Geoscience Reference
In-Depth Information
1
0
1
q
+
2
0.1
4
10
20
0.01
0
0.1
0.2
0.3
t
+
Fig. 10.28 Scaled outflow hydrograph
q
+
=
q
+
(
t
+
) from a linearized sloping hydraulic aquifer into an
adjoining open channel as given by Equation (10.147), for the values of the hillslope flow
number Hi
=
0
,
1
,
2
,
4
,
10 and 20. Hi
=
0 represents the horizontal case (see Example 10.3).
The outflow rate is scaled with the initial outflow rate
q
=
(
I
c
B
x
cos
α
), so that
q
+
=
q
/
(
I
c
B
x
cos
α
); this initial outflow rate results from a steady input
I
c
prior to
t
+
=
0, as
described in Example 10.5. The time is scaled as indicated in Equation (10.137).
[by analogy with (10.130)]
⎨
t
2
∞
−
2
k
0
η
0
P
c
cos
2
α
z
n
[1
−
2 cos(
z
n
)exp(Hi
/
2)]
z
n
+
Hi
2
q
=
q
(
t
)
=
0
.
2
/
4
+
Hi
/
2
n
e
B
x
⎩
n
=
1
,
2
,...
t
1
×
exp
−
z
n
+
Hi
2
d
τ
/
4
[
k
0
η
0
cos
α
] (
t
−
τ
)
n
e
B
x
z
n
1
−
2 cos(
z
n
)exp(Hi
/
2)
z
n
+
Hi
2
t
3
∞
/
4
+
Hi
/
2
+
0
.
9
n
=
1
,
2
,...
t
2
×
exp
d
τ
−
z
n
+
Hi
2
/
4
[
k
0
η
0
cos
α
] (
t
−
τ
)
n
e
B
x
(10.148)
This result can be readily integrated to yield (by analogy with Equation (10.131) for the
horizontal case)
z
n
1
2)
∞
−
2 cos(
z
n
)exp(Hi
/
q
=−
2
B
x
P
c
cos
α
z
n
+
Hi
2
/
4
z
n
+
Hi
2
/
4
+
Hi
/
2
n
=
1
,
2
,..
×
0
2
exp
−
z
n
+
4
(
t
+
−
t
+
2
)
−
exp
−
z
n
+
4
(
t
+
−
t
+
1
)
Hi
2
Hi
2
.
/
/
9
exp
−
z
n
+
4
(
t
+
−
t
+
3
)
−
exp
−
z
n
+
4
(
t
+
−
t
+
2
)
Hi
2
Hi
2
+
0
.
/
/
(10.149)
in which the scaled time variable is defined in Equation (10.137).
