Geoscience Reference
In-Depth Information
of Horton (1941) and Rockwood (1958); Mein
et al
. (1974) concluded that
m
71
could be used as a typical value. As indicated in Equation (12.50),
K
n
is proportional to
the time of travel; accordingly, this parameter was estimated by putting
=
0
.
K
n
=
C
1
K
1
(12.51)
in which
K
1
was taken to be the value of the relative travel time through the subarea
represented by that particular storage tank, i.e. (
L
S
1
/
0
)or
L
mentioned earlier. With
m
and
K
1
known,
C
1
remained as the only unknown parameter of the model; this was
estimated by trial-and-error routings with available data. A more optimal estimation of
the parameters in this model was subsequently developed by Kuczera (1990) and Kuczera
and Williams (1992). A similar nonlinear storage routing procedure was developed by
Boyd
et al.
(1979). However, following Askew's (1970) findings, the storage coefficient
(or lag) was made to depend also on the area
A
represented by the storage element,
namely as
K
/
=
aA
b
y
m
−
1
; in their case the constants were taken as
b
=
.
0
57 and the
slightly different value
m
=
0
.
77.
Physical justification of nonlinear tanks
In the past, the nonlinear storage relationship (12.48) has been justified on physical
grounds, mostly by considering open channel storage. The argument usually follows that
originally developed by Horton (1936; 1937), based on a lumped kinematic analysis for
quasi-steady, quasi-uniform flow. Thus the volume of water stored in a channel reach of
length
L
is assumed to be given by
S
c
=
A
c
L
, in which
A
c
is the average cross-sectional
area in the reach. The channel is assumed to be wide enough, so that the hydraulic radius
equals the mean water depth, or
R
h
=
h
, and the cross-sectional area of the channel
equals the depth times the width, or
A
c
=
hB
c
. For steady uniform conditions, (5.39) (or
(5.43)) produces then the outflow rate from the reach as
Q
C
r
B
c
S
0
h
a
+
1
; the channel
=
storage is in terms of the outflow from the reach
B
c
L
a
+
1
C
r
S
0
1
/
(
a
+
1)
Q
1
/
(
a
+
1)
S
c
=
(12.52)
in which
a
and
b
are the parameters in the open channel equation (5.39). Hence, if it is
assumed that all the storm runoff water in the catchment is stored in the stream channels,
so that
S
=
S
c
/
A
, one has
B
c
L
a
+
1
C
r
S
0
A
a
1
/
(
a
+
1)
y
1
/
(
a
+
1)
S
=
(12.53)
in which, as before,
y
A
,
L
is now the length of all stream channels in the catchment
upstream from the point where
Q
is determined, and in which the other variables are
assumed to be averages over the catchment area
A
. This result is in the form of (12.48),
with
m
=
Q
/
1)
−
1
67 for the Chezy equation.
But this derivation of Equation (12.53) is not wholly convincing, first, because obvi-
ously not all the storm runoff water is stored in channels, and second, because it is
well known that a major part of most storm flows is generated by subsurface runoff,
as explained in Chapter 11. Some estimate of groundwater storage can be obtained
=
(
a
+
=
0
.
60 for the GM equation, and
m
=
0
.
