EPA/600/R-15/162A | January 2016 | www2.epa.gov/water-research
United States
Enviromental Protection
Agency
Storm Water Management Model
Reference Manual
Volume I - Hydrology (Revised)
/
Office of Research and Development
Water Supply and Water Resources Division
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EPA/600/R-15/162A
Revised January 2016
Storm Water Management Model
Reference Manual
Volume I - Hydrology (Revised)
By:
Lewis A. Rossman
National Risk Management Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati, OH 45268
and
Wayne C. Huber
School of Civil and Construction Engineering
Oregon State University
Corvallis, OR 97331
National Risk Management Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
26 Martin Luther King Drive
Cincinnati, OH 45268
January 2016
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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.
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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.
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Acknowledgments
This report was written by Lewis A. Rossman, Environmental Scientist Emeritus, U.S.
Environmental Protection Agency, Cincinnati, OH and Wayne C. Huber, Professor Emeritus,
School of Civil and Construction Engineering, Oregon State University, Corvallis, OR.
The authors would like to acknowledge the contributions made by the following individuals to
previous versions of SWMM that we drew 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, Lewis Rossman, Charles Rowney, and Robert Shubinsky. Finally, we wish to thank
Lewis Rossman, Wayne Huber, Thomas Barnwell (US EPA retired), Richard Field (US EPA
retired), Harry Torno (US EPA retired) and William James (University of Guelph) for their
continuing efforts to support and maintain the program over the past several decades.
Portions of this document were prepared under Purchase Order 2C-R095-NAEX to Oregon State
University.
IV
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Table of Contents
DISCLAIMER II
ABSTRACT Ill
ACKNOWLEDGMENTS IV
LIST OF FIGURES VIM
LIST OF TABLES X
ACRONYMS AND ABBREVIATIONS XII
CHAPTER 1 - OVERVIEW 14
1.1 Introduction 14
1.2 SWMM's Object Model 15
1.3 SWMM's Process Models 20
1.4 Simulation Process Overview 22
1.5 Interpolation and Units 26
CHAPTER 2 - METEOROLOGY 29
2.1 Precipitation 29
2.2 Precipitation Data Sources 32
2.3 Temperature Data 39
2.4 Continuous Temperature Records 44
2.5 Evaporation Data 48
2.6 Wind Speed Data 50
CHAPTER 3 - SURFACE RUNOFF 51
3.1 Introduction 51
3.2 Governing Equations 51
3.3 Subcatchment Partitioning 54
3.4 Computational Scheme 56
3.5 Time Step Considerations 58
3.6 Overland Flow Re-Routing 59
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3.7 Subcatchment Discretization 61
3.8 Parameter Estimates 64
3.9 Numerical Example 79
3.10 Approximating Other Runoff Methods 80
CHAPTER 4 - INFILTRATION 86
4.1 Introduction 86
4.2 Horton's Method 88
4.3 Modified Horton Method 100
4.4 Green-Ampt Method 104
4.5 Curve Number Method 117
4.6 Numerical Example 125
CHAPTER 5 - GROUNDWATER 128
5.1 Introduction 128
5.2 Governing Equations 129
5.3 Groundwater Flux Terms 133
5.4 Computational Scheme 140
5.5 Parameter Estimates 143
5.6 Numerical Example 160
CHAPTER 6 - SNOWMELT 163
6.1 Introduction 163
6.2 Preliminaries 164
6.3 Governing Equations 169
6.4 Areal Depletion 177
6.5 Net Runoff 182
6.6 Computational Scheme 183
6.7 Parameter Estimates 188
6.8 Numerical Example 190
CHAPTER 7 - RAINFALL DEPENDENT INFLOW AND INFILTRATION 196
7.1 Introduction 196
7.2 Governing Equations 196
7.3 Computational Scheme 202
7.4 Parameter Estimates 204
7.5 Numerical Example 207
VI
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REFERENCES 210
GLOSSARY 226
VII
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List of Figures
Figure 1-1 Elements of a typical urban drainage system 16
Figure 1-2 SWMM's conceptual model of a stormwater drainage system 17
Figure 1-3 Processes modeled by SWMM 20
Figure 1-4 Block diagram of SWMM's state transition process 22
Figure 1-5 Flow chart of SWMM's simulation procedure 25
Figure 1-6 Interpolation of reported values from computed values 27
Figure 2-1 Sinusoidal interpolation of hourly temperatures 44
Figure 3-1 Idealized representation of a subcatchment 51
Figure 3-2 Nonlinear reservoir model of a subcatchment 52
Figure 3-3 Types of subareas within a subcatchment 54
Figure 3-4 Idealized subcatchment partitioning for overland flow 55
Figure 3-5 Re-routing of overland flow (Huber, 2001) 60
Figure 3-6 Fisk B catchment, Portland, Oregon (Portland BES, 1996) 62
Figure 3-7 Detailed view of two Fisk B subcatchments (Portland BES, 1996) 63
Figure 3-8 Idealized representation of a subcatchment 68
Figure 3-9 Rectangular subcatchments for illustration of shape and width effects 69
Figure 3-10 Subcatchment hydrographs for different shapes of Figure 3-9 70
Figure 3-11 Irregular subcatchment shape for width calculations (DiGiano et al., 1977, p.
165) 72
Figure 3-12 Runoff results for illustrative example 80
Figure 3-13 SCS (NRCS) triangular unit hydrograph (NRCS, 2007) 83
Figure 4-1 Physical properties for Woodburn silt loam, Benton County, Oregon 88
Figure 4-2 The Horton infiltration curve 89
Figure 4-3 Cumulative infiltration F as the area under the Horton curve 90
Figure 4-4 Regeneration (recovery) of infiltration capacity during dry time steps 92
Figure 4-5 Two-zone representation of the Green-Ampt infiltration model (after Nicklow et
al.,2006) 104
Figure 4-6 Illustration of infiltration capacity as function of cumulative infiltration for the
Green-Ampt method 106
Figure 4-7 Green-Ampt recovery parameters as functions of hydraulic conductivity 109
Figure 4-8 Infiltration rates produced by different methods for a 2-inch rainfall event... 127
Figure 5-1 Definitional sketch of the two-zone groundwater model 129
Figure 5-2 Heights used to compute lateral groundwater flow rate 138
Figure 5-3 Relation between soil moisture limits and soil texture class (Schroeder et al., 1994).
144
Figure 5-4 SPAW's soil water characteristics calculator 150
viii
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Figure 5-5 Measured hydraulic conductivity for three soils 152
Figure 5-6 Fitting SWMM's hydraulic conductivity equation to a power law equation. 154
Figure 5-7 Definition sketch for Dupuit-Forcheimer seepage to an adjacent channel 158
Figure 5-8 Definition sketch for Hooghoudt's method for flow to circular drains 159
Figure 5-9 Surface runoff and groundwater flow for the illustrative groundwater example.
162
Figure 6-1 Typical gage catch deficiency correction (Anderson, 2006, p. 8) 166
Figure 6-2 Subcatchment partitionings used for snowmelt and runoff. 167
Figure 6-3 Seasonal variation of melt coefficients 175
Figure 6-4 Typical areal depletion curve for natural area (Anderson, 1973, p. 3-15) and
temporary curve for new snow 178
Figure 6-5 Effect of snow cover on areal depletion curves 180
Figure 6-6 Schematic of liquid water routing through snow pack 182
Figure 6-7 Continuous air temperature for illustrative snowmelt example 193
Figure 6-8 Precipitation amounts for illustrative snowmelt example 194
Figure 6-9 Snow pack depth for illustrative snowmelt example 194
Figure 6-10 Runoff time series for illustrative snowmelt example 195
Figure 7-1 Components of wet-weather wastewater flow 197
Figure 7-2 Example of an RDII triangular unit hydrograph 198
Figure 7-3 Application of a unit hydrograph to a storm event 199
Figure 7-4 Use of three unit hydrographs to represent RDII (Vallabhaneni et al., 2007). 201
Figure 7-5 Sewershed delineation (Vallabhaneni et al., 2007) 205
Figure 7-6 Extracting RDII flow from a continuous flow monitor (Vallabhaneni et al., 2007).
206
Figure 7-7 Fitting unit hydrographs to an RDII flow record (Vallabhaneni et al., 2007). 206
Figure 7-8 Unit hydrographs used for the illustrative RDII example 208
Figure 7-9 Time series of RDII flows for the illustrative RDII example 208
Figure 7-10 Excerpt from the RDII interface file for the illustrative RDII example 209
IX
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List of Tables
Table 1-1 Development history of SWMM 15
Table 1-2 SWMM's modeling objects 18
Table 1-3 State variables used by SWMM 23
Table 1-4 Units of expression used by SWMM 28
Table 2-1 15-minute precipitation data from NCDC Climate Data Online 35
Table 2-2 15-minute precipitation data in NCDC FTP file format 35
Table 2-3 15-minute precipitation data in comma-delimited format 36
Table 2-4 15-minute precipitation data in space-delimited format 37
Table 2-5 15-minute precipitation data in fixed-length format 37
Table 2-6 Record layout of Canadian HYLO and HLY21 hourly precipitation files 38
Table 2-7 Record layout of Canadian FIF21 15-minute precipitation files 38
Table 2-8 Contents of an NCDC GHCN-Daily climate file 41
Table 2-9 Contents of an NCDC DS3200 climate file 41
Table 2-10 Layout of the ID portion of an NCDC DS3200 climate file record 42
Table 2-11 Layout of the data portion of an NCDC DS3200 climate file record 42
Table 2-12 Record layout of Canadian DLY daily climatologic files 43
Table 2-13 Example user-prepared climate file 43
Table 2-14 Time zones and standard meridians (degrees west longitude) 47
Table 3-1 Impervious area as a percentage of land use 66
Table 3-2 Coefficients for Southerland's EIA equations 67
Table 3-3 Data for example of effect of subcatchment width 69
Table 3-4 Width computations for Portland example 71
Table 3-5 Estimates of Manning's roughness coefficient for overland flow 75
Table 3-6 Sensitivity of runoff volume and peak flow to surface runoff parameters 78
Table 3-7 Parameters used for illustrative runoff example 79
Table 3-8 Contents of a typical Routing Interface file 85
Table 4-1 Hydrologic soil group meanings (NRCS, 2009, Chapter 7) 87
Table 4-2 Horton parameters for selected Georgia soils (Rawls et al., 1976) 95
Table 4-3 Horton parameters provided by Horton (1940) 96
Table 4-4 Values of f» for Hydrologic Soil Groups (Musgrave, 1955) 97
Table 4-5 Rate of decay of infiltration capacity for different values of kd 98
Table 4-6 Representative values for fo 99
Table 4-7 Green-Ampt parameters for different soil classes (Rawls et al., 1983) 114
Table 4-8 Typical values of #dmax for various soil types 116
Table 4-9 Runoff curve numbers for selected land uses (NRCS, 2004a) 124
Table 4-10 Parameters used in example comparison of infiltration methods 126
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Table 5-1 Volumetric moisture content at field capacity and wilting point (derived from
Linsley et al., 1982) 145
Table 5-2 Volumetric moisture content at field capacity and wilting point (U.S. Army Corps
of Engineers, 1956) 145
Table 5-3 Average moisture limits and saturated hydraulic conductivity for different soil
types (Rawls et al., 1983) 146
Table 5-4 Default properties of low-density soils used in the EPA HELP model (from Rawls
et al. (1982) as reported in Schroeder et al. (1994)) 147
Table 5-5 Default properties of moderate-density soils used in the EPA HELP model
(Schroeder et al. (1994)) 148
Table 5-6 Soil texture abbreviations 148
Table 5-7 Regression equations for soil moisture limits (Saxton and Rawls, 2006) 150
Table 5-8 Regression estimates of soil moisture limits from the SPAW calculator* 151
Table 5-9 Estimated HCO for different soil types 155
Table 5-10 DET (in feet) for different soil types and land cover (Shah et al., 2007) 156
Table 5-11 Parameters used in groundwater example 161
Table 6-1 Guidelines for level of service in snow and ice control (Richardson et al., 1974)
170
Table 6-2 Summary of snowmelt parameters (in US customary units) 188
Table 6-3 Typical areal depletion curve for natural areas 189
Table 6-4 Subcatchment and snow pack parameters for illustrative snowmelt example.. 190
Table 6-5 Daily temperatures for illustrative snowmelt example 191
Table 6-6 Periods of precipitation for illustrative snowmelt example 192
Table 7-1 Rainfall time series for the illustrative RDII example 207
XI
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Acronyms and Abbreviations
AASHTO American Association of State Highway and Transportation Officials
ADC areal depletion curve
ADT average daily traffic
AMC antecedent moisture condition
ASCE American Society of Civil Engineers
AWND average daily wind speed
BES Bureau of Environmental Services
BMP best management practice
BWF base wastewater flow
CDO Climate Data Online
CFS cubic feet per second
CMS cubic meters per second
CSO combined sewer overflow
DCIA directly connected impervious area
EIA effective impervious area
EPA Environmental Protection Agency
ET evapotranspiration
EVAP daily pan evaporation
FTP file transfer protocol
GHCN Global Historical Climatology Network
GIS geographic information system
GPM gallons per minute
GWI groundwater infiltration
HELP Hydrological Evaluation of Landfill Performance
HSPF Hydrologic Simulation Program - Fortran
IDF intensity- duration-frequency
ILLUDAS Illinois Urban Drainage Area Simulator
LID low impact development
LPS liters per second
MGD million gallons per day
MLD million liters per day
NCDC National Climatic Data Center
NOAA National Oceanic and Atmospheric Administration
NRCS Natural Resources Conservation Service
NWS National Weather Service
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PRMS Precipitation-Runoff Modeling System
RDII rainfall dependent inflow and infiltration
SCF Snow Catch Factor
SCS Soil Conservation Service
SFWMD South Florida Water Management District
SPAW Soil-Plant-Air-Water
STORM Storage, Treatment, Overflow, Runoff Model
SWMM Storm Water Management Model
TMAX maximum daily temperature
TMIN minimum daily temperature
TVA Tennessee Valley Authority
UDFCD Urban Drainage and Flood Control District
UH unit hydrograph
USCS Unified Soil Classification System
USDA United States Department of Agriculture
USGS United States Geological Survey
WDMV 24-hour wind movement
WE water equivalent
XIII
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Chapter 1 - 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., 197la,
1971b, 1971c, 197Id), 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., 1988) 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 available
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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
SWMM I
SWMM II
SWMM 3
SWMM 3. 3
SWMM 4
SWMM 5
Year
1971
1975
1981
1983
1988
2005
Contributors
Metcalf& Eddy, Inc.
Water Resources
Engineers
University of Florida
University of Florida
University of Florida
Camp Dresser & McKee
US EPA
Oregon State University
Camp Dresser & McKee
US EPA
COM- Smith
Comments
First version of SWMM; focus
was CSO modeling; few of its
methods are still used today.
First widely distributed
version of SWMM.
Full dynamic wave flow
routine, Green-Ampt
infiltration, snow melt, and
continuous simulation added.
First PC version of SWMM.
Groundwater, RDII, irregular
channel cross-sections and
other refinements added over
a series of updates throughout
the 1990's.
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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eathcr Fkws &EPA
Separate-
Storm Sewer
System
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 collection
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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.
Divider
Storage Unit
Subtatchment
Outfall
Orifice
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
Hydrology
Hydraulics
Object Type
Rain Gage
Subcatchment
Aquifer
Snow Pack
Unit
Hydrograph
Junction
Outfall
Divider
Storage Unit
Conduit
Pump
Regulator
Description
Source of precipitation data to one or more
subcatchments.
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.
A subsurface area that receives infiltration from the
subcatchment above it and exchanges groundwater
flow with a conveyance system node.
Accumulated snow that covers a subcatchment.
A response function that describes the amount of
sewer inflow/infiltration generated over time per
unit of instantaneous rainfall.
A point in the conveyance system where conduits
connect to one another with negligible storage
volume (e.g., manholes, pipe fittings, or stream
junctions).
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.
A point in the conveyance system where the inflow
splits into two outflow conduits according to a
known relationship.
A pond, lake, impoundment, or chamber that
provides water storage.
A channel or pipe that conveys water from one
conveyance system node to another.
A device that raises the hydraulic head of water.
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 SWMM's modeling objects (continued)
Category
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).
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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:
Figure 1-3 Processes modeled by SWMM.
• time-varying precipitation
• snow accumulation and melting
• rainfall interception from depression storage (initial abstraction)
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• evaporation of standing surface water
• infiltration of rainfall into unsaturated soil layers
• 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.
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The numerical procedures that SWMM uses to model the hydrologic processes listed above are
discussed in detail in subsequent chapters of this volume. SWMM's hydraulic, water quality,
treatment and low impact development processes are described in subsequent volumes of this
manual.
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 proceeds:
Yt=g(Xt,P} (1-2)
where
Xt = a vector of state variables at time t,
Yt = a vector of output variables at time t,
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.
g(Xt, P)
Figure 1-4 Block diagram of SWMM's state transition process.
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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 later chapters.
Table 1-3 State variables used by SWMM
Process
Runoff
Infiltration*
Groundwater
Snowmelt
Flow Routing
Water Quality
Variable
d
tp
Fe
Fu
T
P
S
0u
dL
W snow
>
ati
cc
y
q
a
tsweep
mB
mp
CN
CL
Description
Depth of runoff on a subcatchment surface
Equivalent time on the Horton curve
Cumulative excess infiltration volume
Upper zone moisture content
Time until the next rainfall event
Cumulative rainfall for current event
Soil moisture storage capacity remaining
Unsaturated zone moisture content
Depth of saturated zone
Snow pack depth
Snow pack free water depth
Snow pack surface temperature
Snow pack cold content
Depth of water at a node
Flow rate in a link
Flow area in a link
Time since a subcatchment was last swept
Mass of pollutant on subcatchment surface
Mass of pollutant ponded on subcatchment
Concentration of pollutant at a node
Concentration of pollutant in a link
Initial Value
0
0
0
0
0
0
User supplied
User supplied
User supplied
User supplied
User supplied
User supplied
0
User supplied
User supplied
Inferred from q
User supplied
User supplied
0
User supplied
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:
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• meteorological conditions, such as precipitation, air temperature, potential evaporation rate
and wind speed
• externally imposed inflow hydrographs and pollutographs at specific nodes of the
conveyance system
• 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. 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 UsersManual (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 (Trpt).
24
-------
Read Input
Parameters
Initialize State
Variables
Legend:
T = current elapsed time
Tl = new elapsed time
Troff = current runoff time
Trpt = current reporting time
ATrout = routing time step
ATroff = runoff time step
ATrpt = reporting time step
DUR = simulation duration
Stop
Compute Runoff
Troff = Troff + ATroff
Route Flows and Water Quality
Save Output Results
Trpt = Trpt+ ATrpt
Figure 1-5 Flow chart of SWMM's simulation procedure.
25
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The program then enters a loop that first determines the time Tl 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 Trpt 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.
26
-------
F
L
O
w
O Computed
Interpolated
rpt
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 and then converted back to the user's
choice of unit system.
27
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Table 1-4 Units of expression used by SWMM
Variable or Parameter
Area (sub catchment)
Area (storage surface area)
Depression Storage
Depth
Elevation
Evaporation
Flow Rate
Hydraulic Conductivity
Hydraulic Head
Infiltration Rate
Length
Manning's n
Pollutant Buildup
Pollutant Concentration
Rainfall Intensity
Rainfall Volume
Storage Volume
Temperature
Velocity
Width
Wind Speed
acres
square feet
inches
feet
feet
inches/day
cubic feet/sec (cfs)
gallons/min (gpm)
106 gallons/day (mgd)
inches/hour
feet
inches/hour
feet
seconds/meter1/3
mass/acre
milligrams/liter (mg/L)
micrograms/liter (ng/L)
organism counts/liter
inches/hour
inches
cubic feet
degrees Fahrenheit
feet/second
feet
miles/hour
hectares
square meters
millimeters
meters
meters
millimeters/day
cubic meters/sec (cms)
liters/sec (Ips)
106 liters/day (mid)
millimeters/hour
meters
millimeters/hour
meters
seconds/meter1/3
mass/hectare
milligrams/liter (mg/L)
micrograms/liter (ng/L)
organism counts/liter
millimeters/hour
millimeters
cubic meters
degrees Celsius
meters/second
meters
kilometers/hour
28
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Chapter 2 - Meteorology
2.1 Precipitation
2.1.1 Representation
Precipitation is the principal driving force in rainfall-runoff-quality simulation. Stormwater runoff
and nonpoint source runoff quality are directly dependent on the precipitation time series. These
time series can range from just a few time periods for a single event to thousands of time periods
used for a multi-year simulation. Within SWMM, the Rain Gage object is used to represent a
source of precipitation data. Any number of Rain Gages may be used, data permitting, to represent
spatial variability in precipitation patterns. Precipitation data for a specific Rain Gage is supplied
either as a user-defined Time Series or through an external data file. Several different file formats
are supported for data distributed by the U.S. National Climatic Data Center and Environment
Canada as well as a standard user-prepared format. Because SWMM is a fully dynamic model that
accounts for physical processes whose time scales are on the order of minutes or less, SWMM
should not be run with either daily average or storm-averaged precipitation data.
Note that precipitation is often used synonymously with rainfall, but precipitation data may also
include snowfall. Because both are simply reported as incremental intensities or depths, the
SWMM program differentiates between rainfall and snowfall by a user-supplied dividing
temperature. In natural areas, a surface temperature of 34° to 35° F (1-2° C) provides the dividing
line between equal probabilities of rain and snow (Eagleson, 1970; Corps of Engineers, 1956).
However, this separation temperature might need to be somewhat lower in urban areas due to
warmer surface temperatures.
2.1.2 Single Event v. Continuous Simulation
Models might be used to aid in urban drainage design for protection against flooding for a certain
return period (e.g., five or ten years), or to protect against pollution of receiving waters at a certain
frequency (e.g., only one combined sewer overflow per year). In these contexts, the frequency or
return period needs to be associated with a very specific parameter. That is, for rainfall one may
speak of frequency distributions of inter-event times, total storm depth, total storm duration or
average storm intensity, all of which are different (Eagleson, 1970, pp. 183-190). But for the
aforementioned objectives, and in fact, for almost all urban hydrology work, the frequencies of
runoff and quality parameters are required, not those of rainfall. Thus, one may speak of the
frequencies of maximum flow rate, total runoff volume, or total pollutant loads. These
distributions are in no way the same as for similar rainfall parameters, although they may be related
29
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through analytical methods (Howard, 1976; Chan and Bras, 1979; Hydroscience, 1979; Adams
and Papa, 2000). Finally, for pollution control, the real interest may lie in the frequency of water
quality standards violations in the receiving water, which leads to further complications.
SWMM is capable of simulating both single rainfall events as well as long-term time histories (e.g.
several years or more) of a continuous precipitation record. In fact, the only distinctions between
the two as far as SWMM is concerned is the simulation duration requested by the user and the
need to supply meaningful initial conditions when only a single event is simulated.
Continuous simulation offers an excellent, if not the only method for obtaining the frequency of
events of interest, be they related to quantity or quality. But it has the disadvantages of a higher
run time and the need for a continuous rainfall record. This has led to the use of a "design storm"
or "design rainfall" or "design event" in a single event simulation instead. Of course, this idea long
preceded continuous simulation, before the advent of modern computers. However, because of
inherent simplifications, the choice of a design event leads to problems.
2.1.3 Temporal Rainfall Variations
The required time interval used to describe rainfall variations over time is a function of the
catchment response to rainfall input. Small, steep, smooth, impervious catchments have fast
response times, while large, flat, pervious catchments have slower response times. As a generality,
shorter time increment data are preferable to longer time increment data, but for a large (e.g., 10
mi2 or 26 km2) subcatchment (coarse schematization), even the hourly inputs usually used for
continuous simulation may be appropriate. Rainfall data with intervals larger than 1-hour (such as
average daily rainfall or event-averaged rainfall) must be suitably disaggregated (Socolofsky et
al., 2001) before they can be used in SWMM.
The rain gage itself is usually the limiting factor. It is possible to reduce data from 24-hour charts
from standard 24-hour, weighing-bucket gages to obtain 7.5-minute or 5-minute increment data,
and some USGS float gages produce no better than 5-minute values. Shorter time increment data
may usually be obtained only from tipping bucket gage installations.
The rainfall records obtained from a gage may be of mixed quality. It may be possible to define
some storms down to 1 to 5 minute rainfall intensities, while other events may be of such poor
quality (because of poor reproduction of charts or blurred traces of ink) that only 1-hour increments
can be obtained.
30
-------
2.1.4 Spatial Rainfall Variations
Even for small catchments, runoff and consequent model predictions (and prototype
measurements) may be very sensitive to spatial variations of the rainfall. For instance,
thunderstorms (convective rainfall) may be highly localized, and nearby gages may have very
dissimilar readings. For modeling accuracy (or even more specifically, for a successful calibration
of SWMM), it is essential that rain gages be located within and adjacent to the catchment.
SWMM accounts for the spatial variability of rainfall by allowing the user to define any number
of Rain Gage objects along with their individual data sources, and assign any rain gage to a
particular SWMM Subcatchment object (i.e., land parcel) from which runoff is computed. If
multiple gages are available, this is a much better procedure than is the use of spatially averaged
(e.g., Thiessen weighted) data, because averaged data tend to have short-term time variations
removed (i.e., rainfall pulses are "lowered" and "spread out"). In general, if the rainfall is uniform
spatially, as might be expected from cyclonic (e.g., frontal) systems, these spatial considerations
are not as important. In making this judgment, the storm size and speed in relation to the total study
area size must be considered.
Storm movement can significantly affect hydrographs computed at the catchment outlet (Yen and
Chow, 1968; Surkan, 1974; James and Drake, 1980; James and Shtifter, 1981).When more than
one gage is available to apply to the simulation, it is possible to simulate moving storms, as rainfall
in one part of the basin may be different from rainfall in another part of the basin. Movement of a
storm in the downstream direction increases the hydrograph peak, while movement upstream tends
to level out the hydrograph (Surkan, 1974; James and Drake, 1980; James and Shtifter, 1981).
For detailed simulation of large cities, radar rainfall data are very useful. Commercial firms
specializing in provision of radar rainfall values may be able to place highly spatially and
temporally variable rainfall data into a time series format easily input to SWMM (e.g., Hoblit and
Curtis, 2002; Meeneghan et al., 2002,2003; Vallabhaneni, 2002). Radar data are spatially averaged
over uniform grid cells of 1 km2 or larger and therefore each cell would cover a number of runoff
subcatchments. In this case one could simply use a separate Rain Gage object for each grid cell
that overlaps the study area, and assign the nearest cell as the subcatchment's source of rainfall
data. A more sophisticated approach is to define a separate Rain Gage for each subcatchment along
with a weighting matrix W whose entries Wij represent the fraction of area from subcatchment /'
that is contained in grid cell j. Then at any time t the vector of subcatchment rainfalls It would
equal the vector of cell rainfall values Rt multiplied by the weighting matrix W These data for
each time period could be placed in a standard SWMM user-prepared rainfall file for direct use by
SWMM (see below).
31
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2.2 Precipitation Data Sources
2.2.7 User-Supplied Data
Many SWMM analyses will rely upon rainfall data supplied by the user, on the basis of
measurements made at the closest rain gages to the catchment, or on an assumed design storm,
either "real" (that is, derived from actual measurements) or "synthetic" (derived from an assumed
duration and temporal distribution). Construction of synthetic design storms is described in many
texts and manuals, e.g., Chow et al. (1988), King County Department of Public Works (1995),
Bedient et al. (2013); SWMM does not supply synthetic design storms automatically, since the
emphasis is more properly on use of measured data. Measured data may be from National Weather
Service (NWS) or Environment Canada sites, as described below, from local agencies (e.g.,
utilities), from special monitoring programs (e.g., by the USGS or at a university), or from several
other sources, even from home weather stations. Naturally, the quality of any data source should
be investigated.
User-supplied rainfall data are provided to SWMM using a Rain Gage object. The user specifies
the format in which the rainfall data were recorded (as intensity, volume, or cumulative volume),
the time interval associated with each rainfall reading (e.g., 15 minutes, 1 hour, etc.), the source of
the data (the name of a Time Series object or name of a Rainfall file), and the ID name of the
recording station or data source if a file is being used.
For rainfall time series, only periods with non-zero precipitation need be included in the series.
Using a Time Series object for user-supplied rainfall data makes sense for single-event or short
duration simulation periods where there are a limited number of Rain Gage objects. In fact it is
possible to create several different time series for a given rain gage in a SWMM project, where
each contains a different rainfall event to be analyzed. Then all one needs to do is select the
appropriate time series for the scenario of interest.
If a Rainfall file is used for user-supplied rainfall data then it must follow SWMM's standard user-
prepared format. Each line of the file contains the station ID, year, month, day, hour, minute, and
non-zero precipitation reading, each separated by one or more spaces. There is no need to include
time periods with zero readings. An excerpt from a sample user-prepared Rainfall data file might
look as follows (i.e., Station STA01 recorded 0.12 inches of rainfall between midnight and one am
on June 12, 2004):
32
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STA01 2004 6 12 00 00 0.12
STA01 2004 6 12 01 00 0.04
STA01 2004 6 22 16 00 0.07
Using a Rainfall file to provide precipitation data is more convenient when a long-term continuous
simulation is being made or when there are many rain gages in a project. Note that it is possible
for a single user-prepared Rainfall file to contain data from more than one recording station or
external data source as would be the case in the radar data example discussed previously.
SWMM's rainfall Time Series and user-prepared Rainfall files treat the data as "start-of-interval"
values, meaning that each rainfall intensity or depth is assumed to occur at the start of its associated
date/time value and last for a period of time equal to the gage's recording interval. Most rainfall
recording devices report their readings as "end-of-interval" values, meaning that the time stamp
associated with a rainfall value is for the end of the recording interval. If such data are being used
to populate a SWMM rainfall time series or user-prepared rainfall file then their date/time values
should be shifted back one recording interval to make them represent "start-of-interval" values
(e.g., for hourly rainfall, a reading with a time stamp of 10:00 am should be entered into the time
series or file as a 9:00 am value).
2.2.2 Data from Government Agencies
SWMM can also use rainfall data from files provided directly from US and Canadian government
agencies. The National Weather Service (NWS) makes available historical hourly precipitation
values (including water equivalent of snowfall depths) for about 5,500 observational stations
around the U.S., with the periods of record usually beginning in the late 1940s. Fifteen-minute
data are available for over 2,400 stations, with records typically beginning in the early 1970s. The
repository for U. S. weather data is the National Oceanic and Atmospheric Administration (NOAA)
National Climatic Data Center (NCDC), located in Asheville, North Carolina. Key access
information is provided below:
National Climatic Data Center
Climate Services Branch
151 Patton Avenue
Asheville, NC 28801
Telephone: 828-271-4800
Web: http://www.ncdc.noaa.gov/
33
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The NCDC digital data bases that house the precipitation data are designated as DSI-3240 for
hourly precipitation and DSI-3260 for 15-minute precipitation. NOAA's Climate Data Online
(CDO) service at http://www.ncdc.noaa.gov/cdo-web provides free access to these archives in
addition to station history information. It features an interactive map application that helps locate
a recording station closest to a site of interest and allows one to request precipitation data for a
stipulated period of record. After a data request has been made through CDO the user receives an
email with a link to a web page where the data can be viewed with a web browser. The page can
then be saved to file for future use with SWMM.
When requesting data from CDO be sure to specify the TEXT format option and not the CSV
option so that SWMM can automatically recognize the file format and parse its contents. In
addition, select the QPCP precipitation option, not the QGAG option, for 15-minute precipitation
and make sure that the data flags are included.
Table 2.1 shows 15-minute precipitation data downloaded for station 410427 from Austin, Texas.
The column headings represent:
Station: cooperative recording station identifier.
Date: date and time at end of fifteen minute recording period.
QPCP: precipitation amount in hundredths of an inch (where 9999 or 99999
indicates a missing value).
Measurement if present, a flag that denotes either the start or end of an accumulation
Flag: period, a deleted period or a missing period.
Quality Flag: if present, a flag that indicates if the data value is erroneous.
Units: a flag indicating the precision of the recorded value where HI is for
hundredths and HT for tenths of an inch.
Hourly precipitation has a similar format except that the label 'HPCP' (for hourly precipitation)
replaces 'QPCP' and there is no Units column since the data precision is always HT. These data
sets only include periods with non-zero precipitation, use time stamps that mark the end of the
recording interval, and use a time of '00:00' to refer to midnight of the previous day. SWMM
recognizes these conventions, as well as missing value codes, when it reads a precipitation data
file.
34
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Table 2-1 15-minute precipitation data from NCDC Climate Data Online
STATION DATE QPCP Measurement Flag Quality Flag Units
COOP:410427 19970729 07:45 10 HT
COOP:410427 19970730 16:15 70 HT
COOP:410427 19970730 16:30 20 HT
COOP:410427 19970730 16:45 30 HT
COOP:410427 19970730 17:00 50 HT
COOP:410427 19970730 17:15 30 HT
COOP:410427 19970730 17:30 10 HT
COOP:410427 19970730 18:00 20 HT
COOP:410427 19970730 18:15 20 HT
COOP:410427 19970730 18:45 10 HT
COOP:410427 19970730 19:30 10 HT
COOP:410427 19970731 08:30 10 HT
The NOAA-NCDC web site also allows one to access the complete set of hourly and 15-minute
precipitation data for a particular station through an FTP server (see
http://www.ncdc.noaa.gov/cdo-web/datasets). For each station, there is one file that houses data
from 1948 (1971 for 15-minute data) to 1998 and then separate files for each year afterward. Each
line in these files contains one day's worth of precipitation data using the format shown in Table
2.2. Note that the third and fourth lines are "wrapped around" as a continuation of the long second
line. These are the same Austin, Texas data listed in Table 2.1 with the addition of an hour '2500'
entry on each line that contains the daily total. Also these files use hour '2400' to represent
midnight unlike hour '00:00' used in the Climate Data Online format.
