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Guo, James C. Y.(2011) “Field Test on Conversion of Natural Watershed into Kinematic Wave
Rectangular Planes, second review, ASCE J. of Hydrologic Engineering, May 16, 2011
FIELD TEST ON CONVERSION OF NATURAL WATERSHED INTO KINEMATIC
WAVE RECTANGULAR PLANE
James C.Y Guo
Professor and Director, Civil Engineering, U of Colorado Denver, E-mail:
James.Guo@UCDenver.edu
Abstract
Conversion of a natural watershed into its equivalent kinematic wave rectangular plane has long
been a concern in the practice of stormwater numerical simulations. Based on the principles of
mass and energy, the actual watershed and its virtual kinematic wave plane can be related by the
watershed shape factor that involves the waterway length and slope, and watershed area. In this
study, two dimensionless watershed shape functions are derived to use parabolic function and
trigonometric Sine curve for watershed conversion. These two watershed shape functions
produce good agreements with the maximum overland flow length method for hypothetical square
watersheds. Also, these two watershed shape functions are able to re-produce similar kinematic
wave plane widths as reported in a calibrated model. Furthermore, in this study, these two
watershed shape functions are tested by nine observed rainfall events and three levels of
modeling details. These 54 case studies reveal that the parabolic shape function consistently
produces better agreements with the observed runoff hydrographs. Also, a model with more
drainage details results in more concentrated flows or higher peak flows. On the contrary, a
model with a low resolution tends to decrease the peak flow because of the significant surface
detention volume spread in the overland flow.
Key Words: Kinematic Wave, SWMM, Watershed Shape, Stormwater, Overland Flow
INTRODUCTION
The numerical procedure for the kinematic wave overland flow (KWOF) method requires the
conversion of an irregular watershed into its equivalent rectangular sloping plane on which the
storm hydrograph can be simulated using the unit-width overland flow multiplied by the plane
width (Lighthill and Whitham 1955, Wooding 1965, Singh 1966). Among all necessary watershed
parameters, the plane width is a pre-knowledge for runoff numerical modeling when using the
KWOF method (Guo 1998). However, in the development of kinematic wave theories, the width of
KW sloping plane is not defined clearly. As a result, there has been a wide variability in how a
plane width is assessed. For instance, as recommended (UDSWMM Manual 2000), the plane
width is twice the channel length. The comparison study between the EPA Storm Water
Management Model Version 5 (Rossman 2005) and the Colorado Urban Hydrograph Procedure
(CHUP 2005) suggests that the ratio of 2.2 be used between waterway length and KW plane
width. As reported (Guo and Urbonas 2009), such a ratio can vary from 2.0 for the watershed
symmetric to its collector channel to 1.0 for the watershed skewed with a side channel. Based on
the calibration study of a watershed in Ontario, Canada, a ratio of 1.67 was adopted between the
waterway length and its KW plane width (Proctor and Redfern 1976). The EPA SWMM5 user
manual (Rossman 2005) suggests that an initial estimate of the plane width be given by the
watershed area divided by the average maximum overland flow length and then the model needs
a calibration to confirm the selections of KW plane width and other parameters. As reported, the
model-calibration approach was successful in the study of Fox Hollow Watershed, Centre County,
PA (Zhang and Hamlett 2006). All these reports amount to the fact that a reliable application of
the KWOF method depends on the model calibration against the historic data or modeler’s
experience with the watershed. It implies that the KWOF method is only applicable to an existing
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watershed and it has no warranty when predicting stormwater movement for a proposed future
condition.
Urban master drainage planning is a common practice to outline the best strategy to mitigate the
future flood potentials before the regional development. When using EPA SWMM5 and HEC
HMS (2010) computer models to predict flood flows for the future alternatives, the selection of
plane width for each subarea used in the computer model has become a fundamental challenge.
