DETERMINISTIC APPROACH FOR PREDICTION AND MANAGEMENT OF FLOOD
FLOW ON ALLUVIAL FANS
by
SHOUCHING HSU
B.S., National Chiao Tung University, 1990
M.S., University of Colorado, 1996
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Civil Engineering Program
2016
This thesis for the Doctor of Philosophy degree by
ShouChing Hsu
has been approved for the
Civil Engineering Program
by
David C. Mays, Chair
James C. Y. Guo, Advisor
Arunprakash Karunanithi
Balaji Rajagopalan
Zhiyong Jason Ren
Date: 04/06/2016
Hsu, ShouChing (Ph.D., Civil Engineering Program)
Deterministic Approach on Prediction and Management of Flood Flow on Alluvial Fans
Thesis directed by Professor James C. Y. Guo
ABSTRACT
Alluvial fans are the most prominent landscape features in the semiarid and arid
regions of the world. This geographic feature is often observed in the American
Southwest. From the viewpoint of engineering practices, the hydraulic behavior of
overland flows on alluvial fans is very different from open channel flows. Alluvial fan
flows are subject to high roughness effects due to wide and shallow surface flow. In
current practices, many floodplain studies on alluvial fans rely on qualitative approaches
to provide general approximations of flow depth, width and velocity. Since 1980s, the
Federal Emergency Management Agency (FEMA) has recommended a stochastic
approach for assessing flood risks and hazards on alluvial fans. The method applies a
conditional probability based on the risk level of the selected flood event, and the site
location on the fan area. This probabilistic approach has been accepted as a reasonable
approximation for alluvial floodplain studies, according to FEMA regulations. From a
regulatory standpoint, this approach provides a conservative assessment to delineate the
boundaries of flood hazard areas. But, it has been long recognized that the rainfallrunoff
process should be included as primary input parameters to determine the flood flows on
an alluvial fan. As urban areas continued to expand and invaded into alluvial fans in the
recent years, it is critically important to quantitatively determine the flood flow
characteristics with a higher accuracy and consistency. By understandings the hydrologic
processes on alluvial fans, it will definitely assist engineers to better manage the risks and
uncertainties in land use planning and mitigate flood hazard when conducting a regional
master drainage plan for alluvial fan areas.
In this study, the Kinematic Wave (KW) overland flow approach on a rectangular
plane was reviewed, and then expended into converging and diverging planes to simulate
overland flow hydraulics on fan shape geometry. The continuity principle was applied to
balance the runoff volumes among excess rainfall amount, runoff inflow and outflow and
surface detention volume. New governing equations were derived to describe the
accumulations of overland flows, according to the geometry of the KW plane 
rectangular, converging, or diverging. These equations were also expanded to provide
analytical and numerical solutions to present overland flows on diverging or converging
planes. The results of the numerical simulations are well agreed with several reported
laboratory studies and field observations. It is believed that the method presented in this
study can improve the current FEMA flood flow study procedure and minimize the
environmental impacts as the urbanization process encroaches into alluvial fans in the
southwest States of the US.
The form and content of this abstract are approved. I recommend its publication.
Approved: James C. Y. Guo
IV
DEDICATION
I dedicate this work to my wife, Hsuehling, and my daughter, Trinity, for their
supports and companies during the low and high moments in this academic adventure.
They constantly remind me that life is more appreciated when a family is supportive.
Also, I like to dedicate this work to my parents, brother and sister back in Taipei, Taiwan,
who taught me the value and the importance of the family.
v
ACKNOWLEDGMENTS
Education is not the learning of facts but the training of
the mind to think
I never teach my pupil, I only provide the conditions in
which they can learn
Albert Einstein
I would like to express my sincere appreciation to Professor James C. Y. Guo for
his mentorship, patience, encouragement and guidance during my graduate studies for
both M.S. and Ph.D. degrees. Five years ago, I thought I had committed to a long and
lonely academic journey. But during this journey, Professor Guo offered me many
different aspects to investigate the selected topics. In return, my five years of Ph.D. study
has been rewarding and I am proud to be named his pupil. Although I was not sure if I
have become a scholarly hydrologist, but I definitely learned how to formulate, present,
and defend my creative thoughts and ideas. I do agree that my Ph.D. training is not
limited to the contents of my study, but the process and the methodology.
Of course, I also wish to express my appreciation to the members of the
committee Dr. Rajagopalan, Dr. Mays, Dr. Rens, and Dr. Pal, for their comments and
helps. Special thanks to Maroun Ghanem for helping me with graphics in the exhibits.
Thanks to Jill Reilly to help me with editorial comments in my writings.
vi
TABLE OF CONTENTS
Chapter
1. Introduction..................................................................1
1.1 Urbanization Impacts to Alluvial Fan......................................1
1.2 Alluvial Fan Location.....................................................5
1.3 Alluvial Landform Descriptions and General Characteristics................7
1.4 FEMA Flood Flow Approach for Alluvial Fans...............................12
1.5 Objective................................................................17
2. Review Of Kinematic Wave Overland flow......................................20
2.1 Introduction.............................................................20
2.2 Basic Principles.........................................................20
2.3 Kinematic Wave Application Limit.........................................23
2.4 Overland Flow on a Rectangular Pervious Plane............................27
2.5 Overland Flow Hydrograph Analysis........................................29
2.6 Overland Flow Travel Time on a Pervious Surface..........................36
2.7 Closing..................................................................41
3. Derivations of Governing Equations for Diverging and Converging Kinematic Wave
Overland Flows..............................................................42
3.1 Introduction.............................................................42
3.2 Kinematic Wave Overland Flow on a Diverging Plane........................43
3.3 Rising and Recession Hydrograph for Kinematic Wave Overland Flow on a
Diverging Plane.........................................................49
3.4 Kinematic Wave Overland Flow on a Converging Plane.......................51
Rising and Recession Hydrograph for Kinematic Wave Overland Flow on a
Converging Plane........................................................58
vii
3.5
3.6 Numerical Approach.......................................................59
3.7 Closing..................................................................62
4. Geometric Transformation of Diverging and Converging Fan Areas into Kinematic
Wave Rectangular Planes......................................................64
4.1 Introduction.............................................................64
4.2 Transformation Procedure Development.....................................66
4.3 Development of Geometric Transformation Using SINE Function..............69
4.3.1 Geometric Transformation of Diverging Fan Area.......................72
4.3.2 Diverging Fan Areas Transformation Case Study........................73
4.4 Development of Geometric Transformation Using Parabolic Function.........75
4.4.1 Geometric Transformation of Converging Fan Area......................77
4.4.2 Converging Fan Area Transformation Case Study........................78
4.5 Closing..................................................................81
5. Model Verification and Case Studies..........................................83
5.1 Introduction.............................................................83
5.2 Verification of Diverging KW Model and Case Studies......................85
5.3 Rising and Recession Hydrograph for Diverging Flow.......................92
5.4 Verification of Converging KW Model and Case Studies.....................94
5.5 Rising Hydrograph for Converging Flow...................................100
5.6 Shape Factor Method Verification and Sensitivity Test...................103
5.6.1 Diverging Plane Shape Factor Method Case Study and Sensitivity Test.. 103
5.6.2 Converging Plane Shape Factor Method Verification...................105
viii
5.6.3 Converging Plane Shape Factor Method Case Study and Sensitivity Test....
....................................................................108
5.7 Closing.................................................................Ill
6. Conclusion..................................................................113
Reference........................................................................118
Appendix
A ASCE Journal Papers..........................................................124
B Diverging Plane Governing Equation Derivative................................137
C Converging Plane Governing Equation Derivative...............................141
D Verification of Diverging KW Model and Case Studies..........................145
E Verification of converging KW Model and Case Studies.........................163
F Verification of Diverging KW Shape factor and Case Studies...................177
IX
LIST OF TABLE
Table
4.1 Testing for Diverging Fan Shape Plane to KW Rectangular Planes..........74
4.2  Testing for Converging Fan Shape Plane to KW Rectangular Planes.........79
5.1 Comparison between Predictions and Measurements for Converging Flows....96
5.2  SWMM 5 Peak Flow Sensitivity Test for KW Plane..........................105
5.3  Conversion Relationship between Converging Plane and KW Rectangular Virtual
Plane.....................................................................107
5.4  W2 and W6 Sample Watershed Geometry Conversion........................107
x
LIST OF FIGURES
Figure
1.1 Population Percent Change By State (2010 Census Bureau).....................1
1.2 Population Percent Change By County (2010 Census Bureau)....................2
1.3  Example of Hillside Development Scottsdale, Arizona (Google Image, 2012).3
1.4 Example of Urban Development Encroaching onto Alluvial Fans, Las Vegas,
Nevada (2008 Las Vegas Valley Master Plan Update, CCRFCD)..................3
1.5 Alluvial Fan Formation Death Valley, California (Geomorphology of Desert
Environments, 2009)..........................................................4
1.6  Alluvial Fan Geographic Locations (USGS, 2000).............................5
1.7 Alluvial Fan Boise, Idaho (Google Image, 2012)............................6
1.8 Alluvial Fan Glenwood Springs, Colorado (Google Image, 2012)..............7
1.9 Alluvial Fan Formation (Google Image, 2012).................................8
1.10 Alluvial Plane Formation (USGS, 2000)....................................11
1.11 Temporal Distribution Regions (NOAA, 2011)...............................13
1.12 Diagram of the Pearson System showing distributions of Types I, III, VI, V, and
IV in terms of pi (squared skewness) and P2 (traditional kurtosis).......15
1.13 Isometric view (French, 1992).............................................16
1.14 Typical Engineering Application Geometry Layout on an Alluvial Fan.......17
2.1 Monoclinal Wave Movement....................................................23
2.2  Monoclinal Wave Rating Curve..............................................24
2.3  Dynamic Wave, Diffusion Wave and Kinematic Wave Applicability Zone........27
2.4  Onedimensional overland unsteady flow profile............................30
2.5  Dimensionless Overland Flow Hydrograph....................................33
2.6  Rising Water Surface Profile when t < Te.................................34
xi
2.7  Recession Hydrograph when t > Td........................................35
2.8  KW Integration Domain Illustration for Rainfall Excess considering Soil Infiltration
38
3.1  Diverging Surface and Parameters..........................................44
3.2  Fan Shape and Fan Area for KW Flow......................................46
3.3  Typical Transportation Alignment with Culvert Crossing Layout on Alluvail Fans
52
3.4  Water Balance in an Arc Segment within Converging Tributary Area........53
3.5  Water Balance over Entire Converging Tributary Area.....................56
3.6  KW Flow Field Mesh Network...............................................61
3.7  Illustration of Implicit and Explicit Finite Difference Method..........62
4.1 Conversion of Actual Watershed into Virtual KW Sloping Plane..............67
4.2  KW Models for Square Watershed...........................................69
4.3  Conversion of Diverging Plan into KW Rectangular Plan....................72
4.4  Watershed Factor vs. KW Shape Factor Trigonometry Sine Function........75
4.5  Shape Factor vs. Interior Expansion Angle Diverging Plane..............75
4.6  Conversion of Converging Plan into KW Rectangular Plane..................78
4.7  Radius Ratio vs. KW Shape Factor (15 degree) Parabolic Function........80
4.8  Radius Ratio vs. KW Shape Factor (30 degree) Parabolic Function........80
4.9  Radius Ratio vs. KW Shape Factor (90 degree) Parabolic Function........81
4.10  Radius Ratio vs. KW Shape Factor (179 degree) Parabolic Function.....81
5.1 USGS Stream Network Location Exhibits......................................85
5.2  Comparison of Laboratory and Numerical Solutions for KW Overland Flow
Hydrographs under Rainfall Intensity of 78 mm/hr for 50 Seconds...........87
xii
5.3  Comparison of Laboratory and Numerical Solutions for KW Overland Flow
Hydrographs under Rainfall Intensity of 115 mm/hr for 50 Seconds........88
5.4  Diverging KW Overland Flow Hydrograph with Various Expansion Angles....89
5.5  KW analytical solution vs FAN program outputs...........................91
5.6  NonDimensional Rising Hydrograph Analytical and Numerical Solution
Comparison...............................................................93
5.7  NonDimensional Recession Hydrograph Analytical and Numerical Solution
Comparison...............................................................94
5.8  Topographical Layout for Watershed W2..................................97
5.9  Topographical Layout for Watershed W6..................................98
5.10 W2 Watershed Surface Runoff Hydrograph Prediction and Comparison......99
5.11 W6 Watershed Surface Runoff Hydrograph Prediction and Comparison.....100
5.12 Converging KW Rising Portion Hydrograph Comaprison {a = 0.1)..........101
5.13  Converging KW Rising Portion Hydrograph Comaprison (Interior Angle =104
degrees, a = 0.5).....................................................102
5.14  Converging KW Rising Portion Hydrograph Comaprison (Interior Angle = 104
degrees, a = 0.8).....................................................102
5.15  Comparison of Laboratory, Analytical and Numerical Solutions for KW Flows
under Rainfall Intensity of 98 mm/hr for 10 minutes...................104
5.16 W2 Watershed Predictions and Field Data Comparison...................109
5.17 W6 Watershed Predictions and Field Data Comparison...................110
xiii
1.
Introduction
1.1 Urbanization Impacts to Alluvial Fan
Stimulated by overall economic growth, population in the western United States has
increased significantly since the 1990s. The 2010 Census reported a 9.7 percent growth in
the countrys overall population from 2000, however, regional growth was 14.3 percent in
the South and 13.8 percent for the West. Nevada grew by 35.1 percent making it the fastest
growing state followed by Arizona, which grew by 24.6 percent (Figure 1.1). Major growth
occurred in metropolitan areas such as Las Vegas, Nevada (over 40 percent), and county
wide such as Maricopa County, Arizona, which accounted for 59.1 percent of the states
growth (Figure 1.2) (U.S. Census Bureau, 2011).
1
CENSUS: POPULATION CHANGE: 200010
Figure 1.2 Population Percent Change By County (2010 Census Bureau)
In response to the rapid increase in population, urbanization spread onto more
challenging terrain throughout the Southwest as desirable, level land became less available.
Development encroached onto hillsides (Figure 1.3) and floodplain areas known as alluvial
fans (Figure 1.4).
2
Figure 1.4 Example of Urban Development Encroaching onto Alluvial Fans, Las
Vegas, Nevada (2008 Las Vegas Valley Master Plan Update, CCRFCD)
3
In the semiarid Southwest where valley land can be soft and saline and mountain
hills steep and rocky, alluvial fans provide attractive development sites. The relatively gentle
slopes, good drainage, and wellgraded material composites can invite transportation,
agriculture, and urban development uses (Anstey 1965). When the alluvial fan is located
near metropolitan areas such as Las Vegas, Nevada, Phoenix, Arizona, and Las Cruces, New
Mexico, rapid growth encourages development to encroach onto alluvial fans. Even with
intensified development pressure, some local jurisdictions such as the City of Las Vegas,
limit site disturbance to 50 percent when the longitudinal slope is greater than 15 percent due
to planning, public safety, and construction concerns (City of Las Vegas 2011).
Conversely, some alluvial fans are concentrated in areas where infrastructure is not
feasible and is, therefore, unlikely to be developed. For instance, over 70 percent of Death
Valley, California, is covered by natural alluvial fans (Figure 1.5).
Figure 1.5 Alluvial Fan Formation Death Valley, California (Geomorphology of
Desert Environments, 2009)
4
1.2 Alluvial Fan Location
Alluvial fan landform is commonly found along the base of mountain fronts in the
western states of Washington, Oregon, California, Idaho, Nevada, Arizona, New Mexico,
Utah, Colorado, Montana, and Wyoming. It was estimated that approximately 30 percent of
the landforms in the semiarid southwestern United States (Figure 1.6) consists of alluvial
fans (Anstey 1965).
The average longitudinal slope of alluvial fans exhibits a wide range, from 2 to 70
percent, with most between 2 and 36 percent (Blair and McPherson 2009). Typically, the
crosssurface slopes on fans vary from 1 to 2 percent for fans where sediment and water
production in the watershed is relatively low. This type of alluvial fan is commonly found in
Boise, Idaho (Figure 1.7).
5
Fan slopes ranging from 20 to 30 percent, primarily formed by debris flow with
approximately 55 percent sediment concentration, are best represented by Glenwood Springs,
Colorado (FEMA 1981) (Figure 1.8).
6
1.3 Alluvial Landform Descriptions and General Characteristics
Alluvial fans are aggradational sedimentary deposits shaped like a cone segment
radiating downslope from a point where a channel emerges from a mountainous catchment
(Figure 1.9) (Drew 1873, Bull 1977).
7
Canyon
Figure 1.9 Alluvial Fan Formation (Google Image, 2012)
Three types of alluvial landforms are often involved in flood flow studies, yet they
have strikingly different flood behavior. All three landforms develop at the base of steep,
highly erodible mountain masses. Upstream of the valley openings are subjected to high
intensity, short duration rainfall events (FEMA 1981). Such rainfall events generate runoff
with sufficient velocity to pick up sediment as it travels through ravines and channels, and
transports it to the canyon opening. Once it leaves its steep mountainous watercourse, the
runoff spreads out, slows down, and deposits its sediment load as it heads toward the valley
floor.
8
The following is a summary describing the differences among the three alluvial
landforms (Blair and McPherson 2009, FEMA 1981):
Alluvial Fans As the sedimentladen flood flow leaves the confinement of the
ravine walls, the flow spreads out and becomes shallower. The majority of the
suspended sediment settles forming a cone at the mountain front. Lines of
maximum topographic slope radiate away from the apex of the fan and terminate
at the valley floor where the original valley slope resumes domination. Fan slopes
depend on factors such as the size of the sediment particles and the sediment
concentration in the runoff of the upstream tributary area. Heavier, larger
sediment drop out quickly with smaller and finer particles depositing farther down
the slope.
Alluvial Aprons Often a series of fans form along the front of steep mountain
ranges where numerous small watersheds are drained by individual streams. As
the fans expand out on the valley floor, the toes of the fan merge into an alluvial
apron. This apron area is characterized by nearly linear contours and a series of
parallel ravines, or arroyos, which drain the aprons.
Washes Washes are typically long, narrow formations with contours
perpendicular to the confined canyon walls. They are mostly found in the section
of ravine immediately connected to alluvial apron where the channel draining the
mountain watershed remains confined until it connects to a large river.
Figure 1.10 presents these landforms in the northern part of Las Vegas Valley, Nevada, to
illustrate the major features of the alluvial floodplain.
9
Because of obvious differences in morphology, flood flow behaviors on the three
landforms are very different. Flood flows in a wash are confined by canyon walls and the
path of flow tends to be stable and predictable. If the wash is wide, braiding and meandering
of the base flows might still occur but it should be within the floodplain limit. Flooding on
an alluvial apron is generally limited to the arroyos that drain the apron. The characteristics
of flow in the arroyos depend on local drainage and on the discharges from the upstream
alluvial fans.
On alluvial fans, overland flows are distributed as a twodimensional sheet flow
under a storm event (Mukhopadhyay et. al. 2003). The flow pattern is sensitive to the radius
of fan area, the angle of apex, the longitudinal slope, and fan surface roughness. The
overland flow may be gradually concentrated through a converging plane or spread out more
over a diverging plane. The sudden expansion or contraction of the fan width does not create
a spatiallyvaried Kinematic Wave (KW) flow that can be solved by conventional KW
solutions, nor does the numerical scheme, because the current KW solutions are only valid
for a rectangular plane formation.
10
TERIOR
Y
STATE OF NEVADA
NEVADA BUREAU OF MINES AND QEOLOOY
0\SS PEAK SW QUADRANOLE , ^
NEVADACLARK CO. jMp
7,5 MINUTE SERIES < TOPOGRAPH 1C}
Figure 1.10 Alluvial Plane Formation (USGS, 2000)
Overland flow on a fan surface is unsteady due to changes in rainfall distribution,
and with each storm event, the shallow gullies and washes can change flow paths and
patterns. The flow converging and diverging processes alter the distribution of surface
runoff on a fan area. When transportation infrastructure crosses an alluvial fan, a culvert
is often proposed to convey storm runoff. However, without the proper governing
equations of motion for both converging and diverging KW flows, the impact to a
drainage crossing along a transportation alignment on the natural alluvial floodplain
cannot be quantified. Currently, the predictions of overland flows on alluvial fans rely on
a probabilistic estimation.
1.4 FEMA Flood Flow Approach for Alluvial Fans
Flood hazards are frequently underestimated in the semiarid Southwest United
States (FEMA 1981). The semiarid Southwest area includes Southeast California,
Nevada, Arizona, New Mexico, and Utah (NOAA 2011). These areas often experience
intense rainfall and subsequent flash flooding. A special report prepared for Clark
County, Nevada, documented 184 different flooding events resulting in damages to
private properties and public facilities from 1905 to 1975 (U.S. Soil Conservation Service
1975). While flooding can and has occurred in almost every month of the year, the most
damaging storms typically occur in the form of convective storms (Figure 1.11) during
the warm months of July through September (NOAA 2011). During these summer
months, moist unstable air from the Gulf of Mexico is forced rapidly upward by hot air
currents, which causes severe thunderstorms with intense rainfall over a very short
duration. For example, a powerful gulf surge brought abundant moisture into Clark
12
County, Nevada, in August 2013. There were numerous isolated, locally intense showers
throughout the southern Nevada area on August 23rd, however, rainfall in the northwest
part of Las Vegas Valley fell during the afternoon of August 25th. During this event, ten
rain gages maintained by Clark County Regional Flood Control District (District)
reported 1.2 to 4.1 inches of rain fell over about four hours with intensities exceeding the
design rainfall standard adopted by the District and local governmental entities. The
average total annual rainfall depth for Las Vegas is 4.19 inches according to National
Weather Service records.
Figure 1.11 Temporal Distribution Regions (NOAA, 2011)
When such intense storms occur on steep mountain terrain and alluvial fantype
desert slopes, surface runoff accumulates and concentrates quickly. Storm runoff often
13
demonstrates high flow velocities and unpredictable flow patterns across alluvial fan
surfaces making advance warning efforts difficult to provide adequate response time.
These flood events also have the capacity to transport a considerable volume of sediment
resulting in erosion and deposition in various areas (FEMA 1989).
Acknowledging the complexity of flooding problems on alluvial fans, Federal
Emergency Management Agency (FEMA) developed a stochastic approach to identify
flood hazards and flood boundaries on an alluvial fan (Anon 1981, Dawdy 1979). This
approach imposes a conditional probability using logPerson Type III distribution (Figure
1.12), and is adjusted by the probability of the flow path being developed at a specific
location on an alluvial fan. This approach later became the foundation of the FAN
computer model (1990) developed by FEMA to assist the flood hazard zone mapping
process to establish a National Flood Insurance Program (NFIP) on this type of landform
(FEMA 1989, 2000).
This approach solves for the 1 percent chance of flood occurrence with the basic
probability function in combination with the conditional probability function to compute
the width of the area subject to alluvial fan flooding for various flow depths and
velocities. The same mathematic approach was used by U.S. Army Corps of Engineers
(USACE) to analyze the flood risk during levee failure while dealing directly with the
uncertainties inherent in such an occurrence (USACE 1993).
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Figure 1.12 Diagram of the Pearson System showing distributions of Types I, III,
VI, V, and IV in terms of pi (squared skewness) and P2 (traditional kurtosis)
In the FEMA approach to assess flood hazards on alluvial fans, the stochastic
procedure was derived based on the following assumptions:
The peak flow frequency must be known at the fan apex.
At any distance below the apex, the path of the wash (channel) will be
developed with a chance that is proportional to the ratio of wash width to
total width of the fan area.
The flood flow is conveyed at critical depth.
The widthtodepth ratio of the wash is approximately 200.
15
These assumptions have been applied to calculations of conditional risk of flood
flow to establish FEMAs NFIP on alluvial fans. Although the methodology has been
criticized since it was first established in 1979, this procedure is the officially acceptable
method for most FEMA floodplain studies on alluvial fans (Burkham 1988, McGinn
1979, French 1992, Fuller 1990, 2011). Even with minor reformations to include channel
deviation from the medial radial line (Figure 1.13) along the fan area for a better
calculation procedure and prediction (French 1992), the overall methodology is still
heavily relying on statistical analysis.
Figure 1.13 Isometric view (French, 1992)
Since its creation in 1990, FEMAs FAN computer model has been a key
component to define and delineate flood hazard boundaries on alluvial fan landforms. Its
use was mandated by numerous communities in southern California. However, the flood
hazard zones published for alluvial fan areas near Scottsdale, Arizona, were criticized as
being too conservative when compared to a twodimensional flow simulation such as the
FLO2D computer model (Fuller 2012). But from a regulatory standpoint, the current
FEMA approach does provide basic information for flood protection and urban planning
purposes (French 1993). It was long recognized that a rainfallrunoff approach should be
explored to delineate flood hazards on an alluvial fan (National Research Council 1996).
16
1.5 Objective
An alluvial fan may be composed of a diverging plane downstream of the apex, a
rectangular plane along a collection ditch, a converging plane immediately upstream of a
culvert, and a diverging plane immediately downstream of a culvert (Figures 1.10 and
1.14). These flows may form a cascading plane system to simulate the generation of
overland flows.
The KW approach for overland flow is a simplified solution for the Dynamic
Wave (DW) methodology. KW governing equations consist of flow continuity and flow
momentum in terms of a rating curve relationship between flow depth and flow discharge,
17
therefore, they are sensitive to the geometry of overland flow plane. In this research
work, the objective is to explore new governing equations and their analytical solutions,
as well as numerical schemes to solve diverging, converging, and rectangular plane KW
flows. The known inputs are the rainfall excess on the fan area and the flow rate at the
fan apex. The process is to generate overland flow rates and the outputs are the flood
flow hydrograph and floodplain boundary limits for the fan area.
It is proposed that the KW approach be expanded to model an alluvial fan as a
hydrologic system. To achieve this goal, this research proposes to:
Conduct a literature review of Kinematic Wave theory.
Derive the governing equations for converging, diverging, and rectangular
plane KW flows using a polar coordinate system to better represent the
shape of alluvial fan.
Expand the KW numerical simulation scheme from a single plane to a
cascading system to simulate overland flow on an alluvial fan area.
Verify the theoretical derivation with published laboratory or field data.
Develop a deterministic approach to quantify flood flow magnitudes and
delineate floodplain boundaries to improve the current FEMA approach on
alluvial fan flood studies.
Develop a green approach to design crossing culverts to preserve alluvial
fan flow patterns upstream and downstream of the culvert under a
highway.
Application of KW cascading plane methodology is directly related to the design
of collector ditches and swales along highways traversing alluvial fan areas, and culverts
18
conveying surface runoff under these highways. This study will greatly improve the
current procedure for FEMA flood flow studies and reduce highway impacts on the
continuity of alluvial fan flows.
19
2.
Review Of Kinematic Wave Overland flow
2.1 Introduction
The SaintVenant equations, derived by Barre de SaintVenant (1871), were
developed to express onedimensional unsteady open channel flow by fulfilling the
continuity and momentum principles simultaneously. There were various simplified
forms, each expressing a onedimensional distributed routing model. The momentum
equation, also named Dynamic Wave Flow Model, consisted of local and convective
acceleration terms as well as pressure, gravity, and friction force terms. The equations
were then simplified to Diffusion Wave Flow Model when flow accelerations were
ignored. It was further reduced to Kinematic Wave Flow Model when flow friction was
balanced with gravitational force (Lighthill and Whitham, 1955). Woolhiser and Liggett
(1967) applied Kinematic Wave to describe overland flow by finitedifference integration
of the characteristic equations. Chen (1970) recommended the first analytical solution to
represent overland flow on an irrigation porous surface. Guo (1998) reported the
analytical solution for overland flow on a pervious surface under a uniform rainfall to
represent the soil infiltration effect on porous ground.
2.2 Basic Principles
DaluzVieira (1983) and Yen (1973) suggested that shallow wave propagation in
open channel or overland flow could be best described by the complete SaintVenant
equations by applying the following assumptions:
Gradually varied flow
20
Onedimensional flow
Flow is incompressible and of constant density
Uniform flow velocity exists within cross sections
Mannings equation is applicable
Hydrostatic pressure is evenly distributed through cross sections
These assumptions were shown to be satisfactory for overland flow in most of the
cases (Lighthill and Whitham, 1955).
Open channel flow is expressed in three dimensions in nature. As the longitudinal
flow velocity governs the flow characteristics, open channel flow can be simplified and
expressed in onedimensional format. And the continuity and momentum principles for
unsteady nonuniform open channel flow can be described as:
8A 80
+ ^ = q
8t 8x
1 dQ.+L d fQ2^
A 8t A 8k
\aj
(2.1)
(2.2)
Where A = flow area in [L2], Q = channel flow in [L3/T], q = lateral inflow per unit length
of the reach in [L2/T], t = time in [T], x = distance measured in flow direction in [T], g =
specific gravity in [L/T2], So = bottom slope, and S/ = friction slope. Eq. (2.1) is suitable
for all channel geometries.
Aided with channel geometry, Eq. (2.2) can be rewritten as:
K+vK+g^g(s^s)=0
8t 8x 8x v 0
(2.3)
Where V= channel cross section velocity in [L/T], Eq. (2.3) describes the diffusion wave
in the channel. It contains local acceleration, convective acceleration, backwater effect,
21
flow friction, and gravitational force. Assuming that the convective and local
acceleration are negligible compared to friction, gravity, and pressure terms, Eq. (2.3) is
further reduced to:
(2.4)
Eq. (2.4) is named Diffusion Wave Model; it is also called the noninertial model.
Without further effect from the upstream and downstream sections, open channel
flow can be viewed as a uniform flow that is mainly dominated by gravitational and
friction forces. Assuming a balance between gravitational and friction forces, Eq. (2.4)
can be reduced to:
The correlation demonstrated in Eq. (2.5) shows a singlevalue rating curve
relationship between flow rate and flow depth in Kinematic Wave flow. It can be
presented as:
Where a and m are constants that can be determined by the channel cross section and
channel roughness. Several empirical formulas have been developed and are summarized
as followed:
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
22
Where n = Mannings roughness, fi = 1.486 for English units, C = Chezys conductivity
coefficient, and/= friction factor. The value of m varies between 3/2 and 5/3 for general
practices (HEC, 1993).
Without any backwater effect, the solution of Eq. (2.6) offers an instantaneous
uniform flow pattern at a specific time. Additionally, the normal flow depth solved by
Mannings equation is a special case of the Kinematic Wave model when the flow rate is
constant.
2.3 Kinematic Wave Application Limit
One of the simplest wave forms, a monoclinal flood wave, can be generated by a
sudden lift of a sluice gate (Figure 2.1). This phenomenon can represent the sudden
increase in channel runoff in the urban environment when responding to rainfall.
Sluice Gate
23
Since the wave height is insignificant, this type of unsteady open channel flow
can be converted to a steady state flow by introducing a negative wave speed, Vw, to the
entire flow field.
The steady flow system continuity principle is shown as:
(KK)A=(v,vM (2io)
Rearranging Eq. 2.10 yields:
v Anw g2a (211)
w a2a, a2a,
As demonstrated in Figure 2.2, a monoclinal flood wave can be illustrated as a single
value rating curve.
Area (A)
Figure 2.2 Monoclinal Wave Rating Curve
24
To make a comparison between kinematic and dynamic waves, Eq. (2.11) is
simplified by applying it to a wide rectangular channel. With a constant bottom width, B,
Eq. (2.11) is presented as:
E =
1 dQ
B dy
Combining Eq. (2.6) and Eq. (2.12) yields the following:
y CClfl m\
W 73
D
(2.12)
(2.13)
Considering the principle of continuity, the cross section velocity, V, can be expressed as:
v =Q_=a_ mi (2.14)
By B
Combining Eq. (2.13) and (2.14) yields:
Vw=mV (2.15)
Eq. (2.15) describes the flow velocity and KW speed relationship when the backwater
effect is ignored. Comparably, the dynamic wave speed, Vd, in a wide rectangular
channel is expressed as:
Vs=VjgY (2 16)
The expresses the wave direction and speed downstream and upstream,
respectively. When the backwater effect is irrelevant, dynamic wave and kinematic wave
coexist in the flood channel because, numerically, these two wave speeds share the same
hydraulic characteristics.
As presented in Eq. (2.2), the Dynamic Wave model takes the local and
convective flow acceleration into consideration. By considering wave speed, yJgY ,
25
Dynamic Wave moves faster than Kinematic Wave thereby presenting an application
limit between the use of kinematic and dynamic wave models.
V
w a
Combining Eq. (2.15) and (2.16) results in:
JgY (m1)
For Mannings Equation,
m= F <1.5
3
For Chezys and DarcyWeisbach equations,
(2.17)
(2.18)
(2.19)
Vieira (1983) combined his work with Liggett and Woolhiser (1967), Overton and
Meadows (1976), and Morris and Woolhiser (1980) by applying nondimensional
Kinematic Wave number, K, and Froude number, Fr, as the coordinate to distinguish the
relationship among Dynamic Wave, Diffusion Wave, and Kinematic Wave (Figure 2.3).
Where K = ^r (2.20)
YFr
26
K
Figure 2.3 Dynamic Wave, Diffusion Wave and Kinematic Wave Applicability
Zone
It is concluded that the Kinematic Wave model is applicable when the flow
Froude number is less than 2 (Badient and Huber 1992)
2.4 Overland Flow on a Rectangular Pervious Plane
Based on the assumption that the gravitational force offsets the friction force, then
it follows that the longitudinal slope is equal to the friction slope. Thus, the Dynamic
Wave equation for open channel flow under uniform rainfall intensity and decayed
infiltration losses is reduced to the kinematic wave equation for unitwidth flow (Yen and
Chow, 1974). The resulting equation is:
dy dq .
+ = h
dt dx
(2.21)
Where ie = excess rainfall intensity in [L/T]
Considering infiltration losses, rainfall excess can be expressed as (Horton, 1938):
27
kt
(2.22)
4W=//W
/(0=X+(./(,/>
In which i = rainfall intensity in [L/T],/(7) = infiltration rate at time t in [L/T],/c = final
infiltration rate in [L/T],/0 = initial infiltration rate in [L/T], and k = soil infiltration decay
constant in [1/T],
The rating curve for an overland unitwidth overland flow is expressed as:
q = qy
Taking the first derivative of Eq. (2.23) with respect to x yields:
dq m1 ky
= amy
dx dx
(2.23)
(2.24)
Substituting Eq. (2.24) into Eq. (2.21), the total derivative for flow depth is derived as:
(2.25)
dy m_i dy .
+ amy = i
dt dx
Eq. (2.25) can be converted to:
dy dy
+ u = i
dt dx
(2.26)
Eq. (2.26) is the total derivative of the flow depth and is presented as:
dy
dt
= z
(2.27)
Similarly, taking the partial derivative of Eq. (2.23) with respect to t and
rearranging the terms yields:
dq
& dt c
dt amyml
Substituting Eq. (2.28) into (2.21), the total derivative for flow rate is concluded as:
28
dt
tdq=i
dx
(2.29)
Eq. (2.29) is rewritten as a total derivative of flow rate:
dq
dt
(2.30)
Where the KW speed is defined as:
dx
= amy
dt
m1
(2.31)
For a unitwidth flow, the flow velocity is calculated as:
V = = ay'
y
(2.32)
Kinematic wave dominates in the propagation of a flood wave when the local and
convective flow accelerations are insignificant. This implies that the friction slope (Sf) is
approximately equal to bed slope (So). KW model can be described as a singlevalue
rating curve similar to the normal flow developed in a prismatic channel. The rating
curve represents the mathematical relationship between flow rate and flow depth.