Table 2-2 15-minute precipitation data in NCDC FTP file format
15M41042707QPCPHT19970700290020745 00010 2500 00010
15M41042707QPCPHT19970700300111615 00070 1630 00020 1645 00030 1700
00050 1715 00030 1730 00010 1800 00020 1815 00020 1845 00010 1930
00010 2500 00270
15M41042707QPCPHT19970700310020830 00010 2500 00010
Earlier online data formats used by NCDC can also be recognized by SWMM. Examples of these
formats, for the 15-minute Austin, Texas data, are shown in Tables 2.3 through 2.5. The formats
for hourly data are identical, except that FtPCP replaces QPCP and time stamps are always for
hours only.
35
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Long precipitation records are subject to meter malfunctions and missing data (for any reason).
The NWS has special codes for its DSI-3240 and DSI-3260 formats denoting these conditions.
They are explained in the NCDC documentation for each type. SWMM will note the number of
recording periods with missing data, often denoted with a 9999 in the rainfall column. Rainfall
time series used by the subcatchment object contain only good, non-zero precipitation data.
Table 2-3 15-minute precipitation data in comma-delimited format
COOPID ,
i
410427,
410427,
410427,
410427,
410427,
410427,
410427,
410427,
410427,
410427,
410427,
410427,
410427,
410427,
410427,
CD,
i
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
ELEM,
i
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
QPCP,
UN,
i
HT,
HT,
HT,
HT,
HT,
HT,
HT,
HT,
HT,
HT,
HT,
HT,
HT,
HT,
HT,
YEAR,
i
1997,
1997,
1997,
1997,
1997,
1997,
1997,
1997,
1997,
1997,
1997,
1997,
1997,
1997,
1997,
MO,
i
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
07,
DA,
i
29,
29,
30,
30,
30,
30,
30,
30,
30,
30,
30,
30,
30,
31,
31,
TIME,
i
0745,
2500,
1615,
1630,
1645,
1700,
1715,
1730,
1800,
1815,
1845,
1930,
2500,
0830,
2500,
VALUE , F , F
i i
00010, ,
00010, ,
00070, ,
00020, ,
00030, ,
00050, ,
00030, ,
00010, ,
00020, ,
00020, ,
00010, ,
00010, ,
00270, ,
00010, ,
00010, ,
36
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Table 2-4 15-minute precipitation data in space-delimited format
COOP ID
410427
410427
410427
410427
410427
410427
410427
410427
410427
410427
410427
410427
410427
410427
410427
CD
07
07
07
07
07
07
07
07
07
07
07
07
07
07
07
ELEM
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
QPCP
UN
HT
HT
HT
HT
HT
HT
HT
HT
HT
HT
HT
HT
HT
HT
HT
YEAR
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
MO
07
07
07
07
07
07
07
07
07
07
07
07
07
07
07
DA
29
29
30
30
30
30
30
30
30
30
30
30
30
31
31
TIME
0745
2500
1615
1630
1645
1700
1715
1730
1800
1815
1845
1930
2500
0830
2500
VALUE F F
00010
00010
00070
00020
00030
00050
00030
00010
00020
00020
00010
00010
00270
00010
00010
Table 2-5 15-minute precipitation data in fixed-length format
15M41042707QPCPHT19970700290020745 00010
15M41042707QPCPHT19970700290022500 00010
15M41042707QPCPHT19970700300111615 00070
15M41042707QPCPHT19970700300111630 00020
15M41042707QPCPHT19970700300111645 00030
15M41042707QPCPHT19970700300111700 00050
15M41042707QPCPHT19970700300111715 00030
15M41042707QPCPHT19970700300111730 00010
15M41042707QPCPHT19970700300111800 00020
15M41042707QPCPHT19970700300111815 00020
15M41042707QPCPHT19970700300111845 00010
15M41042707QPCPHT19970700300111930 00010
15M41042707QPCPHT19970700300112500 00270
15M41042707QPCPHT19970700310020830 00010
15M41042707QPCPHT19970700310022500 00010
37
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SWMM can also automatically recognize and read Canadian precipitation data that are stored in
climatologic files available from Environment Canada: (http://www.climate.weather.gc.ca).
SWMM accepts hourly data from HLY03 and HLY21 files and 15-minute data from FIF21 files:
(http://climate.weather.gc.ca/prods_servs/documentation_index_e.html). Tables 2-6 and 2-7
show the layout of the data records in these files, respectively. The "ELEM" field would contain
the code 123 for rainfall, the "S" field is for a numerical sign, the "VALUE" field has units of 0.1
mm, and the "F' and "FLG" fields are for data quality flags. SWMM makes the proper adjustment
from "end-of-interval" to "start-of-interval" when processing the Canadian precipitation files. As
of this writing, these files are only available through custom requests made to Environment Canada
for a fee.
Table 2-6 Record layout of Canadian HYLO and HLY21 hourly precipitation files
Daily Record of Hourly Data (HLY) - Length 186
I STN ID | YEAR | MO | DY | ELEM | S | VALUE | F |
l_l_l_l_l_l_l_l_l_l_l
These fields are repeated 24 times.
Table 2-7 Record layout of Canadian FIF21 15-minute precipitation files
Daily Record of 15 Minute Data (FIF) - Length 691
STN ID YEAR MO DY ELEM S VALUE F |FLG|
I I I I I I I I I I I I I I I
These fields are repeated 96 times.
When a SWMM rain gage object utilizes any of the standard NCDC or Canadian formatted files,
the only information required from the user is the name of file that contains the data and a station
ID. The latter need not be the same as the station ID referenced in the file. Other user-editable rain
gage properties, such as data format, interval, and units are overridden by the values associated
with the particular data file. SWMM will also convert the depth units used in the file to the user's
choice of unit system. For example, if an NCDC fifteen-minute rainfall file is used in a SWMM
project that employs SI metric units then SWMM knows that the file's data must first be converted
from tenths of an inch per fifteen minute period to mm/hr before they are used for any runoff
calculations.
38
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2.2.3 Rainfall Interface File
When precipitation data are supplied to SWMM from one or more external data files, the program
first collates the data from these files into a single binary formatted Rainfall Interface file. It is this
file that is accessed during the time steps of a SWMM simulation rather than the original rainfall
data files. The Rainfall Interface file can be saved to disk and re-used in subsequent runs should
the user care to do so. The layout of the interface file is as follows:
File stamp ("SWMM5-RAIN") (10 bytes)
Number of SWMM rain gages in file (4-byte integer)
For each rain gage:
recording station ID (80 bytes)
gage recording interval (seconds) (4-byte integer)
starting byte of rainfall data in file (4-byte integer)
ending byte+1 of rainfall data in file (4-byte integer)
For each rain gage:
For each time period with non-zero rainfall:
date/time for start of period (8-byte double)
rain depth (inches) (4-byte float)
The date/time value used here represents the number of decimal days from midnight of December
31, 1899 (i.e., the start of year 1900) expressed as a double precision floating point number. This
is the same representation that SWMM uses internally for all date/time values.
2.3 Temperature Data
SWMM requires representative air temperature data when simulating snow melt or when using the
Hargreaves method to compute potential evapotranspiration. A single set of time-dependent
temperatures is applied throughout the study area. These data can come either from a user-
generated time series or from a climate file. If a time series is used, then linear interpolation is used
to obtain temperature values for times that fall in between those recorded in the time series. The
first recorded temperature in the series is used for dates prior to the beginning date of the series
while the last recorded temperature is used for dates beyond the end of the series. Temperatures
should be in degrees F for SWMM projects built in US units or in degrees C for projects built in
metric units.
39
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A SWMM climate file contains values for minimum and maximum daily temperatures, (and
optionally, evaporation and wind speed). Three climate file formats are supported:
• the current NCDC GHCN-Daily Climate Data Online format
• the older NCDC DS3200 (aka TD-3200) format,
• Environment Canada's DLY daily climatologic file format, and
• a standard user-prepared format.
The National Climatic Data Center's Global Historical Climatology Network - Daily (GHCN-
Daily) dataset integrates daily climate observations from approximately 30 different data sources
for about 30,000 stations across the globe. As with precipitation data, NOAA's Climate Data
Online (CDO) service (http://www.ncdc.noaa.gov/cdo-web) provides free access to these archives.
When making an online request for data to be used with SWMM users should do the following:
• select the "Daily Summaries" dataset
• select a range of dates to retrieve data from
• use the interactive search feature to identify the recording station of interest
• select the "Custom GHCN-Daily Text" output format
• do not select any of the Station Detail and Data Flag options
• select the maximum (TMAX) and minimum (TMIN) air temperature data types
• select the average daily wind speed (AWND) and pan evaporation rate (EVAP) data types
if available and if so desired.
Some stations will offer 24-hour wind movement (WDMV) instead of average daily wind speed
which can be also be selected.
Table 2-8 shows the format of the data retrieved for Austin, Texas using the steps listed above.
Note that the pan evaporation has units of tenths of millimeters, temperatures are in tenths of a
degree Celsius, and 24-hour wind movement is in kilometers. (Had average daily wind speed
(AWND) been available it would have units of tenths of meters per second). Data fields with all
9's in them indicate missing values. SWMM automatically makes the necessary unit conversions
when reading this type of climate file.
The DS3200 (aka TD-3200) dataset was a predecessor to the GHCN that was discontinued in 2011.
SWMM is able to read data files in this older format, an example of which is shown in Table 2-9
for June 1997 for Austin, Texas. Each line of the file begins with "DLY" and contains daily data
for an entire month for a specific variable; hence the lines in the table are displayed in wrap around
fashion. Table 2-10 describes the format of the ID portion of each record while Table 2-11 does
the same for the data portion of the record.
40
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Table 2-8 Contents of an NCDC GHCN-Daily climate file
STATION
GHCND
GHCND
GHCND
GHCND
GHCND
GHCND
GHCND
:USC00410427
:USC00410427
:USC00410427
:USC00410427
:USC00410427
:USC00410427
:USC00410427
DATE
19970706
19970707
19970708
19970709
19970710
19970711
19970712
EVAP
13
15
10
18
61
30
41
TMAX
350
356
344
356
361
356
356
TMIN
228
233
239
217
222
222
222
WDMV
0
0
1
2
1
1
0
.7
.8
.0
.5
.9
.0
.8
Table 2-9 Contents of an NCDC DS3200 climate file
DLY41042707EVAPHI19970699990060319 00004 00419 00043 00519 00000
00619 00036 01919 00075 03019 00018 0
DLY41042707TMAX F19970699990300119 00086 00219 00091 00319 00091
00419 00091 00519 00089 00619 00088 00719 00083 00819 00087 00919
00088 01019 00087 01119 00090 01219 00091 01319 00092 01419 00093
01519 00094 01619 00092 01719 00093 01819 00094)N1919 00095 02019
00092 02119 00089 02219 00085 02319 00090 02419 00090 02519 00093
02619 00092 02719 00092 02819 00094 02919 00093 03019 00096 0
DLY41042707TMIN F19970699990330119 00067 00219 00055 00319 00062
00419 00063 00519 00069 00619 00068 00719 00063 00819 00067 00919
00066 01019 00068 01119 00069 01219 00072 01319 00079 01419 00077
01519 00076 01619 00074 01719 00075 01819 00070)N1919 00074 02019
00073 02119 00069 02219 00067 02319 00085 22319 00077)32419 00082
22419 00073 S2519 00089 22519 00069)N2619 00067 02719 00072 02819
00073 02919 00080 03019 00077 0
DLY41042707WDMV M19970699990300119 00027 00219 00025 00319 00017
00419 00016 00519 00022 00619 00022 00719 00018 00819 00016 00919
00020 01019 00050 01119 00022 01219 00018 01319 00053 01419 00039
01519 00037 01619 00005 01719 00051 01819 00079 01919 99999SS2019
00065A02119 00045 02219 00036 02319 00072 02419 00027 02519 00013
02619 00025 02719 00022 02819 00045 02919 00015 03019 00037 0
41
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Table 2-10 Layout of the ID portion of an NCDC DS3200 climate file record
Field
Record Type (always = DLY)
Station ID
Element Type. Possible types used by SWMM are:
TMAX = daily maximum temperature, deg. F
TMIN = daily minimum temperature, deg. F
EVAP = daily evaporation, in or 1/100 in
WDMV = daily wind movement, miles
Element Measurement Units Code
Year
Month
Filler (= 9999)
Number of data portions that follow
Width
3
8
4
2
4
2
4
3
Table 2-11 Layout of the data portion of an NCDC DS3200 climate file record
(Repeated as many times as needed to contain one month of data).
Field
Day of Month
Hour of Observation
Sign of Measured Value
Measured Value
Quality Control Flag 1
Quality Control Flag 2
Width
2
2
1
5
1
1
The record layout of the Canadian DLY daily climatologic files is depicted in Table 2-12. The
"ELEM" field contains 001 for daily maximum temperature and 002 for daily minimum
temperature, the "5" field is for a numerical sign, the "VALUE' field has units of 0.1 deg C, and
the "F" field is for a data quality flag. Note that only a single temperature file is used containing
records for both daily maximum and daily minimum temperatures. More information on how to
obtain these files from Environment Canada can be found at http://www.climate.weather.gc.ca.
42
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Table 2-12 Record layout of Canadian DLY daily climatologic files
Monthly Record of Daily Data (DLY) - Length 233
STN ID YEAR MO ELEM S VALUE
These two fields are repeated 31 times.
A user-prepared climate file is a plain text file where each line contains the following items, each
separated by one or more spaces:
• recording station name (no spaces allowed)
• 4-digit year,
• 2-digit month (Jan =1, Feb = 2, etc),
• day of the month,
• maximum temperature (deg F or C ),
• minimum temperature (deg F or C),
• evaporation rate (optional, in/day or mm/day),
• wind speed (optional, miles/hr or km/hr).
The units used for the various data items must be compatible with the unit system being used in
the SWMM project. For temperatures, this means degrees F for US units or degrees C for metric
units. If no data are available for a given item on a particular date, then an asterisk should be
entered as its value. Table 2-13 is an example of how the contents of the GHCN-Daily file of Table
2-1 would look in user-prepared format under US units.
Table 2-13 Example user-prepared climate file
410427
410427
410427
410427
410427
410427
410427
1997
1997
1997
1997
1997
1997
1997
07
07
07
07
07
07
07
06
07
08
09
10
11
12
95
96
93
96
97
96
96
.0
.1
.9
.1
.0
.1
.1
73.
73.
75.
71.
72.
72.
72.
0
9
0
1
0
0
0
0
0
0
0
0
0
0
.051
.059
.039
.071
.240
.118
.161
0.
0.
1.
2.
1.
1.
0.
7
8
0
5
9
0
8
Whenever a climate file is used in SWMM the user can specify a date, different from the simulation
starting date, where the program begins reading from. From this date on the daily values are read
from the file sequentially, without regard for what date the simulation clock is actually at. This
43
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feature is useful if one wants to use a rainfall file that covers one span of years and a climate file
that covers another. An error message is issued and the program terminates if this starting date
does not fall within the dates contained in the file. The same holds true if no file start date was
supplied and the simulation start date does not fall within the dates contained in the climate file.
When the simulation reaches a date that falls outside the last date in the file, then he program will
keep using the temperature values that were last read from the file. The same convention applies
whenever there is a gap of missing days or missing data in the file.
2.4 Continuous Temperature Records
When temperature data come from a climate file, a mechanism is needed to convert the daily max-
min readings into instantaneous values at any point in time during the day. To do this, the minimum
temperature is assumed to occur at sunrise each day, and the maximum is assumed to occur three
hours prior to sunset. This scheme obviously cannot account for many meteorological phenomena
that would create other temperature-time distributions but is apparently an appropriate one under
the circumstances. Given the max-min temperatures and their assumed hours of occurrence,
temperatures at any other time between these are found by sinusoidal interpolation, as sketched in
Figure 2-2. The interpolation is performed, using three different periods: 1) between the maximum
of the previous day and the minimum of the present, 2) between the minimum and maximum of
the present, and 3) between the maximum of the present and minimum of the following day.
SINUSOIDAL INTERPOLATION
MIDNIGHT
NOON
MIDNIGHT
Figure 2-1 Sinusoidal interpolation of hourly temperatures.
44
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The time of day of sunrise and sunset are easily obtained as a function of latitude and longitude
of the catchment and the date. Techniques for these computations are explained, for example, by
List (1966) and by the TVA (1972). Approximate (but sufficiently accurate) formulas used in
SWMM are given in the latter reference. (Snowmelt computations that utilize temperatures are
generally insensitive to these effects in SWMM.) Their use is explained briefly below.
The hour angle of the sun, //, is the angular distance between the instantaneous meridian of the sun
(i.e., the meridian through which passes a line from the center of the earth to the sun) and the
meridian of the observer (i.e., the meridian of the catchment). It may be measured in degrees or
radians or readily converted to hours, since 24 hours is equivalent to 360 degrees or 2 71 radians.
The hour angle is a function of latitude, declination of the earth, and time of day and is zero at
noon, true solar time, and positive in the afternoon. However, at sunrise and sunset, the solar
altitude of the sun (vertical angle of the sun measured from the earth's surface) is zero, and the
hour angle is computed only as a function of latitude and declination,
cos h = - tan 8 • tan (2-1)
where
h = hour angle at sunrise or sunset, radians,
8 = earth's declination, a function of season (date), radians, and
(f> = latitude of observer, radians.
The earth's declination is provided in tables (e.g., List, 1966), but for programming purposes an
approximate formula is used (TVA, 1972):
23.45 ?r\ r 2n "I
180 / 1.365 J
where D is number of the day of the year (no leap year correction is warranted) and 6 is in radians.
Having the latitude as an input parameter, the hour angle is thus computed in hours, positive for
sunset, negative for sunrise, as
h = (12/7r)cos~1(- tan 8 • tan
-------
astronomical origin and causes a correction that varies seasonally between approximately ±15
minutes; it is neglected here. The longitude correction accounts for the time difference due to the
separation of the meridian of the observer and the meridian of the standard time zone. These are
listed in Table 2-14. Note that time zone boundaries are very irregular and often are quite displaced
from what might be expected on the basis of the local longitude, e.g., most of Alaska is much
further west than the standard meridian for Alaska time of 135°W. The longitude correction is
readily computed as
minutes
(2'4)
where ATLONG = longitude correction, minutes (of time), 6= longitude of the observer, degrees,
and SM = standard meridian of the time zone, degrees, from Table 2-14.
Note that ATLONG can be either positive or negative, and the sign should be retained. For instance,
Boston at approximately 71°W has ATLONG = -16 minutes, meaning that mean solar noon precedes
EST noon by 16 minutes. (Mean solar time differs from true solar time by the neglected "equation
of time.")
The time of day of sunrise is then
HSR = 12-h + &TLONG/60 (2-5)
and the time of day of sunset is
(2-6)
From these times, the hours at which the minimum (Tmin) and maximum (Tmax) temperatures
occur are Hmm = HSR and Hmax = Hss - 3, respectively.
46
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Table 2-14 Time zones and standard meridians (degrees west longitude)
Time Zone
Newfoundland Std. Time
Atlantic Std. Time
Eastern Std. Time
Central Std. Time
Mountain Std. Time
Pacific Std. Time
Alaska Std. Time
Aleutian Std. Time
Hawaiian Std. Time
Example Cities
St. Johns' s, Newfoundland
Halifax, Nova Scotia
San Juan, Puerto Rico
New York, New York
Toronto, Ontario
Chicago, Illinois
Winnipeg, Manitoba
Saskatoon, Saskatchewan13
Denver, Colorado
Edmonton, Alberta
San Francisco, California
Vancouver, British Columbia
Whitehorse, Yukon
Anchorage, Alaska
Atka, Alaska
Honolulu, Hawaii
Standard Meridian
52.5a
60
75
90
105
120
135
150
aThe time zone of the island of Newfoundland is offset one half hour from other zones.
bSaskatchewan summer time is Mountain, winter is Central.
The temperature Tat any hour H of the day can now be computed as follows:
1. If H < Hmn then
T — T • 4- '•
1 l min ^
AT;
sin
n(Hmin-H)
+ 24 — Hm
(2-7)
where ATi is the difference between the previous day's maximum temperature and the current
day's minimum temperature.
2. lfHmm
-------
AT1
(2-8)
where ravg is the average of Tmin and rmtt)c, /iris the difference between Tmax and rOT«, and Havg
is the average oiHmin and //mar.
3. If H>Hmax then
• / n(H-Hmax) \
^min + ^4 - Hmax
(2-9)
2.5 Evaporation Data
Evaporation can occur in SWMM for standing water on subcatchment surfaces, for subsurface
water in groundwater aquifers, for water flowing in open channels, for water held in storage units,
and for water held in low impact development controls (e.g., green roofs, rain gardens, etc.). Single
event simulations are usually insensitive to the evaporation rate, but evaporation can make up a
significant component of the water budget during continuous simulation. SWMM allows
evaporation rates to be stated as:
• a single constant value,
• a set of monthly average values,
• a user-defined time series of daily values,
• daily values read from an external climate file,
• daily values computed from the daily temperatures in an external climate file.
Monthly and seasonal averages for evaporation are available in NOAA (1974) and Farnsworth and
Thompson (1982). Another source of evaporation and evapotranspiration data in the U.S. is the
AgriMet program of the U.S. Bureau of Reclamation:
(http://www.usbr.gov/pn/agrimet/proginfo.html).
However, AgriMet is aimed primarily at agricultural use, containing information on crop water
use requirements, for instance. Generally, local evaporation data are difficult to obtain.
Fortunately, totals are likely to represent large spatial areas more so than for precipitation. State
climate agencies are often useful when searching for weather data. For instance, the Oregon
48
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Climate Service (http://www.PCS.orst.edu) includes daily pan evaporation data among its weather
archives, and links are provided to other climate agencies regionally and nationwide.
The climate file source of evaporation data is the same climate file used to supply daily max-min
temperatures that was described in section 2.3. For NCDC GHCN-Daily files one would request
that records for the element EVAP be included in the file while for the Canadian DLY files one
would do the same for daily pan evaporation (element code 151). For the user-supplied climate
file, one simply adds an evaporation rate value after the daily minimum temperature entry in each
record. If the file were only being used to supply evaporation and not temperatures one still has
to enter asterisks (*) in the max and min temperature fields so that the file is read correctly.
Note that both the NCDC and Canadian DLY files report pan evaporation while SWMM expects
actual evaporation. SWMM will accept a set of monthly pan coefficients, typically on the order of
0.7, used to convert pan evaporation to actual evaporation (Chow et al., 1988; Bedient et al., 2013).
Also SWMM will automatically convert the units used for evaporation in these files into the ft/sec
units used internally by SWMM. For all other data sources, the evaporation rate values must be in
the same unit system as the rest of the data in a project. For US standard units this is inches/day
while for SI metric units it is mm/day.
SWMM can also use the Hargreaves method (Hargreaves and Samani, 1985) to compute
evaporation rates from the daily max-min temperatures contained in a climate file and the study
area's latitude. The governing equation is:
E = 0.0023(/?a/l)rr1/2(ra + 17.8) (2-10)
where:
E = evaporation rate (mm/day)
Ra = water equivalent of incoming extraterrestrial radiation (MJm^d"1)
Tr = average daily temperature range for a period of days (deg C)
Ta = average daily temperature for a period of days (deg C)
X = latent heat of vaporization (MJkg"1)
= 2.50-0.002361Ta
As noted in Hargreaves and Merkley (1998), for the equation to provide satisfactory results Tr and
Ta must be averaged over a period of 5 or more days. SWMM therefore uses a 7-day running
average of these variables derived from the record of daily max-min temperatures. The
extraterrestrial radiation Ra is computed as:
49
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Ra = 37.6dr(wssin(cp) sin(6) + cos(cp) cos(6) sin(ws)) (2-11)
where:
dr = relative earth-sun distance
= 1 + 0.033cos
J = Julian day (1 to 365)
ws = sunset hour angle (radians)
= cos~1(— tan (p tan <5)
#> = latitude (radians)
S = solar declination (radians)
/2?r(284+/)
365
2.6 Wind Speed Data
SWMM uses wind speed to refine the calculation of a melting rate for accumulated snow during
times when there is precipitation in the form of rainfall (see Section 6.3.2). There are two options
for providing wind speed data to SWMM:
• as an average value for each month of the year (January - December)
• from the same climate file used to supply daily max-min temperature and evaporation.
For the first option the same monthly average applies no matter which year is being simulated.
The wind speed units are miles/hour for US units or km/hour for metric units. The default monthly
values are all 0. The NCDC has compiled average monthly wind speeds for various locations
throughout the US which can be found at:
http://www.ncdc.noaa.gov/sites/default/files/attachments/windl996.pdf
For the NCDC GHCN-Daily climate file, one can request that records for the average daily wind
speed data element AWND or the 24-hour wind movement data element WDMV, whichever is
available, be included in the file. For the user-supplied file, wind speed is added after the field for
evaporation in each daily record (remember to place a * in the evaporation field if evaporation data
is being supplied from some other source). SWMM automatically converts the units used for wind
speed by the NCDC file, but for the user-supplied file they must be in miles/hour for US unit
system data sets or in km/hour for metric data sets. The Canadian DLY file does not report daily
wind speed.
50
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Chapter 3 - Surface Runoff
3.1 Introduction
This chapter describes how SWMM converts precipitation excess (rainfall and/or snowmelt less
infiltration, evaporation, and initial abstraction) into surface runoff (overland flow). Because
SWMM is a distributed model it allows a study area to be subdivided into any number of
irregularly shaped subcatchment areas to best capture the effect that spatial variability in
topography, drainage pathways, land cover, and soil characteristics have on runoff generation.
Generation of runoff is therefore computed on a subcatchment by subcatchment basis.
SWMM uses a nonlinear reservoir model to estimate surface runoff produced by rainfall over a
subcatchment. The model was first published by Chen and Shubinski (1971) and included in the
original release of SWMM (Metcalf and Eddy et al., 1971a). Discussions of ancillary processes
that serve as components of the runoff model, such as infiltration and snowmelt, are covered
elsewhere in this manual.
3.2 Governing Equations
SWMM conceptualizes a subcatchment as a rectangular surface that has a uniform slope S and a
width Wthat drains to a single outlet channel as shown in Figure 3-1. Overland flow is generated
by modeling the subcatchment as a nonlinear reservoir, as sketched in Figure 3-2.
Figure 3-1 Idealized representation of a subcatchment.
51
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Precipitation
Evaporation
^
^
d
, t>
J V i
is
A
I
Runoff
Infiltration
Figure 3-2 Nonlinear reservoir model of a subcatchment.
In this representation, the subcatchment experiences inflow from precipitation (rainfall and
snowmelt) and losses from evaporation and infiltration. The net excess ponds atop the
subcatchment surface to a depth d. Ponded water above the depression storage depth ds can become
runoff outflow q. Depression storage accounts for initial rainfall abstractions such as surface
ponding, interception by flat roofs and vegetation, and surface wetting.
From conservation of mass, the net change in depth d per unit of time t is simply the difference
between inflow and outflow rates over the subcatchment:
dd
(3-1)
where:
i = rate of rainfall + snowmelt (ft/s)
e = surface evaporation rate (ft/s)
/ = infiltration rate (ft/s)
q = runoff rate (ft/s).
Note that the fluxes /', e, f, and q are expressed as flow rates per unit area (cfs/ft = ft/s).
Assuming that flow across the subcatchment's surface behaves as if it were uniform flow within a
rectangular channel of width W (ft), height d-ds, and slope S, the Manning equation can be used
to express the runoffs volumetric flow rate Q (cfs) as:
52
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1 49
Q = — SV2R2X'*AX (3-2)
Here n is a surface roughness coefficient, S the apparent or average slope of the subcatchment
(ft/ft), Ax the area across the subcatchment' s width through which the runoff flows (ft2), and Rx is
the hydraulic radius associated with this area (ft). Referring to Figures 3-1 and 3-2, Ax is a
rectangular area with width W and height d-ds. Because fFwill always be much larger than d\i
follows that Ax = W(d — ds) and Rx = d — ds. Substituting these expressions into Equation 3-2
gives:
149
Q = — VKS1/2^ - d5)5/3 (3-3)
To obtain a runoff flow rate per unit of surface area, q, Equation 3-3 is divided by the surface area
of the subcatchment, A (which should not be confused with the cross-section area^4x through which
the runoff passes):
1.49 VKS1/2 _., ,. ..
q= (rf~rf5)5/3 (3-4)
Substituting this equation into the original mass balance relation 3-1 results in:
d-^ = i-e-f-cc(d-ds}^ (3-5)
where a is defined as:
1.49VKS1/2
a =
An
(3-6)
Equation 3-5 is an ordinary nonlinear differential equation. For known values of/', e,f, ds and ait
can be solved numerically over each time step for ponded depth d. Once d is known, values of the
runoff rate q can be found from Equation 3-4. Note that Equation 3-5 only applies when Jis greater
than ds. When d<=ds, runoff g is zero and the mass balance on d becomes simply:
.
- = t-e-/ (3-7)
53
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3.3 Subcatchment Partitioning
The equation used to generate surface runoff was developed on the basis of an idealized rectangular
subcatchment area with uniform properties. Urban areas usually contain a mix of land surface
types which can conveniently be divided into two primary categories: pervious surfaces (e.g.,
lawns, fields, and forested areas) which allow rainfall to infiltrate into the soil and impervious
surfaces (e.g., roofs, roads, and parking lots) over which no infiltration occurs. Therefore SWMM
allows each subcatchment to have both a pervious and impervious subarea over which Equation
3-5 is solved. The user-supplied parameter Percent Imperviousness determines how much of the
total subcatchment is devoted to each type of surface.
In addition, it is not uncommon for impervious surfaces to begin generating runoff almost
immediately after a rainfall event occurs, well before its depression storage depth fills up. To
model this behavior, SWMM allows the impervious area of a subcatchment to be further divided
into two subareas: one with depression storage and one without. The input parameter % Zero-
Imperv determines what fraction of a subcatchment's impervious area has no depression storage.
Thus overall, a subcatchment can contain three types of subareas as shown in Figure 3-3. Note that
under these definitions all impervious area is directly connected to the subcatchment's outlet point
(typically a drainage pipe or channel). How to model indirectly connected areas, such as roof drains
that discharge to pervious lawn areas, is discussed in section 3.6 below.
Pervious
Impervious
Impervious wo
Depiession
Storage
Figure 3-3 Types of subareas within a subcatchment.
Conceptually, these three sub-areas are incorporated into the idealized subcatchment as shown in
Figure 3-4. Of course in reality the areas will not align in this fashion nor will they necessarily be
compact and connected. The arrangement used here is merely a modeling convenience. Symbols
Al, A2, and A3 refer to the pervious subarea and two types of impervious subareas (with and
without depression storage), respectively, and they discharge their runoff independently of one
another to the same outlet location.
54
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Width
To inlet node of
pipe or channel
A3:
pervious area
impervious area
w/depression storage
impervious area w/o
depression storage
Figure 3-4 Idealized subcatchment partitioning for overland flow.
With this refinement the governing differential equation 3-5 for subcatchment runoff is solved
individually for each subarea. Thus a separate accounting of the ponded depth d over each subarea
is maintained. At the end of each time step, the runoff flows from each subarea are combined
together to determine a total runoff flow for the entire subcatchment. The following conventions
apply when solving the runoff equation for each subarea individually:
• The same precipitation and evaporation rate applies to each subarea.
• The contribution from snowmelt will vary by subarea. See Chapter 6 for details.
• The infiltration rate/is always zero for the two impervious subareas.
• Different values of depression storage ds can be assigned to the pervious (Al) and
impervious area (A2), where by definition ds is zero for the impervious area with no
depression storage (A3).
• Different values of the Manning roughness n can be used for the pervious (Al) and
impervious areas (A2 and A3).
• The same values of Wand S apply for all subareas.
55
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The applicable a-terms to be used in Equation 3-5 for each subarea are:
1.49VK51/2
for the pervious subarea Al (3-8)
1.49VK51/2
ap = - for both impervious subareas A2 and A3 (3-9)
(A2 + A3~)n,
where np is the roughness for the pervious area, ni is the roughness for both impervious areas, and
Ai is the surface area (ft2) associated with sub-area /'.
The reason that the same a applies to both impervious subareas even though their areas are
different arises from how the W/A term is evaluated for the idealized arrangement shown in Figure
3-4. For area A2, W2 = A2W / (A2 + A3) so that W2 / A2 = W/ (A2 + A3). For A3, W3 = A3W / (A2
+ A3) which results in W3 / A3 = W / (A2 + As). Thus both types of impervious areas use the same
factor W/ (A2 + A3).
3.4 Computational Scheme
The detailed computational scheme for computing the runoff generated from each subcatchment
within a study area over a single time step of a simulation is presented in the sidebar below.
56
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Computational Scheme for Runoff
1. If currently there is no precipitation, no snowmelt, and no runoff occurring within the
entire study area then set the current time step At equal to the user-specified dry time step.
Otherwise set it to the user-specified wet time step. If necessary, reduce the time step to
the next time at which either rainfall or evaporation changes. Guidance on time step
selection is provided in section 3.5.
2. For each subcatchment, retrieve its current precipitation rate /' and evaporation rate e from
the data sources described in Chapter 2.
3. For each subarea within each subcatchment:
a. If snow melt is being simulated, use the procedures described in Chapter 6 to
adjust the precipitation rate /' to reflect any snow accumulation (which decreases /')
or snow melt (which increases /').
b. Set the available moisture volume da to iAt + d where d is the current ponded
depth and limit the evaporation rate e to be no greater than d/At.
c. If the subarea is pervious, then determine the infiltration rate/using the methods
described in Chapter 4 and if groundwater is being simulated consider the possible
reduction in/that can occur due to fully saturated conditions (see Chapter 5).
Otherwise set/= 0.
d. If losses exceed the available moisture volume (i.e.,(e + /)At > da) then d = 0
and the runoff rate q is 0. Otherwise, compute the rainfall excess ix as:
ix = i-e-f
e. If the rainfall excess is not enough to fill the depression storage depth ds over the
time step (i.e.,d + i^At ^ ds) then update dio d + i^At and set q = 0. Otherwise
update d and q by solving Equation 3-5 as described below.
