Storm Water Management Model
Reference Manual
Volume II -Hydraulics
&EPA
United States
Environmental Protection
Agency
EPA/600/ -17/111
www.epa.gov/water-research
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EPA/600/R-17/111
May 2017
Storm Water Management Model
Reference Manual
Volume II - Hydraulics
By:
Lewis A. Rossman
Office of Research and Development
National Risk Management Laboratory
Cincinnati, OH 45268
National Risk Management Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
26 Martin Luther King Drive
Cincinnati, OH 45268
May 2017
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Disclaimer
The information in this document has been funded wholly or in part by the U.S. Environmental
Protection Agency (EPA). It has been subjected to the Agency's peer and administrative review,
and has been approved for publication as an EPA document. Mention of trade names or
commercial products does not constitute endorsement or recommendation for use.
Although a reasonable effort has been made to assure that the results obtained are correct, the
computer programs described in this manual are experimental. Therefore the author and the U.S.
Environmental Protection Agency are not responsible and assume no liability whatsoever for any
results or any use made of the results obtained from these programs, nor for any damages or
litigation that result from the use of these programs for any purpose.
11
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Abstract
SWMM is a dynamic rainfall-runoff simulation model used for single event or long-term
(continuous) simulation of runoff quantity and quality from primarily urban areas. The runoff
component of SWMM operates on a collection of subcatchment areas that receive precipitation
and generate runoff and pollutant loads. The routing portion of SWMM transports this runoff
through a system of pipes, channels, storage/treatment devices, pumps, and regulators. SWMM
tracks the quantity and quality of runoff generated within each subcatchment, and the flow rate,
flow depth, and quality of water in each pipe and channel during a simulation period comprised
of multiple time steps. The reference manual for this edition of SWMM is comprised of three
volumes. Volume I describes SWMM's hydrologic models, Volume II its hydraulic models, and
Volume III its water quality and low impact development models.
111
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Acknowledgements
This report was written by Lewis A. Rossman, Environmental Scientist Emeritus, U.S.
Environmental Protection Agency, Cincinnati, OH.
The author would like to acknowledge the contributions made by the following individuals to
previous versions of SWMM that were drawn heavily upon in writing this report: John Aldrich,
Douglas Ammon, Carl W. Chen, Brett Cunningham, Robert Dickinson, James Heaney, Wayne
Huber, Miguel Medina, Russell Mein, Charles Moore, Stephan Nix, Alan Peltz, Don Polmann,
Larry Roesner, Charles Rowney, and Robert Shubinsky. The efforts of Wayne Huber (Oregon
State University emeritus), Thomas Barnwell (US EPA retired), Richard Field (US EPA retired),
Harry Torno (US EPA retired) and William James (University of Guelph emeritus) to support
and maintain the program over the past several decades are also gratefully acknowledged.
iv
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Table of Contents
Disclaimer ii
Abstract iii
Acknowledgements iv
List of Figures vii
List of Tables ix
List of Symbols xi
Chapter 1 - SWMM Overview 15
1.1 Introduction 15
1.2 SWMM's Object Model 16
1.3 SWMM's Process Models 21
1.4 Simulation Process Overview 23
1.5 Interpolation and Units 27
Chapter 2 - SWMM's Hydraulic Model 30
2.1 Network Components 31
2.2 Analysis Methods 35
2.3 Boundary and Initial Conditions 38
Chapter 3 - Dynamic Wave Analysis 40
3.1 Governing Equations 40
3.2 S oluti on Method 44
3.3 Computational Details 46
3.4 Numerical Stability 57
Chapter 4 - Kinematic Wave Analysis 63
4.1 Governing Equations 63
4.2 S oluti on Method 65
4.3 Computational Details 67
4.4 Numerical Stability 72
Chapter 5 - Cross-Section Geometry 74
5.1 Standard Conduit Shapes 74
5.2 Custom Conduit Shapes 92
v
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5.3 Irregul ar Natural Channel s 94
5.4 Storage Unit Geometry 97
5.5 Critical and Normal Depths 100
Chapter 6 - Pumps and Regulators 104
6.1 Pumps 104
6.2 Orifices 107
6.3 Weirs 114
6.4 Outlets 124
Chapter 7 - Advanced Features 127
7.1 Evaporation and Seepage 127
7.2 Minor Losses 136
7.3 Force Mains 138
7.4 Culverts 142
7.5 Roadway Weirs 147
Appendix 151
References 185
vi
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List of Figures
Figure 1-1 Elements of a typical urban drainage system 17
Figure 1-2 SWMM's conceptual model of a stormwater drainage system 18
Figure 1-3 Processes modeled by SWMM 21
Figure 1-4 Block diagram of SWMM's state transition process 23
Figure 1-5 Flow chart of SWMM's simulation procedure 26
Figure 1-6 Interpolation of reported values from computed values 28
Figure 2-1 Node-link representation of a sewer system 30
Figure 2-2 Comparison of dynamic wave and kinematic wave solutions 38
Figure 3-1 Node-link representation of a conveyance network in SWMM 43
Figure 3-2 Special flow conditions for dynamic wave analysis 50
Figure 3-3 Illustration of a surcharged node 53
Figure 3-4 Ponding of excess water above a junction 56
Figure 3-5 Profile view of example rectangular conduit (not to scale) 60
Figure 3-6 Outflow hydrographs for example conduit -I 61
Figure 3-7 Outflow hydrographs for example conduit - II 62
Figure 4-1 Section factor versus area for a circular shape 64
Figure 4-2 Space-time grid for kinematic wave analysis 65
Figure 4-3 Outflow hydrograph for example conduit 73
Figure 5-1 Power law cross section shape 77
Figure 5-2 Geometric properties of a partly filled circular shape based on depth 80
Figure 5-3 Geometric properties of a partly filled circular shape based on area 80
Figure 5-4 Ellipsoid and arch pipe cross sectional shapes 81
Figure 5-5 Masonry sewer shapes 84
Figure 5-6 Composite cross section shapes 86
Figure 5-7 A Shape Curve with a depth segment shown 93
Figure 5-8 A natural channel transect 94
Figure 5-9 A transect depth increment with three compound segments 95
Figure 5-10 Example of a storage curve and its section view 98
Figure 5-11 Finding the volume at a given depth for a storage curve 99
Figure 6-1 Orifice orientations 107
Figure 6-2 Determination of effective head for an orifice 110
Figure 6-3 Orifice with unsubmerged inlet 110
Figure 6-4 Transverse weir shapes 115
Figure 6-5 Coefficient for triangular weirs (from Brater and King, 1976) 119
Figure 6-6 Definitions of submerged and surcharged weir flow 121
Figure 6-7 Rating curve for a vortex device compared to an orifice 125
Figure 7-1 Depths used for computing seepage in storage units 135
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Figure 7-2 Concrete box culvert (from FHWA, 2012) 142
Figure 7-3 Example of a culvert rating curve (from FHWA, 2012) 143
Figure 7-4 Roadway overtopping (from FHWA, 2012) 148
Figure 7-5 SWMM node-link representation of a culvert with a roadway weir 148
Figure 7-6 Discharge coefficients for roadway weirs (from FHWA, 2012) 149
viii
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List of Tables
Table 1-1 Development history of SWMM 16
Table 1-2 SWMM's modeling objects 19
Table 1-3 State variables used by SWMM 24
Table 1-4 Units of expression used by SWMM 29
Table 2-1 Features and limitations of dynamic wave and kinematic wave solutions 37
Table 3-1 Surface area adjustments for various dynamic wave flow conditions 51
Table 5-1 Geometric properties for open channel shapes as functions of water depth 75
Table 5-2 Geometric properties for open channel shapes as functions of flow area 76
Table 5-3 Geometric properties for the power law shape 78
Table 5-4 Geometric properties of a full circular cross section 79
Table 5-5 Full area and hydraulic radius of custom ellipsoid and arch pipe sections 83
Table 5-6 Number of entries in geometric property tables for masonry sewer shapes 85
Table 5-7 Geometric parameters of masonry sewer sections 85
Table 5-8 Geometric properties for a sediment filled circular cross section 87
Table 5-9 Properties of the rectangular section of a rectangular-triangular shape 88
Table 5-10 Geometric parameters for rectangular-round shapes 89
Table 5-11 Geometric properties for rectangular-round shapes 90
Table 5-12 Properties in the rounded top section of a modified basket handle shape 91
Table 5-13 Area at maximum flow to full area for standard closed conduits shapes 91
Table 5-14 Critical depth formulas for simple section shapes 101
Table 6-1 Pump curves recognized by SWMM 105
Table 6-2 Kindsvater-Carter constants for rectangular weir coefficient 118
1/2
Table 6-3 Rectangular broad-crested weir coefficients (ft /sec) 119
Table 6-4 Formulas for flow derivatives of various types of weirs 123
Table 7-1 Relative depth at maximum width for select cross section shapes 129
Table 7-2 Types of minor losses in drainage systems (from Frost, 2006) 137
Table 7-3 Hazen-Williams C-factors for different pipe materials 139
Table 7-4 Darcy-Weisbach roughness heights for different pipe materials 140
Table C-l Circular section properties as function of depth 153
Table C-2 Circular section properties as function of area 154
Table D-l Standard elliptical pipe sizes 155
Table D-2 Elliptical section properties as function of depth 156
Table E-l Standard arch pipe sizes 157
Table E-2 Arch pipe section properties as function of depth 160
Table F-l Area of masonry sewers as function of depth 161
Table F-2 Width of masonry sewers as function of depth -1 162
Table F-3 Width of masonry sewers as function of depth - II 163
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Table F-4 Hydraulic radius of masonry sewers as function of depth 164
Table F-5 Depth of masonry sewers as function of area -1 165
Table F-6 Depth of masonry sewers as function of area - II 167
Table F-7 Section factor for masonry sewers as function of area -1 169
Table F-8 Section factor for masonry sewers as function of area - II 171
Table G-l Manning's roughness coefficient n for open channels 173
Table G-2 Manning's roughness coefficient n for closed conduits 177
Table G-3 Manning's roughness coefficient n for corrugated steel pipe 179
Table H-l Culvert codes 180
Table H-2 Culvert coefficients 183
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List of Symbols
A
2
cross section flow area within a conduit (ft )
A
average cross section flow area along a conduit (ft)
A
average cross section flow area along a conduit over a time period (ft)
Ami
full cross section area of a conduit (ft)
Amax
cross section area at depth where a conduit's section factor is a maximum (ft)
Ao
area of an orifice opening (ft2)
Asp
surface area of water ponded above a node (ft)
As
surface area of a node and its connected links (ft)
Asl
2
surface area of flow within a link (ft )
Ajast
2
surface area of a node the last time it was not surcharged (ft)
Asmin
minimum surface area associated with a node (ft)
Asn
surface area associated with a storage node (ft)
Aw
area of a weir opening (ft)
b
bottom or top width (depending on shape) of a conduit's cross section (ft)
c
wave celerity (ft/sec)
Cl
inlet control constant for submerged culverts
cw
1/2
coefficient for a weir-type flow divider (ft /sec)
cd
orifice discharge coefficient (dimensionless)
Chw
Hazen-Williams C-factor coefficient (dimensionless)
Co
5/2
equivalent orifice constant for a surcharged weir (ft /sec)
Cr
Courant number (dimensionless)
Cw
1/2
weir coefficient (ft /sec)
D
circular pipe diameter (ft)
et
potential evaporation rate at time t (ft/sec)
E
elevation of a node's invert (ft)
EC
specific head at critical depth (ft)
f
Darcy-Weisbach friction factor (dimensionless)
fc
monthly climate adjustment factor (dimensionless)
XI
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fE storage node evaporation factor (dimensionless)
fs weir submergence adjustment factor (dimensionless)
F cumulative depth of infiltrated water (ft)
Fr Froude number (dimensionless)
g acceleration of gravity (ft/sec )
/?£ minor head loss per unit length of a conduit (ft/ft)
hw height of the opening for a weir-type flow divider node (ft)
H hydraulic head (ft)
Hcrown elevation of the crown of the highest conduit at a node (ft)
He effective head seen by an orifice or weir (ft)
His minimum head at a culvert's inlet for it to be submerged (ft)
Hiu maximum head at a culvert's inlet for it to be unsubmerged (ft)
Hmax maximum head at a node before flooding occurs (ft)
Houtfaii head assigned to an outfall node (ft)
K cross section flow conductance (cfs) (equal to nAR2/3)
Ki inlet control constant for unsubmerged culverts
Km minor loss coefficient (dimensionless)
Ks soil saturated hydraulic conductivity (ft/sec)
L conduit length or weir crest length (ft)
Le effective weir crest length (ft)
Mi inlet control exponent for unsubmerged culverts
1/3
n Manning roughness coefficient (sec/m )
P wetted perimeter of a conduit's cross section (ft)
qE uniformly distributed evaporation rate along a channel (cfs/ft)
qi total uniformly distributed outflow rate along a conduit (cfs/ft)
qMiN minimum flow needed to activate a flow divider node (cfs)
qs uniformly distributed seepage rate along a conduit (cfs/ft)
qsN seepage rate per unit area for a storage node (cfs/ft2)
Q flow rate within a conduit, pump, or regulator link (cfs)
Qdiv flow rate diverted to a second outflow conduit from a flow divider node (cfs)
Xll
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Qen evaporation loss rate from a storage unit node (cfs)
Qfuii normal uniform flow rate for a full conduit (cfs)
Qic culvert flow rate under inlet control (cfs)
Qin total inflow rate to a node (cfs)
Qln total loss rate from a storage unit node (cfs)
Qnorm normal uniform flow rate (cfs)
Qout total outflow rate leaving a node (cfs)
Qovs excess flow that overflows a node (cfs)
Qnet average net inflow minus outflow over a time step (cfs)
Qsn seepage loss rate from a storage node (cfs)
R hydraulic radius of flow cross section in a conduit (ft)
R average hydraulic radius of flow cross sections along a conduit (ft)
Re Reynolds number (dimensionless)
Rfuii hydraulic radius of a conduit cross section when full (ft)
s seepage rate per unit area for a conduit (ft/sec)
Scf culvert slope correction factor
Sf friction slope (ft/ft)
So conduit slope (ft/ft)
t time (sec)
U flow velocity at a point along a conduit (ft/sec)
U average flow velocity along a conduit (ft/sec)
"3
V node assembly volume (ft)
"3
Vp ponded volume (ft)
Vn storage node volume (ft)
"3
Vnmi volume of a storage node when full (ft)
W top width of the water surface at a point along a conduit (ft)
W average top width of the water surface along a conduit (ft)
Wmax maximum width of a conduit cross section (ft)
x horizontal distance (ft)
y vertical distance (ft)
Xlll
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yi
inlet control constant for submerged culverts
Y
depth of flow within a conduit or of water in a storage unit (ft)
Y
average depth of flow along a conduit (ft)
Yc
critical depth within a conduit at a given flow rate (ft)
Yfuii
full depth of a conduit, orifice opening or weir height (ft)
Yn
normal flow depth (ft)
Y'*
smaller of the critical and normal flow depth in a conduit (ft)
z
elevation of a conduit's invert (ft)
Zo
elevation of the bottom of an orifice's opening (ft)
Zw
elevation of a weir's crest in its lowest position (ft)
a
generic coefficient
P
the square root of a conduit's slope divided by its roughness
At
time step (sec)
s
convergence tolerance
e
Darcy-Weisbach roughness length (ft)
7
exponent in power law cross section shape
n
Manning's roughness coefficient (sec/ft13) (equal to n/1.486)
a
inertial damping factor
e
time weighting factor, relaxation factor, or subtended angle
s
soil capillary suction head (ft)
conduit section factor (equal to AR2/3) (ft8 3)
¥fiii
8/3
section factor of a conduit at full depth (ft )
¥max
8/3
maximum section factor for a conduit (ft )
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Chapter 1 - SWMM Overview
1.1 Introduction
Urban runoff quantity and quality constitute problems of both a historical and current nature.
Cities have long assumed the responsibility of control of stormwater flooding and treatment of
point sources (e.g., municipal sewage) of wastewater. Since the 1960s, the severe pollution
potential of urban nonpoint sources, principally combined sewer overflows and stormwater
discharges, has been recognized, both through field observation and federal legislation. The
advent of modern computers has led to the development of complex, sophisticated tools for
analysis of both quantity and quality pollution problems in urban areas and elsewhere (Singh,
1995). The EPA Storm Water Management Model, SWMM, first developed in 1969-71, was one
of the first such models. It has been continually maintained and updated and is perhaps the best
known and most widely used of the available urban runoff quantity/quality models (Huber and
Roesner, 2013).
SWMM is a dynamic rainfall-runoff simulation model used for single event or long-term
(continuous) simulation of runoff quantity and quality from primarily urban areas. The runoff
component of SWMM operates on a collection of subcatchment areas that receive precipitation
and generate runoff and pollutant loads. The routing portion of SWMM transports this runoff
through a system of pipes, channels, storage/treatment devices, pumps, and regulators. SWMM
tracks the quantity and quality of runoff generated within each subcatchment, and the flow rate,
flow depth, and quality of water in each pipe and channel during a simulation period comprised
of multiple time steps.
Table 1-1 summarizes the development history of SWMM. The current edition, Version 5, is a
complete re-write of the previous releases. The reference manual for this edition of SWMM is
comprised of three volumes. Volume I describes SWMM's hydrologic models, Volume II its
hydraulic models, and Volume III its water quality and low impact development models. These
manuals complement the SWMM 5 User's Manual (US EPA, 2010), which explains how to run
the program, and the SWMM 5 Applications Manual (US EPA, 2009) which presents a number
of worked-out examples. The procedures described in this reference manual are based on earlier
descriptions included in the original SWMM documentation (Metcalf and Eddy et al., 1971a,
1971b, 1971c, 1971d), intermediate reports (Huber et al., 1975; Heaney et al., 1975; Huber et al.,
1981), plus new material. This information supersedes the Version 4.0 documentation (Huber
and Dickinson, 1988; Roesner et al., 1992) and includes descriptions of some newer procedures
implemented since 1988. More information on current documentation and the general status of
the EPA Storm Water Management Model as well as the full program and its source code is
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available on the EPA SWMM web site:, http://www2.epa.gov/water-research/storm-water-
management-model-swmm.
Table 1-1 Development history of SWMM
Version
Year
Contributors
Comments
SWMM I
1971
Metcalf & Eddy, Inc.
Water Resources
Engineers
University of Florida
First version of SWMM; focus
was CSO modeling; Few of its
methods are still used today.
SWMM II
1975
University of Florida
First widely distributed
version of SWMM.
SWMM 3
1981
University of Florida
Camp Dresser & McKee
Full dynamic wave flow
routine, Green-Ampt
infiltration, snow melt, and
continuous simulation added.
SWMM 3.3
1983
US EPA
First PC version of SWMM.
SWMM 4
1988
Oregon State University
Camp Dresser & McKee
Groundwater, RDII, irregular
channel cross-sections and
other refinements added over a
series of updates throughout
the 1990's.
SWMM 5
2005
US EPA
CDM-Smith
Complete re-write of the
SWMM engine in C; graphical
user interface added; improved
algorithms and new features
(e.g., LID modeling) added.
1.2 SWMM's Object Model
Figure 1-1 depicts the elements included in a typical urban drainage system. SWMM
conceptualizes this system as a series of water and material flows between several major
environmental compartments. These compartments include:
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Urban Wet Weather Ffcws
Storm Water
Sanitary
WusxuwiUer
Separate*
Storm Sewer
System
Sanitary Wastewater
Combined
Sewer System
I Sanitary
Wastewater
I Interceptor
Sewer
Separate Sanitary
Sewer System
Sanitary
Kirwitr
Orcifwrs
Sanitary
Sewer
Ottsrlows
V&steuiler
Treatment Plant
Storm Water
Ptrinl Source
Fully
Treated
EJfUiiftl
Figure 1-1 Elements of a typical urban drainage system
• The Atmosphere compartment, which generates precipitation and deposits pollutants onto the
Land Surface compartment.
• The Land Surface compartment receives precipitation from the Atmosphere compartment in
the form of rain or snow. It sends outflow in the forms of 1) evaporation back to the
Atmosphere compartment, 2) infiltration into the Sub-Surface compartment and 3) surface
runoff and pollutant loadings on to the Conveyance compartment.
• The Sub-Surface compartment receives infiltration from the Land Surface compartment and
transfers a portion of this inflow to the Conveyance compartment as groundwater interflow.
• The Conveyance compartment contains a network of elements (channels, pipes, pumps, and
regulators) and storage/treatment units that convey water to outfalls or to treatment facilities.
Inflows to this compartment can come from surface runoff, groundwater interflow, sanitary
dry weather flow, or from user-defined time series.
Not all compartments need appear in a particular SWMM model. For example, one could model
just the Conveyance compartment, using pre-defined hydrographs and pollutographs as inputs.
As illustrated in Figure 1-1, SWMM can be used to model any combination of stormwater
17
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collection systems, both separate and combined sanitary sewer systems, as well as natural
catchment and river channel systems.
Figure 1-2 shows how SWMM conceptualizes the physical elements of the actual system
depicted in Figure 1-1 with a standard set of modeling objects. The principal objects used to
model the rainfall/runoff process are Rain Gages and Subcatchments. Snowmelt is modeled with
Snow Pack objects placed on top of subcatchments while Aquifer objects placed below
subcatchments are used to model groundwater flow. The conveyance portion of the drainage
system is modeled with a network of Nodes and Links. Nodes are points that represent simple
junctions, flow dividers, storage units, or outfalls. Links connect nodes to one another with
conduits (pipes and channels), pumps, or flow regulators (orifices, weirs, or outlets). Land Use
and Pollutant objects are used to describe water quality. Finally, a group of data objects that
includes Curves, Time Series, Time Patterns, and Control Rules, are used to characterize the
inflows and operating behavior of the various physical objects in a SWMM model. Table 1-2
provides a summary of the various objects used in SWMM. Their properties and functions will
be described in more detail throughout the course of this manual.
Rain gage ^
-ff Junction
Subcatchment
Conduit
Divider 4k
* Outfall
[ Regulator
Storage Unit
Pump
Figure 1-2 SWMM's conceptual model of a stormwater drainage system
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Table 1-2 SWMM's modeling objects
Category
Object Type
Description
Hydrology
Rain Gage
Source of precipitation data to one or more
subcatchments.
Subcatchment
A land parcel that receives precipitation associated
with a rain gage and generates runoff that flows into
a drainage system node or to another subcatchment.
Aquifer
A subsurface area that receives infiltration from the
subcatchment above it and exchanges groundwater
flow with a conveyance system node.
Snow Pack
Accumulated snow that covers a subcatchment.
Unit Hydrograph
A response function that describes the amount of
sewer inflow/infiltration (RDII) generated over time
per unit of instantaneous rainfall.
Hydraulics
Junction
A point in the conveyance system where conduits
connect to one another with negligible storage
volume (e.g., manholes, pipe fittings, or stream
junctions).
Outfall
An end point of the conveyance system where water
is discharged to a receptor (such as a receiving
stream or treatment plant) with known water surface
elevation.
Divider
A point in the conveyance system where the inflow
splits into two outflow conduits according to a
known relationship.
Storage Unit
A pond, lake, impoundment, or chamber that
provides water storage.
Conduit
A channel or pipe that conveys water from one
conveyance system node to another.
Pump
A device that raises the hydraulic head of water.
Regulator
A weir, orifice or outlet used to direct and regulate
flow between two nodes of the conveyance system.
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Table 1-2 Continued
Category
Object Type
Description
Water Quality
Pollutant
A contaminant that can build up and be washed off
of the land surface or be introduced directly into the
conveyance system.
Land Use
A classification used to characterize the functions
that describe pollutant buildup and washoff
Treatment
LID Control
A low impact development control, such as a bio-
retention cell, porous pavement, or vegetative swale,
used to reduce surface runoff through enhanced
infiltration.
Treatment Function
A user-defined function that describes how pollutant
concentrations are reduced at a conveyance system
node as a function of certain variables, such as
concentration, flow rate, water depth, etc.
Data Object
Curve
A tabular function that defines the relationship
between two quantities (e.g., flow rate and hydraulic
head for a pump, surface area and depth for a storage
node, etc.).
Time Series
A tabular function that describes how a quantity
varies with time (e.g., rainfall, outfall surface
elevation, etc.).
Time Pattern
A set of factors that repeats over a period of time
(e.g., diurnal hourly pattern, weekly daily pattern,
etc.).
Control Rules
IF-THEN-ELSE statements that determine when
specific control actions are taken (e.g., turn a pump
on or off when the flow depth at a given node is
above or below a certain value).
20
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1.3 SWMM's Process Models
Figure 1-3 depicts the processes that SWMM models using the objects described previously and
how they are tied to one another. The hydrological processes depicted in this diagram include:
Precipitation
Channel, Pipe & \
Storage Routing J
~r
RDII
Buildup
Washoff
Groundwater
Sanitary
Flows
Surface Runoff
Initial
Abstraction
Treatment / Diversion
Evaporation/
Infiltration
Figure 1-3 Processes modeled by SWMM
• time-varying precipitation
• snow accumulation and melting
• rainfall interception from depression storage (initial abstraction)
• evaporation of standing surface water
• infiltration of rainfall into unsaturated soil layers
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• percolation of infiltrated water into groundwater layers
• interflow between groundwater and the drainage system
• nonlinear reservoir routing of overland flow
• infiltration and evaporation of rainfall/runoff captured by Low Impact Development
controls.
The hydraulic processes occurring within SWMM's conveyance compartment include:
• external inflow of surface runoff, groundwater interflow, rainfall-dependent
infiltration/inflow, dry weather sanitary flow, and user-defined inflows
• unsteady, non-uniform flow routing through any configuration of open channels, pipes
and storage units
• various possible flow regimes such as backwater, surcharging, reverse flow, and surface
ponding
• flow regulation via pumps, weirs, and orifices including time- and state-dependent
control rules that govern their operation.
Regarding water quality, the following processes can be modeled for any number of user-defined
water quality constituents:
• dry-weather pollutant buildup over different land uses
• pollutant washoff from specific land uses during storm events
• direct contribution of rainfall deposition
• reduction in dry-weather buildup due to street cleaning
• reduction in washoff loads due to BMPs
• entry of dry weather sanitary flows and user-specified external inflows at any point in the
drainage system
• routing of water quality constituents through the drainage system
• reduction in constituent concentration through treatment in storage units or by natural
processes in pipes and channels.
The numerical procedures that SWMM uses to model the hydraulic processes listed above are
discussed in detail in subsequent chapters of this volume. SWMM's hydrologic and water quality
processes are described in volumes I and III of this manual.
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1.4 Simulation Process Overview
SWMM is a distributed discrete time simulation model. It computes new values of its state
variables over a sequence of time steps, where at each time step the system is subjected to a new
set of external inputs. As its state variables are updated, other output variables of interest are
computed and reported. This process is represented mathematically with the following general
set of equations that are solved at each time step as the simulation unfolds:
xt = nxt_ltit,p) (i-i)
Yt = g(.Xt,P) (1-2)
where
Xt = a vector of state variables at time t,
Yt = a vector of output variables at time /,
It = a vector of inputs at time t,
P = a vector of constant parameters,
/ = a vector-valued state transition function,
g = a vector-valued output transform function,
Figure 1-4 depicts the simulation process in block diagram fashion.
1
f(Xt1„lt„P)
t-1
9(X
Figure 1-4 Block diagram of SWMM's state transition process
The variables that make up the state vector Xt are listed in Table 1-3. This is a surprisingly small
number given the comprehensive nature of SWMM. All other quantities can be computed from
these variables, external inputs, and fixed input parameters. The meaning of some of the less
obvious state variables, such as those used for snow melt, is discussed in other sections of this set
of manuals.
23
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Table 1-3 State variables used by SWMM
Process
Variable
Description
Initial Value
Runoff
d
Depth of runoff on a sub catchment surface
0
Infiltration*
tp
Equivalent time on the Horton curve
0
Fe
Cumulative excess infiltration volume
0
Fu
Upper zone moisture content
0
T
Time until the next rainfall event
0
P
Cumulative rainfall for current event
0
S
Soil moisture storage capacity remaining
User supplied
Groundwater
Ou
Unsaturated zone moisture content
User supplied
di
Depth of saturated zone
User supplied
Snowmelt
wsnow
Snow pack depth
User supplied
jw
Snow pack free water depth
User supplied
ati
Snow pack surface temperature
User supplied
cc
Snow pack cold content
0
Flow Routing
H
Hydraulic head of water at a node
User supplied
Q
Flow rate in a link
User supplied
A
Flow area in a link
Inferred from Q
Water Quality
tsweep
Time since a subcatchment was last swept
User supplied
mB
Pollutant buildup on subcatchment surface
User supplied
mp
Pollutant mass ponded on subcatchment
0
CN
Concentration of pollutant at a node
User supplied
Cl
Concentration of pollutant in a link
User supplied
*Only a sub-set of these variables is used, depending on the user's choice of infiltration method.
Examples of user-supplied input variables It that produce changes to these state variables
include:
• meteorological conditions, such as precipitation, air temperature, evaporation rate and
wind speed
• externally imposed inflow hydrographs and pollutographs at specific nodes of the
conveyance system
24
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• dry weather sanitary inflows to specific nodes of the conveyance system
• water surface elevations at specific outfalls of the conveyance system
• control settings for pumps and regulators.
The output vector Yt that SWMM computes from its updated state variables contains such
reportable quantities as:
• runoff flow rate and pollutant concentrations from each subcatchment
• snow depth, infiltration rate and evaporation losses from each subcatchment
• groundwater table elevation and lateral groundwater outflow for each subcatchment
• total lateral inflow (from runoff, groundwater flow, dry weather flow, etc.), water depth,
and pollutant concentration for each conveyance system node
• overflow rate and ponded volume at each flooded node
• flow rate, velocity, depth and pollutant concentration for each conveyance system link.
