Geoscience Reference
In-Depth Information
B
δ
s
s=0
stream
s=L
Fig. 10.32 Simplified schematic representation of the catchment shown in Figure 10.29, illustrating the use of the
spatially constant effective parameters
q
and
B
to describe the catchment-scale groundwater outflow
rate
Q
. Thus it is assumed that
Q
=
2
qL
and
A
=
2
LB
.
Application with available groundwater outflow solutions
For several well-known solutions of the Boussinesq equation, describing groundwa-
ter outflow from an unconfined aquifer based on the hydraulic approach, it can be
shown (Brutsaert and Nieber, 1977), that Equation (10.155) can be expressed as a power
function,
dQ
dt
=
aQ
b
(10.157)
where
a
and
b
are constants.
Equation (10.157) is obtainable from each of these solutions by assuming geometric
similarity of the drainage pattern and the channel network within the catchment. With
this assumption, and by defining an equivalent or effective lateral inflow rate
q
into the
stream, one can immediately integrate (10.152) as follows
Q
=
2
L
|
q
|
(10.158)
As before,
L
is the total length of all tributary and main channel sections upstream from
the gauging station where the stream flow is
Q
. Likewise one can define an effective
aquifer breadth
B
, as the distance from channel to divide (see Figure 10.32) by
B
=
A
/
(2
L
)
(10.159)
in which
A
is the drainage area of the catchment, and (
L
A
) is known as the drainage
density. Equation (10.159) is the same as the relationship proposed by Horton (1945) for
the average overland flow distance in a catchment whose channel slope is much smaller
than the land surface slopes.
The first of these solutions, that can be put in the form of (10.157), is the short-time
outflow rate (10.64), which was obtained by Boltzmann similarity and which exhibits
the characteristic
t
−
1
/
2
behavior. Upon substitution of (10.158), (10.64) yields the basin-
scale outflow rate
/
0.664 12
k
0
n
e
D
3
L
2
1
/
2
t
−
1
/
2
Q
=
(10.160)