4. Compute the total runoff Q from the subcatchment at the end of the time step:
3
Q = Z qiAi
7 = 1
where qj is the runoff per unit area in subareay found in step 3 and Aj is the area of
subareay.
(Continued on next page)
57
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The solution of Equation 3-5 at step 3.e of this process proceeds as follows:
1. If ponded depth is currently below the depression storage depth (d < ds) and the rainfall
excess is positive then determine the time step Atx during which the depth will exceed ds:
Mx = At — (ds — d)/ix and set d = ds. Otherwise set Atx =At.
2. Use a standard fifth-order Runge-Kutta integration routine with adaptive step size control
(Press et al., 1992) to solve the equivalent of Equation 3-5,
dd 5/3
— = ix — ocdJ
dt x x
for d over the time step Atx. Here dx = d — ds for d > ds and is 0 otherwise while a is
ap (Equation 3-8) if the subarea is pervious or is ai (Equation 3-9) if the subarea is
impervious.
3. Compute the runoff per unit area q at the end of the time step: q = adx'3 where a and dx
are defined as above.
Recall that the depression storage ds can have different user-supplied values for subareas Al
(pervious) and A2 (impervious) while it is zero by definition for subarea A3. Also note that
initially at time zero the ponded depth don each subarea of each subcatchment is zero.
3.5 Time Step Considerations
SWMM allows the user to specify two different time steps that will be used when evaluating
surface runoff during a simulation: a "wet" step and a "dry" step. The wet time step is used when
there is precipitation or overland flow on any subcatchment within the study area. The longer dry
time step applies when there is both no precipitation input and all depression storage remains
unfilled.
Typically the wet time step will be an integer fraction of the rainfall interval. Five-minute rainfall
might have wet time steps of 1, 2.5 or 5.0 min, for example. If the wet time step is not an integer
fraction of or is larger than the rainfall interval, SWMM will automatically reduce the time step so
that the rainfall intensity remains constant over the adjusted time step. A smaller wet time step
would be desirable when the subcatchment is small and the time of concentration is a fraction of
the rainfall interval. When using 1-hour rainfall, wet time steps of 10 min, 15 min or longer can be
used by the model, unless subcatchments are very small. The key concept is that the wet step
should be less than or equal to the response time of a subcatchment. Time of concentration, tc, is
one measure of response time (Eagleson, 1970; Bedient et al., 2013); hence, the wet step should
58
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be no greater than tc. For subcatchments of a few to several acres, wet steps of 1 to 5 min or longer
should suffice. But for simulation of very small rain gardens or runoff from individual roofs onto
lawns, for instance, values less than 1 min might be necessary. The latter situation could be
encountered when simulating low-impact development (LID) options.
The dry time step is typically several hours or even days. It is used to update the infiltration
parameters, generate groundwater flow, and provide hydrograph continuity for inflow to channels
and conduits (i.e., for downstream flow objects) when there is no rainfall or standing water
anywhere on the study area. The dry time step may be hours to a day in wet climates and a day or
more in very dry climates.
Substantial time savings can be achieved with judicious usage of wet and dry time steps for longer
simulations. As an example consider the execution time saving using a wet step of 15 min and a
dry step of 1 day versus using a single time step of 1 hr for a year. Using Florida rainfall as input
(average annual rainfall between 50 and 60 in. [1250 to 1500 mm]) gives 300 wet hours per year,
flow for approximately 60 days per year, and 205 completely dry days per year. Assuming
overland flow only occurs when it is raining (an underestimate of wet time steps), this translates
to 300 x 4 = 1200 wet time steps, plus at least 60 transition (wet) time steps, plus 205 dry time
steps for a total of 1465. A constant hourly time step for one year requires 8760 time steps. This is
greater than a 500 percent savings in computer time with a better representation of the flow
hydrograph due to the 15 min wet time step.
A separate, usually much smaller time step is used in SWMM for hydraulic flow routing.
Typically, flow routing through channels and conduits requires a much shorter time step than for
overland flow, often down to a few seconds when using dynamic wave routing. SWMM will
linearly interpolate surface runoff hydrographs computed at longer time steps to obtain the inflows
at shorter time steps needed during flow routing.
3.6 Overland Flow Re-Routing
Huber (2001) extended SWMM's traditional surface runoff model to allow overland flow to be re-
routed in three different ways:
1. a specified fraction of the runoff from a subcatchment's impervious areas A2 and A3 can be
routed onto its pervious area Al,
2. a specified fraction of runoff from the pervious area Al can be routed onto the impervious area
with depression storage A2,
3. the total runoff from the subcatchment can be routed onto another subcatchment.
59
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The first of these schemes is illustrated in Figure 3-5.
To channel/pipe, inlet or
another subcatchment
Figure 3-5 Re-routing of overland flow (Huber, 2001).
For a given subcatchment, schemes 1 and 2 are mutually exclusive, while scheme 3 can be
combined with either 1 or 2 if desired. For internal re-routing, the fraction to be routed is a user-
specified input parameter. When flows are re-routed in this manner, the re-routed flow is
distributed uniformly over the downstream subarea or subcatchment, in the same manner as
rainfall. The flow is also delayed at least one time step longer than it would have been without this
extra routing.
The modified overland flow algorithm permits routing of flow from the impervious subarea over
the pervious subarea of the subcatchment, or vice versa. In the first instance, runoff from a rooftop
might flow over a lawn. In the second instance, runoff from a lawn might flow over a sidewalk.
This option is especially useful for simulation of "low impact development" (LID) practices
(Wright and Heaney, 2001; Wright et al., 2000; Lee, 2003).
By routing flow from one subcatchment to another subcatchment, buffer strips or riparian zones
may be simulated. Inflow to the downstream subcatchment is distributed uniformly over the
downstream subcatchment in the same manner as rainfall. This can be done because of the
nonlinear reservoir flow routing method in which there is no spatial variation through the
subcatchment. However, it also means that outflow from one subcatchment cannot be directed just
60
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to the pervious area of a downstream subcatchment that contains both pervious and impervious
sub-areas.
If such routing were desired, the downstream subcatchment should be separated into two: a
pervious subcatchment and an impervious subcatchment. There is no limit on the length of the
overland flow "chain" that can be assembled. Outflow from the most downstream subcatchment
will flow into a pipe or channel inlet (node), or directly to an outfall node, as usual.
To accommodate these options the computational scheme described in section 3.4 is modified as
follows:
1. For each subcatchment that receives runoff from one or more other subcatchments, the
precipitation rate /' for each of its subareas has Qr /A added to it, where Qr is the total runoff
(cfs) routed onto it from the contributing subcatchments, as computed at the end of the previous
time step, and^4 is the total surface area of the receiving subcatchment.
2. For subcatchments where a fraction/of impervious runoff is routed internally to the pervious
area, the precipitation rate /' for the pervious area has /((fe^z + ^s^sV^i added to it, where
qj is the runoff per unit area (ft/sec) from subareay at the end of the previous time step and Aj
is the area of subareay.
3. For subcatchments where a fraction/ of the pervious runoff is routed internally to the
impervious area with depression storage, the precipitation rate /' for the latter subarea has
qiA1/A2 added to it.
After the runoff from each of its subareas is computed, the total runoff reported for the
subcatchment is the flow that actually exits the subcatchment. For example, if 100% of the
impervious runoff was directed onto the pervious area, then the reported runoff for the
subcatchment would consist only of the computed runoff from the pervious area.
3.7 Subcatchment Discretization
Most study areas will require some level of discretization into multiple subcatchments in order to
properly characterize the spatial variability in overland drainage pathways, surface properties, and
connections into drainage pipes and channels. Discretization begins with the identification of
drainage boundaries (drainage divides) using a topographic map, the location of major sewer inlets
using a sewer system map, and the selection of channel/pipes to be simulated "downstream" in the
model. In an urban area, drainage divides based strictly on topography might not apply, since the
subsurface drainage network might transport water in a direction opposite to the surface gradient.
61
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Hence, drainage boundaries must be determined with the aid of both a topographic map and sewer
plans.
For instance, consider the Fisk B Catchment in Portland, Oregon, shown in Figure 3-6 (Portland
BES, 1996). The discretization relies upon both surface contours and invert slopes of the collection
sewers. Additional detail of Subcatchments 8412 and 9412 (highlighted in Figure 3-6) is shown in
Figure 3-7. The surface drainage in Subcatchment 9412 is to the south, but the pipe connecting
junctions 412 and 712 drains north! If only the surface contours were considered a quite different
catchment response to rainfall would result than what actually exists.
8112 Subcatchment #
SI2 I"
New Storm Sewer -,
Existing Combined Sewer Converted to Storm Only
—"" •"— Existing Combined Sewer
----- Subcatchment Boundary
Subcatchment Row Direction
FIGURE 18
Overview of Combined
and New Storm System
with Inlet and Subcatchment Numbers
Figure 3-6 Fisk B catchment, Portland, Oregon (Portland BES, 1996).
62
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SCALE =
0 50
L
412
8412
Manhole Number
Subcatchment Number
New Storm Sewer
Sample Subcatchments
Subcatchment Flow Direction'
FIGURE 16
Sample Subcatchments
with Node and Subcatchment Numbers
Figure 3-7 Detailed view of two Fisk B Subcatchments (Portland BES, 1996).
It is possible with SWMM to provide detail down to the parcel (individual lot) level, if desired and
to simulate virtually every drainage pipe or channel (e.g., Huber and Cannon, 2002). The amount
of detail actually required depends upon the purpose of the simulation. For screening purposes
with continuous simulation, a coarse discretization with a few or just one Subcatchment will
generally suffice, with one or no channel/pipes. On the other hand, if hydraulic conditions are
being studied within the catchment, enough detail in the drainage system and in the Subcatchments
that feed it must be provided. That is, obviously, a pipe must be simulated in order to study it, and
every channel or pipe must have a source of inflow (subcatchment or channel/pipe) at the upstream
end. The most upstream end of a series of channel/pipes must have a subcatchment draining to it
or it will remain dry (and useless) during the simulation. If the principal interest is in flow at the
outlet of the catchment, it is usually acceptable to provide minimal detail (e.g., few or one
subcatchment and one or no channel/pipes). The trade-off, however, is that the coarser the
schematization, the more decisions must be made on how to aggregate catchment properties.
For both single-event and continuous simulations, the amount of detail should be the minimum
consistent with requirements for within-catchment information. Obviously, no information can be
63
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obtained about upstream surcharging if the upstream conduits are not simulated and subcatchments
are not provided to feed them. In addition, sufficient detail needs to be provided to allow within-
system control options to be tried for different areas and land uses. If, however, the primary
objective is simply to produce a hydrograph and pollutograph at the outlet, using a single rain gage,
then one subcatchment will often (but not always) serve as well as many.
3.8 Parameter Estimates
3.8.1 Subcatchment Conceptualization
Each subcatchment is schematized as in Figure 3-4, in which three sub-areas Al, A2, and A3 are
used to represent different pervious and impervious surfaces. The slope of the idealized
subcatchment is in the direction perpendicular to the flow width. The normal option is for outflow
from each subarea to move directly to an inlet node of a drainage pipe or channel and not pass over
any other subarea. That is, the impervious area is assumed to be directly connected impervious
area (DCIA) or hydraulically effective impervious area. Rooftops or other surfaces that drain onto
adjacent pervious areas are not directly connected and, if the user wishes, runoff from such non-
DCIA surfaces may be directed to the pervious area of the subcatchment and vice versa. All sub-
areas are assumed to have the same width perpendicular to the overland flow path. If desired, any
subcatchment may consist entirely of any one (or more) types of the three subarea categories.
Actual subcatchments seldom exhibit the uniform rectangular geometries shown in Figure 3-4. In
terms of runoff generation, all geometrical properties are merely parameters (as explained below)
and no inherent "shape" can be assumed in the nonlinear reservoir technique. Parameter selection
is aided with reference to Figure 3-2 and Equation 3-5 in which the subcatchment "reservoir" is
shown in relation to inflows and outflows (or losses). Subcatchment outflow is a function of the
1 49VKS1/2
coefficient a = and the excess in ponded depth above depression storage. Note that the
relative area A, width W, slope S, and roughness n are combined into the single parameter a.
Equivalent changes in computed runoff may be caused by appropriate alteration of any of these
parameters. Note also that the width and slope are the same for both the pervious and impervious
subareas. Manning's roughness and relative area are the only parameters available to the modeler
to characterize the relative contributions of pervious and impervious areas to the outlet hydrograph.
(However, see further comments below on the subcatchment width.)
The following subsections discuss how values for subcatchment area, imperviousness, width,
slope, roughness, and depression storage can be assigned and the implications they entail.
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3.8.2 Subcatchment Area
In principle, the catchment and subcatchment area can be defined by constructing drainage divides
on topographic maps. In practice, this may or may not be easy because of the lack of detailed
contour information and the presence of unknown inflows and outflows. This may be most
noticeably brought to the modeler's attention when the measured runoff volume exceeds the
measured rainfall volume, if the latter is correct. Actual storm rainfall is seldom accurately
measured over all subcatchments.
From the modeling standpoint, there are no upper or lower bounds on subcatchment area.
Subcatchments are usually chosen to coincide with different land uses, with drainage divides, and
to ease parameter estimation, i.e., homogeneous slopes, soils, etc.
3.8.3 Imperviousness
The percent imperviousness of a subcatchment is another parameter that can, in principle, be
measured accurately from aerial photos or land use maps. In practice, unless impervious layers are
included in a GIS representation of the basin, such work tends to be tedious, and it is common to
make careful measurements for only a few representative areas and extrapolate to the rest. Runoff
volume and flow rates are strongly sensitive to estimates of imperviousness; hence, care should be
taken in imperviousness estimates.
One approach to estimating impervious area across large areas with multiple land uses is to
associate a percent impervious area with each category of land use. Then by knowing the
percentage of each land use within a subcatchment one can calculate its percentage impervious
area. Table 3-1 lists estimates of percent impervious area for different land uses taken from EPA's
Rouge River Project (Kluitenberg 1994) and incorporated into EPA technical guidance for MS4
stormwater permitting in Region I (US EPA, 2014).
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Table 3-1 Impervious area as a percentage of land use.
Land Use
Commercial
Industrial
High density residential
Medium density residential
Low density residential
Institutional
Agricultural
Forest
Open Urban Land
Percent Impervious Area
56
76
51
38
19
34
2
1.9
11
As mentioned earlier, impervious areas in SWMM are hydraulically (directly) connected to the
drainage system - called directly connected impervious areas (DCIA). For instance, if rooftops
drain onto adjacent pervious lawn areas, they should not be treated as a hydraulically effective
impervious area. Such areas are non-effective impervious areas (Doyle and Miller, 1980). On the
other hand, if a driveway drains to a street and then to a stormwater inlet, the driveway would be
considered hydraulically connected. Rooftops with downspouts connected directly to a sewer are
clearly hydraulically connected. An example of careful measurements and statistics on
imperviousness may be found in Field et al. (2000), Lee (2003), and Roy and Shuster (2007). Lee
and Heaney (2003) provide detailed comparisons of imperviousness computations and their
implications for modeling.
Should rooftops be treated as "pervious," the real surrounding pervious area is subject to more
incoming water than rainfall alone and thus might produce runoff sooner than if rainfall alone were
considered. In the possible event that this effect is important (a judgment based on infiltration
parameters) it can be modeled using the overland flow re-routing option discussed earlier in
Section 3.7. For example, if disconnected rooftops comprised 25 percent of the total impervious
area of a subcatchment (as opposed to the total DCIA) then one could tell SWMM that this
percentage of impervious area should be internally routed onto the pervious sub-area of the
subcatchment.
Another method of estimating the effective impervious area given measured data is to plot the
runoff (in. or mm) vs. rainfall (in. or mm) for small storms. The slope of the regression line is a
good estimate of the effective impervious area (Doyle and Miller, 1980).
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Southerland (2000) has proposed a series of regression equations relating effective impervious
area (EIA) to total impervious area (TIA) based on data from over 40 sub-basins collected by the
USGS in Oregon. Each equation has the form EIA = aTIAb where the coefficients a and b are
listed in Table 3-1. Further information on the concept of directly connected (or "hydraulically
effective") impervious areas is contained in the review article by Shuster et al. (2005).
Table 3-2 Coefficients for Southerland's EIA equations.
a
0.1
0.4
1.0
0.04
0.01
b
1.5
1.2
1.0
1.7
2.0
Condition
Average basins served by storm sewers and residential rooftops are not directly
connected to sewers.
Highly connected basins with residential rooftops directly connected to storm
sewers.
Totally connected basins that are completely served by storm sewers to which all
impervious surfaces are directly connected.
Partly disconnected basins where more 50% of the area is served by grassy swales
or roadside ditches instead of storm sewers and residential rooftops are not directly
connected to sewers.
Highly disconnected basins where only a small percentage of area is served by
storm sewers or has 70 percent or more draining to infiltration areas.
3.8.4 Subcatchment Width
If overland flow is visualized as running down-slope off of an idealized, rectangular catchment,
then the width of the subcatchment is the physical width of overland flow. This may be seen for
the idealized catchment shown once again in Figure 3-8 in which the lateral flow per unit width,
-------
w
Figure 3-8 Idealized representation of a subcatchment.
Because real subcatchments will not be rectangular with properties of symmetry and uniformity,
it is necessary to adopt other procedures to obtain the width for more general cases. This is of
special importance because if the slope and roughness are fixed (see Equation 3-4), the width can
be used to alter the hydrograph shape.
For example, consider the five different subcatchment shapes shown on Figure 3-9. Catchment
hydraulic properties, routing parameters are given in Table 3-3. Outflow hydrographs for
continuous rainfall and for rainfall of duration 20 min are shown on Figure 3-10. These were
computed using the nonlinear reservoir equation (Section 3.1) with a time step of 5 min. Clearly,
as the subcatchment width is narrowed (i.e., the outlet is constricted), the time to equilibrium
outflow increases. Thus, equilibrium is achieved quite rapidly for cases A and B and more slowly
for cases C, D and E.
Two routing effects may be observed. A storage effect is very noticeable, especially when
comparing hydrographs A and E for duration of 20 minutes. The subcatchment thus behaves in the
familiar manner of a reservoir. For case E, the outflow is constricted (narrow); hence, for the same
amount of inflow (rainfall) more water is stored and less released. For case A, on the other hand,
water is released rapidly and little is stored. Thus case A has both the fastest rising and recession
limbs of the hydrographs.
A shape effect is also evident. Theoretically, all the hydrographs peak simultaneously (at the
cessation of rainfall). However, a large width (e.g., case A) will cause equilibrium outflow to be
achieved rapidly, producing a flat-topped hydrograph for the remainder of the (constant) rainfall.
Thus, for a catchment schematized with several subcatchments and subject to variable rainfall,
increasing the widths tends to cause peak flows to occur sooner. In general, however, shifting
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hydrograph peaks in time is difficult to achieve through adjustment of subcatchment flow routing
parameters. The time distribution of runoff is by far most sensitive to the time distribution of
rainfall.
A
* '
B
1
1
D
T
I
W
^H
Figure 3-9 Rectangular subcatchments for illustration of shape and width effects.
Table 3-3 Data for example of effect of subcatchment width.
Slope = 1%
Imperviousness = 100%
Depression Storage = 0
n = 0.02
Equilibrium outflow = i*A = 0.926 cfs
At = 5 min = 300 sec
i* = Rainfall excess
= 1.0 in./hr = 0.000023148 ft/sec
Shape
A
B
C
D
E
A
(ft2)
40,000
40,000
40,000
40,000
40,000
W
(ft)
800
400
200
100
50
L
(ft)
50
100
200
400
800
69
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r i i 1 —T 1 1 1,1 1
EQUILIBRIUM OUTFLOW* 0.926 cfs
DURATION" 00
1.0
0.9
0.8
0,7
m 0.6
5°-4
u!«
0.2
0.1
0 5 10 15 20 25 30 35 40 ~45 50 55 60 65
TIME,mln
Figure 3-10 Subcatchment hydrographs for different shapes of Figure 3-9.
So what is the best estimate of subcatchment width? If the subcatchment has the appearance of
Figure 3-8, then the width is approximately twice the length of the main drainage channel through
the catchment. However, if the drainage channel is on the side of the catchment as in Figure 3-9,
the width is just the length of the channel. A good estimate for the width can be obtained by
determining the average maximum length of overland flow and dividing the area by this length.
For example, consider Subcatchment 8412 of the Fisk B Catchment, shown in Figure 3-7. The area
of Subcatchment 8412 is approximately 72,820 ft2 (1.67 ac). A crude estimate of the average
distance from the street to the drainage divide for overland flow is made by measuring the length
on the map ten times (Table 3-4). The street in the lower part of the subcatchment is divided into
six equal segments, approximately 57 ft in length. Distances to the boundary (drainage divide)
from the centerline of the street are then measured normal to the contours from each of the five
internal locations along the street:
The width is then estimated as W~ 72,820 7119 = 612 ft. Clearly, the average length estimate can
be improved with several additional measurements off the figure. But in practice, this may even
be done "by eye," since width is sometimes used as a calibration parameter. The distances are
measured to each side of the street under the assumptions that travel times along the street are
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much less than off the lots. This may not be true if roof drains are directly connected to the street
(unknown for this example).
Table 3-4 Width computations for Portland example.
North side of street to
boundary, ft
247
247
232
103
74
Sum:
South side of street to
boundary, ft
31
74
74
74
60
1,186ft; Average: 119ft
When assigning an overland flow path length, particularly for sites with natural land cover, one
must recognize that there is a maximum distance over which true sheet flow prevails. Beyond this,
runoff consolidates into rivulet flow with much faster travel times and less opportunity for
infiltration. There is no general agreement on what distance should be used as a maximum overland
flow path length. The Natural Resources Conservation Service recommends a maximum length of
100 ft (NRCS, 2010) while Denver's Urban Drainage and Flood Control District uses a maximum
of 500 ft. (UDFCD, 2007).
Another estimate for the width is twice the length of the main drainage channel, the street in this
instance. The street is approximately 360 ft long, which would give an estimate of about 720 ft for
the Subcatchment 8412. However, this estimate assumes approximately equal areas on both sides
of the drainage channel whereas most real subcatchments will be irregular in shape and have a
drainage channel that is off center, as in Figure 3-11. This is especially true of rural or undeveloped
catchments. A simple way of handling this case is given by DiGiano et al. (1977). A skew factor
may be computed,
where:
Z = Am/A
Z = skew factor, 0.5 < Z < 1,
Am = larger of the two areas on each side of the channel
A = total area.
(3-10)
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DIRECTION
OF OVERLAND
FLOW
MAIN
DRAINAGE
CHANNEL
Figure 3-11 Irregular subcatchment shape for width calculations (DiGiano et al., 1977, p.
165).
If L is the length of the main drainage channel then the width Wis simply weighted sum between
the two limits of L and 2L:
W = L + 2L(1-Z)
(3-11)
Applying this idea to Subcatchment 8412 of Figure 3-7, the area north of the street centerline is
approximately 1.19 ac, and the area of the street and south is approximately 0.48 ac. Hence,
7=1.19/1.67 = 0.71
and an estimate for the width is,
W= 360 + 2 • 360 • (1 - 0.71) = 567 ft
This estimate is not far from the estimate of roughly 610 ft obtained by dividing the area by the
average maximum flow length.
A more fundamental approach to estimating both subcatchment width and slope has recently been
developed by Guo and Urbonas (2007). The idea is to use "shape factors" to convert a natural
watershed as pictured in Figure 3-11 into the idealized overland flow plane of Figure 3-8. A shape
factor is an index that reflects how overland flows are collected in a watershed. The shape factor
X for the actual watershed is defined as A/L2 where A is the watershed area and L is the length of
the watershed's main drainage channel (not necessarily the length of overland flow). The shape
factor Y for the idealized watershed is W /L. Requiring that the areas of the actual and idealized
watersheds be the same and that the potential energy in terms of the vertical fall along the drainage
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channel be preserved, Guo and Urbonas (2007) derive the following expression for the shape factor
Y of the idealized watershed:
Y = 2X(1.5- Z}(2K-X}/(2K- 1) (3-12)
where K is an upper limit on the watershed shape factor. Guo and Urbonas (2007) recommend that
K be between 4 and 6 and note that a value of 4 is used by Denver's Urban Drainage and Flood
Control District. Once Y is determined, the equivalent width W for the idealized watershed is
computed as YL.
Applying this approach to Subcatchment 8412 (using K = 4) produces the following:
X = (1.67 acres • 43,560 ft2/acre) / (3602) = 0.56
Z = l.19/1.67 = 0.71
Y = (2 • 0.56) • (1.5 - 0.71) • (0.56 - 2-4) / (1 - 2-4) = 0.94
W = 360'0.94 = 338 ft.
This width value is considerably lower than those derived from direct estimates of either the
longest flow path length or the drainage channel length. As a result, it would most likely produce
a longer time to peak for the runoff hydrograph.
To reiterate, changing the subcatchment width changes the routing parameter a of Equation 3-5.
Thus, identical effects to those discussed above may be created by appropriate variation of the
roughness and/or slope.
3.5.5 Slope
The subcatchment slope should reflect the average slope along the pathway of overland flow to
inlet locations. For a simple geometry (e.g., Figures 3-8 and 3-9) the calculation is simply the
elevation difference divided by the length of flow. For more complex geometries, several overland
flow pathways may be delineated, their slopes determined, and a weighted slope computed using
a path-length weighted average. Such a procedure is described by DiGiano et al. (1977, pp. 101-
102).
Alternatively it may be sufficient to assume that overland flow occurs along what the user
considers to be the hydrological dominant slope for the conditions being simulated. One would
then choose the appropriate overland flow length, slope, and roughness for this equivalent plane.
The Guo and Urbonas (2007) Shape Factor approach discussed in the previous section computes
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the slope of this equivalent plane as S0L/(A/YL + YU) where S0 is the slope of the drainage
channel and the other variables are as defined in Section 3.8.4.
Finally, if there are clearly two different slopes to consider for the subcatchment, it may be
subdivided into two subcatchments and the overland flow re-routing option be used to route flow
from the upper subcatchment onto the lower subcatchment.
3.8.6 Manning's Roughness Coefficient, n
Values of Manning's roughness coefficient, w, are not as well known for overland flow as for
channel flow because of the considerable variability in landscape features, transitions between
laminar and turbulent flow, very small flow depths, etc. Most studies indicate that for a given
surface cover, n varies inversely in proportion to depth, discharge or Reynold's number. Such
studies may be consulted for guidance (e.g., Petryk and Bosmajian, 1975; Chen, 1976; Christensen,
1976; Graf and Chun, 1976; Turner et al., 1978; Emmett, 1978), or generalized values used (e.g.,
Chow, 1959; Crawford and Linsley, 1966; Huggins and Burney, 1982; French, 1985; Engman,
1986; Yen, 2001).
Roughness values used in the Stanford Watershed Model (Crawford and Linsley, 1966) are given
in Table 3-5 along with values from Engman (1986) and Yen (2001). Engman also provides values
for other agricultural land uses and a good literature review. There is no consensus among the three
sources of data in the table, reflecting the uncertainty in these estimates. However, recall the
discussion of Equation 3-5 in Section 3.8.1. For SWMM, it is common to fix estimates of slope
and Manning's n and calibrate with the subcatchment width.
3.8.7 Depression Storage
Depression (retention) storage (depth ds in Figure 3-2) is a volume that must be filled prior to the
occurrence of runoff on both pervious and impervious areas (Viessman and Lewis, 2003). It
represents a loss or "initial abstraction" caused by such phenomena as surface ponding, surface
wetting, interception and evaporation. In the SWMM rainfall-runoff algorithm (Section 3.1), water
stored as depression storage on pervious areas is subject to infiltration (and evaporation), so that
available storage capacity is continuously and rapidly replenished. Water stored in depression
storage on impervious areas is depleted only by evaporation and therefore it takes much longer to
restore such storage to its full capacity.
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Table 3-5 Estimates of Manning's roughness coefficient for overland flow
Source
Crawford and Linsley
(1966)a
Engman (1986)b
Yen (200 l)c
Ground Cover
Smooth asphalt
Asphalt of concrete paving
Packed clay
Light turf
Dense turf
Dense shrubbery and forest litter
Concrete or asphalt
Bare sand
Graveled surface
Bare clay-loam (eroded)
Range (natural)
Bluegrass sod
Short grass prairie
Bermuda grass
Smooth asphalt pavement
Smooth impervious surface
Tar and sand pavement
Concrete pavement
Rough impervious surface
Smooth bare packed soil
Moderate bare packed soil
Rough bare packed soil
Gravel soil
Mowed poor grass
Average grass, closely clipped sod
Pasture
Timberland
Dense grass
Shrubs and bushes
Business land use
Semi-business land use
Industrial land use
Dense residential land use
Suburban residential land use
Parks and lawns
n
0.01
0.014
0.03
0.20
0.35
0.4
0.011
0.010
0.02
0.02
0.13
0.45
0.15
0.41
0.012
0.013
0.014
0.017
0.019
0.021
0.030
0.038
0.032
0.038
0.050
0.055
0.090
0.090
0.120
0.022
0.035
0.035
0.040
0.055
0.075
Range
0.010-0.013
0.01-0.016
0.012-0.03
0.012-0.033
0.01-0.32
0.39-0.63
0.10-0.20
0.30-0.48
0.010-0.015
0.011-0.015
0.012-0.016
0.014-0.020
0.015-0.023
0.017-0.025
0.025-0.035
0.032-0.045
0.025-0.045
0.030-0.045
0.040-0.060
0.040-0.070
0.060-0.120
0.060-0.120
0.080-0.180
0.014-0.035
0.022-0.050
0.020-0.050
0.025-0.060
0.030-0.080
0.040-0.120
aObtained by calibration of Stanford Watershed Model.
bComputed by Engman (1986) by kinematic wave and storage analysis of measured
rainfall-runoff data.
'Computed on basis of kinematic wave analysis.
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Depression storage may be used to simulate interception, the storage of rainfall on vegetation.
Perhaps counter-intuitively, a tree, for instance, that intercepts rainfall can be simulated as an
impervious surface, with depression storage (interception), whose runoff is onto an adjacent or
underlying pervious surface. In this way, the interception capacity is regenerated only by
evaporation.
As described earlier, a percent"% Zero-Imperv" of the impervious area is assigned zero depression
storage in order to promote immediate runoff. Another option to achieve zero depression storage
on impervious areas (and thus immediate runoff) is to set % Zero-Imperv to zero, and enter zero
values for depression storage for the impervious area of each subcatchment, as desired.
Depression storage may be derived from rainfall-runoff data for impervious areas by plotting
runoff volume F(depth) as the ordinate against rainfall volume/1 as the abscissa for several storms.
The rainfall intercept at zero runoff is the depth of depression storage ds, i.e., a regression of the
form
V = C(P- ds) (3-13)
where C is a coefficient. This kind of analysis tends to work better for longer averaging periods
than individual storm events, but for storm events will work better for small, more impervious
catchments than for larger mixed catchments. The reason is that even for small rainfall amounts,
impervious surfaces (DCIA) will generate some runoff (one reason for the % Zero-Imperv
parameter). Hence, a depression storage value found as the intercept may be appropriate for a
longer term water balance than for simulation of hydrographs.
Data obtained in this manner from 18 urban European catchments (Falk and Niemczynowicz,
1978, Kidd, 1978a, Van den Berg, 1978) showed that depression storages ranged between 0.005
and 0.059 inches, depending on slope, with an average of 0.023 inches. Kidd (1978b) presented
the following regression for these data:
ds = 0.303S049 (3-14)
where ds is depression storage (inches) and S is catchment slope (percent).
Viessman and Lewis (2003, p. 140) present a linear relation between depression storage and slope
based on four small impervious areas near Baltimore, MD:
ds = 0.136 - 0.0325 (3-15)
where the observed values of pranged from 0.06 to 0.11 inches.
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Separate values of depression storage can be used for the pervious and impervious subareas within
a subcatchment. Representative values for the latter can probably be obtained from the European
data just discussed. Pervious area measurements are lacking; most reported values are derived from
successful simulation of measured runoff hydrographs. Although pervious area values are
expected to exceed those for impervious areas, it must be remembered that the infiltration loss,
often included as an initial abstraction in simpler models, is computed explicitly in SWMM.
Hence, pervious area depression storage might best be represented as an interception loss, based
on the type of surface vegetation. Many interception estimates are available for natural and
agricultural areas (Linsley et al., 1949; Maidment, 1993; Viessman and Lewis, 2003). For grassed
urban surfaces a value of 0.10 inches (2.5 mm) may be appropriate.
As mentioned earlier, several studies have determined depression storage values in order to
achieve successful modeling results. For instance, Hicks (1944) in Los Angeles used values of
0.20, 0.15 and 0.10 inches (5.1, 3.8, 2.5 mm) for sand, loam and clay soils, respectively, in the
urban area. Tholin and Keifer (1960) used values of 0.25 and 0.0625 inches (6.4 and 1.6 mm) for
pervious and impervious areas, respectively, for their Chicago hydrograph method. Brater (1968)
found a value of 0.2 inches (5.1 mm) for three basins in metropolitan Detroit. Miller and Viessman
(1972) give an initial abstraction (depression storage) of between 0.10 and 0.15 inches (2.5 and
3.8 mm) for four composite urban catchments. The American Society of Civil Engineers (1992)
suggests depression storage of 1/16 inch for impervious areas and 1/4 inch for pervious areas. The
Denver Urban Drainage and Flood Control District (UDFCD, 2007) recommends depression
storage losses of 0.1 inches for large paved areas and flat roofs, 0.05 inches for sloped roofs, 0.35
inches for lawn grass, and 0.4 inches for open fields.
In SWMM, depression storage may be treated as a calibration parameter, particularly to adjust
runoff volumes. If so, extensive preliminary work to obtain an accurate a priori value may be
unnecessary since the value will be changed during calibration anyway. Depression storage is
most sensitive for small storms; as the depth increases it becomes a smaller and smaller relative
component of the water budget.
3.8.8 Parameter Sensitivity
Sensitivity of surface runoff volume and peak flow estimates to key surface runoff parameters is
listed in Table 3-6. The influence of storm depth is not represented in the table.