Since the width of the virtual rectangular KW plane is related to the channel length of the actual
irregular watershed, it was an attempt to derive a KW parabolic shape function that could directly
help the engineer calculate the plane width based on the watershed shape defined as the ratio of
width to length (Guo and Urbana 2009). Although this KW parabolic shape function was
successfully tested for its consistency among several hypothetical watersheds, its application to
real watersheds needs to be verified. Secondly, a question was raised among reviewers and
users as to if a periodical function such as, Sine and Cosine, may be better than the parabolic
function for watershed shape conversion. In this study, a new periodical KW shape function was
derived using trigonometric sine function. Both the periodical sine curve and the parabolic KW
shape function are then tested for real watersheds under the observed rainfall cases. The
comparison with the observed runoff hydrographs provides a basis to evaluation the applicability
of these two KW shape functions. It is expected that the KW shape function tested in this paper
will serve as a guide on the selection of KW plane width when developing EPA SWMM5 and HEC
HMS computer models for purpose of regional master drainage planning.
DERIVATION OF SINE SHAPE FUNCTION
The KWOF method is widely utilized to predict storm hydrographs from natural watersheds (HEC
HMS 2010, EPA SWMM5 2005). As illustrated in Figure 1, the natural watershed has to be
converted into its equivalent rectangular KW plane. The conservation of runoff volume between
these two flow systems is essentially the equality of the surface drainage area because the
amount of rainfall excess remains the same.
A
X w Lw
(1)
in which A = watershed tributary area, Xw = length of overland flow on KW sloping plane, and Lw =
width of KW sloping plane. Between these two flow systems, the potential energy in terms of the
vertical fall along the waterway must be conserved as:
So L
Sw ( X w
Lw )
(2)
In which L = natural waterway length, So = watershed slope and Sw = KW plane slope.
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Figure 1 Conversion of Natural Watershed into KW Rectangular Sloping Plane
Consider the natural waterway length to be the characteristic length for both flow systems. Eq’s (1)
and (2) can be normalized as:
A
L2
B
L
X w Lw
L L
(3)
In which B=average watershed width. In this study, the watershed shape factor is defined as the
ratio of B/L or A/L2.
So
Sw
Xw
L
Lw
L
(4)
Eq (3) represents the geometric relationship between these two watershed systems in Figure 1.
The ratio, Lw/L, is termed the KW shape factor (Guo and Urbonas 2009).
As indicated in Eq (3), the KW shape factor, L/Lw, is related to A/L2. For convenience, in this
study, the watershed shape factors are derived as:
X
A
L2
(5)
Y
Lw
L
(6)
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In which X = watershed shape factor for actual system, and Y = KW shape factor for the KW
sloping plane. The general functional relationship between these two shape factors is described
as:
Y
f (X )
(7)
In which f (X)= mathematical functional relationship. This mathematic function needs to satisfy
three special cases as follows (Guo and Urbonas 2009):
Case 1: When the natural watershed is extremely small, then the KW shape factor will approach
zero as:
X
Y
0
when A
0
(8)
Case 2: As illustrated in Figure 2, the square-shaped watershed provides a unique relationship
between the actual watershed and virtual KW plane as follows:
Y
1 when X
1 for square watershed with a side channel
Y
2 when X
1 for square watershed with a central channel
(9)
(10)
Figure 2 KW Conversion of Square Shaped Watershed
The two cases in Figure 2 represent two extreme locations of channel alignment. In most cases,
the collector channel divides the watershed into two uneven sub-areas. In this study, the area
skewness coefficient, Z, is incorporated into the KW shape function as:
Y
(1.5 Z ) f ( X ) for all watersheds
Z
Am
A
(11)
(12)
in which Z= area skewness coefficient between 0.5 and 1.0, and Am = dominating area that is the
larger one between the right and left sub-areas separated by the collector channel. As illustrated
in Figure 3, Z= 1.0 for a watershed with a side channel along its boundary while Z=0.5 for a
watershed symmetric to its central channel.