2.5 Overland Flow Hydrograph Analysis
KW overland flow takes place when the dynamic term in the Momentum
Equation is negligible. This dominance implies that the friction slope associated with the
KW flow is equivalent to the ground slope. This assumption is justified when backwater
effect is not significant. The KW overland flow model is a onedimensional (1D) unit
width flow approach. This approach demands a conversion of the catchment into its
rectangular sloping plane (Guo and Urbanos, 2009).
29
As presented on Figure 2.4, on a rectangular KW plane, the KW vertical water
surface profile represents the surface detention volume which varies with respect to time.
Uniform Rainfall
Upper
B0Undary Water Surface
Figure 2.4 Onedimensional overland unsteady flow profile
As depicted in Figure 2.4, the initial condition for KW overland flow shall be a
dry bed condition. It implies that no flow is on the surface before the rainfall event starts.
y{t,x) = y{ o,x) = o
(2.33)
q(t,X)=q(0,X)=0
(2.34)
Where t = time in [T] and X= station for flow calculation in [L],
The boundary conditions at the upstream end of the overland flow plane are:
o II o) (2.35)
o II (2.36)
The boundary conditions at the downstream end of the overflow plane can be set
at critical or normal flow depth. In this work, the downstream boundary condition is
assumed to be:
y(t,L) = y(t,L Ax) (2.37)
30
q(t,L)=q(t,L Ax) (2 38)
Where L = total length of the overflow in [L], and Ax is the incremental along the surface
plane. Based on boundary and initial conditions, Eq. (2.27) can be integrated and yields:
y=y t=t
^dy = j iedt
(2.39)
^=0 t=0
y=ij
(2.40)
According to the KW speed described in Eq. (2.31), the KW travel length is
expressed as:
x=X
J dhc a Jmym ldt = a Jm(iet)m 1 dt
m\,r
x = ai t
(2.41)
(2.42)
Aided with Eq. (2.40), Eq. (2.42) is converted to:
y =
y
XI
a
(2.43)
As the kinematic wave propagates from the very upstream where x = 0 to the
outlet at x = L, the flow travel time is termed as the time of equilibrium, Te, of the
catchment. Thus, Eq. (2.42) can be rewritten as:
T. =
f L >
. m1
Vme J
(2.44)
Combined with Eq. (2.23) and (2.40), the flow depth and discharge at Te are (Wooding,
1965):
ye=i&
(2.45)
m fflrrr Ttl T
qe=aye =meTe =ieL
(2.46)
31
Using Mannings Equation with m = 5/3, Eq. (2.44) is expressed as:
3
Using Chezys Equation with m = 3/2, Eq. (2.44) becomes:
2
Using DarcyWeisbachs Equation with m = 3/2, Eq. (2.44) is expressed as:
(2.47)
(2.48)
T =
JJL '
(2.49)
On the KW rectangular plane, the unitwidth peak flow, qe, occurs at x = L. The
peak flow, Qp, on the KW plane with a width, B, and a length, /., can be expressed as:
Qp = ieBL = CiA (2.50)
It is concluded that the Rational Method is a special case of the KW method. The
rainfall intensity, ze, in the Rational Method is the rainfall average rate over the time of
equilibrium of the catchment. The value of runoff coefficient, C, represents the
percentage of impervious area within the catchment. Eq. (2.50) suggests that the KW
approach provides a linear solution between peak flow and tributary area.
To be conservative, it is important to ensure that the design rainfall duration
applied in the stormwater simulation is longer than the Te of the catchment area. The
peak runoff will reach its maximum potential value to reflect the climatologic and
hydrologic conditions imposed to the catchment. Under a long duration rainfall event,
the overland flow will reach its equilibrium state when the surface discharge from the
32
catchment area equals the rate of the rainfall precipitated. As the rainfall event ceases,
surface runoff depth starts tapering off.
The overland runoff hydro graph is consisted with 3 segments rising, peaking
and recession portions (Figure 2.5). The rising portion is the section before reaching time
of equilibrium (Te), the peaking portion is the section between Te and the time the rainfall
stops (TJ), and the recession portion is when the accumulated runoff depth begins to taper
off (Guo 2006a, 2006b).
t = 0 t = Te t = Td
Figure 2.5 Dimensionless Overland Flow Hydrograph
Applying Eq. (2.31), Eq. (2.45), and Eq. (2.46) to a unitwidth catchment under a
long, uniform rainfall excess, ie, the overland flow hydrograph is presented in Figure 2.5.
All flow depths and times are normalized by equilibrium flow rate, qe, and time, Te,
respectively.
(a) Rising Hydrograph (0 < t < Te)
33
At the elapsed time, t, the flow depth, y, and the associated travel length, x, can be
calculated by Eq. (2.31). On the rising hydrograph (Figure 2.6) the elapsed time, t,
is normalized as:
T*= (2.51)
T.
And the flow depth will travel through a distance x and can be expressed as:
x = Vt
(2.52)
Rainfall Distribution
Equilibrium Water Profile
Where T* = dimensionless elapsed time. Aided by Eq. (2.46) and Eq. (2.51), the
flow depth at the rising portion of the hydrograph can be described as:
y(r*)=zM = r* (2.53)
y e
Where y = the dimensionless flow depth. Aided by Eq. (2.23), Eq. (2.46) and Eq.
(2.51), the normalized unit flow rate per unit width, q*, is derived as:
g*(r*)=^ = r*m (2.54)
34
Eq. (2.53) and Eq. (2.54) portrait that the flow depth and flow rate on the rising
hydrograph are increased along the 45degree line between zero and unity. This
rising runoff hydrograph is also discussed elsewhere (Eagleson 1970, Wooding
1965).
(b) Peaking Hydrograph (Te < t < Td)
During the peaking portion of the runoff hydrograph, the inflow and outflow
volumes are balanced. As a result, the normalized peak flow depth and flow rate
during the peaking period are expressed as:
(c) Recession Hydrograph (t > Td)
After the time of equilibrium, the equilibrium water surface profile is formed and
the surface detention is reached to its maximum. As soon as the rain stops, the
flow depths under the equilibrium water surface profile begins to propagate
toward to the outfall location as presented in Figure 2.7.
/(r)=i.o
?'(r*)=i.o
(2.55)
(2.56)
Equilibrium Water Profile
y
t Td > Te
t > Td
Figure 2.7 Recession Hydrograph when t > Td
35
As mentioned previously, the flow depth, y, will travel through a distance from x
to L during the time period from Td to t. The KW movement must satisfy the
relationship as:
L=X + VjtTll) for t>Td (2.57)
Inserting Eq. (2.31) into Eq. (2.57) and normalizing the end results by Te and jy, it
is concluded:
jr* 1 1 y*  mY*m1 m Te (2.58)
And 7* = 7,
For a given t, the corresponding flow depth in Eq. (2.58) can be solved iteratively
The recession hydro graph is formulated as:
y(r)=Y (2.59)
q{r)=rm (2.60)
The recession hydrograph ends when the elapsed time is long enough for the flow
depth to vanish.
2.6 Overland Flow Travel Time on a Pervious Surface
As described in Section 2.3, rainfall excess on a previous surface is subject to
infiltration losses that can be described by a decay curve. After subtracting the
infiltration loss, the uniform rainfall distribution becomes an exponential decay curve
which is nonuniform condition. The kinematic wave travel time through the catchment
is no longer constant and is ranging between time of concentration, Tc, and time of
36
equilibrium, Te. Time of concentration, Tc, is defined as the time for surface runoff to
travel through waterway and reaches outlet location.
At the beginning of the rainfall event, a longer time to produce surface runoff is
expected due to higher soil infiltration rates in the early stages. The higher the infiltration
rate is, the less surface runoff depth. As a result, the flow travel time is gradually
decreased to Te (Guo 1998). For a wide overland flow formation, m = 2 is adopted (Guo
1998, Guo et. al. 2012). Eq. (2.23) is expressed as:
q = ay2 (2.61)
Inserting Eq. (2.61) into Eq. (2.27) and Eq. (2.31), it is concluded:
f = (2.62)
dx ,
= 2 ay (2.63)
at
Under a long duration rainfall event, the pervious surface overland flow runoff
hydrograph can also be categorized into three segments rising, peaking and recession
hydrograph.
(a) Rising Portion Hydrograph
When the rainfall depth exceeds hydrologic losses, surface runoff occurs. As
shown on Figure 2.8, the rising portion hydrograph begins from t= Ts,to t= (Ts +
Tc) where Ts, is termed ponding time which is the period of time for the soil
infiltration rate to be decayed till the surface runoff occurs. The rising hydrograph
is derived by integrating Eq. (2.62) from t = Tsto t = (Ts + Tc) as:
y = (ifjl  eT) (2.64)
37
Parallel to Eq. (2.64), the KW flow will travel a distance, x. Integrating Eq. (2.63)
yields:
=  Ts)2 + (/~o/c)1 k(f Ts)ykT eh } for 0 < x < L (65)
2 a 2 k
Eq. (2.64) and Eq. (2.65) provide the direct solution to define the water surface
profile, (x,y), for t<(Ts+ Tc) where Tc is the time of concentration of the
catchment.
/
Infiltration Decay Curvef(t)
Contributing Rainfall
\
E
I \
I
Rainfall Intensity i
.V'=0 x=L x=0 x=L
Figure 2.8 KW Integration Domain Illustration for Rainfall Excess considering
Soil Infiltration
(b) Peaking Portion Hydrograph
After time (Ts + Tc), the flow reaches the peaking hydrograph because the entire
catchment has become contributing to the outflow. The peaking process ends at
38
time, Td, when the rain ceases. Substituting t= (Ts + Tc) and x = L into Eq. (2.64)
and Eq. (2.65) yields:
(2.66)
(2.67)
Where yc = flow depth in [L] at t = (T, + Tc). For the given distance, /., the value
of Tc in Eq. (2.67) can be iteratively solved. Having Tc known, the corresponding
flow depth, yc, can be solved by Eq. (2.66).
A KW speed is sensitive to the rainfall access on the previous surface. As time
goes on, the soil infiltration rate continues decaying. Consequently, the higher the
rainfall excess is, the faster the KW flow travels. At t > (Ts + Tc), the flow depth is
generated by the net rainfall amount from (/'/,) to t where Tv is the travel time for
the KW flow to go through the flow length, /., as illustrated in Figure 2.8.
Integrating Eq. (2.62) and Eq. (2.63) over the domains from x = 0 to x = L and t =
(t Tv) to time, t, yields:
For a given time t and t> Ts+ Tc, its travel time, Tv, can be iteratively solved by
Eq. (2.68) and Eq. (2.69). The maximum overland flow depth, ymax, on a pervious
surface occurs at t = Td when the rain ceases.
(2.68)
(2.69)
39
(2.70)
yn
={/X
(UL).
k(TdTv
\ekT' l)
When time is long enough and the infiltration rate is reduced to its constant rate.
As a result, Eq. (2.71) and Eq. (2.72) represent the equilibrium condition as:
ye=(ifc)Te=ieTe
y i f c y1 2 ig^ rp 2
2a ~ 2 e 2 e
(2.71)
(2.72)
The peaking hydrograph varies from t= (Ts + Tc) to t= Td. During the peaking
process, the travel time is decreased from Tc towards Te while the flow depth is
increased from yc toward ye.
(c) Recession Portion Hydrograph
The recession on the runoff hydrograph starts when the rain stops at t = Td.
During the recession process, the flow depth at the outlet can be formulated by
integrating Eq. (2.62) with i = 0 as:
f dy = l[fc(fllfyb\lt (2.73)
bmax jTd
Eq. (2.73) is integrated as:
y = y^fXtL) + ^jT{eliefor t>Td (2 74)
At t > Td, Eq. (2.74) provides a direct solution to the overland flow depth at the
outlet.
40
2.7 Closing
Since 1950, the KW theory has been widely applied to simulate overland flows
hydraulic characteristics under a uniform or nonuniform rainfall distribution. The semi
analytical KW solutions were derived under the major assumption that the KW overland
flows are generated from a rectangular plane. Obviously, this assumption has become a
restriction to the application of the KW theory to engineering practices. In this study, the
major effort is to expand the understanding of KW overland flow theory from a
rectangular plane to a fan shape surface. This effort involves the derivation of different
governing equations as well as the associated semianalytical and numerical solutions.
41
3. Derivations of Governing Equations for Diverging and Converging
Kinematic Wave Overland Flows
3.1 Introduction
Under a severe storm event, overland flows generated from an alluvial fan area
follow numerous flow paths that are characterized by networks of shallow grooves
accompanied with high flow velocities due to steep gradients. Because of the complexity
of surface hydrologic and hydraulic characteristics, the Federal Emergency Management
Agency (FEMA) recommended a stochastic approach to delineate flood hazard zones on
an alluvial fan because the pattern of overland flows is uncertain and unpredictable
(FEMA 1989, 2000). With many uncertainties in predicting flood flow magnitudes and
flood flow inundation boundaries, tasks for flood hazard predictions on alluvial fans has
been a challenge.
As reported, the complicated hydrologic and hydraulic processes on alluvial fans
may be best modeled with the aid of the Kinematic Wave (KW) theory of overland flow
(Mukhopadhyay, et. al. 2003). As described in Chapter 2, it was illustrated that the
solution of the KW theory under a uniform rainfall on an impervious surface is well
documented in many studies (Woodings 1965, Woolhiser and Liggett 1969). On a
pervious surface, the travel of KW movement was verified from the time of concentration
to the time of equilibrium. The solution of KW flow on a pervious surface was further
modified to include Hortons formula for soil infiltration losses (Guo 1998). Both
solutions were derived for the KW overland flow generated on an ideal unitwidth
rectangular KW plane using the Cartesian coordinates.
42
In this study, the diverging and converging KW flows on a fan shape plane can be
better described by the polar coordinate systems using the radial distance, r, and the polar
angle, 6. Such KW governing equations were first derived and discussed using the finite
difference over a circular segment across an alluvial fan at a selected radius, then the
semianalytical solution was attempted by integrating over the selected apex angle to
predict the total overland flow generated from the entire fan area (Agiralioglu and Singh,
1980). This approach involves a double integration and aided with a binominal processes
The complicated methematical approach has become a barrier to achieve neither
analytical nor numerical solutions (Singh and Agiralioglu 1981). In this study, a new
approach was developed to derive the diverging and converging planes KW governing
equations using the polar coordinate system. Both new governing equations is compared
with the previous studies and will be further exaimed with laboratory data from different
sources.
3.2 Kinematic Wave Overland Flow on a Diverging Plane
In this study, the entire alluival fan area is treated as a shallow reservoir to derive
the diverging KW governing equation. Based on the water volume balance among excess
rainfall amount, inflow and outflow discharges, and the surface detention volume, a new
governing equation can be derived for the diverging KW flow. As expected, this new
governing equation shall provide the peak flow rate at the time of equilibrium for the
purpose of floodplain delineations (Guo and Hsu, 2014). Furthermore, this research work
will attemp to derive both analytical and numerical solutions for the entire overland
43
runoff hydrograph including rising portion of the hydrograph and compare these
solutions to a numerical solution.
Figure 3.1 is a typical geometry of a diverging alluvial fan. A diverging fan area
is described by internal or apex angle, 6, and the radical distance, R. The apex in Figure
3.1 is the origin of the polar coordinate system. The runoff starts from the apex and the
spreads out as it flow downstream.
t+At, y+Ay
Figure 3.1 Diverging Surface and Parameters
In this study, the KW flow on a diverging fan area is analyzed with an arc
segment approach as presented on Figure 3.1. The water volume balance is determined
by the inflow, outflow, and storage depth within the arc segment. According to the
continuity principle, the water volume balanced within the arc segment is a summation of
the following items:
44
[ieM + vy(R r)9] {(v + Av\y + Ay\(R r)+Arty = ^ = ^AA (3.1)
At dt
AA = (Rr)dAr (3.2)
Inserting Eq. (3.2) into Eq. (3.1) yields:
[ie(Rr)9Ar\\yyAr + vAy(Rr)+ Avy(Rr)^ = ^(Rr)dAr (3.3)
dt
Eq. (3.3) can be rearranged and shown as:
q dq + dy (3.4)
(*o dr dt
q=vy (3.5)
Where ie = excess rainfall intensity in [L/T], A = diverging surface area in [L2], R =
radius for the diverging area in [L], r = location of the segment or representing the
distance in [L], v = radial flow velocity in [L/T], y = flow depth in [L], q = unitwidth
discharge in [L2/T], t = elapsed time in [T] and 6 = angle of the apex of the diverging area.
Eq. (3.4) symbolizes the unit width flow rate and the basic equation to define
diverging overland flow through the continuity equation and KW approximations. Eq.
(3.4) agrees with the previous study (Agiralioglu and Singh, 1980) using various symbols
which is presented as:
9(r,,)_eM=Â£GM+aftM (36)
v r dr dt
In comparison with Eq. (3.4), q(r, t) = ie, Q(r, t) = q, h(r, t) =y, and r = Rr. Eq.
(3.6) cannot be integrated directly from r = 0 to r = R, but can be approximated by
applying a binomial function to include the apex expansion angle and radius of the fan
area (Agiralioglu and Singh, 1980).
45
Both Eq. (3.4) and Eq. (3.6) do not explicitly include the apex expansion angle.
In fact, the apex expansion angle is implicitly involved in the calculation of the tributary
area which is outlined from the locations from Rr to R as illustrated in Figure 3.2. The
geometric parameters for the tributary fan area are described as:
A
2
R2 (Rr)2])
\r6{2Rr)
(3.7)
lr=R9
(3.8)
lr = arc length for outflow released from the fan area in [L],
Considering that the tributary area functions as a shallow water reservoir, the
rainfall excess is the inflow, and the outflow is the sheet flow released through the
downstream arc length.
46
The balance of water volume for the fan area is summarized as:
AS AV .
cUn ^lout = =
At At
Aided with Eq. (3.7) and (3.8), Eq. (3.9) can be rearranged as:
i AqRQ =A
e dt
Inserting Eq. (3.7) into Eq. (3.10) yields:
qRd dy
^r(2Rr)9 dt
(3.9)
(3.10)
(3.11)
Let = a
R
(3.12)
Inserting Eq. (3.12) into Eq. (3.11) yields:
2 q dy
/
(3.13)
>*(2 a) dt
Where a = location ratio for the upper boundary.
Eq. (3.13) serves as the governing equation for KW overland flow on a diverging
plane. If the tributary area starts from the apex of the diverging plane shape, then r = R
or a = 7, and Eq. (3.13) can be further reduced to:
dy . 2q
~dt~h~~R
(3.14)
Combined with Eq. (3.8) and Eq. (2.23), total discharge from the fan area, Q, can
be expressed as:
Q = qlr = qRd = ocymR6 (3.15)
Eq. (3.15) describes the relationship between flow depth and unitwidth flow rate
over the diverging surface area. Analytical or numerical integration of Eq. (3.14) can
47
provide direct solutions to the flow depth of surface runoff on a fanshaped plane.
Application of Eq. (3.14) to an existing alluvial fan with known geometry will satisfy
FEMAs floodplain delineation procedure for flood hazard zone identification.
Under a long duration rainfall event, the runoff flow continually increases to
reach its peak flow rate. Under the equilibrium condition, the rainfall amount is balanced
with the outflow volume from the alluvial fan area, or the surface detention remains
unchanged. Under the equilibrium condition, dy/dt is equal to 0 in Eq. (3.14). The
corresponding peak flow for a diverging alluvial fan is derived as:
cle=\ieR (316)
v.=ayrl (317)
Te==yl;m (3.18)
U a
Qe=\hR20 (3.19)
Where qe = unitwidth peak flow in [L2/T], ve = peak flow velocity in [L/T] and Te = time
to equilibrium in [T].
Eq. (3.19) agrees with the Rational method with a runoff coefficient of unity.
When the rainfall event duration is shorter then Te, it represents a short rainfall event,
otherwise, it is treated as a long rainfall event.
48
3.3 Rising and Recession Hydrograph for Kinematic Wave Overland Flow on a
Diverging Plane
Under a long uniform rainfall excess, the integration of Eq. (3.14) from t = 0 to t
= Tc and y = 0 to y=ye represents the rising hydrograph limb (Guo and Hsu, 2015a).
Cd, = n L,dy where 0
in which Tc = time of concentration in [T], The peaking hydrograph occurs between t =
l'c to l = Td in which Td = rainfall duration in [T] during peaking time, dy/dt = 0 in Eq.
(3.14). Aided with Eq. (3.15), the peak flow rate, peak flow velocity, and time of
concentration are derived as:
ye =
A
2v
E = E
(3.21)
(3.22)
Whereye = peak flow depth in [L] and ve = KW velocity. With Eq. (3.22), by the
definition, the time of concentration is displayed as:
Tc= = y';m (3 23)
E a
In practice, most hydraulic designs are directly related to sizing a structure to
convey the peak flow. Eq. (3.21) and Eq. (3.23) provide the basic information for such
hydraulic designs and can represent the rising hydrograph. It is well understood that KW
speed varies with respect to flow depth. Aided with Eq. (3.22), Eq. (3.20) can be
approximated with a constant KW speed as:
t = f dy where 0
Jy=o 2 vy
i
e R
(3.24)
49
Let the integration variable be:
Y = i ~y
e R y
Taking the first derivative of Eq. (3.25) and then substituting dYfor dy yields:
 R rT=T 1  R T 2v
(3.25)
t =
2v,
J'i=i 1 JX /V
dY =Ln( 1y) + K
T=1Y 2v iR
(3.26)
In which K = integration constant that is to be determined by the condition of 0 < t < Tc.
The solution for Eq. (3.26) is then:
f \
21 1
Tc lf I
y = le R =\e  JcY
ye
(3.27)
Similarly, recession hydrograph can be expressed by integrating Eq. (3.14) with the
boundary condition of t>Td which can be integrated fromy =ye toy = 0. Eq. (3.20) can
be rewritten as:
Jt=t (*
dt= f
t=T. h
>=0
y=ye
i
i 
2 ay'
R
dy where t>Td
(3.28)
Following the same integration procedure and aided with Eq. (3.25) yields:
R rr=ie 1 R T 2v
t = dY =Ln(y) + K
Jr=o V ~ "
2v J^= Y
2v
LR
(3.29)
In which K = integration constant that is to be determined by the condition of I > Tll The
solution for Eq. (3.29) is:
f ^
21 1
2vet Tc !_[
>L = eir=e l U22
(3.30)
50
Eq. (3.30) represents the rising hydrograph as an exponential function that is
leveled when time, t, approaches, Tc.
3.4 Kinematic Wave Overland Flow on a Converging Plane
Wherever a transportation alignement crossing an alluvial fan, excess flood water
tends to innudate any sag points along the alignment (FEMA 2000). Assessing the level
of flood protection required at sag point poses a challenge for designing a culvert
crossing at these locations. In some of the engineering practices, levees are utilized to
collect and direct overland flow toward a sag point culvert, and discharge to the
downstream side of the transportation alignment. As illustrated in Figure 3.3, the
locations of crossing culverts divide the transportation alignement is into segments with
the levees serving as manmade drainage boundaries. Overland flows generated from the
upper fan areas form a converging plane towards the entrance of the culvert (French
1992).
51
Topographic
Contours
Figure 3.3 Typical Transportation Alignment with Culvert Crossing Layout on
Alluvail Fans
The amount of flow concentrating at a culvert depends on the geometry and size
of the upstream converging tributary area. Once the peak flow rate has been determined
per segment, levees and berms are often designed to guide overland flow to plung into the
entrace pool at a crossing culvert (French 1992). This method has been implemented in
the Great Basin area in Southwest US (French et. al. 1993).
The primary uncertainty in assessing flood flows on a converging alluival fan area
is directly related to the direction and magnitude of the flow. Overland flows on a
converging surface are not a simple unitwidth flow relationship as suggested in
Woodings solution (Wooding 1965) for a retangular plane shape. Rather, the unitwidth
flow rate increases as it moves downstream. Therefore, Woodings KW solution is not
applicable for converging overland flows.
52
In this research, the entire converging tributary area is considered as a shallow
reservoir, a similar approach described in the diverging plane governing equation
derivitive (Guo and Hsu, 2015). At a time step, the flow generated from the tributary
area is a water volume balance among rainfall excess, inflow, outflow, and change in
storage volume. Using this approach, the governing equation can then be derived to
describe the movement of converging overland flows. On a converging surface, excess
rainfall produces overland flows on a sloping surface. Assuming there are no backwater
effects, gravitational force is balanced with the friction force on the surface (Chow 1964).
Figure 3.4 illustrates the general geometry of a converging plane in terms of elevation
contours on the surface.
Figure 3.4 Water Balance in an Arc Segment within Converging Tributary Area
53
Between two adjacent contour elevations, KW flow can be modeled using the
inflow and outflow crossing the arc section. The mass balance relationship is derived and
described as:
[ieAA + vy(R r)d] (v + Av)(j/ + Ay\(R r) Ar]6> = = ^AA
At dt
(3.31)
q = vy (3.32)
AA = (Rr)ArO (3.33)
Where ie = excess rainfall intensity in [L/T], A = surface area in [L2], v = flow velocity in
[L/T], y = flow depth in [L], R = radius of converging plane in [L], r = location of KW
flow on converging plane in [L], 6 = interior angle of converging area, V= surface
storage volume in [L3], t = elapsed time in [T], and q = unitwidth discharge in [L2/T],
Combining Eq. (3.31), (3.32) and (3.33), and eliminating the double and triple
incremental terms, the mass balance relationship is derived as:
vy dy (v + Aj + Ayy)
(R r) dt A r
Combined Eq. (3.34) with Eq. (3.32) yields:
fy + &L = i g
dt dr (R r)
(3.34)
(3.35)
Eq. (3.35) is identical to the flow equation reported in the previous study (Singh
1975). This equation does not involve the converging angle because it only represents
the water volume balance within the arc area. However, the angle of the converging area
is a key factor in the determination of the concentrated flow from the entire area. As
suggested in previous work (Agiralioglu and Singh 1980), Eq. (3.35) can be further
integrated using binominal function through the interior angle, 6, as shown in Figure 3.4.
54
Such an indirect solution is cumbersome for engineering applications because of lengthy
numerical iterations. In this study, the entire converging surface area is taking into
consideration. Assuming there is no inflow from the upper boundary, the storage volume
difference between rainfall excess and outflow released from the entire converging plane
can be directly related to the change in flow depth. Thus, the principle of continuity is re
arranged as:
ieAqir
At At
(3.36)
As illustrated in Figure 3.5, the converging tributary area is defined by interior
angle, 6, and radius, R. The flow crossing over a converging plane changes from one
location to another. Therefore, in this study, the governing equation for converging flow
is formed using the location of the outlet and interior angle, 6. The total tributary area
upstream of a specified location is represented as:
A = ^rd(2Rr) (3.37)
lr =(R r)0 (3.38)
Where r is the location of the outlet to collect overland flows, and lr is the arc length at
the selected location to collect overland flows.
55
Upstream boundary
Figure 3.5 Water Balance over Entire Converging Tributary Area
Inserting Eq. (3.37) and Eq. (3.38) to Eq. (3.39) yields the governing equation for
converging overland flow as:
2 qR\ 1
d)i = i l R
dt e r
rR 2
R
(3.39)
Eq. (3.39) can be further reduced as:
T r
Let a =
R
(3.40)
Combining Eq. (3.40) into Eq. (3.39) yields:
dy . 2q{\ a)
dt e r(2 a)
(3.41)
56
Where a = location ratio to define the selected location to collect overland flows. Eq.
(3.41) represents the governing equation applicable to collection of converging KW flows
at a location defined by the spatial ratio. Since rainfall duration is longer than the time of
concentration, KW flow reaches an equilibrium condition. With dy/dt = 0, Eq. (3.41) is
reduced to:
q =i
1 e
. r{la)
2(l a)
Aided with Eq. (3.35), the total flow accumulated is:
(3.42)
Qe=qlr=rR(la)9 (3.43)
It is noted that Eq. (3.43) agrees with Rational method when r approaches R. Eq.
(3.43) can be further reduced to:
y. (3.44)
4 =\r2 (3 45)
Where Qr = equilibrium discharge from the entire converging plane in [L3/T] and Ar =
entire converging surface area in [L2]
By definition, the time of concentration for the converging plane is the required
travel time for the flow from the upper boundary to reach the outlet. Combining Eq.
(2.23) and Eq. (3.41), the time of concentration can be expressed as:
T,=(3 46)
ay,
Eq. (3.46) is critically important to the application of the Rational Method. The
converging plane is subject to a long rainfall event when the rainfall duration is longer
57
than the time of concentration. Otherwise, the rainfall event is too short to cover the
entire converging plane.
3.5 Rising and Recession Hydrograph for Kinematic Wave Overland Flow on a
Converging Plane
Under a long uniform rainfall excess, the rising hydrograph limb on a converging
plane can be derived by following the same procedure stated in the Section 3.3. By
integrating Eq. (3.38) from t = Oto t= Tc, andy = 0 toy = ye, yields:
f=7; dt = f
Jt=0 Jj
y=y
y=o
l 
2q(l a)
r{2 a)
dy where 0
(3.47)
in which Tc = time of concentration in [T], The hydrograph is peaking from t = Tc to t =
Td where Td = rainfall duration in [T], During the peaking time, dy/dt = 0 in Eq. (3.41).
Aided with Eq. (3.42), the peak flow rate, peak flow velocity, and time of concentration
are derived as:
//(2 a)
ye =
2ve(la)
(3.48)
m1
(3.49)
Whereye = peak flow depth in [L] and ve = KW velocity. Combining Eq. (3.46) and Eq.
(3.49), the time of concentration can be described as:
f
T =
<*ye
a
(2r
2ve(ta)y
(3.50)
Eq. (3.48) represents the rising hydrograph. It is well understood that KW speed
varies with respect to flow depth. Using Eq. (3.48), Eq. (3.47) can be approximated with
a constant KW speed as:
58
1
(3.49)
t =
ey=y
Jy=0
l 
2vey{\a)dy
r(l a)
where 0
Let the integration variable be:
2v(la)
(2a)
(3.50)
Taking the first derivative of Eq. (3.50) and then substituting dYfor dy yields:
t =
zdtAr'Yir
2ve(l a) ^r=,e Y
 r(l a)
2vg(l ~a)
Ln 1
2Ve(l~a)v
hr(2~a)
+ K
(3.51)
In which K = integration constant that is to be determined by the condition of 0 < t < Tc.
Attention must be paid as the converging plane approaches the point of singularity
when r is approaching R, which describes the entire converging plane. The solution for
Eq. (3.51) for rising portion hydrograph is presented as:
2ve(la} 2{\a)Y 2(2a)t
JL = \e r(2a) =ie{2a)Tc L A*A _
(3.52)
Following the same procedure described in Section 3.3 by integrating Eq. (3.49)
but replacing the boundary condition of t > Td which can be integrated from y=yetoy =
0, the recession hydrograph can also be derived and can be presented as:
2ve(la)t 2(la)f 2(2a)t
= e r(2a) = e (2a)Tc [ (la)Tc
Ye
(3.53)
3.6 Numerical Approach
Many governing equations are not directly solvable analytically. However, with
assistance from computer computational ability, numerical methods are developed using
the finitedifference approach to approximate the discrete solutions. These numerical
59
solutions require specific boundary conditions to describe the physical condition of the
flow field. Therefore, successful numerical modeling is a combination of computation
ability and a proper mathematical representation of the physical constraints. Roache
(1976) stated that computational fluid dynamics is not just pure theoretical analysis.
Instead, it is closer to an experiment in the sense that each particular calculation of a
numerical simulation closely resembles the performance of a physical experiment.
There are several approaches to transform a partial differential equation into a
numerical arrangement to derive discrete solutions when the flow field is divided into
mesh network. As demonstrated in Figure 3.6, the mesh cell is formed by Ax and At that
represent a small increment in both distance and time, respectively. Each mesh cell is
identified by the coordinates of (z, /) where i = z'th distance and j = /th time step. The
solutions at each mesh include the flow depth and the discharge at a given time.
Generally, a linear relationship is assumed between two adjacent points when estimating
the rate change in time and the spatial rate change between adjacent mesh cells. For
better computational accuracy and numerical stability, it is necessary to use forward,
backward, or central finite difference, according to the boundary and initial conditions.
60
7
6
5
4
3
2
At
3 4 5 6 7
KW Flow Field Mesh Network
As presented in Figure 3.7, an implicit method utilizes all variables at the current
time step to compute Ax/At whereas an explicit method utilizes Ax from the previous time
step to compute the convective term at the current time step. With only one unknown
variable, the explicit method provides a direct solution. The implicit method will require
iteration to obtain converged solution that meets the predetermined tolerance (Guo and
Hinds, 2013).
Ax
i = 1 2
Figure 3.6 
61
Explicit Method
Implicit Method
j AAA
il i i+1
A = Known Parameter / \ = Unknown Parameter
Figure 3.7 Illustration of Implicit and Explicit Finite Difference Method
For this study, Eq. (3.14) is converted into its central finite difference equation as:
y(t + At)y{t)
At
= n
ic(l + Al) + ic(l)
a
\y(t + At)m+y(t)m\
2 R
(3.54)
Similarly, Eq. (3.41) is shown as:
y(t +At)y(t)
At
= n
ie(t +At)+ie(t)
2
k\y(t + Atf + y{tf (1 a)
r(2a)
(3.55)
Where n = 0 or 1, depending on rainfall input. The initial condition for both equations is
a dry bed condition or y(t, r) = 0 everywhere at t = 0, The upstream boundary remains
dry at all times, while the downstream boundary condition can be described as the normal
flow depth. It can be progressively solved for every time step with n= 1 until the rain
stops. Without excess rainfall, n = 0.
3.7 Closing
In this chapter, new governing equations were derived for both diverging and
converging types KW overland flows. These equations are presented in the polar
62
coordinate system that is more flexible and better representing the geometry of fan shape
landform. Additional efforts are required to verify these derived equations with
laboratory data or field records. With these equations, further studies on the sensitivity of
soil infiltration and surface roughness on an overland runoff hydrograph can be
conducted numerically.
63
4. Geometric Transformation of Diverging and Converging Fan Areas into
Kinematic Wave Rectangular Planes
4.1 Introduction
As previously mentioned, in acknowledgment of flooding problems that have
occurred on alluvial fans, the Federal Emergency Management Agency (FEMA) developed a
stochastic methodology that was first developed to identify flood hazards on an alluvial fan
(Anon 1981 and Dawdy 1979). This method later become the foundation of the FAN
computer model (1990) developed by FEMA to establish the National Flood Insurance
Program (NFIP) for this type of landform (FEMA 1989, 2000). One of the deficiencies of
the existing methodology is that it does not clearly define a procedure to conduct the fan
shape drainage basin analyses. Based on previous studies, it was recommended that the
rainfall runoff hydrologic processes on alluvial fan to be best simulated utilizing the
Kinematic Wave (KW) overland flow theory (Mukhopadhyay et. al. 2003).
One of the major computational algorithms developed to simulate surface runoff
utilizing KW overland flow theory is the EPA Storm Water Management Model Version 5.0
(SWMM5). The KW application in SWMM5 was formulated to predict the flows on a
virtual rectangular sloping plane. To apply SWMM5 to either a diverging or converging fan
planes, the fan shape surface must first be converted into its equivalent rectangular plane. In
current practices, the engineer will have to estimate such a shape conversion based on
experience or calibration if any observed data is available. For example, the application of
SWMM5 for the Fox Hollow Watershed located in Centre County, PA was calibrated using
field data (Zang and Hamlett 2006). However, for areas without rainfall and runoff gages,
the estimation of the required KW rectangular plane width relies on engineering judgement.