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Table 3-6 Sensitivity of runoff volume and peak flow to surface runoff parameters.
Parameter
Area
Imperviousness
Width
Slope
Roughness
Depression
storage
Typical
effect on
hydrograph
Significant
Significant
Affects shape
Affects shape
Affects shape
Moderate
Effect of
increase
on runoff
volume
Increase
Increase
Decrease
Decrease
Increase
Decrease
Effect of
increase
on runoff
peak
Increase
Increase
Increase
Increase
Decrease
Decrease
Comments
Less effect for a highly porous
catchment
Less effect when pervious areas
have low infiltration capacity.
For storms of varying intensity,
increasing the width tends to
produce higher and earlier
hydrograph peaks, a generally
faster response. Only affects
volume to the extent that
reduced width on pervious areas
provides more time for
infiltration.
Same as for width, but less
sensitive, since flow is
proportional to square root of
slope.
Inverse effect as for width.
Significant effect only for low-
depth storms.
Losses (ET, depression storage, infiltration) are relatively less important as the storm depth
increases. That is, for flooding the land surface behaves more and more like an impervious surface,
which is one reason why urbanization has less impact on high-return period events than on
common events. If ground saturation is an important consideration, then the groundwater routines
(Chapter 5) might be invoked to allow the water table to rise to the surface, or the maximum
infiltration volume option used (Chapter 4). When calibrating for more common (lower depth)
events, depression storage becomes more important, especially as the storm depth drops to just a
few tenths of an inch. Calibration for small storms is often difficult, since depression storage is
difficult to estimate and dependent on initial conditions.
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3.9 Numerical Example
Earlier in section 3.8.4 a numerical example was presented showing the effect that the width
parameter had on the runoff hydrographs from a completely impervious subcatchment subjected
to constant rainfall intensity. This section presents a more realistic example that highlights the
difference in runoff responses between impervious and pervious subcatchments that are subjected
to the same design storm hyetograph. Table 3-7 lists the parameters used for each subcatchment.
Note that normally a single subcatchment could be used to contain both of these sub-areas, but
they are represented here as separate subcatchments so that the runoff from each can be compared
more readily.
Table 3-7 Parameters used for illustrative runoff example
Item
Subcatchment
Horton Infiltration
Design Storm
Parameter
Area (acres)
Percent Impervious
Percent Slope
Width (ft)
Roughness
Depression Storage (in)
Percent with No Depression
Storage
Evaporation (in/hr)
Initial Capacity (in/hr)
Final Capacity (in/hr)
Decay Coefficient (hr"1)
Duration (hr)
Total Depth (in)
Time-to-Peak / Duration
Impervious
Subcatchment
5
100
0.5
140
0.01
0.05
25
0
N/A
N/A
N/A
6.0
2.0
0.375
Pervious
Subcatchment
5
0
0.5
140
0.1
0.05
0
0
1.2
0.1
2.0
6.0
2.0
0.375
The two subcatchments were given identical area, slope, width, and depression storage. The
roughness of the pervious subcatchment was made ten times higher than the impervious roughness
as reflected in Table 3-7. The infiltration parameters for the pervious area are representative of a
well-drained sandy loam soil. A description of the Horton infiltration method used in SWMM is
supplied in the next chapter. The design storm is a 6-hour, 2-inch event with a triangular-shaped
hyetograph.
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Figure 3-12 shows the runoff hydrographs that result for the example design event. Flow rates are
represented on a per unit area basis so that they can be compared against the rainfall intensities.
For the impervious area, runoff from the 25% of the area with no depression storage begins
immediately, while runoff from the remaining area is delayed by the available depression storage
at the start of the storm. After this storage is filled, the impervious runoff hydrograph follows that
of the storm hyetograph. About 97% of the rain that falls on the impervious area becomes runoff
and there is a slight reduction in peak runoff rate. For the pervious area there is no runoff at all for
the first 2 hours of the storm, as the depression storage and available infiltration capacity are
sufficient to capture all of the rainfall volume during this period. After this, the remaining
infiltration capacity is such that only 30% of the total storm volume becomes runoff. The peak
runoff rate is only one third of the peak rainfall rate. When taken together, the total hydrograph
(equal to half the sum of the two sub-area hydrographs, since flows are expressed per unit area)
reduces the peak storm intensity by 50% and the total storm volume by 64%.
o.oo
0:00
Rainfall
• Impervious
Pervious
•Total
2:00
4:00
Time (hours)
6:00
8:00
Figure 3-12 Runoff results for illustrative example.
3.10 Approximating Other Runoff Methods
To varying degrees it is possible to have the results of SWMM's runoff computations approximate
those obtained from other well known methods. The following sub-sections describe how to do
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this for the runoff coefficient method, the SCS Curve Number method, and the unit hydrograph
method.
3.10.1 Runoff Coefficient Method
This method is sometimes used in preliminary screening-level models to generate runoff flows
from long-term rainfall records or rainfall probability distributions with a minimum of site-specific
data required (see STORM (Corps of Engineers, 1977); NetSTORM (Heineman, 2004); Adams
and Papa, 2000). It computes runoff Q (cfs) after all depression storage has been filled as:
Q = CiA (3-16)
where C is a runoff coefficient, /' is the rainfall rate (ft/s), and A is the subcatchment area (ft2). If
infiltration over the pervious area is considered then
Q = [Ci + (1 - C)max(0, i - f)]A (3-17)
where/is a constant infiltration rate (ft/s) and C can be interpreted as the fraction of impervious
area. Values of C have been tabulated for various types of land uses (see ASCE, 1992 or UDFCD,
2007).
To implement this approach in SWMM one could do the following:
1. Set the subcatchment's percent imperviousness to 100C and its percent of imperviousness with
no depression storage to 0.
2. Assign the same depression storage depth to both the pervious and impervious areas.
3. Use any values for slope and width, and 0 for both the pervious and impervious Manning's n.
4. Use the Horton infiltration option (discussed in Chapter 4) and let its maximum and minimum
infiltration rates be the same (either a very large value if 3-16 is being used or to/for 3-17).
Setting up a model in this fashion will produce exactly the same results as if Equations 3-16 or 3-
17 were implemented directly. When the Manning roughness n is 0, SWMM bypasses Equation
3-1 and simply converts all rainfall excess at each time step into instantaneous runoff.
Note that this method completely ignores any storage or delay that overland flow contributes to
the shape of a runoff hydrograph as well as the declining rate of infiltration that occurs over time.
It can, however, allow one to perform preliminary screening types of analyses relatively quickly
with a minimum of site data required.
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3.10.2 SCS Curve Number Method
The SCS (Soil Conservation Service, now known as the Natural Resources Conservation Service)
Curve Number method is a widely used procedure for computing runoff from single-event design
storms. As implemented in NRCS's TR-55 manual (NRCS, 1986) it consists of three separate
runoff-related computations: one computes total runoff volume for any given rainfall event while
the other two estimate a peak discharge and a runoff hydrograph for a synthetic 24-hour design
storm with a given return period. These latter two computations utilize a kinematic wave approach
to overland flow as well as a standard 24-hour design storm time distribution and are therefore
incompatible with SWMM's approach to generating runoff hydrographs. SWMM can however,
approximate the Curve Number method's estimate of total runoff volume from a subcatchment by
doing the following:
1 . Set the percent impervious area of the subcatchment to zero.
2. Select the Curve Number method for computing infiltration (see Chapter 4) and use the same
curve number that one would use with the SCS method.
3. Set the pervious area depression storage equal to the initial abstraction depth that one would
otherwise use with the SCS method.
4. Set the pervious area roughness coefficient to 0 to prevent any delay in runoff flow.
As an example, consider a residential area with a Curve Number of 80 subj ected to a uniform storm
of 4 inches over 4 hours. The SCS method for computing runoff volume (for US units) is:
(3-18)
P - la + S
where
- 10 (3-19)
uv
and
R = cumulative runoff volume (inches)
P = cumulative rainfall (inches)
la = initial abstraction (inches)
S = soil moisture storage capacity (inches)
CN = curve number.
Using the SCS recommended initial abstraction of 0.2S = 0.5 inches, the resulting SCS runoff
volume is 2.04 inches. Running SWMM for a single subcatchment in the manner prescribed above
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produces a total runoff volume of 1.98 inches. When a roughness of 0.1 (along with a width of 100
ft and slope of 0.5%) is used to allow SWMM to produce a more realistic runoff hydrograph, the
total runoff volume drops to 1.67 inches due to the increased time available for ponded water to
infiltrate as it flows over the surface.
3.10.3 Unit Hydrograph Method
A unit hydrograph (UH) is a linear transfer function used to convert a time series of rainfall excess
into a runoff hydrograph. A unit hydrograph can be derived from observed rainfall-runoff records
within a specific catchment or be chosen from a number of synthetic unit hydrographs that have
been developed over the years. The shapes of synthetic hydrographs have been parameterized with
respect to certain geographic and land cover variables. Specific examples include Snyder's UH,
Clark's UH, the Espey-Altman UH, the SCS (NRCS) Dimensionless UH, the SCS (NRCS)
Triangular UH, the Santa Barbara Urban Hydrograph, and the Colorado Urban Hydrograph (see
Nicklow et al, 2006 for further details). As an example, the SCS (NRCS) triangular UH is shown
in Figure 3-13. The parameters Qp and tp are functions of the catchment's time of concentration
and its area.
Flow
*! Time
Figure 3-13 SCS (NRCS) triangular unit hydrograph (NRCS, 2007).
SWMM normally uses a unit hydrograph approach to empirically model the process by which
rainfall causes subsurface inflow into leaky sewer pipes, otherwise known as rainfall dependent
inflow/infiltration (RDII). The details are described in Chapter 7. Any location within the drainage
system can have a set of RDII UH's assigned to it. Each set of UH's can consist of up to three
individual triangular UH's (like the one shown in Figure 3-13). One can therefore use an RDII-
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type analysis to replace SWMM's normal rainfall-runoff computational scheme by doing the
following:
1. For each subcatchment, define a triangular unit hydrograph that represents the subcatchment's
runoff response to rainfall (as opposed to inflow/infiltration into leaky sewer pipes as would
normally be the case). Assign the same rain gage to the unit hydrograph as one would otherwise
use for the subcatchment. The same unit hydrograph object can be used by multiple
subcatchments.
2. Specify the appropriate depression storage (i.e., initial abstraction) as part of each
subcatchment's unit hydrograph description.
3. If the SWMM data set already had subcatchments delineated in it, either delete them or create
a dummy rain gage that has no rainfall associated with it and assign this gage to all
subcatchments.
4. For each drainage system node that is the outflow point of a subcatchment, designate an
external RDII inflow for it that uses the subcatchment's unit hydrograph and its full area as the
sewershed area that contributes to RDII at the node.
After running SWMM with these modifications, the runoff hydrographs for each subcatchment are
equivalent to the Lateral Inflow results produced at each subcatchment's outlet node.
There are clearly some limitations to keep in mind when considering this approach. First, SWMM
can only utilize triangular shaped unit hydrographs. This might require some approximation if one
wishes to use one of the standard synthetic unit hydrographs whose shape is not triangular. Second,
any losses from infiltration must be taken into account when the unit hydrographs are constructed.
SWMM's RDII procedure does not account for the details of soil infiltration in the manner that
SWMM's normal runoff modeling does. Finally, in by-passing SWMM's normal runoff procedure
one also loses the ability to model other subcatchment-related phenomena, such as overland flow
re-routing as described in section 3.7 or pollutant buildup and washoff
3.10.4 Using Externally-Generated Runoff Data
Finally, it should be mentioned that it is possible to use any set of externally generated runoff data
to drive SWMM's flow and pollutant routing routines. This can be done by placing the runoff time
series data in a specially formatted Routing Interface file. This is a text file whose format is
described in the SWMM 5.0 Users Manual (EPA, 2013). An excerpt from such a file that supplies
runoff hydrographs to two nodes within a drainage network is reproduced in Table 3-8. A Routing
Interface file can be used in lieu of defining any subcatchments and rainfall data for a study area.
Or it can be used as a replacement for the runoff that would have been generated by SWMM for
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the subcatchments and rainfall data already defined for the study area. In this case SWMM' s Ignore
Rainfall/Runoff option must be invoked to prevent the program from adding any internally
computed runoff to that being provided by the interface file.
Table 3-8 Contents of a typical Routing Interface file
File Entry
Remarks
SWMM5
Example File
300
1
FLOW CFS
2
Nl
N2
Node Year Mon Day Hr Min Sec Flow
Nl 200204 01 0020 00 0.000000
N2 200204 01 0020 00 0.002549
Nl 200204 01 0025 00 0.000000
N2 200204 01 0025 00 0.002549
etc.
Required identifier
Data file description (can be blank)
Time step for all data ( seconds)
Number of variables provided by the file
Name and units of each variable (one per
line)
Number of nodes with inflow data
Name of each node (one per line)
Column headings for data to follow (can be
blank)
Node, year, month, day, hour, minute,
second, and value for each time step
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Chapter 4 - Infiltration
4.1 Introduction
Infiltration is the process by which rainfall penetrates the ground surface and fills the pores of the
underlying soil (Akan and Houghtalen, 2003). It often accounts for the largest portion of rainfall
losses over pervious areas. Theoretically, infiltration is governed by the Richards equation
(Richards, 1931) which requires that the relationship between soil permeability and pore water
tension as a function of soil moisture content be known. The difficulty in solving this highly
nonlinear partial differential equation makes it unsuitable for use in a general purpose model like
SWMM, especially for long-term continuous simulations. Engineers have developed several
simpler algebraic infiltration models that capture the general dependence of infiltration capacity
on soil characteristics and the volume of previously infiltrated water during the course of a storm
event. Because there is no universal agreement as to which model is best, SWMM allows the user
to choose from among four of the most widely used methods: Horton's method, a modified Horton
method, the Green-Ampt method, and the Curve Number method.
No matter which infiltration method is used, the parameters that define the method are highly
dependent on the type and condition of the soil being infiltrated. The NRCS (Natural Resources
Conservation Service, formerly the Soil Conservation Service or SCS) has classified most soils
into Hydrologic Soil Groups, A, B, C, and D, depending on their limiting infiltration capacities.
Well drained, sandy soils are "A"; poorly drained, clayey soils are "D," as described in Table 4-1.
Every soil in the United States has an A-D classification, or sometimes a dual classification, such
as B/D, meaning drained (artificially) and undrained (natural) condition.
The group assigned to specific types of soils and locations can be found by consulting:
• the Natural Resources Conservation Service's (NRCS) Field Office Technical Guide
• the NRCS Soil Data Access Web site: http://sdmdataaccess.nrcs.usda.gov/
• the Web Soil Survey Web site: http://websoilsurvey.nrcs.usda.gov/.
Additional soil characterization (physics and chemical) data are available at the aforementioned
web sites.
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Table 4-1 Hydrologic soil group meanings (NRCS, 2009, Chapter 7)
Group
Meaning
A
Low runoff potential. Soils having high infiltration rates even when thoroughly
wetted and consisting chiefly of deep, well to excessively drained sands or gravels.
B
Soils having moderate infiltration rates when thoroughly wetted and consisting
chiefly of moderately deep to deep, moderately well to well-drained soils with
moderately fine to moderately coarse textures. E.g., shallow loess, sandy loam.
Soils having slow infiltration rates when thoroughly wetted and consisting chiefly of
soils with a layer that impedes downward movement of water, or soils with
moderately fine to fine textures. E.g., clay loams, shallow sandy loam.
D
High runoff potential. Soils having very slow infiltration rates when thoroughly
wetted and consisting chiefly of clay soils with a high swelling potential, soils with
a permanent high water table, soils with a clay-pan or clay layer at or near the
surface, and shallow soils over nearly impervious material.
The best source of information about a particular soil type is the Soil Survey Interpretation,
available from a local NRCS office in the U.S. Data for soils in each county are often summarized
in a county soil survey document; the latter is often available in a local Soil and Water
Conservation District. Because printed versions of these documents are increasingly difficult to
obtain, on-line access is more likely (http://soils.usda.gov/survey/). Of particular interest is the
"Physical Properties" report that includes parameters of interest regarding infiltration. This report
may be downloaded for any soil, as illustrated in Figure 4-1. These data include saturated hydraulic
conductivity, for instance. Other potentially useful reports include:
• Water Features, including information such as hydrologic soil group (B for the Woodburn
Silt Loam), water table depth, and ponding frequency.
• RUSLE2 Related Attributes, with data for application of the Universal Soil Loss Equation.
• Engineering Properties, including soil horizon depths, soil classifications (USDA, Unified,
AASHTO), sieve analysis, liquid limit, and plasticity index.
In short, the NRCS provides an invaluable resource for information on soils and drainage of soils.
The agency's data are ever more valuable as they increasingly reside on-line on the Web.
87
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Physical Soil Properties
iEnlilH unce-- 'EruflDn Farore-"
data were • :: * : : •• .y* :;
nun^, Origan
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e sur-a:1; a.>.er. Atserce a" a-i entry i
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Index
17':
Wocdsuri
3-1T
1T-2S
re!
3-25
3-25
2-23
2-23
2-25
2-25
5-25
2C-B3
3C-53
6C-SC
6C-8C
60-TE
SC-^5
6D-T5
6D-T5
5D-T5
15-2i
1S-2S
20-35
23-35.
20-35
2O-35
15-3O
10-20
10-20
£ :c
1.24-1.45
I-- .Is
•C- = =
:c- ==
!=• .4=
.;=• .4=
;=• .4=
;=• ==
;;• ==
4JKM4.03
J 03-'4 33
' 4>-433
43- 4 J3
43- 4 33
43- 4 33
4 C 3- • 4 3D
4 KM2 33
4 C3-42 D3
0.19-O^i
0.1 9-0 .13
0.19-OJ1
C I3-C21
c ;.-: : i
L 13-Ci1
C 5-2.S
C 5-2.S
1XK5.S
c-s.s
= •55
1.S-5.3
= •5 =
CO-2.5
C C-2.5
1-S-4.S
OL3-1J)
: i-c =
: :-c4
0.1-OJ
EE
EE-
EE
.5'
.49
.55
.55
D. o-o.i
c :-:
Figure 4-1 Physical properties for Woodburn silt loam, Benton County, Oregon.
4.2 Horton's Method
Horton's method is empirical in nature and is perhaps the best known of the infiltration equations.
Many hydrologists have a "feel" for the best values of its three parameters despite the fact that
little published information is available. In its usual form it is applicable only to events for which
the rainfall intensity always exceeds the infiltration capacity; however, the modified form used in
SWMM is intended to overcome this limitation. The Horton method has been a part of SWMM
since the program was first released (Metcalf and Eddy et al., 197la).
4.2.1 Governing Equations
Horton (1933, 1940) proposed the following exponential equation to predict the reduction in
infiltration capacity over time as observed from field measurements:
f — f + (fn —
Jp — Joo i \JO
(4-1)
where:
fp =
fo =
t
kd =
infiltration capacity into soil (ft/sec)
minimum or equilibrium value offp (at t = oo) (ft/sec)
maximum or initial value offp (at t = 0) (ft/sec)
time from beginning of storm (sec)
decay coefficient (sec"1).
88
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Equation 4-1 is sketched in Figure 4-2 and can be derived theoretically from the Richards equation
under the proper set of assumptions (Eagleson, 1970). Note that actual infiltration will be the lesser
of actual rainfall and infiltration capacity:
/(t)=min[/p(t),i(t)]
(4-2)
where:
/ = actual infiltration into the soil (ft/sec)
/' = rainfall intensity (ft/sec).
Thus for the case illustrated in Figure 4-2 runoff would be intermittent.
f — f + (fn — f }e~kdt
Jp — Joo i \JO Joojt'
Typical Rainfall Hyetograph
Runoff (shaded areas)
Time (t)
Figure 4-2 The Horton infiltration curve.
Typical values for parameters/, and/» are usually greater than typical rainfall intensities. Thus,
when Equation 4-1 is used such that/, is a function of time only, the exponential term will cause
fp to decrease even if rainfall intensities are very light, as sketched in Figure 4-2. This results in a
reduction in infiltration capacity regardless of the actual amount of entry of water into the soil.
89
-------
To correct this problem, the integrated form of Horton's equation 4-1 is used in SWMM:
F(tp) = fpdt =
(4-3)
where F is the cumulative infiltration capacity at time tp in feet. This function is plotted in Figure
4-3 where it is assumed that actual infiltration has been equal tofp over all time t. As noted before,
there will in fact be times when infiltration/is less than/,, so that the true cumulative infiltration
will be:
F(t)= mm[fp,i]dT
(4-4)
fp(tP) -S—^-7^-,
tp tpi ti
Equivalent Time
Figure 4-3 Cumulative infiltration F as the area under the Horton curve.
Equations 4-3 and 4-4 can thus be used to define the time tp along the Horton curve at which the
next value offp can be found. That is, F is updated with the actual infiltration/over the current
time step and then the following equation, with tp as the only unknown, is solved:
90
-------
Kd
(4-5)
Once the new tp is known, the infiltration capacity/, for the next time step can be found from
Equation 4-1.
An additional optional parameter Fmax can be specified that limits the total volume of water that
can infiltrate the soil. When cumulative infiltration exceeds this value, saturation conditions exist,
and no more infiltration occurs; the land surface behaves as if it were impermeable. Thus F(t) in
Equation 4-4 is not allowed to exceed Fmax.
4.2.2 Recovery of Infiltration Capacity
For simulations that consist of multiple storm events over a set period of time, infiltration capacity
will be regenerated (recovered) during dry weather periods. With Horton's method, SWMM
performs this function whenever a subcatchment is dry - meaning it receives no precipitation and
has no ponded surface water - according to the hypothetical drying curve sketched in Figure 4-4:
f = f — (f — /oo)e~fer^~tw^ (4'6)
where:
kr = decay coefficient for the recovery curve (sec"1)
tw = hypothetical projected time at which/=/o on the recovery curve (sec).
New values of tp are then generated as indicated in Figure 4-4 as recovery proceeds. For example,
let tpr be the tp value at which recovery begins with/ as the corresponding infiltration capacity.
According to the recovery curve,
one can compute tw as:
(4-8)
91
-------
fo
fr
foo
f — f — (f — f ~\p-kr(t-tw~)
Jp — JO UO JcojV
tpl tw tpr twl
Equivalent Time
Figure 4-4 Regeneration (recovery) of infiltration capacity during dry time steps.
Then after a recovery time to twi = tpr+ At, the new infiltration capacity fi is found from:
(4-9)
Finally, the new equivalent time tpi on the infiltration curve from which the infiltration process
would re-start under a wet condition is:
1 , ffo ~ M
tpl=—Znl- -1
Kd VI ~~ Too'
(4-10)
These steps can be combined into the following equation:
(4-11)
92
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On succeeding time steps, tpi may be substituted for tpr, and tP2 substituted for tpi, etc. Note that
fp has reached its maximum value of/o when tp = 0.
Although this recovery method gives sensible results, it is somewhat unsatisfactory inasmuch as
there is no dependence of infiltration recovery on evapotranspiration (ET). Drying of the soil
through ET and deep infiltration should influence the recovery of infiltration capacity, but these
mechanisms are replaced in SWMM by the more empirical approach just discussed.
4.2.3 Computational Scheme
The detailed computational scheme for computing Horton infiltration for each subcatchment
within a study area over a single time step of a simulation is presented in the sidebar below.
Computational Scheme for Horton Infiltration
The following variables are assumed known at the start of each time step A^ (sec) for the
pervious subarea of each subcatchment:
/' = rainfall rate (ft/sec)
d = depth of ponded surface water (ft)
tp = equivalent time on the Horton curve (sec)
as are the following constants:
fo = maximum (or initial) infiltration capacity (ft/sec)
fx = minimum (or ultimate) infiltration capacity (ft/sec)
kd = infiltration capacity decay coefficient (sec"1)
kr = infiltration capacity recovery coefficient (sec"1)
Fmax = maximum infiltration volume possible (ft).
Initially at time 0, tp = 0.
The computational steps for computing the Horton infiltration rate/for a given subcatchment
over a single time step of a simulation proceed as follows:
(Continued on next page)
93
-------
1. Compute the available rainfall rate: ia = i + d/At.
2. If ia = 0, meaning the surface is dry, then update the current time on the Horton infiltration
curve tp as follows:
and set the infiltration rate/to 0.
3. Otherwise compute the cumulative infiltration volume from the integrated form of the
Horton curve at times tp and ti = tp +At(Fp and Fi, respectively) as follows:
a. If tp >= 16/kdthen tp is on the flat portion of the Horton curve so
FP = /ootp +^and F! = Fp + /0At.
Kd
b. Otherwise,
Fp=/Mtp +(1-6-*^) and
d
Limit both Fp and Fy to not exceed Fmax if a value for the latter was supplied.
4. Compute the average infiltration ratefp over the time step: fp = (Fx — Fp)/At.
5 . If £/ > .? <5/fc/ or fp
-------
4.2.4 Parameter Estimates
The parameters that a user must supply for each subcatchment for the Horton infiltration method
are:
fo - the maximum or initial infiltration capacity (in/hr or mm/hr),
/»- the minimum or equilibrium infiltration capacity (in/hr or mm/hr),
kd - the decay coefficient (hr"1),
kr - the regeneration coefficient (days'1), and, optionally,
Fmax - the maximum infiltration volume (in or mm).
Conversions between the user-supplied units of these parameters (such as in, mm or hr) and those
used internally (ft and sec) are handled automatically by the program.
Although the Horton equation is probably the best-known of the several infiltration equations
available, there is little to help the user select values of parameters fo and kd for a particular
application. (Fortunately, some guidance can be found for the value of/«.)- Since the actual values
of/o and kd (and often/to.) depend on the soil, vegetation, and initial moisture content, ideally these
parameters should be estimated using results from field infiltrometer tests for a number of sites of
the watershed and for a number of antecedent wetness conditions. An example of Horton
parameters for Georgia soils is given in Table 4-2 (Rawls et al., 1976). Horton's (1940) estimates
are shown in Table 4-3. Skaggs and Khaleel (1982) provide Horton-type decay curves on the basis
of theoretical estimates.
Table 4-2 Horton parameters for selected Georgia soils (Rawls et al., 1976)
Soil Type
Alpha loamy sand
Carnegie sandy loam
Cowarts loamy sand
Dothan loamy sand
Fuquay pebbly loamy sand
Leefield loamy sand
Robersdale loamy sand
Stilson loamy sand
Tooup sand
foo
in/hr
1.40
1.77
1.95
2.63
2.42
1.73
1.18
1.55
1.80
fo
in/hr
19.0
14.77
15.28
3.47
6.24
11.34
12.41
8.11
23.01
kd
hr1
38.29
19.64
10.65
1.40
4.70
7.70
21.75
6.55
32.71
95
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Table 4-3 Horton parameters provided by Horton (1940)
Soil and Cover
Standard agricultural (bare)
Standard agricultural (turfed)
Peat
Fine sandy clay (bare)
Fine sandy clay (turfed)
foo
in/hr
0.24-8.9
8.2-11.8
0.82-11.8
0.82-1.0
4.1-1.2
fo
in/hr
11.4
36.7
13.3
8.6
27.4
kd
hr1
96
48
108
120
84
If it is not possible to use field data to find estimates of/o, /», and kd for each subcatchment, the
following guidelines should be helpful. Often, NRCS data may be used directly. For instance, for
the two upper horizons (soil layers) of Woodburn silt loam (Figure 4-1), saturated hydraulic
conductivity is listed as 4 - 14 micrometers per second, or 0.6 - 2.0 in/hr (14 - 50 mm/hr).
Unfortunately, this wide range in values is commonly encountered among soil survey data.
Fortunately, the range also serves as a reminder that infiltration rates are notoriously variable in
space as well as in time and should not be considered "exact." Note that saturated hydraulic
conductivity is the more appropriate word for parameter Ks, also termed "permeability" on older
soil survey interpretation tables.
Minimum Infiltration Capacity (/»)
The Horton parameter /» is essentially equal to saturated hydraulic conductivity, Ks, that is,/o ~
Ks. The/o value is also the limiting infiltration rate when water is ponded on the surface, at low
depths. Generalized estimates for Ks will also be discussed in conjunction with the Green-Ampt
infiltration method later in this chapter and are the best source of values for/oin the absence of
site-specific data.
Alternatively, values for/o according to Musgrave (1955) are given in Table 4-4. To help select a
value within the range given for each soil group, the user should consider the texture of the layer
of least hydraulic conductivity in the profile. Depending on whether that layer is sand, loam, or
clay, the/oo value should be chosen near the top, middle, and bottom of the range respectively. For
example, the data sheet for Woodburn silt loam identifies it as being in Hydrologic Soil Group B,
which puts the estimate of/o into the range of 0.15 - 0.30 in/hr (3.8 -7.6 mm/hr), much lower than
the Ks value discussed above. Examination of the texture of the layers in the soil profile indicates
that they are silty in nature, suggesting that the estimate of the/o value should be in the low end
of the range, say 0.15 - 0.20 in/hr (3.8-5.1 mm/hr). A sensitivity test on the/o value will indicate
96
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the importance of this parameter to the overall result; in fact,/o is usually the most sensitive of the
three Horton curve parameters.
Table 4-4 Values of foo for Hydrologic Soil Groups (Musgrave, 1955)
Hydrologic Soil Group
A
B
C
D
foo
(in/hr)
0.45-0.30
0.30-0.15
0.15-0.05
0.05-0
Caution should be used in applying values from Table 4-4 to sandy soils (group A) since reported
Ks values are often much higher. For instance, sandy soils in Florida can have Ks values from 7 to
18 in/hr (180 - 450 mm/hr) (Carlisle et al., 1981). Unless the water table rises to the surface,
minimum infiltration capacity will be very high, and rainfall rates will almost always be less than
_/», leading to little or no overland flow from such soils.
Decay Coefficient (kd)
For any field infiltration test the rate of decrease (or "decay") of infiltration capacity from the
initial value depends on the initial moisture content. Thus the fo-value determined for the same
soil will vary from test to test. It is postulated here that, if/o is always specified in relation to a
particular soil moisture condition (e.g., dry), and for moisture contents other than this the time
scale is changed accordingly (i.e., time "zero" is adjusted to correspond with the constant fo), then
kd can be considered a constant for the soil independent of initial moisture content. Put another
way, this means that infiltration curves for the same soil, but different antecedent conditions, can
be made coincident if they are moved along the time axis. Butler (1957) makes a similar
assumption.
Values of kd found in the literature (Overton and Meadows, 1976; Wanielista, 1978; Maidment,
1993; ASCE, 1996) range from 0.67 to 120 hr"1. Nevertheless most of the values cited appear to
be in the range 3-6 hr"1. The evidence is not clear as to whether there is any relationship between
soil texture and the kd value although several published curves seem to indicate a lower value for
sandy soils. If no field data are available, an estimate of 4 hr"1 could be used. Use of such an
estimate implies that, under ponded conditions, the infiltration capacity will fall 98 percent of the
way towards its minimum value in the first hour, a not uncommon observation. Rates of decay of
infiltration for several values of kd are shown in Table 4-5.
97
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Table 4-5 Rate of decay of infiltration capacity for different values of kd
kd,
hr1
2
3
4
5
Percent of decline of infiltration
towards limiting value foo after
capacity
1 hour
76
95
98
99
Initial Infiltration Capacity (fo)
The initial infiltration capacity, fo depends primarily on soil type, initial moisture content, and
surface vegetation conditions. For example, Linsley et al. (1982) present data that show, for a
sandy loam soil, a 60 to 70 percent reduction in the/o value due to wet initial conditions. They
also show that lower fo values apply for a loam soil than for a sandy loam soil. As to the effect of
vegetation, Jens and McPherson (1964, pp. 20.20-20.38) list data that show that dense grass
vegetation nearly doubles the infiltration capacities over those measured for bare soil surfaces.
For the assumption to hold that the decay coefficient kd is independent of initial moisture content,
fo must be specified for the dry soil condition. For long-term continuous simulations SWMM
automatically adjusts the effective/o value as part of the infiltration capacity regeneration routine.
However, for a single-event simulation, the user must specify the/o value for the storm in question,
which may be less than the value for dry soil conditions.
Published values of fo vary depending on the soil, moisture, and vegetation conditions for the
particular test measurement. The fo values listed in Table 4-6 can be used as a rough guide.
Interpolation between the values may be required.
98
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Table 4-6 Representative values for fo
A. DRY soils (with little or no vegetation):
Sandy soils: 5 in/hr
Loam soils: 3 in/hr
Clay soils: 1 in/hr
B. DRY soils (with dense vegetation):
Multiply values given in A by 2 (after Jens and McPherson, 1964).
C. MOIST soils (change from dry/o value required for single event simulation only):
Soils which have drained but not dried out (i.e., field capacity): divide values from A
and B by 3.
Soils close to saturation: Choose value close to/o value.
Soils which have partially dried out: divide values from A and B by 1.5-2.5.
Regeneration Coefficient (kr}
For continuous simulation, infiltration capacity will be regenerated (recovered) during dry weather
according to Equation 4-6. Instead of asking the user to supply a value for kr, SWMM instead asks
for an estimate of drying time Tdry in days. This is the time it takes for a saturated soil to fully
recover to a dry state. Drying times are typically longer than wetting times, implying kr < kd. On
well-drained porous soils (e.g., medium to coarse sands), recovery of infiltration capacity is quite
rapid and could well be complete in a couple of days. For heavier soils, the recovery rate is likely
to be slower, say 7 to 14 days. The choice of the value can also be related to the interval between
a heavy storm and wilting of vegetation.
The Green-Ampt method (discussed below in Section 4.4), bases its recovery time solely on the
soil's saturated hydraulic conductivity Ks. Adopting its approach produces the following estimate
for Tdry in days:
3.125
Tdry = -=- (4-12)
where Ks is expressed in in/hr. Thus this equation predicts a drying time of 2 days for a sandy soil
with Ks = 2.0 in/hr versus 10 days for a clay soil with Ks of 0. 1 in/hr.