Regarding the constant parameter vector P, SWMM contains over 150 different user-supplied
constants and coefficients within its collection of process models. Most of these are either
physical dimensions (e.g., land areas, pipe diameters, invert elevations) or quantities that can be
obtained from field observation (e.g., percent impervious cover), laboratory testing (e.g., various
soil properties), or previously published data tables (e.g., pipe roughness based on pipe material).
A smaller remaining number might require some degree of model calibration to determine their
proper values. Of course not all parameters are required for every project (e.g., the 14
groundwater parameters for each subcatchment are not needed if groundwater is not being
modeled). The subsequent chapters of this manual carefully define each parameter and make
suggestions on how to estimate its value.
A flowchart of the overall simulation process is shown in Figure 1-5. The process begins by
reading a description of each object and its parameters from an input file whose format is
described in the SWMM 5 Users Manual (US EPA, 2010). Next the values of all state variables
are initialized, as is the current simulation time (T), runoff time (Troff), and reporting time OV).
25
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Read Input
Parameters
1
r
Initialize State
Vana
ales
r
T =
0
T,rff-
= 0
t^at^
Route Flows and Water Quality
T = T1
Legend:
T = current elapsed time
T1 = new elapsed time
current runoff time
T1T:t — current reporting time
= routing time step
ATjrff = runoff tim e step
AT,j;t = reporting time step'
DUR = simulation duration
T >= DUR
T1 = T + AT
Trf < T1
Compute Runoff
Traff Ttoff i" ATrnff
YES J Save Output Result
T,pt = Tipt+ ATjpt
Figure 1-5 Flow chart of SWMM's simulation procedure
26
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The program then enters a loop that first determines the time T1 at the end of the current routing
time step (ATrout). If the current runoff time Troff is less than Tl, then new runoff calculations are
repeatedly made and the runoff time updated until it equals or exceeds time Tl. Each set of
runoff calculations accounts for any precipitation, evaporation, snowmelt, infiltration, ground
water seepage, overland flow, and pollutant buildup and washoff that can contribute flow and
pollutant loads into the conveyance system.
Once the runoff time is current, all inflows and pollutant loads occurring at time T are routed
through the conveyance system over the time interval from T to Tl. This process updates the
flow, depth and velocity in each conduit, the water elevation at each node, the pumping rate for
each pump, and the water level and volume in each storage unit. In addition, new values for the
concentrations of all pollutants at each node and within each conduit are computed. Next a check
is made to see if the current reporting time T^t falls within the interval from T to Tl. If it does,
then a new set of output results at time Trpt are interpolated from the results at times T and Tl
and are saved to an output file. The reporting time is also advanced by the reporting time step
ATrpt. The simulation time T is then updated to Tl and the process continues until T reaches the
desired total duration. SWMM's Windows-based user interface provides graphical tools for
building the aforementioned input file and for viewing the computed output.
1.5 Interpolation and Units
SWMM uses linear interpolation to obtain values for quantities at times that fall in between
times at which input time series are recorded or at which output results are computed. The
concept is illustrated in Figure 1-6 which shows how reported flow values are derived from the
computed flow values on either side of it for the typical case where the reporting time step is
larger than the routing time step. One exception to this convention is for precipitation and
infiltration rates. These remain constant within a runoff time step and no interpolation is made
when these values are used within SWMM's runoff algorithms or for reporting purposes. In
other words, if a reporting time falls within a runoff time step the reported rainfall intensity is the
value associated with the start of the runoff time step.
27
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F
L
O
w
O Computed
~ Interpolated
Time
Figure 1-6 Interpolation of reported values from computed values
The units of expression used by SWMM's input variables, parameters, and output variables
depend on the user's choice of flow units. If flow rate is expressed in US customary units then so
are all other quantities; if SI metric units are used for flow rate then all other quantities use SI
metric units. Table 1-4 lists the units associated with each of SWMM's major variables and
parameters, for both US and SI systems. Internally within the computer code all calculations are
carried out using feet as the unit of length and seconds as the unit of time.
28
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Table 1-4 Units of expression used by SWMM
Variable or Parameter
US Customary Units
SI Metric Units
Area (subcatchment)
acres
hectares
Area (storage surface area)
square feet
square meters
Depression Storage
inches
millimeters
Depth
feet
meters
Elevation
feet
meters
Evaporation
inches/day
millimeters/day
Flow Rate
cubic feet/sec (cfs)
gallons/min (gpm)
106 gallons/day (mgd)
cubic meters/sec (cms)
liters/sec (lps)
106 liters/day (mid)
Hydraulic Conductivity
inches/hour
millimeters/hour
Hydraulic Head
feet
meters
Infiltration Rate
inches/hour
millimeters/hour
Length
feet
meters
Manning's n
1/3
seconds/meter
1/3
seconds/meter
Pollutant Buildup
mass/acre
mass/hectare
Pollutant Concentration
milligrams/liter (mg/L)
micrograms/liter (|~ig/L)
organism counts/liter
milligrams/liter (mg/L)
micrograms/liter (|_ig/L)
organism counts/liter
Rainfall Intensity
inches/hour
millimeters/hour
Rainfall Volume
inches
millimeters
Storage Volume
cubic feet
cubic meters
Temperature
degrees Fahrenheit
degrees Celsius
Velocity
feet/second
meters/second
Width
feet
meters
Wind Speed
miles/hour
kilometers/hour
29
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Chapter 2 - SWMM's Hydraulic Model
As mentioned in Chapter 1, SWMM models the conveyance portion of a drainage system as a
network of links connected together at nodes. External flows from various sources enter the
network at specific nodes, are transported along links, are combined together and split apart at
internal nodes while filling and emptying the volume of storage nodes, and exit the system at
terminal nodes. Figure 2-1 shows how a physical system of sewer lines and their appurtenances
are abstracted into a network of nodes and links of different types (pipe and pump links;
junction, storage and outfall nodes for this particular example).
Catch Basins
Manhole
Wastewater
Treatment Plant
Junction
Conduit
Pumping
Station
Outfall
Storage Unit
Figure 2-1 Node-link representation of a sewer system
(Background from http://www.sewerhistory.org/photosgraphics/japan/)
30
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Table 1-2 has already summarized the different types of node and link objects that can appear in
a SWMM conveyance network model. The remainder of this chapter provides more details on
the properties of network objects, briefly describes and compares the capabilities of the two
principal methods used for analyzing the unsteady hydraulic behavior of a network, and
discusses the boundary and initial conditions needed to compute network hydraulics.
2.1 Network Components
The two principal components of a SWMM conveyance system network are nodes and links.
Nodes represent the end points of conveyance links that form the connection between two or
more links. They are also the points where external inflows (runoff, dry weather flows, etc.) can
enter the network or where internal flows leave the network. Links are conveyance elements that
transport flow between nodes. The following paragraphs describe the different types of nodes
and links that SWMM can model.
2.1.1 Junction Nodes
Junction nodes are points in the drainage system where conveyance links join together.
Physically they can represent the confluence of natural surface channels, manholes in a sewer
system, or pipe connection fittings. Excess water at a junction can become partially pressurized
when connecting conduits are surcharged and can either be lost from the system or be allowed to
pond atop the junction and subsequently drain back into the junction.
The principal input parameters for a junction node are:
• invert (channel or manhole bottom) elevation
• height between its invert and the ground surface
• additional pressure head that can be accepted before flooding occurs
• ponded surface area when flooded.
2.1.2 Outfall Nodes
Outfall nodes are terminal nodes of the drainage system used to define final downstream
boundary locations. The boundary conditions at an outfall can be described by any one of the
following stage relationships:
• the critical or normal flow depth in the connecting conduit
• a fixed stage elevation
• a tidal stage described in a table of tide height versus hour of the day
• a user-defined time series of stage versus time.
31
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The principal input parameters for an outfall node are:
• invert elevation
• type of boundary condition and its associated stage data
• presence of a flap gate to prevent backflow through the outfall.
2.1.3 Flow Divider Nodes
Flow divider nodes divert inflows to a specific link in a prescribed manner. A flow divider can
have no more than two conduit links on its discharge side. There are four types of flow dividers,
defined by the manner in which inflows are diverted:
• Cutoff diverts all inflow above a defined cutoff value.
• Overflow diverts all inflow above the flow capacity of the non-diverted conduit.
• Tabular uses a table that expresses diverted flow as a function of total inflow.
• Weir uses a weir equation to compute diverted flow.
The principal input parameters for a flow divider node are:
• junction parameters (see above)
• name of the link receiving the diverted flow
• method used for computing the amount of diverted flow.
2.1.4 Storage Unit Nodes
Storage unit nodes are the only type of node that can provide storage volume and possess surface
area. Physically they could represent storage facilities as small as a catch basin or as large as a
lake. The volumetric properties of a storage unit are described by a function or table of surface
area versus height. In addition to receiving inflows and discharging outflows to other nodes in
the drainage network, storage nodes can also lose water from surface evaporation and from
seepage into native soil. Unlike other nodes, storage nodes are not allowed to pressurize (i.e.,
they always maintain a free surface).
The principal input parameters for a storage unit are:
• invert (bottom) elevation
• maximum depth
• depth-surface area data
• evaporation potential
• seepage parameters.
32
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2.1.5 Conduit Links
Conduit links are pipes or channels that move water from one node to another in the conveyance
network. Their cross-sectional shapes can be selected from a variety of standard open and closed
geometries. Custom closed shapes for pipes and irregular cross-section profiles for open
channels can also be specified. Conduit geometry is discussed in more detail in Chapter 5.
The required input parameters for a conduit link are:
• identities of the inlet and outlet nodes
• offset height or elevation above the inlet and outlet node inverts
• conduit length
• Manning's roughness coefficient
• cross-section shape and dimensions.
SWMM allows conduits to be offset some distance above
the invert of their connecting end nodes as shown in the
figure on the right. The offset can be specified as either a
distance above the invert (i.e., the distance between points
1 and 2 in the figure) or as the elevation of the conduit's
invert (i.e., the elevation of point 1). Internally the offset is
maintained as an elevation.
SWMM also makes use of a conduit's slope in its
hydraulic calculations. Slope is not provided directly as an
input variable but is instead computed from the elevation
of a conduit's end node inverts and its offsets. Let L be the
length of the conduit, 4/be the difference in elevation and
Ar the horizontal distance between the invert at each end
of the conduit. Then from the diagram on the right:
Ax = yj L? — Ay 2 (2-1)
and the conduit slope 5*0 is:
S0 = Ay/Ax (2-2)
Tl
Ax
33
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SWMM does not allow a slope of 0. Therefore it imposes a minimum value of 0.001 ft on Ay. It
also allows the user to set a non-zero value for minimum slope which will override any smaller
computed slope.
SWMM uses the Manning equation to relate conduit flow rate to flow depth and conduit bed or
friction slope. It therefore requires the user to supply a Manning's "/?" coefficient that represents
the roughness characteristics of the conduit's surface. Values of the coefficient for a wide range
of channel types and pipe materials can be found in Appendix G.
Conduits can also include the following optional parameters:
• presence of a flap gate to prevent reverse flow
• entrance/exit loss coefficients
• seepage rate
• inlet geometry code number if the conduit acts as a culvert.
The latter three properties are employed by the advanced modeling features covered in Chapter 7
of this manual.
2.1.6 Pump Links
Pump links are used to lift water from an inlet node to an outlet node at higher elevation. The
principal input parameters for a pump include:
• identities of its inlet and outlet nodes
• pump curve data
• initial on/off status
• startup and shutoff depths.
A pump curve describes the relation between a pump's flow rate and the head at its inlet and
outlet nodes. The inlet node's startup and shutoff water depths are monitored continuously
during the course of a simulation to allow for automated control of the pump's on/off status.
Pumps are directional devices that are not allowed to have reverse flow through them. Their
hydraulic performance is described in more detail in Chapter 6.
2.1.7 Flow Regulator Links
Flow regulator links model structures or devices used to control and divert flows within a
conveyance system. They are typically used to control releases from storage facilities, prevent
unacceptable surcharging, and divert flow to treatment facilities and interceptors.
34
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SWMM can model the following types of flow regulators: orifices, weirs, and outlets. The
hydraulic behavior of orifices and weirs is modeled using standard rating curves (the nonlinear
relation between hydraulic head applied to the regulator and the flow rate through it). Outlets
utilize a user-supplied rating curve.
The principal input parameters for a flow regulator link include:
• identities of its inlet and outlet nodes
• offset above the invert of its inlet node
• dimensions of its opening (for orifices and weirs)
• parameters that describe its rating curve
• presence of a flap gate to prevent reverse flow.
The hydraulic performance of regulator links is described in more detail in Chapter 6.
2.1.8 Control Rules
Each pump and flow regulator has a setting property that can adjust:
• a pump's on/off status
• a pump's speed
• the size of an orifice opening
• the crest height of a weir
• the flow through an outlet link
The setting can be changed during a simulation by using control rules. These specify conditions,
such as water elevation at certain nodes, flow in certain links, and simulation time, that trigger a
specified change in a link's setting. SWMM's hydraulic analysis methods take into account the
current setting for each pump and flow regulator in the conveyance network. More details on the
formats used for control rules can be found in the SWMM 5 Users Manual (US EPA, 2010).
2.2 Analysis Methods
SWMM's hydraulics solves the equations of one-dimensional, gradually varied, unsteady flow
throughout a node-link network to determine the water level at each node and the flow rate and
flow depth within each link at each time step of an extended simulation period. Flow routing of
inflow hydrographs along channels and sewers entails wave dispersion, wave attenuation or
amplification, and wave retardation or acceleration. These wave characteristics constitute the
hydraulics of flow routing or propagation and are greatly affected by the geometric
characteristics of the conduits, the characteristics of sources and/or sinks, and by initial and
boundary conditions.
35
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The hydraulics of unsteady non-uniform flow is represented in SWMM by a pair of partial
differential equations of conservation of mass and momentum known as the St. Venant
equations. Simultaneous solution of these equations for each conduit, coupled with a
conservation of volume at each node, provides information on the spatial and temporal variation
of water levels and discharge rates throughout the network. SWMM offers the user two principal
alternative methods for solving these equations - dynamic wave or kinematic wave analysis
Dynamic wave analysis solves the complete form of the St. Venant flow equations and therefore
produces the most theoretically accurate results. It can account for channel storage, backwater
effects, entrance/exit losses, culvert flow, flow reversal, and pressurized flow. Because it couples
together the solution for both water levels at nodes and flow in conduits it can be applied to any
general network layout, even those containing multiple downstream diversions and loops. It is
the method of choice for systems subjected to significant backwater due to downstream flow
restrictions and with flow regulation via weirs and orifices. This generality comes at a price of
having to use small time steps to maintain numerical stability.
Kinematic wave analysis solves the continuity equation along with a simplified form of the
momentum equation in each conduit. It cannot account for backwater effects, entrance/exit
losses, flow reversal, or pressurized flow. It is most applicable to steeply sloped (e.g., > 0.1%)
conduits with shallow flow with high velocity. It can usually maintain numerical stability with
much larger large time steps than are required for dynamic wave analysis. If the aforementioned
effects are not expected to be significant then this alternative can be an accurate and efficient
hydraulic analysis method, especially for long-term simulations.
Because kinematic wave analysis ignores both inertial and pressure forces there are limits on its
applicability:
1. It can only analyze directed acyclic networks (where conduits are oriented in the
direction of positive slope and there are no paths that start and end at the same node).
2. Junction nodes can only have at most one outlet link which must be a conduit.
3. Divider nodes must have two outlet links which must be conduits.
4. Storage nodes can have any number of outlet links of any type.
5. Upstream offsets for conduits are ignored except at storage nodes.
SWMM also offers a steady flow analysis option which assumes that within each computational
time step flow is uniform and steady. It simply translates inflow hydrographs at the upstream end
of a conduit to its downstream end, with no delay or change in shape. The Manning equation is
used to relate flow rate to flow area (or depth). It is subject to the same limitations as the
kinematic wave method. Because it ignores the dynamics of free surface wave propagation it is
only appropriate for rough preliminary analysis of long-term continuous simulations.
36
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Table 2-1 compares the features and limitations of the dynamic wave and kinematic wave
methods of hydraulic analysis. Dynamic wave solutions tend to attenuate and disperse an inflow
hydrograph as it routed downstream through a series of conduits while kinematic wave solutions
show no attenuation, no dispersion, and some distortion of the hydrograph shape. This behavior
is depicted in Figure 2-2 from Miller (1984) which shows the results of routing an inflow
hydrograph down a 100-foot wide rectangular channel of 1% slope with a Manning's n of 0.06.
Table 2-1 Features and limitations of dynamic wave and kinematic wave solutions
Feature
Dynamic Wave
Kinematic Wave
Network topology
branched and looped
branched only
Flow splits
yes
with flow divider nodes
Adverse slopes
yes
no
Invert offsets
yes
ignored
Pumping
yes
only from storage nodes
Weirs and orifices
yes
only from storage nodes
Ponded overflows
yes
yes
Lateral seepage
yes
yes
Evaporation
yes
yes
Minor losses
yes
no
Culvert analysis
yes
no
Hydrograph attenuation
yes
no
Backwater effects
yes
no
Surcharge / Pressurization
yes
no
Reverse flow
yes
no
Tidal effects
yes
no
37
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700
Q
em
v>
s
if
£
m
u.
u
1
o
z
f \ Straam
1 80.000 % diswmw
I t Cfeetl
403
5
200
0
20,000
10,000
TIME, IN SECONDS
Figure 2-2 Comparison of dynamic wave and kinematic wave solutions
(from Miller, 1984)
2.3 Boundary and Initial Conditions
2.3.1 Boundary Conditions
There are two types of boundary conditions that a user must supply to a SWMM conveyance
network model:
1. the hydraulic head to be maintained at each outfall node of the network,
2. the external inflow received by specific nodes of the network.
Both types of conditions can vary with time. Outfall node heads are only required for dynamic
wave analysis. The options available for specifying their values were described in Section 2.1.2.
External inflows can originate from any of the following sources:
• subcatchment runoff
• groundwater discharges
• rainfall-dependent infiltration/inflow (RDII)
• user-defined values.
38
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Time-dependent runoff, groundwater, and RDII inflows are normally provided by SWMM's
hydrology module (see Volume I). It automatically links the computed flow from each of these
sources at each time period to their designated receiving node. (Each SWMM subcatchment
object that generates runoff is assigned a conveyance system node that receives this runoff. See
Figure 1-2.)
User-defined external inflows can be attached to any node of the network. They are typically
used to describe dry weather sewage flows in sanitary sewer systems, base flows in natural
stream channels, or inflows in the absence of any hydrologic modeling. They are expressed in
the following general format:
Flow rate at time t = (baseline value) x (baseline pattern factor) +
(scale factor) x (time series value at time t)
The baseline value is some constant. The baseline pattern is a combination of repeating hourly,
daily, and monthly multiplier factors applied to the baseline value. The time series value is a time
varying value and the scale factor is a constant multiplier applied to each time series value. Time
series values can be specified at unequal intervals of time with interpolation used to obtain
values at intermediate times.
2.3.2 Initial Conditions
A set of initial conditions at time 0 for all node heads and link flows in the conveyance network
must be specified before a hydraulic analysis can begin. The default is to set all these values to 0,
with the user having the option to specify initial heads at selected nodes and initial flow rates in
selected conduit links.
Any initial flow rate assigned to a conduit link is assumed to represent a uniform steady flow.
Therefore its flow depth can be set to the normal depth determined by the Manning equation as
described in Section 5.5.2. From this depth an initial cross-section flow area for the conduit can
be found which is required for kinematic wave analysis.
For dynamic wave analysis, if a non-storage, non-outfall node has not had an initial head
assigned to it then it's initial head is set equal to the average elevation of the initial flow depths
in the conduits that deliver flow into it.
39
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Chapter 3 - Dynamic Wave Analysis
The movement of water through a conveyance network of channels and pipes is governed by the
conservation of mass and momentum equations for gradually varied, unsteady free surface flow.
Dynamic wave analysis solves the complete form of these equations and therefore produces the
most theoretically accurate results. It can account for channel storage, backwater effects,
entrance/exit losses, flow reversal, and pressurized flow. Because it couples together the solution
for both water levels at nodes and flow in conduits it can be applied to any general network
layout, even those containing multiple downstream diversions and loops. It is the method of
choice for systems subjected to significant backwater due to downstream flow restrictions and
with flow regulation via weirs and orifices. This generality comes at a price of having to use
small time steps to maintain numerical stability.
Dynamic wave modeling was first introduced into version 3 of SWMM in 1981 as a separate
program module known as EXTRAN (Extended Transport) (Roesner et al., 1983). The node-link
solution method it uses had its origins in the Sacramento-San Joaquin Delta Model (Shubinski et
al., 1965) and the WRE Transport Model (Kibler et al., 1975). Although more powerful solution
techniques are available (such as implicit finite difference schemes (Cunge et al., 1980) and
shock-capturing finite volume schemes (Toro, 2001)), SWMM 5 continues to use EXTRAN's
node-link approach, with modifications made to enhance its stability, because of its simplicity
and versatility.
3.1 Governing Equations
The conservation of mass and momentum for unsteady free surface flow through a channel or
pipe are known as the St. Venant equations and can be expressed as:
dA dQ „ . .
1 = 0 Continuity (3-1)
dt dx
dQ d(Q2/A) dH A;r + _
— + + g A—+ gASf = 0 Momentum (3-2)
at ox ox '
where
x = distance (ft)
t = time (sec)
A = flow cross-sectional area (ft2)
Q = flow rate (cfs)
40
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H = hydraulic head of water in the conduit (Z + Y) (ft)
Z = conduit invert elevation (ft)
Y = conduit water depth (ft)
Sf = friction slope (head loss per unit length)
g = acceleration of gravity (ft/sec2)
The derivation of these equations can be found in standard texts such as Henderson (1966),
Cunge et al. (1980) and French (1985). The assumptions on which they are based are:
1. flow is one dimensional
2. pressure is hydrostatic
3. the cosine of the channel bed slope angle is close to unity
4. boundary friction can be represented in the same manner as for steady flow.
The friction slope Sf can be expressed in terms of the Manning equation used to model steady
uniform flow:
Sf=^2m 0-3)
f Vl.486/ AR4/3
where
n = the Manning roughness coefficient (sec/m13)
R = the hydraulic radius of the flow cross-section (ft)
U = flow velocity, equal to Q/A (ft/sec).
1/3 1/3
and 1.486 converts from m to ft . Use of the absolute value sign on the velocity term makes 5>
a directional quantity (since Q can be either positive or negative) and ensures that the frictional
force always opposes the flow. Manning roughness coefficients for wide range of channel
surfaces and pipe materials can be found in Appendix G.
For a specific cross-sectional geometry, the flow area A is a known function of water depth Y
which in turn can be obtained from the head H. Thus the dependent variables in these equations
are flow rate Q and head //, which are functions of distance x and time t. To solve these
equations over a single conduit of length L, one needs a set of initial conditions for H and Q at
time 0 as well as boundary conditions at x= 0 and x= L for all times t.
The continuity equation 3-1 can be combined with the momentum equation 3-2 to produce the
following form of the momentum equation for a conduit (see sidebar below for details):
dQ dA . dA dH
"wt^u t,-^a-x-^as' (M)
41
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Combining the Continuity and Momentum Equations
The d(Q2/ A)/dx term in the momentum equation 3-2 can be re-expressed as:
d(Q2/A) d(U2A) dU , dA
y = ' =2AU—+U2 —
dx ox ox ox
(a)
Using Q = UA, the continuity equation 3-1 can be written as:
dA dU dA
Tt+A^ + u^-°
(b)
Multiplying both sides of (b) by Uand re-arranging terms leads to:
dU dA , dA
AU —— = -U—-U2 —
dx dt dx
(c)
Substituting this into the first term on the right hand side of (a) produces:
d(Q2/A) dA , dA
= 2U U2
dx dt dx
(d)
Substituting (d) into 3-2 and re-arranging terms gives the final result:
dQ dA , dA dH
Tt=wTt + u2-x-sATx-9AS'
(e)
While this equation can be used to compute the time trajectory of flow in a conduit, another
relationship is needed to do likewise for heads. SWMM's node - link representation of the
conveyance network, conceptualized in Figure 3-1, does this by providing a continuity
relationship at junction nodes that connect conduits together within a conveyance network. As
shown in the figure, a continuous water surface is assumed to exist between the water elevation
at a node and in the conduits that enter and leave it. Two types of nodes are possible. Non-
storage junction nodes are assumed to be points with zero volume and surface area while storage
nodes (such as ponds and tanks) contain both volume and surface area.
42
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LINK N-
NODE J
NODE J+l
B
Figure 3-1 Node-link representation of a conveyance network in SWMM
(from Roesner et al, 1992).
Each "node assembly" consists of the node itself and half the length of each link connected to it.
Conservation of flow for the assembly requires that the change in volume with respect to time
equal the difference between inflow and outflow. In equation terms:
ZQ = net flow into the node assembly (inflow - outflow) (cfs)
The £ Q term includes the flow in the conduits connected to the node as well as any externally
imposed inflows such as wet weather runoff or dry weather sanitary flow.
Each node assembly's surface area consists of the node's storage surface area Asn (if it's a
storage node) plus the surface area contributed by the links connected to it, £i45i, where Asl 'is
the surface area contributed by a connecting link. Thus the node continuity equation can be
written as:
(3-5)
where:
V = node assembly volume (ft )
As = node assembly surface area (ft )
dH _ ZQ
dt Asn + £i4 SL
(3-6)
43
-------
The flow depth at the end of a conduit connected to a node can be computed as the difference
between the head at the node and the invert elevation of the conduit. The node and link surface
areas are computed as functions of their respective flow depths.
Equations 3-4 and 3-6 provide a coupled set of partial differential equations that solve for flow Q
in the conduits and head H at the nodes of the conveyance network. Because they cannot be
solved analytically a numerical solution procedure must be used instead.
3.2 Solution Method
The material that follows applies to networks containing only conduits. Inclusion of flow control
devices (pumps, orifices, and weirs) and other processes (seepage, evaporation, and minor
losses) will be covered in subsequent chapters of this manual.
The spatial and temporal derivatives in equations 3-4 and 3-6 can be replaced with the following
finite difference approximations:
dA = (A2-A1)
dx L
(3-7)
dH (//2 - HO
(3-8)
dx L
dA _ AA
dt At
(3-9)
dQ_ _ AQ
dt At
(3-10)
dH _ AH
dt At
(3-11)
where
Ai = flow area at the upstream end of the conduit (ft2)
A2 = flow area at the downstream end of the conduit (ft)
Hi = hydraulic head at the upstream end of the conduit (ft)
H2 = hydraulic head at the downstream end of the conduit (ft)
L = conduit length (ft)
At = time step (sec)
44
-------
A A = change in average flow area, (i4t+At — At), over time step At{ft2)
AQ = change in conduit flow, (Qt+At — Q1), over time step ,d£(cfs)
AH = change in nodal head, (Ht+At — //t), over time step At (ft),
with the superscripts referring to time periods.
Substituting these finite difference approximations into the link momentum Equation 3-4,
replacing Sf with Equation 3-3, and replacing A, U, and R with their average values over the
conduit length (as indicated by over scores) allows the finite difference form of the link
momentum equation to be written as:
A
-------
and now H and the quantities A, A, U, and R derived from it are all evaluated at the new time
t+At. The finite difference form of the nodal continuity equation 3-12 can be expressed as:
Ht+At = Ht +
(Asn + Z ASLy+At
for non-outfall nodes
(3-15a)
Ht+At = H,
Outfall
for outfall nodes
(3-15b)
Houtfaii is a user-supplied value that sets the head at a terminal outfall node. It can be a constant
value, a value extracted from a user-supplied time series, or the elevation of the critical or normal
flow depth in the connecting conduit. For the latter option, critical or normal depth is computed
internally as a function of the conduit's flow rate and geometry as described in Chapter 5.
Equations 3-14 and 3-15 can be solved implicitly over a given time step ^fusing functional
iteration (also known as successive approximations or Picard's method). The method is
described in the sidebar titled "Dynamic Wave Solution Procedure". Because flows and heads
are updated one conduit and node at a time and not simultaneously, the results at each time step
are invariant to the order in which the conduits and links are evaluated. This allows Steps 2 and 4
of the solution procedure to be implemented using separate threads running in parallel on multi-
processor computers which can offer a significant reduction in computation time.
3.3 Computational Details
3.3.1 Average Cross-Section Properties
Evaluation of the flow updating formula 3-14 requires values for the average area (A), hydraulic
radius (R), and velocity (U) for the conduit in question. These values are computed using heads
Hi and H2 belonging to the most recently computed head estimates Hhlst at either end of the
conduit. The flow depth Yi at the upstream end of the conduit is computed as:
where Zi is the elevation of the invert of the upstream end of the conduit and Ymi is the full
depth of the conduit. A similar expression using H2 and Z2 applies to Y2 at the downstream end
of the conduit.
(3-16)
46
-------
Dynamic Wave Solution Procedure
The following steps are used to update link flows and nodal heads over a given time step from
t to t + At for dynamic wave analysis:
1. Initially let Qast and Hlast be the flow in each link and the head at each node, respectively,
computed at time t. At time 0 these values are provided by the user-supplied initial
conditions.
2. Solve Equation 3-14 for each link producing a new flow estimate Q"ew for time t + At,
basing the values of A, A, U, and R on Hh,st.
