FLOOD05499

• County: Statewide
• Community: State of Colorado
• Basin: Statewide
• Title: Urban Storm Runoff Inlet Hydrograph Study - Volume 5
• Date: 5/1/1975
• Prepared By: Cheng-lung Chen
• Floodplain - Doc Type: Flood Mitigation/Flood Warning/Watershed Restoration

where Lp is the L value at t; ty' Because L in Eq, 47
is not expressed as an explicit function of t, the value
of L for given t must be determined numerically, for
instance, by the Newton-Raphson method, After L is
computed, the value of L will be substituted either
into Eq, 38 or 44 for the computation of the f value,
Note that use of Eq. 44 has a slight advantage over
that of Eq, 38 because any enor involved in the
evaluation of dL/dt in Eq, 38 can be avoided,
However, use of either Eq, 38 or 44 requires the prior
computation of the values of tp and Lp'

Determination of the tp and Lp values, Before
ponding (t<O;tp), the cumulative inmtration, F(t), is rt
for constant rainfall intensity, r. Thus, from Eq, 38,
after integration and rearrangement, one obtains at t
;tp

(r - K )
L 0 t
P - as - 00 p

, , , , ' , , .(48)

and from Eq. 44, at t; tp

~KS(h - ~o)
L -
P (r - Kg)

, , ' , , . .(49)

Equating Eqs, 48 and 49 yields

t =
p

(8 - 8 )(h - ~ )
K s 0 0
~ s (r - K )(r - K )
o s

, , , , ,(50)

The values of Lp and tp can be determined from Eqs,
49 and 50 respectively, Like Eq, 47, the fl value must
be evaluated or given first in order for Eqs, 49 and 50
to be useful,

EvalWltion of the fl value, As defined in Eq, 42,
the fl value depends on the K(z} and ",(z}
distributions, which are, of course, unknown. How-
ever, if one knows the K-"'relationship for the soil,
the fl value can be readily evaluated from Eq, 42 and
the K. '" relation, It is understood that the K.w
relation is hysteretic, but will be assumed unique
herein as long as the wetting process is only con.
sidered, Several forms of the K.",relation have been
assumed by different investigators in an attempt to
best fit the field or experimental data, Because the
soil after ponding (for h>O) has both the saturated
and unsaturated zones, all the forms of the K.",
relation must be modified to include K; K, for ",;;.0.

The simplest of the available K-'" relations is a
linear relationship between K and "', proposed by
Richards (1931),

K .. a;' + Ks

for ;. :OS 0

(5la)

K = K
s

for ;. ~ 0

(Sib)

where "an is simply a constant.

Raats (1971) and Braester (1973) among others
used an exponential form in K and "'.

K" K ealjJ
s

for 1/1 :S 0

(52a)
(52b)

K ,., K
s

for I/J 2:: 0

It should be noted that Eqs, 51 and 52 have only one
parameter, a, needed to be determined from the K-",
data,

Several researchers proposed multi-parameter
models which should fit the K-'" data better than the
one-parameter model. For example, Gardner's (1958)
original three.parameter model for "'<0 may be
extended to include "'; 0 so that

K -

a

(53a)

for I/J :S 0

(_~)n + b

K - K
s

for tjI ~ 0

(53b)

where a, b, and n are all constants. However, the
necessity of having K ; K, at "'; 0 results in the loss
of freedom for Eq, 53a to choose the a value other
than bI(,. Conversely, if K ; K, at '" ; 0 is merely
considered as a data point in the K-", relationship, the
values of a, b, and n in Eq, 53a can be determined by
using a least squares optimization procedure. Mter
that, one should take the value of K, equal to a/b, In
view of the difficulty in integrating Eq, 53a with
respe<:t to '" upon substitution of Eq, 53a into Eq, 42
for the evaluation of the fl value, the following
compatible form similar to Eq, 53a is proposed
herein:

K-

a
(_". + b)n

(54a)

for 1/J:5 0 . . . . .

K - K
s

for ~~ 0 . , , , , (54b)

By the same token, the values of a, b, and n in Eq,
54a can be estimated by using a least squares
optimization technique, and hence the value of K,
may be taken to be equal to a/b".

The values of a, b, and n determined from Eqs.
53 and 54 are not all dimensionless, In order to make
these dimensionally different parameters consistent
(i.e" dimensionless), some investigators proposed
dimensionless forms of Eq, 53 with the same number
of parameters used in Eq, 53, Of them, Wei (1971)
has used the most general one that is nevertheless not
adopted in this study due to the same reason as given
for Eq, 53,

Substituting Eqs, 51, 52, and 54 into Eq, 42
yields

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