99
-------
Since mathematically, the exponential term in Equation 4-6 would require an infinite amount of
time to allow infiltration capacity to return to its initial value/o, SWMM considers "full recovery"
to occur when 98 percent of the difference between the initial and minimum capacities has been
achieved. Thus from Equation 4-6 (for kr in days"1),
0.02(/0 - /«,) = (/o - f^e-Wtry (4-13)
which leads to the following estimate of kr expressed in days"1:
-ln(0.02) 3.912
dry 1 dry
(4-14)
This computation of kr from a user-supplied value of Tdry and its subsequent conversion from days"
1 to sec"1 is done internally by SWMM.
4.3 Modified Horton Method
A. O. Akan developed a modified version of the Horton infiltration method (Akan, 1992; Akan
and Houghtalen, 2003) that has been added as a separate infiltration option in SWMM 5. The
method uses the same parameters as the original Horton method but instead of tracking the time
along the Horton decay curve it uses the cumulative infiltration volume in excess of the minimum
infiltration rate as its state variable. It assumes that part of the infiltrating water will percolate
deeper into the soil at the minimum infiltration rate (commonly taken as the soil's saturated
hydraulic conductivity). As a result, it is the difference between the actual and minimum
infiltration rates that accumulates just below the surface that causes infiltration capacity to decrease
with time. This method is purported to give more accurate infiltration estimates when low rainfall
intensities occur.
4.3.1 Governing Equations
The modified method starts with the same exponential decay equation as the original Horton
method:
/P=/oo + (/o-/oo)e-fcdt (4-15)
where all symbols have been previously defined.
100
-------
As with the original Horton method, the actual infiltration rate/is the smaller offp and the rainfall
rate /'. Integrating Equation 4-15 from 0 to time t produces the following equation for the
cumulative infiltration through time t:
F = /cot + (/°~/co) (1 - e-k«) (4-16)
Solving for e kdt from (4-15) and substituting into (4-16) gives:
F = fmt + f-^ (4-17)
Kd
and solving forfp gives:
fp=fo-kd(F-f00f) (4-18)
The last term in parenthesis is equivalent to/Q (/ — fm}dt. So one can approximate Eq. (4-18) by
fp=fo-kdFe (4-19)
where Fe = £i(/i — /oo)Atj and fa is the actual infiltration over a previous time interval Atj.
4.3.2 Recovery of Infiltration Capacity
Regarding recovery of infiltration capacity during dry periods, one can assume that the
instantaneous recovery rate is proportional to the difference between the current capacity and the
maximum capacity:
df /dt = k (f f 1 (4-20)
where fr represents the infiltration capacity during recovery and kr is the same regeneration
coefficient (I/sec) used in the conventional Horton method. Integrating this equation starting at
some time where the infiltration capacity is fr0 produces the following result for the capacity after
a recovery time of t:
/r=/o-(/o-/ro)e-M (4-21)
101
-------
From Eq. 4-19, the cumulative excess infiltration volume corresponding to this capacity, Fer, would
be:
(4-22)
and substituting 4-21 for fr gives:
Fer = (/0~/r0V^ (4-23)
kd
But again from 4-19,
\JO Jro) I^d = ^e v"-^4)
so the new cumulative volume after recovery is simply:
Fer = Fee~krt (4-25)
4.3.3 Computational Scheme
The detailed computational scheme for computing the Modified Horton infiltration rate for each
subcatchment within a study area over a single time step of a simulation is presented in the sidebar
titled Computational Scheme for Modified Horton Infiltration
4.3.4 Parameter Estimates
Because the modified Horton method utilizes the same parameters as the original Horton method,
the description in section 4.2.4 of how to estimate their values also applies to the modified method.
102
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Computational Scheme for Modified Horton Infiltration
The following variables are assumed known at the start of each time step At (sec) for the
pervious sub-area of each SWMM subcatchment:
/' = rainfall rate (ft/sec)
d = depth of ponded surface water (ft)
Fe = excess infiltrated volume (ft)
as are the following constants:
fo = maximum (or initial) infiltration capacity (ft/sec)
fca = minimum (or ultimate) infiltration capacity (ft/sec)
kd = infiltration capacity decay coefficient (sec-1)
kr = infiltration capacity recovery coefficient (sec-1)
Fmax= maximum infiltration volume possible (optional) (ft).
Initially at time 0, Fe = 0.
The following steps are used to compute the modified Horton infiltration rate/over a single
time step of a simulation:
1. Compute the available rainfall rate: ia = i + d/At
2. If ia = 0, meaning the surface is dry, then update the current excess infiltrated volume as
follows:
p — p p-fcrAt
re — reK
and set the infiltration rate/to 0.
3. Else if Fe > Fmax, set fp to 0. Otherwise compute a potential infiltration rate fp from
fp = max(/0 - kdFe, /«,)
4. Compute the actual infiltration rate /as the lesser of fp and the available rainfall rate:
/ = min(/p, ia)
5. If f> foo then update the cumulative excess infiltration volume:
Fe <- min(Fe + (/ - /oo)At, Fmax~)
103
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4.4 Green-Ampt Method
The Green-Ampt equation (Green and Ampt, 1911) has received considerable attention in recent
years. The original equation was for infiltration with excess water at the surface at all times. Mein
and Larson (1973) showed how it could be adapted to a steady rainfall input and proposed a way
in which the capillary suction parameter could be determined. Chu (1978) has shown the
applicability of the equation to the unsteady rainfall situation, using data for a field catchment. The
Green-Ampt method was added into SWMM III in 1981 by R.G. Mein and W. Huber (Huber et
al., 1981).
4.4.1 Governing Equations
The Green-Ampt conceptualization of the infiltration process is one in which infiltrated water
moves vertically downward in a saturated layer, beginning at the surface (Figure 4-5). In the wetted
zone the moisture content #is at saturation 6 s while the moisture content in the un-wetted zone is
at some known initial level 9i.
Ground Surface
Wetting Front
— >
— >
Wetted Zone
! (e e.)
\t^^^^^^*^^^^^^^
i
t
Non-wetted Zone
(e = e>)
^^ ^
i
i
Ls
Figure 4-5 Two-zone representation of the Green-Ampt infiltration model (after Nicklow et
al., 2006).
The water velocity within the wetted zone is given by Darcy's Law as a function of the saturated
hydraulic conductivity Ks, the capillary suction head along the wetting front i//s, the depth of
ponded water at the surface d, and the depth of the saturated layer below the surface Ls:
104
-------
d
The depth of the saturated layer Ls can be expressed in terms of the cumulative infiltration, F, and
the initial moisture deficit to be filled below the wetting front, 6 d = 9 s - 6i as Ls = F/0^ .
Substituting this into Equation 4-26 and assuming that d is small compared to the other depths
gives the Green-Ampt equation for saturated conditions:
(4-27)
Equation 4-27 applies only after a saturated layer develops at the ground surface. Prior to this point
in time the infiltration capacity will equal the rainfall intensity:
fp = i (4-28)
As time increases, one can test whether saturation has been reached by solving 4-27 for F (which
will be denoted as Fs~) with/, set equal to / and check if this value equals or exceeds the actual
cumulative infiltration F:
I - Ks
(4-29)
Note that there is no calculation of Fs when / <= Ks, although /"still gets updated during such
periods. Finally, in this scheme the actual infiltration /is the same as the potential value fp.
f = fp (4-30)
The two equations are illustrated in Figure 4-6 for the situation Ks = 0.25 in/hr, i//s = 6.5 in, and 6d
= 0.20. The initial, flat portion of the curve corresponds to/ = /', up to the point where F = Fs
(Equation 4-29). The remainder of the curve corresponds to the potential rate computed with
Equation 4-27. Note that the infiltration rate approaches Ks (0.25 in/hr) asymptotically.
105
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0.8 9-
0.6
0.4
0.2
4 6
F(in)
10
Figure 4-6 Illustration of infiltration capacity as function of cumulative infiltration for the
Green-Ampt method.
Equation 4-27 shows that the infiltration capacity after surface saturation depends on the infiltrated
volume, which in turn depends on the infiltration rates in previous time steps. To avoid numerical
errors over long time steps, the integrated form of the Green-Ampt equation is more suitable. That
is, fp is replaced by dF/dtand integrated to obtain:
(4-31)
If Fi is the known cumulative infiltration at the start of the time step and ¥2 the unknown
cumulative infiltration at the end of the time step then one can write:
F2 = C + ijJsedln(F2 +
(4-32)
where C = Ksht + Fx — i/js6dln(F^ + i/Js9d) is a known constant. Equation 4-32 can be solved
numerically for />. The average infiltration capacity fpover the time step can then be computed as
(F2 - Fj/At.
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4. 4. 2 Recovery of Infiltration Capacity
Evaporation, subsurface drainage, and moisture redistribution between rainfall events decrease the
soil moisture content in the upper soil zone and increase the infiltration capacity of the soil. The
processes involved are complex and depend on many factors. In SWMM a simple empirical routine
(Huber et al., 1981) is used as outlined below; commonly used units are given in the equations to
make the description easier to understand. Note that this procedure suffers from the same lack of
relationship to ET as does the Horton recovery, discussed earlier.
Infiltration is usually dominated by conditions in the uppermost layer of the soil. The thickness of
this layer depends on the soil type; for a sandy soil it could be several inches, for heavy clay it
would be less. The equation used to determine the thickness of the layer Lu is:
Lu = 4 (4-33)
where Lu has units of inches and Ks is expressed in in/hr. Thus for a high Ks of 0.5 in/hr (12.7
mm/hr) the thickness computed by Equation 4-33 is 2.83 inches (71.8 mm). For a soil with a low
hydraulic conductivity, say Ks = 0.1 in/hr (2.5 mm/hr), the computed thickness is 1.26 inches (32.1
mm). This constant thickness is different from the saturated zone thickness Ls shown in Figure 4-
5 which grows over time as infiltration proceeds.
In the Green- Ampt model, the initial soil moisture deficit at the start of a rainfall event determines
how much infiltration capacity is available during the event itself. Recall that the moisture deficit
6d is the difference between the saturated moisture content ft and the initial moisture content ft.
During a dry period the moisture deficit in the upper soil zone, ft/H, is regenerated, i.e., its value is
increased. Thus SWMM keeps continuous track of this quantity. At the start of a simulation, ft/H
is set equal to the user-supplied initial value of Odmax. During a wet period when infiltration occurs
at a rate/over a time step of At, Odu is decreased according to:
(4-34)
down to a possible limiting value of 0. During a dry period it increases as follows:
0 *— 0 -\- k 0 At (4-35^
up to a maximum possible value of Odmax , where kr is a recovery constant (hr"1).
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One can assume that the recovery constant is also dependent on Ks, such that tight, clay soils with
low Ks take longer to recover than do loose, sandy soils with high Ks. The following relationship
is used for kr:
(4-36)
75
where the constant 75 has units of (in-hr)1/2. Note that the time it would take a fully saturated soil
to recovery to its maximum capacity is simply:
1 75
hours (or 3.125/^]KS days).
To complete the recovery process it is necessary to define the minimum amount of time that a soil
must remain in recovery before any further rainfall would be considered as an independent event.
This time Tr (hr) is computed as:
0.06 4.5
(4-37)
Thus when a new period of rainfall occurs after a recovery interval of at least Tr hours, the two-
stage Green-Ampt infiltration process is re-started with Od = Odu and F = 0. Figure 4-7 summarizes
the functional dependence of the three internally computed recovery parameters Lu, kr, and Tr on
the saturated hydraulic conductivity Ks.
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14
12
10
1 8
.c
•s.
Q 6
4
2 H
0
0.01
0.10 1.00
Ks On/hi)
1000
100
p
10
1
10.00
Figure 4-7 Green-Ampt recovery parameters as functions of hydraulic conductivity.
4.4.3 Computational Scheme
The detailed computational scheme for computing the Green-Ampt infiltration rate for each
subcatchment within a study area over a single time step of a simulation is presented in the sidebar
below.
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Computational Scheme for Green-Ampt Infiltration
Note: For ease of presentation the following description uses length units of inches and time
units of hours rather than feet and seconds which are used internally in SWMM.
The following variables are assumed known at the start of each time step At (hr) for the
pervious subarea of each subcatchment:
/' = rainfall rate (in/hr)
d = depth of ponded surface water (in)
0d = soil moisture deficit at the start of the current rainfall event
9du = soil moisture deficit in the upper soil recovery zone
F = cumulative infiltrated volume (in)
T = recovery time remaining before the next event can begin (hr).
as are the following constants:
Ks = saturated hydraulic conductivity (in/hr)
i//s = suction head at the wetting front (in)
Odmax = maximum soil moisture deficit
Lu = depth of upper soil recovery zone (in)
kr = moisture deficit recovery constant (hr"1)
Tr = minimum recovery time before a new rainfall event occurs (hr).
The latter three constants are derived from Ks as described previously. Initially at time 0, Od =
Odu = Odmax, F = 0, T = 0 and the surface is in an unsaturated state.
The computational scheme for evaluating the Green-Ampt infiltration rate/over each time
step follows two separate paths, depending on whether the surface layer is in a saturated state
or not. The scheme for the unsaturated state proceeds as follows:
1. Compute the available rainfall rate: ia = i + d/At.
2. Decrease the recovery time remaining before the next storm event: T <- T — At.
(Continued on next page)
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3. If the available rainfall rate is zero (ia = 0) then:
a. Set/= 0.
b. Recover upper zone moisture deficit ddu and cumulative infiltration F:
A0 = kredmaxkt
Qdu <~ Qdu + AS
c. If the minimum recovery time has expired (T <0\ then set Od = Odu and F = 0 to
mark the beginning of a new rainfall event.
4. If the available rainfall rate does not exceed the saturated hydraulic conductivity (ia Fs) then use the procedure for
saturated conditions described below to compute/
d. If F + taAt < Fs (i.e., the surface remains unsaturated) then set/= ia and update F
and Odu as in Step 4b above.
(Continued on next page)
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e. Otherwise the surface becomes saturated during the time step. Solve Equation 4-
32 for F2 using Fs for Fi and At - (Fs - F) / ia for At (i.e., the portion of the
original time step over which saturated conditions exist). Then set:
AF = F2 - F
f = AF/At
The computational steps used for saturated conditions are as follows:
1. Compute the available rainfall rate: ia = i + d/At.
2. Reset T = Tr.
3 . Solve Equation 4-32 for ¥2 with Fi = F.
4. SetAF = F2-F.
5. If AF > iaAt then set AF = iaAt and change the current surface layer condition to
unsaturated.
6. Update the following:
F <- F + AF
f = AF/At
Note that in both of these paths, Odu is not allowed to drop below 0 nor exceed Odmax. Also, a
Newton-Raphson procedure (Press et al., 1992) is used to solve the integrated form of the
Green-Ampt Equation 4-32 at Step 5e of the unsaturated procedure and at Step 3 of the
saturated procedure.
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4.4.4 Parameter Estimates
The soil parameters that a user must supply for each subcatchment for the Green-Ampt infiltration
method are:
• Ks- the saturated hydraulic conductivity (in/hr or mm/hr),
• i//s- the suction head at the wetting front (in or mm),
• Odmax - the maximum moisture deficit available (volume of dry voids per volume of soil).
Conversions between the user-supplied units of these parameters (in (or mm) and hr) and those
used internally (ft and sec) are handled automatically by the program.
Saturated Hydraulic Conductivity (Ks~)
Probably the best single source for estimates of saturated hydraulic conductivity (Ks~) and suction
head (i//s) for a wide range of soils - and one that makes use of the Green-Ampt method relatively
attractive - is the data by Rawls et al. (1983), shown in Table 4-7. These data were derived from
measurements made on roughly 5000 soils across the United States and while they will never be
truly site specific, they are certainly consistent and defensible. Although there is considerable
variation in the parameter estimates, a good first approximation may be made using the table.
Values of hydraulic conductivity may also be used for estimates of the Horton parameter/«,. But
the range of values shown for porosity and suction head (the authors do not provide ranges for ^)
should be a warning about placing too much faith in such generalized estimates.
The NRCS Soil Survey Physical Data (see Figure 4-1) values for hydraulic conductivity could also
be used as a preliminary estimate. A better guide for the Ks values is as given for parameter _/» for
the Horton equation; theoretically these parameters (i.e.,/™ and Ks~) should be equal for the same
soil. Note that, in general, the range of Ks values encountered will be of the order of tenths of an
inch per hour.
Another source of conductivity estimates is the regression equation developed by Saxton and
Rawls (2006) that predicts Ks from the sand, clay and organic matter content of a soil. See Section
5.5.2 of the Groundwater chapter for more details.
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Table 4-7 Green-Ampt parameters for different soil classes (Rawls et al., 1983)
(Numbers in parentheses are ± one standard deviation from the parameter value shown.)
Soil Class
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Sandy clay
loam
Clay loam
Silty clay
loam
Sandy clay
Silty clay
Clay
Porosity, §
0.437
(0.374-0.500)
0.437
(0.363-0.506)
0.453
(0.351-0.555)
0.463
(0.375-0.551)
0.501
(0.420-0.582)
0.398
(0.332-0.464)
0.464
(0.409-0.519)
0.471
(0.418-0.524)
0.430
(0.370-0.490)
0.479
(0.425-0.533)
0.475
(0.427-0.523)
Effective
Porosity, §e*
0.417
(0.354-0.480)
0.401
(0.329-0.473)
0.412
(0.283-0.541)
0.434
(0.334-0.534)
0.486
(0.394-0.578)
0.330
(0.235-0.425)
0.309
(0.279-0.501)
0.432
(0.347-0.517)
0.321
(0.207-0.435)
0.423
(0.334-0.512)
0.385
(0.269-0.501)
Wetting Front
Suction Head,
\l/s (in)
1.95
(0.38-9.98)
2.41
(0.53-11.00)
4.33
(1.05-17.90)
3.50
(0.52-23.38)
6.57
(1.15-37.56)
8.60
(1.74-42.52)
8.22
(1.89-35.87)
10.75
(2.23-51.77)
9.41
(1.61-55.20)
11.50
(2.41-54.88)
12.45
(2.52-61.61)
Saturated
Hydraulic
Conductivity,
Ks (in/hr)
4.74
1.18
0.43
0.13
0.26
0.06
0.04
0.04
0.02
0.02
0.01
*Effective porosity
that remains after a
is the difference between the porosity and the residual moisture content
saturated soil is allowed to drain thoroughly.
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Urban soils are usually highly disturbed (Pitt et al., 1999, 2001; Pitt and Voorhees, 2000).
Construction has often occurred on or nearby the locations in question, and soils may be compacted
from their natural state. Alternatively, soils are sometimes imported for horticultural purposes.
Such imported soils (e.g., for lawns) may exhibit relatively high infiltration rates. The parameter
estimates discussed previously are based on data for undisturbed soils, e.g., using Natural
Resources Conservation Service (NRCS) data. Parameters for natural, undisturbed soils are likely
to overestimate the infiltration characteristics for urban soils. Modelers should bear in mind that
only site-specific infiltrometer and/or soil physics tests can determine local infiltration properties,
and that high spatial variability is the rule, rather than the exception.
Suction Head
The suction head, y/s (also referred to as capillary tension), is perhaps the most difficult parameter
to measure. It can be derived from soil moisture - conductivity data (Mein and Larsen, 1973) of
the type shown in Figures 5-5 in Chapter 5 for groundwater. Unfortunately, such detailed data are
rare for most soils. Fortunately the results obtained for Green-Ampt infiltration are not highly
sensitive to the estimate of i//s (Brakensiek and Onstad, 1977).
An excellent local data source can often be found in Soil Science departments at state universities.
Tests are run on a variety of soils found within the state, including soil moisture versus soil tension
data, from which i//s can be derived. For example, Carlisle et al. (1981) provide such data for
Florida soils along with information on Ks, bulk density, and other physical and chemical
properties.
Approximate values may also be found from several authors: Mein and Larsen (1973), Brakensiek
and Onstad (1977), Clapp and Hornberger (1978), Chu (1978), Rawls et al. (1983). Published
values vary considerably and conflict; however, a range of 2 to 15 inches (50 to 380 mm) covers
virtually all soil textures. But as with Ks, probably the best single source for estimates for capillary
suction (y/s) is the data by Rawls et al. (1983) listed in Table 4-7. Brakensiek et al. (1981) noted
that i//s was highly correlated with hydraulic conductivity over all soil classes. Using nonlinear
regression on the average values for these two variables listed in Table 4-7 produces the following
relationship for Ks in in/hr and y/s in inches:
°'328 (R2 = 0.9) (4-38)
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Maximum Moisture Deficit
The maximum moisture deficit, Odmax is defined as the difference between the moisture content at
saturation and at the start of the simulation. Because this parameter is the most sensitive of the
three parameters for estimates of runoff from pervious areas (Brakensiek and Onstad, 1977), some
care should be taken in determining the best Odmax value to use. The saturated moisture content is
approximately equal to the soil's porosity ^(i.e., the fraction of voids), assuming one ignores the
5 - 10% of trapped air that typically exists at saturation. After a saturated soil is allowed to drain
thoroughly, the residual moisture content that remains is fa. The effective porosity fa is defined as
- r and can be used to represent Odmax for dry antecedent conditions. Typical values of fa are
included in the Rawls et al. (1983) data set listed in Table 4-7.
Sandy soils tend to have lower porosities than clay soils, but drain to lower moisture contents
between storms because the water is not held so strongly in the soil pores. Consequently, values
of Odmax for dry antecedent conditions tend to be higher for sandy soils than for clay soils. Table 4-
8, derived from Clapp and Hornberger (1973), is another source of Odmax values for various soil
types.
Table 4-8 Typical values of ftimax for various soil types.
Soil Texture
Sand
Sandy Loam
Silt Loam
Loam
Sandy Clay Loam
Clay Loam
Clay
Typical Odmax at Soil Wilting Point
0.34
0.33
0.32
0.31
0.26
0.24
0.21
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These Odmax values would be suitable for input for long term continuous simulation; the soil type
selected should correspond to the surface layer for the particular subcatchment. For single event
simulation the values of Table 4-8 would apply only to very dry antecedent conditions. For moist
or wet antecedent conditions lower values of Odmax should be used. When estimating the particular
value it should be borne in mind that sandy soils drain more quickly than clayey soils, i.e., for the
same time since the previous event, the Odmax value for a sandy soil will be closer in value to that
of Table 4-8 than it would be for a clayey soil.
Another estimate for Odmax may be based on the NRCS Soil Survey Physical Data as "Available
Moisture Capacity" in/in of soil (dimensionless fraction), which is defined as the difference
between field capacity and the wilting point. Thus, it is an underestimate of the maximum Od value.
Furthermore, Available Moisture Capacity values listed may exhibit similar variability (or lack
thereof) as for hydraulic conductivity estimates discussed earlier, but these values are at least
specific to the soil in question. For instance, for the Woodburn silt loam illustrated in Figure 4-1,
Odmax might be at the high end of the range of 0.19 - 0.24 for the surface layer (considerably less
than the generic value of 0.32 for silt loam in Table 4-8 or the range of 0.394 to 0.578 given in
Table 4-7).
Finally, the initial moisture deficit can be related to another very general measure of a soil: its
storage capacity, S, which can be expressed as:
S = dwt9dmax (4-39)
where dwt is the depth to the sub-surface water table. Estimates of soil storage capacity, S, are
available using the Curve Number method, discussed below. That is, S is a function of the curve
number (Section 4.5.4), for which a vast literature is available. If depth to water table is known, or
if typical depths are given for a soil on its Soil Survey Interpretation data, then Equation 4-39 may
be solved for Odmax.
4.5 Curve Number Method
The Curve Number infiltration method is new to SWMM 5. It is based on the widely used SCS
(Soil Conservation Service, now known as the NRCS - Natural Resource Conservation Service)
curve number method for evaluating rainfall excess. First developed in 1954, the method is
embodied in the widely used TR-20 and TR-55 computer models (NRCS, 1986) as well as most
hydrology handbooks and textbooks (e.g., Bedient et al., 2013). It was added into SWMM to take
advantage of its familiarity to most practicing engineers and the availability of tabulated curve
117
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numbers for a wide range of land use and soil groups. The original curve number method is a
combined loss method that lumps together all losses due to interception, depression storage, and
infiltration to predict the total rainfall excess from a rainfall event. The SWMM uses a modified,
incremental form of the method that accounts only for infiltration losses, since the other
abstractions are modeled separately. Other incremental applications of the curve number method
have been proposed by Chen (1975), Aron et al. (1977) and Akan and Houghtalen (2003).
4.5.1 Governing Equations
In its classic form, the Curve Number model uses the following equation to relate total event runoff
Q (in) to total event precipitation P (in) (Haan et al., 1994; McCuen, 1998; Bedient et al., 2013;
NRCS, 2004b):
p2
Q = (4-40)
v p <-, \ ;
r T '-'max
where Smax = the soil's maximum moisture storage capacity (inches). Smax can also be thought of
as the difference in water volume contained in a fully saturated soil versus a fully drained soil. In
this sense it is similar to the maximum moisture deficit parameter Odmax used in the Green-Ampt
model, except it is expressed on a volumetric basis rather than as a fraction (see Equation 4-39).
Smax is derived from a tabulated "curve number" CN that varies with soil type and antecedent
conditions:
1000
CN
It should be emphasized that Equation 4-40 and subsequent equations use units of inches. Curve
numbers for various soil types and land covers are tabulated in the NRCS's National Engineering
Handbook (NRCS, 2004a) and in many text books.
In the formal SCS method, Equation 4-40 is written with P replaced byP-Ia where Ia is an initial
abstraction (in) that accounts for the volume of rainfall captured by vegetative interception, filling
of depression storage, and initial soil wetting. Because SWMM already accounts for these
phenomena through its depression storage parameter, dp, this refinement is not included here.
Assuming that all rainfall that does not run off is lost to infiltration (i.e., P - Q = F), Equation 4-
40 can be extended to predict total (cumulative) infiltration F (in) as:
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F = P -- (4-42)
DlC v '
r T '-'max
For a continuous model like SWMM, Equation 4-42 can be applied in an incremental fashion to
compute an infiltration rate/at each time step. Let Pi and Fi be the cumulative precipitation and
infiltration, respectively, at the start of the time step. At the end of the time step:
P2 = P! + iAt (4-43)
and
where Pi and ¥2 are the cumulative precipitation and infiltration values, respectively, at the end of
a time step At (hr), / (in/hr) is the rainfall rate over the time step, and Se is the moisture storage
capacity at the start of the rainfall event to which the time step belongs. For a single event
simulation, Se equals Smaxbut may be lower when moisture storage capacity depletion and recovery
occur over a longer simulation period as discussed in the next section.
The infiltration rate/(ft/sec) can then be computed as:
(4-45)
and the cumulative values get updated to Pi = ?2 and Fi = ¥2 to prepare for the next time step.
Note that as it stands, this model would not allow for any infiltration of ponded water when there
is a period of no rainfall within an event. To overcome this limitation it is assumed that the
infiltration rate for such periods remains the same as in the immediately preceding period. Also,
when overland flow re-routing occurs (see Section 3.6), the rainfall rate /' in Equation 4-43 does
not include the additional re-routed flow.
¥.5.2 Recovery of Storage Capacity
As with the other infiltration methods discussed, a soil's moisture storage capacity is depleted
during wet periods and replenished during dry periods. To model this behavior with the Curve
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Number method, the variable S is introduced to track the remaining storage capacity (i.e., moisture
deficit) over time. It is analogous to the state variable Odu used in the Green-Ampt method. Initially,
S = Smax. Whenever infiltration at rate/occurs over a time step At, S is reduced byfAt. During a
period with no infiltration S is assumed to be replenished at a rate proportional to Smax'.
S^S + krSmaxkt (4-46)
where kr is a storage capacity recovery constant (hr"1). This recovery expression has the same form
as used in the Green-Ampt model and the coefficient kr has a similar meaning in both models.
Because the Curve Number method was originally meant to be applied to single, discrete rainfall
events, a mechanism is needed to define when separate events occur. At the start of a new event,
the cumulative variables P and F are reset to 0 and Se is set equal to the current remaining storage
capacity S. Once again borrowing from the Green-Ampt method, a period of Tr hours without
rainfall must occur before the next rainfall period is deemed to begin a new event. Tr is assumed
to be related to the recovery constant kr through Equation 4-25 which is repeated here:
0.06
Tr = -7— (4-47)
4.5.3 Computational Scheme
The detailed computational scheme for computing Curve Number infiltration for each
subcatchment within a study area over a single time step of a simulation is presented in the sidebar
below.
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Computational Scheme for Curve Number Infiltration
Note: For ease of presentation the following description uses length units of inches and time
units of hours rather than feet and seconds which are used internally in SWMM.
The following variables are assumed known at the start of each time step At (hr) for the
pervious subarea of each subcatchment:
/' = rainfall rate over the current time step (in/hr)
d = depth of ponded surface water (in)
Pi = cumulative rainfall for the current rainfall event (in)
Se = soil moisture storage capacity at the start of the current rainfall event (in)
S = soil moisture storage capacity remaining (in)
Fi = cumulative infiltration volume (in)
T = time since the last period with rainfall (hr).
as are the following constants:
Smax = maximum moisture storage capacity as computed from the curve number
(in)
kr = storage capacity recovery constant (hr"1)
Tr = minimum recovery time before a new rainfall event can occur (hr).
Initially at time 0,Pj = 0,Se=S = Smax, Fi = 0, and T = Tr.
The computational steps for computing the Curve Number infiltration rate/for a given
subcatchment over a single time step of a simulation proceed as follows:
1. If there is rainfall (/' > 0} then:
a. If a new event has begun (T > Tr) then reset the following variables: Pi = 0,Fi =
0, and Se = S.
b. Reset the time since the last rainfall: T = 0.
c. Compute cumulative rainfall (PI) and infiltration (^2) at the end of the time step:
P2 = P! + iAt
(Continued on next page)
121
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d. Compute a potential infiltration rate:
fp = (P2 ~ FJM.
e. Update cumulative rainfall and infiltration:
Pi = P2
Fi = F2.
2. If there is no rainfall then increase the inter-event time (T <- T + At) and set the potential
infiltration rate to the rate from the previous time period (fp =f).
3. If there is some potential infiltration (fp > 0} then:
a. Limit the actual infiltration rate to the maximum available rate:
b. Reduce the soil moisture storage capacity:
5<-mox[5-/At,0].
4. Otherwise regenerate soil moisture storage capacity:
5 <- min[S + krSmaxkt,Smax]
4.5.4 Parameter Estimates
There are only two parameters required for each subcatchment using the Curve Number infiltration
method:
• the curve number
• the drying time (i.e., the time it takes a fully saturated soil to recover to a dry state).
The curve number is used to compute the maximum soil moisture storage capacity (Smax) using
Equation 4-41. The drying time Tdry in days is used to compute the regeneration constant kr in
hours"1 as:
(4-48)
The minimum inter-event recovery time Tr is then computed from kr using Equation 4-47.
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A highly structured method for estimating curve numbers is provided by the NRCS (NRCS, 2004a;
McCuen, 1998, Bedient et al., 2013 and virtually every hydrology text). Such estimates are
embedded in engineering practice through Table 4-9 in which curve number values are given as
function of land use and soil Hydrologic Soil Group (A through D). Hydrologic Soil Group is
provided on the NRCS Soil Survey data discussed in Section 4. 1 . For instance, the Woodburn silt
loam of Figure 4-1 is in Hydrologic Soil Group B.
There are several things to keep in mind when using curve numbers from Table 4-9. First, these
curve numbers apply only to normal antecedent moisture conditions (AMC II). For AMC I (low
moisture) or AMC III (high moisture) the following adjustments can be made to the tabulated
values (NRCS, 2004a):
4.2CN,,
10 -0.0580V,,
23CN,,
where CNi refers to the curve number for antecedent moisture condition /'. For long-term
simulations the AMC I curve number should be used to allow the soil to reach its maximum
possible moisture retention capacity during extended dry periods.
Second, the urban land use descriptions included in Table 4-9 lump together the pervious and
impervious portions of the subcatchment area to which a curve number is assigned. This means
that the subcatchment in question must be modeled as being completely pervious, with no
partitioning into separate pervious and impervious areas as is normally done in SWMM (refer to
Section 3.3). Otherwise too much runoff will be generated. If one wants to continue to partition
their subcatchments into pervious and impervious areas, they will have to either adjust the curve
numbers taken from Table 4-9 to remove the effects of imperviousness or find another source of
curve numbers, such as from calibration against field measurements (see Shuster and Pappas,
2011).
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Table 4-9 Runoff curve numbers for selected land uses (NRCS, 2004a)
(For antecedent moisture condition II)
Land Use Description
Cultivated land1
Without conservation treatment
With conservation treatment
Pasture or range land
Poor condition
Good condition
Meadow
Good condition
Wood or forest land
Thin stand, poor cover, no mulch
Good cover2
Open spaces, lawns, parks, golf courses, cemeteries, etc.
Good condition: grass
cover on 75% or more of the area
Fair condition: grass cover on 50 - 75% of the area
Commercial and business areas (85% impervious)
Industrial districts (72% impervious)
Residential3
Average lot size
1/8 ac or less
1/4 ac
1/3 ac
1/2 ac
1 ac
Average % impervious4
65
38
30
25
20
Paved parking lots, roofs, driveways, etc.5
Streets and roads
Paved with curbs and storm sewers5
Gravel
Dirt
Hydrologic Soil Group
A
72
62
68
39
30
45
25
39
49
89
81
77
61
57
54
51
98
98
76
72
B
81
71
79
61
58
66
55
61
69
92
88
85
75
72
70
68
98
98
85
82
C
88
78
86
74
71
77
70
74
79
94
91
90
83
81
80
79
98
98
89
87
D
91
81
89
80
78
83
77
80
84
95
93
92
87
86
85
84
98
98
91
89
(Footnotes appear on following page)
124
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Footnotes for Table 4-9:
1. For a more detailed description of agricultural land use curve numbers, refer to the
NRCS (2004a) National Engineering Handbook, Chapter 9, "Hydrologic Soil-Cover
Complexes".
2. Good cover is protected from grazing and litter and brush cover soil.
3. Curve numbers are computed assuming that the runoff from the house and driveway is
directed toward the street with a minimum of roof water directed to lawns where
additional infiltration could occur.