64
Consequently, overland flow predictions are varied with respect to different KW rectangular
plane widths estimated for the same study area. For instance, a factor of 2.0 as the ratio
between the KW width and the waterway length was recommended for a square watershed
while a different factor of 2.2 was also recommended for shapes other than a square
(UDSWMM Manual 2000). In Ontario, Canada, a factor of 1.67 was adopted for watershed
studies (Proctor and Redfern 1976).
Determination of KW plane width has been a challenge when applying the KW
overland flow theory to hydrologic studies. Without proper guidance, determination of this
key modeling parameter depends on individuals experience and selection of empirical
formulas. Therefore, it is urgent to acquire a standard procedure for converting a watershed
from its natural shape into its equivalent rectangular shape when using the KW flow theory
for overland runoff predictions.
The effort of this Chapter is to derive a oneonone geometric transformation
procedure to convert the derived new KW governing equation from the polar coordinate
system into Cartesian coordinate system. This transformation shall satisfy the basic
principles of mass and energy conservation. As a result, an alluvial fan watershed can be
described using its apex angle and fan radius into Cartesian coordinate system. With the
specified longitudinal surface slope and rainfall excess, the KW flows generated from the fan
area can be determined with flow depths, flow velocities, and flow discharge at any given
point on the alluvial fan.
65
4.2 Transformation Procedure Development
For a unitwidth approach, the computational methods developed for KW flows
require a geometric conversion of the actual, irregularshaped alluvial fan plane into its
virtual rectangular plane (Huber 1988, Rossman 2005). To maintain the principle of
continuity, the surface area between the two planes must be preserved. Correspondingly, the
potential energy difference caused by elevation variance between highest and lowest point
along the waterway must be also conserved (Guo and Urbonas 2009).
A conformal mapping technique employed in the potential flow theory (Finnemore
and Franzini 2002) is adopted to develop a transformation procedure for converting an
irregularshaped watershed into a virtual KW rectangular sloping plane. To maintain the
watershed geographic characteristics, the continuity and energy principles for KW shape
transform are described as:
A = AW (4.1)
H = HW (4.2)
where A = actual watershed area in [L2], Aw = virtual area on KW sloping plane in [L2], H =
vertical drop in actual watershed in [L], and Hw = vertical drop in virtual KW plane in [L],
Between the two flow systems illustrated in Figure 4.1, the dimensional analysis
indicates that characteristic parameters are average watershed width, flow length, and width
of KW plane. Their dimensionless functional relationship is derived as:
iM!
(4.3)
66
in which Lw = width of KW plane in [L], B = average watershed width in [L], and L = length
of collector channel in [L], Eq. (4.3) basically describes the geometric relationship between
the shapes of these two tributary areas.
Natural Watershed Rectangular Watershed
Figure 4.1 Conversion of Actual Watershed into Virtual KW Sloping Plane
To satisfy Eq. (4.3), the total watershed area must be preserved and can be presented
as:
(44)
Furthermore, the elevation difference over the waterway between the highest points
of the watershed to the lowest point at the outlet location must be also preserved as:
S0L Sw (Xw + Lw ) (4.5)
Normalizing the relationships with watershed length, /., Eq. (4.4) and Eq. (4.5) can be
converted into:
67
(4.6)
L1 L L
L L
(4.7)
The watershed shape factor serves as an index to describe how overland flow is
collected within the watershed. It suggests that the watershed length to width ratio and
elongation and circularity ratios can be engaged to represent the shape of a watershed
(McCuen 2005). As suggested in Figure 4.1, the shape factor for a natural watershed to be
converted into KW virtual plane can be expressed and approximated as:
Where X = watershed shape factor, 7 = KW shape factor for the KW sloping rectangular
plane, and K = upper limit for shape factor. For a rectangular watershed such as a parking lot,
Eq. (4.8) is simplified as a width to length ratio. In engineering practices, it is recommended
that a large watershed to be divided into smaller subareas with a shape factor not to exceed
upper limit K. Commonly, K= 4 is adopted when conducting hydrologic simulation
(UDFCD 2001).
Assisted by Eq. (4.8) and Eq. (4.9), Eq. (4.6) and Eq. (4.7) can be combined as:
^ = + 7 when X
Sw Y
The two cases in Figure 4.2 represent two extreme location of the channel alignment since
the centerline and the boundary line in the watershed sets the limits for all possible channel
alignments (DiGiano et. al. 1976).
X = = when X < K
L L2
(4.8)
L
(4.9)
68
 Overland
Flow
 Channel
Flow
Upstream
High Point
Downstream
Low Point
Outlet Location
Upstream
High Point
Lw=L
Downstream
Low Point
Outlet Location
Square Plane with Side Channel
KW Cascading Plane with Side Channel
Upstream
High Point j
Sx
" /,. So
Downstream
Low Point
Outlet Location
Upstream
High Point _____________Lw=2L
1 k. Sw
= o
Xw=A/Lw
Downstream
Outflow * Low Point
Outlet Location
Square Plane with Central Channel
KW Cascading Plane with Central Channel
Figure 4.2 KW Models for Square Watershed
4.3 Development of Geometric Transformation Using SINE Function
In this section, the trigonometric Sine function is adopted to derive the conversion
parameters for preserving the apex angle on a diverging type alluvial watershed. In this
study, Eq. (4.10) is defaulted to be:
Y = asinbX+c (4.11)
In which a, b and c are the constants to be developed separately for symmetric and
asymmetric watersheds.
Mathematically, this geometric transformation function in Eq. (4.11) is a onetoone
single value relationship between variables X and Y. This functional relationship shall cover
the range from Y = 0 to Y < 4 to 6, depending on the watershed model used in the stormwater
69
studies. With the support of Figure 4.2, in general, this functional curve shall pass three
distinct points as:
Under the Condition 1, X~ 0 a very small watershed, 7~ 0
Eq. (4.11) is reduced to:
7=0+c=0 or c = 0 (412)
Under the Condition 2, X= 1 and 7= 1, it represents a square watershed with a
collecting channel along the side:
Y=asmb=l (4.13)
Similarly, under the condition that X= 1 and 7= 2, it represents a square watershed
with a central collecting channel,
Y=asinb = 2 (4.14)
Under the Condition 3, X=K in which K represents the highest acceptable ratio of
watersheds width to its waterway length. As expected, the higher the value of K is,
the higher the peak flow is. In practice, the value of K shall not exceed 4. As
indicated in Eq. (4.14), the maximum value for 7 shall satisfy:
= abcosbK = 0
dX
(4.15)
Considering the Sine function as the default geometric transform operator, the
n
maximum value in Eq. (4.15) occurs when bK= thus:
cos bK = cos
2
b=
2 K
Inserting Eq. (4.17) into Eq. (4.13) and Eq. (4.14) yields:
(4.16)
(4.17)
70
o
for a sidecollecting channel,
a =
sin
n
2K
and Y =
sin
n
2K
sin
n
2K
X
o for a centercollecting channel,
2 , 2 . f
a = 
sin
n
2 K
and Y =
sin
n
2 K
sin
n
2K
X
(4.18)
(4.19)
Combining Eq. (4.11), Eq. (4.12), Eq. (4.18), andEq. (4.19) yields:
7 = (1.5
Z)
sin()
2 K
sin
n
2K
X
for X
(4.20)
Y
4,
L
(4.21)
Z
An
A
(4.22)
Where Y= KW shape factor, Z = area skewness factor (depending on location of the collector
channel through the watershed), Am = larger half area after the watershed is divided by the
collector channel, A = total watershed area, and K = maximum allowable watershed shape
factor (CUHP 2005). And when the watershed is divided into two parts by the waterway, the
value of Z is defined as the ratio of the larger half to the total watershed area. For instance,
Z = 0.5 is for the case with a centralcollecting waterway, while Z = 1.0 is for the case with a
sidecollecting waterway. Based on Eq. (4.1), the length of KW overland flow on the KW
rectangular cascading plane is defined as:
w
L
(4.23)
71
According to Eq. (4.2), the slope on the KW plane is the vertical fall divided by the
total flow length represented as:
= S
L
X + L
(4.24)
4.3.1 Geometric Transformation of Diverging Fan Area
As illustrated in Figure 4.3, the location of collector channel on a diverging plane area
is along the downstream boundary line. Therefore, this is a case of side channel, or Z, should
be equal to 1.0. The shape factor for a diverging plane area is defined by its tributary area,
1 2
A =~R 6, and the length of the collector channel length, L = R9 .
Thus, Eq. (4.8) is converted into:
A
L2
20
(4.25)
Actual Watershed Virtual KW Plane Watershed
Figure 4.3 Conversion of Diverging Plan into KW Rectangular Plan
72
In hydrologic analyses, a large watershed is suggested to be divided into smaller sub
areas. It is advisable that the watersheds width to length ratio not exceed 4.0 to avoid a
skewed runoff estimation (McCuen, 2005). In this study, the recommendation of K < 4 is
adopted (Guo and Urbonas 2009). Substituting Eq. (4.25) into Eq. (4.20) with K = 4 and Z =
1, the width of the virtual KW rectangular plane is derived as:
L =
R6
sin()
8
sin
for 6 > 1/8
(426)
Referring to Figure 4.3, the vertical drop height is conserved along the flow path, a b c, on
the actual watershed or a' b' c' on the virtual KW plane. Therefore, Eq. (4.24) is converted to:
S
w
R( 1 + 0)
XW+LW
(427)
4.3.2 Diverging Fan Areas Transformation Case Study
The KW shape factor, LJL, is sensitive to watershed geometry and location of the
collecting channel or waterway. Table 1 is a set of various fan shape parameters showing
different fan areas, radii of flow paths, interior apex angles, and surface longitudinal slopes.
Eq. (4.26) and Eq. (4.27) are applied to these hypothetical cases to conduct a sensitivity test
of the KW shape factor in a diverging geometry. As expected, all conversions reflect the
original watersheds geometry, and none is repeated and the results are summarized as
followed.
Figure 4.4 was generated to display the shape factor relationship between the fan
shape and KW virtual rectangular sloping plane by plotting the X and Y values for Case 4 to
Case 8. Figure 4.4 shows a linear relationship when Yranges between 0 and 1 demonstrating
73
a onetoone conversion relationship between a fanshaped tributary area and KW virtual
rectangular plane.
Table 4.1 Testing for Diverging Fan Shape Plane to KW Rectangular Planes
Case ID R Interior Angle S Area L Fan Shape Factor KW Shape Factor
(ft) Radian Degree (%) (ft2) (ft) X = A/L2 Y = Lw/L
1 100 2.09 119.7 5.00 10450 209 0.24 0.25
2 300 2.09 119.7 5.00 94050 627 0.24 0.25
3 500 2.09 119.7 5.00 261250 1045 0.24 0.25
4 500 0.52 29.8 5.00 65000 260 0.96 0.96
5 500 1.05 60.2 5.00 131250 525 0.48 0.49
6 500 1.57 90.0 5.00 196250 785 0.32 0.33
7 500 2.09 119.7 5.00 261250 1045 0.24 0.25
8 500 2.62 150.1 5.00 327500 1310 0.19 0.20
9 500 3.14 179.9 3.00 392500 1570 0.16 0.16
Furthermore, as presented in Figure 4.5, as the interior expansion angle increases, the
width of the KW virtual plane decreases and the effect of K diminish. It reflects the
fundamental concept of basin delineation for keeping the watershed width and length ratio
closer to unity to avoid skewing the flow prediction. In engineering practices, the value of X
is predetermined based on the size of the tributary area and location of the collecting
channel. Eq. (4.20) provides a general conversion from a fan shape tributary area to a KW
virtual rectangular plane to assist the determination of KW rectangular cascading plane width
during modeling efforts.
74
Figure 4.4 Watershed Factor vs. KW Shape Factor Trigonometry Sine Function
Figure 4.5 Shape Factor vs. Interior Expansion Angle Diverging Plane
4.4 Development of Geometric Transformation Using Parabolic Function
In this section, a parabolic function is adopted to derive the functional relationship
between shape factors Xand Yas:
75
Y = aX2+bX + c
(4.28)
In which a, b, and c are constants to be developed separately for symmetric and asymmetric
watersheds as previously demonstrated. Following the same procedure described in the
former discussion, mathematically, this function curve will pass three distinct conditions and
satisfy the following:
Under the Condition 1, X~ 0 a very small watershed, 7~ 0
Eq. (4.28) can be reduced to:
Y= aX2 + bX= 0 or c = 0 (4.29)
Under the Condition 2, X= 1 and Y= 1, it represents a square watershed with a
collecting channel along the side:
Y=a+b= 1 (4.30)
Similarly, under the condition that X= 1 and 7=2, represents a square watershed
with a central collecting channel:
Y=a+b = 2 (4.31)
Under the Condition 3, X=K in which K represents the highest acceptable ratio of
watersheds width to its waterway length. As expected, the higher the value of K is,
the higher the peak flow is. In practice, the value of K shall not exceed 4. As
indicated in Eq. (4.31), the maximum value for 7 shall satisfy ,
= 2 aX + b = 0
dX
(4.32)
Considering the parabolic function as the default geometric transformation function
operator, inserting Eq. (4.32) into Eq. (4.29) and Eq. (4.30) yields:
o for a sidecollecting channel
76
a =
1
2 K
(4.33)
12 K
and b =
12 K
Y =X2 ^X
12 K
12 K
o
for a centercollecting channel
2 , , ~4K
a = and b =
12 K
12 K
Y =X2 ^X
12 K 12 K
(4.34)
(4.35)
(4.36)
Combining Eq. (4.34) and Eq. (4.36), the general conversion relationship and can be
presented as:
Y
JX2X
12K 12 K
(4.37)
Where Y= KW shape factor, Am = larger half area after the watershed is divided by
the collector channel, A = total watershed area, and K = maximum allowable
watershed shape factor (CUHP 2005).
4.4.1 Geometric Transformation of Converging Fan Area
Overland flows on a fanshaped area are collected by a side channel that is located
along boundary line, be, as shown in Figure 4.6. Therefore, Z = 1 for this case. The shape
factor for a converging fanshaped area is defined by its tributary area, A, which equals to
1
r
2
B(lR r), and length, L, of the collector channel equals L = {Rr)d With these
relationships, Eq. (4.8) yields:
x r(2Rr)
20(R r)2
(4.38)
77
Actual Watershed Virtual KW Plane Watershed
Figure 4.6 Conversion of Converging Plan into KW Rectangular Plane
A converging area tends to be wide in shape. In this study, K = 10 and Z = 1 are
adopted (Guo and Urbonas 2009). As a result, the KW shape factor in Eq. (4.35) is reduced
to:
Y = 1.053X 0.053X2 where X< 10 (4.39)
Referring to Figure 4.6, the vertical drop is conserved along flow path, abc, on the
converging watershed or a'b'c' on the virtual KW plane. Therefore, Eq. (4.24) is revised to:
r + (Rr)0 . _ .. ...
= ba1 for converging flow (4.40)
Xw +Lw
Eq. (4.38) through Eq. (4.40) were derived to convert a converging area into the equivalent
rectangular KW plane.
4.4.2 Converging Fan Area Transformation Case Study
Table 4.2 is a set of converging areas, radii of flow paths, interior angles, and
longitudinal surface slopes. With an angle of 1.21 radians (69.3 degrees), the watershed
78
shape factor is 9.92, close to the selected limit for K. Eq. (4.37), Eq. (4.38), and Eq. (4.39)
are applied to these hypothetical cases to test the sensitivity of KW shape factor to the
converging geometry. This table also provides a summary of the corresponding fan shape
verses KW rectangular planes shape factor. As expected, all conversions reflect the actual
watersheds geometry, and none is repeated.
Table 4.2 Testing for Converging Fan Shape Plane to KW Rectangular Planes
Case ID R Interior Angle r S Area L Fan Shape Factor KW Shape Factor
(ft) Radian Degree (ft) (%) (ft2) (ft) X = A/L2 Y = Lw/L
1 500 1.21 69.3 400 5.00 145200 121 9.92 5.23
2 500 1.57 90.0 400 5.00 188400 157 7.64 4.95
3 500 2.10 120.3 400 5.00 252000 210 5.71 4.29
4 500 2.35 134.6 400 5.00 282000 235 5.11 4.00
5 500 2.62 150.1 400 5.00 314400 262 4.58 3.71
6 500 3.14 179.9 400 5.00 376800 314 3.82 3.25
7 100 2.09 119.7 80 5.00 10032 42 5.74 4.30
8 300 2.09 119.7 240 2.00 90288 125 5.74 4.30
9 500 2.09 119.7 400 5.00 250800 209 5.74 4.30
10 500 2.62 150.1 400 7.00 314400 262 4.58 3.71
A series of exhibits of Eq. (4.39) and converging plane radius ratio, r/R, ranging from
0.1 to 0.9 are prepared with K = 10, 6, and 4 and presented as Figures 4.7 to 4.10. These
figures show the sensitivity of the location of the outlet point in relation to the projected
origin of the converging plane. As expected, when the interior expansion angle is increasing,
the KW virtual cascading rectangular plane width decreases to satisfy the area continuity
between converging fan shape plane and KW virtual plane. Also, between 7=0 and 1, under
different interior expansion angles, it suggests K value is not as sensitive, which benefits the
79
selection of the outlet location to determine the alignment of the conveying channel during
engineering design and application.
Converging Plane Radius Ratio vs. KW Shape Factor Y
w/lnterior Expansion Angle, 0 = 15 degree
Figure 4.7 Radius Ratio vs. KW Shape Factor (15 degree) Parabolic Function
Converging Plane Radius Ratio vs. KW Shape Factor Y
w/lnterior Expansion Angle, 0 = 30 degree
V = L/L
*K = 10 a<K = 6 K = 4
Figure 4.8 Radius Ratio vs. KW Shape Factor (30 degree) Parabolic Function
80
Figure 4.9 Radius Ratio vs. KW Shape Factor (90 degree) Parabolic Function
Figure 4.10 Radius Ratio vs. KW Shape Factor (179 degree) Parabolic Function
4.5 Closing
It is critically important to understand the transformation relationships between an
irregular watershed and its virtual KW rectangular plane. Applying the conformal mapping
81
technique to project the actual flow motion onto a virtual rectangular surface, it preserves the
major geometric parameters between the actual and virtual watershed. The concept of
watershed shape factor allows the collection of surface runoff from an actual watershed to be
transformed onto a KW overland flow rectangular plane. This oneonone conversion
relationship can be executed as a preprocess before the hydrologic simulation when working
with a computer model such as HECHMS and SWMM5. Furthermore, with this
transformation procedure, the maximum allowable overland flow length of 300 feet to 500
feet recommended by the current engineering practices is no longer applicable because the
KW flow on the rectangular plane is virtual or mathematical only.
82
5.
Model Verification and Case Studies
5.1 Introduction
A numerical model for rainfall and runoff simulations is best illustrated by the
concept of system which produces an output according to the given input. The reliability of a
numerical model depends on how well the model is calibrated with observed data. Model
verification is a system identification process. Applicability of a numerical algorithm
depends on their application limits and identifications of system constants and parameters. A
good numerical model provides consistency between the input and output relationship while
a reliable numerical model needs a high level of accuracy.
Differential equations derived for a physical process are mostly aimed at providing
solutions to engineering problems. In most cases, these equations cannot be directly solved
for an analytical solution, but their applications rely on numerical approach to generate an
approximated solution. With advanced computer technology and numerical techniques in
both software and hardware, many differential equations can be formulated into numerical
methods. A numerical method is to simulate a physical process by solving a set of governing
equations which depict a realworld event. Correspondingly, numerical solutions are always
generated for a specified initial and boundary conditions. In practice, most of boundary and
initial conditions require special numerical treatments to warrant stable computations.
Although numerical solutions can be effectively applied to engineering designs and
applications, it is not intended to replace the analytical solution if it is available. A numerical
approach is highly valuable when coping with multiple scenario analyses which are subject
to various boundary conditions. It is important to understand that numerical solutions are
83
discrete and only derived at grids while an analytical solution is continuous in time and in
space (Guo 1982). One of the major benefits of the numerical approach is to offer
predictions of a physical process and provide good guidance prior to actual construction of
the physical experiment.
Care must be taken that all numerical procedures require calibrations and all
calibrations require observed data. In order to collect field data, U.S. Geological Survey
(USGS) installed the first stream gage on the Rio Grande River in New Mexico in 1889.
Currently, USGS operates approximately 7,400 stream gages nationwide with approximately
91 percent of these stations are transmitting data in nearly a realtime scale (USGS 2007).
As presented in Figure 5.1, the distribution and density of stream gages in the Southwest U.S.
is less than the rest of the U.S. Continent. This is mainly due to not well defined waterway,
severe erosion and scour along natural washes in the arid climate region.
Due to the lack of stream records on alluvial fan type landforms, for this study, it is
proposed to populate overland flow hydrographs numerically for the selected cases according
to both diverging and converging fan geometries. With a proper surface roughness
coefficient assigned, overland flow depths and the associated hydraulic characteristics on the
fan surface will then be compared with available field record or documented laboratory data.
84
Figure 5.1 USGS Stream Network Location Exhibits
The following sections will discuss the numerical procedures and results for case
studies and provide comparisons with the historical data if available.
5.2 Verification of Diverging KW Model and Case Studies
The laboratory study for diverging KW flows on an impervious surface was
documented by Izzard (Izzard, 1946). It was then followed by other investigators (Yu and
McNown, 1964, Langford and Turner 1973) along with some field observations (Singh 1975,
1976). As reported (Muzik 1973), the KW flows were produced from a rectangular
galvanized surface with 0.61m in width and 0.91m in length and laid on a longitudinal slope
of 0.2079 (m/m). This laboratory layout was evaluated to be equivalent to a diverging KW
plane with a radius, R, of 0.91m (Singh and Agiralioglu 1981). For this case study, to satisfy
the same surface area, it was determined that the apex angle is 1.34 radians (76.75 degrees).
85
Under a laboratory shower simulating, a rainfall event was conducted with an
intensity of 78 mm/hr and with duration of 50 seconds. The surface runoff was produced and
measured with a total simulation time of 80 seconds. The parameters on the KW rating curve
are determined to be:
(5.1)
n
5
(5.2)
m=
3
Where n = Mannings roughness coefficient and is equal to 0.01 for galvanized surface
(Muzik 1973).
Figure 5.2 presents the comparisons between the laboratory data and the analytical
solutions derived in this study for both rectangular and diverging KW flows. As expected,
Eq. (2.23) for rectangular plane and Eq. (3.16) for diverging plane produce the same peak
flow for the same surface area, but the diverging KW flow is characterized with a slower
movement, or a longer time of equilibrium as predicted in Eq. (3.18). As shown in Figbure
5.2, the KW analytical solution computed by Eq. (3.19) agrees well with the previous study
(Singh and Agiralioglu 1981) .
86
Rainfall Intensity = 78mm/hr
Figure 5.2 Comparison of Laboratory and Numerical Solutions for KW Overland
Flow Hydrographs under Rainfall Intensity of 78 mm/hr for 50 Seconds
A spreadsheet was developed based on Eq. (3.54) with the support from Mannings
equation, Eq. (3.12), Eq. (3.13), Eq. (5.1) and Eq. (5.2) to conduct the computation. Detailed
numerical for this case can be located in Appendix D Case 1
As shown in Figure 5.3, a similar laboratory experiment is carried out using the same
equipment with same apex angles. The peak flow is increased from 12 cm3/s to 18 cm3/s as
the manmade rainfall intensity is increased from 78 mm/hr to 115 mm/hr, respectively.
Both studies verify that the diverging KW flow moves with a shallower depth and at a slower
velocity in comparison with the rectangular KW flow. Again, good agreement is achieved
between the KW analytical solution computed by Eq. (3.16) and the laboratory data from the
previous report (Agiralioglu and Singh 1980). The same spredsheet developed is utilzed by
replacing the rainfall intensity from 78 mm/hr with 115 mm/hr to conduct the computation by
87

PAGE 1
DETERMINISTIC APPROACH FOR PREDICTION AND MANAGEMENT OF FLOOD FLOW ON ALLUVIAL FANS by SHOUCHING HSU B.S., National Chiao Tung University, 1990 M.S., University of Colorado, 1996 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Civil Engineering Program 2016
PAGE 2
ii This thesis for the Doctor of Philosophy degree by ShouChing Hsu has been approved for the Civil Engineering Program by David C. Mays, Chair James C. Y. Guo, Advisor Arunprakash Karunanithi Balaji Rajagopalan Zhiyong Jason Ren Date: 04/06/2016
PAGE 3
iii Hsu, ShouChing (Ph.D., Civil Engineering Program) Deterministic Approach on Prediction and Management of Flood Flow on Alluvial Fans Thesis directed by Professor James C. Y. Guo ABSTRACT Alluvial fans are the most prominent landscape features in the semiarid and arid regions of the world. This geographic feature is often observed in the American Southwest. From the viewpoint of engineering practices, the hydraulic behavior of overland flows on alluvial fans is very different from open channel flows. Alluvial fan flows are subject to high roughness effects due to wide and shallow surface flow. In current practices, many floodplain studies on alluvial fans rely on qualitative approaches to provide general approximations of flow depth, width and velocity. Since 1980s, the Federal Emergency Management Agency (FEMA) has recommended a stochastic approach for assessing flood risks and hazards on alluvial fans. The method applies a conditional probability based on the risk level of the selected flood event, and the site location on the fan area. This probabilistic approach has been accepted as a reasonable approximation for alluvial floodplain studies, according to FEMA regulations. From a regulatory standpoint, this approach provides a conservative assessment to delineate the boundaries of flood hazard areas. But, it has been long recognized that the rainfallrunoff process should be included as primary input parameters to determine the flood flows on an alluvial fan. As urban areas continued to expand and invaded into alluvial fans in the recent years, it is critically important to quantitatively determine the flood flow characteristics with a higher accuracy and consistency. By understandings the hydrologic
PAGE 4
iv processes on alluvial fans, it will definitely assist engineers to better manage the risks and uncertainties in land use planning and mitigate flood hazard when conducting a regional master drainage plan for alluvial fan areas. In this study, the Kinematic Wave (KW) overland flow approach on a rectangular plane was reviewed, and then expended into converging and diverging planes to simulate overland flow hydraulics on fan shape geometry. The continuity principle was applied to balance the runoff volumes among excess rainfall amount, runoff inflow and outflow and surface detention volume. New governing equations were derived to describe the accumulations of overland flows, according to the geometry of the KW plane Â– rectangular, converging, or diverging. These equations were also expanded to provide analytical and numerical solutions to present overland flows on diverging or converging planes. The results of the numerical simulations are well agreed with several reported laboratory studies and field observations. It is believed that the method presented in this study can improve the current FEMA flood flow study procedure and minimize the environmental impacts as the urbanization process encroaches into alluvial fans in the southwest States of the US. The form and content of this abstract are approved. I recommend its publication. Approved: James C. Y. Guo
PAGE 5
v DEDICATION I dedicate this work to my wife, Hsuehling, and my daughter, Trinity, for their supports and companies during the low and high moments in this academic adventure. They constantly remind me that life is more appreciated when a family is supportive. Also, I like to dedicate this work to my parents, brother and sister back in Taipei, Taiwan, who taught me the value and the importance of the family.
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vi ACKNOWLEDGMENTS Â“Education is not the learning of facts but the training of the mind to thinkÂ” Â“I never teach my pupil, I only provide the conditions in which they can learnÂ” Albert Einstein I would like to express my sincere appreciation to Professor James C. Y. Guo for his mentorship, patience, encouragement and guidance during my graduate studies for both M.S. and Ph.D. degrees. Five years ago, I thought I had committed to a long and lonely academic journey. But during this journey, Professor Guo offered me many different aspects to investigate the selected topics. In return, my five years of Ph.D. study has been rewarding and I am proud to be named his pupil. Although I was not sure if I have become a scholarly hydrologist, but I definitely learned how to formulate, present, and defend my creative thoughts and ideas. I do agree that my Ph.D. training is not limited to the contents of my study, but the process and the methodology. Of course, I also wish to express my appreciation to the members of the committee Â– Dr. Rajagopalan, Dr. Mays, Dr. Rens, and Dr. Pal, for their comments and helps. Special thanks to Maroun Ghanem for helping me with graphics in the exhibits. Thanks to Jill Reilly to help me with editorial comments in my writings.
PAGE 7
vii TABLE OF CONTENTS Chapter 1. Introduction .............................................................................................................1 1.1 Urbanization Impacts to Alluvial Fan ................................................................1 1.2 Alluvial Fan Location ........................................................................................5 1.3 Alluvial Landform Descriptions and General Characteristics .............................7 1.4 FEMA Flood Flow Approach for Alluvial Fans ............................................... 12 1.5 Objective ......................................................................................................... 17 2. Review Of Kinematic Wave Overland flow ........................................................... 20 2.1 Introduction ..................................................................................................... 20 2.2 Basic Principles ............................................................................................... 20 2.3 Kinematic Wave Application Limit ................................................................. 23 2.4 Overland Flow on a Rectangular Pervious Plane .............................................. 27 2.5 Overland Flow Hydrograph Analysis ............................................................... 29 2.6 Overland Flow Travel Time on a Pervious Surface .......................................... 36 2.7 Closing ............................................................................................................ 41 3. Derivations of Governing Equations for Diverging and Converging Kinematic Wave Overland Flows ...................................................................................................... 42 3.1 Introduction ..................................................................................................... 42 3.2 Kinematic Wave Overland Flow on a Diverging Plane .................................... 43 3.3 Rising and Recession Hydrograph for Kinematic Wave Overland Flow on a Diverging Plane ............................................................................................... 49 3.4 Kinematic Wave Overland Flow on a Converging Plane .................................. 51 3.5 Rising and Recession Hydrograph for Kinematic Wave Overland Flow on a Converging Plane ............................................................................................ 58
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viii 3.6 Numerical Approach ....................................................................................... 59 3.7 Closing ............................................................................................................ 62 4. Geometric Transformation of Diverging and Converging Fan Areas into Kinematic Wave Rectangular Planes ....................................................................................... 64 4.1 Introduction ..................................................................................................... 64 4.2 Transformation Procedure Development .......................................................... 66 4.3 Development of Geometric Transformation Using SINE Function ................... 69 4.3.1 Geometric Transformation of Diverging Fan Area .................................... 72 4.3.2 Diverging Fan Areas Transformation Case Study ..................................... 73 4.4 Development of Geometric Transformation Using Parabolic Function ............. 75 4.4.1 Geometric Transformation of Converging Fan Area ................................. 77 4.4.2 Converging Fan Area Transformation Case Study .................................... 78 4.5 Closing ............................................................................................................ 81 5. Model Verification and Case Studies ..................................................................... 83 5.1 Introduction ..................................................................................................... 83 5.2 Verification of Diverging KW Model and Case Studies ................................... 85 5.3 Rising and Recession Hydrograph for Diverging Flow .................................... 92 5.4 Verification of Converging KW Model and Case Studies ................................ 94 5.5 Rising Hydrograph for Converging Flow ....................................................... 100 5.6 Shape Factor Method Verification and Sensitivity Test .................................. 103 5.6.1 Diverging Plane Shape Factor Method Case Study and Sensitivity Test .. 103 5.6.2 Converging Plane Shape Factor Method Verification ............................. 105
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ix 5.6.3 Converging Plane Shape Factor Method Case Study and Sensitivity Test .... ............................................................................................................... 108 5.7 Closing .......................................................................................................... 111 6. Conclusion ........................................................................................................... 113 Reference .................................................................................................................... 118 Appendix A Â– ASCE Journal Papers ............................................................................................ 124 B Â– Diverging Plane Governing Equation Derivative ................................................... 137 C Â– Converging Plane Governing Equation Derivative ................................................ 141 D Â– Verification of Diverging KW Model and Case Studies ........................................ 145 E Â– Verification of converging KW Model and Case Studies ....................................... 163 F Â– Verification of Diverging KW Shape factor and Case Studies ................................ 177
PAGE 10
x LIST OF TABLE Table 4.1 Â– Testing for Diverging Fan Shape Plane to KW Rectangular Planes ....................... 74 4.2 Â– Testing for Converging Fan Shape Plane to KW Rectangular Planes ..................... 79 5.1 Â– Comparison between Predictions and Measurements for Converging Flows .......... 96 5.2 Â– SWMM 5 Peak Flow Sensitivity Test for KW Plane ........................................... 105 5.3 Â– Conversion Relationship between Converging Plane and KW Rectangular Virtual Plane ................................................................................................................... 107 5.4 Â– W2 and W6 Sample Watershed Geometry Conversion ..................................... 107
PAGE 11
xi LIST OF FIGURES Figure 1.1 Â– Population Percent Change By State (2010 Census Bureau) ....................................1 1.2 Â– Population Percent Change By County (2010 Census Bureau) ................................2 1.3 Â– Example of Hillside Development Scottsdale, Arizona (Google Image, 2012) ......3 1.4 Â– Example of Urban Development Encroaching onto Alluvial Fans, Las Vegas, Nevada (2008 Las Vegas Valley Master Plan Update, CCRFCD) ...........................3 1.5 Â– Alluvial Fan Formation Â– Death Valley, California (Geomorphology of Desert Environments, 2009) ...............................................................................................4 1.6 Â– Alluvial Fan Geographic Locations (USGS, 2000) ..................................................5 1.7 Â– Alluvial Fan Â– Boise, Idaho (Google Image, 2012) ..................................................6 1.8 Â– Alluvial Fan Â– Glenwood Springs, Colorado (Google Image, 2012) ........................7 1.9 Â– Alluvial Fan Formation (Google Image, 2012) ........................................................8 1.10 Â– Alluvial Plane Formation (USGS, 2000) ............................................................. 11 1.11 Â– Temporal Distribution Regions (NOAA, 2011) ................................................... 13 1.12 Â– Diagram of the Pearson System showing distributions of Types I, III, VI, V, and IV in terms of 1 (squared skewness) and 2 (traditional kurtosis) ..................... 15 1.13 Â– Isometric view (French, 1992)............................................................................. 16 1.14 Â– Typical Engineering Application Geometry Layout on an Alluvial Fan ............... 17 2.1 Â– Monoclinal Wave Movement ................................................................................ 23 2.2 Â– Monoclinal Wave Rating Curve ............................................................................ 24 2.3 Â– Dynamic Wave, Diffusion Wave and Kinematic Wave Applicability Zone ........... 27 2.4 Â– Onedimensional overland unsteady flow profile .................................................. 30 2.5 Â– Dimensionless Overland Flow Hydrograph ........................................................... 33 2.6 Â– Rising Water Surface Profile when t < T e .............................................................. 34
PAGE 12
xii 2.7 Â– Recession Hydrograph when t > T d ....................................................................... 35 2.8 Â– KW Integration Domain Illustration for Rainfall Excess considering Soil Infiltration ...................................................................................................................................... 38 3.1 Â– Diverging Surface and Parameters ........................................................................ 44 3.2 Â– Fan Shape and Fan Area for KW Flow .................................................................. 46 3.3 Â– Typical Transportation Alignment with Culvert Crossing Layout on Alluvail Fans ...................................................................................................................................... 52 3.4 Â– Water Balance in an Arc Segment within Converging Tributary Area ................... 53 3.5 Â– Water Balance over Entire Converging Tributary Area ......................................... 56 3.6 Â– KW Flow Field Mesh Network ............................................................................. 61 3.7 Â– Illustration of Implicit and Explicit Finite Difference Method ............................... 62 4.1 Â– Conversion of Actual Watershed into Virtual KW Sloping Plane .......................... 67 4.2 Â– KW Models for Square Watershed ........................................................................ 69 4.3 Â– Conversion of Diverging Plan into KW Rectangular Plan ..................................... 72 4.4 Â– Watershed Factor vs. KW Shape Factor Â– Trigonometry Sine Function................. 75 4.5 Â– Shape Factor vs. Interior Expansion Angle Â– Diverging Plane ............................... 75 4.6 Â– Conversion of Converging Plan into KW Rectangular Plane ................................. 78 4.7 Â– Radius Ratio vs. KW Shape Factor (15 degree) Â– Parabolic Function .................... 80 4.8 Â– Radius Ratio vs. KW Shape Factor (30 degree) Â– Parabolic Function .................... 80 4.9 Â– Radius Ratio vs. KW Shape Factor (90 degree) Â– Parabolic Function .................... 81 4.10 Â– Radius Ratio vs. KW Shape Factor (179 degree) Â– Parabolic Function ................ 81 5.1 Â– USGS Stream Network Location Exhibits ............................................................. 85 5.2 Â– Comparison of Laboratory and Numerical Solutions for KW Overland Flow Hydrographs under Rainfall Intensity of 78 mm/hr for 50 Seconds ....................... 87
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xiii 5.3 Â– Comparison of Laboratory and Numerical Solutions for KW Overland Flow Hydrographs under Rainfall Intensity of 115 mm/hr for 50 Seconds...................... 88 5.4 Â– Diverging KW Overland Flow Hydrograph with Various Expansion Angles ......... 89 5.5 Â– KW analytical solution vs FAN program outputs .................................................. 91 5.6 Â– NonDimensional Rising Hydrograph Analytical and Numerical Solution Comparison .......................................................................................................... 93 5.7 Â– NonDimensional Recession Hydrograph Analytical and Numerical Solution Comparison .......................................................................................................... 94 5.8 Â– Topographical Layout for Watershed W2 ............................................................ 97 5.9 Â– Topographical Layout for Watershed W6 ............................................................ 98 5.10 Â– W2 Watershed Surface Runoff Hydrograph Prediction and Comparison ............ 99 5.11 Â– W6 Watershed Surface Runoff Hydrograph Prediction and Comparison .......... 100 5.12 Â– Converging KW Rising Portion Hydrograph Comaprison ( a = 0.1) ................... 101 5.13 Â– Converging KW Rising Portion Hydrograph Comaprison (Interior Angle = 104 degrees, a = 0.5) ............................................................................................... 102 5.14 Â– Converging KW Rising Portion Hydrograph Comaprison (Interior Angle = 104 degrees, a = 0.8) ............................................................................................... 102 5.15 Â– Comparison of Laboratory, Analytical and Numerical Solutions for KW Flows under Rainfall Intensity of 98 mm/hr for 10 minutes ........................................ 104 5.16 Â– W2 Watershed Predictions and Field Data Comparison.................................... 109 5.17 Â– W6 Watershed Predictions and Field Data Comparison.................................... 110
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1 1. Introduction 1.1 Urbanization Impacts to Alluvial Fan Stimulated by overall economic growth, population in the western United States has increased significantly since the 1990s. The 2010 Census reported a 9.7 percent growth in the countryÂ’s overall population from 2000, however, regional growth was 14.3 percent in the South and 13.8 percent for the West. Nevada grew by 35.1 percent making it the fastest growing state followed by Arizona, which grew by 24.6 percent (Figure 1.1). Major growth occurred in metropolitan areas such as Las Vegas, Nevada (over 40 percent), and countywide such as Maricopa County, Arizona, which accounted for 59.1 percent of the stateÂ’s growth (Figure 1.2) (U.S. Census Bureau, 2011). Figure 1.1 Â– Population Percent Change By State (2010 Census Bureau)
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2 Figure 1.2 Population Percent Change By County (2010 Census Bureau) In response to the rapid increase in population, urbanization spread onto more challenging terrain throughout the Southwest as desirable, level land became less available. Development encroached onto hillsides (Figure 1.3) and floodplain areas known as alluvial fans (Figure 1.4).
