Geoscience Reference
In-Depth Information
1
Fig. 10.21 Successive scaled positions
of the water table
η
+
=
η/
D
in an
unconfined riparian
aquifer, as calculated with
Boussinesq's solution
η
+
=
F
/
(1
+
at
+
), for
the indicated values
of the scaled time
t
+
=
[
k
0
D
/
(
n
e
B
2
)]
t
. The
variable
x
+
=
x
/
B
is the
scaled distance from
the stream and the
function
F
t
+
=0
0.8
0.2
η
+
0.5
0.6
1
0.4
2
0.2
0
F
(
x
+
)isthe
curve shown for
t
+
=
=
0
0.2
0.4
0.6
0.8
1
x
+
0.
(10.61) and (10.63), for convenient reference the outflow into the stream can now be
expressed in terms of the original values and parameters as
0.332 06(
k
0
n
e
D
3
)
1
/
2
t
−
1
/
2
q
=
(10.64)
10.3.5
Long-time outflow behavior
As formulated in Equation (10.50), the third condition describes an initially saturated
aquifer over the entire width of the aquifer, i.e. for 0
B
. Boussinesq (1904) showed
how, by relaxing this condition and by specifying the initial value of
≤
x
≤
η
=
B
,
it becomes possible to obtain an exact solution of (10.30). Thus instead of (10.50), the
relaxed boundary conditions are the following
only at
x
η
=
0
x
=
0
t
≥
0
∂η
∂
(10.65)
x
=
0
x
=
Bt
≥
0
η
=
Dx
=
Bt
=
0
As will be shown below, Boussinesq's method of solution implicitly requires the assump-
tion of self-preservation of the
x
-dependency of the water table height
η
, so that the shape
of the water table remains the same with time. This water table shape is not arbitrary, but
it results from the solution. As illustrated in Figure 10.21, that solution produces a water
table that is curvilinear throughout the drainage process, from the beginning at
t
0to
the end when the outflow finally ceases. Normally, an initially saturated aquifer (cf. the
third of (10.50)) will exhibit this kind of shape only after drainage has proceeded for
some time; it is for this reason that the solution obtained in what follows is referred to
as a long-time solution.
=
Similarity considerations
As was the case for the short-time solution, useful insight can be gained by making
the formulation dimensionless. Again, it is reasonable to normalize the length variables