4. The remaining pervious areas (lawn) are considered to be in good pasture condition for
these curve numbers.
5. In some warmer climates of the country a curve number of 95 may be used.
Estimates of a soil's drying time have been discussed previously in conjunction with both the
Horton regeneration constant in Section 4.2.4 and the Green-Ampt recovery process in Section
4.3.2. It was suggested that the drying time Tdry in days could be related to a soil's saturated
hydraulic conductivity Ks in in/hr as follows:
3.125
(4-51)
where estimates of Ks based on soil type can be found from Table 4-7.
4.6 Numerical Example
Because the four infiltration methods discussed in this chapter have very different formulations, it
is interesting to compare the results they produce for a specific set of modeling conditions. Each
method was used to simulate infiltration over a relatively flat, completely pervious subcatchment
containing a well-drained Group B soil. The subcatchment properties, rainfall event, and
infiltration parameters for each method are listed in Table 4-10. The infiltration parameters were
chosen to have each method produce about the same amount of runoff for the design storm yet be
within the normal ranges discussed in previous sections of this chapter.
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Table 4-10 Parameters used in example comparison of infiltration methods
Item
Subcatchment
Rainfall Event
Horton Infiltration
Green-Ampt Infiltration
Curve Number Infiltration
Parameter
Percent Impervious
Percent Slope
Width (ft)
Roughness
Depression Storage (in)
Duration (hr)
Total Depth (in)
Time-to-Peak / Duration
Evaporation (in/hr)
Initial Capacity (in/hr)
Ultimate Capacity (in/hr)
Decay Coefficient (hr"1)
Drying Time (days)
Saturated Hydraulic Conductivity (in/hr)
Suction Head (in)
Initial Moisture Deficit
Curve Number
Drying Time (days)
Value
0
0.5
140
0.1
0.05
6.0
2.0
0.375
0
1.2
0.1
2.0
7.0
0.1
2.0
0.2
80
7.0
Figure 4-8 shows the infiltration rates obtained with each infiltration method under these
conditions. The numbers in the chart's legend are the fraction of rainfall that becomes runoff for
each method. Even though similar amounts of runoff are produced, the methods display distinctly
different infiltration patterns over time. These patterns are influenced not only by the parameters
that were chosen for each method, but also by the temporal pattern of rainfall intensity that occurs
during an event.
126
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Rainfall
-Green-Ampt(0.29)
• Morton (0.30)
• Curve Number (0.34)
Modified Morton (0.26)
0.6
0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00
Figure 4-8 Infiltration rates produced by different methods for a 2-inch rainfall event.
(Numbers in parentheses are the fraction of rainfall that becomes runoff.)
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Chapter 5 - Groundwater
5.1 Introduction
Because SWMM was originally developed to simulate combined sewer overflows in urban
catchments, the fate of infiltrated water was considered insignificant. Since its development,
however, SWMM has been used on areas ranging from highly urban to completely rural. Many
undeveloped, and even some developed areas, especially in areas like south Florida, are very flat
with high water tables, and their primary drainage pathway is through the surficial groundwater
aquifer and the unsaturated zone above it, rather than by overland flow. In these areas, underlain
by permeable sub-soils and dynamic water tables, a storm will cause a rise in the water table and
subsequent slow release of groundwater back to the receiving water (Capece et al., 1984). For this
case, the fate of the infiltrated water is a highly significant part of the local water cycle. By
assuming that the infiltration is lost from the system, an important part of the subsurface flow
system is not properly described (Gagliardo, 1986). In unlined channels and natural streams, the
complete water balance in the near surface soils needs to be maintained in order to compute
baseflow. Saturated zone outflow is a critical component of models such as HSPF (Bicknell et al.
1997) for realistic simulation of streamflow in areas in which overland flow rarely exists, which
is characteristic of most non-urban soils except for very clayey areas.
Groundwater discharge accounts for the time-delayed recession curve that is prevalent in most
non-urban watersheds (Fetter, 1980). This process cannot, however, be satisfactorily modeled by
surface runoff methods alone. By modifying infiltration parameters to account for subsurface
storage, attempts have been made to overcome the fact that SWMM assumes infiltration is lost
from the system (Downs et al., 1986). Although the modeled and measured peak flows matched
well, the volumes did not match well, and the values of the infiltration parameters were unrealistic.
Some research on the nature of the soil storage capacity has been done in south Florida (SFWMD,
1984). However, it was directed towards determining an initial storage capacity for the start of a
storm.
Another need is to combine the groundwater discharge hydrograph with the surface runoff
hydrograph and determine when the water table will rise to the surface. Additionally, a threshold
saturated water zone storage is needed (corresponding to the bottom of a stream channel), below
which no saturated zone outflow will occur. This is required to simulate dry watershed conditions.
Finally, it is also desirable to simulate bank storage, the movement of water from a stream channel
into the saturated groundwater zone when the stream water level is higher than the adjacent
groundwater table.
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To address these needs, a simple, two-zone groundwater routine was incorporated into version 4
of SWMM in 1987 by W. Huber and B. Cunningham, based on Gagliardo's (1986) MS thesis. The
intent was to develop a physically-based model whose parameters were based on readily available
soil properties. The current version of SWMM has reformulated and simplified the original
model's governing equations and solution procedure. This section describes the theory and
limitations of these methods.
5.2 Governing Equations
SWMM analyzes groundwater flow for each subcatchment independently. It represents the
subsurface region beneath a subcatchment as consisting of an unsaturated upper zone that lies
above a lower saturated zone, illustrated in Figure 5-1. The height of the water table (i.e., the
boundary between the two zones) changes with time depending on the rates of inflow and outflow
of the saturated lower zone. This variable volume, two-zone configuration is similar to that used
by Dawdy and O'Donnell (1965) and serves as an alternative pathway for infiltrated rainfall to
pass between a subcatchment and a point in the conveyance system in an attenuated and delayed
fashion.
Upper
Zone
Lower
Zone
6r*
ftj
XXXXXXXXXXXXXXXXX
^mmmmm
Figure 5-1 Definitional sketch of the two-zone groundwater model.
Flow from the unsaturated upper zone to the saturated lower zone is controlled by a percolation
equation for which parameters may either be estimated or calibrated, depending on the availability
of the necessary soils data. The upper zone receives vertical inflow from infiltrating rainfall as
described in Chapter 4 and can also lose moisture through evapotranspiration. For time steps where
the water table has risen to the surface (reducing the unsaturated zone volume to zero), infiltration
ceases and runoff is produced by saturation excess.
129
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Losses and outflow from the lower zone consist of deep percolation, saturated zone
evapotranspiration, and lateral groundwater flow. The latter is a user-defined power function of
water table stage and depth of water in the receiving node of the conveyance system. If the water
elevation at the node is higher than the saturated zone water table, back-flow (bank storage) can
occur into the saturated zone.
This two-zone representation of surface runoff-groundwater interaction is modeled as follows
(refer to Figure 5-1). The ground surface has a known elevation (relative to some fixed reference)
of EG (ft) and the bottom of the saturated zone has a known elevation of EB (ft). The unsaturated
upper zone has a varying moisture content denoted as 9. The lower zone is completely saturated,
and therefore its moisture content is fixed at the soil porosity . Aside from 0, the other principal
unknown is di, the depth of the saturated zone (i.e., the water table depth). Because the depth from
the ground surface to the bottom of the lower zone is fixed, the depth of the unsaturated zone du
is simply EG — EB — dL.
The depths of the two zones and the water content of the upper zone are controlled by the
volumetric water fluxes shown in Figure 5-1. These fluxes, expressed as volume per unit horizontal
area per unit time (or ft/sec internally in SWMM), consist of the following:
fi = infiltration from the subcatchment surface, which is the value computed in
Chapter 4 multiplied by the fraction of pervious areaFperv.
fsu = evapotranspiration from the upper zone, which is a fixed fraction of the unused
surface evaporation, e x Fperv.
fu = percolation from the upper to lower zone, which depends on the upper zone
moisture content 6 and upper zone depth du.
/EL = evapotranspiration from the lower zone, which is a function of the depth of the
upper zone du.
/L = percolation from the lower zone to deep groundwater, which depends on the
lower zone depth di.
fo = lateral groundwater seepage to the conveyance network which depends on the
lower zone depth di as well as the water surface elevation in the receiving node.
Computation of these fluxes will be discussed subsequently, but keep in mind that they are either
supplied externally or depend on the unknown variables 6, du and di.
The conservation of mass equation for the upper zone can be written as:
^ = fc (5-D
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where Vu is the volume of water per unit area (ft) in the upper zone andfuz (ft/sec) is the net influx
rate to the upper zone. The latter is equal to:
fuz =fi~ fsu ~ fu (5-2)
The conservation of mass equation for the lower zone is:
dVL
dt
= fiz (5-3)
where VL is the volume of water per unit area (ft) in the lower zone andfiz is the net influx rate
into the lower zone given by:
fiz = fu~ fsi -fi-fo (5-4)
A third equation is needed to express the change in lower zone depth as a function of change in
lower zone volume:
dd, dV, , ^
(*-e)= (5-5)
This equation accounts for the fact that as the lower zone contracts or expands it is consuming or
vacating a portion of upper zone which has a moisture content of 9. For example, if the lower zone
expands it absorbs an amount of moisture # contained in the upper zone. Because the lower zone
always has a fixed moisture content of $, its expansion must be accompanied by a volume increase
of - 9 to make up the difference. Substituting Equation 5-5 into 5-3 results in the following
expression for the rate of change of the lower zone depth:
(5-6)
And since Vu = Ody, Equation 5-1 can be expanded as:
( }
From the relation da = EG — EB — dL and Equation 5-6 one can write:
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dd,, ddj fI7
—- = = —— (5-8)
dt dt (0-0)
Substituting this into 5-7 and solving for 801 dt gives:
d9 9fLZ + (0 - 0)/yz
(5-9}
dt ((b- n^^ " ^ '
Equations 5-6 and 5-9 form a system of ordinary differential equations in 6 and di that can be
solved using a standard fifth-order Runge-Kutta integration routine with adaptive step size control
(Press et al., 1992). The integration is applied over each runoff time step as the calculation of
surface runoff unfolds (see Section 3.4). The initial conditions at time zero are di = dioand 0= Oo
where dio is the initial depth of the saturated zone and Oo is the initial moisture content of the
unsaturated zone. Additional conditions that must be honored during each time step At are:
• The volume of infiltration that enters the upper zone over a time step must not exceed the
available pore volume, i.e., //At < dy(0 — 0) + /yAt. Any excess is subtracted off and
returned to the surface in the form of a reduced infiltration rate.
• The upper zone moisture content cannot be less than the soil's wilting point moisture
content nor greater than its porosity, i.e., 0WP < 0 < 0 where 0m1 is the sub-soil wilting
point moisture content.
• The depth of the lower layer cannot be greater than the distance from the ground surface
to the bottom of the saturated zone, i.e., dL < EG — EB
This simple two-zone groundwater model has a number of limitations that the reader should be
aware of:
• Since the moisture content of the unsaturated zone is taken as an average over the entire
zone, the shape of the moisture profile is totally obscured. Therefore, infiltrated water
cannot be modeled more realistically as an expanding volume of saturated soil moving
downward through the unsaturated zone (which is the Green-Ampt conceptualization).
Furthermore, water from the capillary fringe of the saturated zone cannot move upward by
diffusion or capillary suction into the unsaturated zone.
• The simplistic representation of subsurface storage by one unsaturated "tank" and one
saturated "tank" limits the ability of the user to match non-uniform soil columns.
• The assumption that the infiltrated water is spread uniformly over the entire catchment
area, not just over the pervious area means that mounding under a pervious area cannot be
simulated.
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Groundwater cannot be routed from the saturated zone under one subcatchment to that of
another subcatchment, i.e., lateral groundwater flow within an aquifer system that underlies
several subcatchment areas cannot be simulated.
No attempt is made to model the fate of any water quality constituents entering the
groundwater system. The concentration of all pollutants in the water infiltrating into the
subsurface zone is set to zero. One can, however, assign a constant concentration to the
discharge/? out of the saturated zone. If true quality routing through the subsurface region
is needed, a model such as HSPF (Bicknell et al., 1997) might be considered.
5.3 Groundwater Flux Terms
In order to integrate the groundwater conservation of mass equations over a succession of time
steps one must compute the various flux terms that transport water into and out of the two sub-
surface zones. This section discusses how each of these terms is modeled.
5. 3. 1 Surface Infiltration (fi)
The surface infiltration flux rate/ is set equal to the runoff infiltration rate/computed as described
in Chapter 4, multiplied by the fraction of the subcatchment that is pervious, Fperv. (The
groundwater zones extend over the entire subcatchment area while surface infiltration is computed
only for the pervious portion of this area.)/ is considered a constant quantity over the current
runoff time step At. However, it is not allowed to exceed a rate that would fill up the available pore
volume of the upper unsaturated zone by the end of the time step. This ratefimax can be computed
as:
(5-10)
where fu is an estimate of the percolation flux rate between the upper and lower zones at the start
of the time period and is computed using the equations given in Section 5.3.2 below. Thus if the
infiltration computed from the surface runoff calculation,/, is greater thanfimax then/ is set equal
and the infiltration rate used for surface runoff calculations is reduced to fi/Fperv.
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5. 3.2 Upper Zone Evapotranspiration (fsu)
Evapotranspiration (ET) from the upper zone,_/k/, represents soil moisture lost via cover vegetation
and by direct evaporation from the pervious area of the subcatchment. This ET is a portion of the
overall potential evaporation rate for the study area supplied externally to the program using the
data sources described in Section 2.5. The order in which this overall rate is allocated to the various
types of ET losses is as follows: 1) land surface evaporation, 2) upper zone evapotranspiration,
and 3) lower zone transpiration. Upper zone ET is computed as:
fEU = min(emax - es, UEF x emax~) (5-11)
where UEF is a fraction of available evaporation that is apportioned to the upper zone, emax =
eFperv, e is the maximum potential evaporation rate (ft/s) available for the current time period
supplied externally, Fperv is the fraction of the subcatchment that is pervious, and es is the
evaporation loss (ft/s) seen by any rainfall and ponded water on the pervious subcatchment surface.
The latter is computed as:
es = min(e, da/At)Fperv (5-12)
where da is the depth of available moisture on the pervious area of the subcatchment (ft). The latter
quantity was evaluated at Step 3b of the procedure used to compute surface runoff (see Section
3.4). In addition, fsu is set to 0 whenever the upper zone soil moisture drops below the wilting
point or when the infiltration rate^/ > 0 (since it is assumed that the resulting vapor pressure will
be high enough to prevent any evapotranspiration from the unsaturated zone). Note the need to
adjust the surface evaporation rates by Fperv because although evaporation from the groundwater
zone extends over the entire subsurface area of the subcatchment it can only be released through
the pervious portion of the subcatchment.
5.3.3 Lower Zone Evapotranspiration (/EL)
Lower zone evapotranspiration,^/,, represents the ET, or more properly just the transpiration, lost
from the saturated lower zone. It is assumed to vary in direct proportion to the distance that the
water table sits above some reference level below which no ET can occur. In equation form:
DEL-du
fBL = (1 - UEF)ema, (5-13)
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where DEL is the depth from the ground surface below which no lower zone ET is possible (ft).
The ./EL value computed from (5-12) is constrained to be non-negative and be no greater than
emax ~ es ~ )EU •
5.3.4 Percolation (fu)
Percolation,//, represents the flow of water from the unsaturated zone to the saturated zone, and
apart from possible bank storage is the only inflow for the saturated zone. The percolation equation
is formulated from Darcy's Law for unsaturated flow, in which the hydraulic conductivity, K, is a
function of the moisture content, 9. For one-dimensional, vertical flow, Darcy's Law may be
written as:
dh
V = K(0)— (5-14)
az
where:
v = velocity (specific discharge), positive in the downward direction of z (ft/s),
z = vertical coordinate with respect to the ground surface (ft),
K(9) = hydraulic conductivity (ft/s),
9 = moisture content (dimensionless), and
h = hydraulic potential or head (ft).
The hydraulic potential is the sum of the elevation (gravity) and pressure heads,
h = z + ip (5-15)
where y/ = soil water tension (negative pressure head) in the unsaturated zone. Note that the wetting
front suction, if/s, used in the Green-Ampt equations is simply the average value of if/ along the
wetting front during the infiltration process. Equating vertical velocity to percolation, and
differentiating the hydraulic potential, h, yields:
(5-16)
A choice is customarily made between using the tension, if/, or the moisture content, 9, as
parameters in equations for unsaturated zone water flow. Since the quantity of water in the
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unsaturated zone is identified by 9m previous equations, it is the choice here. Parameter y/ can be
related to # if the characteristics of the unsaturated soil are known. Thus, for use in Equation 5-16,
the derivative is:
(5-17)
dz d9 dz
However, since #is assumed constant throughout the upper zone, dO/dz = 0 and the percolation
flux becomes simply:
fu = K(B} (5-18)
The hydraulic conductivity K as a function of moisture content $is approximated functionally in
the moisture range of interest as:
ff(0) = Kse-^-^HCO (5-19)
where Ks is the saturated hydraulic conductivity (ft/s) and HCO is a calibration parameter.
Estimates of HCO can be made from soil test data and some examples will be given in section 5.4
below. Substituting 5-19 into 5-16 yields the final form of the percolation rate expression:
fu = Kse~^-a)HCO (5-20)
If the moisture content 9 is less than or equal to field capacity 9FC, then the percolation rate
becomes zero. This limit is in accordance with the concept of field capacity as the drainable soil
water that cannot be removed by gravity alone (Hillel, 1982, p. 243). Once 9 drops below field
capacity, it can only be further reduced by upper zone evapotranspiration (to a lower bound of the
wilting point moisture content).
5.3.4 Deep Percolation (fi)
Deep percolation, /L, represents a lumped sink term for un-quantified losses from the saturated
zone. The two primary losses are assumed to be percolation through the confining layer and lateral
outflow to somewhere other than the conveyance system. The arbitrarily chosen equation for deep
percolation is:
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(5-21)
where DP is a recession coefficient derived from inter-event water table recession curves. The
dependence offi on di allows it to be a function of the static pressure head above the confining
layer.
5. 3. 5 Groundwater Discharge (fo)
Groundwater discharge,/?, (lateral flow per horizontal area of the groundwater region or cfs/ft2)
represents lateral flow from the saturated zone to elements in the conveyance system. The latter
can take the form of an adjacent stream or channel or under-drains in the groundwater region, with
the recognition that groundwater discharge in SWMM is actually to (and from) nodes, not directly
to channels or pipes. (If need be, refer back to Section 1.2 for a description of how SWMM
represents a conveyance system as a network of links and nodes.) If a channel receives
groundwater, then its upstream node is used instead. The flux equation for groundwater discharge
takes on the following general form:
/c = Al(dL - /T)B1 - A2(hsw - /T)B2 + A3dLhsw (5-22)
where:
fo = groundwater flow rate (cfs/ft2),
hsw = height of surface water above the bottom of the groundwater zone (ft),
h* = reference height above the bottom of the groundwater zone (ft),
Al, Bl = groundwater flow coefficient and exponent,
A2, B2 = surface water flow coefficient and exponent,
A3 = surface-groundwater interaction coefficient.
Figure 5-2 illustrates the meaning of each of the water depths used in this expression. The reference
height h* is typically chosen as the height to the bottom of the conveyance system node, but other
choices are possible. The coefficients Al, A2, and A3 are units-dependent. As shown here Al has
units of ft(1'S7)/s, A2 has units of ft(1'S2)/s, while A3 is in (ft-s)'1. In an actual SWMM input data set
the user would use coefficients that produce flow rates measured in cfs/ac for US units or cms/ha
for metric units. SWMM automatically converts these input coefficients so that Equation 5-22 is
evaluated internally using cfs/ft2.
The particular function form of Equation 5-22 was selected in order to approximate various
horizontal flow conditions as will be illustrated later. The reference height h* sets the minimum
elevation at which groundwater flow is possible (i.e.,/G becomes 0 when either di or hsw is below
137
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h*). If h* is not explicitly set by the user it defaults to the height of the receiving node's invert as
shown in Figure 5-2. Also note that the conveyance system node receiving groundwater flow need
not be the same node that receives runoff from the subcatchment that lies above the groundwater
zones.
Upper
Zone
Lower
Zone
•
i\.\.' -- -
fc ei
Receiving
Node
V
liiiiiifcssdl^
^^^^^^
flflfli -J>JWJ
•ViViViViViViViViViViViViViViViViViViVi^iViViViViViViViViViViViViViViVjiV'
LV.V.V.V.V.V.V.V.V.V.V.V.V.V.V.V.V.V.V.^-.>^
>.•>.•>.•>.•>.•>.••>.•>.•>.•>.•>.•>.•>.•>.•
^•^•^•^•^ ••^•^•^•^•^•^•^
h*
hsw
dL
Figure 5-2 Heights used to compute lateral groundwater flow rate.
The effects of channel water on groundwater flow can be dealt with in two different ways. The
first option entails setting hsw (water surface height in the receiving node) to a constant value
greater than or equal to h* and A2, B2 and/or A3 to values greater than zero. If this method is
chosen, then the user is specifying an average tailwater influence over the entire run to be used at
each time step.
The second option uses the actual water surface height at the receiving node, as determined during
the flow routing calculations for the conveyance system (flow routing is discussed in Volume II
of this manual). In this case hsw can vary over time and the value used in Equation 5-22 is the flow
routing result at the start of the current time step.
Note that when conditions warrant, the groundwater flux,/G, can be negative, simulating flow into
the aquifer from the channel, in the manner of bank storage. An exception occurs when A3 ^ 0,
since the surface water - groundwater interaction term is usually derived from groundwater flow
models that assume unidirectional flow (examples are provided below). Otherwise, to ensure that
negative fo values will not occur, one can make Al greater than or equal to A2, Bl greater than or
equal to 52, and^43 equal to zero. More examples of adjusting the flow coefficients and exponents
to reproduce specific physical conditions are provided in section 5.5 on Parameter Estimation.
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5. 3. 6 User-Defined Flux Equations
SWMM also has the ability to employ custom user-defined equations for the lateral groundwater
discharge flux (fo) and the deep percolation flux (/L). These can be any well-formed mathematical
expression relating/G (in cfs/acre or cms/ha) or/L (in in/hr or mm/hr) to any of several pre-defined
variables. More details can be found in the SWMM 5 User's Manual (US EPA, 2010).
For example, a two-stage linear reservoir model for lateral groundwater outflow could be
expressed as:
/G = 0.001*Hgw + 0.05* (Hgw-5) *STEP(Hgw-5)
where Hgw is the pre-defined variable name used for height of the groundwater table (i.e., di as
used here) and STEP is a special cutoff function pre-defined as STEP (x) = 0 if x < 0 and is 1
otherwise. The expression says that there is some small background flow out of the aquifer that is
proportional to the height of the saturated zone plus a second larger source of outflow that only
occurs when the saturated zone height exceeds 5. It would not be possible to express this type of
behavior using just the standard discharge equation 5-21.
An example for deep percolation flux might be
fL = 2.5*Hgw - 0.1
which is equivalent to expressing^ through Darcy's Law as:
where Kc is the hydraulic conductivity of the confining layer beneath the shallow aquifer, dc is the
thickness of this layer, and Hc is the hydraulic head below the layer. The values 2.5 and 0.1 in the
user-defined expression would come from knowing specific values of Kc, dc, and Hc.
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5.4 Computational Scheme
Groundwater computations are a sub-procedure implemented as part of SWMM's runoff
calculations. They are made at each runoff time step, for each subcatchment that has a groundwater
component, immediately after infiltration over the subcatchment's pervious area has been
computed. This is at Step 3c of the runoff procedure described in Section 3.4. The detailed steps
involved are described in the sidebar below.
Computational Scheme for Groundwater
The following variables are assumed known at the start of each time step of length At (sec) for
each subcatchment with a defined groundwater component:
Available from surface runoff calculations:
/ = infiltration rate from pervious surface of subcatchment (ft/sec)
e = maximum potential evapotranspiration rate at the land surface (ft/sec)
da = depth of available moisture on the pervious area of the subcatchment (ft)
Fperv = fraction of subcatchment area that is pervious
Available from conveyance system flow routing calculations:
VN = inflow + stored volume of water at the node receiving groundwater flow (ft3)
hsw = water surface elevation at the node receiving groundwater flow (ft)
Groundwater state variables:
9 = moisture content of the upper unsaturated groundwater zone (ratio)
di = depth of the lower saturated groundwater zone (ft)
In addition, the following constants are also assumed known for each subcatchment:
Soil properties:
-------
Elevations:
EG = ground surface elevation (ft)
EB = elevation of bottom of lower groundwater zone (ft)
h* = minimum water table height for groundwater flow to occur (ft)
Groundwater flow constants:
Al, Bl, A2, B2, and A3 as described in section 5.3.5.
Note that at time 0 the state variables #and di are initialized with user-supplied values.
With the above information in hand, the following steps are used to update each subcatchment's
groundwater system:
1 . Determine the maximum limit on the upper zone percolation rate, fumax, as:
where du = EG - EB - di.
2. Compute the portion of evaporation consumed by ponded surface water, es:
es = min[e,da/ht]Fperv
3. Make an initial estimate of the upper zone percolation rate,//, using Equation 5-19 and limit
/t/to be no greater than fu
4. Determine the maximum limit on the infiltration rate fimax as:
du(4> - 0) f
Umax — A j. JU
and set/ to the smaller of / x Fperv (as computed by the infiltration routine) andfimax. If/
= fimax then reduce / to fi/Fperv for use in the runoff routine after it returns from the
groundwater calculations.
5. Estimate maximum and minimum bounds on lateral groundwater flow/? as follows:
fcmax = dL(f>/At (cannot release more than what is stored)
= ~du((p — 0)/At (cannot accept more than can be stored)
(Continued on next page)
141
-------
= (VN/kt)/A (cannot accept more than node can release)
fcmin = max [fGmnl, fGmin2\ (the maximum is used because fGminis negative)
where A is the total area of the subcatchment.
6. Use a standard fifth-order Runge-Kutta integration routine (RK5) with adaptive step size
control (Press et al., 1992) to solve the following equations simultaneously:
eflz + 0
between 0 and emax - es -/EU
0 when 9 <9pc ; otherwise between 0 andfumax
between 0 and DP (for Eq. 5-21)
between fomin and fomax
1. To avoid numerical issues (such as division by zero), adjust the new value of 9 so that it is
no lower than dwp and no higher than - XTOL where XTOL is a tolerance factor of 0.001.
Likewise, adjust di so that it does not drop below 0 and does not exceed EG-£B- XTOL.
8. Re-evaluate the groundwater flow term/G at the updated value of di and save_/b4, where
A is the subcatchment area, for use as lateral inflow to the receiving node when the next
conveyance system flow routing solution is found.
142
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5.5 Parameter Estimates
Estimates of the following constants are required in order to implement the two-zone groundwater
model:
• soil moisture limits ((/), OFC, and OWP)
• percolation parameters (Ks, HCO, and DP)
• ET coefficients (UEF and DEL)
• groundwater discharge constants (Al, Bl, A2, B2, and^3).
SWMM uses an Aquifer object to bundle together a common set of soil moisture limits, ET
coefficients, and percolation parameters that can be shared by any number of subcatchments. This
helps to reduce the amount of input values that must be supplied to the program. Multiple Aquifer
objects can be defined to accommodate variations in subsurface conditions across the study area.
On the other hand, a distinct set of groundwater discharge constants must be supplied for each
subcatchment that experiences groundwater flow.
5.5.1 Soil Moisture Limits
Porosity ($) is defined as the volumetric water content of a soil (volume of water per total volume)
when its pore spaces are at saturation. No distinction is made here between the actual porosity and
the apparent porosity, which includes trapped air, since no mechanism exists for adjusting for the
latter and the difference is usually minor (5-10 %). Porosity is a critical parameter because of its
role in determining moisture storage. Field capacity (6k:) is usually considered to be the amount
of water a well-drained soil holds after free water has drained off, or the maximum amount it can
hold against gravity (Linsley et al., 1982; SCS, 1991). This occurs at soil moisture tensions of from
0.1 to 0.7 atmospheres, depending on soil texture. Moisture content at a tension of 1/3 atmosphere
is often used. The wilting point (or permanent wilting point) (6wp\ is the soil moisture content at
which plants can no longer obtain enough moisture to meet transpiration requirements; they wilt
and die unless water is added to the soil. The moisture content at a tension of 15 atmospheres is
accepted as a good estimate of the wilting point (Linsley et al., 1982; Jensen et al., 1990; SCS,
1991). The field capacity must be greater than the wilting point and less than the porosity. The
general relationship among soil moisture parameters is shown in Figure 5-3.
143
-------
o
V
ff
0.6O
0.5O -
0.4O -
0.30 -
0.20 -
0 10 -
o.oo '•"vv-^vvvv-
•tV5=>T: : : : : : : : : : ! ! : : : : : ! ! : : ; E ! IWATER
HANI)
SANDY
LOAM
L
-------
Table 5-1 Volumetric moisture content at field capacity and wilting point (derived from
Linsley et al., 1982)
Soil Type
Sand
Sandy loam
Loam
Silt loam
Clay loam
Clay
Peat
Field Capacity
(ffVft3)
0.08
0.17
0.26
0.28
0.31
0.36
0.56
Wilting Point
(ffVft3)
0.03
0.07
0.14
0.17
0.19
0.26
0.30
*Fraction moisture content = fraction dry weight x dry density / density of water.
Table 5-2 Volumetric moisture content at field capacity and wilting point (U.S. Army Corps
of Engineers, 1956)
Soil Type
Sand
Fine sand
Sandy loam
Fine sandy loam
Silty loam
Light clay loam
Clay loam
Heavy clay loam
Clay
Field Capacity
(ft3/ft3)
0.10
0.12
0.16
0.22
0.28
0.30
0.32
0.33
0.33
Wilting Point
(ffVft3)
0.03
0.03
0.05
0.07
0.12
0.13
0.15
0.18
0.21
145
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Table 5-3 Average moisture limits and saturated hydraulic conductivity for different soil
types (Rawls et al., 1983)
Soil Type
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Sandy clay loam
Clay loam
Silty clay loam
Sandy clay
Silty clay
Clay
Porosity
(ffVft3)
0.437
0.437
0.453
0.463
0.501
0.398
0.464
0.471
0.430
0.479
0.475
Field Capacity
(ft3/ft3)
0.062
0.105
0.190
0.232
0.284
0.244
0.310
0.342
0.321
0.371
0.378
Wilting Point
(ffVft3)
0.024
0.047
0.085
0.116
0.135
0.136
0.187
0.210
0.221
0.251
0.265
Saturated
Hydraulic
Conductivity
(in/hr)
4.74
1.18
0.43
0.13
0.26
0.06
0.04
0.04
0.02
0.02
0.01
Schroeder et al. (1994) developed more extensive tables of soil moisture limits that were used to
provide default parameter values for the U.S. EPA HELP (Hydrological Evaluation of Landfill
Performance) model. They were derived from the large data base of soil measurements reported
by Rawls et al. (1982). Table 5-4 contains a version of the HELP table for uncompacted, low-
density soils while Table 5-5 does the same for compacted, moderate-density soils. The soils in
these tables are referred to by both their USDA and Unified Soil Classification System (USCS)
textures. Table 5-6 explains the abbreviations used for these classifications.
146
-------
Table 5-4 Default properties of low-density soils used in the EPA HELP model (from Rawls
et al. (1982) as reported in Schroeder et al. (1994))
Soil Texture Class
CoS
S
FS
LS
LFS
SL
FSL
L
SiL
SCL
CL
SiCL
SC
SiC
C
SP
sw
sw
SM
SM
SM
SM
ML
ML
SC
CL
CL
SC
CH
CH
0.417
0.437
0.457
0.437
0.457
0.453
0.473
0.463
0.501
0.398
0.464
0.471
0.430
0.479
0.475
0.045
0.062
0.083
0.105
0.131
0.190
0.222
0.232
0.284
0.244
0.310
0.342
0.321
0.371
0.378
0.018
0.024
0.033
0.047
0.058
0.085
0.104
0.116
0.135
0.136
0.187
0.210
0.221
0.251
0.251
14.173
8.220
4.394
2.409
1.417
1.020
0.737
0.524
0.269
0.170
0.091
0.060
0.047
0.035
0.035
147
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Table 5-5 Default properties of moderate-density soils used in the EPA HELP model
(Schroeder et al. (1994))
Soil Texture Class
USDA
L
SiL
SCL
CL
SiCL
SC
SiC
C
uses
ML
ML
SC
CL
CL
SC
CH
CH
Porosity
(ft3/ft3)
0.419
0.461
0.365
0.437
0.445
0.400
0.452
0.451
Field Capacity
(ft3/ft3)
0.307
0.360
0.305
0.373
0.393
0.366
0.411
0.419
Wilting Point
(ft3/ft3)
0.180
0.203
0.202
0.266
0.277
0.288
0.311
0.332
Saturated
Hydraulic
Conductivity
(in/hr)
0.027
0.013
0.004
0.005
0.003
0.001
0.002
0.001
Table 5-6 Soil texture abbreviations
USDA Soil Texture
Symbol
S
Si
C
L
Co
F
Meaning
Sand
Silt
Clay
Loam (mixture of sand, silt, clay
and humus)
Coarse
Fine
Unified Soil Classification System
Symbol
S
M
C
P
W
H
L
Meaning
Sand
Silt
Clay
Poorly graded
Well graded
High plasticity or compressibility
Low plasticity or compressibility
More specific soil parameter estimates can be obtained from the NRCS Soil Survey reports
available for each county in the U.S. These were discussed previously in Section 4.1. An excerpt
from the Physical Properties portion of one such report was displayed in Figure 4-1. Using the
bulk density pb value provided in these reports, an estimate of the porosity can be derived from:
148
-------
4> = l-pb/ps (5-23)
where:
(f) = porosity,
pb = bulk density (mass of dried soil to total volume of soil and voids), g/cm3,
ps = soil particle density, typically in range 2.6-2.7 g/cm3 for quartz particles.
