Geoscience Reference
In-Depth Information
hydraulic approach with a horizontal aquifer, they become
η
=
D
c
x
=
0
t
≥
0
∂η
∂
(10.50)
x
=
0
x
=
B
t
≥
0
η
=
D
0
≤
x
≤
Bt
=
0
The governing equation of this one-dimensional horizontal flow is the Boussinesq equa-
tion (10.30). However, even in this simple form, the equation is still nonlinear; this means
that, in contrast to linear problems, there are no general solutions available, and that a
specific ad hoc method must be devised for each new problem. Equation (10.30) subject
to (10.50) can, of course, be solved numerically (Verma and Brutsaert, 1971a); in Figure
10.14 some results, obtained this way with (10.30) subject to (10.50), are compared with
calculations with the complete free-surface formulation. There are, however, also two
exact analytical solutions available for boundary conditions, that may be considered as
short-time and long-time cases of Equations (10.50) with
D
c
=
0, and that are of interest
in practical situations. These solutions are treated in the next two sections.
Outflow rate
Once the solution of this problem has been obtained as
η
=
η
(
x
,
t
), the outflow rate
q
from the aquifer into the stream at
x
0 can be determined by applying the hydraulic
form of Darcy's equation (10.27), as follows
=
k
0
η
∂η
∂
q
=−
(10.51)
x
x
=
0
Recall that in this chapter
q
denotes the volume of water per unit time and per unit length
of channel (i.e. per unit span or per unit width of aquifer normal to the main direction
of the flow in the aquifer), so that its dimensions are [L
2
T
−
1
]. In some cases it is more
convenient to follow a procedure, analogous with that used to obtain Equation (9.15)
for infiltration. Accordingly, as sketched in Figure 10.19, from continuity considerations
the cumulative outflow volume per unit span (with dimensions [L
2
]) from the aquifer at
x
=
0 can be written in general as
D
∀=
n
e
xd
η
(10.52)
D
c
which produces the outflow rate as
q
dt
. Both (10.51) and (10.52) are used in
the remainder of this chapter. The second approach to obtain the outflow is especially
useful, whenever the method of solution is based on Boltzmann's transform, as was the
case for sorption and infiltration in Chapter 9.
=
d
∀
/
10.3.4
Short-time outflow behavior
As will become clear below, the short-time outflow behavior of an unconfined aquifer
can be studied by analyzing the case of an infinitely wide aquifer, that is for
B
→∞