3. Combine Q"ew and Q'ast together using a relaxation factor 6 to produce a weighted value
of Q"ew\
Qnew = (1 _ 0)Qlast + QQnew
4. Compute a value for Hnew at each node from Equation 3-15 using the flows Qnewfor Qt+At
and the heads HIast to evaluate
5. As with flows, apply a relaxation factor to combine Hlast and Hnew:
Hnew = (i _ 0)Hlast + 9Hnew
6. If Hnew is close enough to HIast for each node then the process stops with Qiew and Hnew as
the solution for time t At. Otherwise, Hlast and Qbst are set equal to Hnew and Qnew\
respectively, and the process returns to step 2.
Notes:
1. The relaxation factor 0 is set to 0.5.
2. The convergence tolerance and maximum number of trials can be set by the user.
Their default values are 0.005 feet and 8, respectively.
3. For links whose end node heads have already converged, steps 2 and 3 can be skipped
and Qnew can be set equal to Qbst.
47
-------
Values of A and R are computed from the conduit's cross section geometry at the average flow
depth F = (yi + Y2)/2. Formulas for doing so are described in Chapter 5 of this manual. The
average velocity U is found by dividing the most current flow value Qastby the average area A.
In addition, the average area and hydraulic radius used in the pressure and friction terms of
equation 3-14 are upstream weighted to reflect how close a conduit's flow is to being
supercritical. Supercritical flow is influenced only by upstream conditions (i.e., wave
disturbances propagate only in the downstream direction). The weight is derived from the Froude
number Fr for Qast\
\U\
Fr = , (3-17)
JJI/W
where W is the top water surface width at the average depth F. (Fr is set to 0 for closed conduits
flowing full). A factor cris then computed as:
!1 for Fr < 0.5
2(1 — Fr) for 0.5 < Fr < 1 (3-18)
0 for Fr > 1
It is used to modify the average area in Equation 3-14b and the average hydraulic radius in
Equation 3-14c as follows:
A=A1+a(A-A1) (3-19)
R =R1 + a(R - fli) (3-20)
where Ai and Ri are the flow area and hydraulic radius, respectively, based on the upstream flow
depth Yi.
3.3.2 Surface Area Calculations
Under normal conditions the surface area that a conduit contributes to its upstream node (Asli) is
the average top width of the water surface over the upstream half of the conduit times half of the
conduit's length. In equation form:
Asli —
/VKQ9 + W(Y)\L
{ 2 ) 2
(3-21)
48
-------
where W(Y) is the flow cross-section top width at a given flow depth Yand Y = (Y1 + Y2)/2. A
similar expression applies to the downstream surface area Asl2. W(Y) is computed from the
conduit's cross-section geometry as described in Chapter 5.
Because sewer systems are frequently built with pipe invert discontinuities at manholes they can
encounter free-fall conditions where the water elevation in the node receiving flow is below the
pipe's invert elevation or the flow's critical depth. Also during periods of filling or draining,
conduits can have one end or the other dry. These conditions require that adjustments be made to
the way that flow depth is assigned and to how surface area is computed.
Figure 3-2 illustrates the various types of special flow conditions that affect surface area
calculations:
1. Case one is the normal situation of subcritical flow where flow depths and surface areas
are computed as previously described.
2. Case two represents a critical downstream condition. The conduit has a downstream
offset and the water level at the node is below the flow's critical depth. The downstream
depth is set equal to the smaller of the critical depth and normal depth for the current flow
and all of the conduit's surface area is assigned to the upstream node.
3. Case three is a critical upstream condition. There is reverse flow with a free-fall
discharge into the upstream node. Adjustments equivalent to those for case two are made
but with the definitions of upstream and downstream reversed.
4. Case four depicts an upstream dry condition. The upstream end of the conduit is dry and
the water level at the downstream end is below the upstream conduit invert. If there is an
upstream invert offset then no surface area is assigned to the upstream node. A
complementary set of rules applies to the opposite case of a downstream dry condition.
Table 3-1 summarizes the various flow conditions and the adjustments that are made for each.
Procedures for computing the critical depth and normal depth for a given flow rate and cross-
section geometry are discussed in Chapter 5 of this manual.
Finally, to guard against the nodal head change formula 3-15 from becoming unbounded as
surface area becomes vanishingly small, a global minimum surface area Asmm is imposed as
follows:
Ac = max {A
(3-22)
49
-------
Its default value is 12.56 sq ft (i .e., the area of a 4-ft diameter manhole) which can be overridden
by the user. This is strictly a computational device and does not add volume to a junction node
(where Asn = 0) nor change it into a storage node.
Node 1
-Y
Hor*°ntal Datum
1
Figure 3-2 Special flow conditions for dynamic wave analysis
50
-------
Table 3-1 Surface area adjustments for various dynamic wave flow conditions
Condition
Criteria
Adjustments
Upstream Dry
V l|
& ^
Asli = 0 if H2 < Z1
otherwise use Upstream Critical adjustment
Downstream Dry
y2 = o
Z2> E2
Asl2 = 0 if H1 < Z2
otherwise use Downstream Critical adjustment
Upstream Critical
Q< 0
Zi > Ei
Hi - Zi < Y*
Yi = Y*
Hi = Y* + Zi
Asli = 0
Asl2 = L (W + W2)/2
Downstream Critical
Q> 0
Z2> E2
h2-z2
-------
wave formulation which also drops the local acceleration term (dQ/dt) of the momentum
equation as well.) This option can also result in improved stability particularly during periods of
rapid flow change.
3.3.4 Flow Limitations
Each time a new flow is computed using Equation 3-14 it is checked to see if it should be limited
by the normal flow value for the upstream flow depth and conduit slope. The following criteria
are used to perform this check:
1. The computed flow is positive.
2. The conduit is not flowing full.
3. The conduit does not fall into any of the categories listed in Table 3-1 (upstream /
downstream dry or upstream / downstream critical).
4. The water surface slope is less than the conduit's slope or the flow's Froude number
based on upstream velocity and depth is greater than 1.
The last criterion can be limited to just slope, just Froude number or either slope or Froude
number as a program option. When all of these criteria are satisfied the flow is limited to be no
greater than that found by the Manning equation (Qnorm) using upstream conditions:
1.49 ,/o
Qnorm = —^1 V^O (3"23)
n
where ibis the conduit slope. Two other flow limiting conditions are also checked. If the conduit
was assigned an upper flow limit then the flow is not allowed to exceed that value. If the conduit
contains a flap gate and the computed flow is negative then the flow is set to 0.
3.3.5 Surcharge Conditions
SWMM defines a node to be in a surcharged condition when all conduits connected to it are full
or when the node's water level exceeds the crown of the highest conduit connected to it (see
Figure 3-3). It should be noted that surcharged (or pressurized) flow can occur in a closed
conduit without either of its end nodes being surcharged. For example, if the node water level in
Figure 3-3 was above the invert of pipe N+l but below its crown, then pipes N and N-l would
remain pressurized (assuming they were also full at their upstream ends) while the node itself
would no longer be surcharged.
52
-------
STREET SURFACE
mm of
JUNCTION J
(beginning of surcharge)
Figure 3-3 Illustration of a surcharged node
When a node becomes surcharged there is no more volume available in the conduits forming the
node's assembly to absorb the difference between inflow and outflow at the node. Thus dV/dt
in the flow continuity Equation 3-5 is 0 and the surcharged nodal continuity condition becomes:
YjQ = 0 (3"24)
By itself, this equation is insufficient to update nodal heads at the new time step since it only
contains flows. In addition, because the flow and head updating equations for the system are not
solved simultaneously, there is no guarantee that the condition will hold at the surcharged nodes
after a flow solution has been reached.
To enforce the surcharge flow continuity condition, it can be expressed in the form of a
perturbation equation:
53
-------
Z[
dQ
Q+WAH
= 0 (3-25)
where AH is the adjustment to the node's head that must be made to achieve a flow balance.
Solving for ^//yields:
AH = „ (3-26)
ZdQ/dH K J
where the summations are made over all conduits that are connected to the node in question.
The gradient of flow in a conduit with respect to the head at either end node can be evaluated by
differentiating the flow updating equation 3-14 resulting in:
dQ_ = -gAM/L
dH 1 + AQ friction
The numerator of dQ/dH has a negative sign in front of it because when evaluating EQ flow
directed out of a node is considered negative while flow into the node is positive. It is computed
for each link at the same time that the link's flow is updated at Step 2 of the iterative process
described in Section 3.3. The surcharge equation 3-26 is analogous to the head updating formula
used in the Hardy Cross method for pressurized water distribution networks (Bhave, 1991).
To accommodate node surcharging, Step 4 of the iterative process that updates a node's head is
modified as follows. First the node is checked to see if it is in a surcharged state, i.e., that it is not
a storage or outfall node and has Hlast greater than the top of the highest connecting conduit
Hcrown. If it is not surcharged then Equation 3-15 is used as before to update its head. Otherwise
the following modified form of Equation 3-26 is used to estimate the new head Hnew for time t +
At.
ccZQnew
jjnew _ _| ^2 28")
(1 - (J) Z(dQ/dH)last + pAlsast/At K ' J
where
a =0.6 for upstream terminal nodes with only outflow links and 1.0 otherwise
p = exp(—15.0 fH)
fH = (Hlast-E)/(Hcrown-E)- 1
Hcrown = elevation of the crown of the node's highest connecting conduit (ft)
54
-------
E
elevation of the node's invert (ft)
surface area of the node the last time it was not surcharged (ft)
The a factor is used to reduce oscillations in head at upstream terminal nodes that have only
outflow links (Roesner et al., 1992). The /? factor helps to reduce fluctuations in head when the
node first begins to surcharge (Roesner et al., 1980). At low surcharge depths it makes the
denominator in the head update formula be a weighted combination of the pure surcharge
formula 3-26 and the surface area formula 3-15. By the time that the water level rises 25% above
the highest conduit, the equation is 98% pure surcharge.
The flow values used for £ Q are the new flow estimates found from Step 3 of the solution
procedure. The dQ/dH values are those that were last evaluated at Step 2. And finally, empirical
testing has shown that more robust performance is obtained when under-relaxation is not applied
to Hnew at Step 5 of the solution procedure when surcharging occurs.
3.3.6 Flooding and Ponding
Each non-outfall node is assigned a maximum allowable head Hmax by the user. It consists of
both a maximum free water surface elevation that can exist at the node plus an optional
"surcharge" depth that allows for pressurization. For example, if the node were a manhole
junction Hmax would typically be the ground surface elevation. If it were a storage unit it would
be the water surface elevation when the unit is full. For a junction between natural channels it
would be the top of the highest channel. For a fitting that connects pipe segments together it
would be the top of the highest pipe. In the latter case a large surcharge depth (such as several
hundred feet) should be assigned to the fitting junction so that the connected pipes can pressurize
if need be. A manhole junction might also be assigned a surcharge depth if it has a bolted cover.
Normally when the new head estimate Hnew at a node computed at Step 5 of the iterative solution
process exceeds Hmax it is set equal to Hmax and the node becomes flooded. The overflow rate
Qovfi associated with this condition is the average net flow rate (inflow - outflow) seen by the
node over the current time step:
This flow is then lost from the system, the same as the flow entering a terminal outfall node.
The option exists for a junction node with no surcharge depth (and thus always maintaining a
free surface) to have excess flooded water pond atop the node (see Figure 3-4). In this case the
(3-29)
55
-------
user assigns the node a "ponded area" parameter, A p., that creates a virtual storage area on top of
the node and H'"'w is no longer limited to HmM . When Hnew exceeds Hmaa-the ponded node is
treated as a normal storage node whose head is updated using the normal, non-surcharge formula
Equation 3-15 with Asm = A p. The only exception to this is when the node transitions between
having a head below Hmax to a flooded head above Hmax (or vice versa) within a time step. In this
case the updated head is restricted to be just a small value above Hmax (or below it in the opposite
case) to avoid wide swings in head during the transition.
Figure 3-4 Ponding of excess water above a junction
When a node is allowed to pond, flooded water is not lost from the system. The ponded depth
above the node will rise during periods of flow excess (i.e., inflow greater than outflow) and fall
during periods of flow deficit. A node with a large ponded area will see smaller changes in
ponded depth for a given flow excess (or deficit) than will one with a small ponded area.
Selection of which nodes can pond and their respective ponded areas would depend on local
topography, typically occurring along flat sections or at sag points of the drainage system.
GROUND ELEV. ^
viiiiriiiinhinnrrTTTJ/
JUNCTION j
Ponded Water
mm of
JUNCTION J
lbe0mgo(mharyel
56
-------
3.3.7 Summary of Special Conditions
Here is a summary of the special conditions that are applied to the basic iterative solution process
for dynamic wave analysis described earlier in Section 3.2:
1. Upstream weighting, based on the current flow's Froude number, is applied to the
average area in the pressure term and to the average hydraulic radius in the friction term
of the flow updating formula 3-14 (see Section 3.3.1).
2. Optional inertial damping, again based on the Froude number, is applied to the inertial
term of the flow updating formula 3-14 (see Section 3.3.2).
3. The surface area contributed by a conduit to its end nodes in the head updating formula 3-
15 is modified when either critical flow depth or dry conditions occur (see Section 3.3.3).
4. A conduit's updated flow is limited to the Manning normal flow if warranted by water
surface slope and/or Froude number criteria (see Section 3.3.4).
5. The head updating formula 3-15 is replaced with equation 3-28 when a node is in a
surcharged state (see Section 3.3.5).
6. If a node is assigned a ponded area then a virtual storage unit of constant surface area is
used along with equation 3-15 to update its head when it exceeds the node's maximum
value. Otherwise a node's head cannot exceed its maximum value and any excess inflow
it receives is lost from the system (see Section 3.3.6).
3.4 Numerical Stability
The numerical stability of SWMM's dynamic wave results can be affected by the choice of the
simulation time step. Numerical instability is characterized by oscillations in flow and water
surface elevation that do not dampen out over time. Another indicator of numerical instability is
a node which continues to "dry up" on each time-step despite a constant or increasing inflow
from upstream sources.
Aside from examining the results for each conduit and node, SWMM 5 provides two metrics in
its Status Report that can help determine if a solution shows signs of instability. One is the
overall flow continuity error for the system. This is the difference between inflow and outflow
for the entire system over the duration of the simulation. If this number is greater than 5 to 10
57
-------
percent then the cause may be numerical instability (although other factors can affect the
continuity error as well).
A second metric is a link's Flow Instability Index (FII). This index counts the number of times
that the flow value in a link is higher (or lower) than the flow in both the previous and
subsequent time periods. The index is normalized with respect to the expected number of such
'turns' that would occur for a purely random series of values and can range from 0 to 150. The
Status Report identifies the links having the five highest FII's. Unfortunately since the FII does
not take into account the magnitude of the flow fluctuations it cannot determine whether the
instability is of engineering significance or not.
Stable explicit solutions of the St. Venant equations require that the time step be no longer than
the time it takes for a dynamic wave to travel the length of the conduit (Cunge et al., 1980). This
is known as the Courant-Friedrichs-Lewy (CFL) condition and can be expressed as:
At < -= (3-30)
\U + c\
where cis the wave celerity given by:
c = ^JgA/W (3-31)
An equivalent form of this condition can be written as:
L ( Fr \
A t 1) one wishes to be in strictly meeting the CFL condition (Cr= 1).
Although the SWMM 5 solution method uses an iterative implicit procedure in time to update
flows and heads, it does so one conduit and node at a time, not simultaneously. There is no
spatial coupling between elements as would occur in an unconditionally stable implicit solution
scheme. Thus the CFL condition would still apply but perhaps not as strictly (by allowing one to
use a Cr value greater than 1).
58
-------
One can estimate a At for each conduit by using the conduit's full depth Yfun in place of A/W in
Equation 3-31 and ignoring the velocity in Equation 3-30. The solution time step would then be
determined by the conduit with the smallest value of L/^gYfUu . Short conduits lead to small
time steps and longer computational times. Time steps of 10 to 30 seconds should suffice for
conduit lengths of 200 to 400 feet (the typical spacing between sewer manholes) and full depths
from 1 to 4 feet.
An option is available to artificially lengthen short conduits so that the CFL condition for a given
user-supplied time step At is met. The modified length L' is given by
L = max{L,AtQgYfuu + Qfuu/Afull)} (3-33)
where Qmi is the Manning's normal flow value (Equation 3-23) evaluated at full depth Ymi and
Ami is the flow area at full depth. This modified length is used in place of the original length in
the equations presented in section 3.4. To make the artificially lengthened conduit have a flow
resistance equivalent to the original length, its slope So and roughness coefficient n are adjusted
so that the Manning equation produces an equal head loss across both the original and
lengthened conduit for any given flow. The modified slope S0 for the lengthened conduit is:
S0 = SoJl/T (3-34)
while its modified roughness n' is:
n = njL/L' (3-35)
The conduit lengthening option is applied to all conduits whenever the user supplies a non-zero
value for the "lengthening" time step to be used in equation 3-33. This time step does not have to
be the same as the computational time step used to solve the dynamic wave equations.
Another option available in SWMM 5 is to have the program use a variable computational time
step that is adjusted throughout the simulation. The user supplies values of the smallest allowable
time step (AtWm), the largest allowable time step (At,m,x) and a desired Courant number (Cr) to be
met. At any time £, the next time step is computed from the smaller of:
1. The smallest value of
L / Fr \
~\U~\ VI + Fr) Cr
for all conduits with non-negligible Fr.
59
-------
2. The smallest value of
0.25(//crown - E)
A H*
for all non-outfall nodes that are not surcharged.
The second condition guards against an excessive change in node head over a single time step.
Both conditions are evaluated using the flow and head solutions found at time t (AHt is the
change in head found from the prior time step). The resulting time step is not allowed to be less
than Atmin nor greater than Atmax. The initial time step used at time 0 is Atnim.
To illustrate these concepts consider a 2 ft x 2 ft rectangular conduit that is 2,000 ft long with a
0.05% slope and has a Manning's roughness of 0.015 (see Figure 3-5). When divided into 10
equal length sections of 200 ft each the estimated stable time step is 200/V32.2 x 2 = 25
seconds. When analyzed as just a single 2,000 ft long section it increases to 250 seconds.
CM
CO
00
103
102
£.101
c
o
ro
11[>o
200
400
600
800
1.000 1,200
Distance (ft)
1 400
1.600 1.800 2.000
Figure 3-5 Profile view of example rectangular conduit (not to scale)
Figure 3-6 shows the outflow hydrographs for these two analysis options for a 1-hour sinusoidal
inflow hydrograph with peak flow of 10 cfs (the dotted curve in the figure). Both results are
completely stable. The option with the higher spatial resolution produces a more skewed
hydrograph with a slightly lower peak.
60
-------
10
Inflow
10 sections. At = 25 s
1 section, At = 250 s
-------
30
Inflow
25
Fixed Time Step
Variable Time Step
20
10
5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time, hours
Figure 3-7 Outflow hydrographs for example conduit - II
62
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Chapter 4 - Kinematic Wave Analysis
The kinematic wave model is derived from a simplified form of the St. Venant equations that
combines the continuity equation with the uniform flow equation. It cannot model pressurized
flow, reverse flow, or backwater effects. It is most applicable to steeply sloped conduits
subjected to long duration inflow hydrographs that produce shallow flow with high velocity
(Ponce et al., 1978). For these situations its results will not be far off from those of dynamic
wave analysis and can be computed much more efficiently using much larger time steps.
Kinematic wave modeling was included in the original release of SWMM in 1971 before a full
dynamic wave option was available. The original method included an enhancement to
approximate a backwater effect for sub-critical flow. SWMM 5 has dropped this enhancement in
favor of the classical kinematic wave formulation since the code now includes a full dynamic
wave option (described in the previous chapter) that rigorously models backwater effects.
4.1 Governing Equations
The kinematic wave model for unsteady flow in a channel or pipe is derived from the same St.
Venant equations for conservation of mass and momentum that were used for dynamic wave
analysis:
dA dQ „ . .
1 = 0 Continuity (4-1)
dt dx
dQ d(Q2/A) dH
~T + V + 9 A -r- + gASf = 0 Momentum (4-2)
at ox ox J
where all variables were defined previously in Chapter 3. Expressing head //as Z + Y(invert
elevation plus flow depth) and recognizing that dZ/dx = — S0 (the conduit's slope) allows one
to write the momentum equation as:
dQ d(Q2/A) dY
8F + -a^ + ^te = ^5»"^ (4"3)
If one assumes that the terms on the left hand side of Equation 4-3 are negligible one is left with
the relation:
S0 = Sf (4-4)
63
-------
Having the conduit's bottom slope equal the friction slope implies that the fluid motion caused
by gravity is balanced by the frictional resistance to flow. Using the Manning equation to
represent the friction slope allows one to represent the relationship between flow rate Q and flow
area A with Manning's equation for steady uniform flow:
(4-5)
7]
where the hydraulic radius R is an implicit function of flow area A for a specific conduit cross-
sectional shape. (R is defined as area divided by wetted perimeter where the latter can be
determined from flow depth which can be inferred from the flow area A.)
By defining /? = and ¥ = AR2/3 the Manning equation can be expressed as:
Q = mA) (4-6)
!F4s known as the section factor (Chow, 1959) and is a function of the flow area and conduit
geometry. For some closed conduit shapes, such as circular pipes, the section factor achieves a
maximum value at a less than full flow area (as shown in Figure 4-1) resulting in a maximum
flow that is larger than when the conduit flows full.
l.i
1.0
0.9
0.7
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8
0.9 1.0 1.1
^ / Afuii
Figure 4-1 Section factor versus area for a circular shape
64
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Thus the governing equations for kinematic wave modeling along a single conduit are the
continuity equation 4-1 along with the "rating curve" equation 4-6 that relates flow rate to area.
The dependent variables in these equations are flow rate Q and flow area A, which are functions
of distance x and time t. To solve these equations over a single conduit of length L, one needs a
set of initial conditions for either Q or A at time 0 as well as boundary condition for either
variable at x= 0 for all times t.
4.2 Solution Method
The material that follows applies to networks containing only conduits connected by junction
nodes, where there is at most one outflow conduit from each junction. Inclusion of flow control
devices (pumps, orifices, and weirs) and other processes (seepage and evaporation) will be
covered in subsequent chapters of this manual. Allowances for both flow divider nodes and
storage nodes are discussed later in this chapter in Section 4.3.
The continuity equation 4-1 is solved along a space-time grid depicted in Figure 4-2. A weighted
Wendroff implicit finite difference scheme (Smith, 1978) is used to re-express the equation as:
(1 - e)(A{+At - A[) + 0(A
At
t+At
(4-7)
(1 - 0)«?| - Qj) + 0(Qf+At - Qj+At) q
1
2
t + At
E
t
L
Distance
Figure 4-2 Space-time grid for kinematic wave analysis
65
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The subscripts 1 and 2 on A and Q refer to the upstream and downstream ends of the conduit,
respectively. The superscript refers to the time period. 0 and (p are weights chosen to be between
0.5 and 1. At each time step this equation is applied conduit by conduit starting at the most
upstream node and working downstream. The only unknowns will be A2+At and Q|+At. As there
is only one outflow conduit connected to a junction node, Q[+At is known from the sum of the
already computed Q|+At values of the upstream conduits that flow into the conduit being
analyzed, and also includes any externally imposed inflows such as wet weather runoff or dry
weather sanitary flow. The area Ai+At associated with this flow can be found by evaluating the
inverse of the section factor at the value of Q[+At /{3.
The Manning equation 4-6 can be substituted into Equation 4-7 which after some rearrangement
results in the following nonlinear equation with single unknown A*2~At:
/(^+At) = + ClA^ + C2 = 0 (4-8)
where the constants CI and C2 are given by:
L6
C1 = Mi (4"9)
C2=ik I(1 _ - 4) - 'M'l
After solving 4-8 for A2+At equation 4-6 can then be used to find the corresponding flow <2|+At-
Equation 4-8 is solved using a combination of bisection and Newton-Raphson methods (Press et
al., 1992) with both 6 and (p set to 0.6. As a first step, a bracket [Aww, AHIGH] is sought where
/(Atow) and f(AHIGH) are of opposite sign. For conduit shapes whose section factor has a
maximum value at an area Amax below Ami (such as the circular conduit of Figure 4-1), these two
areas are tried first. If these areas do not form a valid bracket then 0 to Amax is used. For shapes
whose section factor always increases with increasing area, (9 to Ami is used.
If a valid bracket is found then the procedure described in Appendix A, "Newton-Raphson-
Bisection Root Finding Method\ is used to find A2+m. Its initial estimate is A\ and a
convergence tolerance s of 0.1 percent of Ami is used. The derivative of f(A) required by the
method is / (i4) = /?f'(A) + CI where ¥ (A) is the derivative of the section factor with respect
to area A. If [Aww, AHIGH] does not form a valid bracket then A*2~At is set to 0 if both f(Aww)
and f(AHIGH) are positive and set to AfuU if both are negative.
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4.3 Computational Details
4.3.1 Order of Network Traversal
The kinematic wave procedure finds new flows, flow areas, and flow depths at the downstream
end of each conduit at the end of each time period of the analysis. At time t each conduit is
examined in its topologically sorted order when updating it to time t + At This allows the
inflows to a conduit at t + At (C{+At) to be determined from the previously computed
downstream outflows (
-------
Cutoff Divider: diverts all inflow above a user-supplied cutoff value qMrn.
Overflow Divider: diverts all inflow above the flow capacity QfUu = of the non-
diversion conduit
Tabular Divider: uses a pre-defined table that expresses diverted flow as a function of total
inflow.
Weir Divider: diverts inflow above a minimum qMrn as flow over a weir of full height
Awwith discharge coefficient cw.
The diverted flow for a weir divider node with total inflow of Qm is computed as:
f 0 for Qin < qMIN
Qdiv = j ^MAXf for qM1N < Qin < qMAX (4-11)
VQmaxJJ for Qin > qMAX
where qMAX = cwHv ar|d / = (Qin — qmw)/(Imax ~ Rmin)-
When the next conduit in sorted order is selected for routing analysis, its upstream node is
checked to see if it is a divider node. If it is, then depending on its type, the diverted flow (Wis
calculated from the node's total inflow Qin. If the conduit is the node's diversion link, then its
inflow Q[+At is set equal to Qdiv. Otherwise its inflow is set to Qin — Qdiv
4.3.4 Storage Nodes
Kinematic wave analysis allows a storage node to have more than one outlet link of any type
connected to it. The flow rate released from the storage node into the upstream end of an outflow
link will be a function of the water level in the node. Thus whenever a storage node is
encountered as the topologically sorted list of conduits is traversed its new water level must be
determined before the routing process can continue. Storage units that are terminal nodes (nodes
with no outflow links) are updated after all conduits have been analyzed.
Storage node updating is carried out using the following mass balance equation:
= Qin ~ Qout (4"12)
dVN
dt
Vn is the volume of stored water in the node, Qm is the total rate of inflow to the node and Qout is
the total rate of outflow from the node. Replacing dVN/dt with its finite difference equivalent
68
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and using the average flow rates over the time step being updated produces the following
expression for V^+At :
V^+At = V< + 0.5(Qfn + Q[n+At)At - 0.5(QoUt + Q'tf^At (4-13)
Once V^+At is known the corresponding water surface elevation Hcan be found from the storage
node's invert elevation and its user-supplied relation between surface area and water depth. A
more detailed discussion of how this is done is provided in Chapter 5.
Equation 4-13 can be re-written with all of the known values grouped together in a constant Cn
as follows:
^+At = Cjv " 0-5Qttf (4-14)
where Cn is
CN = VlN + 0.5(Q[n - Qlut + Q^)At (4-15)
and contains the known volumes and flows from time t as well as the known inflow to the
storage node at time t + At.
)t+At
'out
will be a function of the storage unit's water surface elevation H. For a conduit outflow
link, the flow at its upstream end that contributes to Qout will be determined by its upstream flow
area via Equation 4-6: Q1 = The upstream flow area Ai is determined by the conduit's
upstream water depth where it meets the storage node. This depth, Yi, is given by:
for H Z1 + Yfull
where Zi is the elevation of the conduit's upstream invert and Ymi is the conduit's full depth. A
similar situation exists for other types of outflow links, such as pumps, orifices, and weirs as will
be discussed later in Chapter 6. As an example, the flow through an orifice varies as the square
root of the head across it: Q1 = c^JH — Zx where cis a constant.
As both Qout1 and V^+At depend on H, equation 4-14 must be solved in implicit fashion using
successive approximations. The details are given in the side bar entitled "Updating a Storage
Node".
69
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The hydraulic head at storage nodes is updated one more time after all link flows at the end of a
time step of size At have been found. First a new volume for the node is found from:
Vt+At = V£ + Qnetto (4-17)
Qnet is the average net inflow to the node between times tand t + At:
Qnet = 0.5(Q[n + Q£At) - 0.5(Q'ut + Q%*) (4-18)
with Qm being the total inflow entering the node from all upstream links plus any external
sources (such as runoff flow) and Qout being the total flow rate in the links leaving the node.
Then the new head at the node, Ht+At, can be found from the node's curve of surface area versus
depth as described in Chapter 5.
Updating a Storage Node
1. Let Hlast equal the storage node's water surface elevation found at time t.
2. Using H,ast compute the flow rate into each of the node's outflow links and add these
together to determine Qout-
3. Let VN = CN — O.SQoutAt (where Ovis given by Equation 4-16), not allowing it to be
below 0 or greater than the full storage volume.
4. Find the water surface elevation //corresponding to volume VN from the node's curve of
surface area versus depth.
5. Let Hnew = (1 - 6)Hlast + 6H where £is 0.55.
6. If |Hnew — Hlast | is below 0.005 ft then stop with Hnew as the water surface elevation at
time t + At.