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Figure 3 Illustration of Area Skewness
Case 3: As a common practice, a watershed is often divided into smaller sub-basins before the
numerical simulation. The sub-basin’s shape shall not be too wide or too slender. Mathematically,
it implies that the first derivative of Eq 11 vanishes when Y reaches its allowable maximum as:
dY
dX
0 at X = K
(13)
In which K = maximal value for X. As recommended, the value of X is not to exceed 4 when
dividing a watershed into sub-basins (CUHP 2005). In this study, it is proposed that Eq 11 can be
formulated as:
Y
(1.5 Z )[a sin( bX ) c ] (Trigonometric function)
(14)
And
Y
(1.5 Z )[aX 2
bX
c ] (Parabolic function)
(15)
Substituting Eqs (14) into Eq’s (8), (11), and (13) yields
Y
(1.5 Z )[
2
sin
sin(
X
)] for 0
8
4
(16)
8
Repeat the same process, Eq (15) becomes
Y
(1.5 Z )(2.286 X
0.286 X 2 ) for 0
4
(17)
Figure 4 presents a comparison between the parabolic function and Sine curve for KW plane
width. When X<1.5, there is not any difference between these two KW shape functions, but the
difference grows when X increases.
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Figure 4 Plot of KW Shape Functions for Z=0.5 and 1.0
COMPARISON AMONG THREE KW PLANE-WIDTH METHODS
In this study, the first test is conducted on five square watersheds of 100 feet by 100 feet. As
shown in Figure 5, the collector channel in each square is aligned with a different path. The three
methods are used and then compared in this test, including the parabolic function, Sine curve,
and maximum overland flow length method (MOFL). The maximum overland flow length is the
distance of the flow path from the upper watershed boundary to the collector channel. Maximum
lengths from several different overland flow paths should be averaged. These paths should depict
the slow flows through pervious surfaces, rather than the rapid flows over paved surfaces. To
apply the MOFL method, the overland flow is defined in the direction perpendicular to the channel
alignment. The KW plane width is then equal to the ratio of watershed area to maximum overland
flow length as:
Lwm
A
Lmax
(18)
where Lwm = KW plane width determined by MOFL method in [L], and Lmax = maximum overland
flow length in [L]. Eqs (16) and (17) can be used to directly calculate the KW plane widths. Table
1 is the summary of detailed computations using these three methods. For the first two special
cases, these three methods produce the same exact solutions. For the rest of cases, the
differences among these three methods are negligible. It implies that for these five hypothetic
square watersheds, both parabolic and periodical KW shape functions derived in this study can
select good KW plane widths in case of no experience about the site or no data for model
calibration.
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Figure 5 Channel and Overland Flows in Hypothetical Square Catchments
Case Channel
Area
Shape
KW Plane Width
ID
Length
Skew
Factor Parabolic Function
Sine
Curve Max Overland Flow Method
2
X=A/L
L
Z
Y=Lw/L
Lw
Y=Lw/L
Lw
Lmax
Lwm
ft
ft
ft
ft
ft
100.0
100.0
100.0
1.00
100.0
1
100.00
1.00
1.00
1.00
200.0
200.0
200.0
2
100.00
0.50
1.00
2.00
2.00
50.0
152.0
145.0
141.0
3
141.00
0.50
0.50
1.08
1.03
71.0
171.0
169.0
154.0
4
103.00
0.63
0.94
1.66
1.64
65.0
138.0
135.0
133.0
5
112.00
0.75
0.80
1.23
1.21
75.0
Table 1 KW Plane Widths Determined for Square Watersheds of 100-ft by 100-ft
TEST OF PARABOLIC SHAPE FUNCTION ON CALIBRATED WATERSHED
The next test is to extend the comparison from the hypothetical cases into a real case that
involves the watershed located in the City of Miami, Florida. As reported, this watershed in Figure
6 has a tributary area of 14.7 acres which was divided into 13 sub-basins. The KW plane widths
were calibrated by several observed rainfall-runoff events and then examined for other events
independently (Bedient and Huber 1997). Table 2 presents the calibrated plane widths, Lmw, for
this watershed determined using the averaged maximum overland flow length method.
