Geoscience Reference
In-Depth Information
Operating on this result in the manner of (10.157), one obtains in this case for its constants
b
1
=
3
(10.161)
1336
k
0
n
e
D
3
L
2
−
1
a
1
=−
1
.
in which the subscripts 1 indicate that it is the first solution which is considered here.
The second solution is the long-time outflow rate derived from the nonlinear Boussi-
nesq equation, namely (10.76) or (10.85) with (10.86). After applying (10.158) and
(10.159), this solution can be written in terms of basin-scale parameters as
1
t
−
2
3
.
448
L
2
k
0
D
2
A
4
.
46
L
2
k
0
D
n
e
A
2
Q
=
+
(10.162)
With this expression the resulting constants for Equation (10.157) are
b
2
=
3
/
2
(10.163)
8038
k
1
/
2
0
L
(
n
e
A
3
/
2
)
−
1
a
2
=−
4
.
The third solution of interest is the long-time outflow rate (10.115) or (10.116)
obtained from the fundamental harmonic of the linear solution. In the case of (10.116),
substitution of (10.158) and (10.159) immediately produces the outflow rate in terms of
catchment-scale parameters
exp
2
k
0
pDL
2
t
n
e
A
2
−
π
8
k
0
pD
2
L
2
A
−
1
Q
=
(10.164)
Thus with this result Equation (10.157) has the constants
b
3
=
1
a
3
=−
π
(10.165)
2
k
0
pDL
2
(
n
e
A
2
)
−
1
A fourth expression in the form of (10.157) can be obtained for a sloping aquifer, from
the first term of (10.143). With (10.158) and (10.159) this can be written as
8
k
0
pD
2
L
2
cos
α
A
z
1
[1
−
2 cos(
z
1
)exp(Hi
/
2)]
z
1
+
Hi
2
Q
=
/
4
+
Hi
/
2
×
exp
−
z
1
+
Hi
2
t
/
4
4
k
0
pDL
2
cos
α
(
n
e
A
2
)
(10.166)
In this case the constants of (10.157) are
b
4
=
1
−
z
1
+
Hi
2
/
4
4
k
0
pDL
2
cos
α
(
n
e
A
2
)
(10.167)
a
4
=
The three solutions for horizontal aquifers can also be combined into a single expression,
that can be applied with arbitrary values of
b
. This expression can be obtained by scaling
Equation (10.157) with the dimensionless variables implicit in the Boussinesq equation and