7. Set Hlast = Hnew and return to Step 2.
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4.3.5 Nodal Heads
Kinematic wave analysis does not depend on or even define the hydraulic head that exists at
nodes that are not storage units. To make its reported output compatible with that provided by
dynamic wave analysis the head at a non-storage node is arbitrarily set equal to the highest water
elevation in the links that are connected to it. For inflowing conduits the water surface elevation
at the downstream end of the conduit, as derived from the downstream area, is considered. For
out flowing conduits it would be the water surface elevation at the upstream end. It should be
noted that kinematic wave analysis ignores the presence of any offset that an outflow conduit at a
non-storage node has at its upstream end. Under dynamic wave analysis there would be no flow
into such a conduit until the water level at the node reached the offset elevation. The kinematic
wave model also ignores any surcharge depth that may have been assigned to a node since
neither conduits nor nodes are allowed to pressurize.
4.3.6 Flooding and Ponding
Normally any excess inflow to a node under kinematic wave analysis over what the outflow links
can handle will be lost from the system. For non-storage, non-terminal nodes the flooded
overflow rate at time t + Atwould be:
where VNfUu is the volume of the storage node when full. When (Wis non-zero, the head at the
node is set equal to its elevation at full depth for reporting purposes.
As with dynamic wave analysis, the option exists for a junction or divider node to have the
excess inflow volume over a time step be stored at the node and then released as external inflow
during the next time step. The node's "ponded area" parameter is used to indicate that ponding is
allowed if it is assigned a non-zero value (and does not enter into any computations). In this case,
the node's ponded volume Vp is kept track of as the simulation unfolds. Its initial value is 0. At
time t + Atit is updated as follows:
Qwfi = max(0, Qnet)
(4-19)
where Qnet is given by Equation 4-18. For storage nodes it would be:
Qltfi = max(0, Qnet - (VNfull - V£)/At)
(4-20)
V£+At = max(0, V£ + Qnetbi)
(4-21)
The overflow reported for the time period is given by:
71
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= max(0, (V£+At - V£)/M)
(4-22)
The flow added to the node's total inflow at the start of the next time period is Vp+At/At. And
for reporting purposes, anytime Vpis greater than 0 the node's head is set equal to the elevation
at full depth.
4.4 Numerical Stability
The authors of the original version of SWMM's kinematic wave routine applied the techniques
of O'Brien et al. (1951) to show that the method was unconditionally stable for any choice of 0
and ^both greater than 0.5 ( Metcalf and Eddy et al., 1971a). Smith (1978, p. 188) showed that
the Wendroff implicit scheme using centered differences (6= (p = 0.5) was also unconditionally
stable. Because the scheme is stable it does not employ the variable time step option as does
dynamic wave analysis. And although it is stable, it is still subject to numerical dispersion when
the Courant number differs from 1 and to numerical diffusion (hydrograph attenuation) due to
the discrete grid size (Ponce, 1991).
To illustrate the stability properties of SWMM's kinematic wave method the example of Chapter
3 solved earlier under dynamic wave analysis will now be solved again using the kinematic wave
model. The example consists of a 2,000 ft long, 2 ft x 2 ft rectangular conduit whose slope is
0.05 percent and Manning's roughness is 0.015. When divided into 10 equal sub-conduits of 200
ft each the dynamic wave solution required a 25 second time step to produce a stable outflow
hydrograph (refer to Figure 3-6). At a 120 second time step the solution was highly unstable (see
Figure 3-7).
As shown in Figure 4-3, kinematic wave (KW) is able to produce a stable outflow hydrograph
for the 120 second time step. A stable dynamic wave (DW) solution required a much smaller
time step of 25 seconds. The KW solution exhibits the properties associated with this
approximate method of flow routing, with a very modest attenuation of the inflow hydrograph
peak and some distortion of the outflow hydrograph. The DW solution should be considered the
more accurate one, with its greater reduction in peak flow due to the storage effect provided by
the additional inertia and pressure terms included in the dynamic wave formulation.
72
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10
Inflow
KW, At = 120 s
DW, At = 25 s
o
LL
0.2
0.4
0.6
1.2
1.4
1.8
2
0
0.8
1
1.6
Time, hours
Figure 4-3 Outflow hydrograph for example conduit
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Chapter 5 - Cross-Section Geometry
The hydraulic modeling procedures described in chapters 3 and 4 require calculation of several
cross-section geometric properties for partially full conduits. These include the following
functions:
A(Y) flow area A as a function of flow depth Y
W(Y) top width Was a function of flow depth Y
R(Y) hydraulic radius R as a function of flow depth Y
Y(A) flow depth Fas a function of flow area A
W(A) section factor If as a function of flow area A
W(A) derivative of section factor If with respect to area A
A(W) flow area A as a function of section factor W
as well as the following constants used in evaluating these functions:
Ami
area at full depth
Wfmax
maximum width
Rfull
hydraulic radius at full depth
WfuU
section factor at full depth
Vmax
maximum section factor
Amax
area corresponding to Wmax.
This chapter describes how these properties are computed for the wide range of conduit shapes,
both standard and irregular, included in SWMM. In addition, the procedures used to compute
both the normal and critical flow depths used in dynamic wave analysis are discussed.
5.1 Standard Conduit Shapes
SWMM recognizes a number of standard pre-defined conduit shapes. These include five open
channel shapes (rectangular, trapezoidal, triangular, parabolic and power law), four commonly
used closed pipe shapes (circular, rectangular, ellipsoid and arch), seven closed shapes found
mainly in older masonry sewers, and four closed composite shapes that are combinations of
rectangular, triangular and circular sections.
74
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5.1.1 Open Channel Shapes
SWMM can analyze the following standard open channel shapes:
• Rectangular with bottom width b
• Trapezoidal with bottom width b and side slope (run over rise) 5
• Triangular with side slope 5
• Parabolic with top width b at full depth Yml
Table 5-1 lists the formulas used to compute the geometric properties of these shapes that are
functions of water depth Y: A(Y), W(Y), and R(Y). Table 5-2 lists the formulas used to compute
the properties that are functions of flow area A: Y(A), R(A), and the derivative of the wetted
perimeter P'(A)). The latter quantity is used for computing the derivative of the section factor as
described below.
Table 5-1 Geometric properties for open channel shapes as functions of water depth
Shape
A(Y)
W(Y)
R(Y)
Rectangular
bY
b
bY
b + 2Y
Trapezoidal
(,b+sY)Y
b + 2 sY
(b + zY)Y
b + 2Yy/l + s2
Triangular
sY2
2 sY
sY
2Vl + s2
2A(Y)
Parabolic
c = b2/(4YfUu)
^Y4cY
2 VcF
c(xt + ln(x + t))
X = 2yjY/C
t = V1 + x2
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Table 5-2 Geometric properties for open channel shapes as functions of flow area
Shape
Rectangular
A
b
R(A)
A
b + 2A/b
P'(A)
2
b
Trapezoidal
yjb2 + 4s A
2s
i4Vl + s2
b + Y(A)
2Vl + s2
b2 + 4sA
Triangular
Ja/s
A
2F(i4)Vl + s2
Vl + s2
sA
Parabolic
c = b2/(4YfUu)
(—)
l4v'c'
2/3
2c(xt + Zn(x + t))
x = 2^Y(A)/c
t = Vl + X2
not used
The section factor Wior each of these shapes is given by:
fCA) =AR(A)2/3
(5-1)
With the exception of the parabolic shape, its derivative with respect to area A is:
W\A) = (5/3 -2/3PR)R2/3
(5-2)
where P' and R are evaluated at the desired value of A. For parabolic channels the section factor
derivative is computed using the difference formula:
V(A) =
V(A + AA) - ¥(A - AA)
2AA
(5-3)
where AA is 0.1% of the full cross section area.
In addition to the four open sections just described SWMM can also analyze a cross section
whose side wall shape is described by the power law function:
y = ax
l/Y
(5-4)
76
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where xis horizontal distance from the centerline, y\s vertical distance, 1/y is an exponent and a
is a constant. To use this shape the user supplies values for 1/y, the full depth Ymi and the top
width when full b (see Figure 5-1). Note that the parabolic shape is a special case of this power
function shape where 1/y equals 2.
y
Ax
Figure 5-1 Power law cross section shape
With this shape it is more convenient to work with water surface width Was a function of water
depth Y, which can be done by re-expressing Equation 5-4 as:
W = cYr (5-5)
where cis another constant. Since W= b at Y= Ymu the constant c equals b/Yjull . The full area
Ami is bYfuU/(y + !)• Table 5-3 lists the expressions used to compute the geometric properties
for partially full power law shapes. The wetted perimeter P table entry is evaluated by
approximating each of the curved sides of the shape by a series of 50 line segments whose
lengths up to height Yare added together.
5.1.2 Closed Rectangular Shape
A closed (or covered) rectangular conduit has the same A(Y), W(Y), and Y(A) functions as its
open counterpart. Its R(Y) and (A) functions are also the same up to the point where the
conduit becomes full and the wetted perimeter then includes the top width. This introduces a
discontinuity in the relationship between R and Y as well as between If and A. To avoid this, a
maximum section factor is deemed to occur at 97% full after which it decreases linearly to the
section factor when completely full. These two section factors are given by:
77
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Table 5-3 Geometric properties for the power law shape
Property
Expression
c
b/y/nu
A(Y)
c YY+1/(y + 1)
W(Y)
cYY
P(Y)
N
2 ^ J Ax? + Ay2 where Ay = 0.02 YfUu, N = Y/Ay,
i=1
and Axj = (c/2){(iAy)7 — ((/ — l)Ay)7}
R(Y)
A(Y)/P(X)
Y(A)
[(y+lWc]1^1'
R(A)
A/P(Y(A))
W(A)
AR(A)2/3
V(A)
W{A + Ai4) - V(A - Ai4)
—— where AA = 0.001i4fu;;
2AA '
Vfuii — Afuii(Afuii/Pfuii) (5-6)
^'max = 0-97 Afuii(0.97AfUu! Pmax) (5-7)
where Aj?uh — bYfUn, PfUu — 2(b + YfUii), and Pmax — b -\- 2(0.97YfUii).
When either Y or A do not exceed 97% of their full values, the closed rectangular hydraulic
radius and section factor are computed in the same fashion as for the open rectangular shape
described in section 5.2.1. Above this point the hydraulic radius at a given depth Fis:
ROD = A(Y)/P(Y) (5-8)
where
POO = 2Y + b + b (Y/Yfull - 0.97)/0.03 (5-9)
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and the section factor and its derivative at a given flow area A are:
V(A) ~ ^max QPmax
VMl)(A/Afull-0.97)/0.03
WW = (Vfull ~ ^max)/(0.03^/u„)
(5-10)
(5-11)
5.1.3 Circular Shape
Although analytical formulas are available for the properties of partly full circular cross sections
(see French, 1985), they contain trigonometric functions that are time consuming to compute.
Thus for reasons of efficiency SWMM uses a set of lookup tables that are based on those
published by Chow (1959). The tables consist of the following:
Atbi: A/A mi as a function of Y/Ymi
Wtbi: W/Wmax as a function of Y/Ymi
Rtbi: R/Rmi as a function of Y/Ymi
Ytbi: Y/Ymi as a function of A/Ami
ft/,/.- Y/Ymi as a function of A/Ami
Each table consists of 51 equally spaced values of Y/Ymi or A/Ami between 0 and 1. They are
graphed in Figures 5-2 and 5-3 and are listed in Appendix C. The normalizing factors used in the
tables are for full flow conditions (Y = YfuLL) whose formulas are listed in Table 5-4.
Table 5-4 Geometric properties of a full circular cross section
Property
Value
Depth
Yfuii
Area
Afull = 0.7854Yfulh
Maximum Width
^Knax ~ Yfull
Hydraulic Radius
Rfuii = 0-2 SYfuu
Section Factor
V'full = AfullRfill
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1.4
1.2
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y/Yfu,,
Figure 5-2 Geometric properties of a partly filled circular shape based on depth
1.2
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Afun
Figure 5-3 Geometric properties of a partly filled circular shape based on area
80
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To find A, W, or ^?for a given Fone first evaluates i = (Y/YfUii)(N — 1) rounded down to the
nearest integer value where N= 51, linearly interpolates the appropriate table between the entries
at index / and i+1, and then multiplies by the appropriate normalizing factor (either Ami, Ymu or
Rfuii). A similar procedure is used to evaluate Y or If as a function of A normalized by Ami The
section factor derivative is determined directly from the Wtbi as follows:
where / is the integer value of {A/AfuU)(N — 1) for N = 51. For added accuracy, analytical
functions are used to compute Y, W, and Wfor areas below 4% of Ami They are described in the
side bar entitled "Analytical Functions for Circular Cross Sections'
5.1.4 Ellipsoid and Arch Shapes
Figure 5-4 depicts standard ellipsoid and arch sewer pipe cross sectional shapes. Next to circular
pipes these are the most commonly used shapes for newly installed sewers and culverts. Each
shape is defined by its "rise" which is its full depth Ym, and its "span" which is its maximum
width Wmax. The vertical and horizontal ellipsoids have the same shape but rotated by 90 degrees
(the span of one is the rise of the other and vice versa).
V'(.A) = CVtmli + 1] - Ytw[i])(W - 1 )(Wfun/Afull)
(5-12)
Vertical Ellipse
Horizontal Ellipse
Arch
Figure 5-4 Ellipsoid and arch pipe cross sectional shapes
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Analytical Functions for Circular Cross Sections
The following relation holds between the central angle 6 (in radians)
subtended by the water surface in the conduit's cross section (see figure / \
at right) and flow area A (French, 1985): l~ J
A = Afull(6 - sin6)/2n
Given a value for A, this expression is solved for 6 using the following Newton-Raphson
routine:
1. Let 6 = 0.031715 — 12.79384a + 8.28479-\/a where a = A/AfUu.
2. Compute A6 = 2na — (6 — sin6)/( 1 — cos6).
3. Let 6 = 6 + A6.
4. If |A0| < 0.0001 then stop. Otherwise return to step 2.
Once 0 is known the
remaining cross section variables can be found as follows:
Flow Depth:
Y = Yfull (1 - cos(6/2))/2
Section Factor:
Vfuii(0 - sind)5'3
2nd2/3
Wetted Perimeter:
P = 0Yfull/2
Wetted Perimeter
4
P' =
Yfuiii 1 - cosO)
Derivative:
Hydraulic Radius:
II
"a
Section Factor
Derivative:
y = [(5/3) - (2/3)P'R]R2/3
SWMM contains a list of 23 standard ellipsoid pipe sizes and 102 standard arch pipe sizes taken
from the American Concrete Pipe Association's and the American Iron and Steel Institute's
design manuals (American Concrete Pipe Association, 2011; American Iron and Steel Institute,
1999). The standard ellipsoid and arch pipe sizes are tabulated in Appendixes D and E,
respectively. Each size is characterized by its rise, span, full area, and full hydraulic radius. Users
can either select from one of these standard sizes or supply their own values for rise Ymi and
span Wmax, both in feet. In the latter case the corresponding full area Ami and hydraulic radius
Rfiiiiare estimated using the formulas in Table 5-5.
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Table 5-5 Full area and hydraulic radius of custom ellipsoid and arch pipe sections
Property
Ellipsoid Shape
Arch Shape
Full Area AfuU
1.2692$,,
0.7879YfUuWmax
Full Hydraulic Radius RfUu
0.3061^,,,,
0.2991Yfull
Information in the aforementioned design manuals was used to construct the following tables for
both the ellipsoid and arch shapes (only a single set of tables is needed for the two ellipsoid
shapes since they are just rotated versions of one another):
Atbi: A/A mi as a function of Y/Ymi
Wtbi: W/Wmax as a function of Y/Ymi
Rtbi: R/Rmi as a function of Y/Ymi
Each table contains entries for N = 26 equally spaced values of Y/Ymi between 0 and 1. The
tables for ellipsoid pipes are in Appendix D and those for arch pipes are in Appendix E. To find
A, W, or R for a given Y one first determines the integer portion of (JV — 1)(Z/YfuLL), linearly
interpolates the appropriate table between the entries at this and the next higher index, and then
multiplies by the appropriate normalizing factor (either Ami, W,m,x, or Rmi).
To find the depth associated with a given area Y(A), a bisection (or interval halving) procedure is
first used on the appropriate (either ellipsoid or arch) area table Am to find the position / so that
Am[i] ^ A/Afuii ^ Atm[i + 1], Then the desired depth Yis interpolated from this position in
the table using the following expression with N= 26:
Y'uu iA/Aruii --W']) ^
( ' (W-l)l GWi + l]-i4tw[i]))
The following steps are used to find the section factor associated with a given area W(A):
1. Use the aforementioned procedure to find the depth Ycorresponding to area A from the
appropriate Am
2. Use the shape's hydraulic radius table Rtbi to find the hydraulic radius R for this depth.
3. Compute the section factor as: V(A) = AR2/3.
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The section factor derivative for a given area W(A) is found using the following central
difference equation:
where AA = 0.001AfUu
(5-15)
5.1.5 Older Masonry Sewer Shapes
SWMM contains seven pre-defined closed conduit shapes shown in Figure 5-5 that were used
primarily in older masonry sewers built over a century ago. Their geometric properties have been
derived from information and drawings found in Metcalf and Eddy (1914) and Davis (1952).
These properties are represented using the same type of lookup tables discussed previously for
circular cross sections (see section 5.2.3). The number of entries TVin each table for each shape is
listed in Table 5-6. The full tables are provided in Appendix F. The values of Ami, Rmi; and Wmax
used to normalize the entries in the tables for each shape are listed in Table 5-7. The full section
2 /3
factor Wfuiiused to normalize the section factor table is computed as ^fUiiRf Uli •
Basket Handle Catenary
Egg
Gothic
Horseshoe Semi-Circular Semi-Elliptical
Figure 5-5 Masonry sewer shapes
84
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Table 5-6 Number of entries in geometric property tables for masonry sewer shapes
Shape
Atbi
Rtbl
Wwi
Ytbl
Basket Handle
26
26
26
51
51
Egg
26
26
26
51
51
Horseshoe
26
26
26
51
51
Catenary
-
-
21
51
51
Gothic
-
-
21
51
51
Semi-Circular
-
-
21
51
51
Semi-Elliptical
-
-
21
51
51
Table 5-7 Geometric parameters of masonry sewer sections
Shape
Ami
Rfuii
Wmax
Wmax
Basket Handle
0-7862 Yfull
0.2464 YfUu
0-944 Yfull
1.06078 Vfull
Egg
0-5105 Yfull
0.1931 YfUll
0.667 Yfull
1-065 Wfull
Horseshoe
0-8293 Yfull
0.2538 Yfua
Yfull
1-077 Wfull
Catenary
0.70277 Yfull
0.23172 Yfull
0-9 Yfull
1-05 Wfull
Gothic
0-6554 Yfull
0.2269 Yfull
0-84 YfUll
1-065 Wfull
Semi-Circular
1-2697 Yfull
0.2946 Yfull
1-64 YfUll
1.06637 Vfull
Semi-Elliptical
0-785 Yfull
0.242 Yfull
Yfull
1-045 Wfull
The tables are used in the same manner as the ones for a circular shape to directly evaluate A(Y),
W(Y), R(Y), Y(A), W(rA), and ^'(A). For the shapes that do not have an At/,/, A(Y) is determined
using the inverse lookup method on the Ytbi described in section 5.2.4 for ellipsoids and arches.
For shapes without an Rtbi, R(Y) is found by first finding A(Y) as just described, then finding
W(A) for the resulting area A, and finally evaluating^/7/M)3/2. Equation 5-15 is used to compute
W'(A).
85
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5.1.6 Composite Shapes
Figure 5-6 shows four cross section shapes that
triangular sections. The formulas for computing t
following paragraphs.
btm
Filled Circular
Rectangular - Round
'e combinations of circular, rectangular, and
ir geometrical properties are presented in the
htm
Rectangular - Triangular
Modified Basket Handle
Figure 5-6 Composite cross section shapes
Sediment Filled Circular Shape
This is a circular cross section that is partially filled with immobile sediment to a specified depth
Ybtm. (This filled depth remains constant - SWMM does not model how it might change over
time due to sediment transport processes.) The depth available for flow is YfuU — Ybtm. To
compute the geometric properties of this shape one first uses the circular shape functions to
compute the area Abtm, top width Wbtm, and hydraulic radius Rbtm at a depth of Ybtm for the full
circular shape with diameter Yml The wetted perimeter at this depth, Pbtm, is Abtm/Rbtm . Then
the expressions listed in Table 5-8 can be used to find the section properties for a specific flow
depth Kabove Ybtm or area A above Abtm .
86
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Table 5-8 Geometric properties for a sediment filled circular cross section
Property
Value Based on Full Circular Shape Functions
A(Y)
A(Y + Ybtm) Abtm
W(Y)
W(Y + Ybtm)
R(Y)
A(Y + Ybtm) — Abtm
(.A(Y + Ybtm)/R(Y + Ybtmj) - Pbtm + Wbtm
Y(A)
F(i4 + Abtm) — Ybtm
W(A)
AR(AY)2/3 where AY = Y(A + Abtm) - Ybtm
^'(A)
V{A + AA) - W(A - M)
2AA where AA = 0.001(i4/u„ Abtm)
Rectangular-Triangular Shape
This shape consists of a triangular bottom section of height Ybtm connected to a closed
rectangular top section of width b and height Ymi- Ybtm. The slope of the triangular section's
sidewalls 5 is b/2Ybtm . For depths below Ybtm (or areas below Ybtm b/2) the geometric
properties are computed in the same manner as for the open triangular shape of section 5.2.1. At
higher depths (or areas) the methods used for the closed rectangular shape of section 5.2.2 are
applied with some adjustments made to accommodate the filled triangular section. The
applicable formulas are listed in Table 5-9.
Rectangular-Round Shape
This composite shape consists of a closed rectangular top with a rounded bottom section. It has
full height Ymi, top width b, and bottom radius of curvature r(see Figure 5-6). Table 5-10 lists
the parameters used to compute the section's properties whose formulas are given in Table 5-11.
87
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Table 5-9 Properties of the rectangular section of a rectangular-triangular shape
Property
Expression
s
b/(2Ybtm)
Abtm
bYbtm/2
Afull
biXfull ~ Ybtm/2)
Rfull
Afull/(^Yfitxnyl 1 + S2 + 2(Yfuu — Ybtrri) ^
Vfull
Afull*fuu
A(Y)
Abtm (X ~ Ybtm)b
Y(A)
Ybtm (A — Abtm)/b
W(Y)
b
P(Y)
2 WV(1 + S2) + 2(Y - Ybtm)
if A(Y) > 0.98Afua add on (A(Y)/Afua - 0.98)^/0.02
R(Y)
A(Y)/P(Y)
R(A)
A/P(Y(A))
Tmax
0.98AfuuR(0.98Afuu)2/3
V(A)
AR(A)2/3 for A < 0.98Afua
Vmax + (Trull - Vmax) (A/AfuU - 0.98)/0.02 for A > 0.98AfuU
V\A)
(5/3 - (2/3)(2/b)R(A))R(A)2/3 for A < 0.98Afua
(Truii ~ x¥max)l^®2AfUii for A > 0.98AfuU
Modified Basket Handle Shape
The modified basket handle shape is the reverse of the rectangular-round shape with a
rectangular bottom section below a rounded top section. It has full height Ymi; bottom width b,
and top section radius of curvature r(see Figure 5-6). The central angle # formed by the rounded
top section is:
6 = 2sin~1(b/2r) (5-14)
The depth Ybtm of the bottom rectangular section is:
88
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Ybtm = Yfuu ~ r(l - cos(B/2))
(5-15)
and its area Abtm is bYhtm . The shape's full area Ami is:
Afuii = Abtm + r2/{2(0 - sin6)} (5-16)
For depths up to Ybtm and areas up to Abtm the open rectangular shape functions of Tables 5-1 and
5-2, respectively, are used to compute the modified basket handle's section properties. For
depths and areas above this the functions listed in Table 5-12 are used.
Table 5-10 Geometric parameters for rectangular-round shapes
Parameter
Value
Central Angle 0
2sm_1(Z)/2r)
Bottom Section Height Ybtm
r( 1 — cos{6 / 2))
Bottom Section Area Abtm
(r2/2)(0 — sin(0))
Full Area Ami
b(Yfull ~ Ahtm
Full Hydraulic Radius Rmi
Afull/{r6 2(Yfull ~ ^btm)
Full Section Factor f/,,//
AfullRfull
Maximum Hydraulic Radius Rmax
0.98i4fUu/{r6 + 2(0.98AfuU — Abtm)/b}
Maximum Section Factor Wmax
0.98AfUiiRr/mx
89
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Table 5-11 Geometric properties for rectangular-round shapes
Property
Formula
Applicable Region
A(Y)
0.5r2(0 —sin(0)) where Abtm
2rcos~1(l — Y(A)/r)
A < Abtm
P(A)
2rsin~1(b/2r) + 2 (A — Abtm)/b
Abtm < A < 0.98AfUn
2rsin~1(b/2r) + 2(A — Abtm)/b
A > Q.9QAfUu
+ (A/Afull-0.98)b/0.02
R(A)
A/P(A)
W(A)for circular shape with YfuU = 2r
A < Abtm
W(A)
AR(A)2/3
Abtm < A < 0.98AfUn
Vmax + (Vfull ~ ^max) (A/AfuU ~ 0.98)/0.02
A > Q.9QAfUu
CFG4 + Ai4) - f (i4 - Ai4)}/2Ai4
A < Abtm
V(A)
(5/3 - (2/3)(2/b)R(A))R(A)2/3
Abtm < A < 0.98Afuii
(V/iM ~ Vmax^/iP-QZAfull)
A > Q.9QAfUu
90
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Table 5-12 Properties in the rounded top section of a modified basket handle shape
Property
Expression
A(Y)
Afuii ~ O2/2)(0 - sin0) where 0 = 2cos 1(l - (Yfull - Y)/r)
W(Y)
2 J (fan -Y) (2r - (fan - 0)
R(Y)
R(A(Y)) using R(A)function below
Y(A)
Yfuii ~ Y(Afuu — A) using Y(A)for circular shape with YfuU = 2r
P(A)
C0 - 0)r + 2(Yfull - Y(A)) + b where 0 = 2cos_1(l - (Yfull - Y(A))/r)
R(A)
A/P(A)
V(A)
AR(A)2/3
V\A)
{V(A + AA) - ¥(A - AA)}/(2AA) where AA = 0.001i4/u„
5.1.7 Area at Maximum Flow
The solution method for kinematic wave analysis in a closed conduit needs to know what cross-
sectional area corresponds to the flow depth where the section factor and hence the Manning
equation flow rate is a maximum (see Sections 4.2 and 4.3.2). Below this point the section factor
is an increasing function of area, after which it decreases until the conduit becomes full. Table 5-
13 lists the ratio of the area at maximum flow (denoted as A,m,x) to the full area {Ami) for the
standard closed conduit shapes recognized by SWMM. For open shapes Amax is the same as Ami
Table 5-13 Area at maximum flow to full area for standard closed conduits shapes
Shape
Amax/Afull
Shape
Amax/Afull
Rectangular
0.97
Circular
0.9756
Elliptical
0.96
Arch
0.92
Basket Handle
0.96
Egg
0.96
Horseshoe
0.96
Catenary
0.98
Gothic
0.96
Semi-Circular
0.96
Semi-Elliptical
0.98
Rectangul ar-Tri angul ar
0.98
Rectangul ar-Round
0.98
Modified Basket Handle
0.96
91
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5.1.8 Area from Section Factor
Kinematic wave analysis also needs to know the area A corresponding to a given normal flow
seven masonry sewer shapes discussed in section 5.2.5 the following "reverse" lookup method is
used with the shape's ^versus A table (Wtbi) to determine A given W.
Let Y*be the section factor value whose area is sought and let TV be the number of entries in
Wtbi. First the interval in the table that brackets f is located. Since these are all closed
shapes, there will be a table entry index imax after which the f /Wfun values begin to decrease. If
XV*l^fuii is between and fft/f/V] then this portion of the table is examined to find the
index i*so that f is between Ytb{i*\ and lf/tb{i*+l\. Otherwise a bisection search is used
between index 0 and imax to find the interval starting at /*that brackets f Then the area
A*corresponding to IP* is computed as:
For all other shapes the Newton-Raphson-Bisection method (see Appendix A) is used to find the
solution of
where If* is the section factor value whose corresponding area is sought. The derivative of f(A)
required by the method is the shape's W(A) function. If the shape is closed with Amax < Ami and
W* is between f/,,// and Wmax then the search interval is [Ami, Armtx\. Otherwise it is [0, Amax\.
The convergence criterion is 0.01 percent of Ami .
5.2 Custom Conduit Shapes
In addition to its catalog of standard pre-defined shapes, SWMM can also utilize custom closed
shapes that are defined by a Shape Curve supplied by the user. This curve specifies how the
width of the cross-section varies with height, where both width and height are scaled relative to
the section's full height. This allows the same shape curve to be used for conduits of differing
sizes. An example shape curve along with its table of width versus height is shown in Figure 5-7.
rate Q from its associated section factor W, where ¥ = Q^JSo/rj. For circular shapes and the
(5-17)
/CA) = fCA) - V* = 0
(5-18)
92
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Y/Yfuii W/Yfuii
0.000
0.928
0.667
0.874
0.930
0.798
1.000
0.697
0.997
0.567
0.988
0.342
0.967
0.000
Figure 5-7 A Shape Curve with a depth segment shown
The flow area A, top width Wand hydraulic radius R of a custom shape are pre-computed at 51
equally spaced vertical values between 0 and 1 along the shape curve and stored in tables Atbi,
Wtbi, and Rtbi, respectively. The tables are constructed by analyzing each depth segment of size
1/50 = 0.02 starting at 0 and working upwards. As shown in Figure 5-7, each depth segment
forms a trapezoid within the cross section. The area of this trapezoid is added to the shape's total
area ASUm and the length of its side walls is added to the total wetted perimeter PSUm. If the depth
segment straddles more than one shape curve segment, then additional trapezoids are formed at
the shape curve's vertices, each of which contributes to ASUm and PSum. The Atbi entry for the
segment is set to ASUm, the Rtbi entry to Asum/Psum, and the Wtbi entry to the segment's top
width.
