Geoscience Reference
In-Depth Information
Since
h
(
x
,η
)
=
η
(
x
) defines the free surface, (10.45) can be integrated to yield
η
2
2
+
C
h
(
x
,
z
)
dz
−
k
0
η
−
qx
=
k
0
(10.46)
0
The constant C can be determined by applying the boundary condition at
x
=
0, where
h
=
D
c1
for 0
≤
z
≤
D
c1
and
h
=
z
for
D
c1
≤
z
≤
η
. (The lower part of this boundary
condition describes the hydrostatic conditions and constant hydraulic head in the canal, and
the higher part the seepage surface; thus this condition differs from the first of (10.40), i.e.
the corresponding condition of the hydraulic approach, which is incapable of incorporating
a seepage surface.) After breaking up the integral in Equation (10.46) over the two parts of
its range, one obtains
D
c1
η
0
2
0
2
zdz
+
k
0
η
C
=−
k
0
D
c1
dz
−
k
0
(10.47)
0
D
c1
in which
η
0
is the value of
η
at
x
=
0, or finally upon integration
C
=−
k
0
D
c1
2
(10.48)
Application of the second boundary condition, namely at
x
=
B
, where
h
=
η
=
D
c2
over
the whole range 0
≤
z
≤
D
c2
, with insertion of (10.48), changes (10.46) into
D
c2
k
0
D
c2
2
k
0
D
c1
2
−
qB
=
k
0
D
c2
dz
−
−
(10.49)
0
Upon integration of (10.49), the desired result, namely the Dupuit formula (10.43), follows
immediately. This confirms that, even though the Dupuit formula was originally obtained
by means of the hydraulic approximation, it is in fact identical to the result obtainable with
a more rigorous derivation. This fact should instill some confidence in the more general
applicability of the hydraulic approach, as a very close approximation to describe flow rates
in other situations as well.
The exact derivation of the Dupuit formula was probably first presented by I. A. Charnii,
and it can also be found in the topic by Polubarinova-Kochina (1952, p. 281). Later a similar
proof was presented by Hantush (1962; 1963).
10.3.3
Unsteady flow described with standard hydraulic theory
It is again instructive to consider the phenomenon of outflow from an unconfined aquifer
on a horizontal bed into a stream. This situation is still like the one described schemat-
ically in Figure 10.11; however, because now the hydraulic approximation is made use
of, the two-dimensional problem has been reduced to a one-dimensional problem, and
the
z
-coordinate is no longer part of the formulation.
Basic formulation
This flow problem was shown to be subject to boundary conditions (10.2) and (10.3)
or, after a first approximation, to (10.13); when these conditions are translated to the
