Geoscience Reference
In-Depth Information
can then be used to predict the evolution of the water table resulting from arbitrary inputs
(e.g. steady rainfall as in the following example) by convolution, i.e. with the analog of
(10.118) applied to
η
instead of
q
. However, as noted earlier, the position of the water table
in hydraulic groundwater theory is quite unreliable, and need therefore not be of further
concern here.
Example 10.2
The application of (10.118) can be readily illustrated by considering the case of a constant
input of unit intensity
I
(
t
)
=
I
c
. The resulting outflow rate becomes steady when this input
has been applied for a very long time. Thus one has from (10.118) with (10.117)
exp
−
(2
n
−
1)
2
d
τ
t
∞
2
k
0
η
0
(
t
−
τ
)
4
n
e
B
2
−
2
k
0
η
0
n
e
B
π
q
=
I
c
(10.119)
n
=
1
,
2
, ...
−∞
which upon integration becomes
exp
−
(2
n
−
1)
2
exp
(2
n
−
1)
2
t
∞
−
8
BI
c
(2
n
−
1)
2
π
2
k
0
η
0
t
2
k
0
η
0
τ
4
n
e
B
2
π
q
=
π
2
4
n
e
B
2
−∞
(10.120)
n
=
1
,
2
2
Applying the integration limits, and recalling that 1
+
1
/
9
+
1
/
25
+···=
π
/
8, one
obtains finally
q
=−
BI
c
(10.121)
as expected. It goes without saying, that this case is the linearized version of the case already
treated earlier, for which the position of the water table was shown to be given by Equations
(10.35) and (10.37), illustrated in Figures 10.16 and 10.17.
Example 10.3
The next case to be considered is the outflow rate some time after a steady input
I
c
has
ceased. This represents the outflow rate from the aquifer, with the initial shape of the water
table given by (10.37) resulting from a steady infiltration, rather than by the third of (10.50)
describing fully saturated conditions. This case is of practical interest, as the steady input
may represent prolonged rainfall or irrigation. Indeed, the onset of drainage after prolonged
rainfall or irrigation, which does not fully saturate the aquifer, is a common occurrence in
humid regions. If
t
=
0 is the time when the steady input stops, the input
I
=
I
(
t
) can be
formulated as follows
I
=
I
c
for
−∞
<
t
<
0
(10.122)
I
=
0
for 0
≤
t
Thus (10.118) can be written as
0
t
q
=
I
c
u
(
t
−
τ
)
d
τ
+
0
u
(
t
−
τ
)
d
τ
(10.123)
−∞
0