Based on the aerial photo from Google Earth, this watershed is a high-density residential area
with roadside ditches to serve as the side channel. As a result, Z=1 for all sub-basins. Table 2 is
the summary of the watershed parameters. Again, the parabolic KW shape function almost
reproduces the calibrated plane widths without on-site experience. As can be seen, the
differences between Lw and Lmw are negligible. As shown in Figure 6, the EPA SWMM5
computer model was developed in this study to reproduce the observed runoff hydrographs when
applying the parabolic KW shape function to plane widths. Details can be found elsewhere
(Cheng 2010). This case study further confirms that the parabolic shape function provides
reasonably good guidance to select the KW plane widths for use in the EPA SWMM5 models.
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Figure 6 Layout using EPA-SWMM Model for Miami Watershed
Subarea
ID
KW
Plane Width Calibrated KW Plane Width Predicted
Lwm
Lw
A
L
So
Z
X
Y
Sw
ft
ft
acre
ft
%
%
254
244.0
1.157
198
3.1
1
1.29
1.23
1.36
201
87
93.4
0.352 176
3.5
1
0.50
0.53
2.39
202
148
161.4
2.9
1
0.36
0.39
2.22
1.412 416
203
150
162.4
2.0
1
0.42
0.45
1.45
1.236 359
204
241
220.9
0.842
152
2.7
1
1.59
1.45
1.06
205
88
94.7
0.395 196
3.0
1
0.45
0.48
2.13
206
81
91.2
1.204 647
1.8
1
0.13
0.14
1.75
207
65
73.4
1.006
674
2.0
1
0.10
0.11
2.01
208
126
135.4
0.761 263
3.1
1
0.48
0.51
2.14
209
175
193.8
2.798 696
2.1
1
0.25
0.28
1.78
210
89
99.6
1.4
1
0.17
0.19
1.29
1.049 513
211
112
124.7
565
1.3
1
0.20
0.22
1.16
1.452
212
145
156.4
1.079 324
2.8
1
0.45
0.48
1.99
213
Tabel 2 Plane Widths Predcited by Parabolic Shape Function for Miami Watershed.
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TEST OF WATERSHED SHAPE FUNCTIONS ON MONITORED WATERSHED
In this study, the derived periodic and parabolic KW shape functions are further tested for the
observed rainfall and runoff events recorded at two rain gages and one stream gage, USGS
06711570, installed in Harvard Gulch located in the City of Denver, Colorado. The Harvard Gulch
Watershed is one of the matured urban watersheds managed under a long term rainfall and
runoff monitor programs. The drainage system consists of grassy waterway, concrete channels,
and closed conduits. The details of its hydrologic parameters are well documented as public
information (FHAD 1979, Zarriello 1998, OSP 2010). Referring to Figure 7, the tributary area is
approximately 728 acres (or 295 ha.) and has been developed into a mixed land use among
commercial, high-density apartments, low-density residential, parks and open space.
Figure 7 Topographic Map for Harvard Gulch Watershed
The rain gages and stream gage operated in this watershed provide a continuous record for 18
years. After an extensive review of more than 150 events, it was found that most of the observed
events could not even satisfy the basic principle of volume conservation, or the runoff volume
under the direct runoff hydrograph is greater than the rainfall volume under the recorded
hyetograph. Such a volumetric discrepancy is often caused by the rain under-catch at the rain
gages due to wind and vegetation canopy effects (Guo et al. 2001). As recommended in EPS
SWMM manual, four major water volumes in storm water simulation shall satisfy the principle of
continuity as:
Dv
VP VF
VR
(19)
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In which Dv = depression volume in [L], VP = rainfall volume in [L], VF = infiltration volume in [L],
and VR = direct runoff volume in [L]. All these volumes are computed as unit depth per watershed
area. The infiltration volume can be estimated by the Horton formula for Type C and D soils. The
unknown depression volume is set to be within 0.1 to 0.7 inch based on many field investigations
(UDFCD 2001). Using Eq 19 as a screening criteria, there were only 9 events identified for
modeling tests. These selected events had a total rainfall depth greater than 1.0 inch. Table 3
summarizes these 9 events identified in this study.
