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where D c1 and D c2 are the depths in the two channels. Integrating (10.39) once, one
obtains
η ∂η
x =
C 1
(10.41)
where C 1 is an integration constant. Comparison with Equation (10.27) for a horizontal
bed shows immediately that C 1 =−
q x ). The variable q is the
rate of flow in the aquifer between the two channels per unit length of channel [L 2
q
/
k 0 , in which q
=
(
η
T].
Because the flow is steady, in the present situation q is a constant, that is, independent of x .
A second integration, with the first of (10.40), yields the position of the free surface
/
2 q
k 0
2
D c1
η
=−
x
+
(10.42)
Application of the second of (10.40) yields the rate of flow between the two channels, in
terms of the hydraulic conductivity and the known water levels in the two channels, or
k 0 D c2
D c1
q
=−
(10.43)
2 B
Again, the negative sign in Equation (10.43) merely indicates that the flow is taking
place in the minus x direction. Equation (10.43) is known as the Dupuit formula (see
also Dupuit, 1863, p. 236). This result is of considerable theoretical interest, because
it can be shown to be exact. In other words, even though in the derivation of (10.43)
use is made of the hydraulic assumptions, it has the same form as the solution for the
same free surface problem, obtained when no use is made of the hydraulic assumptions.
The fact that, in some cases, it produces the exact result, suggests that the hydraulic
approach can be a powerful and reliable tool in the derivation of the groundwater flow
rates. This has been confirmed in other instances as well. However, it is now also known
that the hydraulic approach is not as accurate in the prediction of the geometry of the free
surface. One obvious reason for this is that an inherently two-dimensional flow pattern
is being described by a one-dimensional formulation. This precludes then, for example,
the inclusion of a seepage surface in the boundary conditions, as was done in the second
of (10.13). In hydraulic groundwater theory, there is no way to include the existence of
a seepage surface and the first two of (10.13) must of necessity be combined into one
condition, namely the first of (10.40) (or of (10.34)).
Exactness of the Dupuit formula
The proof proceeds as follows, for the situation sketched in Figure 10.18. Without consid-
eration of the hydraulic approximation, the rate of flow through a vertical section at any
point x between the two open channels is given by
η
h
x dz
q =− k 0
(10.44)
0
Recall that h = h ( x , z ) and
η = η
( x ); application of Leibniz's formula (see Appendix,
Equation (A2)) therefore allows (10.44) to be rewritten as
η
d
dx
) d dx
q = k 0
hdz k 0 h ( x
(10.45)
0
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