Geoscience Reference
In-Depth Information
100
q
+
20
4
10
10
2
1
1
0
0.1
0
0.1
0.2
0.3
0.4
0.5
t
+
Fig. 10.27 Scaled outflow hydrograph
q
+
=
q
+
(
t
+
) from a linearized sloping hydraulic aquifer into an
adjoining open channel as given by (10.143), for the values of the hillslope flow number
Hi
=
0
,
1
,
2
,
4
,
10 and 20; Hi
=
0 represents the horizontal case. The scaled time variable is
defined in (10.137) and the aquifer is initially fully saturated.
where the scaled outflow rate is defined as
q
+
=
B
x
q
/
(
k
0
η
0
D
cos
α
) and the scaled time
t
+
is
defined in (10.137). The reader can verify that, when the hillslope flow number vanishes, or
Hi
=
0, this expression reduces immediately to the solution for the horizontal case (10.113)
with
D
c
=
0.
The outflow hydrograph from a sloping aquifer (10.143) is illustrated in Figure 10.27
for different values of the hillslope flow number Hi. The long-time behavior of this outflow
rate displays the typical exponential decay with time of linear systems, but the exponential
function has two extinction coefficients; the first (
z
n
) reflects the diffusive character of the
flow, and the second (Hi
2
/
4) reflects its kinematic character, that is the effect of the steepness
of the slope. As a result, the outflow rate displays two features which are worth noting. First,
as is the case with (10.101), the values of
z
n
increase rapidly with
n
in the higher-order
terms, so that these terms decay rapidly, regardless of the value of Hi. This means that for
large values of
t
only the first term in the series survives, producing the straight lines in the
semi-logarithmic plot of Figure 10.27; thus for larger values of
t
the rate of flow
q
decays
exponentially in approximately the same way as in the horizontal case as a result of diffusion,
but this rate is increased by the presence of the term containing Hi. The second feature is
that, as the hillslope flow number Hi increases, the outflow hydrograph gradually displays
a “hump,” or rather a “pause” in its progress. Mathematically, this phenomenon results
from the fact that, as Hi becomes larger, the relative importance of the diffusion term in
Equation (10.138) (the first term on the right) decreases compared to the advective term (the
second term on the right). This means that the nature of the flow in the aquifer becomes less
diffusive and more kinematic, that is, increasingly driven by gravity on account of the slope.
As will be discussed further in Section 10.5, kinematic motion is purely translatory without
change in shape of the water table. In the present case described by Equation (10.138),
in the early stages, while all the higher order terms in the series solution are still active,
diffusion causes the water table near the outlet to spread out, a behavior not unlike that of
flood waves in open channels, discussed in Chapter 7; but later on, as the kinematic effect