As an example, the bulk density for the Woodburn silt loam listed in Figure 4-1 is 7.35 g/cm3 and
using a ps = 2.65 g/cm3 in Equation 5-23 yields a
-------
Table 5-7 Regression equations for soil moisture limits (Saxton and Rawls, 2006)
Soil Moisture Limit1
Wilting Point (6 WP)
Field Capacity (Ope)
Porosity (0)
Equation2
OWP = 0i5oot + (0.140150ot - 0.02) where
0isoot = -0.0245 + 0.487C + 0.0060M + 0.005(5 X OM)
- 0.013(C X 0M) + 0.068(5 X C) + 0.031
0FC = 033t + (1.2830f3t - 0.374033t - 0.015) where
033t = -0.2515 + 0.195C + 0.0110M + 0.006(5 X 0M)
- 0.027(C X 0M) + 0.452(5 X C) + 0.299
-------
Table 5-8 Regression estimates of soil moisture limits from the SPAW calculator*
Soil Type
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Sandy clay loam
Clay loam
Silty clay loam
Sandy clay
Silty clay
Clay
Porosity
(ft3/ft3)
0.463
0.457
0.450
0.458
0.482
0.432
0.472
0.510
0.440
0.532
0.488
Field Capacity
(ft3/ft3)
0.094
0.121
0.179
0.267
0.321
0.283
0.350
0.379
0.371
0.416
0.420
Wilting Point
(ft3/ft3)
0.050
0.057
0.081
0.126
0.137
0.183
0.213
0.210
0.260
0.278
0.299
Saturated
Hydraulic
Conductivity
(in/hr)
4.49
3.59
1.98
0.73
0.48
0.31
0.18
0.23
0.03
0.15
0.03
*For 2.5% organic matter content by weight.
5.5.2 Percolation Parameters
The two parameters that govern the percolation rate between the upper and lower groundwater
zones are the soil's saturated hydraulic conductivity Ks and the coefficient HCO that characterizes
the exponential decrease in hydraulic conductivity with decreasing moisture content. The most
accurate way of estimating these parameters is from laboratory tests that measure hydraulic
conductivity K as a function of soil moisture content #for the particular soil under consideration.
Such data for three particular soils - sand, sandy loam, and silty loam - are shown in Figure 5-5.
They were generated from disturbed soil samples under desaturation (draining) conditions (see
Brooks and Corey (1964) and Laliberte et al. (1966)). In some cases (e.g., sand), K(6) may range
through several orders of magnitude. Soils data of this type are becoming more readily available;
for example, soil science departments at universities often publish such information (e.g., Carlisle
etal., 1981).
151
-------
Touchet Silt Loam
0.5
-0.4
I 0.3
•c
| 0.2
Q 0.1
I 0
B
0.1 0.2 0.3 0.4
Moisture Content (fraction)
0.5
0.6
_0.7
;= 0.6
J=-0.5
.•|0.4
"G 0.3
"= 02
Columbia Sandy Loam
z
0.1 0.2 0.3 0.4
Moisture Content (fraction)
0.5
0.6
Unconsolidated Sand
Ivdraulk: Conductivity (in/hr)
fin -
/Eh-ea
/
/
^^
LJILLJI *- 1 1 1 1
0 0.1 0.2 O.3 0.4 0.5
Moisture Content (fraction)
Figure 5-5 Measured hydraulic conductivity for three soils.
When soil data like this are available, Ks and HCO can be estimated by fitting Equation 5-20 to
the data, i.e., fitting a straight line to the plot of the logarithm of K versus 9. The fits are not
152
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optimal over the entire data range because the fit is only performed for the high moisture content
region between field capacity and porosity.
When laboratory data are not available general estimates of Ks based on soil texture class can be
obtained from Tables 5-3, 5-4, and 5-5. Another alternative is the regression equation derived by
Saxton and Rawls (2006) from the same soils data base used to derive the moisture limit equations
listed in Table 5-7. The equation for Ks (in/hr) is:
Ks = 76(0 - 6FC^~V (5-24)
(n \
-^-) and = soil porosity, OFC = field capacity and OWP = the wilting point.
»WP/
This equation is also included in the SPAW soil water characteristics calculator described in the
previous section and shown in Figure 5-4. The estimates of Ks it produces for the different soil
classes are shown in Table 5-8. For the Woodburn silt loam soil in Section 5.5.1, it estimates a
saturated hydraulic conductivity of 0.48 in/hr (see Figure 5-4). This value falls within the range of
0.2 - 2.0 in/hr (1.4 - 14 jim/sec) listed in the Physical Properties report of Figure 4-1.
HCO can be estimated by utilizing Campbell's theoretical power law relation (Campbell, 1974) as
described in Saxton and Rawls (2006):
3+2/A
Ks - (5-25)
One can then estimate a value for HCO that gives a best fit between Equation 5-19 and Equation
5-25 as 6 ranges between and OFC. Figure 5-6 shows one such fit for the soil limits associated
with the Woodburn Silt Loam discussed earlier (= 0.482, 9pc= 0.321, and OWP= 0.137). The
data points come from evaluating Equation 5-25 for a series of different moisture levels 9. The
line of best fit that passes through the origin has a slope of 28.864 which would be the estimate of
HCO for this soil.
153
-------
Fit of exp(-HCO((^9) to
y=28.864x
R2= 0.9945
0.05
0.1
x=(4>-0)
0.15
0.2
Figure 5-6 Fitting SWMM's hydraulic conductivity equation to a power law equation.
Repeating this fitting process for the sand and clay content of the various standard soil classes
under a variety of organic contents, using the SPAW calculator to estimate the associated moisture
limits, produced the following regression estimate for HCO:
ECO = 0.48(%Sand) + 0.85(%CZay) R2 = 0.99
The resulting HCO values for the different soil classes are shown in Table 5-9.
(5-26)
A third percolation parameter, DP, governs the rate of at which water is lost from the lower
saturated zone by seepage through a confining layer into a deeper groundwater aquifer. DP
essentially represents the saturated hydraulic conductivity of this confining bottom layer and will
therefore typically have very low values, similar to those for compacted clay soils. If water table
measurements are available, DP can also be estimated from the rate at which the water table
elevation drops over a prolonged dry period.
154
-------
Table 5-9 Estimated HCO for different soil types
Soil Type
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Sandy clay loam
Clay loam
Silty clay loam
Sandy clay
Silty clay
Clay
Percent Sand
92
82
65
42
20
60
33
10
52
7
30
Percent Clay
5
6
10
18
20
28
34
34
42
47
50
HCO
48
44
40
35
27
53
45
34
61
43
57
5.5.3 ET Parameters
The two evapotranspiration parameters used by the groundwater routine are GET, the fraction of
the available evaporation apportioned to the upper unsaturated zone, and DET, the depth from the
ground surface below which no lower zone ET is possible (ft). The total rate available for
subsurface evaporation is the external evaporation rate supplied to the program for the current
month or day (see Section 2.5) minus the rate used for surface evaporation (Equation 5-12). The
GET parameter determines what fraction of this remaining evaporation rate is used in the upper
subsurface zone. In general, higher GET values will be associated with looser soils, lower water
table elevations, and surface vegetation with shallow root zones.
The amount of ET available to the saturated lower zone is 1 - GET of the total subsurface available
ET. The fraction of this amount actually utilized is proportional to the height that the water table
rises above a depth DET measured from the ground surface. DET is the maximum depth from
which water may be removed by evapotranspiration. Because the lower zone is saturated, ET
losses reflect mainly plant transpiration. Where surface vegetation is present, DET should at least
equal the expected average depth of root penetration. The influence of plant roots usually extends
somewhat below the depth of root penetration because of capillary suction to the roots. The depth
of capillary draw to the surface without vegetation may be 4 to 8 inches for sands, about 8 to 18
inches in silts, and in clays about 12 to 60 inches. Rooting depth is dependent on many factors —
155
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species, moisture availability, maturation, soil type, and plant density. In humid areas where
moisture is readily available near the surface, grasses may have rooting depths of 6 to 24 inches.
In drier areas, the rooting depth is very sensitive to plant species and to the depth to which moisture
is stored and may range from 6 to 48 inches. The evaporative zone depth would be somewhat
greater than the rooting depth. The local Agricultural Extension Service office can provide
information on characteristic rooting depths for vegetation in specific areas. Table 5-10 presents
values of DET for different combinations of soil type and ground cover that were derived from
unsaturated-saturated flow simulations (Shah et al., 2007).
Table 5-10 DET (in feet) for different soil types and land cover (Shah et al., 2007)
Soil Type
Sand
Loamy Sand
Sandy Loam
Sandy clay loam
Sandy clay
Loam
Silty clay
Clay loam
Silt loam
Silt
Silty clay loam
Clay loam
Bare Soil
2
2
4
7
7
9
11
13
14
14
15
20
Grass
5
6
8
10
10
12
14
17
17
17
18
23
Forest
8
9
11
13
13
15
17
20
20
21
21
27
5.5.4 Groundwater Discharge Constants
The groundwater discharge constants Al, Bl, A2, B2, and A3 appear in Equation 5-22 and
determine the rate of groundwater exchange with a specific node in the conveyance system. The
equation is repeated here for easy reference:
/c = Al(dL -
-A2(hsw-
A3dLhsw
(5-27)
156
-------
where the heights di, hsw, and h* are defined in Figure 5-2. Because of its general nature this
equation can assume a variety of functional forms. Several specific examples will now be
discussed.
Linear Reservoir
The saturated groundwater zone can be thought of as a storage reservoir whose lateral outflow is
linearly proportional to the water table depth di. Two cases are possible - with and without surface
water interaction. Without surface water interaction, the groundwater flow rate is simply:
/G=41(dL-/i') (5-28)
In terms of Equation 5-27 this implies thatAl > 0, Bl = 7, and A2 = A3 = 0. Note that the user-
supplied value of Al would be expressed as cfs/ac-ft for US units and cms/ha-m for metric units.
With surface water interaction, the groundwater flow rate is proportional to the difference between
the groundwater table height and the surface water height:
fG=Al(dL-hsw) (5-29)
which can be achieved with Al = A2 > 0, Bl = B2 = 7, and A3 = O.A1 would have the same units
as before (cfs/ac-ft or cms/ha-m). Because both of these cases are empirical simplifications, Al
would have to be determined through model calibration against observed groundwater table and
conveyance system head measurements.
Dupuit-Forcheimer Lateral Seepage
Under the assumption of uniform infiltration and horizontal flow by the Dupuit-Forcheimer
approximation, the relationship between water table elevation and groundwater flow rate for the
configuration shown in Figure 5-7 is (Bouwer, 1978, p. 51):
where Ks is the saturated hydraulic conductivity and the other parameters are defined in Figure 5-
7.
While h2 is the same as the surface water height hsw, hi is the maximum groundwater table height.
The height di that SWMM computes is only an average over the catchment. One can, however,
assume this average is equivalent to the average of hi and fc, i.e.:
157
-------
dL =
i +h2
(5-31)
so that hi = 2dL — h2. Substituting this and h2 = hsw into Equation 5-30 and simplifying terms
results in:
Flow
Impermeable
Figure 5-7 Definition sketch for Dupuit-Forcheimer seepage to an adjacent channel.
/2KS\ ,
= br«'-
(5-32)
Comparing Equation 5-32 with Equation 5-27 shows that the two will be equivalent if Al = -A3 =
2Ks/L2, A2 = 0, Bl = 2, and h* = 0. Note that Equation 5-30 is only valid for unidirectional flow
into the receiving node, but because A3 ^ o, SWMM will set/? to 0 should di drop below hsw.
Hooghoudt's Equation for Tile Drainage
The geometry of a tile drainage installation is illustrated in Figure 5-8. Hooghoudt's relationship
(Bouwer, 1978, p. 295) among the indicated parameters is
= (2De+m)4Ksm/L2
(5-33)
158
-------
where De = effective depth of the impermeable layer below the drain center, and the other
parameters are defined in Figure 5-8. De is less than or equal to bo in Figure 5-8 and is a function
of bo, drain diameter, and drain spacing L; the complicated relationship is given by Bear (1972, p.
412) and graphed by Bouwer (1978, p. 296).
t * . k t t
t t * t l J
^
// 1\\ \
^x.
Impermeable
7 / > t\ ^ V;
I ^ ^
*
'/ \ ^ \\i/t\\\
i. >-
\
/ / / \ \/ / /
Figure 5-8 Definition sketch for Hooghoudt's method for flow to circular drains.
From Figure 5-8, the maximum rise of the water table, m, is:
m = hi-b0 (5-34)
Once again approximating the average water table depth above the impermeable layer by:
dL =
(5-35)
results in:
m = 2(dL - b0}
Substituting 5-36 into 5-33 gives:
(5-36)
fc =
- b0)2 - Deb0 + DedL]
(5-37)
159
-------
This can be written in a format compatible with the general groundwater discharge equation 5-27
as follows:
fG = Al(dL - /i*)2 -A2+ A3dLhsw (5-38)
where
Al = 16KS/L2,
51 = 2,
A2 = AWeb0,
B2 = 0,
A3 = Al(De/b0l
h* is set equal to bo and a constant value ofhsw only slightly higher than bo is used.
The internal units of both Al and A3 are (ft-s)"1 while A2 has units of ft/s. In terms of the program
input though, where fo is expressed as flow per acre (or per hectare), the units on^47 and^3 would
be would be ft/s/ac (or m/s/ha) and for A2 would be ft3/s/ac (or m3/s/ha). Since A3 7= 0, flow back
into the groundwater zone would not be allowed should di drop below bo. The mathematics of
drainage to ditches or circular drains is complex; several alternative formulations are described by
van Schilfgaarde (1974).
5.6 Numerical Example
A simple numerical example will help illustrate the effect that groundwater can have on the runoff
generated from a subcatchment. It is a variation of the runoff example used in Section 3.10 which
consists of a single relatively flat, completely pervious subcatchment containing a well-drained
Group B soil that is subjected to a 2-inch, 6-hour rain event. The subsurface zone beneath the
subcatchment extends to a depth of 6 feet and its initial water table height is 3.5 feet. Because a
conveyance node is required to complete a groundwater model, a single such node is included that
receives both the surface runoff and the groundwater flow from the subcatchment. Its invert
elevation is 0.5 feet above the initial water table level. The Linear Reservoir form of the
groundwater discharge equation without surface water interaction is used. Table 5-11 summarizes
the pertinent parameters for this example. The initial moisture content of the unsaturated zone is
0.4, midway between fully saturated and fully drained.
160
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Table 5-11 Parameters used in groundwater example
Item
Subcatchment
Rainfall Event
Horton Infiltration
Groundwater
Parameter
Percent Impervious
Percent Slope
Width (ft)
Roughness
Depression Storage (in)
Duration (hr)
Total Depth (in)
Time-to-Peak / Duration
Evaporation Rate (in/hr)
Initial Capacity (in/hr)
Ultimate Capacity (in/hr)
Decay Coefficient (hr"1)
Porosity
Field Capacity
Wilting Point
Saturated Hydraulic Conductivity (in/hr)
Conductivity Curve Parameter (HCO)
Deep Percolation Constant (DP)
Reference Depth (A*) (ft)
GW Flow Coefficient (Al) (cfs/ac-ft)
GW Flow Exponent (Bl)
A2, A3 and B2
Value
0
0.5
140
0.1
0.05
6.0
2.0
0.375
0.0
1.2
0.1
2.0
0.5
0.3
0.15
0.1
12.0
0.002
4.0
0.5
1.0
0.0
The surface runoff and the groundwater flow seen by the outlet node over a 24-hour simulation
period are shown in Figure 5-9. The surface runoff is unaffected by inclusion of the subsurface
zones, since the upper zone never fully saturates. Its hydrograph looks the same as for the example
in Section 3.10 (see the pervious curve in Figure 3-12). However, as the infiltrated water percolates
through the upper soil zone, the depth of the lower saturated zone rises and begins to produce
groundwater outflow into the receiving node. This outflow continues long after the surface runoff
ceases, creating an extended recession limb on the total outflow hydrograph.
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Rainfall
Surface Runoff
•Groundwater Flow
•Total Outflow
0.7
0.6
g 0.4
_o
LL.
° 0.3
Figure 5-9 Surface runoff and groundwater flow for the illustrative groundwater example.
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Chapter 6 - Snowmelt
6.1 Introduction
Snowmelt is an additional mechanism by which urban runoff may be generated. Although flow
rates are typically low, they may be sustained over several days and remove a significant fraction
of pollutants deposited during the winter. Rainfall events superimposed upon snowmelt baseflow
may produce higher runoff peaks and volumes as well as add to the melt rate of the snow. In the
context of long term continuous simulation, runoff and pollutant loads are distributed quite
differently in time between the cases when snowmelt is and is not simulated. The water and
pollutant storage that occurs during winter months in colder climates cannot be simulated without
including snowmelt.
As part of a broad program of testing and adaptation to Canadian conditions, a snowmelt routine
was placed in SWMM for single event simulation by Proctor and Redfern, Ltd. and James F.
MacLaren, Ltd., abbreviated PR-JFM (1976a, 1976b, 1977), during 1974-1976. The basic melt
computations were based on routines developed by the U.S. National Weather Service, NWS
(Anderson, 1973). The current SWMM implementation utilizes the Canadian SWMM snowmelt
routines as a starting point and extends their capabilities to model long term continuous
simulations. In addition, features were added to adapt the snowmelt process to urban conditions,
since the snowmelt routines used in other watershed runoff models are aimed primarily at
simulation of spring melts in large river basins. The work of the National Weather Service
(Anderson, 1973, 2006) as reflected in their SNOW-17 model was heavily utilized, especially for
the extension to continuous simulation and the resulting inclusion of cold content, variable melt
coefficients, and areal depletion.
Several hydrologic models include snowmelt computations, e.g., Stanford Watershed Model
(Crawford and Linsley, 1966), HSPF (Bicknell et al., 1997), NWS (Anderson, 1973, 1976),
STORM (Corps of Engineers, 1977; Roesner et al., 1974), SSARR (Corps of Engineers, 1971),
and PRMS (Leavesley et al., 1983). Useful summaries of snowmelt modeling techniques are
available in texts by Eagleson (1970), Gray (1970), Fleming (1975), Linsley et al. (1975), Bedient
et al. (2013), and Viessman and Lewis (2003). All of these draw upon the classic work, Snow
Hydrology, of the Corps of Engineers (1956).
A review of snowmelt components of urban drainage models has been performed by Semadeni-
Davies (2000). Three models were reviewed in some detail: SWMM (version 4), MouseNAM
(Danish Hydraulic Institute, 1994), and HBV (Bergstrom, 1976; Lindstrom et al., 1997).
Semadeni-Davies (2000) concludes that urban snowmelt routines (including those in SWMM)
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have been adapted directly from models developed for rural situations and therefore may not
represent urban conditions well. Degree-day methods are used in all three models that she
reviewed, and only limited information is available regarding coefficients in urban areas. Plowing
and piling of snow in urban areas, and the change in the nature of its albedo and density are also
important considerations, for which SWMM includes options for their representation. Overall,
SWMM appears to be no better - and no worse - than the other two models reviewed. The
descriptions of SWMM snowmelt algorithms that follow do not reflect any general improvements
recommended by Semadeni-Davies (2000).
6.2 Preliminaries
6.2.1 Snow Depth
SWMM treats all snow depths as "depth of water equivalent" to avoid specification of the specific
gravity of the snow pack, which is highly variable with time. The specific gravity of new snow is
of the order of 0.09; an 11:1 or 10:1 ratio of snow pack depth to water equivalent depth is often
used as a rule of thumb. With time, the pack compresses until the specific gravity can be
considerably greater, to 0.5 and above. In urban areas, lingering snow piles may resemble ice more
than snow with specific gravities approaching 1.0. Although snow pack heat conduction and
storage depend on specific gravity, sufficient accuracy may be obtained without involving specific
gravity. It is adequate to maintain continuity through the use of depth of water equivalent. Most
input parameters are in units of inches or mm of water equivalent (in w.e., or mm w.e.). For all
internal computations, conversions are made to feet of water equivalent.
6.2.2 Me teorological Inputs
Snowfall rates are determined directly from precipitation inputs by using a dividing temperature
SNOTMP. If the current air temperature is at or below SNOTMP, the precipitation falls as snow.
Otherwise it falls as rain. In natural areas, a surface temperature of 34° to 35°F (1-2°C) provides
the dividing line between equal probabilities of rain and snow (Eagleson, 1970; Corps of
Engineers, 1956). However, this separation temperature might need to be somewhat lower in urban
areas due to warmer surface temperatures.
Precipitation gages tend to produce inaccurate snowfall measurements because of the complicated
aerodynamics of snowflakes falling into the gage. Snowfall totals are generally underestimated as
a result, by a factor that varies considerably depending upon gage exposure, wind velocity and
whether or not the gage has a wind shield. The program includes a multiplier for each Rain gage
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object, the Snow Catch Factor (SCF), which adjusts for these effects. The SCF is only applied
when precipitation falls as snow.
Although it will vary considerably from storm to storm, SCF acts as a mean correction factor over
a season in the model. Anderson (1973) provides typical values of ^CFas a function of wind speed,
as shown in Figure 6-1, which may be helpful in establishing an initial estimate. The value of SCF
can also be used to account for other factors, such as losses of snow due to interception and
sublimation not accounted for in the model. Anderson (1973) states that both losses are usually
small compared to the gage catch deficiency.
As discussed in Section 2.3, air temperature data is supplied to a SWMM data set from either a
user-generated time series or from a climate file. If a time series is used, the entries represent
instantaneous temperature readings at given points in time. Linear interpolation is used to obtain
temperature values for times that fall in between those recorded in the time series. If a climate file
is used, then a continuous record of maximum and minimum daily temperatures is provided. The
sinusoidal interpolation method described in Section 2.4 is used to obtain an instantaneous value
at any point in time during a day based on the day's max-min values. (Any missing days in the
record are filled in with the max-min values from the previous day).
During the simulation, melt is generated at each time step using a degree-day type equation during
dry weather and a heat balance equation during rainfall periods. The latter equation makes an
adjustment for wind speed (higher melt rates at higher wind speeds). The input of wind speed data
to the program was discussed in Section 2.4. There are two options: 1) as average values for each
month of the year, or 2) as daily values from the same climate file used to supply daily max-min
temperatures. Should wind speed data not be available, the adjustment to the melt equation is
simply ignored.
The coefficients used in the degree-day melt equation vary sinusoidally, from a maximum on June
21 to a minimum on December 21. In addition, a record of the cold content of the snow is
maintained. Thus, before melt can occur, the pack must be "ripened," that is, heated to a specified
base temperature. Specified, constant areas of each subcatchment may be designated as snow
covered, or, following the practice of melt computations in natural basins, "areal depletion curves"
may be used to describe the spatial extent of snow cover as the pack melts. For instance, shaded
areas would be expected to retain a snow cover longer than exposed areas. Thus, the snow covered
area of each subcatchment changes with time during the simulation. Melt, after routing through
the remaining snow pack, is combined with rainfall to form the spatially weighted "effective
rainfall" for overland flow routing.
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Snow-17 Model
Precipitation Catch Deficiency
CD
O
-------
Snowmelt Sub-Areas:
SA1 = pervious
SA2 = plowable impervious
SA3 = remaining impervious
Runoff Sub-Areas:
Al = pervious (= SA1)
A2 = impervious w/
depression storage
A3 = impervious w/o
depression storage
Figure 6-2 Subcatchment partitionings used for snowmelt and runoff.
A separate accounting is kept for snow accumulation and melting from each of these fractions
(pervious, plowable impervious, and remaining impervious). After snowmelt calculations are
made at the start of each time step, the net precipitation over the plowable and remaining
impervious areas are summed together and then, for the purpose of computing runoff, are re-
distributed between the fractions of impervious areas with and without depression storage. Because
the pervious areas for runoff and snowmelt are the same, the snowmelt result over this sub-area
can be used directly for computing pervious area runoff.
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6.2.4 Redistribution and Snow Removal
Snow removal practices form a major difference between the snow hydrology of urban and rural
areas. Much of the snow cover may be completely removed from heavily urbanized areas, or
plowed into windrows or piles, with melt characteristics that differ markedly from those of
undisturbed snow. Management practices in cities vary according to location, climate, topography
and the storm itself; they are summarized in Table 6.1. It is probably not possible to treat them all
in a simulation model. However, provision is made to simulate approximately some of these
practices.
6.2.5 Effect on Infiltration
A snow pack tends to insulate the surface beneath it. If ground has frozen prior to snowfall, it will
tend to remain so, even as the snow begins to melt. Conversely, unfrozen ground is generally not
frozen by subsequent snowfall. The infiltration characteristics of frozen versus unfrozen ground
are not well understood and depend upon the moisture content at the time of freezing. For these
and other reasons, SWMM assumes that snow has no effect on infiltration or other parameters,
such as surface roughness or detention storage (although the latter is altered in a sense through the
use of the free water holding capacity of the snow). In addition, all heat transfer calculations cease
when the water becomes "net runoff. Thus, water in temporary surface storage during the
overland flow routing will not refreeze as the temperature drops and is also subject to evaporation
beneath the snow pack.
It is assumed that all snow subject to "redistribution", (e.g., plowing) resides on a user-specified
fraction of the total impervious area (area SA2 in Figure 6-2) that might consist of streets,
sidewalks, parking lots, etc. (The desired degree of definition may be obtained by using several
subcatchments, although a coarse schematization, e.g., one or two subcatchments, may be
sufficient for some continuous simulations.) The following five parameters, which can vary by
subcatchment, govern how snow is removed or re-distributed from this sub-area:
Fimp: fraction of current snow transferred to the remaining impervious sub-area (SA3)
Fperv. fraction of current snow transferred to the pervious area (SA1)
Fsub: fraction of current snow transferred to the pervious area of another designated
subcatchment.
Font: fraction of current snow transferred out of the watershed
Fimelt: fraction of current snow converted into immediate melt
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An instantaneous redistribution of the current snow depth begins when the latter exceeds the user-
supplied parameter WEPLOW.
Fimp orFperv are used if snow is usually windrowed onto adjacent impervious or pervious areas.
If it is trucked to the pervious area of another subcatchment, the fraction Fsub will so indicate, or
Font can be used if the snow is removed entirely from the simulated watershed. In the latter case,
such removals are tabulated and included in the final continuity check. Finally, excess snow may
be immediately "melted" (i.e., treated as rainfall), using Fimelt. The five fractions can sum to less
than 1.0 in which case some residual snow will remain on the surface. See Table 6-1 for guidelines
on typical levels of service for snow and ice control (Richardson et al., 1974). The snow
redistribution process does not account for snow management practices that use chemicals, such
as roadway salting. This is handled using the melt equations, as described subsequently.
No pollutants are transferred with the snow. The transfers listed above are assumed to have no
effect on pollutant washoff and regeneration. In addition, all the redistribution parameters remain
constant throughout the simulation and can only represent averages over a snow season.
6.3 Governing Equations
6.3.1 Overview
Excellent descriptions of the processes of snowmelt and accumulation are available in several texts
and simulation model reports and in the well-known 1956 Snow Hydrology report by the Corps of
Engineers. The important heat budget and melt components are mentioned briefly here; any of the
above sources may be consulted for detailed explanations. A brief justification for the techniques
adopted for snowmelt calculations in SWMM is presented below.
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Table 6-1 Guidelines for level of service in snow and ice control (Richardson et al., 1974)
Road Classification
Low-Speed
Multilane Urban
Expressway
Level of Service
Roadway
storms
All traffic
routinely patrolled during
lanes treated with chemicals
Snow
Depth to
Start
Plowing
(Inches)
0.5 to 1
Max.
Snow
Depth on
Pavement
(Inches)
1
Full
Pavement
Clear of
Snow
After
Storm
(Hours)
1
Full
Pavement
Clear of
Ice
After
Storm
(Hours)
12
All lanes (including breakdown lanes)
operable at all times but at reduced
speeds
Occasional patches of well sanded
snow pack
Roadway repeatedly cleared by
echelons of plows to minimize traffic
disruption
Clear pavement obtained as soon as
possible
High Speed 4-Lane
Divided Highways;
Interstate System;
ADT greater than
10,000a
Roadway routinely patrolled during 1
storms
Driving and passing lanes treated with
chemicals
Driving lane operable at all times at
reduced speeds
Passing lane operable depending on
equipment availability
Clear pavement obtained as soon as
possible
1.5
12
Primary Highways;
Undivided 2 and 3
lanes;
ADT 500-50003
Roadway is routinely patrolled during 1
storms
Mostly clear pavement after storm
stops
Hazardous areas receive treatment of
chemicals or abrasive
Remaining snow and ice removed
when thawing occurs
2.5
24
Secondary Roads
ADT less than 500a
Roadway is patrolled at least once
during a storm
Bare left-wheel track with intermittent
snow cover
Hazardous areas are plowed and
treated with chemicals or abrasives as
a first order of work
Full width of road is cleared as
equipment becomes available
48
aADT - average daily traffic
Snowpack Heat Budget
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Heat may be added or removed from a snowpack by the following processes:
• Absorbed solar radiation (addition).
• Net long wave radiation exchange with the surrounding environment (addition or removal).
• Convective (diffusive) transfer of sensible heat from/to air (addition or removal).
• Release of latent heat of vaporization by condensate (addition) or, the opposite, its removal
by sublimation (removing the latent heat of vaporization plus the latent heat of fusion).
• Advection of heat by rain (addition) plus addition of the heat of fusion if the rain freezes.
• Conduction of heat from underlying ground (removal or addition).
The terms may be summed, with appropriate signs, and equated to the change of heat stored in the
snowpack to form a conservation of heat equation. All of the processes listed above vary in relative
importance with topography, season, climate, local meteorological conditions, etc., but items 1-4
are the most important. Item 5 is of less importance on a seasonal basis, and item 6 is often
neglected. A snow pack is termed "ripe" when any additional heat will produce liquid runoff.
Rainfall (item 5) will rapidly ripen a snowpack by release of its latent heat of fusion as it freezes
in subfreezing snow, followed by quickly filling the free water holding capacity of the snow.
Melt Prediction Techniques
Prediction of melt follows from prediction of the heat storage of the snow pack. Energy budget
techniques are the most exact formulation since they evaluate each of the heat budget terms
individually, requiring as meteorological input quantities such as solar radiation, air temperature,
dew point or relative humidity, wind speed, and precipitation. Assumptions must be made about
the density, surface roughness and heat and water storage (mass balance) of the snow pack as well
as on related topographical and vegetative parameters. Further complications arise in dealing with
heat conduction and roughness of the underlying ground and whether or not it is permeable.
Several models treat some or all of these effects individually, for instance, the NWS river forecast
system developed by Anderson (1976). Interestingly, under many conditions he found that results
obtained using his energy balance model were not significantly better than those obtained using
simpler (e.g., degree-day or temperature-index) techniques in his earlier model (Anderson, 1973).
The more open and variable the condition, the better is the energy balance technique. Closest
agreement between his two models was for heavily forested watersheds.
The minimal data needed to apply an energy balance model are a good estimate of incoming solar
radiation, plus measurements of air temperature, vapor pressure (or dew point or relative humidity)
and wind speed. All of these data, except possibly solar radiation, are available for at least one
location (e.g., the airport) for almost all reasonably sized cities. Even solar radiation measurements
171
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are taken at several locations in most states. Predictive techniques are also available, for solar
radiation and other parameters, based on available measurements (TVA, 1972; Franz, 1974).
Choice of Predictive Method
Two major reasons suggest that simpler, e.g., temperature-index, techniques should be used for
simulation of snowmelt and accumulation in urban areas. First, even though required
meteorological data for energy balance models are likely to be available, there is a large local
variation in the magnitude of these parameters due to the urbanization itself. For example, radiation
melt will be influenced heavily by shading of buildings and albedo (reflection coefficient) reduced
by urban pollutants. In view of the many unknown properties of the snowpack itself in urban areas,
it may be overly ambitious to attempt to predict melt at all! But at the least, simpler techniques are
probably all that are warranted. They have the added advantage of considerably reducing the
already extensive input data to a model such as SWMM.
Second, the objective of the modeling should be examined. Although it may contribute, snowmelt
seldom causes flooding or hydrologic extremes in an urban area itself. Hence, exact prediction of
flow magnitudes does not assume nearly the importance it has in the models of, say, the NWS, in
which river flood forecasting for large mountainous catchments is of paramount importance. For
planning purposes in urban areas, exact quantity (or quality) prediction is not the objective in any
event; rather, these efforts produce a statistical evaluation of a complex system and help identify
critical time periods for more detailed analysis.
For these and other reasons, simple snowmelt prediction techniques are incorporated into SWMM.
Anderson's NWS (1973) temperature-index method is also well documented and tested, and is
used in SWMM. As described subsequently, the snowmelt modeling follows Anderson's work in
several areas, not just in the melt equations. It may be noted that the STORM model (Corps of
Engineers, 1977; Roesner et al., 1974) also uses the temperature-index method for snowmelt
prediction, in a considerably less complex manner than is programmed in SWMM.
6.3.2 Melt Equations
Anderson's NWS model (1973) treats two different melt situations: with and without rainfall.
When there is rainfall (greater than 0.1 in/hr or 2.5 mm/hr in the NWS model, (greater than 0.02
in/hr or 0.51 mm/hr in SWMM), accurate assumptions can be made about several energy budget
terms. These are: zero solar radiation, incoming long wave radiation equals blackbody radiation at
the ambient air temperature, the snow surface temperature is 32° F (0° C), and the dew point and
172
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rain water temperatures equal the ambient air temperature. Anderson combines the appropriate
terms for each heat budget component into one equation for the melt rate SMELT:
SMELT = (0.001167 + 7.5yUA + 0.007i)(Ta - 32) + B.5UA(ea . .
(6-1)
- 0.18) V '
where
SMELT = melt rate (in/hr)
Ta = air temperature (° F)
y = psychrometric constant (in Hg/° F)
UA = wind speed adjustment factor (in/in Hg - hr)
/' = rainfall intensity (in/hr)
ea = saturation vapor pressure at air temperature (in Hg).
