Geoscience Reference
In-Depth Information
WT
D
c2
D
D
c1
z
x
x=B
Fig. 10.18 Schematic representation of the cross section of an unconfined aquifer, lying on a horizontal
impermeable layer between two open channels with constant water levels. If the water table (WT) is
assumed to be a free surface, the resulting steady flow rate
q
between the two channels is given exactly
by the Dupuit formula (10.43). The solid curve represents the true WT with a seepage surface and the
dashed curve the WT obtained with the hydraulic approach.
in fact, with a number of subsequent improvements it still provides the basis for many
of the soil drainage design procedures in use today. In order to apply it in its original
form, the variables on the right-hand side of Equation (10.38) must be known or decided
upon. Thus
k
0
is the hydraulic conductivity of the soil,
I
c
is taken as the average rate
of precipitation or other input during the period when drainage is needed most,
D
c
is
the depth of the water in the drainage channel or, to a first approximation, the height
of the drainage pipe above the impermeable layer, and
η
max
is the main design vari-
able, namely the maximal allowable height of the water table above the impermeable
layer.
Equation (10.38) has a long history. It is now often referred to as the ellipse equation,
on account of the shape of the water table given by Equation (10.35) (see Figure 10.17).
It was probably first derived by A. Colding in Denmark before 1872 for the case
D
c
=
0,
after he became aware of earlier experimental results published in 1859 by S. C. Delacroix
in France; interestingly, he also recommended a 10% reduction of any
B
value obtained
with Equation (10.38), to make it agree better with these experimental data. Hooghoudt
(1937), who knew indirectly of Colding's result through the work of others, was proba-
bly the first to derive (10.38) for arbitrary values of
D
c
; he later (Hooghoudt, 1940)
adjusted it to make it more suitable for drainage with pipes. A detailed history of
the equation and its more recent derivatives has been presented by VanderPloeg
et al
.
(1999).
Steady flow between two parallel channels without precipitation
In this problem, as shown in Figure 10.18, the flow in the unconfined aquifer is described
by the one-dimensional Laplace equation
2
2
∂
η
=
0
(10.39)
x
2
∂
The boundary conditions are
η
=
D
c1
x
=
0
(10.40)
η
=
D
c2
x
=
B
