Geoscience Reference
In-Depth Information
I
c
WT
D
D
c
x=0
x=B
Fig. 10.16 Schematic representation of the cross section of an unconfined riparian aquifer, lying on a horizontal
impermeable layer, under steady flow conditions. The position of the water table results from a steady
and constant recharge rate
I
c
, and it has the shape of an ellipse without seepage surface at
x
=
0, when
determined with the hydraulic approach.
η
=
η
(
x
,
t
). Similarly, three-dimensional flow is simplified to a two-dimensional problem
in Equation (10.31), as the unknown hydraulic head
h
=
h
(
x
,
y
,
z
,
t
) is replaced by the
unknown height of the water table
η
=
η
(
x
,
y
,
t
).
10.3.2
Steady flow described with hydraulic theory
Over the years hydraulic groundwater theory has been a powerful tool to solve a large
number of important practical problems under steady state conditions. The main reason
for its wide use is that under steady state conditions the Boussinesq equation becomes
linear in
2
, which greatly simplifies the mathematical analysis. For instance, under
conditions of steady flow over a horizontal bed and in the absence of lateral inflow,
Equation (10.31) reduces to
η
2
2
2
2
∂
η
+
∂
η
=
0
(10.32)
∂
x
2
∂
y
2
This is Laplace's equation in
2
, for which many solution methods are available. More-
over, because the problem is linear, known solutions for
η
2
obtained for relatively simple
boundary conditions can be extended to more complicated situations by the application
of image methods and other methods of superposition. Two examples of steady aquifer
outflow are presented in what follows.
η
Steady outflow resulting from a uniform precipitation
Under steady conditions and for an aquifer cross section with horizontal bed, as shown
in Figure 10.16, one can write Equation (10.29) as
∂
∂
η
∂η
∂
I
c
k
0
=−
(10.33)
x
x
where
I
c
is a constant recharge rate; this is usually taken as a climatological average
rainfall for design purposes, but it may also represent irrigation or a negative rate of
