Geoscience Reference
In-Depth Information
water table has a certain shape, it keeps that shape; the height of the water table only becomes
lower with time.
The outflow rate from the aquifer, Equation (10.51), can be written in terms of the
dimensionless variables defined in (10.66) as
x
+
=
0
k
0
D
2
B
η
+
∂η
+
∂
x
+
q
=−
(10.73)
Observe that this suggests that the outflow rate be scaled with (
k
0
D
2
/
B
), defining a dimen-
sionless outflow rate
Bq
k
0
D
2
q
+
=
(10.74)
With the solution (10.72) and putting
x
+
=
0
=
b
dF
dx
+
F
(10.75)
one obtains in dimensionless form
−
b
(1
+
at
+
)
2
q
+
=
(10.76)
In (10.76)
a
and
b
are dimensionless constants, whose values depend on the solution of
(10.70) for
F
(
x
+
).
Again, as in the previous section, this brief derivation shows how, save for these two
constants, it is possible to obtain the exact form of the outflow rate, without actually solving
for
F
(
x
+
), and mainly on the basis of two types of similarity considerations. The first type
involves self-preservation of the shape of the water table; this follows from the fact that the
solution can be obtained by separation of variables. The second type involves dimensional
analysis to scale the variables.
Solution
As mentioned, Boussinesq (1904) presented the exact solution to this problem. This solution
is greatly simplified by the use of the scaled variables and it proceeds as follows. The function
F
2
(
t
+
) in Equation (10.70) has been solved for, and only
F
1
(
x
+
) remains to be determined.
With the transformations given below Equation (10.71) this requires the solution of the
following ordinary differential equation
d
2
dx
2
+
(
F
2
/
2)
=−
aF
(10.77)
Putting
p
=
d
(
F
2
/
dx
, one can readily check that the left-hand side of Equation (10.77)
can be written as
pdp
/
2)
/
d
(
F
2
/
2). A first integration of this result yields
p
2
2
=−
aF
3
3
+
C
3
(10.78)
where
C
3
is a third constant of integration. Performing a second integration one obtains
from (10.78)
F
ydy
(2
C
3
−
(2
a
/
3)
y
3
)
1
/
2
x
+
=
(10.79)
0