Time
min
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
Rainfall
Depth
8/4/1988 7/20/1991 7/23/1992 9/18/1993 7/25/1998 8/17/00 7/8/2001
cm
cm
cm
cm
cm
cm
cm
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.20
0.00
0.03
0.36
0.08
0.15
0.10
0.46
0.00
0.05
0.30
0.38
0.20
1.02
0.64
0.18
0.23
0.18
0.23
0.08
0.41
0.51
0.41
0.79
0.13
0.28
0.08
0.25
0.56
0.46
0.41
0.08
0.25
0.18
0.91
0.13
0.46
0.15
0.18
0.36
0.20
1.70
0.13
0.51
0.03
0.64
0.28
0.25
0.20
0.30
0.33
0.03
0.33
0.38
0.20
1.88
0.58
0.13
0.00
0.10
0.48
0.15
0.33
0.64
0.23
0.00
0.38
0.23
0.10
0.46
0.28
0.10
0.00
0.05
0.13
0.08
0.36
0.15
0.05
0.00
0.00
0.13
0.05
0.23
0.20
0.03
0.00
0.03
0.10
0.03
0.48
0.15
0.00
0.00
0.00
0.05
0.03
0.33
0.15
0.03
0.00
0.00
0.05
0.03
0.20
0.15
0.00
0.00
0.00
0.08
0.03
0.05
0.05
0.03
0.00
0.00
0.03
0.03
0.03
0.05
0.03
0.00
0.00
0.03
0.03
0.03
0.05
0.03
0.00
0.00
0.03
0.03
0.05
0.03
0.00
0.00
0.00
0.03
0.05
0.00
0.05
0.00
0.00
0.00
0.10
0.03
0.00
0.03
0.00
0.00
0.00
0.03
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.03
0.00
Table 3 Rainfall Events Selected for Harvard Gulch Watershed Tests
9/12/2002
cm
0.00
0.01
0.41
0.69
0.91
0.38
0.15
0.08
0.10
0.05
0.05
0.01
0.01
0.01
0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
The computer model, EPA-SWMM5, is employed to simulate these 9 recorded rainfall and runoff
events. The watershed was divided into 23 smaller sub-areas with an average size about 30
acres each. As shown in Table 4, the KW plane widths are determined using the parabolic and
periodical shape functions. Minor differences are detected between these two functions. Aided
with Eq (4), the slopes on the KW planes are determined to preserve the potential energy along
the waterways in the two flow systems. The KW plane widths for the sub-basins in Table 4 were
directly computed with deterministic formulas. No specific calibration or empirical adjustment was
involved. These parameters were then entered into the EPA SWMM5 computer model to predict
the storm hydrographs at the location of USGS Stream Gage, ID 671157. Figure 8 presents the
predicted and observed hydrographs for the 7/21/1991 rainfall event. In comparison, the Sine
shape function underestimates the peak flow while the parabolic shape function produces good
Page 10
6/18/2003
cm
0.00
0.15
0.79
0.66
0.18
0.28
0.38
0.30
0.13
0.05
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
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agreement with the observed. This tendency is consistently revealed throughout these 9 rainfall
events (Cheng 2010).