When a conduit with full depth Ymi is assigned a shape curve for its cross section, the curve's
geometry tables are used in the same manner as the tables for ellipsoid and arch shapes described
in section 5.1.4 for evaluating A(Y), W(Y), R(Y), Y(A), W(A), and ^'(A). The values of Ami,
Rfnii, and Wmax used to convert the normalized values in the tables to actual dimensions are as
follows:
Afuii = Atbi[50]YfUu (5-19)
Rfuii = Rtbi\5Q\Yfuii (5-20)
^Knax ~ {o
-------
Amax = {^.^0(^tbiU]RtbLU]2/3)}Yf2u[i
(5-22)
The Newton-Raphson-Bisection method described in section 5.1.8 is used to evaluate A(W).
5.3 Irregular Natural Channels
SWMM also has the ability to model natural channels with irregular shaped cross sections. The
cross sectional shape is represented by a transect that begins at the top of the left bank of the
channel (looking downstream) and extends transversely across the channel to the top of its right
bank. The channel's bed elevation (y) relative to a known elevation is recorded at a series of
measurement stations (x) across the transect (see Figure 5-8). A single transect is used to
represent a channel's cross section along its entire length. This might require that longer
channels with varying cross section profiles be broken into smaller more uniform segments.
As shown in Figure 5-8, transects can contain two overbank areas on either side. Each is optional
and is used to specify a different Manning's roughness coefficient than that assigned to the main
channel. Each overbank boundary location must coincide with one of the transect's measurement
stations.
Bed Elevation
Measurement x
X Coordinate
HWH
Left Overbank iiSs
,
Right Overbank
Main Channel
Figure 5-8 A natural channel transect
The flow area A, top width W and hydraulic radius R of a transect are pre-computed at 51
equally spaced values of flow depth relative to full depth (Y/YfllU) and stored in tables Am Wtbi;
and Rtbi, respectively. The table values are normalized with respect to the full section area Ami,
the maximum width Wmax, and the full section hydraulic radius Rfuii, respectively. These tables
94
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are used in the same manner as the tables for ellipsoid and arch shapes described in section 5.1.4
for evaluating A(Y), W(Y), R(Y). Y(A), W(A), and W(A). A(XF) is found using the Newton-
Raphson-Bisection method as described in section 5.1.8.
The first step in constructing the geometric property tables for a transect is to find the
measurement stations with the lowest and highest elevations. The full channel depth Yfun is set
equal to the difference between these values. If necessary, a new station is added at either end of
the transect so that both ends are at the highest elevation. Then all station elevation values y are
converted to the height above the lowest elevation station.
Next table entries are generated for a series of depths that divide the full depth into 50 equal
increments starting at 0 (whose table entries are set to 0). The procedure for finding each table's
entry for the k-th depth interval is described in the side bar entitled "Computing Geometry Table
Entries for Irregular Cross Sections". It traverses the cross section's transect, computing the
area, width, and wetted perimeter for each measurement station segment that lays above the
current depth increment. It also finds the conductance (the section factor times roughness) of
compound segments that separate regions of differing roughness or where valleys occur in the
transect's profile. (Figure 5-9 shows a flow depth increment with three compound segments.)
After the end of the transect is reached the sum of the compound conductances is used along
with the main channel roughness to find the hydraulic radius for the current depth increment.
\
\
•' - - - -
. .• • '
- , : .
, •• ¦
SH8
Figure 5-9 A transect depth increment with three compound segments
Once table entries for all depth increments have been generated, the following quantities are
assigned and used to normalize the entries in their respective tables: AfUU = v4tw[50], Wmax =
Wlbl[50J, RfUu — Run[50] Another adjustment is to set WtM[0] = Wlbl\t\ since the above
procedure does not calculate a width at zero depth.
95
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Computing Geometry Table Entries for Irregular Cross Sections
1. To find the k-th entry in an irregular cross section's geometry tables first initialize the
following:
Flow depth: Y = kYfUli/50
Table entries for index k. Atbl[k] = 0,Wtbl[k] = 0,Rtbl[k] = 0
Compound segment area: Asum = 0
Compound wetted perimeter: Psum = 0
Total flow conductance: K = 0
Transect station index: i = 1
2. Select the cross section segment between transect stations at x-i and xu
3. If the flow depth is below the channel bottom ( Y < min(yi_ltyi ) ) go to step 10.
4. Compute the width wand wetted perimeter p of the full segment:
W = X£ - Xj_!
p = Vw2 + Ay2 where Ay = |y£ - yt_x \
5. If the segment is completely submerged (Y > max(yj_1( y j)) compute its area a as:
a = w(Y - (yj_! + yd/2)
Otherwise let a = (Y — min(y£_1,y£))/Ay and set a = a2wAy.
6. Adjust the width and wetted perimeter for partial submergence:
w = aw, p = ap
7. Update the table entries for area and top width:
Am[k] = Am[k] + a; Wm[k] = Wm[k] + w
8. Update the area and wetted perimeter of the current compound segment:
A-sum ~ A sum Psum ~ ^'sum P
9. Let /?/be the roughness coefficient between stations i-1 and i. If station /marks the end of
a compound segment (y£ > Y or n£ =£ n£+1) then update the total conductance:
K = K + (1.486/n£)i4sum(i4sum/Psum)2/3
and begin a new compound segment by setting Asum and Psum to 0.
10. If more transect stations remain, increment the station index,/ = i + 1 and go to Step 2.
11. Compute the hydraulic radius table entry:
/ ncK \3/2
Ral[k] ~(i486Aw[fc])
where nc is the main channel roughness.
96
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An irregular natural channel can also be assigned a meander modifier. This is the ratio of the
length of a meandering main channel to the length of the overbank area that surrounds it. While
the user-supplied length for the overall channel is that of the longer main channel, SWMM will
use the shorter overbank length in its calculations. The Manning's n of the main channel will be
increased by the square root of the meander modifier to provide an equivalent friction head loss
over the reduced main channel length.
5.4 Storage Unit Geometry
SWMM's hydraulic modeling procedures require knowledge of how a storage unit's surface area
A and volume V vary with surface depth Y above the bottom of the unit. It is not necessary to
describe the actual shape of the unit (round, rectangular, etc.), just a relationship between either
surface area or volume and depth as one can be derived from the other
(.A = dV/dY and V = / AdY). SWMM asks the user to supply the relationship between surface
area and depth using a Storage Curve. There are two types of curves that can be used: functional
and tabular.
The functional storage curve has the general form:
where cO, cl, and c2 are user-supplied constants. The surface area at a given depth is found
directly from this equation. The relation between volume V and depth Y (required for dynamic
wave analysis) is:
To find the depth associated with a given volume (required for kinematic wave analysis) one
solves the following nonlinear equation for Y:
It is solved using the Newton-Raphson-Bisection method described in Appendix A over the
interval [0, Yfuii\ with initial estimate Y = V/(cO + cl), convergence tolerance of 0.001 ft and
derivative / (Y*) given by Equation 5-23.
A = cO + clYc2
(5-23)
c2+l
(5-24)
(c0r+(&h:)rc'+1)
(5-25)
97
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The tabular storage curve is a series of user-supplied data pairs Y, A that define the vertices of a
piecewise linear curve of surface area versus surface depth for the storage unit. An example of
this type of curve is shown in Figure 5-10.
Storage Curve
60000
50000
j40000
£ 30000
>r 20000
10000
0
1
2
3
4
5
6
7
Depth (ft)
Section View
Figure 5-10 Example of a storage curve and its section view
To find the area associated with a given storage depth one interpolates between the data points
that bracket the depth value on the storage curve. Determining the storage volume Fat a given
depth Yis equivalent to finding the area under the storage curve from depth 0 to Y This can be
done by using the Trapezoidal Rule (Atkinson, 1989) which results in:
98
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!n
in-*.
i=1
V = ^\> (Yi-Yi^XAi+Ai.Jl+^iY-YNXA+An) (5-26)
where n is the largest data point index with Yn < Y and A is the surface area associated with
depth Fas found from the storage curve itself. The shaded rectangles in Figure 5-11 illustrate
how the trapezoidal rule is applied to a storage curve to find the stored volume at a particular
depth.
60000
50000
cr 40000
l/l
ra
2 30000
<
u
u
£ 20000
=i
10000
0
01234567
Depth (ft)
Figure 5-11 Finding the volume at a given depth for a storage curve
Storage Curve
Y, A
1
1 1
1 1
The depth that corresponds to a particular volume for a storage curve can be found as follows.
Using the trapezoidal rule, sum the volumes contributed by each curve segment starting from 0
until the accumulated volume Vsum exceeds the target volume V. Let the data point index at the
start of this segment be denoted by /. Then the depth Fthat results in volume Fis:
Y = Yt +
Aj+2a(V-Vsum)-Ai
/
a
(5-27)
where a = (Ai+1 - Ai)/(Yi+1 - Yt).
99
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5.5 Critical and Normal Depths
SWMM needs to calculate the critical and normal flow depths in a conduit for dynamic wave
analysis whenever:
1. the conduit is connected to a free outfall node
2. a discontinuity exists between the water level in the conduit and in its connecting node
(i.e. a free fall condition exists).
These depths are functions of flow rate and cross section shape. For all but the simplest shapes,
iterative numerical methods are required to compute them.
5.5.1 Critical Depth
Critical depth is defined as the depth Y where the specific energy at a given flow rate Q is a
minimum and the Froude number Fr equals 1 (Chow, 1959). From the latter condition
where £/is flow velocity and g\s the acceleration of gravity. Since U = Q/A and both area and
width are functions of flow depth, at the critical flow depth Fcthe following relation holds:
Yc can be computed explicitly for several simple conduit shapes. The formulas are listed in Table
5-14. Other shapes require that an iterative root finding procedure be applied to the following re-
arranged form of Equation 5-29:
Because analytical derivatives of f(Y) are not available for most shapes, derivative-free methods
are used instead of the Newton-Raphson method. Two such methods are interval enumeration
and Ridder's method (Press et al., 1992). Ridder's method is a variation on the method of false
position. The user supplies a set of depths Yi and X? that bracket Yc along with a stopping
tolerance s. The full algorithm is described in Appendix B.
Fr = U/JgA/W = 1
(5-28)
A(Yc)3/W(Yc) = Q2 / g
(5-29)
f(Y)=A(Y)3/W(Y)-Q2/g = 0
(5-30)
100
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Table 5-14 Critical depth formulas for simple section shapes
Shape
Formula
Remarks
Rectangular1
*¦&)"
b = width
Triangular1
*¦©)"
5= side slope
Parabolic
Perimeter Equation:
y = ax2
Power Law2
„ /ci + r)3cr2r<32\1/t3+2rl
c"l 4a j
Perimeter Equation:
y = axxly
French (1985).
2Swamee (1993).
With interval enumeration the full depth of the cross section is divided into N equal intervals
(SWMM 5 currently uses N= 25). Given a flow Q and an initial estimate of its critical depth Yc,
the following steps are used to calculate its actual value:
1. Let /be the integer part of N Yc/Yfua and set Y = i YfUll/N.
2. Find Q0 = yjgAiYY/WiY).
3. If <2o Q then stop with = [«? -
-------
1. The ratio of the section's full area to that of a circular section of same full depth is
between 0.5 and 2.0
2. The initial estimate of Yc is computed from the following approximation for circular
sections (French 1985):
«?2/s)
0.25
rc = 1.01 • 26 (5-31)
Ifull
Therefore interval enumeration is used when the first condition listed above holds, with the
second condition used to set the initial estimate of Yc. Otherwise Ridder's method is used with
Equation 5-30 as the function whose root Yc is to be found with a convergence tolerance of 0.001
feet. The initial bracket [ Yi, X?] on Ycis determined as follows:
1. Let *1/2 — 0 ¦5YfuU and Yobe the value computed by Equation 5-31 above.
2. Compute Q0 = Vg A(Y0)*/W(Y0) and Q1/2 = Jg A(Y1/2)3/W(Y1/2) .
3. If Q0 > Q then:
a. Setr2=r0.
b. If Q1/2 < Q then set Y1 = Yy2, otherwise set Y1 = 0.
4. Otherwise:
a. Set Y1 = Yq.
b. If Q1/2 > Q then set Y2 = yi/2, otherwise set Y2 = 0.99YfuR.
5.5.2 Normal Depth
Normal depth is defined as the flow depth that results in a given uniform flow rate along a
conduit. When the Manning equation is used to describe uniform flow, the relation between flow
rate Q and normal depth F/vis:
A(Yn)R(Yn)273 = Qfj/JTo (5-32)
where tj /sthe Manning roughness expressed in US units and 5}; is the conduit's slope. From the
definition of the section factor ^introduced in Chapter 4, Equation 5-32 can be written as:
V = Qn/4s~o (5-33)
102
-------
To find Yn for flow rate Q one first computes If from Equation 5-33, then finds the flow area A
that produces this value of fusing the methods described in section 5.1.8 and finally evaluates
the depth that produces this area using the Y(A) function for the particular shape being analyzed.
In equation terms:
Yff = Y (a{¥ = Qi(5-34)
103
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Chapter 6 - Pumps and Regulators
Pumps are used in stormwater and wastewater collection systems to lift water to a higher
elevation so that gravity flow at a reasonable velocity can be maintained. They are also used to
produce pressurized flow within force mains. Regulators act like valves that restrict the flow rate
along a conduit or out of a storage unit. They can also serve as diverters that split flow between
different branches of a conveyance system (e.g., between the interceptor and an overflow pipe in
a combined sewer system). Specific types of regulators include orifices, weirs, or general outlets
that differ in their geometry and relationship between flow and head. This chapter describes how
the flow rate through pumps and regulators is computed for both the dynamic and kinematic
wave models.
6.1 Pumps
SWMM treats pumps as links that have a pre-defined relationship between flow rate Q and head
H or some suitable surrogate. This relationship is defined by a user-supplied Pump Curve. Table
6-1 depicts the four types of pump curves that SWMM recognizes. Although not a requirement, a
pump's inlet node would typically be a storage node that represents a pump station's wet well.
An exception would be an inline booster pump placed inside a force main line under dynamic
wave analysis. A fifth type of pump, called an Ideal pump, does not use a pump curve but instead
has its flow rate equal the inflow rate into its inlet node. It must be the only outflow link from its
inlet node and is used mainly for preliminary design.
A single point on a Typel or Type2 curve would typically represent an operating point for a
constant flow positive displacement pump. Additional points might represent flow rates at
different pump speeds or contributions from additional constant speed pumps running in parallel.
The Type3 curve represents the characteristic curve of a centrifugal pump operating at some
fixed speed, where there is a continuous range of flows available depending on the head
required. The Type4 curve could be a positive displacement pump with continuous speed control
or a centrifugal pump that lifts water to a more or less fixed elevation so that the required head
depends only on the water level at its inlet node.
Whenever a pump link is encountered in either the dynamic wave or kinematic wave methods its
new flow is found directly from its pump curve using whatever values were last computed for
nodal heads and volumes.
104
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Table 6-1 Pump curves recognized by SWMM
Tvpel
Consists of a series of constant flow rates that
apply over a corresponding series of volume
intervals at the pump's inlet node.
o
Ll_
! Q
Volume
Tvpe2
Similar to a Typel pump except that the fixed
flow rate levels vary over a set of depth
intervals at the pump's inlet node.
£
O
Ll_
Depth
Tvpe3
A centrifugal pump characteristic curve at
some nominal impeller speed represented in a
piecewise linear fashion. Flow is a function of
the head difference between the inlet and
outlet nodes.
CTJ
Oj
X
Flow
Tvoe4
A variable speed in-line pump where flow
varies continuously with inlet node depth.
o
Ll.
Depth
For Typel and Type 2 curves, the curve is searched in step-wise fashion for the first point whose
volume or depth exceeds the volume or depth at the pump's inlet node. The pump's flow is the
flow associated with that point. For the Type3 curve, the flow is determined by first finding the
pair of adjacent data points that bracket the difference in head between the pump's outlet and
inlet nodes and then interpolating a flow between these points for the given head difference. A
similar lookup procedure is used for the Type4 curve except that water level at the pump's inlet
node is used instead of head difference. A pump's flow is not allowed to be outside the minimum
and maximum values defined by its pump curve and is not allowed to be negative.
105
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The flow value found from the pump curve is multiplied by whatever speed setting the pump
is currently operating under. Speed settings can be changed during a simulation by using control
rules. The setting can also be used to control pump operation based on wet well level (e.g., set a
to 1 when the level is above a startup depth and to 0 when below a shutoff depth). The adjusted
pump flow is checked to insure it does not cause the water level at the inlet node to drop below 0
over the current time step. If the node is a storage node then the pumping rate cannot exceed
Qmax where
and Qw is the most recently computed total inflow to the node, Vn is the node volume at the start
of the time step, and At is the current time step interval. If the inlet node is not a storage node
and dynamic wave analysis is being made, Equation 3-15a is used with the current pumping rate
to estimate what the inlet node head at the end of the time step would be. If this head is below the
node's invert elevation then the pumping rate is set equal to the node's current inflow.
Some additional computational details regarding pumps are as follows:
1. If the inlet node of a Typel (flow v. volume) pump is not a storage node then it is
assigned a virtual wet well whose volume varies linearly with depth up to the highest
volume on the pump curve at full node depth. While the normal non-storage node
methods are used to update the node's water level, the virtual wet well volume
corresponding to the node's water level is used to determine the pumping rate. Equation
6-1 is also used to limit the pump flow to the maximum flow that the node can release.
2. For dynamic wave modeling:
a. Pumps do not contribute any surface area to the node-link assemblies at their inlet
and outlet nodes.
b. For Type3 and Type4 pump curves the dQ/dH term used for evaluating a
surcharged node is the negative of the slope of the line segment on which the
pumping rate lies. For the other pump types it is zero since their line segments
have zero slope.
c. No under-relaxation is applied to consecutive pump flows at Step 3 of the
iterative solution method described in Section 3.2.
3. SWMM computes the power consumed in kilowatt-hours by each pump over each time
step zlfas:
Q:
max
— Qin + VN/At
(6-1)
Kwh = 0.7457(tf2 - HJ
-------
where heads Hi and H2 are in feet, flow Q is in cfs, and time step At is in seconds. The
pump's wire to water efficiency is not included in this calculation. The power
consumption in each time period is totaled up and reported for each pump in SWMM's
Pumping Summary Report. Also reported are the percent of time each pump is online and
operates at either the lower or upper end of its pump curve.
6.2 Orifices
Orifices are regularly shaped, submerged openings through which flow is proportional to the
square root of the head across the opening. Orifices are typically used to:
• regulate flow out of detention ponds and other storage facilities
• regulate flow through channels in the form of sluice gates
• divert excess flow from interceptor sewers to overflow structures
• model storm drain inlets.
6.2.1 Representation
SWMM represents an orifice as a link between two nodes. The opening can be oriented either in
a vertical plane for a side orifice or in a horizontal plane for a bottom orifice (see Figure 6-1) and
be elevated some distance above the inlet node's invert. A riser pipe or inlet box used as an
outlet structure in a detention pond can be modeled as a bottom orifice with a vertical offset. For
kinematic wave analysis the inlet node must be a storage node since this is the only type of node
for which a true hydraulic head is computed. For dynamic wave analysis it can be any type of
node.
Side
Orifice
Bottom
Orifice
Figure 6-1 Orifice orientations
The properties of an orifice link include:
• the height of its opening above the invert of its upstream node
• the shape of its opening which can be either circular or rectangular
107
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• the dimensions of its opening (the diameter for a circular orifice or the height and width
for a rectangular orifice)
• its discharge coefficient (described in more detail below)
• whether or not it contains a flap gate that prevents reverse flow.
The size of the orifice opening can be changed during a simulation by having its setting adjusted
using control rules. An orifice's setting is the fraction of its full height that remains open (such as
would occur due to the action of lowering or raising a sluice gate above a side orifice). In this
case an additional optional parameter is the time it takes to fully close a completely open (or
fully open a completely closed) orifice.
6.2.2 Flow Rate for Submerged Inlet
Whenever an orifice link is encountered in the dynamic wave or kinematic wave solution
procedure with its inlet side fully submerged its flow rate Q (cfs) can be found using Torricelli's
equation (Brater et al., 1996):
Q = CdA0y[2gH~e (6-3)
where C/is a dimensionless orifice discharge coefficient, Ao is the area of the opening (ft), and
He is the effective head seen by the orifice (ft). The following paragraphs describe how each of
these parameters is evaluated.
Discharge Coefficient (Cd)
The most commonly cited value for C/is 0.6 while 0.4 is recommended for ragged edge orifices
(Federal Highway Administration, 2009). Brater et al. (1996) review a number of experimental
studies that show the coefficient varying between 0.59 and 0.67 depending on orifice shape, size,
and effective head.
Area of Opening (Ao)
The area of the orifice's opening depends on what its setting is. Let co be the orifice setting
(between 0 and 1) in place at the end of the previous routing time step and co* be the target
setting that was established the last time that a control rule involving the orifice was activated. If
the time to close/open the orifice, Ato, is 0 then for the current time step co = co*. Otherwise let
Act) be defined as co*- co and co gets updated as follows:
108
-------
M _ r<^ + sgn(/3oo)At/At0 ifAt/At0 Yfull/2
e { H1 — H2 otherwise
2. For a bottom orifice:
= (H1- Z0 for H2 < Z0
e Wl — H2 otherwise
Figure 6-2 illustrates how He is evaluated for a side orifice.
6.2.3 Flow Rate for Unsubmerged Inlet
When the water level at the inlet to a side orifice is below the top of its opening the orifice
behaves more like a weir and Equation 6-3 no longer applies (see Figure 6-3). A similar situation
occurs when the head above a bottom orifice is below some threshold level. For these cases
SWMM determines what the threshold head for weir behavior is and what the equivalent weir
coefficient and crest length should be when using the standard rectangular weir formula to
compute the orifice's flow rate. The details differ for side and bottom orifices as described
below.
109
-------
coYfull
A /
Hi
Submerged Upstream Only
Submerged Both Up and Downstream
Figure 6-2 Determination of effective head for an orifice
v
Figure 6-3 Orifice with unsubmerged inlet
Side Orifices
For a side orifice, weir behavior occurs when the inlet water level is below the top of the orifice
opening. Thus the threshold head H*is:
H* = Z0 + 0iYfUii (6-7)
110
-------
When the inlet head Hi is below this height the flow through the orifice can be approximated by
using the general weir formula:
Q = CWL(H1-Z0)1-5 (6-8)
1/2
where Cw is a weir coefficient (ft /sec) and L is the crest length of the equivalent weir (ft).
Equating the flow from this equation to that from the orifice equation 6-3 when H1 = H* and
solving for CwL results in:
CWL = CdA°^ (6-9)
full
Thus whenever the upstream head Hi is below H*, flow through the side orifice can be found
using the weir formula 6-8 with CwL given by Equation 6-9.
Bottom Orifices
For a bottom orifice it is assumed that the threshold inlet head H*for weir flow will be at a point
where the flow through the orifice using both the orifice and general weir equations will be the
same. In equation terms:
CdA0j2^(H* - Z0)0,5 = CWL(H* - Z0)1-5 (6-10)
Solving for H*results in:
H* = Z0 + CdA°^ (6-11)
In order to evaluate H* values for Cw and L must be assigned. Cw can be set to the commonly
cited value of 3.33 ft°'5/sec used for sharp crested weirs (Mays, 2001). L can be set to the
circumference of the opening as follows:
f no)YfUu for a circular opening
L = \ , , (6-12)
[Z^b + (oYfUu) for a rectangular opening
where b is the fixed width of the rectangular opening. Now H* can be determined for a given
orifice coefficient and opening dimensions. Whenever the upstream head Hi is below H*, flow
111
-------
through the bottom orifice can be found using the general weir formula 6-8 with Cw= 3.33 and L
given by Equation 6-12.
Tailwater Submergence Correction
As described later on in Section 6.3, whenever the downstream water level is above a weir's
crest the Villemonte equation is applied to account for the effects of submergence (Brater et al.,
1996). So when the general weir equation 6-8 is used to compute orifice flow and the
downstream head H2 is above the bottom of the orifice opening Zo, the following submergence
adjustment factor Tjis applied to the computed flow value:
fs = [ 1 - (tf2M)1-5]0 385 (6-13)
6.2.4 Flap Gate Head Loss Adjustment
When an orifice has a flap gate it adds a small amount of head loss for flow through the gate. An
empirical formula for this head loss was derived from experiments performed at Iowa State
University in the 1930's and published by Armco (1978):
4 U2
AH = exp(—1.15 U/JtQ (6-14)
9
where AH \s the head loss added by the flap gate (ft) and £/is the velocity through the orifice
(ft/sec) which equals Q/A0. After the orifice's flow is first computed without this additional
head loss, AH is computed with Equation 6-14 and subtracted from He. Then the flow is re-
computed, this time using the adjusted value of effective head.
6.2.5 Dynamic Wave Considerations
Dynamic wave modeling uses the surface area of the links attached to a node to update the
node's head when it is not in a surcharged state (see Chapter 3). As an orifice has no length, its
contribution to a node's surface area should be zero. However in older versions of SWMM an
orifice was represented as an equivalent pipe that contributed surface area to its end nodes just as
a real conduit did. To maintain compatibility with previous versions, SWMM 5 computes a
surface area Asl for an orifice as
112
-------
f W(Y0)L0 for a side orifice
SL \A{o)YfUi{) for a bottom orifice ^ ^
where:
Fb = depth of flow through the orifice (ft), equal to min(H1 — Z0 ,
-------
Equation 6-9 to compute its equivalent weir constant CwL. For bottom weirs, use
Equation 6-12 to find an equivalent weir length L, Equation 6-11 to find the critical head
H* and set the equivalent weir constant to 3.33L.
For each iteration within a time step that requires computing flow through the orifice:
1. Let Hi denote the most recently computed head at the orifice's upstream node and //? be
the same at the downstream node. (For kinematic wave analysis H2 is the downstream
node's invert elevation.)
2. If Hi < H2 reverse the values so that Hi as the higher head and note that reverse flow
will occur. If the orifice has a flap gate or Hi is below the orifice opening then set its flow
to 0.
3. If the orifice is not submerged on its upstream side {HI < H*) then use Equation 6-8 to
find its flow rate along with Equation 6-13 to correct for any tailwater submergence.
Otherwise use Equation 6-5 (for side orifices) or 6-6 (for bottom orifices) to find the
effective head He on the orifice and then use Equation 6-3 to compute its flow rate.
4. If the orifice has a flap gate then use Equation 6-14 to reduce its effective head and repeat
the flow calculation of step 2.
5. If the orifice has reverse flow then make the computed flow negative.
6. Under dynamic wave analysis use Equation 6-15 to assign a surface area to the orifice
and use Equation 6-16 (for side orifices) or 6-17 (for bottom orifices) to compute dQ/dH
for the orifice.
6.3 Weirs
A transverse weir is a barrier with a cut-out placed across a conduit perpendicular to the direction
of flow. A side weir is a cut-out along the side wall of a conduit parallel to the direction of flow.
Flow through a weir is proportional to the height of water above the weir's crest raised to a
power greater than one. Weirs are used for the same types of reasons as orifices: to regulate flow
out of storage facilities, to regulate flow through channels, and to divert excess flow from
interceptor sewers to overflow structures. While orifices normally operate with their inlet sides
submerged, weirs normally maintain a free surface above them.
114
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6.3.1 Representation
SWMM represents a weir as a link between two nodes. For kinematic wave analysis the inlet
node must be a storage node since this is the only type of node for which a true hydraulic head is
computed. For dynamic wave analysis it can be any type of node.
The properties of a weir link include:
the height of its crest above the invert of its upstream node
its orientation (transverse or side flow)
the shape and dimensions of its opening
the number of end contractions
its effective weir coefficient
whether or not it contains a flap gate that prevents reverse flow.
Figure 6-4 shows the different shapes of transverse weirs modeled by SWMM. The only shape
allowed for a side weir is rectangular.
Suppressed Rectangular Weir
Crest
Yfull
/
J )
*1 \
\
ZW
f 1
rf
L
1
/
k hj
t Yfuii f
Crest
\
ZW
(
Contracted Rectangular Weir
Crest
Triangular Weir
Yfull ^
v y1
Crest s
/
ZW
>
^ < >
L
' 1
Trapezoidal Weir
Figure 6-4 Transverse weir shapes
A suppressed rectangular weir has its opening extended across the entire channel while a
contracted weir does not. Weirs are also classified as being sharp-crested or broad-crested.
Sharp-crested weirs have a relatively short crest thickness so that water springs clear of the crest
as it flows over the weir. The crest of a broad-crested weir is thick enough so that the overflow
remains in contact with the crest surface.
115
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The elevation of a weir's crest can be changed during a simulation by having its setting adjusted
using control rules. A weir's setting ct> is the fraction of its full height that remains open after it's
crest is moved up or down, as might occur with a downward opening weir gate or inflatable dam.
At a setting of 1 the weir's crest position is at its lowest possible value and the full height of its
opening is available for flow. At a value of 0 the crest has been raised so that no opening height
remains and no flow can pass through the weir. At intermediate settings the crest elevation
equals its lowest possible value plus 1 - co times its full opening height.
