Geoscience Reference
In-Depth Information
Comparison between the first and second term in the series of (10.105) shows that this
long-time solution can be assumed to be valid when
t
+
>
0
.
2, where the error drops well
below 1%. In terms of the original variables the long-time solution is from (10.107) and
(10.94)
2
B
exp
−
π
4
n
e
B
2
π
−
1
sin
π
x
2
k
0
η
0
t
η
=
D
c
+
4(
D
−
D
c
)
(10.108)
Outflow rate
After linearization, the outflow rate (10.51) from a hydraulic aquifer into the adjoining
open water body becomes
x
=
0
k
0
η
0
∂η
∂
q
=−
(10.109)
x
In terms of the scaled variables of (10.94) this can be written as
q
+
=−
∂η
+
∂
x
=
0
(10.110)
x
+
in which the rate of outflow is scaled with
k
0
η
0
(
D
−
D
c
)
/
B
, so that by definition
Bq
k
0
η
0
(
D
q
+
=
(10.111)
−
D
c
)
Application of (10.110) with the general solution (10.105) yields
exp
∞
1)
2
2
(2
n
−
π
q
+
=−
2
−
t
+
(10.112)
4
n
=
1
,
2
,...
This result is illustrated in Figure 10.23. In terms of the original variables, after trans-
formation with (10.94), the rate of outflow from the unconfined aquifer (10.112) can be
written as
exp
−
∞
1)
2
2
k
0
η
0
t
(2
n
−
π
D
c
)
B
−
1
q
=−
2
k
0
η
0
(
D
−
(10.113)
4
n
e
B
2
n
=
1
,
2
,...
As already noted, eventually with increasing time only the first term of the series remains,
as the terms in
n
become negligible. Therefore, the long-time expression of
the outflow rate is from (10.112)
=
2
,
3
,...
2exp
2
−
π
q
+
=−
t
+
(10.114)
4
As shown in Figure 10.23, (10.114) becomes applicable for
t
+
>
0
.
2, that is for
t
>
2
n
e
B
2
0
(
k
0
η
0
). Thus, when this criterion is satisfied, (10.113) yields the long-time
outflow rate in terms of the original variables
.
/
D
c
)
B
−
1
exp
2
k
0
η
0
t
4
n
e
B
2
−
π
q
=−
2
k
0
η
0
(
D
−
(10.115)
If, as is often the case in small upland catchments, the water in the stream is shallow
compared to the water table levels in the aquifer, it can be assumed that
D
c
=
0. With
