(y)
dy
k
r
(ψ)
=
y
2
y
2
ψ
ψ
aev
(8.20)
where:
ψ
aev
=
air-entry value of the soil under consideration
(i.e., the suction where air starts to enter the
largest pores in the soil),
ln
ψ
aev
=
ln
10
6
. Let us divide the range [
a, b
] into
N
subin-
tervals of the same size and let
y
denote the length of each
subinterval. Then,
and
b
=
ψ
r
=
suction corresponding to the residual water con-
tent,
θ
r
,
y
=
dummy variable of integration representing suc-
tion, and
b
−
a
a
=
y
1
<y
2
<
···
s<y
N
<y
N
+
1
=
by
=
N
(8.22)
The denominator of Eq. 8.21 can be evaluated as follows:
θ
=
derivative of Eq. 8.19.
Equation 8.19 fits the experimental data over the entire
suction range and the integrations in Eq. 8.20 can be per-
formed from
ψ
aev
to 10
6
for all types of soils. This simplifies
the prediction procedure for the coefficient of permeabil-
ity since the residual value (
θ
r
or
ψ
r
) does not have to be
accurately determined for each soil.
It is necessary to independently determine the air-entry
value
ψ
aev
of the soil. The air-entry value can be deter-
mined using the empirical construction procedure shown in
Chapter 5. Consequently, the soil is assumed to remain at the
saturated coefficient of permeability at suction values below
the air-entry value of the soil. The coefficient of permeabil-
ity of the soil then decreases at soil suctions higher than the
air-entry value in accordance with the proposed integration
procedure. A series of discrete data points are calculated
corresponding to the relationship between the coefficient of
permeability and soil suction. It is also possible to best fit
the permeability data points with another empirical equation,
if so desired.
To avoid numerical difficulties associated with performing
integration over the soil suction range from
ψ
aev
to 10
6
kPa
on an arithmetic scale, it is more convenient to perform the
integration on a logarithmic scale. Therefore, the following
variation of Eq. 8.20 is preferred:
θ
e
y
i
−
b
N
θ(e
y
)
θ
e
y
dy
θ
s
θ
e
y
i
−
θ
s
≈
y
e
y
e
y
i
i
=
1
ln
(ψ
aev
)
(8.23)
y
i
is the midpoint of the
i
th interval,
y
i
,y
i
+
1
, and
where
¯
θ
is the derivative of Eq. 8.19 given as follows:
θ
s
θ
(ψ)
C
(ψ)
=
ln
e
+
ψ/a
f
n
f
m
f
θ
s
−
C (ψ)
ln
e
+
ψ/a
f
n
f
m
f
+
1
m
f
n
f
ψ
a
f
n
f
−
1
a
e
×
+
ψ/a
f
n
f
(8.24)
where
−
1
C
(ψ)
=
ψ
r
+
ψ
ln
1
10
6
ψ
r
(8.25)
+
For any suction value
ψ
in the range between the air-entry
value,
ψ
aev
, and 10
6
, the value of ln
(ψ)
is between
a
and
b
.
Let us assume that ln
(ψ)
is in the
j
th interval,
y
j
,y
j
+
1
.
Then, the numerator of Eq. 8.21 can be evaluated as
follows:
θ
e
y
dy
b
b
θ(e
y
)
−
θ(ψ)
θ(e
y
)
θ
e
y
dy
−
θ(ψ)
k
r
(ψ)
=
e
y
e
y
ln
(ψ)
ln
(ψ)
θ
e
y
i
−
b
N
θ(e
y
)
θ
e
y
dy
−
θ
s
θ
e
y
i
θ(ψ)
(8.21)
≈
y
(8.26)
e
y
e
y
i
i
=
j
ln
(ψ
aev
)
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