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3 Figure 1.3 Â– Example of Hillside Development Scottsdale, Arizona (Google Image, 2012) Figure 1.4 Â– Example of Urban Development Encroaching onto Alluvial Fans, Las Vegas, Nevada (2008 Las Vegas Valley Master Plan Update, CCRFCD)
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4 In the semiarid Southwest where valley land can be soft and saline and mountain hills steep and rocky, alluvial fans provide attractive development sites. The relatively gentle slopes, good drainage, and wellgraded material composites can invite transportation, agriculture, and urban development uses (Anstey 1965). When the alluvial fan is located near metropolitan areas such as Las Vegas, Nevada, Phoenix, Arizona, and Las Cruces, New Mexico, rapid growth encourages development to encroach onto alluvial fans. Even with intensified development pressure, some local jurisdictions such as the City of Las Vegas, limit site disturbance to 50 percent when the longitudinal slope is greater than 15 percent due to planning, public safety, and construction concerns (City of Las Vegas 2011). Conversely, some alluvial fans are concentrated in areas where infrastructure is not feasible and is, therefore, unlikely to be developed. For instance, over 70 percent of Death Valley, California, is covered by natural alluvial fans (Figure 1.5). Figure 1.5 Â– Alluvial Fan Formation Â– Death Valley, California (Geomorphology of Desert Environments, 2009)
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5 1.2 Alluvial Fan Location Alluvial fan landform is commonly found along the base of mountain fronts in the western states of Washington, Oregon, California, Idaho, Nevada, Arizona, New Mexico, Utah, Colorado, Montana, and Wyoming. It was estimated that approximately 30 percent of the landforms in the semiarid southwestern United States (Figure 1.6) consists of alluvial fans (Anstey 1965). Figure 1.6 Â– Alluvial Fan Geographic Locations (USGS, 2000) The average longitudinal slope of alluvial fans exhibits a wide range, from 2 to 70 percent, with most between 2 and 36 percent (Blair and McPherson 2009). Typically, the crosssurface slopes on fans vary from 1 to 2 percent for fans where sediment and water production in the watershed is relatively low. This type of alluvial fan is commonly found in Boise, Idaho (Figure 1.7).
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6 Figure 1.7 Â– Alluvial Fan Â– Boise, Idaho (Google Image, 2012) Fan slopes ranging from 20 to 30 percent, primarily formed by debris flow with approximately 55 percent sediment concentration, are best represented by Glenwood Springs, Colorado (FEMA 1981) (Figure 1.8).
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7 Figure 1.8 Â– Alluvial Fan Â– Glenwood Springs, Colorado (Google Image, 2012) 1.3 Alluvial Landform Descriptions and General Characteristics Alluvial fans are aggradational sedimentary deposits shaped like a cone segment radiating downslope from a point where a channel emerges from a mountainous catchment (Figure 1.9) (Drew 1873, Bull 1977)
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8 Figure 1.9 Â– Alluvial Fan Formation (Google Image, 2012) Three types of alluvial landforms are often involved in flood flow studies, yet they have strikingly different flood behavior. All three landforms develop at the base of steep, highly erodible mountain masses. Upstream of the valley openings are subjected to high intensity, short duration rainfall events (FEMA 1981). Such rainfall events generate runoff with sufficient velocity to pick up sediment as it travels through ravines and channels, and transports it to the canyon opening. Once it leaves its steep mountainous watercourse, the runoff spreads out, slows down, and deposits its sediment load as it heads toward the valley floor.
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9 The following is a summary describing the differences among the three alluvial landforms (Blair and McPherson 2009, FEMA 1981): Alluvial Fans As the sedimentladen flood flow leaves the confinement of the ravine walls, the flow spreads out and becomes shallower. The majority of the suspended sediment settles forming a cone at the mountain front. Lines of maximum topographic slope radiate away from the apex of the fan and terminate at the valley floor where the original valley slope resumes domination. Fan slopes depend on factors such as the size of the sediment particles and the sediment concentration in the runoff of the upstream tributary area. Heavier, larger sediment drop out quickly with smaller and finer particles depositing farther down the slope. Alluvial Aprons Often a series of fans form along the front of steep mountain ranges where numerous small watersheds are drained by individual streams. As the fans expand out on the valley floor, the toes of the fan merge into an alluvial apron. This apron area is characterized by nearly linear contours and a series of parallel ravines, or arroyos, which drain the aprons. Washes Washes are typically long, narrow formations with contours perpendicular to the confined canyon walls. They are mostly found in the section of ravine immediately connected to alluvial apron where the channel draining the mountain watershed remains confined until it connects to a large river. Figure 1.10 presents these landforms in the northern part of Las Vegas Valley, Nevada, to illustrate the major features of the alluvial floodplain.
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10 Because of obvious differences in morphology, flood flow behaviors on the three landforms are very different. Flood flows in a wash are confined by canyon walls and the path of flow tends to be stable and predictable. If the wash is wide, braiding and meandering of the base flows might still occur but it should be within the floodplain limit. Flooding on an alluvial apron is generally limited to the arroyos that drain the apron. The characteristics of flow in the arroyos depend on local drainage and on the discharges from the upstream alluvial fans. On alluvial fans, overland flows are distributed as a twodimensional sheet flow under a storm event (Mukhopadhyay et. al. 2003). The flow pattern is sensitive to the radius of fan area, the angle of apex, the longitudinal slope, and fan surface roughness. The overland flow may be gradually concentrated through a converging plane or spread out more over a diverging plane. The sudden expansion or contraction of the fan width does not create a spatiallyvaried Kinematic Wave (KW) flow that can be solved by conventional KW solutions, nor does the numerical scheme, because the current KW solutions are only valid for a rectangular plane formation.
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11 Figure 1.10 Alluvial Plane Formation (USGS, 2000)
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12 Overland flow on a fan surface is unsteady due to changes in rainfall distribution, and with each storm event, the shallow gullies and washes can change flow paths and patterns. The flow converging and diverging processes alter the distribution of surface runoff on a fan area. When transportation infrastructure crosses an alluvial fan, a culvert is often proposed to convey storm runoff. However, without the proper governing equations of motion for both converging and diverging KW flows, the impact to a drainage crossing along a transportation alignment on the natural alluvial floodplain cannot be quantified. Currently, the predictions of overland flows on alluvial fans rely on a probabilistic estimation. 1.4 FEMA Flood Flow Approach for Alluvial Fans Flood hazards are frequently underestimated in the semiarid Southwest United States (FEMA 1981). The semiarid Southwest area includes Southeast California, Nevada, Arizona, New Mexico, and Utah (NOAA 2011). These areas often experience intense rainfall and subsequent flash flooding. A special report prepared for Clark County, Nevada, documented 184 different flooding events resulting in damages to private properties and public facilities from 1905 to 1975 (U.S. Soil Conservation Service 1975). While flooding can and has occurred in almost every month of the year, the most damaging storms typically occur in the form of convective storms (Figure 1.11) during the warm months of July through September (NOAA 2011). During these summer months, moist unstable air from the Gulf of Mexico is forced rapidly upward by hot air currents, which causes severe thunderstorms with intense rainfall over a very short duration. For example, a powerful Â“gulf surgeÂ” brought abundant moisture into Clark
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13 County, Nevada, in August 2013. There were numerous isolated, locally intense showers throughout the southern Nevada area on August 23 rd however, rainfall in the northwest part of Las Vegas Valley fell during the afternoon of August 25 th During this event, ten rain gages maintained by Clark County Regional Flood Control District (District) reported 1.2 to 4.1 inches of rain fell over about four hours with intensities exceeding the design rainfall standard adopted by the District and local governmental entities. The average total annual rainfall depth for Las Vegas is 4.19 inches according to National Weather Service records. Figure 1.11 Â– Temporal Distribution Regions (NOAA, 2011) When such intense storms occur on steep mountain terrain and alluvial fantype desert slopes, surface runoff accumulates and concentrates quickly. Storm runoff often
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14 demonstrates high flow velocities and unpredictable flow patterns across alluvial fan surfaces making advance warning efforts difficult to provide adequate response time. These flood events also have the capacity to transport a considerable volume of sediment resulting in erosion and deposition in various areas (FEMA 1989). Acknowledging the complexity of flooding problems on alluvial fans, Federal Emergency Management Agency (FEMA) developed a stochastic approach to identify flood hazards and flood boundaries on an alluvial fan (Anon 1981, Dawdy 1979). This approach imposes a conditional probability using logPerson Type III distribution (Figure 1.12), and is adjusted by the probability of the flow path being developed at a specific location on an alluvial fan. This approach later became the foundation of the FAN computer model (1990) developed by FEMA to assist the flood hazard zone mapping process to establish a National Flood Insurance Program (NFIP) on this type of landform (FEMA 1989, 2000). This approach solves for the 1 percent chance of flood occurrence with the basic probability function in combination with the conditional probability function to compute the width of the area subject to alluvial fan flooding for various flow depths and velocities. The same mathematic approach was used by U.S. Army Corps of Engineers (USACE) to analyze the flood risk during levee failure while dealing directly with the uncertainties inherent in such an occurrence (USACE 1993).
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15 Figure 1.12 Â– Diagram of the Pearson System showing distributions of Types I, III, VI, V, and IV in terms of 1 (squared skewness) and 2 (traditional kurtosis) In the FEMA approach to assess flood hazards on alluvial fans, the stochastic procedure was derived based on the following assumptions: The peak flow frequency must be known at the fan apex. At any distance below the apex, the path of the wash (channel) will be developed with a chance that is proportional to the ratio of wash width to total width of the fan area. The flood flow is conveyed at critical depth. The widthtodepth ratio of the wash is approximately 200.
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16 These assumptions have been applied to calculations of conditional risk of flood flow to establish FEMAÂ’s NFIP on alluvial fans. Although the methodology has been criticized since it was first established in 1979, this procedure is the officially acceptable method for most FEMA floodplain studies on alluvial fans (Burkham 1988, McGinn 1979, French 1992, Fuller 1990, 2011). Even with minor reformations to include channel deviation from the medial radial line (Figure 1.13) along the fan area for a better calculation procedure and prediction (French 1992), the overall methodology is still heavily relying on statistical analysis. Figure 1.13 Â– Isometric view (French, 1992) Since its creation in 1990, FEMAÂ’s FAN computer model has been a key component to define and delineate flood hazard boundaries on alluvial fan landforms. Its use was mandated by numerous communities in southern California. However, the flood hazard zones published for alluvial fan areas near Scottsdale, Arizona, were criticized as being too conservative when compared to a twodimensional flow simulation such as the FLO2D computer model (Fuller 2012). But from a regulatory standpoint, the current FEMA approach does provide basic information for flood protection and urban planning purposes (French 1993). It was long recognized that a rainfallrunoff approach should be explored to delineate flood hazards on an alluvial fan (National Research Council 1996).
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17 1.5 Objective An alluvial fan may be composed of a diverging plane downstream of the apex, a rectangular plane along a collection ditch, a converging plane immediately upstream of a culvert, and a diverging plane immediately downstream of a culvert (Figures 1.10 and 1.14). These flows may form a cascading plane system to simulate the generation of overland flows. Figure 1.14 Â– Typical Engineering Application Geometry Layout on an Alluvial Fan The KW approach for overland flow is a simplified solution for the Dynamic Wave (DW) methodology. KW governing equations consist of flow continuity and flow momentum in terms of a rating curve relationship between flow depth and flow discharge,
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18 therefore, they are sensitive to the geometry of overland flow plane. In this research work, the objective is to explore new governing equations and their analytical solutions, as well as numerical schemes to solve diverging, converging, and rectangular plane KW flows. The known inputs are the rainfall excess on the fan area and the flow rate at the fan apex. The process is to generate overland flow rates and the outputs are the flood flow hydrograph and floodplain boundary limits for the fan area. It is proposed that the KW approach be expanded to model an alluvial fan as a hydrologic system. To achieve this goal, this research proposes to: Conduct a literature review of Kinematic Wave theory. Derive the governing equations for converging, diverging, and rectangular plane KW flows using a polar coordinate system to better represent the shape of alluvial fan. Expand the KW numerical simulation scheme from a single plane to a cascading system to simulate overland flow on an alluvial fan area. Verify the theoretical derivation with published laboratory or field data. Develop a deterministic approach to quantify flood flow magnitudes and delineate floodplain boundaries to improve the current FEMA approach on alluvial fan flood studies. Develop a green approach to design crossing culverts to preserve alluvial fan flow patterns upstream and downstream of the culvert under a highway. Application of KW cascading plane methodology is directly related to the design of collector ditches and swales along highways traversing alluvial fan areas, and culverts
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19 conveying surface runoff under these highways. This study will greatly improve the current procedure for FEMA flood flow studies and reduce highway impacts on the continuity of alluvial fan flows.
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20 2. Review Of Kinematic Wave Overland flow 2.1 Introduction The SaintVenant equations, derived by BarrÂ’e de SaintVenant (1871), were developed to express onedimensional unsteady open channel flow by fulfilling the continuity and momentum principles simultaneously. There were various simplified forms, each expressing a onedimensional distributed routing model. The momentum equation, also named Dynamic Wave Flow Model, consisted of local and convective acceleration terms as well as pressure, gravity, and friction force terms. The equations were then simplified to Diffusion Wave Flow Model when flow accelerations were ignored. It was further reduced to Kinematic Wave Flow Model when flow friction was balanced with gravitational force (Lighthill and Whitham, 1955). Woolhiser and Liggett (1967) applied Kinematic Wave to describe overland flow by finitedifference integration of the characteristic equations. Chen (1970) recommended the first analytical solution to represent overland flow on an irrigation porous surface. Guo (1998) reported the analytical solution for overland flow on a pervious surface under a uniform rainfall to represent the soil infiltration effect on porous ground. 2.2 Basic Principles DaluzVieira (1983) and Yen (1973) suggested that shallow wave propagation in open channel or overland flow could be best described by the complete SaintVenant equations by applying the following assumptions: Gradually varied flow
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21 Onedimensional flow Flow is incompressible and of constant density Uniform flow velocity exists within cross sections ManningÂ’s equation is applicable Hydrostatic pressure is evenly distributed through cross sections These assumptions were shown to be satisfactory for overland flow in most of the cases (Lighthill and Whitham, 1955). Open channel flow is expressed in three dimensions in nature. As the longitudinal flow velocity governs the flow characteristics, open channel flow can be simplified and expressed in onedimensional format. And the continuity and momentum principles for unsteady nonuniform open channel flow can be described as: q x Q t A (2.1) 0 1 1 0 2 f S S g x y g A Q x A t Q A (2.2) Where A = flow area in [L 2 ], Q = channel flow in [L 3 /T], q = lateral inflow per unit length of the reach in [L 2 /T], t = time in [T], x = distance measured in flow direction in [T], g = specific gravity in [L/T 2 ], S 0 = bottom slope, and S f = friction slope. Eq. (2.1) is suitable for all channel geometries. Aided with channel geometry, Eq. (2.2) can be rewritten as: 0 0 f S S g x y g x V V t V (2.3) Where V = channel cross section velocity in [L/T]. Eq. (2.3) describes the diffusion wave in the channel. It contains local acceleration, convective acceleration, backwater effect,
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22 flow friction, and gravitational force. Assuming that the convective and local acceleration are negligible compared to friction, gravity, and pressure terms, Eq. (2.3) is further reduced to: f S S x y 0 (2.4) Eq. (2.4) is named Diffusion Wave Model; it is also called the noninertial model. Without further effect from the upstream and downstream sections, open channel flow can be viewed as a uniform flow that is mainly dominated by gravitational and friction forces. Assuming a balance between gravitational and friction forces, Eq. (2.4) can be reduced to: f o S S (2.5) The correlation demonstrated in Eq. (2.5) shows a singlevalue rating curve relationship between flow rate and flow depth in Kinematic Wave flow. It can be presented as: m y Q (2.6) Where and m are constants that can be determined by the channel cross section and channel roughness. Several empirical formulas have been developed and are summarized as followed: 0 S n and 3 5 m for ManningÂ’s formula (2.7) 0 S C and 2 3 m for ChezyÂ’s formula (2.8) 0 8 S f g and 2 3 m for Darcy Â– WeisbachÂ’s formula (2.9)
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23 Where n = ManningÂ’s roughness, = 1.486 for English units, C = ChezyÂ’s conductivity coefficient, and f = friction factor. The value of m varies between 3/2 and 5/3 for general practices (HEC, 1993). Without any backwater effect, the solution of Eq. (2.6) offers an instantaneous uniform flow pattern at a specific time. Additionally, the normal flow depth solved by ManningÂ’s equation is a special case of the Kinematic Wave model when the flow rate is constant. 2.3 Kinematic Wave Application Limit One of the simplest wave forms, a monoclinal flood wave, can be generated by a sudden lift of a sluice gate (Figure 2.1). This phenomenon can represent the sudden increase in channel runoff in the urban environment when responding to rainfall. Figure 2.1 Â– Monoclinal Wave Movement
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24 Since the wave height is insignificant, this type of unsteady open channel flow can be converted to a steady state flow by introducing a negative wave speed, V w to the entire flow field. The steady flow system continuity principle is shown as: 1 1 2 2 A V V A V V w w (2.10) Rearranging Eq. 2.10 yields: 1 2 1 2 1 2 1 1 2 2 A A Q Q A A V A V A V w (2.11) As demonstrated in Figure 2.2, a monoclinal flood wave can be illustrated as a singlevalue rating curve. Figure 2.2 Â– Monoclinal Wave Rating Curve
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25 To make a comparison between kinematic and dynamic waves, Eq. (2.11) is simplified by applying it to a wide rectangular channel. With a constant bottom width, B, Eq. (2.11) is presented as: dy dQ B V w 1 (2.12) Combining Eq. (2.6) and Eq. (2.12) yields the following: 1 m w y B m V (2.13) Considering the principle of continuity, the cross section velocity, V can be expressed as: 1 m y B By Q V (2.14) Combining Eq. (2.13) and (2.14) yields: mV V w (2.15) Eq. (2.15) describes the flow velocity and KW speed relationship when the backwater effect is ignored. Comparably, the dynamic wave speed, V d in a wide rectangular channel is expressed as: gY V V d (2.16) The Â“Â” expresses the wave direction and speed downstream and upstream, respectively. When the backwater effect is irrelevant, dynamic wave and kinematic wave coexist in the flood channel because, numerically, these two wave speeds share the same hydraulic characteristics. As presented in Eq. (2.2), the Dynamic Wave model takes the local and convective flow acceleration into consideration. By considering wave speed, gY
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26 Dynamic Wave moves faster than Kinematic Wave thereby presenting an application limit between the use of kinematic and dynamic wave models. d w V V Combining Eq. (2.15) and (2.16) results in: 1 1 m gY V F r (2.17) For ManningÂ’s Equation, 3 5 m 5 1 r F (2.18) For ChezyÂ’s and DarcyWeisbach equations, 2 3 m 0 2 r F (2.19) Vieira (1983) combined his work with Liggett and Woolhiser (1967), Overton and Meadows (1976), and Morris and Woolhiser (1980) by applying nondimensional Kinematic Wave number, K and Froude number, F r as the coordinate to distinguish the relationship among Dynamic Wave, Diffusion Wave, and Kinematic Wave (Figure 2.3). Where 2 0 r YF L S K (2.20)
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27 Figure 2.3 Â– Dynamic Wave, Diffusion Wave and Kinematic Wave Applicability Zone It is concluded that the Kinematic Wave model is applicable when the flow Froude number is less than 2 (Badient and Huber 1992) 2.4 Overland Flow on a Rectangular Pervious Plane Based on the assumption that the gravitational force offsets the friction force, then it follows that the longitudinal slope is equal to the friction slope. Thus, the Dynamic Wave equation for open channel flow under uniform rainfall intensity and decayed infiltration losses is reduced to the kinematic wave equation for unitwidth flow (Yen and Chow, 1974). The resulting equation is: e i x q t y (2.21) Where i e = excess rainfall intensity in [L/T] Considering infiltration losses, rainfall excess can be expressed as (Horton, 1938):
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28 t f i t i e kt c c e f f f t f 0 (2.22) In which i = rainfall intensity in [L/T], f(t) = infiltration rate at time t in [L/T], f c = final infiltration rate in [L/T], f 0 = initial infiltration rate in [L/T], and k = soil infiltration decay constant in [1/T]. The rating curve for an overland unitwidth overland flow is expressed as: m y q (2.23) Taking the first derivative of Eq. (2.23) with respect to x yields: x y my x q m 1 (2.24) Substituting Eq. (2.24) into Eq. (2.21), the total derivative for flow depth is derived as: e m i x y my t y 1 (2.25) Eq. (2.25) can be converted to: e i x y u t y (2.26) Eq. (2.26) is the total derivative of the flow depth and is presented as: e i dt dy (2.27) Similarly, taking the partial derivative of Eq. (2.23) with respect to t and rearranging the terms yields: 1 m my t q t y (2.28) Substituting Eq. (2.28) into (2.21), the total derivative for flow rate is concluded as:
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29 e m i x q my t q 1 (2.29) Eq. (2.29) is rewritten as a total derivative of flow rate: e w i V dt dq (2.30) Where the KW speed is defined as: 1 m w my dt dx V (2.31) For a unitwidth flow, the flow velocity is calculated as: 1 m y y q V (2.32) Kinematic wave dominates in the propagation of a flood wave when the local and convective flow accelerations are insignificant. This implies that the friction slope ( S f ) is approximately equal to bed slope ( S 0 ). KW model can be described as a singlevalue rating curve similar to the normal flow developed in a prismatic channel. The rating curve represents the mathematical relationship between flow rate and flow depth. 2.5 Overland Flow Hydrograph Analysis KW overland flow takes place when the dynamic term in the Momentum Equation is negligible. This dominance implies that the friction slope associated with the KW flow is equivalent to the ground slope. This assumption is justified when backwater effect is not significant. The KW overland flow model is a onedimensional (1D) unitwidth flow approach. This approach demands a conversion of the catchment into its rectangular sloping plane (Guo and Urbanos, 2009).
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30 As presented on Figure 2.4, on a rectangular KW plane, the KW vertical water surface profile represents the surface detention volume which varies with respect to time. Figure 2.4 Â– Onedimensional overland unsteady flow profile As depicted in Figure 2.4, the initial condition for KW overland flow shall be a dry bed condition. It implies that no flow is on the surface before the rainfall event starts. 0 0 X y X t y (2.33) 0 0 X q X t q (2.34) Where t = time in [T] and X = station for flow calculation in [L]. The boundary conditions at the upstream end of the overland flow plane are: 0 0 t y (2.35) 0 0 t q (2.36) The boundary conditions at the downstream end of the overflow plane can be set at critical or normal flow depth. In this work, the downstream boundary condition is assumed to be: x L t y L t y (2.37)
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31 x L t q L t q (2.38) Where L = total length of the overflow in [L], and x is the incremental along the surface plane. Based on boundary and initial conditions, Eq. (2.27) can be integrated and yields: t t t e y y y dt i dy 0 0 (2.39) t i y e (2.40) According to the KW speed described in Eq. (2.31), the KW travel length is expressed as: X x x t t t m e t t t m dt t i m dt my dx 0 0 1 0 1 (2.41) m m e t i x 1 (2.42) Aided with Eq. (2.40), Eq. (2.42) is converted to: m e xi y 1 (2.43) As the kinematic wave propagates from the very upstream where x = 0 to the outlet at x = L the flow travel time is termed as the time of equilibrium, T e of the catchment. Thus, Eq. (2.42) can be rewritten as: m m e e i L T 1 1 (2.44) Combined with Eq. (2.23) and (2.40), the flow depth and discharge at T e are (Wooding, 1965): e e e T i y (2.45) L i T i y q e m e m e m e e (2.46)
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32 Using ManningÂ’s Equation with m = 5/3, Eq. (2.44) is expressed as: 5 3 67 0 0 e e i S NL T (2.47) Using ChezyÂ’s Equation with m = 3/2, Eq. (2.44) becomes: 3 2 0 e e i S C L T (2.48) Using DarcyWeisbachÂ’s Equation with m = 3/2, Eq. (2.44) is expressed as: 3 2 0 8 e e i S g L f T (2.49) On the KW rectangular plane, the unitwidth peak flow, q e occurs at x = L The peak flow, Q p on the KW plane with a width, B and a length, L can be expressed as: CiA BL i Q e p (2.50) It is concluded that the Rational Method is a special case of the KW method. The rainfall intensity, i e in the Rational Method is the rainfall average rate over the time of equilibrium of the catchment. The value of runoff coefficient, C, represents the percentage of impervious area within the catchment. Eq. (2.50) suggests that the KW approach provides a linear solution between peak flow and tributary area. To be conservative, it is important to ensure that the design rainfall duration applied in the stormwater simulation is longer than the T e of the catchment area. The peak runoff will reach its maximum potential value to reflect the climatologic and hydrologic conditions imposed to the catchment. Under a long duration rainfall event, the overland flow will reach its equilibrium state when the surface discharge from the
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33 catchment area equals the rate of the rainfall precipitated. As the rainfall event ceases, surface runoff depth starts tapering off. The overland runoff hydrograph is consisted with 3 segments Â– rising, peaking and recession portions (Figure 2.5). The rising portion is the section before reaching time of equilibrium ( T e ), the peaking portion is the section between T e and the time the rainfall stops ( T d ), and the recession portion is when the accumulated runoff depth begins to taper off (Guo 2006a, 2006b). Figure 2.5 Â– Dimensionless Overland Flow Hydrograph Applying Eq. (2.31), Eq. (2.45), and Eq. (2.46) to a unitwidth catchment under a long, uniform rainfall excess, i e the overland flow hydrograph is presented in Figure 2.5. All flow depths and times are normalized by equilibrium flow rate, q e and time, T e respectively. (a) Rising Hydrograph Â–
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34 At the elapsed time, t the flow depth, y and the associated travel length, x can be calculated by Eq. (2.31). On the rising hydrograph (Figure 2.6) the elapsed time, t is normalized as: e T t T (2.51) And the flow depth will travel through a distance x and can be expressed as: t V x (2.52) Figure 2.6 Â– Rising Water Surface Profile when t < T e Where T = dimensionless elapsed time. Aided by Eq. (2.46) and Eq. (2.51), the flow depth at the rising portion of the hydrograph can be described as: T y t y T y e (2.53) Where y = the dimensionless flow depth. Aided by Eq. (2.23), Eq. (2.46) and Eq. (2.51), the normalized unit flow rate per unit width, q is derived as: m e T q t q T q (2.54)
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35 Eq. (2.53) and Eq. (2.54) portrait that the flow depth and flow rate on the rising hydrograph are increased along the 45degree line between zero and unity. This rising runoff hydrograph is also discussed elsewhere (Eagleson 1970, Wooding 1965). (b) Peaking Hydrograph During the peaking portion of the runoff hydrograph, the inflow and outflow volumes are balanced. As a result, the normalized peak flow depth and flow rate during the peaking period are expressed as: 0 1 T y (2.55) 0 1 T q (2.56) (c) Recession Hydrograph After the time of equilibrium, the equilibrium water surface profile is formed and the surface detention is reached to its maximum. As soon as the rain stops, the flow depths under the equilibrium water surface profile begins to propagate toward to the outfall location as presented in Figure 2.7. Figure 2.7 Â– Recession Hydrograph when t > T d
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36 As mentioned previously, the flow depth, y will travel through a distance from x to L during the time period from T d to t The KW movement must satisfy the relationship as: d w T t V X L for d T t (2.57) Inserting Eq. (2.31) into Eq. (2.57) and normalizing the end results by T e and y e it is concluded: e d m T T Y m mY T 1 1 1 (2.58) And e y Y Y For a given t the corresponding flow depth in Eq. (2.58) can be solved iteratively. The recession hydrograph is formulated as: Y T y (2.59) m Y T q (2.60) The recession hydrograph ends when the elapsed time is long enough for the flow depth to vanish. 2.6 Overland Flow Travel Time on a Pervious Surface As described in Section 2.3, rainfall excess on a previous surface is subject to infiltration losses that can be described by a decay curve. After subtracting the infiltration loss, the uniform rainfall distribution becomes an exponential decay curve which is nonuniform condition. The kinematic wave travel time through the catchment is no longer constant and is ranging between time of concentration, T c and time of
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37 equilibrium, T e Time of concentration, T c is defined as the time for surface runoff to travel through waterway and reaches outlet location. At the beginning of the rainfall event, a longer time to produce surface runoff is expected due to higher soil infiltration rates in the early stages. The higher the infiltration rate is, the less surface runoff depth. As a result, the flow travel time is gradually decreased to T e (Guo 1998). For a wide overland flow formation, m = 2 is adopted (Guo 1998, Guo et. al. 2012). Eq. (2.23) is expressed as: 2 y q (2.61) Inserting Eq. (2.61) into Eq. (2.27) and Eq. (2.31), it is concluded: kt c c e f f f i dt dy 0 (2.62) y dt dx 2 (2.63) Under a long duration rainfall event, the pervious surface overland flow runoff hydrograph can also be categorized into three segments Â– rising, peaking and recession hydrograph. (a) Rising Portion Hydrograph When the rainfall depth exceeds hydrologic losses, surface runoff occurs. As shown on Figure 2.8, the rising portion hydrograph begins from t = T s to t = ( T s + T c ) where T s is termed ponding time which is the period of time for the soil infiltration rate to be decayed till the surface runoff occurs. The rising hydrograph is derived by integrating Eq. (2.62) from t = T s to t = ( T s + T c ) as: s kT kt c s c e e k f f T t f i y 0 (2.64)
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38 Parallel to Eq. (2.64), the KW flow will travel a distance, x Integrating Eq. (2.63) yields: kt kT s c s c e e T t k k f f T t f i x s 1 2 2 2 0 2 for 0 x L (65) Eq. (2.64) and Eq. (2.65) provide the direct solution to define the water surface profile, ( x y ), for t ( T s + T c ) where T c is the time of concentration of the catchment. Figure 2.8 Â– KW Integration Domain Illustration for Rainfall Excess considering Soil Infiltration (b) Peaking Portion Hydrograph After time ( T s + T c ), the flow reaches the peaking hydrograph because the entire catchment has become contributing to the outflow. The peaking process ends at
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39 time, T d when the rain ceases. Substituting t = ( T s + T c ) and x = L into Eq. (2.64) and Eq. (2.65) yields: 1 0 c s kT kT c c c c e e k f f T f i y (2.66) c s kT c kT c c c e kT e k f f T f i L 1 2 2 2 0 2 (2.67) Where y c = flow depth in [L] at t = ( T s + T c ). For the given distance, L the value of T c in Eq. (2.67) can be iteratively solved. Having T c known, the corresponding flow depth, y c can be solved by Eq. (2.66). A KW speed is sensitive to the rainfall access on the previous surface. As time goes on, the soil infiltration rate continues decaying. Consequently, the higher the rainfall excess is, the faster the KW flow travels. At t > ( T s + T c ), the flow depth is generated by the net rainfall amount from ( t T v ) to t where T v is the travel time for the KW flow to go through the flow length, L as illustrated in Figure 2.8. Integrating Eq. (2.62) and Eq. (2.63) over the domains from x = 0 to x = L and t = ( t T v ) to time, t yields: 1 0 v v kT T t k c v c e e k f f T f i t y for ( T s + T c ) t T d (2.68) v v kT v T t k c v c e kT e k f f T f i L 1 2 2 2 0 2 (2.69) For a given time t and t > T s + T c its travel time T v can be iteratively solved by Eq. (2.68) and Eq. (2.69). The maximum overland flow depth, y max on a pervious surface occurs at t = T d when the rain ceases.