6.3.2 Transverse Weirs
General Equations
The general equation for free flow over a transverse rectangular weir is (Brater et al., 1996):
Q = CwLeHg/2 (6-18)
and for a triangular weir is (Brater et al., 1996):
Q = Cwtan(9/2)Hg/2 (6-19)
In these equations Qis the flow rate (cfs), Le is the effective crest length (ft), $ is the slot angle of
a triangular weir, He is the effective head seen by the inflow side of the weir (ft), and Cwis a weir
1/2
coefficient (ft /sec). A trapezoidal weir can be treated as a combination of a rectangular weir
and two half-triangular weirs (Featherstone and Nalluri, 1982) leading to the equations:
Q = Qr + Qt (6-20a)
Qr = CWRLeH3e'2 (6-20b)
Qt = CWTsHg'2 (6-20c)
where 5 is the slope (run / rise) of the trapezoidal side wall and Cwr and Cwt are the coefficients
that apply to the rectangular and triangular portions of the weir, respectively.
116
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Effective Head (//e)
The effective head seen by a weir, accounting for its current setting, is:
He = H1- (Zw + (1 - co)Yfull) (6-21)
where Hi is the higher of the heads at the weir's end nodes, Zw is the elevation of the weir's crest
when fully open (i.e., when a = 1), and Ymi is the full height of the weir's opening. If Hi
corresponds to the downstream node of the weir then reverse flow occurs through the weir unless
a flap gate is present in which case the flow is 0. Flow will also be 0 if He < 0.
Effective Crest Length (Le)
The effective crest length of a rectangular weir
follows (Mays, 2001):
Le = L — 0.1 nHe
is reduced by the number of end contractions as
(6-22)
where L is the actual crest length and n = 1 if the weir is placed away from one side wall, n = 2 if
it is placed away from both side walls and n = 0 if it occupies the entire width of the conduit (see
Figure 6-4).
When the setting ct> for a triangular weir is less than 1 its opening takes the shape of a trapezoidal
weir. In this case the trapezoidal weir equation 6-18 is used with both Cwr and Cwt set equal to
the weir's original coefficient, the side wall slope 5 set equal to tan{6/2) and the effective
length becomes:
Le = 2s(l - a))Yfull (6-23)
This equation is also used for a trapezoidal weir whose setting is less than 1.
Weir Coefficient (Cw)
1/2
The standard weir coefficient Cw for a sharp crested rectangular weir is 3.33 ft /sec (Mays,
2001). For Hw/L > 1/3 the coefficient has been found to vary with effective head and weir
sizing and placement (Bureau of Reclamation, 2001). The Kindsvater-Carter method (Bureau of
Reclamation, 2001) expresses this dependence with the following formula:
117
-------
Cw ~ d(.Hy\r/Zyy^ + c2
(6-24)
where the constants cl and c2 vary with the ratio of the crest length L to the full width b of the
cross section in which the weir is placed as listed in Table 6-2.
Table 6-2 Kindsvater-Carter constants for rectangular weir coefficient
L/b
cl (ft12/sec)
c2 (ft'2/sec)
0.2
-0.0087
3.152
0.4
0.0317
3.164
0.5
0.0612
3.173
0.6
0.0995
3.178
0.7
0.1602
3.182
0.8
0.2376
3.189
0.9
0.3447
3.205
1.0
0.4000
3.220
Broad crested weir behavior is considered to occur when the ratio of the water level above the
crest to the crest thickness exceeds a certain limit. Limits of 1 to 2 have been proposed by Brater
et al. (1996), 15 by French (1985), and 2 to 20 by the Bureau of Reclamation (2001). Table 6-3 is
a compilation of broad-crested weir coefficients synthesized by Brater and King (1976) from
several different experimental studies. It shows the dependence of the coefficient on both head
and breadth of crest. Above a ratio of about 2 the weir behaves as sharp-crested with a
coefficient of 3.32. For ratios below 0.5 the coefficient approaches 2.63.
1/2
The standard value for the triangular weir coefficient Cwis 2.5 ft /sec (Mays, 2001). Figure 6-5
1/2
shows the variation of Cw(in ft /sec ) with head over the weir Hw (in feet) presented by Brater
and King (1976). The range of coefficients is rather small, from 2.5 up to 2.8.
118
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2,4 i l —i
' 0.2 0.4 0.6 0.8
Hw
Figure 6-5 Coefficient for triangular weirs (from Brater and King, 1976)
1/2
Table 6-3 Rectangular broad-crested weir coefficients (ft /sec)
0.2
2.80
2.75
2.69
2.62
2.54
2.48
2.44
2.38
2.34
2.49
2.68
0.4
2.92
2.80
2.72
2.64
2.61
2.60
2.58
2.54
2.50
2.56
2.70
0.6
3.08
2.89
2.75
2.64
2.61
2.60
2.68
2.69
2.70
2.70
2.70
0.8
3.30
3.04
2.85
2.68
2.60
2.60
2.67
2.68
2.68
2.69
2.64
1.0
3.32
3.14
2.98
2.75
2.66
2.64
2.65
2.67
2.68
2.68
2.63
1.2
3.32
3.20
3.08
2.86
2.70
2.65
2.64
2.67
2.66
2.69
2.64
1.4
3.32
3.26
3.20
2.92
2.77
2.68
2.64
2.65
2.65
2.67
2.64
1.6
3.32
3.29
3.28
3.07
2.89
2.75
2.68
2.66
2.65
2.64
2.63
1.8
3.32
3.31
3.31
3.07
2.88
2.74
2.68
2.66
2.65
2.64
2.63
2.0
3.32
3.30
3.30
3.03
2.85
2.76
2.72
2.68
2.65
2.64
2.63
2.5
3.32
3.31
3.31
3.28
3.07
2.89
2.81
2.72
2.67
2.64
2.63
3.0
3.32
3.32
3.32
3.32
3.20
3.05
2.92
2.73
2.66
2.64
2.63
3.5
3.32
3.32
3.32
3.32
3.32
3.19
2.97
2.76
2.68
2.64
2.63
4.0
3.32
3.32
3.32
3.32
3.32
3.32
3.07
2.79
2.70
2.64
2.63
4.5
3.32
3.32
3.32
3.32
3.32
3.32
3.32
2.88
2.74
2.64
2.63
5.0
3.32
3.32
3.32
3.32
3.32
3.32
3.32
3.07
2.79
2.64
2.63
5.5
3.32
3.32
3.32
3.32
3.32
3.32
3.32
3.32
2.88
2.64
2.63
Breadth of Weir Crest (ft)
0.5
0.75
1.00
1.5 2.0 2.5 3.00 4.00
5.00 10.00 15.00
119
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6.3.3 Rectangular Side Weirs
Flow through a rectangular side weir is a case of spatially varied flow with decreasing discharge
and varying flow depth with distance along the weir. Mays (2001) cites a number of different
studies that have developed discharge equations for side weirs where both the head and weir
coefficient vary spatially. Unfortunately these approaches are too complex to implement in a
program like SWMM. The empirical Engels equation (Metcalf & Eddy, Inc. 1972) is used
instead:
Q = CwL°e83 Hi67 (6-25)
1/2
Where flow Q is in cfs, length Le and head He are in feet, and Cw is in ft /sec. (It should be
noted that previous versions of SWMM used an incorrect form of this equation that had the
exponent on Le equal to 1.0)
Equation 6-25 applies to positive flow through the weir. For reverse flow the standard
rectangular weir equation 6-18 is used. Cw was assigned a value of 3.32 in the original Engels
equation. Brunner (2014) notes that side weir coefficients should be lower than the typical values
used for transverse weirs, and suggests a range of 1.5 to 2.6 for weirs that model levees or
roadways along natural channels.
6.3.4 Submerged Weir Flow
As shown in Figure 6-6, submerged weir flow occurs when the water level on the downstream
side of the weir (H2) is above the crest elevation (Zw). Under this condition weir flow is related
not only to the head on the upstream side of the weir (Hi) but also to H2 and (Brater et al.,
1996). These effects are commonly accounted for by applying an adjustment factor fs developed
by Villemonte (1947) to the flow computed using the free flow equation:
fs = [1 - (H2/H1)"]0385 (6-26)
where n is the exponent on head used in the weir flow equation and the heads Hi and H2 are in
feet. For transverse rectangular weirs (Equation 6-18) it is 3/2, for side weirs (Equation 6-25) it is
1.67, and for triangular weirs (Equation 6-19) it is 5/2. For trapezoidal weirs separate
submergence factors are computed for the rectangular flow portion (Qr in Equation 6-20b with n
= 3/2) and for the triangular flow portion (Qtin Equation 6-20c with n = 5/2).
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Submerged Weir Flow Surcharged Weir Flow
Figure 6-6 Definitions of submerged and surcharged weir flow
6.3.5 Surcharged Weir Flow
As shown in Figure 6-4, the weirs modeled by SWMM assume that the top of the flow opening
extends to the top of the structure that houses the weir. If this structure is an open channel then
the highest head that the weir can see is — Qwi^Yfuii)/^jti)YfUu/2 (6-28)
where Qwi^Yfuii) is the flow in cfs from the relevant weir equation for a head of
-------
Thus if the user indicates that a weir is allowed to surcharge, then whenever the upstream head
Hi is above Zw + Ymi its flow is computed using Equation 6-27. The head He to be used in this
equation is computed as follows. Let H* be the head corresponding to the elevation at half of the
weir's opening height, i.e.:
H* = Zw + (1 — w)YfUu + 0)YfUu/2 (6-29)
Then
_
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6.3.7 Dynamic Wave Considerations
A weir does not contribute any surface area to its end nodes under dynamic wave modeling. The
derivative of its flow rate with respect to head (dQ/dH), used when updating the head of a
surcharged end node (see section 3.3.5), is computed using the formulas in Table 6-4.
Table 6-4 Formulas for flow derivatives of various types of weirs
Weir Type
Flow Derivative (dQ/dH)
Transverse Rectangular
1.5\Q\/He
Side Flow Rectangular:
a. Q > 0
1-67 \Q\/He
b . <2 < 0
1.5\Q\/He
Transverse Triangular:
a. Fully open (ct> = 1)
2-5\Q\/He
b. Partly open (ct> < 1)
1-5 \QR\/He + 2.5 \QT\/He
Transverse Trapezoidal
1-5 \QR\/He + 2.5 \QT\/He
Note: For trapezoidal openings, Qr is the flow through the central rectangular portion and Qtis
the flow through the triangular end portions (see Equation 6-20).
6.3.8 Summary of Weir Computations
The computational steps used to compute flow through a weir link can be summarized as
follows. If the weir is allowed to surcharge and its setting a changes at the start of a time step
then use Equation 6-28 to compute an equivalent orifice coefficient Co to use during surcharge
conditions. For each iteration within a time step that requires computing flow through the weir:
1. Let Hi denote the most recently computed head at the weir's upstream node and H2 be the
same at the downstream node. (For kinematic wave analysis /feis the downstream node's
invert elevation.)
2. If Hi < H2 reverse the values so that Hi as the higher head and note that reverse flow
will occur. If the weir has a flap gate or Hi is below the weir crest then set its flow to 0.
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3. If the head Hi is above the top of the weir's opening and the weir is allowed to surcharge
then use the equivalent orifice equation 6-27 to find its flow where the effective head is
found from Equations 6-29 and 6-30.
4. Otherwise use Equation 6-21 to find the effective head on the weir and either Equation 6-
18, 6-19, 6-20, or 6-25, depending on weir type, to find its flow rate.
5. If the weir has a flap gate then use Equation 6-31 to adjust its effective head and repeat
the flow calculation of steps 3 and 4.
6. If the weir is not surcharged use Equation 6-26 to correct the flow for any tailwater
submergence.
7. If the weir has reverse flow then make the computed flow negative.
8. Under dynamic wave analysis use the formulas in Table 6-4 to compute dQ/dH for the
weir.
6.4 Outlets
SWMM's outlet link is a generic type of flow regulator with a user defined rating curve that
relates flow rate to effective head. It can be used in cases where the head-flow relationships that
SWMM uses for orifice or weir links do not apply. Some examples would be:
• a side orifice using the Smith and Coleman weir equation, where flow rate varies with
head raised to the 1.645 power (Metcalf & Eddy, Inc., 1972),
• a perforated riser pipe with a grate on top used as a detention pond outlet structure,
• a vortex-type flow regulator (Hydro International, 2009; Faram et al., 2010) that provides
more precise flow control than do standard orifices (see Figure 6-7).
For kinematic wave analysis the outlet's upstream node must be a storage node since this is the
only type of node for which a true hydraulic head is computed. For dynamic wave analysis it can
be any type of node.
The properties of an outlet link include:
• the height of its offset above the invert of its upstream node
• a rating curve that defines the relationship between head and the resulting flow rate
• whether head is defined by just the water level at the upstream node of the link or by the
difference in head between its upstream and downstream nodes
• whether or not it contains a flap gate that prevents reverse flow.
An outlet link can also have a flow setting between 0 and 1 that can be modified by control rules.
The setting serves as a multiplier applied to the flow determined from the outlet's rating curve.
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LL
Vortex Device
-- Orifice
Head
Figure 6-7 Rating curve for a vortex device compared to an orifice
The rating curve can be defined either as an analytical power law function or a tabular listing of
points on the curve. The analytical power function has the form:
Q = aHg (6-33)
where Q is flow rate (cfs), He is the effective head (ft), and a and b are user-supplied constants.
The tabular rating curve consists of pairs of head (He) and flow ( Q) values for points that the user
chooses to represent the shape of the outlet's rating curve.
The following steps are used whenever the flow through an outlet link must be computed:
1. Let Hi denote the most recently computed head at the outlet's upstream node and //? be
the same at the downstream node. (For kinematic wave analysis H2 is the downstream
node's invert elevation.)
2. If Hi < H2 reverse the values so that Hi has the higher head and note that reverse flow
will occur. If the outlet has a flap gate or Hi is below the outlet's offset elevation then set
its flow to 0.
3. For dynamic wave modeling, if the outlet's rating curve is based on head difference then
compute an effective head on the outlet as He = H1 — max(H2, Z0) where Zo is the
outlet's offset elevation. Otherwise He = H1 — Z0.
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4. For an analytical rating curve use Equation 6-33 to compute the outlet's flow rate Q. For
a tabular rating curve find the adjacent head values in the table that bracket He and use
linear interpolation to find a corresponding flow rate Q. (If He is below the first entry in
the table then use the first entry's flow value. If it is above the last entry then use the last
entry's flow value.)
5. Multiply Q by whatever outlet setting is currently in effect and change its sign if reverse
flow occurs.
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Chapter 7 - Advanced Features
7.1 Evaporation and Seepage
7.1.1 Conduits
SWMM can model evaporation and seepage losses from conduits as a uniformly distributed
lateral outflow along the length of the conduit. The following paragraphs explain how user-
supplied evaporation and seepage rates per unit area are converted into a distributed loss per unit
length for a conduit and how this loss rate is factored into the solution of the governing equations
for both dynamic and kinematic wave modeling.
Distributed Uniform Evaporation Rate
SWMM can use time varying evaporation data from several different types of external sources.
These include historical daily values from the National Weather Service, values computed from
an historical record of daily temperatures, user-supplied monthly average values or a user-
supplied hourly time series. The details are described in Volume I (Hydrology) of this reference
manual. These data express the potential evaporation rate over the entire study area as a
volumetric loss per unit of area per unit of time, which SWMM converts to internal units of
cfs/ft . The following expression converts the rate per unit area into a rate per unit length of
channel (only open channels can evaporate) over the time period t to t+At.
W(Y) = width of water surface at depth of flow K(ft).
The program automatically extracts the appropriate rate et from the evaporation data source for
the current time period being analyzed. The water surface width W is computed using the
procedures described in Chapter 5 for a particular channel's cross sectional shape.
The average depth of flow in the channel is computed differently depending on the hydraulic
modeling procedure used. For kinematic wave modeling,
qE = etW{Y)
(7-1)
where
-------
Y = (Y(A[) + Y{A\y)l2
(7-2)
where A[ is the flow area at the upstream end of the channel previously computed for time £, A2
is the same at the downstream end of the channel, and Y(A) is the flow depth associated with
flow area A . The latter function is evaluated using the procedures described in Chapter 5.
For dynamic wave modeling the average channel depth is computed as:
Y = (Yt + Ft+At)/2 (7-3)
where f = (yi +Y2)/2. The Yi and K? values for Yt+At are evaluated with Equation 3-16 for the
most recently computed nodal head solution Hlast in the iterative procedure used to solve the
dynamic wave equations. Thus Y and therefore cje can change as the dynamic wave iterations
unfold within a time step.
Distributed Uniform Seepage Rate
The seepage loss from a conduit could be due to infiltration into the soil beneath an unlined or
natural channel or result from a leaking or perforated pipe. In theory the rate would depend upon
such factors as the flow depth in the conduit, the hydraulic conductivity of the surrounding soil,
the variation in moisture content of this soil and the depth to groundwater. Rather than try to
rigorously model the dynamics of soil infiltration beneath the seeping conduit SWMM uses a
user-supplied constant seepage rate per unit area that can be different for each conduit. At any
given time period this rate is converted to a uniformly distributed rate per length of conduit as
follows:
qs = sfcW (?) (7-4)
where
qs = uniformly distributed seepage rate per length of conduit (cfs/ft)
5 = user-supplied seepage rate per unit area for the conduit (cfs/ft2)
fc = monthly climate adjustment factor for the current time step (dimensionless)
? = average depth of flow in the conduit over the current time period (ft).
The monthly climate adjustment factors are a set of 12 user-supplied constants that apply to the
study area as a whole. They allow the intensity of infiltration-based processes to vary on a
128
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seasonal basis. The average flow depth is computed using Equation 7-2 or 7-3, depending on the
choice of flow routing method.
Equation 7-4 assumes that seepage occurs only in the vertical direction so that the area over
which it takes place is limited by the largest horizontal extent of the flow cross section. Thus the
average depth of flow F is limited by the depth at which the cross section width is a maximum.
Table 7-1 lists the depth at maximum width, as a fraction of full conduit depth, for several
different cross section shapes recognized by SWMM. For other shapes not listed the depth at
maximum width is as follows:
• For the Modified Basket Handle shape it is the height of the bottom rectangular portion
of the shape (see Equation 5-15).
• For irregular channels and custom conduit shapes it is the entry in the table of width
versus depth just prior to where width starts to decrease with depth. (If width always
keeps increasing with depth then it is the full depth.)
• For all other shapes it is the full depth.
Table 7-1 Relative depth at maximum width for select cross section shapes
Shape
Relative Depth
Shape
Relative Depth
Circular
0.50
Horseshoe
0.50
Ellipsoid
0.48
Catenary
0.25
Arch
0.28
Gothic
0.45
Basket Handle
0.20
Semi-Circular
0.15
Egg
0.64
Semi-Elliptical
0.15
Total Uniform Loss Rate
The total uniform outflow rate along a conduit qL is the sum of the evaporative and seepage loss
rates:
Rl = Re + Rs (7"5)
Over any given time step At the total volume lost to this outflow cannot exceed the average
volume contained in the conduit:
129
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qLLAt < AL (7-6)
where A is the average flow area over the time step and L is the conduit length. Thus
qL = mm(qL,A/At) (7-7)
For kinematic wave analysis the average flow area over the time step from t to t+At is:
A = (A\+A\)/2 (7-8)
where A\ is the flow area at the upstream end of the conduit computed for time t and A\ is the
same at the downstream end of the conduit. For dynamic wave analysis:
A = (At+At+At)/2 (7-9)
where At = (i4(li) + A(Y2))/2 for Yi and K? computed at time t.; with a similar expression used
for i4t+At. In the latter case the flow depths Y are computed using the most recently computed
nodal heads (see Equation 3-16) as the iterative dynamic wave solution unfolds.
An additional constraint on qL is that it cannot be greater than the inflow rate Q[+At to the
conduit under kinematic wave analysis or the last computed flow Qast under dynamic wave
analysis.
Dynamic Wave Modifications
For dynamic wave analysis, including a uniform loss rate adds an additional term AQiaterai to
Equation 3-14 used to update a conduit's flow rate over a time step. The revised equation is:
_ _ Qt ^Qinertia ^Qpressure AQiaterai
Vt+At _ 1 . KQ (7"10)
1 ^ friction
where AQiaterai = 2.5UqL and all other AQ terms were defined previously in Section 3.2. See
the sidebar below for the derivation of this modified equation.
130
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The St. Venant Equations with Uniform Loss Rate
The governing conservation of mass and momentum equations for dynamic wave analysis
along a conduit, first discussed in Section 3.1, are modified as follows to include a distributed
lateral loss rate qL (Strelkoff,1969):
dA dQ „ . .
— + — + q, = 0 Continuity (a)
at ox
dQ d(Q2/A) dH A;r +
— + - + gA — + gASf - U qj2 = 0 Momentum (b)
(_/ L kJ J\, kj a
where all variables were defined previously in Section 3.1. As in Section 3.1, the continuity
equation is used to re-express the d(Q2 / A)/dx term of the momentum equation as:
d(Q2/A) dA , dA
^^=-2Um-u2-x-w^ (c)
Substituting (c) into (b) and re-arranging terms produces:
dQ dA , dA dH
m=2Um + u2^-^-eASr + 2-5u^
The finite difference form of this equation is:
^i) r(tf2-ffi) ,q\u\
m = 2Um+u —I gA—I gi
where all notation is the same as used in Section 3.2. After re-arrangement, the following
equation to update conduit flow Q over a time step results:
n _ Qt inertia ^Qpressure lateral
t+M ~ 1 + AQfriction
with AQiaterai = 2.SUqL and all other AQterms defined previously in Section 3.2.
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Another modification needed when including a uniform loss rate is to add qLL (the total flow lost
over the length of the conduit) to the total outflow from the upstream node of a conduit with
positive flow or add it to the total inflow of the downstream node of a conduit with negative
flow. This modifies the £ Qt+At term (the net inflow minus outflow to a node) in Equation 3-15a
which is used to update nodal heads.
Kinematic Wave Modifications
For kinematic wave analysis, adding a uniform loss term modifies the original continuity
equation 4-1 as follows:
^+^+qL = 0 <7"n)
Carrying the qi term over into the original finite difference form of this equation (Equation 4-7)
produces:
(1 - 0)(i4£+At - A[) + 0(i4|+At - A\)
^ (7-12)
, (1 - 0)«2! - Qi) + 0(Q2t+At - ,
+ } + = 0
where all notation was defined previously in Section 4.2. After substituting the Manning
equation Q = /?f (/I) into this expression the same non-linear equation for A2+At results as
before (equation 4-8):
pif/(At+At^ + clAt+At + C2 = o (7-13)
with CI given by Equation 4-9 and C2 by Equation 4-10 but with the additional term qLL/
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processes is then subtracted from the net inflow to the storage unit so that its new water depth at
the end of the time step can be computed.
Evaporation Loss
The evaporation loss rate from the surface of a storage unit during a time period is based on the
surface area in the unit at the start of the time period using the following equation:
Asn(Y) = storage unit surface area at water depth F(ft).
The potential evaporation rate et is the same quantity discussed in the previous section on
conduit evaporation and is automatically retrieved from a study area's evaporation data source as
the simulation unfolds over time. The fraction of this rate realized, fE; is a user-supplied value for
each storage unit that allows the rate to be adjusted for specific local conditions. It would
normally be 1.0 but could be 0 if the storage unit has a roof over it. The depth of stored water Yt
is the difference between the water surface elevation Hl at time t and the storage unit's invert
elevation E. The A(Y) function represents the user-supplied curve of surface area versus depth as
described in Section 5.4.
Seepage Loss
The seepage loss from a storage unit is modeled as infiltration of ponded water into the native
soil beneath the unit. The Green-Ampt soil infiltration method is used to compute the rate of
seepage per unit area over time. Its fundamental formula is:
Qen — etfE^sN(yt)
(7-14)
where
et
fE
F
Qen
t
evaporation loss rate from a storage unit node (cfs)
starting time for the current computational time period (sec)
potential evaporation rate per unit area at time t (cfs/ft)
fraction of evaporation rate realized
depth of stored water at time £(ft)
(7-15)
where
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2
qsN = seepage rate per unit area from the storage unit node (cfs/ft)
Ks = soil saturated hydraulic conductivity (ft/sec)
fc = monthly climate adjustment factor for the current time step (dimensionless)
d = depth of stored water above the area undergoing seepage (ft)
ips = soil capillary suction head (ft)
0d = soil moisture deficit (dimensionless)
F = cumulative depth of infiltrated water (ft).
The monthly seepage adjustment factor is the same user-supplied set of multipliers used for
conduit seepage described previously for equation 7-4. Ks, i//s, and the initial value of 6d are all
parameters associated with the Green-Ampt model. Both Od and Fare modified by the model
over time. Equation 7-15 makes the seepage rate dependent on the ratio of stored water depth to
cumulative infiltrated depth, both of which will vary over time
The details of SWMM's implementation of the Green-Ampt infiltration model are covered in
Chapter 4 of Volume I (Hydrology) of this manual. The only difference when using it for a
storage unit is that the quantity xps in the original formulation is replaced with xps + d. Volume I
also provides guidance on selecting values of Ks, ips, and an initial Od based on soil type. If
either xps or Od are 0 then SWMM assumes a constant seepage rate equal to Ks that is
independent of storage depth. If Ks is 0 then no seepage occurs.
The depth of water to use in the Green-Ampt formula will vary across the top surface of a
storage unit if it has sloped sides as shown in Figure 7-1. SWMM accounts for this by applying
the Green-Ampt infiltration method independently to two separate seepage areas - one for water
in contact with the flat bottom portion of the unit and a second for water in contact with the
sloped sides. The total seepage loss rate Qsn{in cfs) can be expressed as:
Qsn ~ QbtmiAbtm)^btm ^sideiAside)^side (^~16)
where:
dbtm
qbtm
Abtm
dside
Qside
Aside
= depth of stored water above bottom of unit (ft)
= Green-Ampt infiltration rate based on d = dbtm (cfs/ft)
= surface area over which bottom seepage occurs (ft)
= average depth of stored water above sloped sides of unit (ft)
= Green-Ampt infiltration rate based on d = dside (cfs/ft)
= surface area of sloped sides over which seepage occurs (ft).
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m a:<
/\
Jhtm
y V V v .• V • v
Figure 7-1 Depths used for computing seepage in storage units
As noted previously, the depth above the bottom of the storage unit (dbtm) is Yt = Hl - E. The
bottom surface area is found from the unit's storage curve at a depth of 0 (see Section 5.4 for a
discussion of storage curves). The average depth above the sloped sides is computed as:
d-side
o
(Yt ~ dmin)/2
^ (d-max
fol Y d
-------
where Vn(Y) is the storage unit's volume at depth Y(see section 5.4 for how it is computed) and
At is the size of the current time step.
For a given storage node, Qln is computed once at the start of the current time step based on the
known stored water level. For dynamic wave analysis it is subtracted from the £ Qt+At term of
Equation 3-15a (i.e., is treated as a nodal outflow) each time the node's head is updated at step 4
of the solution procedure described in Section 3.2. For kinematic wave analysis, after all link
flows have been found, Qln is added to the node's total outflow at the end of the time step (see
Equation 4-18) which is used to update the node's volume and subsequently its hydraulic head
(see Section 4.3.5).
7.2 Minor Losses
Energy losses caused by rapid changes in magnitude or direction of velocity are called minor or
local losses. They can occur at bends, contractions, or enlargements in conduit geometry and also
be associated with flows entering a conduit from a larger water body or flows exiting a conduit
to a larger water body. Table 7-2 lists the types of minor losses most frequently considered in
storm water conveyance networks.
A minor loss is represented as the product of a loss coefficient and the local velocity head for a
specific location /along a conduit:
where AHL is the minor head loss (ft), Km,i is a loss coefficient, and Uj is flow velocity (ft/sec).
The location index / is 1 for an entrance loss based on the conduit's upstream velocity, 2 for an
exit loss based on its downstream velocity, or 3 for an average loss based on its average velocity.
Minor losses can be included in the St. Venant momentum equation for a conduit by treating
them as a loss per unit length, /?£, in the same way that the friction slope S/ is treated. This
modified version of the momentum equation (originally equation 3-2 with uniform lateral
outflow rate qL included) is:
Ut
2 g
(7-21)
dQ_ d(Q2/A)
dt + dx
dH
+ gA— + gASf - U qj2 + gAhL = 0
(7-22)
where hL = £?=1 Km iUf /(2gL) with L being the conduit length.
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Table 7-2 Types of minor losses in drainage systems (from Frost, 2006)
Type of Loss
Frequently
Modeled
Occasionally
Modeled
Rarely
Modeled
Pipes (Full or Partially Full)
Entrance
X
Exit
X
Expansion and Contraction
X
Inlet on branch
X
Curves or bends
X
Outfall
X
Junctions (Full or Partially Full)
Flow through junction
X
Bend within junction
X
Junction with lateral
X
Junction with inlet
X
Channels
Expansion and Contractions
X
Curves or bends
X
X
Culvert entrance
X
Culvert exit
X
Outfall
X
For dynamic wave hydraulics the finite difference form of equation 7-22 can be found by
following the same derivation used earlier in Section 3.2. This results in:
AQ
At
. Ai4
At
-AAz-Aj _ (//2 — //i)
^- = 2U — +U2K , , 1 -ff772^7T+2.5t/gL
Q\U\
R4/3
(7-23)
2L
After re-arranging terms, the following revised form of the flow updating equation 3-14 used in
step 2 of the dynamic wave solution procedure of Section 3.2 is:
137
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Qt+At —
Qt ^Qinertia AQ:
pressure
^Qlateral
(7-24)
1 friction AQloss
where
(7-25)
and all other AQterms are as defined previously in Sections 3.2 and 7.1.1. Minor losses are not
computed for kinematic wave analysis since it uses a simplified version of the momentum
equation that only accounts for gravitational and friction forces. Frost (2006) provides guidance
on selecting values for the loss coefficient Km.
