Geoscience Reference
In-Depth Information
design of urban and airport storm drainage facilities (see Horner and Jens, 1942; Hathaway,
1945; Jens, 1948). The study by Izzard (1946), which built on Horton's approach, is also
well known and its results have been widely applied (Linsley
et al.
, 1975). On the basis of
extensive experiments, Izzard (1946) concluded that for
iL
<
3
.
8m
2
h
−
1
the flow can be
described by the lumped kinematic formulation with Equation (6.40) in which (
a
+
1)
=
3
for laminar flow. The value of the other parameter, namely
K
l
, was derived from his experi-
mental data as a function of surface roughness and of rainfall intensity. For the rising phase
of the outflow he obtained the solution presumably by numerical means, and presented it
graphically as a dimensionless hydrograph. The time variable
t
was scaled with the time
to equilibrium, which Izzard took as the time required to produce an outflow rate which
is 97% of the equilibrium outflow rate, or
q
L
=
0
.
97(
iL
). By assuming (see the curve for
a
=
2 in Figure 6.11) that the volume of water detained in surface storage on the plane is
roughly half of the volume of inflow during the time required to reach equilibrium
t
e
,he
was able to propose the following expression
2
h
s
i
t
e
=
(6.45)
in which
t
e
is the time to equilibrium after the start of the rain and
h
s
is the average water
depth after equilibrium has been reached.
Figure 6.11 illustrates that the lumped kinematic approach does not really produce a
very good mathematical description of overland flow, as compared with the kinematic wave
solution, which is known to provide a close approximation to the exact solution. Thus
the question can be raised how the experimental results could be described so well by
the lumped approach in Izzard's (1946) study. The explanation of this discrepancy proba-
bly lies in the method used to scale the experimental rising hydrographs. As illustrated in
Figure 6.11,
q
L
+
(
≡
q
L
/
iL
) approaches unity asymptotically in the lumped kinematic solu-
tion, so that with noisy data the identification of
q
L
+
=
0
.
97, to determine the time to
equilibrium, is not easy. However, Izzard (1946, p. 148) noted that with the above definition
of
t
e
in (6.45), for
a
=
2 the lumped kinematic solution indicates that at the time
t
=
0
.
5
t
e
the outflow rate is roughly
q
L
+
=
0
.
55; therefore, he decided instead to non-dimensionalize
the experimental rising hydrographs with the criterion
t
e
=
2
t
0
.
55
, in which
t
0
.
55
is the time
at which the outflow is 0.55 the equilibrium value. As shown by Woolhiser and Liggett
(1967; Fig. 8), with this time scaling the agreement is improved considerably. This should
not be surprising because this way the curves are forced to coincide at
t
/
t
e
equal to 0 and
to 0.55.
The recession hydrograph
After the rain stops
i
=
0, and Equation (6.41) can immediately be integrated for any value
of
a
. Again, in dimensionless terms, this can be written as
a
(
a
(
q
Li
+
)
−
a
/(
a
+
1
)
−
(
a
+
1
)/
a
2)
(
a
+
1)
+
q
L
+
=
t
+
+
(6.46)
in which the subscript
i
indicates the initial value of the dimensionless outflow rate, that
is at
t
=
0, when the rain stops; for the case
D
>>
t
s
, when the rainfall duration is much
larger than
t
s
(see Equation (6.20)) this initial outflow rate can be taken equal to one. Izzard
(1946) used a recession function, which is essentially the same as (6.46) with
a
=
2, but
with different scaling, to analyze his experimental data.