2
X=A/L
Watershed
Subarea
A
L
Z Parameters
So
ID acres
feet
%
28 25.45 6688.2
1.0
0.70
29 25.70 3567.2
0.6
1.00
30 15.81 1189.2
1.0
0.80
31 35.58 3864.3
1.0
1.50
32 21.74 2972.5
1.0
1.50
33 38.55 1189.2
0.8
2.10
34 26.69 7431.5
0.5
1.80
36 48.43 1486.2
1.0
1.20
37 26.44 2378.0
1.0
0.50
38 59.30 4756.1
1.0
0.40
39 23.23 1664.7
0.9
0.90
40 22.24 1307.8
0.5
1.20
41 34.10 2021.5
0.9
0.80
42 28.91 1664.7
1.0
2.00
43 36.08 1605.2
1.0
1.20
44 15.81 1307.8
0.9
1.00
45 38.05 1545.7
0.9
2.00
48
6.18 6391.1
1.0
2.50
134 35.58 7802.9
0.8
1.80
135 15.57 1394.3
0.8
1.50
136 12.11
743.3
0.8
1.50
137 67.71 8174.4
0.6
2.00
138 69.19 9475.1
0.6
2.00
Note: 1 ha= 2.47 acres, 1meter=3.25 feet
0.02
0.09
0.49
0.10
0.11
1.19
0.02
0.96
0.20
0.11
0.37
0.57
0.36
0.45
0.61
0.40
0.69
0.01
0.03
0.35
0.95
0.04
0.03
Sine Curve
Lw
Sw
feet
%
170.10
0.70
579.53
1.42
590.83
0.54
411.49
1.39
326.90
1.38
1955.8
0.89
4
321.04
3.39
1422.7
0.61
3
496.46
0.42
557.19
0.37
745.90
0.71
1507.7
0.73
4
901.76
0.63
772.21
1.39
995.06
0.75
645.93
0.76
1304.2
1.20
8
43.21
2.55
285.37
2.46
696.55
1.25
995.71
0.73
666.39
3.21
587.51
3.31
Parabolic
Lw Function
Sw
feet
%
188.80
0.77
638.24
1.49
621.48
0.55
452.31
1.49
359.18
1.49
1922.86
0.89
356.49
3.70
1427.89
0.61
539.22
0.44
611.69
0.39
795.22
0.72
1572.58
0.72
961.57
0.65
815.16
1.41
1033.14
0.75
685.76
0.77
1342.58
1.20
48.06
2.83
316.71
2.70
743.97
1.26
999.37
0.73
737.87
3.45
651.38
3.59
Table 4 Hydrologic Parameters for 23 Sub-basins Harvard Gulch Watershed
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Figure 8 Hydrographs from 23 Sub-basins under 7/20/1991 Rainfall Event in Harvard Gulch
Watershed
TESTS OF WATERSHED SHAPE FUNCTIONS ON MODELING DETAILS
As stated in SWMM4 User’s manual (Huber and Dickson 1988), the level of watershed’s
modeling details using the KWOF method leads to a variation in runoff predictions. In this study,
the two KW shape functions are tested for 3 levels of modeling details. They are Model A using
23 sub-basins or approximately 30 acres per sub-basin, Model B using 4 sub-basins or
approximately 180 acres for each sub-basin, and Model C using a single sub-basin of 730 acres.
For instance, Figures 9 and 10 present the predicted hydrographs for 7/20/1991 event. This case
reveals that the larger size the sub-basins, the lower peak flow the watershed produces. This
tendency reflects the fact that each sub-basin is numerically treated as a shallow reservoir. The
longer the overland flow, the higher the surface detention and the lower the peak flow.