7.3 Force Mains
For dynamic wave modeling SWMM allows the user to designate particular circular pipes as
force mains. These pipes will use either the Hazen-Williams or the Darcy-Weisbach equation to
compute their friction losses when pressurized conditions occur. For free surface flow the
Manning equation continues to be used.
7.3.1 Hazen- Williams Force Mains
The standard form of the Hazen-Williams equation in US units is (Clark et al., 1977):
where £/is velocity (ft/sec), Rmi is the full pipe hydraulic radius (ft), 5/ is the friction slope (head
loss per unit length) (ft/ft), and Chw is the user-supplied Hazen-Williams C-factor coefficient.
Typical values of the C-factor are listed in Table 7-3.
Solving Equation 7-26 for Sf and putting the result in a form similar to the Manning equation
(see Equation 3-3) gives:
U = 1.318 C
(7-26)
0.6\U\°-852Q
(7-27)
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Table 7-3 Hazen-Williams C-factors for different pipe materials
Pipe Material
C-factor
Pipe material
C-Factor
Asbestos Cement
140
Corrugated Steel
60
Brick Sewer
100
Ductile Iron
140
Cast Iron:
Galvanized Iron
120
Unlined
Asphalt Coated
Cement Lined
130
Plastic PVC
130
100
140
Polyethylene
140
Vitrified Clay
110
Concrete
120
Welded Steel
100
This expression replaces the Manning formula for Sf when a force main flows full. As a result,
the friction term in the equation used to update the conduit's flow in the iterative dynamic wave
solution procedure (Equation 3-14) becomes:
117|0 852 At
AQfriction = ^-^9 ri.852 nl.667 (7-28)
lHW Kfull
where U = Qlast / Afull.
7.3.2 Darcy- Weisbach Force Mains
The standard form of the Darcy-Weisbach head loss equation is (Clark et al., 1977):
fU2
Sf=TT; (7"29)
; 2 gD
where Sf is the friction slope (head loss per unit length) (ft/ft), £/is flow velocity (ft/sec), D is
pipe diameter (ft), and f is a dimensionless friction factor. Noting that D = 4RfUu for a circular
pipe allows this equation to be expressed in a form similar to the Manning formula:
f\u\Q
SgAfuiiRfuii
Sf = o;A » (7-30)
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As a result, the friction term in the equation used to update a pressurized force main's flow in the
iterative dynamic wave solution procedure (Equation 3-14) becomes:
/" | U | At
A Q friction =-7^~ (7-31)
°Kfull
The friction factor f can be determined graphically from the Moody diagram as a function of the
flow's Reynolds number (Re) and the pipe's relative roughness (Bhave, 1991). For laminar flow
(Re > 2000) the friction factor is:
64
/ = — (7-32)
J Re
where Re = D\U\/fx with //being the kinematic viscosity of water taken as l.lxlO"5 ft2/sec. For
transition and rough turbulent flow (Re > 4000) the Swamee and Jain approximation to the
Colebrook-White formula is used (Bhave, 1991):
0.25
f~u ( e , 5.74m2 (?-33>
\iog\3.7D+ Re°V\
where e is the equivalent surface roughness height (ft) of the pipe wall as supplied by the user.
This roughness height serves the same purpose as the Manning roughness coefficient or the
Hazen-Williams C-factor. Typical values for different pipe materials are given in Table 7-4. For
Re between 2000 and 4000 linear interpolation is used between the friction factor at Re = 2000
(equal to 0.032) and that at Re= 4000 (which will also depend on e/D).
Table 7-4 Darcy-Weisbach roughness heights for different pipe materials
Material
e (inches)
Material
e (inches)
Concrete
0.012-0.12
Asphalted Cast Iron
0.0048
Cast Iron
0.010
Welded Steel
0.0018
Galvanized iron
0.006
PVC
0.00006
140
-------
7.3.3 Equivalent Manning's n
For pipes designated as force mains a C-factor or roughness height will be specified instead of a
Manning roughness coefficient. Since the Manning equation will still be used to analyze free
surface flow when the pipe is only partly full a method is needed to determine a Manning's n
that will be consistent with the other forms of pipe roughness. SWMM does this by equating the
Manning full pipe flow to either the Hazen-Williams or Darcy-Weisbach flow computed under
fully turbulent conditions for a friction slope equal to the pipe's bottom slope.
When the Manning full normal flow is equated to the Hazen-Williams formula flow the result is:
2
(^) = (1-318C„wR%f,S0o™)2 <7"34)
1/3
where So is the pipe's bottom slope (ft/ft), Rmi is in feet, and n has units of sec/m . Expressing
Rmi as D/4 and solving for n gives:
1.067(D/S0)om
n = (7-35)
lhw
Doing the same for the Darcy-Weisbach formula produces:
where
2
fie, °°) = 0.25/[log (j^)] C7"37)
Expressing Rmi as D/4 and solving for n gives:
„ = 1/6 (7-38)
185
To summarize, when flowing full under dynamic wave analysis, a pipe designated as a force
main uses Equation 7-28 (for Hazen-Williams) or 7-31 (for Darcy-Weisbach) in place of
Equation 3-14c to evaluate AQfriction m the iterative flow updating Equation 3-14. For free
141
-------
surface flow it uses the Manning form of AQfriction (Equation 3-14c) with an /7-value given by
Equation 7-35 (for Hazen-Williams) or 7-38 (for Darcy-Weisbach).
7.4 Culverts
Culverts are closed conduits that allow water from an open stream or channel to flow under a
road, railroad, trail, or similar obstruction from one side to the other side (see Figure 7-2). A
complete description of culverts and their hydraulic performance is provided by the Federal
Highway Administration in their Hydraulic Design of Highway Culverts manual (FHWA, 2012).
The equations used by SWMM to model culverts are taken from this publication.
Figure 7-2 Concrete box culvert (from FIIWA, 2012)
Culvert flow can be controlled either by the inlet or the outlet. Inlet control occurs when the
conveyance capacity of the culvert's barrel is higher than what the inlet will accept. Otherwise
outlet control occurs, with the possibility that flow may be limited by backwater effects. Culverts
are usually analyzed under a steady design flow condition to determine if the resulting inlet
water depth will be acceptable. However for SWMM's unsteady dynamic wave analysis they are
analyzed to find the flow corresponding to known inlet and outlet depths. (Culvert analysis is not
made under kinematic wave analysis.)
142
-------
Any SWMM conduit link can be designated as a culvert by assigning it one of the code numbers
associated with a particular shape, material, and inlet configuration listed in Table H-l in
Appendix H. The choice made from the table should be consistent with the conduit's designated
shape (circular, rectangular, ellipsoid, or arch). At any given time step of a simulation the flow
through the culvert is first determined using SWMM's usual dynamic wave procedure. This flow
represents the outlet control condition. Then an inlet controlled flow is computed to see if it
becomes the limiting flow rate.
7.4.1 Inlet Control Flow
Under inlet control, a rating curve establishes the relationship between culvert flow rate and inlet
head. The shape of the curve varies depending on the culvert's shape, material, and geometry of
its inlet opening. Figure 7-3 shows a typical inlet control rating curve in normalized form, where
inlet headwater depth (Yi) is normalized by the full barrel depth (Yfuii) and flow rate (Q) is
normalized by AfUu^jYfUu where Ami is the full cross-section area of the barrel. When the inlet is
submerged it performs as an orifice while when unsubmerged it performs as a weir.
3.0
2.0
h
Yfuii
1.0
0
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Q1'A fun Yfuif)
Figure 7-3 Example of a culvert rating curve (from FHWA, 2012)
143
SUBMERGED
EQUATION
TRANSITION
ZONE
UNSUBMERGED
EQUATION
-------
Based on extensive experimental testing done by the National Bureau of Standards, FHWA has
developed inlet rating curves for a number of different types of culverts and their inlet
configurations. The curves have been fitted to analytical functions that describe both their
submerged and unsubmerged portions. Table H-l in Appendix H lists the different types of
culverts for which parameterized rating curves have been developed. Table H-2 lists the
parameter values used for both the unsubmerged and submerged portions of each curve.
7.4.2 Unsubmerged Inlet Control Curves
The FHWA procedures identify two types of unsubmerged curves used to compute inlet control
for a culvert. The form 1 equation is:
H1~Z1
Yfuii
Yfuii
+ Kj
Qic
Afully[Yf
ull
Mi
+ ScfS0
(7-39)
while the form 2 equation is:
H1~Z1
Yfuii
= Kj
Qic
AfullyjYfull
Mi
(7-40)
The following definitions apply to these equations:
Hi = hydraulic head at the inlet node of the culvert link (ft)
Zi = elevation of the culvert invert at its inlet end (ft)
Qic = inlet controlled flow rate through the culvert (cfs)
Ec = specific head at critical depth for flow Qic (ft)
Yfuii = full depth of the culvert barrel (ft)
A fun = full area of the culvert barrel cross section (ft)
So = culvert barrel slope (ft/ft)
Scf = slope correction factor (0.7 for mitered inlets and -0.5 for all others)
Ki, Mi = constants from Table H-2 for corresponding culvert type in Table H-l.
Table H-2 also specifies which equation is used for each type of culvert. It should be noted that
Ki has a factor of g~M>/2 embedded in it so that equations 7-39 and 7-40 are dimensionally
consistent.
The form 2 equation can be solved directly to determine Qic for a given inlet head Hr.
144
-------
Qic — AfuujY,
(lh-Z£
fuU V KiYfun ,
1 /Mi
(7-41)
For the form 1 equation, the specific head at critical depth is defined as:
Uc
Ec = Yc+-t
2 g
(7-42)
where Yc i s the critical depth for flow Qic and He i s the velocity at this depth. From the definition
of critical depth given by Equation 5-28 in Section 5.5.1:
U2C = gA(Yc)/W(Yc)
(7-43)
and from Equation 5-29:
Qic = A(YcygA(Yc)/W(Yc)
(7-44)
The area A and top width W values in these equations are evaluated at flow depth Yc using the
methods described in Chapter 5 that depend on the culvert's shape and dimensions.
Substituting these relations into the form 1 equation 7-39 results in the following nonlinear
equation in the single unknown Yc:
Yfuii
Hi ~ Z1 — Yhc/2
Yfuii
A(YC)
A
lfull
\d Ync/Yf
ull
Mi
~ ScfS0
(7-45)
where YHC is the critical hydraulic depth defined as A(YC)/W(YC). This equation is solved using
Ridder's root finding method (see Appendix B) with an initial bracket on Yc of 10 to 100 percent
of Hi - Zi and a stopping tolerance of 0.001 ft. After Yc is found the corresponding inlet control
flow rate Qic can be computed using Equation 7-44.
7.4.3 Submerged Inlet Control Curve
The FHWA equation for inlet control of a culvert whose inlet is submerged is:
H1~Z1
Yfuii
= c,
Qic
AfullyfYf
+ yi + Scfs0
(7-46)
ull
145
-------
where ci and yi are constants from Table H-2 for a particular culvert type in Table H-l. The ci
constant has a factor of 1/ g embedded in it so that equation 7-46 is dimensionally consistent.
Solving this expression for Qic results in:
Qic —
-yi- ScfSo
1/2
Afuii-jYf
full
(7-47)
7.4.4 Inlet Control Transition Zone
The FHWA procedure states that the submerged inlet control equation should be used for values
°f Qic/(Afuii^jYfuii) above 4.0. Converting this into a condition on Hi results in:
Hi > HIS = Z1 + YfUu(16cI + yi + ScfS0) (7-48)
When this condition is satisfied SWMM uses the submerged portion of the inlet control curve to
compute the inlet control flow rate Qic.
FHWA states that the unsubmerged inlet control equation applies for QIC/(AfuiijYfuii) values
below 3.5. This is difficult to convert to an a priori limit on Hi because of the Ec term in the
form 1 unsubmerged equation. Therefore SWMM uses an arbitrary criterion of
Hi < Hm = Z1 + 0.95YfUu (7-49)
to determine if the unsubmerged inlet control equations should be used.
When Hi is between Hiu and His linear interpolation is used to compute Qic as follows:
Qic = Qic(Hw) + (Qic(HIS) — QiciHju)^— — - (7-50)
(."is ~ "iu)
where Qic(Hw) is the flow from the unsubmerged equation for a head of Hiu and Qic(HIS) is the
flow from the submerged equation for a head of His.
146
-------
7.4.5 Flow Derivatives
SWMM's dynamic wave procedure needs to evaluate the derivative of a conduit's flow rate with
respect to head (dQ/dH) for use when updating the head of a surcharged end node (see section
3.3.5). The derivatives for the various methods of computing an inlet control flow limit are as
follows:
dQic
~dlf1
= <
Qic
MIH1
® A full
ciQic
Qic~ Qic(Hiu)
His ~ H
unsubmerged
submerged
transition
(7-51)
l u
7.4.6 Summary of Culvert Analysis
The following steps summarize how a culvert conduit is checked for inlet control each time that
a new flow is computed for it under dynamic wave analysis:
1. Equation 3-14 is used as usual to compute a first flow estimate Q. This represents an
outlet control condition.
2. If the conduit is not flowing full at both ends, a flow limit Qic due to inlet control is
computed. Equations 7-41 or 7-45 are used if the head Hi at the culvert's inlet node is
below Hiu, Equation 7-47 is used if Hi is above His, or Equation 7-50 is used if Hi is in
between these limits.
3. If Qic is less than Q then it is replaced with Q/c and Equation 7-48 is used to compute the
conduit's flow derivative with respect to head.
7.5 Roadway Weirs
A culvert will become overtopped when the headwater rises to the elevation of the roadway (see
Figure 7-4). SWMM represents the flow across the road with a special type of weir link
designated as a roadway weir. It is similar to a standard SWMM transverse rectangular weir but
has its own specific methods for computing a weir coefficient and submergence factor based on
road characteristics. Figure 7-5 shows how a roadway weir would be configured with a culvert in
a SWMM node-link layout.
147
-------
Figure 7-4 Roadway overtopping (from FHWA, 2012)
Roadway Weir
Headwater
Tailwater
Outlet
Inlet
Channel
Culvert
Channel
Figure 7-5 SWMM node-link representation of a culvert with a roadway weir
The standard transverse rectangular weir equation can be used to compute the flow across a
roadway weir as follows: (FHWA, 2012):
Q = fsCwLH3/2 (7-52)
where Q is the overtopping flow rate (cfs), H is the height of the upstream water surface above
the roadway crest (ft), L is the length of the roadway crest (ft), Cw is free flow weir discharge
1/2
coefficient (ft /sec) and fs is a submergence adjustment factor.
Values for the flow coefficients Cw and fs have been published by the FHWA as functions of the
headwater depth (//), tailwater depth (ht), the width of the roadway (Lr), and road surface
material. The functions are shown in graphical form in Figure 7-6.
148
-------
H
Lr
3.10
Cw 3.00
2.90
0.16 0 20 0.24 0 28 0.32
H I Lr
A) Discharge Coefficient for H I Lr > 0,15
3.10
3.00
2.90
Cw 2.8Q
2.7D
2,60
2.50
1.0 2.0 3.0 4.0
H
— Paved
Gravel
1.00
0.90
Q-BO
k,
0.70
0.60
0,50
^1
/
\
\\
v\
v \
\\
v\
11
11
M
\\
11
l|
il
0.6
0.7
0.8
0.9
1,0
B) Discharge Coefficient for H I Lr £ 0.15
ht/H
C) Submergence Factor
Figure 7-6 Discharge coefficients for roadway weirs (from FHWA, 2012)
149
-------
To summarize, SWMM includes an additional type of weir, a roadway weir, used to model flow
over the top of a road that sits above a culvert or possibly an embankment. Its properties include
the following:
• crest elevation (typically the elevation of the road surface)
• crest length (determined by the top width of the channel that the road crosses)
• width of the roadway (perpendicular to its crest length).
• whether the road surface is paved or gravel.
Unlike the other weirs discussed in Chapter 6, a roadway weir has neither a control setting nor a
flap gate. Its flow versus head relation is given by equation 7-52, where the head H is the
difference between the head at its inlet node and its crest elevation, the tailwater head ht is the
difference between its outlet node head and its crest elevation, and its flow coefficients are
determined from the curves in Figure 7-6.
150
-------
Appendix
A. Newton-Raphson-Bisection Root Finding Method
The following Newton-Raphson procedure adapted from Press et al. (1992) is used to solve the
equation /(x) = 0 over the interval [xL0W,xHIGH] that is known to bracket the solution with
initial estimate xand convergence tolerance s:
1. Perform the following initial steps:
a. If /(xL0W) > f (xHIGH) then switch xL0W with xHIGH.
b. If A'is outside [xL0W, xH1GH] then set x = (xL0W + xmGH)/2.
c. Set Ax = \xHIGH — xL0W\.
d. Evaluate /(x) and its derivative A'(x).
2. If [(x - xHIGH)f\x) - /(x)][(x - xL0W)f\x) - /(x)] > 0 or |2/(x)| > |Ax/'(x)| then
update xas follows:
Ax = 0.5(xhigh — xL0W)
x = xlow +
3. Otherwise take the Newton step:
Ax = /(x)//'(x)
x = x — Ax
4. If | Ax| is below the tolerance s then stop with the current value of xas the solution.
5. Evaluate /(x) and / (x). If /(x) < 0 then set xL0W = x. Otherwise set xHIGH = x.
6. Return to Step 2.
151
-------
B. Ridder's Root Finding Method
Ridder's method uses the following iterative procedure adapted from Press et al. (1992) to solve
the equation /(x) = 0 over the interval [xt, x2] that is known to bracket the solution with a
convergence tolerance of s:
1. Let
A = /(*i)
A = f(x2)
x3 = (xx +x2)/2
2. Set
h = /Os)
(x3 -x^sgnih -/2)/3
x4 = x3 H ,
V/s2 -fifi
3. If |x4 — x31 < e then stop with solution x;i
4. Set/4=/(x4).
5. If sgn(f3) =£ sgn(f4) then set
Xi = x3
h=h
x2 = x4
f2=h
6. Otherwise if sgntfi) =£ sgn(f4) then set
x2 = x4
f2=h
7. Otherwise if sgn(f2) =£ sgn(f4) then set
xx = x4
A=/4
8. Set x3 = (xx + x2)/2.
9. If |x2 — xx | < £ then stop with solution X3.
10. Return to Step 2.
152
-------
C. Section Properties of Circular Pipes
Table C-l Circular section properties as function of depth
Y/Yfuii
A/Afuii
W/Wmaz
R/Rfuii
Y/Ym
A/Ami
w/wmax
R/Rfuii
0.00
0.00000
0.00000
0.01000
0.52
0.52550
0.99920
1.02400
0.02
0.00471
0.28000
0.05280
0.54
0.55093
0.99680
1.04800
0.04
0.01340
0.39190
0.10480
0.56
0.57630
0.99280
1.07000
0.06
0.024446
0.47500
0.15560
0.58
0.60135
0.98710
1.09120
0.08
0.03740
0.54260
0.20520
0.60
0.62640
0.97980
1.11000
0.10
0.05208
0.60000
0.25400
0.62
0.65126
0.97080
1.12720
0.12
0.06800
0.64990
0.30160
0.64
0.67580
0.96000
1.14400
0.14
0.08505
0.69400
0.34840
0.66
0.70015
0.94740
1.15960
0.16
0.10330
0.73320
0.39440
0.68
0.72410
0.93300
1.17400
0.18
0.12236
0.76840
0.43880
0.70
0.74764
0.91650
1.18480
0.20
0.14230
0.80000
0.48240
0.72
0.77080
0.89800
1.19400
0.22
0.16310
0.82850
0.52480
0.74
0.79335
0.87730
1.20240
0.24
0.18450
0.85420
0.56640
0.76
0.81540
0.85420
1.21000
0.26
0.20665
0.87730
0.60640
0.78
0.83690
0.82850
1.21480
0.28
0.22920
0.89800
0.64560
0.80
0.85760
0.80000
1.21700
0.30
0.25236
0.91650
0.68360
0.82
0.87764
0.76840
1.21720
0.32
0.27590
0.93300
0.72040
0.84
0.89670
0.73320
1.21500
0.34
0.29985
0.94740
0.75640
0.86
0.91495
0.69400
1.21040
0.36
0.32420
0.96000
0.79120
0.88
0.93200
0.64990
1.20300
0.38
0.34874
0.97080
0.82440
0.90
0.94792
0.60000
1.19200
0.40
0.37360
0.97980
0.85680
0.92
0.96260
0.54260
1.17800
0.42
0.39878
0.98710
0.88800
0.94
0.97555
0.47500
1.15840
0.44
0.42370
0.99280
0.91760
0.96
0.98660
0.39190
1.13200
0.46
0.44907
0.99680
0.94640
0.98
0.99516
0.28000
1.09400
0.48
0.47450
0.99920
0.97360
1.00
1.00000
0.00000
1.00000
0.50
0.50000
1.00000
1.00000
153
-------
00
02
04
06
08
10
12
14
16
18
20
.22
24
26
28
30
32
34
36
38
40
42
44
.46
,48
,50
Circular section properties as function of area
Y/Ymi V/Vfuii
0.0000
0.00000
A/Am, Y/Ym V/Vw
0.52
0.51572
0.05236
0.00529
0.54
0.53146
0.08369
0.01432
0.56
0.54723
0.11025
0.02559
0.58
0.56305
0.13423
0.03859
0.60
0.57892
0.15643
0.05304
0.62
0.59487
0.17755
0.06877
0.64
0.61093
0.19772
0.08551
0.66
0.62710
0.21704
0.10326
0.68
0.64342
0.23581
0.12195
0.70
0.65991
0.25412
0.14144
0.72
0.67659
0.27194
0.16162
0.74
0.69350
0.28948
0.18251
0.76
0.71068
0.30653
0.20410
0.78
0.72816
0.32349
0.22636
0.80
0.74602
0.34017
0.24918
0.82
0.76424
0.35666
0.27246
0.84
0.78297
0.37298
0.29614
0.86
0.80235
0.38915
0.32027
0.88
0.82240
0.40521
0.34485
0.90
0.84353
0.42117
0.36989
0.92
0.86563
0.43704
0.39531
0.94
0.88970
0.45284
0.42105
0.96
0.91444
0.46858
0.44704
0.98
0.94749
0.48430
0.47329
1.00
1.0000
0.50000
0.49980
154
-------
D. Section Properties of Elliptical Pipes
Table D-l Standard elliptical pipe sizes
Code
Minor Axis (in)
Major Axis (in)
Ami (ft2)
Rfull (ft)
1
14
23
1.80
0.367
2
19
30
3.30
0.490
3
22
34
4.10
0.546
4
24
38
5.10
0.613
5
27
42
6.30
0.686
6
29
45
7.40
0.736
7
32
49
8.80
0.812
8
34
53
10.20
0.875
9
38
60
12.90
0.969
10
43
68
16.60
1.106
11
48
76
20.50
1.229
12
53
83
24.80
1.352
13
58
91
29.50
1.475
14
63
98
34.60
1.598
15
68
106
40.10
1.721
16
72
113
46.10
1.845
17
77
121
52.40
1.967
18
82
128
59.20
2.091
19
87
136
66.40
2.215
20
92
143
74.00
2.340
21
97
151
82.00
2.461
22
106
166
99.20
2.707
23
116
180
118.60
2.968
Note: The Minor Axis is the maximum width for a vertical ellipse and the full depth for a
horizontal ellipse while the Major Axis is the maximum width for a horizontal ellipse and the full
depth for a vertical ellipse.
Source: American Concrete Pipe Association (2011).