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40 1 0 max v v d kT T T k c v c e e k f f T f i y (2.70) When time is long enough and the infiltration rate is reduced to its constant rate. As a result, Eq. (2.71) and Eq. (2.72) represent the equilibrium condition as: e e e c e T i T f i y (2.71) 2 2 2 2 2 e e e c T i T f i L (2.72) The peaking hydrograph varies from t = ( T s + T c ) to t = T d During the peaking process, the travel time is decreased from T c towards T e while the flow depth is increased from y c toward y e (c) Recession Portion Hydrograph The recession on the runoff hydrograph starts when the rain stops at t = T d During the recession process, the flow depth at the outlet can be formulated by integrating Eq. (2.62) with i = 0 as: t T kt c c y y d dt e f f f dy 0 max (2.73) Eq. (2.73) is integrated as: d kT kt c d c e e k f f T t f y y 0 max for t > T d (2.74) At t > T d Eq. (2.74) provides a direct solution to the overland flow depth at the outlet.
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41 2.7 Closing Since 1950, the KW theory has been widely applied to simulate overland flows hydraulic characteristics under a uniform or nonuniform rainfall distribution. The semianalytical KW solutions were derived under the major assumption that the KW overland flows are generated from a rectangular plane. Obviously, this assumption has become a restriction to the application of the KW theory to engineering practices. In this study, the major effort is to expand the understanding of KW overland flow theory from a rectangular plane to a fan shape surface. This effort involves the derivation of different governing equations as well as the associated semianalytical and numerical solutions.
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42 3. Derivations of Governing Equations for Diverging and Converging Kinematic Wave Overland Flows 3.1 Introduction Under a severe storm event, overland flows generated from an alluvial fan area follow numerous flow paths that are characterized by networks of shallow grooves accompanied with high flow velocities due to steep gradients. Because of the complexity of surface hydrologic and hydraulic characteristics, the Federal Emergency Management Agency (FEMA) recommended a stochastic approach to delineate flood hazard zones on an alluvial fan because the pattern of overland flows is uncertain and unpredictable (FEMA 1989, 2000). With many uncertainties in predicting flood flow magnitudes and flood flow inundation boundaries, tasks for flood hazard predictions on alluvial fans has been a challenge. As reported, the complicated hydrologic and hydraulic processes on alluvial fans may be best modeled with the aid of the Kinematic Wave (KW) theory of overland flow (Mukhopadhyay, et. al. 2003). As described in Chapter 2, it was illustrated that the solution of the KW theory under a uniform rainfall on an impervious surface is well documented in many studies (Woodings 1965, Woolhiser and Liggett 1969). On a pervious surface, the travel of KW movement was verified from the time of concentration to the time of equilibrium. The solution of KW flow on a pervious surface was further modified to include HortonÂ’s formula for soil infiltration losses (Guo 1998). Both solutions were derived for the KW overland flow generated on an ideal unitwidth rectangular KW plane using the Cartesian coordinates.
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43 In this study, the diverging and converging KW flows on a fan shape plane can be better described by the polar coordinate systems using the radial distance, r and the polar angle, Such KW governing equations were first derived and discussed using the finite difference over a circular segment across an alluvial fan at a selected radius, then the semianalytical solution was attempted by integrating over the selected apex angle to predict the total overland flow generated from the entire fan area (Agiralioglu and Singh, 1980). This approach involves a double integration and aided with a binominal processes. The complicated methematical approach has become a barrier to achieve neither analytical nor numerical solutions (Singh and Agiralioglu 1981). In this study, a new approach was developed to derive the diverging and converging planes KW governing equations using the polar coordinate system. Both new governing equations is compared with the previous studies and will be further exaimed with laboratory data from different sources. 3.2 Kinematic Wave Overland Flow on a Diverging Plane In this study, the entire alluival fan area is treated as a shallow reservoir to derive the diverging KW governing equation. Based on the water volume balance among excess rainfall amount, inflow and outflow discharges, and the surface detention volume, a new governing equation can be derived for the diverging KW flow. As expected, this new governing equation shall provide the peak flow rate at the time of equilibrium for the purpose of floodplain delineations (Guo and Hsu, 2014). Furthermore, this research work will attemp to derive both analytical and numerical solutions for the entire overland
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44 runoff hydrograph including rising portion of the hydrograph and compare these solutions to a numerical solution. Figure 3.1 is a typical geometry of a diverging alluvial fan. A diverging fan area is described by internal or apex angle, and the radical distance, R The apex in Figure 3.1 is the origin of the polar coordinate system. The runoff starts from the apex and the spreads out as it flow downstream. Figure 3.1 Â– Diverging Surface and Parameters In this study, the KW flow on a diverging fan area is analyzed with an arc segment approach as presented on Figure 3.1. The water volume balance is determined by the inflow, outflow, and storage depth within the arc segment. According to the continuity principle, the water volume balanced within the arc segment is a summation of the following items:
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45 A t y t V r r R y y v v r R vy A i e (3.1) r r R A (3.2) Inserting Eq. (3.2) into Eq. (3.1) yields: r r R t y r R vy r R y v r vy r r R i e (3.3) Eq. (3.3) can be rearranged and shown as: t y r q r R q i e (3.4) vy q (3.5) Where i e = excess rainfall intensity in [L/T], A = diverging surface area in [L 2 ], R = radius for the diverging area in [L], r = location of the segment or representing the distance in [L], v = radial flow velocity in [L/T], y = flow depth in [L], q = unitwidth discharge in [L 2 /T], t = elapsed time in [T] and = angle of the apex of the diverging area. Eq. (3.4) symbolizes the unit width flow rate and the basic equation to define diverging overland flow through the continuity equation and KW approximations. Eq. (3.4) agrees with the previous study (Agiralioglu and Singh, 1980) using various symbols which is presented as: t t r h r t r Q r t r Q t r q , (3.6) In comparison with Eq. (3.4), q(r, t) = i e Q(r, t) = q h(r, t) = y and r = Rr Eq. (3.6) cannot be integrated directly from r = 0 to r = R, but can be approximated by applying a binomial function to include the apex expansion angle and radius of the fan area (Agiralioglu and Singh, 1980).
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46 Both Eq. (3.4) and Eq. (3.6) do not explicitly include the apex expansion angle. In fact, the apex expansion angle is implicitly involved in the calculation of the tributary area which is outlined from the locations from Rr to R as illustrated in Figure 3.2. The geometric parameters for the tributary fan area are described as: r R r r R R A 2 2 1 2 1 2 2 (3.7) R l r (3.8) l r = arc length for outflow released from the fan area in [L]. Considering that the tributary area functions as a shallow water reservoir, the rainfall excess is the inflow, and the outflow is the sheet flow released through the downstream arc length. Figure 3.2 Â– Fan Shape and Fan Area for KW Flow
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47 The balance of water volume for the fan area is summarized as: A t V t S q q out in (3.9) Aided with Eq. (3.7) and (3.8), Eq. (3.9) can be rearranged as: A dt dy qR A i e (3.10) Inserting Eq. (3.7) into Eq. (3.10) yields: dt dy r R r qR i e 2 2 1 (3.11) Let a R r (3.12) Inserting Eq. (3.12) into Eq. (3.11) yields: dt dy a r q i e 2 2 (3.13) Where a = location ratio for the upper boundary. Eq. (3.13) serves as the governing equation for KW overland flow on a diverging plane. If the tributary area starts from the apex of the diverging plane shape, then r = R or a = 1 and Eq. (3.13) can be further reduced to: R q i dt dy e 2 (3.14) Combined with Eq. (3.8) and Eq. (2.23), total discharge from the fan area, Q can be expressed as: R y qR ql Q m r (3.15) Eq. (3.15) describes the relationship between flow depth and unitwidth flow rate over the diverging surface area. Analytical or numerical integration of Eq. (3.14) can
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48 provide direct solutions to the flow depth of surface runoff on a fanshaped plane. Application of Eq. (3.14) to an existing alluvial fan with known geometry will satisfy FEMAÂ’s floodplain delineation procedure for flood hazard zone identification. Under a long duration rainfall event, the runoff flow continually increases to reach its peak flow rate. Under the equilibrium condition, the rainfall amount is balanced with the outflow volume from the alluvial fan area, or the surface detention remains unchanged. Under the equilibrium condition, dy/dt is equal to 0 in Eq. (3.14). The corresponding peak flow for a diverging alluvial fan is derived as: R i q e e 2 1 (3.16) 1 m e e y v (3.17) m e e e y R v R T 1 (3.18) 2 2 1 R i Q e e (3.19) Where q e = unitwidth peak flow in [L 2 /T], v e = peak flow velocity in [L/T] and T e = time to equilibrium in [T]. Eq. (3.19) agrees with the Rational method with a runoff coefficient of unity. When the rainfall event duration is shorter then T e it represents a short rainfall event, otherwise, it is treated as a long rainfall event.
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49 3.3 Rising and Recession Hydrograph for Kinematic Wave Overland Flow on a Diverging Plane Under a long uniform rainfall excess, the integration of Eq. (3.14) from t = 0 to t = T c and y = 0 to y = y e represents the rising hydrograph limb (Guo and Hsu, 2015a). dy R y i dt y y y m e T t t c 0 0 2 1 where 0 t T c (3.20) in which T c = time of concentration in [T]. The peaking hydrograph occurs between t = T c to t = T d in which T d = rainfall duration in [T] during peaking time, dy/dt = 0 in Eq. (3.14). Aided with Eq. (3.15), the peak flow rate, peak flow velocity, and time of concentration are derived as: e e e v R i y 2 (3.21) 1 m e e y v (3.22) Where y e = peak flow depth in [L] and v e = KW velocity. With Eq. (3.22), by the definition, the time of concentration is displayed as: m e e C y R v R T 1 (3.23) In practice, most hydraulic designs are directly related to sizing a structure to convey the peak flow. Eq. (3.21) and Eq. (3.23) provide the basic information for such hydraulic designs and can represent the rising hydrograph. It is well understood that KW speed varies with respect to flow depth. Aided with Eq. (3.22), Eq. (3.20) can be approximated with a constant KW speed as: y y y e e dy R y v i t 0 2 1 where 0 t T c (3.24)
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50 Let the integration variable be: y R v i Y e e 2 (3.25) Taking the first derivative of Eq. (3.25) and then substituting dY for dy yields: K y R i v Ln v R dY Y v R t e e e Y Y i Y e e ) 2 1 ( 2 1 2 (3.26) In which K = integration constant that is to be determined by the condition of 0 t T c The solution for Eq. (3.26) is then: c c e T t T t R t v e e e y y 1 1 2 2 1 1 (3.27) Similarly, recession hydrograph can be expressed by integrating Eq. (3.14) with the boundary condition of t T d which can be integrated from y = y e to y = 0. Eq. (3.20) can be rewritten as: dy R y i dt y y y m e t t T t e d 0 2 1 where t T d (3.28) Following the same integration procedure and aided with Eq. (3.25) yields: K y R i v Ln v R dY Y v R t e e e i Y Y e e ) 2 ( 2 1 2 0 (3.29) In which K = integration constant that is to be determined by the condition of t T d The solution for Eq. (3.29) is: c c e T t T t R t v e e e y y 1 1 2 2 (3.30)
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51 Eq. (3.30) represents the rising hydrograph as an exponential function that is leveled when time, t approaches, T c 3.4 Kinematic Wave Overland Flow on a Converging Plane Wherever a transportation alignement crossing an alluvial fan, excess flood water tends to innudate any sag points along the alignment (FEMA 2000). Assessing the level of flood protection required at sag point poses a challenge for designing a culvert crossing at these locations. In some of the engineering practices, levees are utilized to collect and direct overland flow toward a sag point culvert, and discharge to the downstream side of the transportation alignment. As illustrated in Figure 3.3, the locations of crossing culverts divide the transportation alignement is into segments with the levees serving as manmade drainage boundaries. Overland flows generated from the upper fan areas form a converging plane towards the entrance of the culvert (French 1992).
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52 Figure 3.3 Â– Typical Transportation Alignment with Culvert Crossing Layout on Alluvail Fans The amount of flow concentrating at a culvert depends on the geometry and size of the upstream converging tributary area. Once the peak flow rate has been determined per segment, levees and berms are often designed to guide overland flow to plung into the entrace pool at a crossing culvert (French 1992). This method has been implemented in the Great Basin area in Southwest US (French et. al. 1993). The primary uncertainty in assessing flood flows on a converging alluival fan area is directly related to the direction and magnitude of the flow. Overland flows on a converging surface are not a simple unitwidth flow relationship as suggested in WoodingÂ’s solution (Wooding 1965) for a retangular plane shape. Rather, the unitwidth flow rate increases as it moves downstream. Therefore, WoodingÂ’s KW solution is not applicable for converging overland flows.
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53 In this research, the entire converging tributary area is considered as a shallow reservoir, a similar approach described in the diverging plane governing equation derivitive (Guo and Hsu, 2015). At a time step, the flow generated from the tributary area is a water volume balance among rainfall excess, inflow, outflow, and change in storage volume. Using this approach, the governing equation can then be derived to describe the movement of converging overland flows. On a converging surface, excess rainfall produces overland flows on a sloping surface. Assuming there are no backwater effects, gravitational force is balanced with the friction force on the surface (Chow 1964). Figure 3.4 illustrates the general geometry of a converging plane in terms of elevation contours on the surface. Figure 3.4 Â– Water Balance in an Arc Segment within Converging Tributary Area
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54 Between two adjacent contour elevations, KW flow can be modeled using the inflow and outflow crossing the arc section. The mass balance relationship is derived and described as: A t y t V r r R y y v v r R vy A i e (3.31) vy q (3.32) r r R A (3.33) Where i e = excess rainfall intensity in [L/T], A = surface area in [L 2 ], v = flow velocity in [L/T], y = flow depth in [L], R = radius of converging plane in [L], r = location of KW flow on converging plane in [L], = interior angle of converging area, V = surface storage volume in [L 3 ], t = elapsed time in [T], and q = unitwidth discharge in [L 2 /T]. Combining Eq. (3.31), (3.32) and (3.33), and eliminating the double and triple incremental terms, the mass balance relationship is derived as: r vy y v t y r R vy i e (3.34) Combined Eq. (3.34) with Eq. (3.32) yields: r R q i r q t y e (3.35) Eq. (3.35) is identical to the flow equation reported in the previous study (Singh 1975). This equation does not involve the converging angle because it only represents the water volume balance within the arc area. However, the angle of the converging area is a key factor in the determination of the concentrated flow from the entire area. As suggested in previous work (Agiralioglu and Singh 1980), Eq. (3.35) can be further integrated using binominal function through the interior angle, as shown in Figure 3.4.
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55 Such an indirect solution is cumbersome for engineering applications because of lengthy numerical iterations. In this study, the entire converging surface area is taking into consideration. Assuming there is no inflow from the upper boundary, the storage volume difference between rainfall excess and outflow released from the entire converging plane can be directly related to the change in flow depth. Thus, the principle of continuity is rearranged as: A t y t V ql A i r e (3.36) As illustrated in Figure 3.5, the converging tributary area is defined by interior angle, and radius, R The flow crossing over a converging plane changes from one location to another. Therefore, in this study, the governing equation for converging flow is formed using the location of the outlet and interior angle, The total tributary area upstream of a specified location is represented as: r R r A 2 2 1 (3.37) r R l r (3.38) Where r is the location of the outlet to collect overland flows, and l r is the arc length at the selected location to collect overland flows.
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56 Figure 3.5 Â– Water Balance over Entire Converging Tributary Area Inserting Eq. (3.37) and Eq. (3.38) to Eq. (3.39) yields the governing equation for converging overland flow as: R r rR R r qR i dt dy e 2 1 2 (3.39) Eq. (3.39) can be further reduced as: Let R r a (3.40) Combining Eq. (3.40) into Eq. (3.39) yields: a r a q i dt dy e 2 1 2 (3.41)
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57 Where a = location ratio to define the selected location to collect overland flows. Eq. (3.41) represents the governing equation applicable to collection of converging KW flows at a location defined by the spatial ratio. Since rainfall duration is longer than the time of concentration, KW flow reaches an equilibrium condition. With dy/dt = 0 Eq. (3.41) is reduced to: a a r i q e e 1 2 2 (3.42) Aided with Eq. (3.35), the total flow accumulated is: a rR i ql Q e r e 2 2 (3.43) It is noted that Eq. (3.43) agrees with Rational method when r approaches R Eq. (3.43) can be further reduced to: R e e R A i R i Q 2 2 1 (3.44) 2 2 1 R A R (3.45) Where Q R = equilibrium discharge from the entire converging plane in [L 3 /T] and A R = entire converging surface area in [L 2 ] By definition, the time of concentration for the converging plane is the required travel time for the flow from the upper boundary to reach the outlet. Combining Eq. (2.23) and Eq. (3.41), the time of concentration can be expressed as: 1 m e c y r T (3.46) Eq. (3.46) is critically important to the application of the Rational Method. The converging plane is subject to a long rainfall event when the rainfall duration is longer
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58 than the time of concentration. Otherwise, the rainfall event is too short to cover the entire converging plane. 3.5 Rising and Recession Hydrograph for Kinematic Wave Overland Flow on a Converging Plane Under a long uniform rainfall excess, the rising hydrograph limb on a converging plane can be derived by following the same procedure stated in the Section 3.3. By integrating Eq. (3.38) from t = 0 to t = T c and y = 0 to y = y e yields: dy a r a q i dt y y y e T t t c 0 0 2 1 2 1 where 0 t T c (3.47) in which T c = time of concentration in [T]. The hydrograph is peaking from t = T c to t = T d where T d = rainfall duration in [T]. During the peaking time, dy/dt = 0 in Eq. (3.41). Aided with Eq. (3.42), the peak flow rate, peak flow velocity, and time of concentration are derived as: a v a r i y e e e 1 2 2 (3.48) 1 m e e y v (3.49) Where y e = peak flow depth in [L] and v e = KW velocity. Combining Eq. (3.46) and Eq. (3.49), the time of concentration can be described as: m e e m e c a v a r i r y r T 1 1 1 2 2 (3.50) Eq. (3.48) represents the rising hydrograph. It is well understood that KW speed varies with respect to flow depth. Using Eq. (3.48), Eq. (3.47) can be approximated with a constant KW speed as:
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59 y y y e e dy a r a y v i t 0 2 1 2 1 where 0 t T c (3.49) Let the integration variable be: y a r a v i Y e e 2 1 2 (3.50) Taking the first derivative of Eq. (3.50) and then substituting dY for dy yields: K y a r i a v Ln a v a r dY Y a v a r t e e e Y Y i Y e e 2 1 2 1 1 2 2 1 1 2 2 (3.51) In which K = integration constant that is to be determined by the condition of 0 t T c Attention must be paid as the converging plane approaches the point of singularity when r is approaching R which describes the entire converging plane. The solution for Eq. (3.51) for rising portion hydrograph is presented as: c c e T a t a T a t a a r t a v e e e y y 1 2 2 1 2 1 2 2 1 2 1 1 (3.52) Following the same procedure described in Section 3.3 by integrating Eq. (3.49) but replacing the boundary condition of t T d which can be integrated from y = y e to y = 0, the recession hydrograph can also be derived and can be presented as: c c e T a t a T a t a a r t a v e e e y y 1 2 2 1 2 1 2 2 1 2 (3.53) 3.6 Numerical Approach Many governing equations are not directly solvable analytically. However, with assistance from computer computational ability, numerical methods are developed using the finitedifference approach to approximate the discrete solutions. These numerical
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60 solutions require specific boundary conditions to describe the physical condition of the flow field. Therefore, successful numerical modeling is a combination of computation ability and a proper mathematical representation of the physical constraints. Roache (1976) stated that computational fluid dynamics is not just pure theoretical analysis. Instead, it is closer to an experiment in the sense that each particular calculation of a numerical simulation closely resembles the performance of a physical experiment. There are several approaches to transform a partial differential equation into a numerical arrangement to derive discrete solutions when the flow field is divided into mesh network. As demonstrated in Figure 3.6, the mesh cell is formed by x and t that represent a small increment in both distance and time, respectively. Each mesh cell is identified by the coordinates of ( i, j ) where i = i th distance and j = j th time step. The solutions at each mesh include the flow depth and the discharge at a given time. Generally, a linear relationship is assumed between two adjacent points when estimating the rate change in time and the spatial rate change between adjacent mesh cells. For better computational accuracy and numerical stability, it is necessary to use forward, backward, or central finite difference, according to the boundary and initial conditions.
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61 Figure 3.6 Â– KW Flow Field Mesh Network As presented in Figure 3.7, an implicit method utilizes all variables at the current time step to compute x/ t whereas an explicit method utilizes x from the previous time step to compute the convective term at the current time step. With only one unknown variable, the explicit method provides a direct solution. The implicit method will require iteration to obtain converged solution that meets the predetermined tolerance (Guo and Hinds, 2013).
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62 Figure 3.7 Â– Illustration of Implicit and Explicit Finite Difference Method For this study, Eq. (3.14) is converted into its central finite difference equation as: R t y t t y t i t t i n t t y t t y m m e e 2 2 (3.54) Similarly, Eq. (3.41) is shown as: ) 2 ( ) 1 ( 2 a r a t y t t y k t i t t i n t t y t t y m m e e (3.55) Where n = 0 or 1, depending on rainfall input. The initial condition for both equations is a dry bed condition or y ( t, r ) = 0 everywhere at t = 0. The upstream boundary remains dry at all times, while the downstream boundary condition can be described as the normal flow depth. It can be progressively solved for every time step with n = 1 until the rain stops. Without excess rainfall, n = 0. 3.7 Closing In this chapter, new governing equations were derived for both diverging and converging types KW overland flows. These equations are presented in the polar
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63 coordinate system that is more flexible and better representing the geometry of fan shape landform. Additional efforts are required to verify these derived equations with laboratory data or field records. With these equations, further studies on the sensitivity of soil infiltration and surface roughness on an overland runoff hydrograph can be conducted numerically.
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64 4. Geometric Transformation of Diverging and Converging Fan Areas into Kinematic Wave Rectangular Planes 4.1 Introduction As previously mentioned, in acknowledgment of flooding problems that have occurred on alluvial fans, the Federal Emergency Management Agency (FEMA) developed a stochastic methodology that was first developed to identify flood hazards on an alluvial fan (Anon 1981 and Dawdy 1979). This method later become the foundation of the FAN computer model (1990) developed by FEMA to establish the National Flood Insurance Program (NFIP) for this type of landform (FEMA 1989, 2000). One of the deficiencies of the existing methodology is that it does not clearly define a procedure to conduct the fan shape drainage basin analyses. Based on previous studies, it was recommended that the rainfall Â– runoff hydrologic processes on alluvial fan to be best simulated utilizing the Kinematic Wave (KW) overland flow theory (Mukhopadhyay et. al. 2003). One of the major computational algorithms developed to simulate surface runoff utilizing KW overland flow theory is the EPA Storm Water Management Model Version 5.0 (SWMM5). The KW application in SWMM5 was formulated to predict the flows on a virtual rectangular sloping plane. To apply SWMM5 to either a diverging or converging fan planes, the fan shape surface must first be converted into its equivalent rectangular plane. In current practices, the engineer will have to estimate such a shape conversion based on experience or calibration if any observed data is available. For example, the application of SWMM5 for the Fox Hollow Watershed located in Centre County, PA was calibrated using field data (Zang and Hamlett 2006). However, for areas without rainfall and runoff gages, the estimation of the required KW rectangular plane width relies on engineering judgement.
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65 Consequently, overland flow predictions are varied with respect to different KW rectangular plane widths estimated for the same study area. For instance, a factor of 2.0 as the ratio between the KW width and the waterway length was recommended for a square watershed while a different factor of 2.2 was also recommended for shapes other than a square (UDSWMM Manual 2000). In Ontario, Canada, a factor of 1.67 was adopted for watershed studies (Proctor and Redfern 1976). Determination of KW plane width has been a challenge when applying the KW overland flow theory to hydrologic studies. Without proper guidance, determination of this key modeling parameter depends on individualÂ’s experience and selection of empirical formulas. Therefore, it is urgent to acquire a standard procedure for converting a watershed from its natural shape into its equivalent rectangular shape when using the KW flow theory for overland runoff predictions. The effort of this Chapter is to derive a oneonone geometric transformation procedure to convert the derived new KW governing equation from the polar coordinate system into Cartesian coordinate system. This transformation shall satisfy the basic principles of mass and energy conservation. As a result, an alluvial fan watershed can be described using its apex angle and fan radius into Cartesian coordinate system. With the specified longitudinal surface slope and rainfall excess, the KW flows generated from the fan area can be determined with flow depths, flow velocities, and flow discharge at any given point on the alluvial fan.
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66 4.2 Transformation Procedure Development For a unitwidth approach, the computational methods developed for KW flows require a geometric conversion of the actual, irregularshaped alluvial fan plane into its virtual rectangular plane (Huber 1988, Rossman 2005). To maintain the principle of continuity, the surface area between the two planes must be preserved. Correspondingly, the potential energy difference caused by elevation variance between highest and lowest point along the waterway must be also conserved (Guo and Urbonas 2009). A conformal mapping technique employed in the potential flow theory (Finnemore and Franzini 2002) is adopted to develop a transformation procedure for converting an irregularshaped watershed into a virtual KW rectangular sloping plane. To maintain the watershed geographic characteristics, the continuity and energy principles for KW shape transform are described as: w A A (4.1) w H H (4.2) where A = actual watershed area in [L 2 ], A w = virtual area on KW sloping plane in [L 2 ], H = vertical drop in actual watershed in [L], and H w = vertical drop in virtual KW plane in [L]. Between the two flow systems illustrated in Figure 4.1, the dimensional analysis indicates that characteristic parameters are average watershed width, flow length, and width of KW plane. Their dimensionless functional relationship is derived as: L B fct L L w (4.3)
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67 in which L w = width of KW plane in [L], B = average watershed width in [L], and L = length of collector channel in [L]. Eq. (4.3) basically describes the geometric relationship between the shapes of these two tributary areas. Figure 4.1 Conversion of Actual Watershed into Virtual KW Sloping Plane To satisfy Eq. (4.3), the total watershed area must be preserved and can be presented as: W W o L X A (4.4) Furthermore, the elevation difference over the waterway between the highest points of the watershed to the lowest point at the outlet location must be also preserved as: W W W L X S L S 0 (4.5) Normalizing the relationships with watershed length, L Eq. (4.4) and Eq. (4.5) can be converted into:
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68 L L L X L A W W 2 (4.6) L L L X S S W W W 0 (4.7) The watershed shape factor serves as an index to describe how overland flow is collected within the watershed. It suggests that the watershed length to width ratio and elongation and circularity ratios can be engaged to represent the shape of a watershed (McCuen 2005). As suggested in Figure 4.1, the shape factor for a natural watershed to be converted into KW virtual plane can be expressed and approximated as: 2 L A L B X when K X (4.8) L L Y W (4.9) Where X = watershed shape factor, Y = KW shape factor for the KW sloping rectangular plane, and K = upper limit for shape factor. For a rectangular watershed such as a parking lot, Eq. (4.8) is simplified as a width to length ratio. In engineering practices, it is recommended that a large watershed to be divided into smaller subareas with a shape factor not to exceed upper limit K Commonly, K = 4 is adopted when conducting hydrologic simulation (UDFCD 2001). Assisted by Eq. (4.8) and Eq. (4.9), Eq. (4.6) and Eq. (4.7) can be combined as: Y Y X S S W 0 when K X (4.10) The two cases in Figure 4.2 represent two extreme location of the channel alignment since the centerline and the boundary line in the watershed sets the limits for all possible channel alignments (DiGiano et. al. 1976).
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69 Figure 4.2 Â– KW Models for Square Watershed 4.3 Development of Geometric Transformation Using SINE Function In this section, the trigonometric Sine function is adopted to derive the conversion parameters for preserving the apex angle on a diverging type alluvial watershed. In this study, Eq. (4.10) is defaulted to be: c bX a Y sin (4.11) In which a b and c are the constants to be developed separately for symmetric and asymmetric watersheds. Mathematically, this geometric transformation function in Eq. (4.11) is a onetoone single value relationship between variables X and Y This functional relationship shall cover the range from Y = 0 to Y 4 to 6, depending on the watershed model used in the stormwater
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70 studies. With the support of Figure 4.2, in general, this functional curve shall pass three distinct points as: Under the Condition 1, X 0 Â– a very small watershed, Y 0 Eq. (4.11) is reduced to: Y = 0 + c = 0 or c = 0 (4.12) Under the Condition 2, X = 1 and Y = 1, it represents a square watershed with a collecting channel along the side: Y = a sin b = 1 (4.13) Similarly, under the condition that X = 1 and Y = 2, it represents a square watershed with a central collecting channel, Y = a sin b = 2 (4.14) Under the Condition 3, X = K in which K represents the highest acceptable ratio of watershedÂ’s width to its waterway length. As expected, the higher the value of K is, the higher the peak flow is. In practice, the value of K shall not exceed 4. As indicated in Eq. (4.14), the maximum value for Y shall satisfy: 0 cos bK ab dX dY (4.15) Considering the Sine function as the default geometric transform operator, the maximum value in Eq. (4.15) occurs when bK = 2 thus: 2 cos cos bK (4.16) K b 2 (4.17) Inserting Eq. (4.17) into Eq. (4.13) and Eq. (4.14) yields:
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71 o for a sidecollecting channel, K a 2 sin 1 and X K K Y 2 sin 2 sin 1 (4.18) o for a centercollecting channel, K a 2 sin 2 and X K K Y 2 sin 2 sin 2 (4.19) Combining Eq. (4.11), Eq. (4.12), Eq. (4.18), and Eq. (4.19) yields: X K K Z Y 2 sin ) 2 sin( 2 5 1 for K X (4.20) L L Y w (4.21) A A Z m (4.22) Where Y = KW shape factor, Z = area skewness factor (depending on location of the collector channel through the watershed), A m = larger half area after the watershed is divided by the collector channel, A = total watershed area, and K = maximum allowable watershed shape factor (CUHP 2005). And when the watershed is divided into two parts by the waterway, the value of Z is defined as the ratio of the larger half to the total watershed area. For instance, Z = 0.5 is for the case with a centralcollecting waterway, while Z = 1.0 is for the case with a sidecollecting waterway. Based on Eq. (4.1), the length of KW overland flow on the KW rectangular cascading plane is defined as: w w L A X (4.23)
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72 According to Eq. (4.2), the slope on the KW plane is the vertical fall divided by the total flow length represented as: w w o w L X L S S (4.24) 4.3.1 Geometric Transformation of Diverging Fan Area As illustrated in Figure 4.3, the location of collector channel on a diverging plane area is along the downstream boundary line. Therefore, this is a case of side channel, or Z, should be equal to 1.0. The shape factor for a diverging plane area is defined by its tributary area, 2 2 1 R A and the length of the collector channel length, R L Thus, Eq. (4.8) is converted into: 2 1 2 L A X (4.25) Figure 4.3 Conversion of Diverging Plan into KW Rectangular Plan
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73 In hydrologic analyses, a large watershed is suggested to be divided into smaller subareas. It is advisable that the watershedÂ’s width to length ratio not exceed 4.0 to avoid a skewed runoff estimation (McCuen, 2005). In this study, the recommendation of K 4 is adopted (Guo and Urbonas 2009). Substituting Eq. (4.25) into Eq. (4.20) with K = 4 and Z = 1, the width of the virtual KW rectangular plane is derived as: 16 sin ) 8 sin( 2 2 R L w for 1/8 (426) Referring to Figure 4.3, the vertical drop height is conserved along the flow path, a b c, on the actual watershed or a b c on the virtual KW plane. Therefore, Eq. (4.24) is converted to: w w o w L X R S S ) 1 ( (427) 4.3.2 Diverging Fan Areas Transformation Case Study The KW shape factor, L w /L is sensitive to watershed geometry and location of the collecting channel or waterway. Table 1 is a set of various fan shape parameters showing different fan areas, radii of flow paths, interior apex angles, and surface longitudinal slopes. Eq. (4.26) and Eq. (4.27) are applied to these hypothetical cases to conduct a sensitivity test of the KW shape factor in a diverging geometry. As expected, all conversions reflect the original watershedÂ’s geometry, and none is repeated and the results are summarized as followed. Figure 4.4 was generated to display the shape factor relationship between the fan shape and KW virtual rectangular sloping plane by plotting the X and Y values for Case 4 to Case 8. Figure 4.4 shows a linear relationship when X ranges between 0 and 1 demonstrating
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74 a onetoone conversion relationship between a fanshaped tributary area and KW virtual rectangular plane. Table 4.1 Â– Testing for Diverging Fan Shape Plane to KW Rectangular Planes Case ID R Interior Angle S o Area L Fan Shape Factor KW Shape Factor (ft) Radian Degree (%) (ft 2 ) (ft) X = A/L 2 Y = L w /L 1 100 2.09 119.7 5.00 10450 209 0.24 0.25 2 300 2.09 119.7 5.00 94050 627 0.24 0.25 3 500 2.09 119.7 5.00 261250 1045 0.24 0.25 4 500 0.52 29.8 5.00 65000 260 0.96 0.96 5 500 1.05 60.2 5.00 131250 525 0.48 0.49 6 500 1.57 90.0 5.00 196250 785 0.32 0.33 7 500 2.09 119.7 5.00 261250 1045 0.24 0.25 8 500 2.62 150.1 5.00 327500 1310 0.19 0.20 9 500 3.14 179.9 3.00 392500 1570 0.16 0.16 Furthermore, as presented in Figure 4.5, as the interior expansion angle increases, the width of the KW virtual plane decreases and the effect of K diminish. It reflects the fundamental concept of basin delineation for keeping the watershed width and length ratio closer to unity to avoid skewing the flow prediction. In engineering practices, the value of X is predetermined based on the size of the tributary area and location of the collecting channel. Eq. (4.20) provides a general conversion from a fan shape tributary area to a KW virtual rectangular plane to assist the determination of KW rectangular cascading plane width during modeling efforts.