Figure 9 Hydrographs from 4 Sub-basins under 7/20/1991 Rainfall Event in Harvard Gulch
Watershed
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Figure 10 Hydrographs from Single Sub-basin under 7/20/1991 Rainfall Event in Harvard
Gulch Watershed
In this study, these three models, A, B, and C are tested for both parabolic and periodical shape
functions under 9 observed rainfall events or a matrix of 54 cases (3x2x9=54) generated for
comparison. For the purpose of comparison, the Nash-Sutcliffe efficiency index is adopted for
case comparisons (Nash and Sutcliffe 1970). The coefficient of model-fit efficiency is defined as:
N
N
(Qoi
E
Qo ) 2
i 1
(Qoi
i 1
(20)
N
(Qoi
Qo )
Qsi ) 2
2
i 1
Where E= model-fit efficiency, QOi =observed runoff at i-th time step; QSi =simulated runoff at i-th
time step; QO =average observed runoff, and N=number of hydrograph ordinates. Using Eq 20,
the 54 cases are evaluated as shown in Table 5. Both the parabolic and Sin shape functions
provide good guidance to conduct the KWOF method using EPA SWMM. In general, a model
with a higher resolution on drainage details tends to produce more concentrated flows. Based on
the comparison of 54 cases, it is concluded that the shape functions derived in this study do not
replace the effort in watershed’s modeling details. The longer the plane width is, the higher the
peak flow will be.
Page 13
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Rainfall
event
8/4/1988
7/20/91
7/23/92
9/18/93
7/25/98
8/17/00
7/8/2001
9/12/2002
6/18/2003
Average
Model A
Model B
Model C
Sine
Parabolic Sine
Parabolic Sine
Parabolic
92%
88%
92%
95%
95%
87%
95%
78%
84%
88%
88%
77%
94%
91%
88%
86%
85%
91%
94%
90%
92%
88%
87%
90%
98%
70%
78%
83%
84%
68%
92%
88%
92%
95%
95%
87%
92%
88%
92%
95%
95%
87%
92%
88%
92%
95%
95%
87%
92%
86%
93%
94%
94%
85%
93%
90%
91%
84%
85%
89%
Table 5 Summary of Model-fit Efficiency for 54 Cases
CONCLUSION
1. The two KW shape functions were presented in this study for the purpose of converting a
natural watershed into its KW rectangular plane. Both parabolic and sine functions
provide same plane widths when the watershed shape factor, X, less than 1.5. The
difference between these two functions increases as the watershed shape factor, X,
becomes greater than 1.5. In this study, the watershed shape factor, X, is set not to
exceed 4.0. In case of wide watersheds, Eq’s 16 and 17 can be revised with a preselected K in Eq 13. In general, it is advisable to maintain X 4 for engineering practices.
2. The comparison with a calibrated model published in the previous report verifies that the
watershed shape functions provide reasonably good guidance to re-produce the on-site
experience on the selection of KW plane widths. Based on 54 case studies, the parabolic
shape function produces a higher model-fit efficiency at all levels of modeling details than
the Sine shape function. Mathematically, both Eq’s (3) and (4) implies that the shape
function is related to L2, or the parabolic function fits the field data better than the
periodical.
3. In this study, it was also confirmed that both parabolic and periodic shape functions for
plane width do not replace the effort in watershed’s modeling details. In comparison, a
higher level of details in watershed model, the more concentrated flows will be generated.
The numerical sensitivity in peak flow prediction is directly related to the surface
detention volume under the overland flow profile. In comparison, a smooth surface and a
longer plane width produce a higher peak flow while a rougher surface or/and a shorter
plane width result in a more surface detention volume.
REFERENCES
Bedient, P.N. and Huber W. C. (1997) “Hydrology and Floodplain Analysis”, Addison Wesley,
New York.
CUHP (2005) “User Manual, Colorado Urban Hydrograph Procedure - CUHP2005, Urban
Drainage and Flood Control District, latest update in 2008,
http://www.udfcd.org/downloads/down_software.htm
Cheng, Y.C. (2010) “Modification of Kinematic Wave Method for Cascading Plane”, PhD
dissertation, Dept of Civil Engineering, U of Colorado Denver, December.
EPA SWMM (2005). “Storm Water Management Model”, EPA, Cincinnati, OH.
Page 14
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PD
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F -X C h a n ge
F -X C h a n ge
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FHAD (1979), Flood Hazard Area Delineation Study for Harvard Gulch, West Harvard Gulch and
Dry Gulch, UDFCD, prepared by Gingery Associates, Inc., December.