155
-------
'-2
.00
.04
,08
,12
,16
,20
,24
,28
,32
,36
,40
,44
,48
,52
,56
,60
,64
,68
,72
,76
,80
,84
,88
,92
,96
,00
HOO
.250
1436
1536
¦474
i484
.366
155
'768
1396
1969
1480
1925
023
053
084
107
130
154
170
177
177
170
162
122
000
Elliptical section properties as function of depth
Horizontal Ellipse
A/Am W/Wmax R/Rfuii
0.0000
0.0100
Vertical Ellipse
A/Am W/Wmax
0.000
0.0000
0.3919
0.0764
0.010
0.3919
0.5426
0.1726
0.040
0.5426
0.6499
0.2389
0.070
0.6499
0.7332
0.3274
0.100
0.7332
0.8000
0.4191
0.140
0.8000
0.8542
0.5120
0.185
0.8542
0.8980
0.5983
0.230
0.8980
0.9330
0.6757
0.280
0.9330
0.9600
0.7630
0.330
0.9600
0.9798
0.8326
0.380
0.9798
0.9928
0.9114
0.430
0.9928
0.9992
0.9702
0.480
0.9992
0.9992
1.030
0.520
0.9992
0.9928
1.091
0.570
0.9928
0.9798
1.146
0.620
0.9798
0.9600
1.185
0.670
0.9600
0.9330
1.225
0.720
0.9330
0.8980
1.257
0.770
0.8980
0.8542
1.274
0.815
0.8542
0.8000
1.290
0.860
0.8000
0.7332
1.282
0.900
0.7332
0.6499
1.274
0.930
0.6499
0.5426
1.257
0.960
0.5426
0.3919
1.185
0.990
0.3919
0.0000
1.000
1.000
0.0000
156
-------
E. Section Properties of Arch Pipes
Table E-l Standard arch pipe sizes
Code
Rise (JW) (in)
Span ( WWj) (in)
(in2)
Rmi( in)
Concrete
1
11
18
1.1
0.25
2
13.5
22
1.65
0.30
3
15.5
26
2.2
0.36
4
18
28.5
2.8
0.45
5
22.5
36.25
4.4
0.56
6
26.625
43.75
6.4
0.68
7
31.3125
51.125
8.8
0.80
8
36
58.5
11.4
0.90
9
40
65
14.3
1.01
10
45
73
17.7
1.13
11
54
88
25.6
1.35
12
62
102
34.6
1.57
13
72
115
44.5
1.77
14
77.5
122
51.7
1.92
15
87.125
138
66.0
2.17
16
96.875
154
81.8
2.42
17
106.5
168.75
99.1
2.65
CoiTii»:iled Steel, 2-2/3 \ 1/2" Corrugation
18
13
17
1.1
0.324
19
15
21
1.6
0.374
20
18
24
2.2
0.449
21
20
28
2.9
0.499
22
24
35
4.5
0.598
23
29
42
6.5
0.723
24
33
49
8.9
0.823
25
38
57
11.6
0.947
26
43
64
14.7
1.072
27
47
71
18.1
1.171
28
52
77
21.9
1.296
29
57
83
26.0
1.421
157
-------
Table E-2 Continued
Code
Rise (Yfuii) (in)
Span (WW) (in)
Ami (in2)
#u//(in)
Corru<>2i(ed Steel, 3 x 1" Corrugation
30
31
40
7.0
0.773
31
36
46
9.4
0.773
32
41
53
12.3
1.022
33
46
60
15.6
1.147
34
51
66
19.3
1.271
35
55
73
23.2
1.371
36
59
81
27.4
1.471
37
63
87
32.1
1.570
38
67
95
37.0
1.670
39
71
103
42.4
1.770
40
75
112
48.0
1.869
41
79
117
54.2
1.969
42
83
128
60.5
2.069
43
87
137
67.4
2.168
44
91
142
74.5
2.268
Structiir
-------
Table E-3 Continued
Code
Rise (?/„//) (in)
Span (Wmax) (in)
Ami (in2)
Rfull (in)
Sirucliir
-------
Table E-4 Arch pipe section properties as function of depth
Y/Yfui,
A/Ami
w/wmax
R/Rtiiii
0.00
0.000
0.0000
0.0100
0.04
0.020
0.6272
0.0983
0.08
0.060
0.8521
0.1965
0.12
0.100
0.9243
0.2948
0.16
0.140
0.9645
0.3940
0.20
0.190
0.9846
0.4962
0.24
0.240
0.9964
0.5911
0.28
0.290
0.9988
0.6796
0.32
0.340
0.9917
0.7615
0.36
0.390
0.9811
0.8364
0.40
0.440
0.9680
0.9044
0.44
0.490
0.9515
0.9640
0.48
0.540
0.9314
1.018
0.52
0.590
0.9101
1.065
0.56
0.640
0.8864
1.106
0.60
0.690
0.8592
1.142
0.64
0.735
0.8284
1.170
0.68
0.780
0.7917
1.192
0.72
0.820
0.7527
1.208
0.76
0.860
0.7065
1.217
0.80
0.895
0.6544
1.220
0.84
0.930
0.5953
1.213
0.88
0.960
0.5231
1.196
0.92
0.985
0.4355
1.168
0.96
0.995
0.3195
1.112
1.00
1.000
0.000
1.000
160
-------
F. Section Properties of Masonry Sewers
Table F-l Area of masonry sewers as function of depth
y/Ymi
Basket Handle
A/AfaU
Egg
Horseshoe
0.00
0.0000
0.000
0.0000
0.04
0.0173
0.015
0.0181
0.08
0.0457
0.040
0.0508
0.12
0.0828
0.055
0.0908
0.16
0.1271
0.085
0.1326
0.20
0.1765
0.120
0.1757
0.24
0.2270
0.155
0.2201
0.28
0.2775
0.190
0.2655
0.32
0.3280
0.225
0.3118
0.36
0.3780
0.275
0.3587
0.40
0.4270
0.320
0.4064
0.44
0.4765
0.370
0.4542
0.48
0.5260
0.420
0.5023
0.52
0.5740
0.470
0.5506
0.56
0.6220
0.515
0.5987
0.60
0.6690
0.570
0.6462
0.64
0.7160
0.620
0.6931
0.68
0.7610
0.680
0.7387
0.72
0.8030
0.730
0.7829
0.76
0.8390
0.780
0.8253
0.80
0.8770
0.835
0.8652
0.84
0.9110
0.885
0.9022
0.88
0.9410
0.925
0.9356
0.92
0.9680
0.955
0.9645
0.96
0.9880
0.980
0.9873
1.00
1.0000
1.000
1.0000
161
-------
Table F-2 Width of masonry sewers as function of depth - I
Y/Yfuii
Basket Handle
w/wmax
Egg
Horseshoe
0.00
0.000
0.000
0.0000
0.04
0.490
0.298
0.5878
0.08
0.667
0.433
0.8772
0.12
0.820
0.508
0.8900
0.16
0.930
0.582
0.9028
0.20
1.000
0.642
0.9156
0.24
1.000
0.696
0.9284
0.28
1.000
0.746
0.9412
0.32
0.997
0.791
0.9540
0.36
0.994
0.836
0.9668
0.40
0.988
0.866
0.9798
0.44
0.982
0.896
0.9928
0.48
0.967
0.926
0.9992
0.52
0.948
0.956
0.9992
0.56
0.928
0.970
0.9928
0.60
0.904
0.985
0.9798
0.64
0.874
1.000
0.9600
0.68
0.842
0.985
0.9330
0.72
0.798
0.970
0.8980
0.76
0.750
0.940
0.8542
0.80
0.697
0.896
0.8000
0.84
0.637
0.836
0.7332
0.88
0.567
0.764
0.6499
0.92
0.467
0.642
0.5426
0.96
0.342
0.310
0.3919
1.00
0.000
0.000
0.0000
-------
Table F-3 Width of masonry sewers as function of depth - II
Y/Yfuii
Gothic
w/wmax
Catenary Semi-Elliptical
Semi-Circular
0
0.000
0.0000
0.00
0.0000
0.05
0.286
0.6667
0.70
0.5488
0.10
0.643
0.8222
0.98
0.8537
0.15
0.762
0.9111
1.00
1.0000
0.20
0.833
0.9778
1.00
1.0000
0.25
0.905
1.0000
1.00
0.9939
0.30
0.952
1.0000
0.99
0.9878
0.35
0.976
0.9889
0.98
0.9756
0.40
0.976
0.9778
0.96
0.9634
0.45
1.000
0.9556
0.94
0.9451
0.50
1.000
0.9333
0.91
0.9207
0.55
0.976
0.8889
0.88
0.8902
0.60
0.976
0.8444
0.84
0.8537
0.65
0.952
0.8000
0.80
0.8171
0.70
0.905
0.7556
0.75
0.7683
0.75
0.833
0.7000
0.70
0.7073
0.80
0.762
0.6333
0.64
0.6463
0.85
0.667
0.5556
0.56
0.5732
0.90
0.524
0.4444
0.46
0.4756
0.95
0.357
0.3333
0.34
0.3354
1.00
0.000
0.0000
0.00
0.0000
163
-------
Table F-4 Hydraulic radius of masonry sewers as function of depth
Y/Yfuii
Basket Handle
R/Rfuii
Egg
Horseshoe
0.00
0.010
0.010
0.0100
0.04
0.0952
0.097
0.1040
0.08
0.189
0.216
0.2065
0.12
0.273
0.302
0.3243
0.16
0.369
0.386
0.4322
0.20
0.463
0.465
0.5284
0.24
0.560
0.536
0.6147
0.28
0.653
0.611
0.6927
0.32
0.743
0.676
0.7636
0.36
0.822
0.735
0.8268
0.40
0.883
0.791
0.8873
0.44
0.949
0.854
0.9417
0.48
0.999
0.904
0.9905
0.52
1.055
0.941
1.036
0.56
1.095
1.008
1.077
0.60
1.141
1.045
1.113
0.64
1.161
1.076
1.143
0.68
1.188
1.115
1.169
0.72
1.206
1.146
1.189
0.76
1.206
1.162
1.202
0.80
1.206
1.186
1.208
0.84
1.205
1.193
1.206
0.88
1.196
1.186
1.195
0.92
1.168
1.162
1.170
0.96
1.127
1.107
1.126
1.00
1.000
1.000
1.000
-------
Table F-5 Depth of masonry sewers as function of area - I
A/Afuii
Basket Handle
Y/Ym
Egg
Horseshoe
0.00
0.00000
0.00000
0.00000
0.02
0.04112
0.04912
0.04146
0.04
0.07380
0.08101
0.07033
0.06
0.10000
0.11128
0.09098
0.08
0.12236
0.14161
0.10962
0.10
0.14141
0.16622
0.12921
0.12
0.15857
0.18811
0.14813
0.14
0.17462
0.21356
0.16701
0.16
0.18946
0.23742
0.18565
0.18
0.20315
0.25742
0.20401
0.20
0.21557
0.27742
0.22211
0.22
0.22833
0.29741
0.23998
0.24
0.24230
0.31742
0.25769
0.26
0.25945
0.33742
0.27524
0.28
0.27936
0.35747
0.29265
0.30
0.30000
0.37364
0.30990
0.32
0.32040
0.40000
0.32704
0.34
0.34034
0.41697
0.34406
0.36
0.35892
0.43372
0.36101
0.38
0.37595
0.45000
0.37790
0.40
0.39214
0.46374
0.39471
0.42
0.40802
0.47747
0.41147
0.44
0.42372
0.49209
0.42818
0.46
0.43894
0.50989
0.44484
0.48
0.45315
0.53015
0.46147
0.50
0.46557
0.55000
0.47807
-------
Table F-5 Continued
A/Afuii
Basket Handle
Y/Ym
Egg
Horseshoe
0.52
0.47833
0.56429
0.49468
0.54
0.49230
0.57675
0.51134
0.56
0.50945
0.58834
0.52803
0.58
0.52936
0.60000
0.54474
0.60
0.55000
0.61441
0.56138
0.62
0.57000
0.62967
0.57804
0.64
0.59000
0.64582
0.59478
0.66
0.61023
0.66368
0.61171
0.68
0.63045
0.68209
0.62881
0.70
0.65000
0.70000
0.64609
0.72
0.66756
0.71463
0.66350
0.74
0.68413
0.72807
0.68111
0.76
0.70000
0.74074
0.69901
0.78
0.71481
0.75296
0.71722
0.80
0.72984
0.76500
0.73583
0.82
0.74579
0.77784
0.75490
0.84
0.76417
0.79212
0.77447
0.86
0.78422
0.80945
0.79471
0.88
0.80477
0.82936
0.81564
0.90
0.82532
0.85000
0.83759
0.92
0.85000
0.86731
0.86067
0.94
0.88277
0.88769
0.88557
0.96
0.91500
0.91400
0.91159
0.98
0.95000
0.95000
0.94520
1.00
1.00000
1.00000
1.00000
166
-------
Table F-6 Depth of masonry sewers as function of area - II
A/Ami
Catenary
Gothic
Y/Yfuii
Semi-Circular
Semi-Elliptical
0.00
0.00000
0.00000
0.00000
0.00000
0.02
0.02974
0.04522
0.04102
0.03075
0.04
0.06439
0.07825
0.07407
0.05137
0.06
0.08433
0.10646
0.10000
0.07032
0.08
0.10549
0.12645
0.11769
0.09000
0.10
0.12064
0.14645
0.13037
0.11323
0.12
0.13952
0.16787
0.14036
0.13037
0.14
0.15560
0.18641
0.15000
0.14519
0.16
0.17032
0.20129
0.16546
0.15968
0.18
0.18512
0.22425
0.18213
0.18459
0.20
0.20057
0.24129
0.20000
0.19531
0.22
0.21995
0.25624
0.22018
0.21354
0.24
0.24011
0.27344
0.24030
0.22694
0.26
0.25892
0.29097
0.25788
0.23947
0.28
0.27595
0.30529
0.27216
0.25296
0.30
0.29214
0.32607
0.28500
0.26500
0.32
0.30802
0.33755
0.29704
0.27784
0.34
0.32372
0.35073
0.30892
0.29212
0.36
0.33894
0.36447
0.32128
0.30970
0.38
0.35315
0.37558
0.33476
0.32982
0.40
0.36557
0.40000
0.35000
0.35000
0.42
0.37833
0.41810
0.36927
0.36738
0.44
0.39230
0.43648
0.38963
0.38390
0.46
0.40970
0.45374
0.41023
0.40000
0.48
0.42982
0.46805
0.43045
0.41667
0.50
0.45000
0.48195
0.45000
0.43333
167
-------
Table F-6 Continued
A/Ami
Catenary
Gothic
Y/Yfuii
Semi-Circular
Semi-Elliptical
0.52
0.46769
0.49626
0.46769
0.45000
0.54
0.48431
0.51352
0.48431
0.46697
0.56
0.50000
0.53190
0.50000
0.48372
0.58
0.51466
0.55000
0.51443
0.50000
0.60
0.52886
0.56416
0.52851
0.51374
0.62
0.54292
0.57787
0.54271
0.52747
0.64
0.55729
0.59224
0.55774
0.54209
0.66
0.57223
0.60950
0.57388
0.55950
0.68
0.58780
0.62941
0.59101
0.57941
0.70
0.60428
0.65000
0.60989
0.60000
0.72
0.62197
0.67064
0.63005
0.62000
0.74
0.64047
0.69055
0.65000
0.64000
0.76
0.65980
0.70721
0.66682
0.66000
0.78
0.67976
0.72031
0.68318
0.68000
0.80
0.70000
0.73286
0.70000
0.70000
0.82
0.71731
0.74632
0.71675
0.71843
0.84
0.73769
0.76432
0.73744
0.73865
0.86
0.76651
0.78448
0.76651
0.76365
0.88
0.80000
0.80421
0.80000
0.79260
0.90
0.82090
0.82199
0.82090
0.82088
0.92
0.84311
0.84363
0.84311
0.85000
0.94
0.87978
0.87423
0.87978
0.88341
0.96
0.91576
0.90617
0.91576
0.90998
0.98
0.95000
0.93827
0.95000
0.93871
1.00
1.00000
1.00000
1.00000
1.00000
168
-------
Table F-7 Section factor for masonry sewers as function of area -1
A/Ami
Basket Handle
V/Vfull
Egg
Horseshoe
0.00
0.00000
0.00000
0.00000
0.02
0.00758
0.00295
0.00467
0.04
0.01812
0.01331
0.01237
0.06
0.03000
0.02629
0.02268
0.08
0.03966
0.04000
0.03515
0.10
0.04957
0.05657
0.04943
0.12
0.06230
0.07500
0.06525
0.14
0.07849
0.09432
0.08212
0.16
0.09618
0.11473
0.10005
0.18
0.11416
0.13657
0.11891
0.20
0.13094
0.15894
0.13856
0.22
0.14808
0.18030
0.15896
0.24
0.16583
0.20036
0.18004
0.26
0.18381
0.22000
0.20172
0.28
0.20294
0.23919
0.22397
0.30
0.22500
0.25896
0.24677
0.32
0.25470
0.28000
0.27006
0.34
0.28532
0.30504
0.29380
0.36
0.31006
0.33082
0.31790
0.38
0.32804
0.35551
0.34237
0.40
0.34555
0.37692
0.36720
0.42
0.36944
0.39809
0.39239
0.44
0.40032
0.42000
0.41792
0.46
0.43203
0.44625
0.44374
0.48
0.46004
0.47321
0.46984
0.50
0.47849
0.50000
0.49619
169
-------
Table F-7 Continued
A/Ami
Basket Handle
V/Vfull
Egg
Horseshoe
0.52
0.49591
0.52255
0.52276
0.54
0.51454
0.54481
0.54950
0.56
0.53810
0.56785
0.57640
0.58
0.56711
0.59466
0.60345
0.60
0.60000
0.62485
0.63065
0.62
0.64092
0.65518
0.65795
0.64
0.68136
0.68181
0.68531
0.66
0.71259
0.70415
0.71271
0.68
0.73438
0.72585
0.74009
0.70
0.75500
0.74819
0.76738
0.72
0.78625
0.77482
0.79451
0.74
0.81880
0.80515
0.82144
0.76
0.85000
0.83534
0.84814
0.78
0.86790
0.86193
0.87450
0.80
0.88483
0.88465
0.90057
0.82
0.90431
0.90690
0.92652
0.84
0.93690
0.93000
0.95244
0.86
0.97388
0.95866
0.97724
0.88
1.00747
0.98673
0.99988
0.90
1.03300
1.01238
1.02048
0.92
1.05000
1.03396
1.03989
0.94
1.05464
1.05000
1.05698
0.96
1.06078
1.06517
1.07694
0.98
1.05500
1.05380
1.07562
1.00
1.00000
1.00000
1.00000
170
-------
Table F-8 Section factor for masonry sewers as function of area - II
A/Ami
Catenary
Gothic
YFfuii
Semi-Circular
Semi-Elliptical
0.00
0.00000
0.00000
0.00000
0.00000
0.02
0.00605
0.00500
0.00757
0.00438
0.04
0.01455
0.01740
0.01815
0.01227
0.06
0.02540
0.03098
0.03000
0.02312
0.08
0.03863
0.04272
0.03580
0.03638
0.10
0.05430
0.05500
0.04037
0.05145
0.12
0.07127
0.06980
0.04601
0.06783
0.14
0.08778
0.08620
0.05500
0.08500
0.16
0.10372
0.10461
0.07475
0.10093
0.18
0.12081
0.12463
0.09834
0.11752
0.20
0.14082
0.14500
0.12500
0.13530
0.22
0.16375
0.16309
0.15570
0.15626
0.24
0.18779
0.18118
0.18588
0.17917
0.26
0.21157
0.20000
0.20883
0.20296
0.28
0.23478
0.22181
0.22300
0.22654
0.30
0.25818
0.24487
0.23472
0.24962
0.32
0.28244
0.26888
0.24667
0.27269
0.34
0.30741
0.29380
0.26758
0.29568
0.36
0.33204
0.31901
0.29346
0.31848
0.38
0.35505
0.34389
0.32124
0.34152
0.40
0.37465
0.36564
0.35000
0.36500
0.42
0.39404
0.38612
0.37720
0.38941
0.44
0.41426
0.40720
0.40540
0.41442
0.46
0.43804
0.43000
0.43541
0.44000
0.48
0.46531
0.45868
0.46722
0.46636
0.50
0.49357
0.48895
0.50000
0.49309
-------
Table F-8 Continued
A/Ami
Catenary
Gothic
yffuii
Semi-Circular
Semi-Elliptical
0.52
0.52187
0.52000
0.53532
0.52000
0.54
0.54925
0.55032
0.56935
0.54628
0.56
0.57647
0.58040
0.60000
0.57285
0.58
0.60321
0.61000
0.61544
0.60000
0.60
0.62964
0.63762
0.62811
0.62949
0.62
0.65639
0.66505
0.64170
0.65877
0.64
0.68472
0.69290
0.66598
0.68624
0.66
0.71425
0.72342
0.70010
0.71017
0.68
0.74303
0.75467
0.73413
0.73304
0.70
0.76827
0.78500
0.76068
0.75578
0.72
0.79168
0.81165
0.78027
0.77925
0.74
0.81500
0.83654
0.80000
0.80368
0.76
0.84094
0.86000
0.82891
0.83114
0.78
0.86707
0.88253
0.85964
0.85950
0.80
0.89213
0.90414
0.89000
0.88592
0.82
0.91607
0.92500
0.91270
0.90848
0.84
0.94000
0.94486
0.93664
0.93000
0.86
0.96604
0.96475
0.96677
0.95292
0.88
0.99000
0.98567
1.00000
0.97481
0.90
1.00714
1.00833
1.02661
0.99374
0.92
1.02158
1.03000
1.04631
1.01084
0.94
1.03814
1.05360
1.05726
1.02858
0.96
1.05000
1.06500
1.06637
1.04543
0.98
1.05000
1.05500
1.06000
1.05000
1.00
1.00000
1.00000
1.00000
1.00000
172
-------
G. Manning's Roughness Coefficients
Table G-l Manning's roughness coefficient n for open channels
Type of Channel and Description
Minimum
Normal
Maximum
1. Natural si minis- minor streams (lop width at Hood stage
< 100 ID
a. clean, straight, full stage, no rifts or deep pools
0.025
0.030
0.033
b. same as above, but more stones and weeds
0.030
0.035
0.040
c. clean, winding, some pools and shoals
0.033
0.040
0.045
d. same as above, but some weeds and stones
0.035
0.045
0.050
e. same as above, lower stages, more ineffective
slopes and sections
0.040
0.048
0.055
f. same as "d" with more stones
0.045
0.050
0.060
g. sluggish reaches, weedy, deep pools
0.050
0.070
0.080
h. very weedy reaches, deep pools, or floodways
with heavy stand of timber and underbrush
0.075
0.100
0.150
2. Mountain streams, no \cgclation in channel, hanks usually sleep, trees and hrush along hanks
submerged at high stages
a. bottom: gravels, cobbles, and few boulders
0.030
0.040
0.050
b. bottom: cobbles with large boulders
0.040
0.050
0.070
3. l-'loodplains
a. Pasture, no brush
1. short grass
0.025
0.030
0.035
2. high grass
0.030
0.035
0.050
b. Cultivated areas
1. no crop
0.020
0.030
0.040
2. mature row crops
0.025
0.035
0.045
3. mature field crops
0.030
0.040
0.050
173
-------
Table G-l Continued
Type of Channel and Description
Minimum
Normal
Maximum
3. l-'loodplains
c. Brush
1. scattered brush, heavy weeds
0.035
0.050
0.070
2. light brush and trees, in winter
0.035
0.050
0.060
3. light brush and trees, in summer
0.040
0.060
0.080
4. medium to dense brush, in winter
0.045
0.070
0.110
5. medium to dense brush, in summer
0.070
0.100
0.160
d. Trees
1. dense willows, summer, straight
0.110
0.150
0.200
2. cleared land with tree stumps, no sprouts
0.030
0.040
0.050
3. same as above, but with heavy growth of sprouts
0.050
0.060
0.080
4. heavy stand of timber, a few down trees, little
undergrowth, flood stage below branches
0.080
0.100
0.120
5. same as 4. with flood stage reaching branches
0.100
0.120
0.160
4. K\c:i\sited or Dredged Channels
a Earth, straight, and uniform
1. clean, recently completed
0.016
0.018
0.020
2. clean, after weathering
0.018
0.022
0.025
3. gravel, uniform section, clean
0.022
0.025
0.030
4. with short grass, few weeds
0.022
0.027
0.033
b. Earth winding and sluggish
1. no vegetation
0.023
0.025
0.030
2. grass, some weeds
0.025
0.030
0.033
3. dense weeds or aquatic plants in deep channels
0.030
0.035
0.040
4. earth bottom and rubble sides
0.028
0.030
0.035
5. stony bottom and weedy banks
0.025
0.035
0.040
6. cobble bottom and clean sides
0.030
0.040
0.050
c. Dragline-excavated or dredged
1. no vegetation
0.025
0.028
0.033
2. light brush on banks
0.035
0.050
0.060
174
-------
Table G-l Continued
Type of Channel and Description
Minimum
Normal
Maximum
d. Rock cuts
1. smooth and uniform
0.025
0.035
0.040
2. jagged and irregular
0.035
0.040
0.050
e. Channels not maintained, weeds and brush uncut
1. dense weeds, high as flow depth
0.050
0.080
0.120
2. clean bottom, brush on sides
0.040
0.050
0.080
3. same as above, highest stage of flow
0.045
0.070
0.110
4. dense brush, high stage
0.080
0.100
0.140
5. Lined or Constructed Channels
a. Cement
1. neat surface
0.010
0.011
0.013
2. mortar
0.011
0.013
0.015
b. Wood
1. planed, untreated
0.010
0.012
0.014
2. planed, creosoted
0.011
0.012
0.015
3. unplaned
0.011
0.013
0.015
4. plank with battens
0.012
0.015
0.018
5. lined with roofing paper
0.010
0.014
0.017
c. Concrete
1. trowel finish
0.011
0.013
0.015
2. float finish
0.013
0.015
0.016
3. finished, with gravel on bottom
0.015
0.017
0.020
4. unfinished
0.014
0.017
0.020
5. gunite, good section
0.016
0.019
0.023
6. gunite, wavy section
0.018
0.022
0.025
7. on good excavated rock
0.017
0.020
8. on irregular excavated rock
0.022
0.027
175
-------
Table G-l Continued
Type of Channel and Description
Minimum
Normal
Maximum
d. Concrete bottom float finish with sides of:
1. dressed stone in mortar
0.015
0.017
0.020
2. random stone in mortar
0.017
0.020
0.024
3. cement rubble masonry, plastered
0.016
0.020
0.024
4. cement rubble masonry
0.020
0.025
0.030
5. dry rubble or riprap
0.020
0.030
0.035
e. Gravel bottom with sides of:
1. formed concrete
0.017
0.020
0.025
2. random stone mortar
0.020
0.023
0.026
3. dry rubble or riprap
0.023
0.033
0.036
f. Brick
1. glazed
0.011
0.013
0.015
2. in cement mortar
0.012
0.015
0.018
g. Masonry
1. cemented rubble
0.017
0.025
0.030
2. dry rubble
0.023
0.032
0.035
h. Dressed ashlar/stone paving
0.013
0.015
0.017
i. Asphalt
1. smooth
0.013
0.013
2. rough
0.016
0.016
j. Vegetal lining
0.030
0.500
Source: Chow, 1959.
176
-------
Table G-2 Manning's roughness coefficient n for closed conduits
Type of Conduit and Description
Minimum
Normal
Maximum
1. Brass, smooth:
0.009
0.010
0.013
2. Steel:
Lockbar and welded
0.010
0.012
0.014
Riveted and spiral
0.013
0.016
0.017
3. Cast Iron:
Coated
0.010
0.013
0.014
Uncoated
0.011
0.014
0.016
4. Wrought Iron:
Black
0.012
0.014
0.015
Galvanized
0.013
0.016
0.017
5. Corrugated Metal:
Subdrain
0.017
0.019
0.021
Stormdrain
0.021
0.024
0.030
6. Cement:
Neat Surface
0.010
0.011
0.013
Mortar
0.011
0.013
0.015
7. Concrete:
Culvert, straight and free of debris
0.010
0.011
0.013
Culvert with bends, connections, and
some debris
0.011
0.013
0.014
Finished
0.011
0.012
0.014
Sewer with manholes, inlet, etc., straight
0.013
0.015
0.017
Unfinished, steel form
0.012
0.013
0.014
Unfinished, smooth wood form
0.012
0.014
0.016
Unfinished, rough wood form
0.015
0.017
0.020
8. Wood:
Stave
0.010
0.012
0.014
Laminated, treated
0.015
0.017
0.020
177
-------
Minimum Normal Maximum
9. Clay:
Common drainage tile
0.011
0.013
0.017
Vitrified sewer
0.011
0.014
0.017
Vitrified sewer with manholes, inlet, etc.
0.013
0.015
0.017
Vitrified subdrain with open joint
0.014
0.016
0.018
10. Brickwork:
Glazed
0.011
0.013
0.015
Lined with cement mortar
0.012
0.015
0.017
Sanitary sewers coated with sewage slime
with bends and connections
0.012
0.013
0.016
Paved invert, sewer, smooth bottom
0.016
0.019
0.020
Rubble masonry, cemented
0.018
0.025
0.030
Source: Chow, 1959.
178
-------
Table G-3 Manning's roughness coefficient n for corrugated steel pipe
Type of Pipe, Diameter and Corrugation Dimension
n
1. Annular 2.67 x 1/2 inch (all diameters)
0.024
2. Helical 1.50 x 1/4 inch
8" diameter
0.012
10" diameter
0.014
3. Helical 2.67 x 1/2 inch
12" diameter
0.011
18" diameter
0.014
24" diameter
0.016
36" diameter
0.019
48" diameter
0.020
60" diameter
0.021
4. Annular 3x1 inch (all diameters)
0.027
5. Helical 3x1 inch
48" diameter
0.023
54" diameter
0.023
60" diameter
0.024
66" diameter
0.025
72" diameter
0.026
78" diameter and larger
0.027
6. Corrugations 6x2 inches
60" diameter
0.033
72" diameter
0.032
120" diameter
0.030
180" diameter
0.028
Source: American Iron and Steel Institute, 1999.
179
-------
H. Culvert Coefficients
Table H-l Culvert codes
Culvert Shape and Material
Inlet Configuration
Code
Circular Concrete
Square edge with headwall
1
Groove end with headwall
2
Groove end projecting
3
Circular Corrugated Metal Pipe
Headwall
4
Mitered to slope
5
Projecting
6
Circular Pipe, Beveled Ring Entrance
45 deg. bevels
7
33.7 deg. bevels
8
Rectangular Box; Flared Wingwalls
30-75 deg. wingwall flares
9
90 or 15 deg. wingwall flares
10
0 deg. wingwall flares (straight sides)
11
Rectangular Box;Flared Wingwalls and
Top Edge Bevel
45 deg flare; 0.43D top edge bevel
12
18-33.7 deg. flare; 0.083D top edge bevel
13
Rectangular Box, 90-deg Headwall,
Chamfered / Beveled Inlet Edges
Chamfered 3/4-in.
14
Beveled 1/2-in/ft at 45 deg (1:1)
15
Beveled 1-in/ft at 33.7 deg (1:1.5)
16
Rectangular Box, Skewed Headwall,
Chamfered / Beveled Inlet Edges
3/4" chamfered edge, 45 deg skewed headwall
17
3/4" chamfered edge, 30 deg skewed headwall
18
3/4" chamfered edge, 15 deg skewed headwall
19
45 deg beveled edge, 10-45 deg skewed headwall
20
Rectangular Box, Non-offset Flared
Wingwalls, 3/4" Chamfer at Top of Inlet
45 deg (1:1) wingwall flare
21
8.4 deg (3:1) wingwall flare
22
18.4 deg (3:1) wingwall flare, 30 deg inlet skew
23
180
-------
Table H-l Continued
Culvert Shape and Material
Inlet Configuration
Code
Rectangular Box, Offset Flared
Wingwalls, Beveled Edge at Inlet Top
45 deg (1:1) flare, 0.042D top edge bevel
24
33.7 deg (1.5:1) flare, 0.083D top edge bevel
25
18.4 deg (3:1) flare, 0.083D top edge bevel
26
Corrugated Metal Box
90 deg headwall
27
Thick wall projecting
28
Thin wall projecting
29
Horizontal Ellipse Concrete
Square edge with headwall
30
Grooved end with headwall
31
Grooved end projecting
32
Vertical Ellipse Concrete
Square edge with headwall
33
Grooved end with headwall
34
Grooved end projecting
35
Pipe Arch, 18" Corner Radius, Corrugated
Metal
90 deg headwall
36
Mitered to slope
37
Projecting
38
Pipe Arch, 18" Corner Radius, Corrugated
Metal
Projecting
39
No bevels
40
33.7 deg bevels
41
Pipe Arch, 31" Corner Radius,Corrugated
Metal
Projecting
42
No bevels
43
33.7 deg. bevels
44
Arch, Corrugated Metal
90 deg headwall
45
Mitered to slope
46
Thin wall projecting
47
Circular Culvert
Smooth tapered inlet throat
48
Rough tapered inlet throat
49
181
-------
Table H-l Continued
Culvert Shape and Material
Inlet Configuration
Code
Tapered inlet, beveled edges
50
Elliptical Inlet Face
Tapered inlet, square edges
51
Tapered inlet, thin edge projecting
52
Rectangular
Tapered inlet throat
53
Side tapered, less favorable edges
54
Rectangular Concrete
Side tapered, more favorable edges
55
Slope tapered, less favorable edges
56
Slope tapered, more favorable edges
57
182
-------
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Culvert coefficients
Equation Unsubmerged Unsubmerged Submerged
Form K M c
0.0098
2.00
0.0398
0.0018
2.00
0.0292
0.0045
2.00
0.0317
0.0078
2.00
0.0379
0.0210
1.33
0.0463
0.0340
1.50
0.0553
0.0018
2.50
0.0300
0.0018
2.50
0.0243
0.026
1.0
0.0347
0.061
0.75
0.0400
0.061
0.75
0.0423
0.510
0.667
0.0309
0.486
0.667
0.0249
0.515
0.667
0.0375
0.495
0.667
0.0314
0.486
0.667
0.0252
0.545
0.667
0.04505
0.533
0.667
0.0425
0.522
0.667
0.0402
0.498
0.667
0.0327
0.497
0.667
0.0339
0.493
0.667
0.0361
0.495
0.667
0.0386
0.497
0.667
0.0302
0.495
0.667
0.0252
0.493
0.667
0.0227
0.0083
2.00
0.0379
0.0145
1.75
0.0419
0.0340
1.50
0.0496
0.0100
2.00
0.0398
0.0018
2.50
0.0292
0.0045
2.00
0.0317
183
-------
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
Continued
Equation Unsubmerged Unsubmerged Submerged
Form K M c
0.0100
2.00
0.0398
0.0018
2.50
0.0292
0.0095
2.00
0.0317
0.0083
2.00
0.0379
0.0300
1.00
0.0463
0.0340
1.50
0.0496
0.0300
1.50
0.0496
0.0088
2.00
0.0368
0.0030
2.00
0.0269
0.0300
1.50
0.0496
0.0088
2.00
0.0368
0.0030
2.00
0.0269
0.0083
2.00
0.0379
0.0300
1.00
0.0473
0.0340
1.50
0.0496
0.534
0.555
0.0196
0.519
0.640
0.0210
0.536
0.622
0.0368
0.5035
0.719
0.0478
0.547
0.800
0.0598
0.475
0.667
0.0179
0.560
0.667
0.0446
0.560
0.667
0.0378
0.500
0.500
0.667
0.667
0.0446
0.0378
184
-------
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