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75 Figure 4.4 Â– Watershed Factor vs. KW Shape Factor Â– Trigonometry Sine Function Figure 4.5 Â– Shape Factor vs. Interior Expansion Angle Â– Diverging Plane 4.4 Development of Geometric Transformation Using Parabolic Function In this section, a parabolic function is adopted to derive the functional relationship between shape factors X and Y as:
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76 c bX aX Y 2 (4.28) In which a b and c are constants to be developed separately for symmetric and asymmetric watersheds as previously demonstrated. Following the same procedure described in the former discussion, mathematically, this function curve will pass three distinct conditions and satisfy the following: Under the Condition 1, X 0 Â– a very small watershed, Y 0 Eq. (4.28) can be reduced to: Y = aX 2 + bX = 0 or c = 0 (4.29) Under the Condition 2, X = 1 and Y = 1, it represents a square watershed with a collecting channel along the side: Y = a + b = 1 (4.30) Similarly, under the condition that X = 1 and Y = 2, represents a square watershed with a central collecting channel: Y = a + b = 2 (4.31) Under the Condition 3, X = K in which K represents the highest acceptable ratio of watershedÂ’s width to its waterway length. As expected, the higher the value of K is, the higher the peak flow is. In practice, the value of K shall not exceed 4. As indicated in Eq. (4.31), the maximum value for Y shall satisfy 0 2 b aX dX dY (4.32) Considering the parabolic function as the default geometric transformation function operator, inserting Eq. (4.32) into Eq. (4.29) and Eq. (4.30) yields: o for a sidecollecting channel
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77 K a 2 1 1 and K K b 2 1 2 (4.33) X K K X K Y 2 1 2 2 1 1 2 (4.34) o for a centercollecting channel K a 2 1 2 and K K b 2 1 4 (4.35) X K K X K Y 2 1 4 2 1 2 2 (4.36) Combining Eq. (4.34) and Eq. (4.36), the general conversion relationship and can be presented as: X K K X K A A Y m 2 1 4 2 1 2 5 1 2 (4.37) Where Y = KW shape factor, A m = larger half area after the watershed is divided by the collector channel, A = total watershed area, and K = maximum allowable watershed shape factor (CUHP 2005). 4.4.1 Geometric Transformation of Converging Fan Area Overland flows on a fanshaped area are collected by a side channel that is located along boundary line, bc as shown in Figure 4.6. Therefore, Z = 1 for this case. The shape factor for a converging fanshaped area is defined by its tributary area, A which equals to r R r 2 2 1 and length, L of the collector channel equals r R L With these relationships, Eq. (4.8) yields: 2 ) ( 2 ) 2 ( r R r R r X (4.38)
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78 Figure 4.6 Conversion of Converging Plan into KW Rectangular Plane A converging area tends to be wide in shape. In this study, K = 10 and Z = 1 are adopted (Guo and Urbonas 2009). As a result, the KW shape factor in Eq. (4.35) is reduced to: 2 053 0 053 1 X X Y where 10 X (4.39) Referring to Figure 4.6, the vertical drop is conserved along flow path, abc on the converging watershed or on the virtual KW plane. Therefore, Eq. (4.24) is revised to: w w o w L X r R r S S ) ( for converging flow (4.40) Eq. (4.38) through Eq. (4.40) were derived to convert a converging area into the equivalent rectangular KW plane. 4.4.2 Converging Fan Area Transformation Case Study Table 4.2 is a set of converging areas, radii of flow paths, interior angles, and longitudinal surface slopes. With an angle of 1.21 radians (69.3 degrees), the watershed
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79 shape factor is 9.92, close to the selected limit for K Eq. (4.37), Eq. (4.38), and Eq. (4.39) are applied to these hypothetical cases to test the sensitivity of KW shape factor to the converging geometry. This table also provides a summary of the corresponding fan shape verses KW rectangular planes shape factor. As expected, all conversions reflect the actual watershedÂ’s geometry, and none is repeated. Table 4.2 Â– Testing for Converging Fan Shape Plane to KW Rectangular Planes Case ID R Interior Angle r S o Area L Fan Shape Factor KW Shape Factor (ft) Radian Degree (ft) (%) (ft 2 ) (ft) X = A/L 2 Y = L w /L 1 500 1.21 69.3 400 5.00 145200 121 9.92 5.23 2 500 1.57 90.0 400 5.00 188400 157 7.64 4.95 3 500 2.10 120.3 400 5.00 252000 210 5.71 4.29 4 500 2.35 134.6 400 5.00 282000 235 5.11 4.00 5 500 2.62 150.1 400 5.00 314400 262 4.58 3.71 6 500 3.14 179.9 400 5.00 376800 314 3.82 3.25 7 100 2.09 119.7 80 5.00 10032 42 5.74 4.30 8 300 2.09 119.7 240 2.00 90288 125 5.74 4.30 9 500 2.09 119.7 400 5.00 250800 209 5.74 4.30 10 500 2.62 150.1 400 7.00 314400 262 4.58 3.71 A series of exhibits of Eq. (4.39) and converging plane radius ratio, r/R ranging from 0.1 to 0.9 are prepared with K = 10, 6, and 4 and presented as Figures 4.7 to 4.10. These figures show the sensitivity of the location of the outlet point in relation to the projected origin of the converging plane. As expected, when the interior expansion angle is increasing, the KW virtual cascading rectangular plane width decreases to satisfy the area continuity between converging fan shape plane and KW virtual plane. Also, between Y = 0 and 1, under different interior expansion angles, it suggests K value is not as sensitive, which benefits the
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80 selection of the outlet location to determine the alignment of the conveying channel during engineering design and application. Figure 4.7 Â– Radius Ratio vs. KW Shape Factor (15 degree) Â– Parabolic Function Figure 4.8 Â– Radius Ratio vs. KW Shape Factor (30 degree) Â– Parabolic Function
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81 Figure 4.9 Â– Radius Ratio vs. KW Shape Factor (90 degree) Â– Parabolic Function Figure 4.10 Â– Radius Ratio vs. KW Shape Factor (179 degree) Â– Parabolic Function 4.5 Closing It is critically important to understand the transformation relationships between an irregular watershed and its virtual KW rectangular plane. Applying the conformal mapping
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82 technique to project the actual flow motion onto a virtual rectangular surface, it preserves the major geometric parameters between the actual and virtual watershed. The concept of watershed shape factor allows the collection of surface runoff from an actual watershed to be transformed onto a KW overland flow rectangular plane. This oneonone conversion relationship can be executed as a preprocess before the hydrologic simulation when working with a computer model such as HECHMS and SWMM5. Furthermore, with this transformation procedure, the maximum allowable overland flow length of 300 feet to 500 feet recommended by the current engineering practices is no longer applicable because the KW flow on the rectangular plane is virtual or mathematical only.
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83 5. Model Verification and Case Studies 5.1 Introduction A numerical model for rainfall and runoff simulations is best illustrated by the concept of system which produces an output according to the given input. The reliability of a numerical model depends on how well the model is calibrated with observed data. Model verification is a system identification process. Applicability of a numerical algorithm depends on their application limits and identifications of system constants and parameters. A good numerical model provides consistency between the input and output relationship while a reliable numerical model needs a high level of accuracy. Differential equations derived for a physical process are mostly aimed at providing solutions to engineering problems. In most cases, these equations cannot be directly solved for an analytical solution, but their applications rely on numerical approach to generate an approximated solution. With advanced computer technology and numerical techniques in both software and hardware, many differential equations can be formulated into numerical methods. A numerical method is to simulate a physical process by solving a set of governing equations which depict a realworld event. Correspondingly, numerical solutions are always generated for a specified initial and boundary conditions. In practice, most of boundary and initial conditions require special numerical treatments to warrant stable computations. Although numerical solutions can be effectively applied to engineering designs and applications, it is not intended to replace the analytical solution if it is available. A numerical approach is highly valuable when coping with multiple scenario analyses which are subject to various boundary conditions. It is important to understand that numerical solutions are
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84 discrete and only derived at grids while an analytical solution is continuous in time and in space (Guo 1982). One of the major benefits of the numerical approach is to offer predictions of a physical process and provide good guidance prior to actual construction of the physical experiment. Care must be taken that all numerical procedures require calibrations and all calibrations require observed data. In order to collect field data, U.S. Geological Survey (USGS) installed the first stream gage on the Rio Grande River in New Mexico in 1889. Currently, USGS operates approximately 7,400 stream gages nationwide with approximately 91 percent of these stations are transmitting data in nearly a realtime scale (USGS 2007). As presented in Figure 5.1, the distribution and density of stream gages in the Southwest U.S. is less than the rest of the U.S. Continent. This is mainly due to not well defined waterway, severe erosion and scour along natural washes in the arid climate region. Due to the lack of stream records on alluvial fan type landforms, for this study, it is proposed to populate overland flow hydrographs numerically for the selected cases according to both diverging and converging fan geometries. With a proper surface roughness coefficient assigned, overland flow depths and the associated hydraulic characteristics on the fan surface will then be compared with available field record or documented laboratory data.
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85 Figure 5.1 Â– USGS Stream Network Location Exhibits The following sections will discuss the numerical procedures and results for case studies and provide comparisons with the historical data if available. 5.2 Verification of Diverging KW Model and Case Studies The laboratory study for diverging KW flows on an impervious surface was documented by Izzard (Izzard, 1946). It was then followed by other investigators (Yu and McNown, 1964, Langford and Turner 1973) along with some field observations (Singh 1975, 1976). As reported (Muzik 1973), the KW flows were produced from a rectangular galvanized surface with 0.61m in width and 0.91m in length and laid on a longitudinal slope of 0.2079 (m/m). This laboratory layout was evaluated to be equivalent to a diverging KW plane with a radius, R of 0.91m (Singh and Agiralioglu 1981). For this case study, to satisfy the same surface area, it was determined that the apex angle is 1.34 radians (76.75 degrees).
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86 Under a laboratory shower simulating, a rainfall event was conducted with an intensity of 78 mm/hr and with duration of 50 seconds. The surface runoff was produced and measured with a total simulation time of 80 seconds. The parameters on the KW rating curve are determined to be: 6 67 486 1 5 0 0 S n (5.1) 3 5 m (5.2) Where n = ManningÂ’s roughness coefficient and is equal to 0.01 for galvanized surface (Muzik 1973). Figure 5.2 presents the comparisons between the laboratory data and the analytical solutions derived in this study for both rectangular and diverging KW flows. As expected, Eq. (2.23) for rectangular plane and Eq. (3.16) for diverging plane produce the same peak flow for the same surface area, but the diverging KW flow is characterized with a slower movement, or a longer time of equilibrium as predicted in Eq. (3.18). As shown in Figbure 5.2, the KW analytical solution computed by Eq. (3.19) agrees well with the previous study (Singh and Agiralioglu 1981)
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87 Figure 5.2 Comparison of Laboratory and Numerical Solutions for KW Overland Flow Hydrographs under Rainfall Intensity of 78 mm/hr for 50 Seconds A spreadsheet was developed based on Eq. (3.54) with the support from ManningÂ’s equation, Eq. (3.12), Eq. (3.13), Eq. (5.1) and Eq. (5.2) to conduct the computation. Detailed numerical for this case can be located in Appendix D Â– Case 1 As shown in Figure 5.3, a similar laboratory experiment is carried out using the same equipment with same apex angles. The peak flow is increased from 12 cm 3 /s to 18 cm 3 /s as the manmade rainfall intensity is increased from 78 mm/hr to 115 mm/hr, respectively. Both studies verify that the diverging KW flow moves with a shallower depth and at a slower velocity in comparison with the rectangular KW flow. Again, good agreement is achieved between the KW analytical solution computed by Eq. (3.16) and the laboratory data from the previous report (Agiralioglu and Singh 1980). The same spredsheet developed is utilzed by replacing the rainfall intensity from 78 mm/hr with 115 mm/hr to conduct the computation by
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88 maintaning the same duration time of 50 seconds and total simulation time of 80 seconds. The detail computation can be located in Appendix D Â– Case 2. Figure 5.3 Comparison of Laboratory and Numerical Solutions for KW Overland Flow Hydrographs under Rainfall Intensity of 115 mm/hr for 50 Seconds Another case study is conducted to assess the hydrograph changes with respect to various apex angles. By keeping the same surface area equal to a rectangular plane with 0.61m in width and 0.91m in length and a longitudinal slope of 0.2079 (m/m), a set of equilibrium diverging planes with various apex angle changing from 60 degrees, 120 degrees and 180 degrees with associated radius is prepared and the rainfall intensity is set to be 115 mm/hr. Figure 5.4 presents the numerical results with various apex angles and are compared to the previous laboratory record (Muzik 1973). It is noticed that the peak flow at the time of equilibrium is preserved by the size of the diverging surface area for all cases. As the apex angle increases, the diverging KW flow becomes more converging to the rectangular KW flow. This suggests that the rectangular KW flow is the limiting case among all diverging
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89 flows with various apex angles. The detail computation can be located in Appendix D Â– Case 3. Figure 5.4 Diverging KW Overland Flow Hydrograph with Various Expansion Angles A case study was conducted to compare the variances for both flow velocity and depth between Eq. (3.14) developed in this study and FEMA FAN (FAN) program output. The detailed runoff frequency analysis and the topographical information for this case study can be located in the FAN UserÂ’s Manual (FAN 1990). In this case, the design discharge under the 100year storm event was determined to be 2,120 cfs for an upstream tributary area of 1 mi 2 (640 acres). The equilibrium rainfall excess, i e is obtained through SCS Method (Soil Conservation Services 1972). S i S i i e 8 0 2 0 2 (5.3) 10 1000 CN S (5.4)
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90 Where CN represent curve number and is equal to 85 to represent alluvial fan surface condition (CCRFCD 1999). The design discharge is released at the apex with an expansion angle of 0.42 radians (25 degrees) and ManningÂ’s roughness coefficient is estimated to be 0.05 to represent the alluvial fan surface. The longitudinal surface slope is approximated at 0.085 (ft/ft) per the topographical and contour elevation difference and the KW rating coefficient, m is determined to be 5/3. The outputs from FAN program are the water spread width, flow velocities and 100year flood depth which was the sum of hydraulic head and dynamic head (FAN 1990). The water spread width was defined as the width of the surface area and is subject to the flood event which has a 1percent chance of being exceeded in any given year. Similar computations are prepared by utilizing Eq. (3.13) to determine the related hydraulic parameters for comparisons. Figure 5.5 presents the flow velocity and depth variances between the derived governing equation, Eq. (3.16), and FAN outputs. In general, the governing equation produces an evenly distributed flow depth and velocity as the fan width is expanded downstream. But, the upper and lower limits for both flow depth and velocity are quite agreed with the distribution of the FAN output results. The detail flow velocity and flow depth calculation can be located in Appendix D Â– Case 4.
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91 Figure 5.5 Â– KW analytical solution vs FAN program outputs
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92 5.3 Rising and Recession Hydrograph for Diverging Flow The rising and recession portions of the overland hydrograph are not as interested as the peak flow. In the engineering practices especially in hydraulic structure sizing such as culvert and storm sewer, peak flow is the most important design variable. However, the latest development in stormwater detention requires the entire hydrograph for runoff volume estimation. In the application, the rising hydrograph is defined with the peak flow rate and the time of concentration, T c In this study, all flow rates are to be normalized by the peak flow rate and all time parameters are to be normalized by its associated time of concentration, T c As previously mentioned, two separate nondimensional hydrograph segments, Eq. (3.20) and Eq. (3.28), are developed to represent rising and recession portions of the hydrograph, respectively. In this section, both equations will be examined and compared with numerical solution and linear approximation. Figure 5.6 presents the rising portion hydrograph segment comparison. In this figure, the numerical solution is extracted from the KW diverging plane solution presented in Figure 5.2 using Eq. (3.14), the analytical solution is calculated by Eq. (3.16) and the linear approximation is calculated Rational Method. As mentioned previously, all flow rates are normalized by peak flow (11.97 cm 3 /sec) and all time parameters are normalized by T c (18 seconds). The comparison shows that the peak flow rates are the same disregard which equation is applied. It suggests that the Rational Method can be a simplify approach during the engineering planning stage when peak flow rate is needed for infrastructure sizing. The rates of rising between numerical and analytical solutions agree well especially when approaching peak flow. The numerical solution using Eq. (3.14) suggests that the rising rate of the hydrograph is not as sensitive under different expansion angles and location ratio, a
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93 since these two variables are not included in Eq. (3.14). But these two variables are incorporated when conducting surface area calculation to describe the fan geometry. Figure 5.6 NonDimensional Rising Hydrograph Analytical and Numerical Solution Comparison A nondimensional recession hydrograph is also presented as shown on Figure 5.7. In the case, the comparison is made between numerical solution applying Eq. (3.14) and analytical solution calculated based on Eq. (3.28). Both analytical and numerical solutions present a decay curve. Both curves demonstrate a faster drawdown at the early stage when the rain ceases. These curves suggest that a linear approximation similar to the rising portion hydrograph might not be as proper to estimate the outflow release rate. In the engineering application, it might post an overestimated runoff volume when conducting detention or retention basin design. But with the development of the analytical solution, it should benefit when approximate inundation duration after the storm event is ended.
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94 Figure 5.7 Â– NonDimensional Recession Hydrograph Analytical and Numerical Solution Comparison 5.4 Verification of Converging KW Model and Case Studies For the converging plane, the performance of the derived governing equation was first compared with a set of laboratory data. The laboratory experiment was documented in the study of converging flow observed and measured at the Rainfall Runoff Experimental Facility, the Colorado State University (Singh 1975). The experimental layout consisted of an upper converging plane and a lower rectangular plane. The converging plane was designed with a constant longitudinal slope of 0.05 (m/m) with a radius, R of 35.36m (116ft) and an interior angle of 104 degrees. The total surface area was 2,322.58m 2 (25,000ft 2 ), including 1,179.87m 2 (12,700ft 2 ) from the converging section and 1,142.71m 2 (12,300ft 2 ) from the rectangular section. The rainfall intensity was simulated and controlled through an array of sprinkler standpipes. In this laboratory documentation, a comparison was conducted for ten (10) sets of rainfall and runoff experimental events. Each data set included rainfall
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95 intensities, durations, and peak flow rates. However, there was no information about where the peak flows were measured on the converging plane. In this research work, the rainfall data sets are used to reproduce the observed runoff flows by calibrating the value of r described in Eq. (3.39) and Eq. (3.40) which represents the location of the flow measurement. These results are summarized in Table 5.1 to present the comparison between the laboratory observation and the result predicted by Eq. (3.41). An error presented in percentage (%) is also included as an index to show the agreement between observation and prediction. It is noted that Eq. (3.41) was able to closely reproduce the observed runoff flows with a 0.81. The outlet in Cases 2 and 7 was set to be very close to the origin of the converging fan area, i.e. r = 33.53m (110ft) and a = r/R = 0.95. As shown in Eq. (3.41), the origin is a singular point because the denominator becomes zero. Based on Table 5.1, Eq. (3.42) performs reasonably well up to a 0.81 under a rainfall intensity of 11.15cm/hr (4.39 inch/hr). It implies that with a higher value, 0.81 a 1.0 Eq. (3.42) may not well represent a converging KW flow because of significant accelerations. Field measurements on natural alluvial watersheds using the KW overland converging flow model were also reported (Singh 1976). The watersheds W2 and W6 were located in Riesel, TX. Both watersheds were covered with Huston black clay soils with low permeability. For both study cases, the infiltration loss was estimated using PhilipÂ’s infiltration equation (Philips 1957) and the details could be found elsewhere (Singh 1976). The upstream tributary area for Watershed W2 was approximated around 530,000m 2 (53.42 hectares) with an average longitudinal slope of 0.025 (m/m) and the radius is estimated at 908m with an interior angle of 96 degrees. Watershed W6 was equipped with an upstream tributary area of 170,000m 2 (17.12 hectares) on an average longitudinal slope of
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96 0.015 (m/m) and the radius is estimated at 478m with an interior angle of 92 degrees. In both watersheds, ManningÂ’s roughness is equal to 0.02 to represent the surface condition (Singh 1976). The general layouts of W2 and W6 watersheds are shown on Figures 5.8 and 5.9. Although both watersheds are not on a perfect converging shape plane, the recorded rainfall and runoff events were the most detailed field data available. Table 5.1 Â– Comparison between Predictions and Measurements for Converging Flows Case Laboratory Observation Predicted by Converging KW Equation i e T d q obs T p R Rr a q pred Error cm/hr sec cm/hr sec m m cm/hr % 1 10.64 33.95 3.96 90.00 18.04 17.31 0.51 3.96 0.00 2 2.74 85.09 1.50 139.90 35.36 0.00 1.00 1.32 10.58 3 2.62 69.59 1.04 151.90 19.74 15.61 0.56 1.04 0.00 4 8.53 42.21 3.86 91.40 25.59 9.77 0.72 3.86 0.00 5 2.29 65.59 0.89 140.50 19.75 15.61 0.56 0.89 0.00 6 11.15 52.98 5.26 93.80 28.60 6.76 0.81 5.26 0.00 7 2.49 353.33 2.18 463.60 35.36 0.00 1.00 1.22 44.21 8 2.77 80.78 1.19 159.90 23.44 11.91 0.66 1.19 0.00 9 2.64 77.79 1.07 102.90 20.47 14.88 0.58 1.07 0.00 10 11.23 32.69 2.69 94.10 10.09 25.26 0.29 2.69 0.00 i e rainfall intensity T d Â– duration q obs Â– observed peak flow T p Â– time to peak q pred Â– predicted peak flow applying Eq. (3.42)
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97 Figure 5.8 Â– Topographical Layout for Watershed W2
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98 Figure 5.9 Â– Topographical Layout for Watershed W6 Similarly, excess rainfall intensity was adopted for rainfallrunoff simulation to be applied in Eq. (3.41). The KW rating curve parameters were determined by Eq. (5.1) and Eq. (5.2). Applying the observed 40minute excess rainfall depth recorded for watershed W2 on June 4, 1957, Eq. (3.43) is used to calculate the surface runoff. Figure 5.10 presents the field observe data, numerical solution from the previous work (Singh, 1976), converging analytical solution by Eq. (3.43) and excess rainfall intensity hyetograph. It is noticed that after the first peak, the predicted hydrograph is fluctuated with the rainfall pattern and agrees well with the observed hydrograph. The deficit between early rainfall excess and runoff volume is due to the surface detention on the converging plane. Expectedly, the predicted hydrograph carried double peaks as recorded in the excess rainfall hyetograph. As understood, the original documented solution for this filed observation case was calculated
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99 using a complicated approach to integrate Eq. (3.35) through the interior angle (Singh 1976). Although Eq. (3.41) is a simplistic approach, it does capture the major threads in the rainfallrunoff simulations, but ignored the backwater and storage effects. A spreadsheet was constructed based on Eq. (3.55) with the support from ManningÂ’s equation, Eq. (3.40), Eq. (3.41), Eq. (5.1) and Eq. (5.2) was developed to conduct the computation. Detailed numerical computation for W2 watershed can be located in Appendix E Â– Case 1 Figure 5.10 Â– W2 Watershed Surface Runoff Hydrograph Prediction and Comparison The second field case study was observed for Watershed W6 on March 29, 1965. The rainfall distribution appears to be a composite event continuous for 4 hours. The developed spreadsheet mentioned above is utilized by replacing the parameters and variables from W2 watershed to W6 watershed to conduct the analysis. The comparison to include field observe data, numerical solution from the previous work (Singh, 1976), converging analytical solution by Eq. (3.43) and excess rainfall intensity hyetograph is presented in
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100 Figure 5.11. As shown in this figure, the rainfall distribution exhibited multiple peaks as the early rainfall excess was also delayed by the surface detention. Surface detention before the peak flow is mostly dictated by ManningÂ’s roughness coefficient (Ponce 1989, Guo 1998 and 2006a). After the watershed reached equilibrium condition, starting from the second peak, the predicted runoff hydrograph responded to the excess rainfall hyetograph quite well. After the rain ceased, the surface detention volume was gradually released. Detailed numerical computation for W6 watershed can be located in Appendix E Â– Case 2. Figure 5.11 W6 Watershed Surface Runoff Hydrograph Prediction and Comparison 5.5 Rising Hydrograph for Converging Flow To determine the effectiveness of location ratio, a affecting the overland flow hydraulic depth, sample geometry converging planes are selected and sensitivity tests are conducted. A hypothetic converging plane is designed for this test. The dimension of the converging plane is R = 3,000ft, a uniform slope of 0.01(ft/ft) with a ManningÂ’s roughness
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101 coefficient of 0.02. The excess rainfall intensity, i e is equal to 1.0 in/hr with duration of 600 seconds. The computed flow rates are normalized by the peak flow rate and all time parameters are normalized by the time of concentration computed by Eq. (3.46). The results are presented in a form of a normalized hydrograph but the discussion in this study is focused only on the rising portion of the hydrograph. Figures 5.12, 5.13 and 5.14 are the nondimensional rising portion hydrographs including numerical solution calculated by the spreadsheet mentioned previously, analytical solutions solved by Eq. (3.47) and the linear approximation. Comparisons are made for different converging interior angles and various a values. Figure 5.12 presents the comparison for numerical and analytical solution with r = 300 ft ( a = 0.1) for both cases but varies the interior angle with 45 degrees and 104 degrees. Figure 5.13 and Figure 5.14 show the comparison under the same interior angle of 104 degrees with different r values ( r = 1500 ft and 2400ft) to detect the impacts from various a values ( a = 0.5 and 0.8). Figure 5.12 Converging KW Rising Portion Hydrograph Comaprison ( a = 0.1)
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102 Figure 5.13 Converging KW Rising Portion Hydrograph Comaprison (Interior Angle = 104 degrees, a = 0.5) Figure 5.14 Converging KW Rising Portion Hydrograph Comaprison (Interior Angle = 104 degrees, a = 0.8)
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103 The figures presented suggest that the rising portion of the hydrograph respond much more sensitively to location ratio, a, rather than to various interior angle. As the study area and location approach the projected point of origin, the rate of hydrograph rising delayed. This suggests a potential to underestimate in flow rate when the study area is small before time of concentration approaching time of equilibrium. 5.6 Shape Factor Method Verification and Sensitivity Test 5.6.1 Diverging Plane Shape Factor Method Case Study and Sensitivity Test Flood hazards on the alluvial fan area are often under estimated due to dry condition, lack of rainfall data and absence of defined watercourses (FEMA, 1989). In the literature review, the hydraulic structure maintenance records and the historical flood records along the transportation alignment major crossings were recorded and adopted as the only validation to the alluvial fan hydrology (French, 1994). In the absence of applicable alluvial fan watershed hydrologic information, this study presents the simulation results by comparing to the fan shape geometry impervious surfaces reported in the laboratory experiments (Izzard 1946, Muzik 1973). A case study was constructed to verify a diverted KW plane applying the shape factor conversion relation to produce surface runoff similar to those generated from the laboratory records. As documented (Izzard 1946), KW flows were produced under a laboratory shower simulation. The detail of this simulation including plane geometry and excess rainfall intensity can be located in Section 5.2 of this study. This sample watershed was then converted into virtual KW planes using Eq. (4.26) and (4.27). The virtual KW watershed suggested a 4.55m (15 ft) by 8.82m (19 ft) with a slope of 0.19 (m/m) virtual rectangular
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104 planes. These parameters were imported into SWMM5 (EPA 2010, USSWMM 2000, DiGiano et. al. 1976) computer model for surface runoff simulation. For this case, the KW rating curve parameters are determined by Eq. (5.1) and Eq. (5.2). is determined as 3.62 and 0.013 is used for ManningÂ’s surface roughness coefficient for smooth asphalt and m is proposed as 5/3. Figure 5.15 Comparison of Laboratory, Analytical and Numerical Solutions for KW Flows under Rainfall Intensity of 98 mm/hr for 10 minutes Figure 5.15 presents the comparisons among laboratory observed data, analytical solutions applying Eq. (3.42), and SWMM5 simulation outputs with the converted virtual KW rectangular plane. As expected, both rectangular and diverging planes produce the same peak flow as predicted by EqÂ’s (3.42). The rising hydrograph, time to peak and peak flow are well predicted by the SWMM5 model with the converted virtual KW rectangluar plane. Even though SWMM5 shows a faster rising in hydrograph but as the rain shower ceased, the two predicted recession hydrographs are parallel to each other.
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105 To better understand the KW plane width and length ratio relationship impacting the peak flow prediction, another sensitivity test is conducted. Adopting the virtual KW plane dimension and predicted results presented on Figure 5.15 as the base while keeping the surface area the same, by increasing L w a set of the derived X w and S w can be generated by using Eq. (4.27). All these dimensions are imported into SWMM5 model for runoff prediction sensitivity test. Table 5.2 presents the paired L w X w and S w with predicted peak flow for all cases. As expected, with L w increases, peak flow decreases. This finding also supports a onetoone singlevalued functional relationship between the actual watershed geometry and the virtual KW plane width and overland width using shape factor geometric conversion relationship. Please see Appendix F Â– case 1 for detail computation and SWMM5 simulation result (due SWMM5 model output limitation, only hydrograph is provided). Table 5.2 Â– SWMM 5 Peak Flow Sensitivity Test for KW Plane KW Width KW Overland Flow Width KW Slope SWMM5 Peak Flow Prediction L w X w S w cm cm cm/cm (cm 3 /sec) 4.55 8.82 0.19 1083.74 4.42 9.08 0.18 1101.53 17.68 2.27 0.13 1070.38 35.36 1.13 0.07 877.82 70.71 0.57 0.04 586.16 5.6.2 Converging Plane Shape Factor Method Verification Similar verification exercise is conducted to confirm whether the KW shape factor concept is applicable to the converging plane type geometry. Table 5.3 presents a group of converging areas, radii of flow paths, interior angles, and surface slopes. With an interior
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106 angle of 1.22 radian (70 degrees) and the shape factor is set at 9.82 which is close to the upper limit for K as recommended in Eq. (4.39). Applying Eq. (4.38), Eq. (4.39) and Eq. (4.40) are applied to these hypothetical cases to conduct the sensitivity test of KW shape factor variances for the converging type geometry. Table 5.4 is a summary of the corresponding KW rectangular planes and the converging type geometry planes. Unsurprisingly, all conversions reflect the actual watershedÂ’s geometry, and none is repeated.
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107 Table 5.3 Â– Conversion Relationship between Converging Plane and KW Rectangular Virtual Plane Converging Plane KW Rectangular Virtual Plane Radius Interior Angle Outlet Location Slope Surface Area Collector Channel Shape Factor Shape Factor Width Length Slope R r S 0 A L X=A/L 2 Y=L w /L L w X w S w Ft radian ft % ft 2 Ft ft ft % 500 1.22 400 5.0 146587.5 122.2 9.8 4.0 494.1 296.7 3.3 500 1.577 400 5.0 188469.6 157.1 7.6 4.3 668.8 281.8 2.9 500 1.83 400 5.0 219881.2 183.2 6.6 4.1 756.1 290.8 2.8 500 2.62 400 5.0 314116.0 261.8 4.6 3.5 913.3 343.9 2.6 500 3.14 400 5.0 376939.2 314.1 3.8 3.1 974.4 386.9 2.6 100 2.09 80 5.0 10051.7 41.9 5.7 3.9 164.3 61.2 2.7 300 2.09 240 2.0 90465.4 125.7 5.7 3.9 492.9 183.5 1.1 500 2.09 400 5.0 251292.8 209.4 5.7 3.9 821.6 305.9 2.8 500 2.62 400 7.0 314116.0 261.8 4.6 3.5 913.3 343.9 3.7 Table 5.4 Â– W2 and W6 Sample Watershed Geometry Conversion Watershed ID Drainage Area Effective Radius Converging Interior Angle Average Surface Slope KW Plan Width (L w ) KW Plane Overland Flow Length KW Plane Slope A R S 0 L w X w S w acre ft degree ft/ft ft ft ft/ft W2 132 2982 96 0.025 2390 2400 0.021 W6 42 1570 92 0.015 2190 842 0.009
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108 5.6.3 Converging Plane Shape Factor Method Case Study and Sensitivity Test The next test is to examine if the virtual KW rectangular planes can produce a KW flow rate similar to the flow rate generated from the actual watershed. Although alluvial fan type surface seems to demonstrate a higher permeability, the hydraulic characteristic of harden alluvial surface performs similarly to an impervious surface (CCRFCD 1997). As the local design criteria suggested (CCRFCD Manual 1999), a higher curve number (CN) ranging between 85 to 95, were recommended for the local master drainage study for Las Vegas Valley. In this test, the selected KW converging planes are the previously reported W2 and W6 Watersheds. The field records for these two watersheds are used to conduct KW plane peak flow comparisons by converting these two watersheds into virtual KW rectangular plane and applying the parameters into SWMM5 model for peak flow prediction. For the general topographical layouts of both watersheds, please refer to Figure 5.8 and Figure 5.9 reported in Section 5.4. Table 5.4 summarizes the conversion of Watersheds W2 and W6 into their virtual KW planes using Eq. (4.38), Eq. (4.39), and Eq. (4.40). Next, the virtual rectangular KW sloping plane parameters were imported into SWMM5 model for peak flow predictions. Figure 5.16 presents the results for the rainfallrunoff event for Watershed W2 on June 4, 1957. The observed rainfall event consisted of double peaks. As shown, there are two predicted hydrographs including the one from the numerical integration along with the converging surface area using Eq. (3.28), and another from SWMM5 output using the KW overland procedure along with the virtual rectangular plane with shape factor theory for geometry conversion. Even though the recorded runoff hydrograph does not reflect the fact of the double peaks in the
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109 rainfall distribution, both predicted hydrographs show good agreement with double peaks, according to the temporal changes in the rainfall distribution. Comparing to the results predicted by using Eq. (3.42), the rising hydrograph and the time to peak are well agreed to the results predicted by SWMM5. After the rain ceases, the two predicted recession hydrographs are parallel to each other. Please see Appendix F Â– case 2 for detail computation and SWMM5 simulation result (due SWMM5 model output limitation, only hydrograph is provided). Figure 5.16 Â– W2 Watershed Predictions and Field Data Comparison
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110 Figure 5.17 Â– W6 Watershed Predictions and Field Data Comparison Another comparison is conducted using Watershed W6. Figure 5.17 presents the comparison results for the event of March 29, 1965 recorded for this watershed. The same process was repeated to predict the runoff hydrographs. The predicted hydrographs populated from SWMM5 model and analytical solutions for this case agree with double peaks shown in the excess rainfall hyetograph. While the predicted hydrographs do not have as much surface detention as the observed, good agreement is obtained among the recession hydrographs. These two cases involve complicated surface geometric conversions and temporally varied rainfall patterns. Figures 5.16 and 5.17 provide good evidences that the peak flow generated by the converted virtual rectangular KW plane using Eq. (4.38) through Eq. (4.40) can almost be reproduced the same peak flow recorded from an actual watershed in converging type plan geometry. Please see Appendix F Â– case 3 for detail computation and SWMM5 simulation result (due SWMM5 model output limitation, only hydrograph is provided).
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111 5.7 Closing With an extensive effort, the exact solution was achieved only for the peak hydrographs under a uniform rainfall distribution on both diverging and converging KW flows. Although the rising hydrographs could be approximated by the constant KW speed determined by the peak flow condition, it would be necessary to detect any limitation of the assumption of constant KW speed. In this Chapter, the effort of validating the governing equations developed in this study was presented by numerical modeling. The finite difference numerical model derived in this study was employed to simulate several cases of diverging and converging flows recorded in the laboratory. For both KW flows, the finite difference model produces good agreements to the observed diverging and converging KW flows in the laboratory, Aided with the concept of shape factor, both diverging and converging surface areas are converted into their KW virtual rectangular planes that are further used in the EPA SWMM5 computer model to produce numerical solutions. The numerical predictions present good agreement with the analytical solution, laboratory data, and field records. The numerical results were also compared with the FEMA FAN model to validate its applicability to delineate the boundaries of floodplains on alluvial fans. The finite difference numerical model developed in this study provides good guidance to define water widths, water depths, flow discharges, and flow velocities on a floodplain. In conclusion, there are 8 (eight) cases reviewed in this study. All cases observed in laboratory or in field are well reproduced by the numerical models based on the new governing equations derived for diverging and converging KW flows. It is recommended
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112 that these new equations of KW flows on diverging and converging surface areas be used for studies of floodplains on alluvial fans.