Guo, James C.Y. and Urbonas, B. (2009) “Conversion of Natural Watershed to Kinematic Wave
Cascading Plane”, ASCE J. of Hydrologic Engineering, Vol 14, No. 8, August.
Guo, James C.Y., Urbonas, Ben. and Stewart, Kevin. (2001). “Rain Catch under Wind and
Vegetal Effects,” ASCE J. of Hydrologic Engineering, Vol 6, No.1, in Jan.
Guo, James C.Y. (1998). "Overland Flow on a Pervious Surface," IWRA International J. of Water,
Vol 23, No 2, June.
HEC HMS (2010) “Hydrologic Modeling System”, U.S. Army Corps of Engineer, Hydrologic
Engineering Center, Davis, California.
Lighthill, M.H. and Whitham, G.B., (1955). “On Kinematic Waves, I, Flood Movement
in Long Rivers, Proc., Royal Soc., of London, Ser. A., Vol 229, (1955), pp 281-316
Huber, W. C., and Dickinson, R. E. (1988). “Storm water management model user’s manual”,
Version 4, EPA, Athens, Ga.
Nash J. E. and Sutchiliffe, J.V. (1970). “River Flow Forecasting through Conceptual Models, Part
I – Discussion of Principles”, J of Hydrology, vol 10, pp 282-290.
OSP (2010),” Outfall System Plan – An Alternative Analysis and Feasibility Study for Harvard
Gulch”, Preliminary Engineering Report, published by Urban Drainage and Flood Control District,
March.
Proctor and Redfern, Ltd, and MacLaren, J.F. Ltd, (1976) “Stormwater Management Model
Study – Vol I, Research Report No. 7, Canada-Ontario Research Program, Environmental
Protection Service, Ottawa, Ontario, September.
Rossman, L. A. (2005) “Storm Water Management Model User’s Manual. Version 5”, Water
Supply and Water Resources Division, National Risk Management Research Laboratory,
Cincinnati, OH.
Singh, V. P. (1996). Kinematic wave modeling in water resources. Wiley, New York.
UDFCD (2001). “Urban Storm Water Drainage Criteria Manual.” vol 1 Urban Drainage and Flood
Control District, Denver, Colorado.
UDSWMM Manual (2000), “Urban Drainage Storm Water Management Model”, Urban Drainage
and Flood Control District, Denver, Colorado.
Wooding, R.A, A Hydraulic Model for a Catchment-Stream Problem, J. of Hydrology, Vol 3,
(1965), pp 254-267.
Zarriello, Phillip J. (1998), Comparison of Nine Uncalibrated Runoff Models to Observed Flows in
Two Small Urban Watersheds, presented in the First Federal Interagency Hydrologic Modeling
Conference, held in Las Vegas on April 19-23, 1998
Zhang, G. S. and Hamlett, J. M., (2006) “Development of the SWMM hydrologic model for the
Fox Hollow Watershed, Centre County, PA”, prepared in cooperation with Office of Physical Plant
Penn State University, University Park, PA 16802.
Page 15
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II. NOTATIONS
A = watershed tributary area,
Am = dominating area or the larger half
B=average watershed width
Dv = depression volume in [L],
E= model-fit efficiency,
f (X)= mathematical functional relationship.
K = maximal value for X
Lwm = KW plane width determined by MOFL method in [L]
Lmax = maximum overland flow length in [L]
L = natural waterway length
Lw = width of KW sloping plane.
N=number of hydrograph ordinates
QOi =observed runoff at i-th time step;
QSi =simulated runoff at i-th time step;
QO=average observed runoff,
So = watershed slope
Sw = KW plane slope
VP = rainfall volume in [L],
VF = infiltration volume in [L],
VR = direct runoff volume in [L]
X = watershed shape factor for actual watershed
Xw = length of overland flow on KW sloping plane
Y = KW shape factor for the KW sloping plane
Z= area skewness coefficient between 0.5 and 1.0
Page 16
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