Geoscience Reference
In-Depth Information
Equation (6.37) indicates how at
t
0, when the rainfall ceases abruptly, the rate of
flow
q
L
immediately becomes larger than
iL
by an amount (
K
rn
/
=
K
rr
); this increase
is caused by the sudden decrease of the flow resistance in the absence of the impact
of the rainfall drops. With the expressions given in Table 5.2, this increased flow rate
is roughly (1
cS
0
P
e
) times the equilibrium flow rate
iL
; for instance, with a slope
+
3cmh
−
1
, this indicates a sudden increase of
38%. But in actual flow situations this sudden increase is unlikely to be that large, and
the value predicted by Equation (6.37) can only be considered as an upper limit. Indeed,
a sudden change in shear stress resulting from the cessation of the rainfall, must also
involve accelerations, which are neglected in the kinematic approach leading to (6.37),
and which will tend to offset this effect. Moreover, even if it were to occur, the spike is
rapidly dissipated. Finally, natural rainfall events never cease suddenly, but they tend to
decrease rather gradually. Brief increases in runoff, upon the cessation of rainfall have
been observed experimentally and reported by Izzard (1946), but they were much smaller
than those predicted here by Equation (6.37).
S
0
=
0
.
001 and a rainfall intensity
P
=
0
.
6.3
Lumped kinematic approach
Although this approach is now dated, as it was developed prior to the more fundamental
analyses described above, the lumped kinematic approach is still of some interest because
it has often been used as the framework to analyze valuable experimental data. It was devel-
oped by Horton (1938) in his pioneering analysis of overland flow; it was subsequently
applied by Izzard (1944) in processing the data from his extensive experimental investiga-
tions on rain runoff from paved and grassy surfaces. In this approach the continuity equation
is replaced by the storage equation (1.10) or (5.125). In the notation of overland flow this
storage equation can be written as
iL
−
q
L
=
L
d
h
dt
(6.38)
where
L
1
L
h
=
hdx
(6.39)
0
denotes the spatial average of the water depth over the plane. To close (6.38),
q
L
must be
related with
; this can be done for steady equilibrium flow conditions by combining (6.8)
and (6.18) to obtain
h
a
+
1
q
L
=
K
l
h
(6.40)
where
K
l
={
[(
a
+
2)
/
(
a
+
1)]
(
a
+
1)
K
r
}
. If it is now assumed that (6.40) is also valid under
non-steady conditions during buildup or subsidence as well, its substitution in (6.38) yields
dq
1
/
(
a
+
1)
L
dt
iL
−
q
L
=
LK
−
1
/
(
a
+
1)
l
(6.41)
To determine the outflow rate
q
L
at the downstream end of the surface, (6.41) must be
integrated for the imposed input
i
=
i
(
t
). The essential features of the problem can again be
obtained readily by considering the buildup phase and the decay phase for a lateral inflow
rate
i
, which is constant in time.