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113 6. Conclusion In this study, new governing equations are derived applying KW overland flow theory for diverging and converging type planes. The equations are presented in the polar coordinate system to better represent the fan shape geometry. By balancing among inflows, outflows, excess rainfall depth and surface detention volume, it leads to the new governing equations that are described by the apex angle, radius of the fan shape plane, and the longitudinal surface slope. With extensive efforts, the exact solutions for both diverging and converging governing equations are achieved under a uniform excess rainfall condition for the peak flow portion hydrograph. Through finite difference numerical models devleoped for this study, both governing equations are valided and compared with the previous studies and are further examined with laboratory data, historical document and field record to verify the applicabilty to delineate floodplain boundaries on alluvial fans. For diverging planes, integration of Eq. (3.13) produces similar solutions to the previous study by employing a double integration with the support of a binomial process to accumulate the flow through the apex angle and the radius of a fan. In comparison, Eq. (3.13), is a much simpler approach to produce numerical solutions for diverging plane KW overland flow. For a floodplain study, the conservative approach is to set the critical design rainfall duration to be the time of equilirium. From Eq. (3.16) to Eq. (3.19), these equations are best to describe the equilirium condition to better understand the time of concentration on this type of fan shape geometry. For convenience, Eq. (3.19) is a special case of the Rational Method and aided by Eq. (3.23), the peak flow can be directly calculated from the local rainfallintensitydurationfrequency (IDF) curve to establish a design discharge for the
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114 selected base flood for design purposes. This direct relationship can benefit hydraulic engineering design to size hydraulic structure to convey peak flow. Eq. (3.21) and Eq. (3.23) provide this basic information and application for such a hydraulic design and can represent the rising hydrograph. The findings presented also suggest that the peak flow can be calculated directly when time of concentration is obtained, disregarding the rate of rising portion hydrograph. It demonstrates that the rate of the rising portion hydrograph is not as sensitive under different apex angles and location ratio, a since these two parameters are not included in Eq. (3.24). The governing equation derived for KW overland flows on a converging plane indicates that the converging KW overland flow can also be described by excess rainfall depth, longitudinal surface slope, surface roughness, and converging plane geometry in terms of interior angle and radius. Similar to the findings presented in the diverging plane, the peak flow in Eq. (3.44) can also be calculated directly from the local rainfall IDF curve to establish a selected base flood. With the support from Eq. (3.38), the hydraulic calculation to determine the minimum length required for a dike or earthen berm serving as a collector channel can be calculated by 19 0 R l r It is important that the origin of a radial converging surface is a sink point to produce a point of singularity within the derived governing equation. Through verification of the laboratory data and historical documents, the ratio of r/R 0.81 is recommended when applying the equilibrium converging flow for engineering application. In fact, the inherent rating curve includes the converging effect on the increase of unitwidth flow as the flow moves toward downstream. Eq. (3.55) is numerically stable as long as t is not exceeding the time step used in the rainfall excess.
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115 This research work also discusses the converging type plane rising portion hydrograph and its effectiveness during different interior angles and the location ratio, a The figures presented suggest that the rising portion of the hydrograph respond more sensitive in location ratio, a, than interior angle, As the study area and location approach the projected point of origin, the rate of hydrograph rising is delayed. This suggests a potential to underestimate the flow rate when the study area is small and before time of concentration approaches time of equilibrium. During flood flow estimation and planning exercise, the location ratio can provide a better guideline when conducting tributary area delineation on a converging type plane layout. But for the peak flow estimation, a linear approximation such as Rational Method and analytical solution can be accepted when time of concentration is reached. Although the general solution for converging plane is not completely achieved, under the equilibrium condition, peak flow and depth can be obtained to support the delineation of the flood flow boundaries on the converging type alluvial fan. This will greatly benefit sizing the drainage facilities such as dike or collector channel and cross culvert to collect surface runoff when a transportation alignment crossing alluvial fans form converging type geometry. FEMA FAN methodology and model was first generated to assist alluvial fan flood flow prediction and to delineate regulatory flood hazard zone boundaries. It provides limited information to hydraulic engineers in need of assessing flood hazard and its associated flood risk protection. With the governing equations derived in this study, it is feasible to calculate the flood flow depth, width and velocity directly at any point of interest on an alluvial fan through a deterministic approach. With the direct application of the local IDF curve to establish the design discharge for the selected base flood on an alluvial fan, the hydrologic
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116 analysis can be consistent and integrated between the regional and local level hydrologic analyses. The flood risk evaluation and the assessment of flood protection can be reviewed simultaneously as a master drainage plan type hydrologic study to include both alluvial and nonalluvial plane types of landform. Supported by the shape factor geometric transformation equations which are presented by Eq. (4.26), Eq. (4.27), Eq. (4.39) and Eq. (4.40), both diverging and converging surface areas can be converted into their virtual KW rectangular sloping planes. This conversion procedure is confirmed with a onetoone, singlevalued functional relationship between the actual irregular watershed and virtual KW rectangular plane. This transformation method offers good guidance and can be directly applied when conducting fan shape surface runoff simulation using available computer model such as EPA SWMM or HECHMS. It is important to understand that although the geometric conversion provides a consistency basis to transform watershed area, waterway length, and flow movement to an equilibrant KW rectangular sloping plane; but it is highly depended on selecting a proper surface roughness coefficient to mimic hydrologic regime and response of the alluvial fan type watershed. Aided with this geometric transform method, the model calibration efforts can be focused on determining ManningÂ’s roughness coefficient to properly represent alluvial fan surface roughness. A proper surface roughness coefficient can significantly improve the accuracy of the predictions of KW flow movement. Over years, watershed shape has been recognized as an important factor affecting the magnitude of the surface runoff. The procedure described in this research work is a good example to quantify such an impact. This procedure is not limited to diverging or converging
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117 geometries, it can be expanded into other applications that involve a geometric transformation between the actual and virtual tributary areas. Alluvial fan type flood hazard in the arid southwest are often underestimated due to absence of welldefined waterways, a longer rainfall interevent time and lack of rainfall and stream gage records. It is an engineering challenge during the master drainage study type planning exercises and detail design phases on this type of land form and climate. In this research work, KW sloping plane overland flow theory is utilized to explore new approaches to respond to alluvial fan type hydrology. The major effort is to expand the understanding of KW overland flow theory from a rectangular plane to a fan shape surface. This effort successfully demonstrates the derivation of different governing equations. These equations are expanded to provide semianalytical and numerical solutions to simulate hydraulic features by solving KW overland flow depth and velocity for both diverging and converging type planes. The simulation results are compared to laboratory data and field records and presents with good agreement. These output parameters such as flow depth and flow velocity can be utilized to generate flood flow hydrograph to assist delineation of the floodplain boundary limits on an alluvial fan type landscape. Aided by the shape factor geometric transformation method, it increases the flexibility to apply computer models for surface runoff simulation on alluvial fans. It is believed that the methods presented in this study can improve not only the current engineering practices but also the FEMA flood flow study procedure in studying alluvial fan. With the use of this research, it can and will benefit and potentially minimize the environmental impact footprint while urbanization continues on the alluvial fan type landform.
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118 REFERENCES Agiralioglu, N.; Singh, V.P. (1980), Â“A mathematical investigation of diverging surfce overland flow, 2. Numerical solutions and applicationsÂ”, Tech. Rep. MSSUEIRSCE804, Eng and Industrial Res. Station, Mississippi State University, Mississippi State, Mississippi Anon (1981), Â“Guidelines for determining flood flow frequencyÂ”, Bulletin 17b, U.S. Water Resources Council, Washington, D.C. Anstey, R.L. (1965), Â“Physical charateristics of alluvial fansÂ”, Natick, MA: Army Natick Laboratory, Technical Report ES20 Bedient, P.B. and Huber, W.C., (1992), "Hydrology and Floodplain Analysis", 2nd Edition, Addison Wesley Publishing Company, New York. Blackler, G. and Guo, James C.Y., (2012), Â“Field Test of Paved Area Reduction Factors using a Storm Water Management Model and Water Quality Test Site," Vol. 17, No. 8, Journal of Irrigation and Drainage Engineering, ASCE, ISSN 07339437/(0). IRENG6569, August. Blair T.C., McPherson J.G. (2009), Â“Process and forms of Alluvial FansÂ”, Chapter 14, Geomorphology of Desert Enviroment, 2 nd Edition. Bull, W.B. (1977), Â“The Alluvial Fan EnviromentÂ”, Progress in Physical Geography 1:222270 Burkham, D. E. (1988), Â“Methods for Delineating FloodProne Areas in the Great Basin of Nevada and Adjacent StatesÂ”, USGS WaterSupply Paper 2316, Washington, D.C. CCRFCD (1998), Â“Clark County Regioanl Flood Control District Hydroloic Criteria and Drainage Design ManualÂ”, Clark County Regioanl Flood Control District, 1998 City of Las Vegas (2011), Â“Unified Development Code Â– Title 19Â”, City of Las Vegas, Nevada Chow, V.T., (1964), "Handbook of Applied Hydrology", McGrawHill Book Company, Chapters 17 and 21. CUHP (2005), Â“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), Â“Modification of Kinematic Wave Method for Cascading PlaneÂ”, PhD dissertation, Dept of Civil Engineering, U of Colorado Denver, December.
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119 Dawdy, D.R. (1979), Â“FloodFrequency Estimates on Alluvail FanÂ”, J. of Hydraulic Division, ASCE, Proceedings, Vol. 105, No. HY11, pp. 14071413 DaluzVieira, J. H. (1983), Â“Conditions Governing the Use of Approximations for the SaintVenant Equations for Shallow Surface Water FlowÂ”, J. of Hydrology, 60, pp. 4358. Drew, F. (1873), Â“Alluvial and lacustrine deposits and glacial records of the Upper Indus BasinÂ”, Geological Society of London Quarterly Journal 29, pp. 441471 DiGiano, F.A., Adrain, D.D. and Mangarella, P.A. (1976), Â“Short Course Proceedings Â– Applications of Stormwater Management ModelsÂ”, EPA600/277065, EPA Cincinnati, OH EPA (2010), Â“Storm Water Management ModelÂ”, EPA, Cincinnati, OH Eagleson, P.S. (1970), Â“Dynamics of Flood FrequencyÂ”, Water Resources Res., 8(4), pp 878898 FAN: An alluvial fan flooding computer program (1990), Federal Emergency Management Agency, Washington, D.C. FEMA (1981), Â“Flood Plain Management Tools for Alluvial Fan Â– Final ReportÂ”, Federal Emergency Management Agency, Washington D.C. FEMA (1989), Â“Alluvial Fans: Hazard and ManagementÂ”, Federal Emergency Management Agency, Washington D.C. FEMA (2000), Â“Guidelines for determine flood hazards on alluvial fansÂ”, Federal Emergency Management Agency, Washington D.C. French, R.H. (1992), Â“Design of Flood Protection for Transportation Alignments on Alluvial FanÂ” J. Irrigation and Draiange Eng. Vol. 118, No. 2, April. French, R.H. (1993), Â“Preferred Directions of Flow on Alluvial FansÂ” J. Hydraul. Eng. Vol. 118, No. 7, July. French, R.H., Fuller, J.E. and Waters, S. (1993), Â“Alluvial Fan: Proposed new processoriented definitions for the Arid SouthwestÂ” J. Water Resources Planning and Management, v. 119, 588598. Fuller, J.E. (1990), Â“Misapplication of the FEMA Alluvial Fan Model: A Case HistoryÂ”, Proceedings of the ASCE Conference on Hydrology and Hydraulics of Arid Land, San Diego, CA Fuller, J.E. (2011), Â“Top Ten Reasons the FEMA FAN Model Gave You the Wrong AnswerÂ”, Floodplain Management Association Annual Conference, San Diego, CA. September 711, 2011
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120 Fuller, J.E. (2012), Â“Theoretical and Practical Deficiencies in the FEMA FAN MethodologyÂ”, Arizona Geological Survey Continued Report CR12B, January. Guo, James C.Y. (1998), "Overland Flow on a Pervious Surface", IWRA International J. of Water, Vol 23, No 2, June. Guo, James C.Y. (2006a), Â“Dimensionless Kinematic Wave Unit Hydrograph for Storm Water PredictionsÂ” ASCE J. of Irrigation and Drainage Engineering, Vol. 132, No. 4, July. Guo, James. C.Y. (2006b), Â“Stormwater Predictions using Dimensionless Unit HydrographÂ”, Vol 132, No. 4, ASCE J. of Irrigation and Drainage Engineering, July/August. Guo, James C.Y. and Urbanos B. (2009), "Conversion of Natural Watershed to Kinematic Wave Cascading Plane" Journal of Hydrologic Engineering, Vol. 14, No 8, August, 839846. Guo, James C. Y., Cheng, Jeff, Wright, L.(2012), Â“Field Test on Conversion of Natural Watershed into Kinematic Wave Rectangular Planes, ASCE J. of Hydrologic Engineering, Vol. 17, No. 8, August. Guo, James C.Y., Hinds, J. W. (2013), Â“Numerical Criteria for Stable Kinematic Wave Overland Flow SolutionÂ”, ASCE J. of Irrigation and Drainage Engineering, JRNIRENGS1300290, August. Guo, James C.Y., Hsu, Eric S.C. (2014), Â“Diverging Kinematic Wave FlowÂ”, ASCE J. of Irrigation and Drainage Engineering, 140(11), 04014036 Guo, James C.Y., Hsu, Eric S.C. (2015a), Â“Closure to Diverging Kinematic Wave FlowÂ”, ASCE J. of Irrigation and Drainage Engineering, 141(8), 07015002 Guo, James C. Y., Hsu, Eric S. C. (2015b), Â“Converging Kinematic Wave FlowÂ”, ASCE J. of Irrigation and Drainage Engineering, 141(8), 04015006 HECHMS (2010), Â“Hydrologic Modeling SystemÂ”, U.S. Army Corps of Engineer, Hydrologic Engineering Center, Davis, California. Henderson, F.M., and Wooding, R.A., "Overland Flow and Groundwater from a Steady Rainfall of Finite Duration," J. Geophys. Res., Vol 69, No 8., 1964, pp 3967. Holden, A.P. and Stephenson, D., "Finite Difference Formulations of Kinematic Equations", J. of Hydraulic Engineering, ASCE, pp 423426, May, 1995 Holden, A.P. and Stephenson, D., "Improved fourpoint Solution of the Kinematic Equations.", J. of Hydro. Res, 26(4), 1988, pp 413423.
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121 Horton, R.E. (1938), "The Interpretation and Application of Runoff Plot Experiments with Reference to Soil Erosion Problems," Proceeding, Coil Science Society of America, Vol 3, pp 340349. Hydrologic Engineering Center (HEC) (1993), Â“Introduction and Application of Kinematic Wave Routing Techniques Using HEC1Â”, United States Army Corps of Engineers, Davis, CA Huber, W.C., and Dickinson, R.E. (1988), Â“Storm water management model userÂ’s manualÂ”, Version 4, EPA, Athens, Ga. Izzard, C.F. (1946), Â“Hydraulics of runoff from developed surfacesÂ” Proceedings of the Highway Research Board, Vol. 26, pp. 129146. Liggett, J.A., and Woolhiser, D.A (1967), "Finite Difference Solution for the Shallow Water Equations." ASCE J. Engrg. Mech. Div., 93(2), pp 3971 Lighthill, M.H. and Whitham, G.B. (1955), Â“On Kinematic Waves, I, Flood Movement in Long RiverÂ”, Proceeding of Royal Society, London, Ser. A., Vol 229, pp 281316 Langford, K.J. and Turner, A.K. (1973), Â“An experimental study of the application of kinematic wave theory to overland flowÂ”, J. Hydrol. 18, 125245 McGinn, R.A. (1980), Â“Flood Frequency Estimates on Alluvial Fans Â– DiscussionÂ”, ASCE J. of Hydraulics Division, HY10, pp. 17181719. McCuen, R. H. (2005), Â“Hydrologic Analysis and DesignÂ”, 3 rd Edition, Pearson Prentice Hall Inc., N.J. Mukhopadhyay, B., Cornelius, J., and Zehner, W. (2003), Â“Application of kinematic wave theory for predicting flash flood hazards on coupled alluvial fanÂ–piedmont plain landformsÂ”, Hydrol. Process. Wiley InterScience (www.interscience.wiley.com Vol 17, pp. 839Â–868 Muzik, I. (1973), Â“State Variable Model of Surface Runoff from a Laboratory CatchmentÂ” PhD Dissertation, University of Alberta, Edmonton, Alberta, Canada. NOAA Atlas 14 (2011), Â“Precipitation Â– Frequency Atlas of United StatesÂ”, Volume 1 Version 5, U.S. Department of Commerce. National Research Council (1996), Â“Alluvial Fan FloodingÂ”, Washington D.C., National Academy Press, p172 Ponce, V. M. (1989), Â“Engineering hydrology, principles and practicesÂ”, PrenticeHall, Englewood Cliffs, N.J.
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122 Proctor and Redfern, Ltd, and MacLaren, J.F. Ltd, (1976), Â“Stormwater Management Model Study Â– Vol I, Research Report No. 7, CanadaOntario Research Program, Environmental Protection Service, Ottawa, Ontario, September. Roache, P. J. (1976), Â“Computational Fluid DynamicsÂ”, Hermosa Publishers Inc., New Mexcio. Rossman, L.A. (2005), Â“Storm Water Management Model UserÂ’s Manual. Version 5Â”, Water Supply and Water Resources Division, National Risk Management Research Laboratory, Cincinnati, OH. Singh, V.P. (1975), Â“A Laboratory Investigation of Surface RunoffÂ” Journal of Hydrology, 25(2), 187200. Singh, V.P. (1976), Â“A Distributed Converging Overland Flow Model 3. Application to Natural WatershedsÂ” Journal of Hydrology, Vol. 31, 221243. Singh, V.P.; Agiralioglu, N. (1981), Â“Diverging Overland FlowÂ”, Adv. Water Resources, Vol 4, September, pp 117124 UDFCD (2001), Â“Urban Storm Water Drainage Criteria Manual.Â” Vol 1 Urban Drainage and Flood Control District, Denver, Colorado. USSWMM (2000), Â“Urban Drainage Storm Water Management ModelÂ”, Urban Drainage and Flood Control District, Denver, Colorado. U.S. Census Bureu (2011), Â“Population Distribution and Change: 2000 to 2010Â”, United States Department of Commerce USACE (1993), Â“Assessment of Structure FloodControl Measures on Alluvial FanÂ”. Davis, California: USACE Hydrologic Engineering Center USGS (2007), Â“Natural Hazard on Alluvial Fans: The Venezuela Debris Flow and Flash Flood DisasterÂ”, United States Department of Interior U.S. Soil Conservation Service (1975), Â“History of Flooding, Clark County, NevadaÂ”, United States Department of Agriculture Wooding, R.A. (1965), Â“A Hydraulic Mode for a CatchmentStream ProblemÂ”, J. of Hydrology, Vol 3, pp 254267 Woolhiser, D.A.and Ligget J.A. (1969), Â“Overland Flow On A Converging SurfaceÂ”, Trans. ASAE 12(4), pp 460462 Yen, B.C. (1973), Â“Open Â–channel flow equation revisitedÂ”, J. Eng. Mech. Div., Am. Soc. Civ. Engr., Vol. 99, no. EM5, pp. 9791009
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123 Yen, B. C. and Chow, V.T. (1974), Â“Experimental Investigation of Watershed Surface RunoffÂ”, Hydraulic Engineer Series, No. 29, Dept of Civil Engineering, U of Illinois at UrbanaChampaign, September, 1974 Yu, U. S. and McNown, J. S. (1964), Â“Runoff patterns and their significanceÂ”, J. Geol. 40, 498521 Zhang, G. S. and Hamlett, J. M., (2006) Â“Development of the SWMM hydrologic model for the Fox Hollow Watershed, Centre County, PAÂ”, prepared in cooperation with Office of Physical Plant Penn State University, University Park, PA 16802.
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124 APPENDIX A Â– ASCE Journal Papers
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DivergingKinematicWaveFlowJamesC.Y.Guo,M.ASCE1;andEricS.C.Hsu,M.ASCE2Abstract: Thegoverningequationforkinematicwave(KW)flowonadivergingalluvialfanisderivedandpresentedwithcomparisonwith observeddata.ThekeyfactorstodescribeadivergingKWflowarefoundtobetheangleofapex,radiusofalluvialfanarea,groundsurface slope,andrainfallintensity.Thesensitivitytestsindicatethatunderaspecifiedcondition,therectangularKWflowistheenvelopingcurvefor alldivergingKWflows.Thewidertheangleofapex,thecloserthedivergingKWflowbecomessimilartotherectangularKWflow.The analyticalsolutionderivedinthisstudyprovidesthetimetoequilibriumandthepeakflowonthedivergingKWhydrograph.Bothparameters areimportantforpeakflowpredictions.Thenewequationderivedinthisstudyoffersadeterministicapproachtodefinethebasefloodflow andthecorrespondingfloodplainboundariesonthealluvialfan.Itisbelievedthattheapproachoutlinedinthispaperpresentsasignificant improvementtothecurrentpracticeonalluvialfloodplainhydrologyandhydraulics. DOI: 10.1061/(ASCE)IR.19434774.0000767 2014 AmericanSocietyofCivilEngineers. Authorkeywords: Kinematicwave;Alluvialfan;Floodplain.IntroductionAlluvialfanisoftenlocatedatatopographicbreak,suchasawash mouthatthebaseofamountainfront.Continualtransportofsedimentsinfluvialflowsshapesthealluvialdepositintoafullorpartialfanarea( NationalResearchCouncil1996 ).Underasevererain storm,thesheetflowsonanalluvialfanareformedalongnumerous flowpathsthatarecharacterizedbynetworksofshallowgrooves withhighvelocitiesassociatedwithsteepgradients.FEMA( 1989 2000 )recommendedthatastochasticmethodologybeappliedto delineatingfloodhazardzonesonalluvialfansoratraditional hydraulicmethodbeusedtodefinethewatersurfaceprofileif thebasefloodflowcanbedeterminedandtraced( French1992 ). Combinedwiththelargeuncertaintiesinthemagnitudesandboundariesoffloodflows,predictionsoffloodhazardsonanalluvialfan arenotaneasytask.Asreported( Mukhopadhyayetal.2003 ),these complicatedhydrologicprocessesmaybebestmodeledwiththe aidofkinematicwave(KW)theoryofoverlandflow.Thesolution ofKWtheoryunderauniformrainfallexcesswaswelldocumented inmanyreports( Woodings1965 WoolhiserandLigget1967 ). Onapervioussurface,thetimeofequilibriumwasexpandedinto thetimeofconcentration,andtheKWsolutionwasmodifiedwith Horton Â’ sformulaforvarioustypesofsoil( Guo1998 ).Forconvenience,theKWgoverningequationswereconvertedfromtherectangularintothecylindricalcoordinatesystemsforKWstudieson alluvialfans.SuchKWgoverningequationswerefirstderivedfor thefiniteflowdifferenceoveracircularsegmentacrossanalluvial fanataselectedradius,andthenintegratedfortheselectedapex angleforthepredictionofthetotalflowovertheentirefanarea ( AgiraliogluandSingh1980 ).Althoughcasestudieswere presentedtodemonstratetheapplicationoftheKWapproachto alluvialfanflows,thedualintegrationprocessesareverycompicatedtoachievetheanalyticalornumericalsolutions( Singhand Agiralioglu1981 ). Inthisstudy,theentirealluivalfanareaistreatedasashallow reservoirtoderivetheKWgoverningequation.Thewatervolume balanceamongrainfallamount,inflowandoutflowrates,andsurfacedetentionvolumeundertheKWprofileprovidesanewgoverningequation.Althoughtheanalyticalsolutionsfortherisingand recedinghydrographshavenotbeendevelopedyet,thepeakflow rateandflowdepthatthetimeofequilibriumcanbeanalytically derivedforthepurposeoffloodplaindelineation.Thenumerical solutionsofoverlandflowhydrographusingthenewKWequation foralluvialfanflowsagreewellwiththecasestudiesinthepreviousreportsandlaboratorydata.Itisbelievedthattheapproach outlinedinthispapercanbeagreatassistancetoimprovethe currentmethodforalluvialfloodplaindelineationprocess.DivergingKinematicWaveFlowTheKWflowonafanareacanbeanalyzedwithanarcsegment approachasshowninFig. 1 .Thewatervolumebalanceisdeterminedbytheinflow,outflow,andstoragedepthinthearcsegment. Onadivergingfanarea,theradiusofthefanareaincreasesdownstream,sodoestheaccumulationofoverlandflow.Accordingto thecontinuityprinciple,thewatervolumebalancewithinthearc segmentissummedofthefollowingitems: ie A vy R r f v v y y R r r g y t A 1 A R r r 2 q vy 3 where ie=excessrainfallintensity, A =divergingsurfacearea, R =radiusforthedivergingarea, r =locationofthesegment orrepresentingthedistance, R r ,fromtheapex, v =radialflow1ProfessorandDirector,HydrologyandHydraulicsGraduateProgram, CivilEngineering,Univ.ofColoradoDenver,P.O.Box173363,Campus Box113,Denver,CO802173363(correspondingauthor).Email:James .Guo@UCDenver.edu 2Ph.D.Candidate,CivilEngineering,Univ.ofColoradoDenver, P.O.Box173363,CampusBox113,Denver,CO802173363.Email: HsuE@pbworld.com Note.ThismanuscriptwassubmittedonNovember10,2013;approved onApril17,2014;publishedonlineonJune13,2014.Discussionperiod openuntilNovember13,2014;separatediscussionsmustbesubmittedfor individualpapers.Thispaperispartofthe JournalofIrrigationandDrainageEngineering ,ASCE,ISSN07339437/04014036(5)/$25.00.ASCE040140361J.Irrig.DrainEng.
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velocity, y =flowdepth, q =unitwidthdischarge, t =elapsedtime, and =angleoftheapexofthedivergingarea.Alltermswiththe symbol, ,inEqs.( 1 )and( 2 )areassociatedwiththeincremental. Aftercancellingtheproductsofdoubleandtripleincremental terms,thegoverningequationfortheflowthroughthearcsegment isreducedto y t q r ie q R r 4 Eq.( 4 )doesnotincludetheapexanglebecauseitonlydescribes thebalanceofflowsthroughthearcsegmentatthelocation, ( R r ),fromtheapexasillustratedinFig. 1 .Eq.( 4 )agreeswith thepreviousreport,andcanbefurtherintegratedfrom r 0 to r R byapplyingabinomialfunctiontotheangleofapexand theradiusofthefanarea( AgiraliogluandSingh1980 ). Inthisstudy,thetributaryfanareaisoutlinedfromthelocation, R r ,to R asillustratedinFig. 2 .Thegeometricparametersforthe tributaryfanareaare A 1 2 R2 R r 2 1 2 r 2 R r 5 lr R 6 where lr=arclengthforoutflowreleasedfromthefanarea. Consideringthatthetributaryareafunctionsasashallowwater reservoir,therainfallexcessistheinflow,whilethesheetflow releasedthroughthedownstreamarclengthistheoutflow.Thebalanceofwatervolumeforthefanareaissummarizedas ieA qlr dy dt A 7 AidedwithEqs.( 5 )and( 6 ),Eq.( 7 )yields dy dt ie 2 q r 2 a 8 r R a 9 where a =spatialratioforupperboundary.Eq.( 8 )servesasthe governingequationfortheKWoverlandflowonadiverging fanarea.Ifthetributaryareastartsfromtheapexofthediverging fanshape,then r R and a 1 orEq.( 8 )isreducedto dy dt ie 2 q R 10 q kym 11 Q qlr qR 12 where k =constantontheKWratingcurve,dependingon Manning Â’ sroughnesscoefficientandsurfaceslope, m =exponent ontheKWratingcurvevariedbetween 3 = 2 and 5 = 3 ( HEC1993 ), and Q =totaldischargefromfanarea.Eq.( 10 )describesthe relationshipbetweenflowdepthandunitwidthflowrateover thedivergingfanarea.Analyticalornumericalintegrationof Eq.( 9 )providesdirectsolutionstothefloodflowonthealluvial fan.ApplicationsofEq.( 9 )toanexistingalluvialfanwithknown geometrywillsatisfytheFEMA Â’ sfloodplaindelineationprocedure foridentifyingfloodhazardzones. Underalongrainfallevent,therunoffflowcontinuallyincreases toitspeak.Undertheequilibriumcondition,therainfallamountis balancedwiththeoutflowvolumefromthealluvialfanarea,orthe surfacedetentionremainsunchanged.With dy = dt 0 inEq.( 10 ), thepeakflowforadivergingalluvialfanundertheequilibrium conditionisderivedas qe 1 2 ieR 13 ve kym 1 e 14 Bydefinition,thetimeofconcentrationisthetimerequiredfor theflowfromtheupperboundarytoreachtheoutletas TC R ve R k y1 m e 15 where TC=timeofconcentrationin T .Whentherainfallevent durationisshorterthan TC,itisashortrainfallevent;otherwiseitis alongevent.Underalongrainfallevent,thepeakflowfromthe divergingplaneis Fig.1. Arcsegmentondivergingfanarea Fig.2. FanshapeandfanareafordivergingKWflowASCE040140362J.Irrig.DrainEng.
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Qe 1 2 ieR2 16 where qe=unitwidthpeakflowin L2= T Q =totalpeakflow fromfanareain L2 ,and ve=peakflowvelocityin L = T Eq.( 16 )agreeswiththerationalmethodwitharunoffcoefficient ofunity.Whentherainfalleventdurationisshorterthanthetimeof concentration,itisashortrainfallevent;otherwiseitisalongevent.NumericalSolutionNumericalschemesprovidediscretesolutionatthegridsina numericalmeshnetwork.Forthisstudy,aidedwithEq.( 11 ), Eq.( 10 )isconvertedintoitscentralfinitedifferenceequationas y t t y t t n ie t t ie t 2 k y t t m y t m 2 R 17 where n 0 or1,dependingonrainfallinput.Theinitialcondition forEq.( 17 )isadrybedconditionor y t ; r 0 everywhereat t 0 .Theupstreamboundaryremainsdryatalltimes,while thedownstreamboundaryconditioncanbedescribedasthenormal flowdepth.Eq.( 17 )canbeprogressivelysolvedforeverytimestep with n 1 iftheraincontinuesor n 0 aftertherainceases.Asa singlevariablefunction,Eq.( 17 )isasimpleaccumulationofflow volumesforagiventimeperiod;thereforethereisnoissueof numericalstability.ModelVerificationandCaseStudyThelaboratorydivergingflowmodelonimpervioussurfaceapplicationwasfirstdocumentedbyIzzard( Izzard1946 ),andthen followedbymanyinvestigators( LangfordandTurner1973 ).As reported( Muzik1973 ),KWflowswereproducedunderalaboratoryshowersimulationofrainfallintensityof 78 mm = honarectangulargalvanizedplaneof0.61by0.91mlaidonaslopeof 0.2079.Thisarrangementwasevaluatedtobeequivalenttoa divergingKWplanewitharadiusof0.91mandanapexangle of1.34rad( AgiraliogluandSingh1980 ).Forthiscase,theparametersontheKWratingcurvearedeterminedas 1 486 n S0 5 0 67 6 18 m 5 3 19 where n =Manning Â’ sfrictioncoefficientequalto0.01forgalvanizedsurface( Muzik1973 ). Fig. 3 presentsthecomparisonsbetweenlaboratorydataand analyticalsolutionsforrectangularanddivergingKWflows. Asexpected,bothrectangularanddivergingplanesproducethe samepeakflowaspredictedbyEq.( 13 ),butthedivergingKW flowischaracterizedwithaslowermovement,oralongertime ofconcentrationaspredictedinEq.( 15 ).Eq.( 17 )agreeswellwith thepreviousstudy( AgiraliogluandSingh1980 ).Detailsare summarizedinTable 1 AsshowninFig. 4 ,thetestinFig. 3 wasrepeatedwithahigher rainfallintensityat 115 mm = h.ThepeakflowinFig. 4 isincreased from12to 18 cm3= s,accordingtotherainfallratioisraisedfrom 78to 115 mm = h.BothstudiesshowthatthedivergingKWflow moveswithashallowerdepthandataslowervelocityincomparisonwiththerectangularKWflow.Again,goodagreementis Fig.3. Comparisonoflaboratoryandnumericalsolutionsfordiverging KWflowsunderrainfallintensityof 78 mm = hfor50s Table1. DivergingKWFlowsunderRainfallIntensityof 78 mm = h for50s Time Muzik ( 1973 ) Rectangular KWanalytical solution Agiralioglu andSingh ( 1980 ) Diverging KWanalytical solution t (s)Dischargeq (cm3= s) 0.00.00.00.00.0 4.01.04.91.02.6 8.02.512.46.08.6 12.06.712.410.811.1 16.011.012.412.011.7 20.012.012.412.011.9 24.012.012.412.011.9 28.012.012.412.012.0 32.012.012.412.011.9 36.012.012.412.011.9 40.012.012.412.011.9 44.012.012.412.011.9 48.012.012.412.011.9 52.08.710.38.610.4 56.04.03.23.94.0 60.03.01.22.41.8 64.01.80.61.41.0 Fig.4. Comparisonoflaboratoryandnumericalsolutionsfordiverging KWflowsunderrainfallintensityof 115 mm = hfor50sASCE040140363J.Irrig.DrainEng.
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achievedbetweenEq.( 17 )andthepreviousreport( Agiraliogluand Singh1980 ).ThedetailsaresummarizedinTable 2 AsshowninFig. 5 ,thetestinFig. 4 wasrepeatedwithvarious apexangles.Therainfallintensityissettobe 115 mm = h.Itis noticedthatthepeakflowatthetimeofconcentrationispreserved bythedivergingsurfaceareaforallcases;andtherectangularKW flowisthelimitingcaseforalldivergingflowswithvariousapex angles.TheresultsarepresentedinFig. 5 .Astheangleincreases, thedivergingKWflowbecomesmoreconvergingtotherectangularKWflow.DetailsforthesedivergingKWflowsaresummarized inTable 3 .ConclusionsTheKWflowonadivergingfanareaisprescribedbytheangleof apex,radiusofthefanarea,groundslope,andrainfallstatistics. Thewatervolumebalanceamonginflows,outflows,andrainfall amountconvertsthegeneralhydrologicequationintothegoverning equationfordivergingKWflowasshowninEq.( 10 ).Numerical integrationofEq.( 10 )producessimilarsolutionstotheprevious studyusingabinomialfunctiontoaccumulatetheflowthroughthe angleofapexandtheradiusofthefanarea( SinghandAgiralioglu 1981 ).Incomparison,Eq.( 10 ),ismuchsimplerandeasiertoproducenumericalsolutionsfordivergingKWflows. Eqs.( 13 ) Â– ( 16 )describetheequiliriumconditionfordiverging KWflow.Eq.( 16 )isaspecialcaseoftherationalmethod. Forfloodplainstudies( FAN1990 ),theconservativeapproachis tosetthecriticaldesignrainfalldurationtobethetimeofequilirium.AidedbyEq.( 15 ),thepeakflowcanbedirectlycalculated fromthelocalrainfallintensitydurationfrequency(IDF)curve fortheselctedbaseflood.Thisstudypresentsanewdeterministic approachtodelineatefloodplainboundariesonanalluvialfan area.Thenumericalsolutionsprovidetheentirewatersurface profileacrossthefanarea.Thisnewmethodwilldefinitelyimprove thecurrentpracticeinhydrologicengineeringdesignsonalluvialfans.ReferencesAgiralioglu,N.,andSingh,V.P.(1980). Â“ Amathematicalinvestigationof divergingsurfaceoverlandflow,2.Numericalsolutionsandapplications. Â” TechnicalRep.MSSUEIRSCE804 ,EngandIndustrialRes. Station,MississippiStateUniv.,MississippiState,MS. FAN:Analluvialfanfloodingcomputerprogram.(1990).Federal EmergencyManagementAgency,Washington,DC. FEMA.(1989). Â“ Alluvialfans:Hazardsandmanagement. Â” Federal EmergencyManagementAgency,Washington,DC. FEMA.(2000). Â“ Guidelinesfordeterminefloodhazardsonalluvialfans. Â” FederalEmergencyManagementAgency,Washington,DC. French,R.H.(1992). Â“ Preferreddirectionsofflowonalluvialfans. Â” J.Hydraul.Eng. ,10.1061/(ASCE)07339429(1992)118:7(1002), 1002 Â– 1013 Guo,J.C.Y.(1998). Â“ Overlandflowonapervioussurface. Â” WaterInt. 23(2),91 Â– 96. HydrologicEngineeringCenter.(1993). Â“ Introductionandapplicationof kinematicwaveroutingtechniquesusingHEC1. Â” UnitedStatesArmy CorpsofEngineers,Davis,CA. Izzard,C.F.(1946). Â“ Hydraulicsofrunofffromdevelopedsurfaces. Â” Proc. HighwayRes.Board ,26,129 Â– 146. Langford,K.J.,andTurner,A.K.(1973). Â“ Anexperimentalstudyofthe applicationofkinematicwavetheorytooverlandflow. Â” J.Hydrol. 18(2),125 Â– 245. Mukhopadhyay,B.,Cornelius,J.,andZehner,W.(2003). Â“ Applicationof kinematicwavetheoryforpredictingflashfloodhazardsoncoupled Fig.5. DivergingKWflowswithvariousapexangles Table3. DivergingKWFlowswithVariousApexAngles TimeMuzik( 1973 )60KWplane120KWplane180KWplane t (s)Dischargeq (cm3= s) 0.00.00.00.00.0 4.01.74.25.36.1 8.09.013.415.015.8 12.017.616.717.217.4 16.017.617.417.517.6 20.017.617.517.617.6 24.017.617.617.617.6 28.017.617.617.617.6 32.017.617.617.617.6 36.017.617.617.617.6 40.017.617.617.617.6 44.017.617.617.617.6 48.017.617.617.617.6 52.08.615.214.914.6 56.03.95.54.43.9 60.02.12.31.71.4 64.01.11.20.90.7 Table2. DivergingKWFlowsunderRainfallIntensityof 115 mm = h for50s Time Muzik ( 1973 ) Rectangular KWanalytical solution Agiralioglu andSingh ( 1980 ) Diverging KWanalytical solution t (s)Dischargeq (cm3= s) 0.00.00.00.00.0 4.01.79.34.64.6 8.09.018.214.814.0 12.017.618.218.016.9 16.017.618.218.017.5 20.017.618.218.017.6 24.017.618.218.017.6 28.017.618.218.017.6 32.017.618.218.017.6 36.017.618.218.017.6 40.017.618.218.017.6 44.017.618.218.017.6 48.017.618.218.017.6 52.08.610.38.615.1 56.03.93.23.95.1 60.02.11.22.12.1 64.01.10.61.11.1ASCE040140364J.Irrig.DrainEng.
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alluvialfan Â– piedmontplainlandforms. Â” Hydrol.Process. ,17(4), 839 Â– 868. Muzik,I.(1973). Â“ Statevariablemodelofsurfacerunofffromalaboratory catchment. Â” Ph.D.dissertation,Univ.ofAlberta,Edmonton,AB, Canada. NationalResearchCouncil.(1996). Â“ Alluvialfanflooding. Â” National AcademyPress,Washington,DC,172. Singh,V.P.,andAgiralioglu,N.(1981). Â“ Divergingoverlandflow. Â” Adv. WaterResour. ,4(3),117 Â– 124. Wooding,R.A.(1965). Â“ Ahydraulicmodeforacatchmentstream problem. Â” J.Hydrol. ,3(3 Â– 4),254 Â– 267. Woolhiser,D.A.,andLigget,J.A.(1967). Â“ Unsteadyonedimensional flowoveraplane:Therisinghydrograph. Â” WaterResour.Res. ,3(3), 753 Â– 771.ASCE040140365J.Irrig.DrainEng.
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DiscussionsandClosuresClosureto Â“ DivergingKinematicWaveFlow Â” byJamesC.Y.GuoandEricS.C.HsuDOI: 10.1061/(ASCE)IR.19434774.0000767JamesC.Y.Guo1andEricS.C.Hsu21ProfessorandDirector,HydrologyandHydraulicsProgram,Univ.of Colorado,P.O.Box173364,CampusBox113,Denver,CO802173363(correspondingauthor).Email:James.Guo@UCDenver.edu2GraduateStudent,CivilEngineering,Univ.ofColorado,Denver,CO 802173363.Email:HsuE@pbworld.comThediscusserhasexpressedaninterestintherisinghydrographof thedivergingkinematicwave(KW)flow.Undertheassumption thatthedivergingKWflowstartsfromtheapexofthediverging fanarea,thegoverningequationisreducedto dy dt ie 2 q R 1 q kym 2 Q qlr qR 3 where y =unitwideoverlandflowdepth(L); t =elapsedtime(T), q =unitwidthoverlandflow(L2= T); ie=excessrainfallintensity (L = T); R =radiusofdivergingfanarea(L); k =constantontheKW ratingcurve,dependingonManning Â’ sroughnesscoefficientand surfaceslope;m=exponentontheKWratingcurvevariedbetween 3 = 2 and 5 = 3 [ HydrologicEngineeringCenter(HEC)1993 ]; lr=arc lengthofthedownstreamboundaryofdivergingfanarea;and Q = totaldischargefromthefanarea(L3= T).Underalonguniform rainfallexcess,theintegrationofEq.( 1 )from t 0 to t Tc, and y 0 to y ye,representstherisinghydrograph( Guo2006 ) Zt t t 0dt Zy y y 0 1 ie 2 kymRdy 4 where 0 t Tc;and Tc=timeofconcentration(T).From t Tcto t Tdisthepeakinghydrograph,inwhich Td=rainfall duration(T).EquilibriumFlowConditionDuringthepeakingtime, dy = dt 0 inEq.( 1 ).AidedwithEq.( 2 ), thepeakflowrate,peakflowvelocity,andtimeofconcentrationare derivedas ye ieR 2 ve 5 ve kym 1 e 6 where ye=peakflowdepth(L);and ve=KWvelocity.Bydefinition,thetimeofconcentrationistheflowtimefromtheupstream boundarytothedesignpointas TC R ve R k y1 m e 7 where Tc=timeofconcentration(T).Inpractice,mosthydraulic structuresaredesignedtopassthepeakflow.Eqs.( 5 )and( 7 ) providethebasicinformationforhydraulicdesigns.RisingHydrographEq.( 4 )representstherisinghydrograph.Itiswellunderstood thattheKWspeedisvariedwithrespecttotheflowdepth.Aided withEq.( 6 ),Eq.( 4 )canbeapproximatedwithaconstantKW speedas Zt t t 0dt Zy y y 0 1 ie 2 vey Rdy 8 where 0 t Tc. Lettheintegrationvariablebechangedto Y ie 2 veR y 9 TakingthefirstderivativeofEq.( 9 )andthensubstituting dY for dy yields t R 2 veZY Y Y ie 1 Y dY R 2 veln 1 2 veieR y K 10 where K isanintegrationconstantthatservesasascalefactorto incorporatetheconditionof 0 t Tcintothesolution.ThesolutionforEq.( 10 )isderivedas y ye 1 exp 2 vet R 1 exp 2 t Tc 1 1 t Tc 11 AsillustratedinFig. 1 ,thenondimensionalrisinghydrograph derivedinEq.( 11 )iscomparedwiththenumericalsolution.The flowdepthsfortheearlyrisinghydrographareshallowerthanthe equilibriumdepth.Eq.( 11 )tendstooverestimatethedivergingKW flow.Astimeapproachesthetimeofconcentration,Eq.( 11 )agrees withthenumericalsolutionverywell. Fig.1. Dimensionlessrisinghydrographfordivergingkinematic waveflowASCE070150021J.Irrig.DrainEng.
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Inthelaboratory,aKWoverlandflowisgeneratedbyan artificialsprinklershowersystem.Thewritersagreewiththe discusser Â’ scommentonflowmeasurement.Thethinoverland flowisnotdirectlymeasureable.Aftertheunitwidthoverland flowsarecollectedintoaflume,thetotalflowinEq.( 3 )is measuredwithacalibratedweirusingthedepthflowratingcurve relationship.ReferencesGuo,J.C.Y.(2006). Â“ Dimensionlesskinematicwaveunithydrographfor stormwaterpredictions. Â” J.Irrig.Drain.Eng. 10.1061/(ASCE)0733 9437(2006)132:4(410) ,410 Â– 417. HEC(HydrologicEngineeringCenter).(1993). IntroductionandapplicationofkinematicwaveroutingtechniquesusingHEC1 ,U.S.Army CorpsofEngineers,Davis,CA.ASCE070150022J.Irrig.DrainEng.
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ConvergingKinematicWaveFlowJamesC.Y.Guo1andEricS.C.Hsu2Abstract: Thegoverningequationforconvergingkinematicwave(KW)flowwasderivedandcomparedwithlaboratorymeasurementand fieldtests.ThekeyfactorsforconvergingKWflowsarefoundtobeexcessrainfalldepth,surfaceslopeintheradialflowdirection,and converginggeometryintermsoffanangleandradius.AlthoughthecompleteanalyticalsolutionsforKWflowrateanddepthonaconverging surfacearenotyetfoundinthisstudy,thepeakflowunderequilibriumconditionhasbeenderivedforthepredictionofdesignflow.The numericalsolutionprovidesadeterministicapproachtopredicttheKWwatersurfaceprofileonaconvergingsurface.Thisnewmethodwill definitelyimprovethecurrentpracticefordetermingtheboundariesofafloodplainonaconvergingalluvialfanandtheradialflowpatterns towardsaculvertentrance. DOI: 10.1061/(ASCE)IR.19434774.0000868 2015AmericanSocietyofCivilEngineers. Authorkeywords: Convergingplane;Kinematicwave;Alluvialfan;Floodplain.IntroductionWhereveratransportationalignmentrunsacrossanalluvialfan,the accessivefloodwatertendstoinnudatethesagpointsalongthe transportationalignment( FEMA1989 2000 ).Itisalwaysachallengeastohowtoassesstheleveloffloodprotectionatacrossing culvertunderthetransportationalignment( French1989 ).AsillustratedinFig. 1 ,thehighwayalignmentisdividedintoseveral segments,accordingtothelocationsofcrossingculverts.Infront ofaculvert,theoverlandflowsgeneratedfromtheupperfan areasformaconvergingflowtowardstheentranceoftheculvert ( French1992 ). Theamountofconcentratedflowataculvertdependsonthe sizeandgeometryoftheupstreamconvergingtributaryarea. Thepeakflowratehavingbeendeterminedpersegment,dikes andbermsarethendesignedtoguidetheoverlandflowstoplung intotheentracepoolinfrontofacrossingculvert( French1992 ). ThisapproachhasbeenimplementedintheGreatBasinareasinthe southwestUnitedStates( Frenchetal.1993 ).Theconvergingflow onanalluivalfanareaisproducedunderanintensestorm.The majoruncertaintyinassessingthefloodflowsonaconvergingalluivalfanareaisdirectlyrelatedtothedirectionandmagnitudeof theflow.Althoughthisisatypicalkinematicwave(KW)overland flow,theanalyticalandnumericalsolutionshavenotbeenclearly developedyet.Inthecurrentpractice,theapplicationoftheKW theoryislimitedtotherectangularslopingplanesorallirregular tributaryareashavetobeconvertedintotheireqivalentretangular planes( Guo2000 2006 ).Theoverlandflowsonaconverging surfacearenotsuchasimpleunitwidthflowassuggestedin Wooding Â’ ssolution( Wooding1965 ).Rather,theunitwidthflow rateincreasesasitmovesdownstream( GuoandUrbano2009 ). Therefore,Wooding Â’ sKWsolutionisnotapplicabletoconverging overlandflows. Inthisstudy,theentireconvergingtributaryareaisconsidered asashallowreservoir.Atatimestep,theconvergingflowgeneratedfromthetributaryareaisanalyzedusingtheprincipleofwater volumebalanceamongrainfallexcess,inflow,outflow,andchange instoragevolumeontheconvergingsurfacearea.Anewgoverning equationwasderivedtodescribethemovementofconvergingoverlandflows.Thenumericalsolutionsdevelopedforseveralcases usingthenewconvergingKWequationagreeswellwiththelaboratorydataandfieldmeasurement( Izzard1946 ; Langfordand Turner1973 ; Muzik1973 ).Theapproachoutlinedinthispaper isdefinitelyagreatassitancetoimprovethefloodflowprediction onalluvialfans.ConvergingKinematicWaveFlowOnaconvergingsurface,rainfallinducedoverlandflowsare generatedfromaslopingsurface.Undertheassumptionoffree frombackwatereffects,thegravitationalforceisbalancedwith thefrictionforcefromthesurface( Chow1964 ).Fig. 2 illustrates thegeneralgeometryofaconvergingplaneintermsofelevation contoursonthesurface.Betweentwoadjacentelevationcontour lines,theKWflowcanbemodeledbasedontheinflowandoutflowcrossingthearcsection.Inaccordancewiththecontinuity principle,thewatervolumeisbalancedamonginflow,outflowand changeinsurfacestoragevolume.Themassbalancerelationshipis describedas ie A vy R r v v y y R r r V t y t A 1 q vy 2 A R r r 3 where ie=excessrainfallintensityin[L = T]; A =surfacearea in L2 ; v =flowvelocityin[L = T]; y =flowdepthin[L]; R =radiusofconvergingplanein[L]; r =locationofKWflow onconvergingplane; =interiorangleofconvergingarea; V =surfacestoragevolumein L3 ; t =elapsedtimein[T]; and q =unitwidthdischargein L2= T .Allincrementaltermsare expressedwithasymbolof .1ProfessorandDirector,HydrologyandHydraulicsProgram,CivilEngineering,Univ.ofColorado,Denver,CO802173363(corresponding author).Email:James.Guo@UCDenver.edu 2Ph.D.Candidate,CivilEngineering,Univ.ofColorado,Denver, CO802173363.Email:HsuE@pbworld.com Note.ThismanuscriptwassubmittedonMarch13,2014;approvedon December15,2014;publishedonlineonJanuary20,2015.DiscussionperiodopenuntilJune20,2015;separatediscussionsmustbesubmittedfor individualpapers.Thispaperispartofthe JournalofIrrigationandDrainageEngineering ,ASCE,ISSN07339437/04015006(5)/$25.00.ASCE040150061J.Irrig.DrainEng.
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CombiningEqs.( 1 ) Â– ( 3 )andeliminatingthedoubleandtriple incrementalterms,themassbalancerelationshipatlocation r is derivedas y t q r ie q R r 4 Eq.( 4 )isidenticaltotheflowequationreportedinthepreviousstudy( Singh1975 ).Eq.( 4 )doesnotinvolvetheangle oftheconvergingareabecauseitonlyrepresentsthewatervolumebalancewithinanarcarea.Obviously,theangleofthe convergingareaisakeyfactorindeterminationoftheconcentratedflowfromtheentirearea.Assuggestedintheprevious report( AgiraliogluandSingh1981 ),Eq.( 4 )canfurtherbe integratedusingabinomialfunctionthroughtheinteriorangle, ,asshowninFig. 2 .Suchanindirectsolutioniscumbersome forengineeringusebecauseit requireslengthynumerical iterations.Inthisstudy,anewapproachwasderivedtotake theentireconvergingsurfacear eaintoconsideration.Under theassumptionthatthereisnoinflowfromtheupperboundary, thevolumedifferencebetweentherainfallexcessandtheoutflowfromtheentireconvergingplanecanbedirectlyrelated tothechangeinflowdepth.Thus,theprincipleofcontinuity isrewrittenas ieA qlr V t y t A 5 AsillustratedinFig. 3 ,theconvergingtributaryareaisdefinedbytheinteriorangle, ,andtheradius, R .Thecrossing flowonaconvergingplanechangesfromonelocationtoanother. Therefore,inthisstudy,thegoverningequationforconverging flowisformedusingthelocationoftheoutletandinteriorangle, .Thetotaltributaryareaupstreamofaspecifiedlocationiscalculatedas A 1 2 r 2 R r 6 lr R r 7 where r =locationofoutlettocollectoverlandflows;and lr=arc lengthattheselectedlocationtocollectoverlandflows. InsertingEqs.( 6 )and( 7 )toEq.( 5 )yieldsthegoverningequationforconvergingoverlandflowas dy dt ie 2 q 1 a r 2 a 8 a r R 9 where a =spatialratiotodefinetheselectedlocationtocollect theoverlandflows.Eq.( 8 )representsthegoverningequation applicabletothecollectionofconvergingKWflowsatthe locationdefinedbythespatialratio.Whenrainfalldurationislongerthanthetimeofconcentration,theKWflow reachesitsequilibriumcondition.With dy = dt 0 ,Eq.( 8 )is reducedto Fig.1. Typicaltransportationlayoutonalluvialfan Fig.2. Waterbalanceinanarcsegmentwithinconvergingtributary area Fig.3. WaterbalanceoverentireconvergingtributaryareaASCE040150062J.Irrig.DrainEng.
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qe ie r 2 a 2 1 a 10 AidedwithEq.( 7 ),thetotalflowisaccumulatedtobe Qe qlr ie2 rR 2 a 11 ItisnotedthatEq.( 11 )agreeswiththerationalmethod.For instance,whenthevalueof r iscloseto R ,Eq.( 11 )isfurther reducedto QR 1 2 ieR2 ieARwhen r R 12 AR 1 2 R2 13 where QR=equilibriumdischargefromtheentireconvergingplane in L3= T ;and AR=entireconvergingsurfaceareain L2 .AccordingtotheratingcurverelationshipforKWflow,theunitwidth dischargeandflowvelocityareformulatedas qe kym e 14 ve kym 1 e 15 where m =exponentrangingbetween 3 = 2 and 5 = 3 ,and k = constantdependingonsurfaceroughnessandslope( HEC1993 ). Bydefinition,thetimeofconcentrationfortheconvergingplaneis therequiredtraveltimefortheflowfromtheupperboundaryto reachtheoutletas TC r kym 1 e 16 Eq.( 16 )iscriticallyimportanttotheapplicationsoftherational method.Theconvergingplaneissubjecttoalongrainfallevent whentherainfalldurationislongerthanthetimeofconcentration. Otherwise,therainfalleventistooshorttocovertheentireconvergingplane.NumericalSolutionNumericalschemesprovidediscretesolutionsatallgridsina numericalmeshnetwork.AidedwithEq.( 14 ),Eq.( 8 )isconverted intoitscentralfinitedifferenceequationas y t t y t t n ie t t ie t 2 k y t t m y t m 1 a r 2 a 17 where n 1 ifrainfallcontinuesoriszeroafterrainfallceases.The initialconditionforEq.( 17 )isadrybedconditionor y t ; r 0 everywhereat t 0 .Theupstreamboundaryremainsdryatall times,whereasthedownstreamboundaryconditioncanbedescribedasthenormalflowdepth.Eq.( 17 )isimplicitfortheunknown y t t .ThesolutionforEq.( 17 )ateachtimestepcanbe iterativelyachievedwith n 1 iftheraincontinuesor n 0 ifrain stops.Thenumericalconvergingtoleranceusedduringtheiterative processwassettobelessthan0.5%.ModelValidationandCaseStudyTheperformanceofthederivedgoverningequationiscompared withasetoflaboratorydata.Thelaboratoryexperimentwasdocumentedinthestudyofconvergingflowobservedandmeasured attheRainfallRunoffExperimentalFacility,theColoradoState University( Singh1975 ).Theexperimentallayoutwascomposed ofanupperconvergingplaneandalowerrectangularplane. Theconvergingplanewasdesignedwithaconstantslopeof 5%,aradiusof35.36m(116ft)andaninteriorangleof104 degrees.Thetotalsurfaceareawas 2,322 58 m2( 25,000 ft2),including 1,179 87 m2( 12,700 ft2)fromtheconvergingsectionand 1,142 71 m2( 12,300 ft2)fromtherectangularsection.Therainfall intensitywassimulatedandcontrolledthroughanarrayofsprinkler standpipes.Inthisstudy,acomparisonwasconductedfor10sets ofrainfallandrunoffexperimentalevents.Eachdatasetincludes rainfallintensity,duration,andpeakflowrates.However,therewas noinformationaboutwherethepeakflowsweremeasuredonthe convergingplane.Inthisstudy,therainfalldatasetswereusedto reproducetheobservedrunoffflowsbycalibratingthevalueof r inEq.( 10 )orthelocationoftheflowmeasurement.Theresultsare summarizedinTable 1 Eq.( 10 )isabletocloselyreproducetheobservedrunoffflows with a 0 81 .TheoutletinCases2and7wassettobeveryclose totheoriginoftheconvergingfanarea,i.e., r 3 61 m(110ft) and a r = R 0 95 .AsshowninEq.( 8 ),theoriginisasingular pointbecausethedenominatorbecomeszero.BasedonTable 1 Eq.( 8 )performsreasonablywellupto a 0 81 underarainfall intensityof 111 5 mm = h( 4 39 in := h).Itimpliesthatwithahigher value, 0 81 a 1 0 ,Eq.( 8 )maynotwellrepresentaconverging KWflowbecauseofsignificantaccelerations. Table1. ComparisonbetweenPredictionsandMeasurementsforConvergingFlows Case LaboratoryobservationPredictedbyconvergingKWequation ie(cm = h) Td(s) q obs(cm = h) Tp(s) r (m) R r (m) aq pred(cm = h)Error(%) 110.6433.953.9690.0018.0417.310.513.960.00 22.7485.091.50139.9035.360.001.001.3410.58 32.6269.591.04151.9019.7415.610.561.040.00 48.5342.213.8691.4025.599.770.723.860.00 52.2965.590.89140.5019.7515.610.560.890.00 611.1552.985.2693.8028.606.760.815.260.00 72.49353.332.18463.6035.360.001.001.2244.21 82.7780.781.19159.9023.4411.910.661.190.00 92.6477.791.07102.9020.4714.880.581.070.00 1011.2332.692.6994.1010.0925.260.292.690.00 Note: ie=rainfallintensity; q obs=observedpeakflow; q pred=predictedpeakflow; Td=duration; Tp=timetopeak.ASCE040150063J.Irrig.DrainEng.
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FieldtestsonnaturalalluvialwatershedsusingtheKWoverland convergingflowmodelwerealsoreported,includingWatersheds W2andW6locatedinRiesel,TX( Singh1976 ).Watershed W2hasatributaryareaofapproximately 530,000 m2(131acres) onanaveragelongitudinalslopeof0.025,whereasWatershedW6 hasatributaryareaof 170,000 m2(42acres)onanaveragelongitudinalslopeof0.015.BothwatershedsarecoveredwithHuston blackclaysoilswithlowpermeability.Forbothcases,theinfiltrationlosswasestimatedusingPhilip Â’ sinfiltrationequation( 1957 ). Detailscanbefoundelsewhere( Singh1976 ).Inthisstudy,theexcessrainfallintensitywasadoptedforrainfallrunoffsimulationusingEq.( 17 ).TheparametersontheKWratingaredeterminedas k Knn S0 5 0 18 m 5 3 19 where Kn 1 486 forftsunitsor1.0formsunits; n =Manning Â’ s roughnesscoefficient;and S0=averagelongitudinalslope.Aided withEqs.( 18 )and( 19 ),asshowninFig. 4 ,Eq.( 17 )wasapplied totheobserved40minrainfallexcessrecordedfromWatershed W2onJune4,1957.Thedeficitbetweenearlyrainfallexcess andrunoffvolumewasattributabletothesurfacedetentionon theconvergingplane.Afterthefirstpeak,thepredictedhydrographisfluctuatedwiththerainfallpatternandagreeswellwith theobservedhydrograph.Asexpected,thepredictedhydrograph carriesdoublepeaksasshownintheexcessrainfallhyetograph. Asunderstood,Singh Â’ ssolutionforthiscasewascalculatedusing acomplicatedapproachtointegrateEq.( 4 )throughtheinterior angle.AlthoughEq.( 8 )isasimplisticapproach,itdoescapture themajortreatsintherainfallrunoffsimulations,butignoresthe backwaterandstorageeffects. ThesecondcaseoffieldstudywasobservedfromWatershed W6onMarch29,1965,forarainfalleventwithdurationof4h. AsshowninFig. 5 ,therainfalldistributionhasmultiplepeaks. Theearlyrainfallexcesswasdelayedbecauseofthesurfacedetentionontheconvergingsurface.Afterthewatershedreachedthe equilibrium,startingfromthesecondpeak,thepredictedrunoff hydrographrespondstotherainfallexcessquitewell.Afterthe rainceased,thesurfacedetentionvolumewasgraduallyreleased. Fig.4. W2watershedsurfacerunoffpredictionandcomparison Fig.5. W6watershedsurfacerunoffpredictionandcomparisonASCE040150064J.Irrig.DrainEng.
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ConclusionThegoverningequationderivedforoverlandflowsonaconverging planeindicatesthattheconvergingKWflowischaracterizedwith rainfallintensity,surfaceslopeandroughness,andconvergingsurfacegeometryintermsofinteriorangleandradius.Althoughthe generalsolutionwasnotachievedyet,theequilibriumflowand depthwerederivedtodelineatetheflowboundariesonalluvial fanareasandtosizethedrainagefacilitiesacrossahighway. Theoriginofaradialconvergingsurfaceisasinkthatproduces asingularpointtotheflowgoverningequation.Basedonthelaboratorydata,theratioof r = R 0 81 isrecommendedwhenusingthe equilibriumconvergingflowfordesigns.AidedwithEq.( 7 ),the dikesandbermsshallbedesignedtoacollectorchannelthathasa minimumlengthas lrR 0 19 20 Thisstudypresentsanewdeterministicapproachtoestimatethe surfacerunoffonanalluvialfan.Delineationofflooplainboundariesrequiresthedetailedanalysisofwatersurfaceprofile.Eq.( 17 ) appearstobeaonedimensionalequation,infact,theinherent ratingcurveincludestheconvergingeffectontheincreaseofunitwidthflowastheflowmovestowardtheoutlet.Eq.( 17 )isnumericallystableaslongas t isnotexceedingthetimestepusedin therainfallexcess.Forsizingaculvert,thepeakflowinEq.( 11 ) canbedirectlycalculatedfromthelocalrainfallintensitydurationfrequency(IDF)curvefortheselectedbaseflood.Thedeterministicapproachpresentedinthisstudyshouldbeveryusefultothe designofhighwayalignmentacrossalluvialfans.ReferencesAgiralioglu,N.,andSingh,V.P.(1981). Â“ Amathematicalinvestigationof divergingsurfaceoverlandflow:2.Numericalsolutionsandapplications. Â” Rep.MSSUEIRSCE804 ,EngineeringandIndustrialResearch Station,MississippiStateUniv.,MississippiState,MS. Chow,V.T.(1964). Handbookofappliedhydrology ,McGrawHill, NewYork. FEMA.(1989). Alluvialfans:Hazardsandmanagement ,Washington,DC. FEMA.(2000). Guidelinesfordeterminingfloodhazardsonalluvialfans Washington,DC. French,R.H.(1989). Hydraulicprocessesonalluvialfans ,Elsevier Science,Amsterdam,Netherlands. French,R.H.(1992). Â“ Preferreddirectionsofflowonalluvialfans. Â” J.Hydraul.Eng. 10.1061/(ASCE)07339429(1992)118:7(1002) 1002 Â– 1013. French,R.H.,Fuller,J.E.,andWaters,S.(1993). Â“ Alluvialfan:Proposed newprocessorienteddefinitionsforthearidsouthwest. Â” J.Water Resour.Plann.Manage. 10.1061/(ASCE)07339496(1993)119:5(588) 588 Â– 598. Guo,J.C.Y.(2000). Â“ Stormhydrographsforsmallcatchments. Â” IWRAInt. J.Water ,25(3),1 Â– 12. Guo,J.C.Y.(2006). Â“ Dimensionlesskinematicwaveunithydrographfor stormwaterpredictions. Â” J.Irrig.Drain.Eng. 10.1061/(ASCE)0733 9437(2006)132:4(410) ,410 Â– 417. Guo,J.C.Y.,andUrbanos,B.(2009). Â“ Conversionofnaturalwatershedto kinematicwavecascadingplane. Â” J.Hydrol.Eng. 10.1061/(ASCE)HE .19435584.0000045 ,839 Â– 846. HEC(HydrologicEngineeringCenter).(1993). Â“ IntroductionandapplicationofkinematicwaveroutingtechniquesusingHEC1. Â” U.S.Army CorpsofEngineers,Davis,CA. Izzard,C.F.(1946). Â“ Hydraulicsofrunofffromdevelopedsurfaces. Â” Proc., HighwayResearchBoard ,Vol.26,Washington,DC,129 Â– 146. Langford,K.J.,andTurner,A.K.(1973). Â“ Anexperimentalstudyof theapplicationofkinematicwavetheorytooverlandflow. Â”J.Hydrol. 18(2),125 Â– 145 Muzik,I.(1973). Â“ Statevariablemodelofsurfacerunofffromalaboratory catchment. Â” Ph.D.dissertation,Univ.ofAlberta,Edmonton,Alberta, Canada. Philip,J.R.(1957). Â“ Thetheoryofinfiltrations:4.Sorptivityandalgebraic equations. Â” SoilSci. ,84(3),257 Â– 264. Singh,V.P.(1975). Â“ Alaboratoryinvestigationofsurfacerunoff. Â” J.Hydrol. ,25(3 Â– 4),187 Â– 200 Singh,V.P.(1976). Â“ Adistributedconvergingoverlandflowmodel Â— 3. Applicationtonaturalwatersheds. Â” J.Hydrol. ,31(3 Â– 4),221 Â– 243 Wooding,R.A.(1965). Â“ Ahydraulicmodeforacatchmentstream problem. Â” J.Hydrol. ,3(3 Â– 4),254 Â– 267 .ASCE040150065J.Irrig.DrainEng.
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137 APPENDIX B Diverging Plane Governing Equation Derivative KW flow on a fan area can be analyzed with an arc segment approach as presented on figure above. The water volume balance is determined by the inflow, outflow, and storage depth within the arc segment. On a diverging plane area, the radius of the fan area increases downstream as well as the flow depth accumulation of overland flow. According to the continuity principle, the water volume balanced within the arc segment is a summation of the following items: A t y t V r r R y y v v r R vy A i e A t y r r R vy y v vy r R vy A i e A t y vyr vyR yr v yR v r vy vyr vyR r R vy A i e A t y r R vy r R y v r vy A i e
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138 Where r r R A r r R t y r R vy r R y v r vy r r R i e t y r v y r y v r R vy i e t y r vy r R vy i e t y r vy r R vy i e t y r q r R q i e t y r q r R q i e (3.1) Where i e = excess rainfall intensity [L/T] A = diverging surface area [L 2 ] R = radius for the diverging area [L] r = location of the segment or representing the distance [L] v = radial flow velocity [L/T] y = flow depth [L] q = unitwidth discharge [L 2 /T] t = elapsed time [T] = angle of the apex of the diverging area
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139 r R r r R R A 2 2 1 2 1 2 2 (3.3) R l r (3.4) Where l r = arc length for outflow released from the fan area [L] The balance of water volume for the fan area is summarized as: A t V t S q q out in (3.5) Aided with Eq. (3.3) and (3.4), Eq. (3.5) can be rearranged as: A dt dy qR A i e dt dy A qR i e
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140 dt dy r R r qR i e 2 2 1 dt dy r R r qR i e 2 2 Let a R r dt dy a r q i e 2 2 (3.6) Where a = location ratio for the upper boundary Eq. (3.6) serves as the governing equation for KW overland flow on a diverging plane. If the tributary area starts from the apex of the diverging plane shape, then r = R or a = 1 and Eq. (3.6) can be further reduced to: R q i dt dy e 2 (3.7)
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141 APPENDIX C Converging Plane Governing Equation Derivative In this research, the entire converging tributary area is considered as a shallow reservoir, a similar approach described in the diverging plane governing equation derivitive (Guo and Hsu, 2015). At a time step, the flow generated from the tributary area is a water volume balance among rainfall excess, inflow, outflow, and change in storage volume. Using this approach, the governing equation can then be derived to describe the movement of converging overland flows. On a converging surface, excess rainfall produces overland flows on a sloping surface. Assuming there are no backwater effects, gravitational force is balanced with the friction force on the surface (Chow 1964). Figure above illustrates the general geometry of a converging plane in terms of elevation contours on the surface.
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142 Between two adjacent contour elevations, KW flow can be modeled using the inflow and outflow crossing the arc section. The mass balance relationship is derived and described as: A t y t V r r R y y v v r R vy A i e (3.21) A t y r r R vy y v vy r R vy A i e A t y r R vy r R y v r vy r R vy r R vy A i e A t y r R vy y v r vy A i e vy q (3.22) r r R A (3.23) Where i e = excess rainfall intensity [L/T] A = surface area [L 2 ] v = flow velocity [L/T] y = flow depth [L] R = radius of converging plane [L] r = location of KW flow on converging plane [L] = interior angle of converging area V = surface storage volume [L 3 ] t = elapsed time [T] q = unitwidth discharge [L 2 /T]
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143 Combining Eq. (3.21), (3.22) and (3.23), and eliminating the double and triple incremental terms, the mass balance relationship is derived as: r vy y v t y r R vy i e r vy t y r R vy i e r R q i r q t y e (3.24) Assuming there is no inflow from the upper boundary, the storage volume difference between rainfall excess and outflow released from the entire converging plane can be directly related to the change in flow depth. Thus, the principle of continuity is rearranged as: A t y t V ql A i r e (3.25)
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144 As illustrated in figure above, the converging tributary area is defined by interior angle, and radius, R The flow crossing over a converging plane changes from one location to another. Therefore, in this study, the governing equation for converging flow is formed using the location of the outlet and interior angle, The total tributary area upstream of a specified location is represented as: r R r A 2 2 1 (3.26) r R l r (3.27) Where r is the location of the outlet to collect overland flows, and l r is the arc length at the selected location to collect overland flows. Inserting Eqs. (3.26) and (3.27) to Eq. (3.25) yields the governing equation for converging overland flow as: R r rR R r qR i dt dy e 2 1 2 a r a q i dt dy e 2 1 2 (3.28) R r a (3.29) Where a = location ratio to define the selected location to collect overland flows.
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145 APPENDIX D Verification of Diverging KW Model and Case Studies Case 1 Comparison of Laboratory and Numerical Solutions for KW Flows under Rainfall Intensity of 78 mm/hr for 50 Seconds
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148 Case 2 Comparison of Laboratory and Numerical Solutions for KW Flows under Rainfall Intensity of 115 mm/hr for 50 Seconds
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151 Case 3 Diverging KW Flows with Various Expansion Angles
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Case 4 Â– KW analytical solution vs FAN program outputs
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163 APPENDIX E Verification of converging KW Model and Case Studies Case 1 W2 Watershed Surface Runoff Hydrograph Prediction and Comparison
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166 Case 2 W6 Watershed Surface Runoff Hydrograph Prediction and Comparison
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170 Case 3 Converging KW Rising Portion Hydrograph Comaprison ( a = 0.1)
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173 Case 4 Converging KW Rising Portion Hydrograph Comaprison (Interior Angle = 104 degrees, a = 0.5)
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176 176 Case 5 Converging KW Rising Portion Hydrograph Comaprison (Interior Angle = 104 degrees, a = 0.8)
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177 APPENDIX F Verification of Diverging KW Shape factor and Case Studies Case 1 Comparison of Laboratory, Analytical and Numerical Solutions for KW Flows under Rainfall Intensity of 98 mm/hr for 10 minutes
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181 181 Case 2 W2 Watershed Predictions and Field Data Comparison
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185 Case 3 W6 Watershed Predictions and Field Data Comparison
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