The origin of the numerical constants found in Equation 6-1 is given by Anderson (1973), and
reflect units conversions as well as U.S. customary units for physical properties. The
psychrometric constant, y, is calculated as:
Y = 0.000359Pa (6-2)
where Pa is the atmospheric pressure (in Hg). The latter, in turn, is calculated as a function of
elevation, z:
Pa = 29.9 - 1.02(z/1000) + 0.0032(z/1000)24 (6-3)
where z is the average catchment elevation (ft). The wind adjustment factor, UA, accounts for
turbulent transport of sensible heat and water vapor. Anderson (1973) gives:
UA = 0.006J7 (6-4)
where C/is the average wind speed 1.64 ft (0.5 m) above the snow surface (mi/hr). In practice,
available wind data are used and are seldom corrected for the actual elevation of the anemometer.
Section 6.2.2 (as well as Section 2.6) discusses how wind data are supplied to SWMM. If no such
data are available on a particular date then UA is set equal to 0. Finally, the saturation vapor
pressure, ea, is given accurately by the convenient exponential approximation:
-7701.544
= 8.1175
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During non-rain periods, melt is calculated as a linear function of the difference between the air
temperature, Ta, and a base temperature, Tbase, using a degree-day or temperature-index type
equation:
SMELT = DHM(Ta - Tbase) (6-6)
where:
SMELT = melt rate (in/hr),
Ta = air temperature (° F)
Tbase = base melt temperature (° F)
DHM = melt coefficient (in/hr-0 F)
Different values of Tbase and DHM may be used for each of the three types of snow surfaces
within a subcatchment. For instance, these parameters may be used to account for street salting,
which lowers the base melt temperature. If desired, rooftops could be simulated using a lower
value of Tbase to reflect heat transfer vertically through the roof. Suggested values for Tbase and
DHM are provided in the Parameter Estimates section (6.7) below.
During the simulation, Tbase remains constant, but DHM is allowed a seasonal variation, as
illustrated in Figure 6-3. Following Anderson (1973), the minimum melt coefficient is assumed to
occur on December 21 and the maximum on June 21. Parameters DHMIN and DHMAX are
supplied as input for the three snowpack areas of each subcatchment, and sinusoidal interpolation
is used to produce a value of DHM that is constant over each day of the year:
DHMAX + DHMIN\ /DHMAX - DHMIN
where
DHMIN = minimum melt coefficient, occurring Dec. 21 (in/hr-°F)
DHMAX = maximum melt coefficient, occurring June 21 (in/hr-°F)
day = number of the day of the year.
No special allowance is made for leap year. However, the correct date (and day number) is
maintained.
174
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DHM
250
300 350
200
Dec, 21
Day of Year
June 2 [
Figure 6-3 Seasonal variation of melt coefficients.
6.3.3 Snow Pack Heat Exchange
During subfreezing weather, the snow pack does not melt, and heat exchange with the atmosphere
can either warm or cool the pack. The difference between the heat content of the subfreezing pack
and the (higher) base melt temperature is taken as positive and termed the "cold content" of the
pack. No melt will occur until the cold content is reduced to zero. It is maintained in inches (or
feet) of water equivalent. That is, a cold content of 0.1 in. (2.5 mm) is equivalent to the heat
required to melt 0.1 in. (2.5 mm) of snow. Following Anderson (1973), the heat exchange altering
the cold content within each 6-hour period is proportional to the difference between the air
temperature, Ta, and an antecedent temperature index, ATI, indicative of the temperature of the
surface layer of the snow pack. The value of^477is updated at the start of each time step as follows:
ATI <- ATI + TIPMt(Ta - ATI)
where TIPMt is given by (Anderson, 2006):
TIPMt = 1 - (1 - 7YPM)At/6
(6-8)
(6-9)
for a time step At in hours. 77PM is a 6-hour weighting factor whose value lies between 0 and 1.0.
The value of ^4 77 is not allowed to exceed Tbase., and when snowfall is occurring, ATI takes on the
current air temperature.
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The weighting factor TIPM is a user-supplied constant that applies over the entire watershed. It is
an indication of the thickness of the "surface" layer of snow. Values less than 0.1 give significant
weight to temperatures over the past week or more and would thus indicate a deeper layer than
values greater than, say, 0.5, which would essentially only give weight to temperatures during the
past day. In other words, the pack will both warm and cool more slowly with low values of TIPM.
Anderson states that TIPM = 0.5 has given reasonable results in natural watersheds, although there
is some evidence that a lower value may be more appropriate. No calibration has been attempted
on urban watersheds.
After the antecedent temperature index is calculated, the cold content COLDC is changed by an
amount
ACC = RNM X DHM X (ATI - Ta~) X At (6-10)
where
ACC = change in cold content (inches water equivalent)
RNM = ratio of negative melt coefficient to melt coefficient,
DHM = melt coefficient (in/hr-° F)
ATI = antecedent temperature index (°F)
At = time step (hr).
Note that the cold content is increased, (ACC is positive) when the air temperature is less (colder)
than the antecedent temperature index. Since heat transfer during non-melt periods is less than
during melt periods, Anderson uses a "negative melt coefficient" in the heat exchange
computation. SWMM computes this simply as a fraction, RNM, of the melt coefficient, DHM.
Hence, the negative melt coefficient, i.e., the product RNMxDHMaho varies seasonally. As
with TIPM, a single user-supplied value of RNMis used throughout the study area. A typical value
is 0.6.
During melting periods, cold content of the pack is reduced by an amount:
ACC = -SMELT X RNM X At (6-11)
with an equal reduction made in SMELT. Thus no liquid melt actually occurs until the snow pack
cold content is reduced to 0. Even then, runoff will not occur, until the "free water holding
capacity" of the snow pack is filled. This is discussed subsequently. The value of COLDC is in
units of inches of water equivalent over the area in question. The cold content "volume," equivalent
to calories or BTUs, is obtained by multiplying by the area. Finally, an adjustment is made to
Equations 6-10 and 6-11 depending on the areal extent of snow cover. This is discussed below.
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6.4 Areal Depletion
The snow pack on a catchment rarely melts uniformly over the total area. Rather, due to shading,
drifting, topography, etc., certain portions of the catchment will become bare before others, and
only a fraction, ASC, will be snow covered. This fraction must be known in order to compute the
snow covered area available for heat exchange and melt, and to know how much rain falls on bare
ground. Because of year to year similarities in topography, vegetation, drift patterns, etc., the
fraction, ASC, is primarily a function of the amount of snow on the catchment at a given time; this
function, called an "areal depletion curve", is discussed below. These functions are used as an
option to describe the seasonal growth and recession of the snow pack. For short, single event
simulation, fractions of snow covered area may be fixed for the pervious and impervious areas of
each subcatchment.
As used in most snowmelt models, it is assumed that there is a depth, ,57, above which there will
always be 100 percent cover. In some models, the value of ,57 is adjusted during the simulation; in
SWMM it remains constant. The amount of snow present at any time is indicated by the state
variable WSNOW, which is the depth (water equivalent) over each of the three possible snow
covered areas of each subcatchment (see Figure 6-2). This depth is made non-dimensional by
dividing it by ,57 for use in calculating ASC. Thus, an areal depletion curve (ADC) is a plot of
WSNOW/ SI versus ASC', a typical ADC for a natural catchment is shown in Figure 6-4. For values
of the ratio A WESI = WSNOW7 SI greater than 1.0, ASC = 1.0, that is, the area is 100 percent snow
covered.
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AWESl*
WSNOW/S1
TYPICAL
ADC FOR
NATURAL AREA
TEMPORARY ADC
FOR NEW SNOW
0.5 -
--SNEW
SBWS
AWE
0.5
ASC= AS/AT
Figure 6-4 Typical areal depletion curve for natural area (Anderson, 1973, p. 3-15) and
temporary curve for new snow.
Some of the implications of different functional forms of the ADC may be seen in Figure 6-5.
Since the program maintains snow quantities, WSNOW, as the depth over the total area, AT, the
actual snow depth, fFS', and actual area covered, AS, are related by continuity:
WSNOW
= WSxAS
(6-12)
where:
WSNOW
AT
WS
AS
depth of snow over total area (inches water equivalent)
total area (ft2),
actual snow depth (inches water equivalent), and
snow covered area (ft2).
In terms of parameters shown on the ADC, this equation may be rearranged to read:
178
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AW ESI = W SNOW/SI = (WS/SIXAS/AT) = (WS/SI)ASC (6-13)
This equation can be used to compute the actual snow depth, WS, from known ADC parameters,
if desired. It is unnecessary to do this in the program, but it is helpful in understanding the curves
of Figure 6-5. Thus:
WS = (AWESI/ASC) SI (6-14)
Consider the three ADC curves B, C and D of Figure 6-5. For curve B, AWESIis always less than
ASC; hence WS is always less than ,57 as shown in Figure 6-5d. For curve C,AWESI = ASC, hence
WS = SI, as shown in Figure 6-5e. Finally, for curve D, A WESIis always greater than ASC', hence,
WS is always greater than 57, as shown in Figure 6-5f Constant values of ASC at 100 percent cover
and 40 percent cover are illustrated in Figure 6-5c, curve A, and Figure 6-5g, curve E, respectively.
At a given time (e.g., tl in Figure 6-5), the area of each snow depth versus area curve is the same
and equal to AWES I X 57, (e.g., 0.8 SI for time tl).
Curve B on Figure 6-5a is the most common type of ADC occurring in nature, as shown in Figure
6-4. The convex curve D requires some mechanism for raising snow levels above their original
depth, SI. In nature, drifting provides such a mechanism; in urban areas, plowing and windrowing
could cause a similar effect. A complex curve could be generated to represent specific snow
removal practices in a city. However, the program uses only one ADC curve for all impervious
areas (e.g., area SA3 of Figure 6-2 for all subcatchments) and only one ADC curve for all pervious
areas (e.g., area SA1 of Figure 6-2 for all subcatchments). This limitation should not hinder an
adequate simulation since the effects of variations in individual locations are averaged out in the
city-wide scope of most continuous simulations.
The program does not require the ADC curves to pass through the origin, A WESI = ASC = 0; they
may intersect the abscissa at a value of ASC > 0 in order to maintain some snow covered area up
until the instant that all snow disappears (see Figure 6-4). However, the curves may not intersect
the ordinate, AWESI > 0 when ASC = 0.
The preceding paragraphs have centered on the situation where a depth of snow greater than or
equal to SI has fallen and is melting. (The ADC curves are not employed until J^WCWbecomes
less than SI.) The situation when there is new snow needs to be discussed, starting from both zero
and non-zero initial cover. The SWMM procedure again follows Anderson's NWS method (1973).
179
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AREAL DEPLETION CURVES
7m LO
AWESN
WSNOW/SI
7c A
7,
SI'
Tfc DEFINITION SKETCH
, -SMOW FftCK BARE
f SI I- if /[ CATCHMENT
SNOW
DEPTH
t
AS AT
AREA
0 Q5 LO
ASC'AS/AT
INITIAL MELT PROGRESSING
*• t|
AWES I* 1.0
WSNOW _ m AS
St SI * AT
si
AT
0.8
t
t
»l
0.6
0,3
1-L
SI
7e
7f D
si
Si-
AT
AT
AT
3 a
Figure 6-5 Effect of snow cover on areal depletion curves.
When there is new snow and WSNOW is already greater than or equal to SI, ASC remains
unchanged at LO. However, when there is new snow on bare or partially bare ground, it is assumed
that the total area is 100 percent covered for a period of time, and a "temporary" ADC is established
as shown in Figure 6-4. This temporary curve returns to the same point on the ADC as the snow
melts. Let the depth of new snow be SNO, measured in equivalent inches of water. Then the value
ofAWESIwill be changed from an initial value of AWE to a new value of SNEWby:
180
-------
SNEW = AWE + SNO/SI (6-15)
It is assumed that the areal snow cover remains at 100 percent until 25 percent of the new snow
melts. This defines the value of SBWS of Figure 6-4 as:
SBWS = AWE + 0.75(SJVO/S/) (6-16)
Anderson (1973) reports low sensitivity of model results to the arbitrary 25 percent assumption.
When melt produces a value of AWESI between SBWS and AWE, linear interpolation of the
temporary curve is used to find ASC until the actual ADC curve is again reached. When new snow
has fallen, the program thus maintains values of AWE, SBA and SBWS (Figure 6-4).
The interactive nature of melt and fraction of snow cover is not accounted for during each time
step. It is sufficient to use the value of ASC at the beginning of each time step, especially with a
short (e.g., one-hour) time step for the simulation.
The fraction of area that is snow covered, ASC, is used to adjust 1) the volume of melt that occurs,
and 2) the "volume" of cold content change, since it is assumed that heat transfer occurs only over
the snow covered area. The melt rate is computed from either of the two equations for SMELT.
The snow depth is then reduced by an amount AWSNOWwhich equals:
kWSNOW = SMELT X ASC X At (6-17)
and includes appropriate continuity checks to avoid melting more snow than is there, etc.
Cold content changes are also adjusted by the value of ASC. Thus, using Equation 6-10, cold
content, COLDC, is changed by an amount ACC given by:
ACC = RNM X DHM X (ATI - Ta~) X At X ASC (6-18)
where variables are as previously defined. Again there are program checks for negative values of
COLDC, etc.
181
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6.5 Net Runoff
Production of melt does not necessarily mean that there will be liquid runoff at a given time step
since a snow pack, acting as a porous medium with a "porosity," has a certain "free water holding
capacity" at a given instant in time. Conway and Benedict (1994) describe the physics of the
various processes underway as melt infiltrates into a snowpack. Following PR-JFM (1976a,
1976b), this capacity is taken to be a constant fraction, FWFRAC, of the variable snow depth,
WSNOW, at each time step. This volume (depth) must be filled before runoff from the snow pack
occurs. The program maintains the depth of free water, FW, inches of water, for use in these
computations. When FW = FWFRAC x WSNOW, the snow pack is fully ripe. The procedure is
sketched in Figure 6-6.
SNOW
PACK
WSNOW
FREE
WATER
FWFRAC'WSNOW
EXCESS = RUNOFF
Figure 6-6 Schematic of liquid water routing through snow pack.
182
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The inclusion of the free water holding capacity via this simple reservoir-type routing delays and
somewhat attenuates the appearance of liquid runoff. When rainfall occurs, it is added to the melt
rate entering storage as free water. No free water is released when melt does not occur, but remains
in storage, available for release when the pack is again ripe. This re-frozen free water is not
included in subsequent cold content or melt computations.
Melt from snow covered areas and rainfall on bare surfaces is area weighted and combined to
produce net runoff onto the surface as follows:
RI = ASC X SMELT + (1.0 - ASQ X i (6-19)
where,/?/is the net equivalent precipitation input onto the subcatchment surface (in/hr) and /' is the
liquid rainfall intensity (in/h). RI is used in place of the externally supplied rainfall value in
subsequent overland flow and infiltration calculations.
If immediate melt is produced through the use of the snow redistribution fraction Fimelt it is added
to the last equation. Furthermore, all melt calculations are ended when the depth of snow water
equivalent becomes less than 0.001 in. (0.025 mm), and any remaining snow and free water are
converted to immediate melt and added to Equation 6-19.
6.6 Computational Scheme
Snowmelt computations are a sub-procedure implemented as part of SWMM's runoff calculations.
They are made at each runoff time step, for each subcatchment that has snow pack parameters
assigned to it, immediately after atmospheric precipitation has been determined. This is at Step 3a
of the runoff procedure described in Section 3.4. The snowmelt routine returns an adjusted
precipitation rate (in/h), consisting of liquid rainfall and/or snowmelt, over each runoff sub-area
of the subcatchment. These rates serve as the actual precipitation input used in the remainder of
the surface runoff computation. The steps used to compute snow accumulation and snowmelt are
listed in the sidebar below.
183
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Computational Scheme for Snowmelt
The following variables are assumed known at the start of the time step of length At (h) for
each subcatchment:
Externally supplied time series variables:
Ta = air temperature (°F)
U = wind speed (mi/h)
/' = precipitation rate (in/h).
State variables for the snow pack on each snow surface:
WSNOW = snow pack depth (inches water equivalent)
COLDC = cold content depth (inches water equivalent)
FW = free water depth (inches water equivalent)
ATI = antecedent temperature index (°F).
In addition, the following constant parameters have been supplied by the user:
Constants defined for each subcatchment assigned a Snow Pack object:
SNN = fraction of impervious area that is plowable (i.e., SA2)
Tbase = temperature at which snow begins to melt (°F)
DHMIN = melt coefficient for December 21 (in/hr-°F)
DHMAX = melt coefficient for June 21 (in/hr-°F)
57 = depth at which surface remains 100% snow covered (inches)
FWFRAC = free water fraction that produces liquid runoff from the snow pack.
Snow redistribution constants for each subcatchment with a plowable sub-area SA2:
WEPLOW = depth that initiates snow redistribution (inches)
Redistribution fractions Fimp, Fperv, Fsub, Fout, andFimelt as defined in Section
6.2.5.
Constants defined for the entire study area:
SNOTMP = dividing temperature between snowfall and rainfall (°F)
SCF = rain gage snow capture factor (ratio)
TIPM = ATI weighting factor (fraction)
RNM = negative melt ratio (fraction)
Areal depletion curves (ASC as a function of AWE) for both pervious and impervious
areas.
Initially (at time 0) COLDC =AWE = 0, ATI = Tbase, and both WSNOW and FW are user-
supplied. The snowmelt computations are comprised of the following 11 steps:
(Continued on next page)
184
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1. Compute the melt coefficient ZIffl/f for each snow pack surface (SA1, SA2, and SA3) for
the current day of the year using Equation 6-7 and set the immediate melt IMELT on each
surface to 0.
2. If Ta < = SNOTMP then precipitation is in the form of snow so update the snow pack
depth on each snow surface:
WSNOW <- WSNOW + i x SCF x At
3. For the plowable impervious snow surface (SA2), if WSNOW > WEPLOWthen WSNOW
is reduced to reflect the redistributions produced by the fractions Fimp, Fperv, Fsub, Font,
andFimelt. IfFimelt > 0 then the immediate melt for surface SA2 is set to:
IMELT = Fimelt x WSNOW /At
4. If the snow pack depth over a snow surface is below 0.001 inches then convert the entire
pack for that surface into immediate melt:
IMELT <- IMELT + (WSNOW + FVK)/At
and reset the pack's state variables to 0.
5. Use the Areal Depletion Curves supplied for the pervious (SA1) and non-plowable
impervious (SA3) snow surfaces to compute a new areal snow coverage ratio ASC for
these surfaces (ASC for the plowable impervious surface is always 1.0). The details are
supplied below.
6. Compute a snowmelt rate SMELT for the snow pack on each surface:
a. If rain is falling (Ta > SNOTMP and / > 0.02 in/h) use the heat budget equation,
Equation 6-1, converted from a 6-hour to a 1-hour time base.
b. Otherwise, if Ta >= Tbase, use the degree-day equation, Equation 6-6.
c. Otherwise set SMELT to 0.
d. Multiply SMELTby its respective surface's ASC value to account for any areal
depletion.
7. For each snow pack surface, if SMELT is 0, then update the pack's cold content as follows:
a. If snow is falling (Ta < = SNOTMP and / > 0), set A TI to Ta. Otherwise set A TI to
the smaller of Tbase and the result of Equation 6-8.
b. Use Equation 6-10 with the updated ATI value to compute ACC and add ACC x
ASC to COLDC.
(Continued on next page)
185
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c. Limit COLDC to be no greater than 0.007 WSNOW (Tbase - ATI) which
assumes a specific heat of snow of 0.007 inches water equivalent per °F.
8. For each snow pack surface under melting conditions (SMELT > 0) reduce both the cold
content COLDC and the melt rate SMELT for each snow surface as follows:
ACC = SMELT X RNM X At
COLDC <- COLDC - ACC
SMELT <- SMELT - ACC
limiting both COLDC and SMELT to be >= 0.
9. Update the snow depth and free water content of the snow pack on each snow surface:
WSNOW <- WSNOW - SMELT X At
FW <- FVK + (SMFL71 + t^/w)At
where IRAIN = i if precipitation falls as rain or 0 otherwise.
10. Check each snow surface to see if the free water content is high enough to produce liquid
runoff, i.e., if FW > FWFRAC x WSNOW then set:
AFF = FW - FWFRAC X WSNOW
FW <- FW - AFF
SMELT = AFF
Otherwise set SMELT = 0.
1 1 . Compute the overall equivalent precipitation input RI (in/h) for each snow surface as:
RI = SMELT + I MELT + iRAIN X (1 - ASC}
Use these values to return an adjusted precipitation rate / (in/h) to each of the sub-areas
used to compute runoff:
i = RI[SAl] for the pervious area Al and
RI[SA2]AS2+RI[SA3]AS3 f u *u • • AOJAO
i = - - — - — for both impervious areas A2 and A3,
where RIfSAjJ is the value ofRI for snow surface SAj, ASJ is the area of snow surfacey, and
is the total impervious area.
186
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Step 5 of the snowmelt process uses Areal Depletion curves to compute the fraction of snow
covered area (ASC) for both the pervious (SA1) and impervious (SA3) areas subject to areal
depletion. Note that at this stage of the calculations any snow that has fallen during the time step
has already been added on to the accumulated snow depth WSNOW. The scheme used to update
the fraction of snow covered area is described in the sidebar below.
Computational Scheme for Snow Covered Area
There are four different cases that can arise when computing the fraction of snow covered area
ASC during the snowmelt calculations at a particular time step:
1. There is no snow accumulation (WSNOW = 0). Set ASC = 0.0 and re-set A WE to 0.
2. The updated snow accumulation WSNOW is greater than ,57. In this case both ASC and
AWE are set to 1.0.
3. There was snowfall during the time step (Ta < = SNOTMP and / > 0). ASC is set to 1.0 and
the parameters of a temporary linear ADC are computed as follows:
a. Find the A WE value for the accumulated depth at the start of the time step:
AWE = WSNOW1/SI
where WSNOW1 is the accumulated depth before the new snowfall was added on.
b. Use the ADC to look up the areal coverage SBA for this prior A WE value.
c. Compute the relative depth SBWS at which 75% of the new snow still remains (i.e.,
25% has melted):
SBWS = AWE + 0.75 (WSNOW - WSNOW1~)/SI
and save^J^E, SBA, and SBWS for use with the fourth case described next.
4. The accumulated snow depth WSNOW is below 57 and there is no snowfall. Define A WESI
as the current ratio of WSNOW to SI. Three conditions are possible:
a. If A WESI < A WE the original ADC applies so set ASC to the curve value for
A WESI and set A WE to 1.0.
b. IfAWESI >= SBWS the limit of the temporary ADC for new snowfall has been
reached so set ASC to 1.0.
c. Otherwise compute ASC from the temporary ADC as follows:
ASC = SBA + (1 - SBA)(AWESI - AWE}/(SBWS - AWE}
187
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6.7 Parameter Estimates
Table 6-2 summarizes the parameters used by the snowmelt routine as well as their typical range
of values. The first four entries (SNOTMP, SCF, 77PM, and RNM) are system-wide parameters
that apply to the entire study area. Values for the remaining parameters are specified for each snow
surface within each subcatchment where snowmelt can occur. SWMM uses a Snow Pack object
to bundle together a common set of these parameters that can be applied to an entire group of
subcatchments. This helps reduce the amount of input that a user must provide.
Table 6-2 Summary of snowmelt parameters (in US customary units)
Parameter
SNOTMP
SCF
77PM
RNM
WEPLOW
Tbase
DHMIN
DHMAX
SI
FWFRAC
Meaning
dividing temperature between snowfall and rainfall (°F)
rain gage snow capture factor (ratio)
ATI weighting factor (fraction)
negative melt ratio (fraction)
depth at which snow redistribution begins (inches)
temperature at which snow begins to melt (°F)
melt coefficient for December 21 (in/hr-°F)
melt coefficient for June 21 (in/hr-°F)
depth at which surface remains 100% snow covered
(inches)
free water fraction to produce liquid runoff from pack
Typical Range
32 to 36
1 to 2
0.5
0.6
0.5 to 2
25 to 32
0.001 to 0.003
0.006 to 0.007
1 to 4
0.02 to 0.10
Snowmelt results will be sensitive to the values used for the degree-day melt coefficient DHM. In
rural areas, the melt coefficient ranges from 0.03 - 0.15 in/day-°F (1.4 - 6.9 mm/day-°C) or from
0.001 - 0.006 in/h-°F (0.057 - 0.29 mm/h-°C). Gray and Prowse (1993) provide a useful summary
of such equations. In urban areas, values may tend toward the higher part of the range due to
compression of the pack by vehicles, pedestrians, etc. and due to reflection of radiation onto the
snow from adjacent buildings (Semadeni-Davies, 2000). Most of the available data are
summarized by Semadeni-Davies (2000). Bengtsson (1981) and Westerstrom (1981) describe
results of urban snowmelt studies in Sweden, including degree-day coefficients, which range from
3 to 8 mm/°C-day (0.07 - 0.17 in/°F-day). Additional data for snowmelt on an asphalt surface
(Westerstrom, 1984) gave degree-day coefficients of 1.7 - 6.5 mm/°C-day (0.04 - 0.14 in/°F-day).
Values of Tbase will probably range between 25 and 32 °F (-4 and 0 °C). Unfortunately, few urban
area data exist to define adequately appropriate modified values for Tbase and DHM, and they
may be considered calibration parameters.
188
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The value of FWFRAC will normally be less than 0.10 and usually between 0.02 - 0.05 for deep
snow packs (WSNOW > 10 inches or 254 mm water equivalent). However, Anderson (1973)
reports that a value of 0.25 is not unreasonable for shallow snow packs that may form a slush layer.
An additional set of parameters not listed in Table 6-2 are those used to characterize the Areal
Depletion Curves (ADCs). An ADC is characterized in SWMM by providing values of ASC
(fraction of area with snow cover) for snow depth ratios (ratio of depth to depth at 100% areal
coverage) that range from 0.0 to 0.9 in 0.1 increments. (By definition ASC is 1.0 for a snow depth
ratio of 1.0). Table 6.3 lists the points of the ADC shown previously in Figure 6-4 that is typical
of natural areas. Two ADC curves, one for pervious area and one for impervious areas, are assumed
to apply across the entire watershed. The curves are not required to pass through the origin, AWE
= ASC = 0; they may intersect the abscissa at a value of ASC > 0 in order to maintain some snow
covered area up until the instant that all snow disappears (see Figure 6-4). However, the curves
may not intersect the ordinate, AWE must be greater than 0 when ASC = 0. A curve whose ASC
values are all 1.0 causes the areal depletion phenomenon to be ignored.
Table 6-3 Typical areal depletion curve for natural areas
Depth Ratio
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ASC
0.10
0.35
0.53
0.66
0.75
0.82
0.87
0.92
0.95
0.98
189
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6.8 Numerical Example
The following numerical example illustrates the dynamic nature of snow accumulation, snow melt,
and subsequent runoff. A one acre, completely impervious subcatchment is modeled over an 18
day period during which temperature fluctuates between 0 and 50 °F. The simulation begins with
1 inch of snow accumulation over the subcatchment. Table 6-4 lists the relevant subcatchment and
snowpack parameters, while Tables 6-5 and 6-6 list the daily temperatures and hourly
precipitation, respectively, used in the simulation. The meteorological conditions are recorded data
for Raleigh, NC. Neither snow removal nor areal depletion is considered.
Table 6-4 Subcatchment and snow pack parameters for illustrative snowmelt example
Parameter
Area (acres)
Width (ft)
Slope (%)
Percent Impervious
Roughness Coefficient
Depression Storage (in)
Minimum Melt Coefficient (in/h/°F)
Maximum Melt Coefficient (in/h/°F)
Base Temperature (Tbase) (°F)
Free Water Fraction (FWFRAC)
Initial Snow Depth (in)
Initial Free Water (in)
Dividing Temperature (SNOJMP) (°F)
ATI Weighting Factor (TIPM)
Negative Melt Ratio (RNM)
Latitude (°)
Value
1
140
0.5
100
0.01
0.25
0.001
0.006
30
0.05
1.0
0.2
34
0.5
0.6
42
190
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Table 6-5 Daily temperatures for illustrative snowmelt example
Month/Day
1/24
1/25
1/26
1/27
1/28
1/29
1/30
1/31
2/1
2/2
2/3
2/4
2/5
2/6
2/7
2/8
2/9
2/10
Maximum
Temperature (°F)
49
50
46
50
45
36
46
51
46
27
29
42
46
54
45
41
51
45
Minimum
30
32
28
27
24
14
21
22
26
-5
-7
27
18
19
28
20
20
25
191
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Table 6-6 Periods of precipitation for illustrative snowmelt example
Date
01/26
01/29
01/29
01/29
02/01
02/02
02/02
02/02
02/02
02/02
02/02
02/02
02/02
02/03
02/03
02/03
02/03
02/03
02/03
02/03
02/09
02/09
02/09
02/09
02/09
02/09
Time
04:00:00
18:00:00
19:00:00
20:00:00
23:00:00
00:00:00
01:00:00
02:00:00
03:00:00
04:00:00
05:00:00
22:00:00
23:00:00
00:00:00
01:00:00
02:00:00
03:00:00
12:00:00
13:00:00
14:00:00
00:00:00
01:00:00
02:00:00
03:00:00
04:00:00
05:00:00
Precipitation (in)
0.26
0.11
0.01
0.08
0.02
0.06
0.08
0.14
0.19
0.09
0.01
0.02
0.06
0.12
0.22
0.17
0.05
0.02
0.00
0.02
0.01
0.02
0.00
0.00
0.00
0.06
192
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Figures 6-7 through 6-10 show the resulting temperature, precipitation, snow depth and runoff
amounts, respectively produced by SWMM for this example. The original inch of snow takes about
four days to melt completely. Runoff during this time is sporadic, due to the fluctuation in
temperature around the base melt temperature. The first storm event arrives just before the end of
day 2 and falls mainly as snow. This bumps up the snow cover during its 3-hour duration as shown
in Figure 6-9. Snow levels rise again with the arrival of the second storm during the morning of
day 5, when temperatures are below freezing. By day 6, temperatures again rise above the base
melt temperature (30 °F) for part of the day and the snow from the second storm is completely
melted by the start of day 7. The next storm arrives at noon of day 8 and lasts for 7 hours. The
runoff spike of 0.15 in/hr seen in Figure 6-10 occurs during the first hour of this event when the
temperature is still above freezing. The remainder of the storm falls as snow and starts the buildup
of a snow pack once again. The next two storms add onto the pack, and no melting occurs until
day 10, when temperatures again rise above the base melt value for portions of the day. Runoff
from the melting pack is delayed until its free water fraction is exceeded. The pack takes another
6 days to melt during which time the runoff is sporadic as the temperature fluctuates above and
below the base melt level.
60.0
50.0
10 12
Elapsed Time (days)
14
16
18
20
Figure 6-7 Continuous air temperature for illustrative snowmelt example.
193
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0.25-
0.2-
0.15-
0)
Dl
0.1-
0.05-
0.0J
8 10 12
Elapsed Time (days)
14
16
18
20
Figure 6-8 Precipitation amounts for illustrative snowmelt example.
1.2-
1.0-
0.8
ti
&
0.6-
0.4-
0.2-
0.0-
10 12
Bapsed Time (days)
14
16
18
20
Figure 6-9 Snow pack depth for illustrative snowmelt example.
194
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0.06
0.05
0.04
8= 0.03
0.02
0.01
10 12
Elapsed Time (days)
16
18
20
Figure 6-10 Runoff time series for illustrative snowmelt example.
195
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Chapter 7 - Rainfall Dependent Inflow and Infiltration
7.1 Introduction
Rainfall dependent (or rainfall-derived) inflow and infiltration (RDII) are stormwater flows that
enter sanitary or combined sewers due to "inflow" from direct connections of downspouts, sump
pumps, foundation drains, etc. as well as "infiltration" of subsurface water through cracked pipes,
leaky joints, poor manhole connections, etc. RDII can be a significant cause of sanitary sewer
overflows (SSOs) of untreated wastewater into basements, streets and other properties, as well as
receiving streams. It can also cause significant flow increases to wastewater treatment plants
resulting in hydraulic overloading and disruption of plant processes.
SWMM treats RDII as a separate category of external inflows that enters the conveyance system
at specific user-designated nodes. It is computed independently of the surface runoff, infiltration,
snowmelt and groundwater processes described in previous chapters of this manual. RDII flow is
added onto the other inflow categories (such as dry weather sanitary flow, overland runoff, and
groundwater interflow) during each time step of a simulation. RDII calculations were added to
version 4 of SWMM by C. Moore of CDM in 1993. This chapter describes how these RDII flows
are computed from the precipitation records supplied to a SWMM data set.
7.2 Governing Equations
Figure 7-1 depicts the three major components of wet-weather wastewater flow within a sanitary
sewer system (Vallabhaneni et al., 2007). These are base sanitary flow (BSF), groundwater
infiltration (GWI), and RDII. BSF is the flow discharged to sanitary sewers by homes, businesses,
institutions, and industrial water users throughout the normal course of a day. It exhibits a typical
diurnal pattern, with higher flows during the morning and early evening hours and lower flows
overnight. The average daily BSF remains more or less constant during the week, but can vary by
both month and season.
GWI consists of groundwater that enters the collection system through cracked pipes, pipe joints
and manhole walls during extended periods of time when water table levels are high, even in the
absence of any rainfall. It is different from RDII because it does not occur as a direct response to
a rainfall event. GWI varies throughout the year, with the highest rates in late winter and spring as
groundwater levels rise, and the lowest rates (or no GWI at all) during late summer or after an
extended dry period.
196
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RDII is the flow that can be directly attributed to a rainfall event. This flow is zero before the start
of the event, increases during the event, and declines back to zero sometime after the event is over.
The start of the RDII response may be delayed during the time it takes for surfaces to capture a
portion of the initial rainfall and for soils to become saturated. If the event is small enough, then
no RDII at all may be generated. The maximum volume of rainfall that does not produce any RDII
response is referred to as "initial abstraction" (Vallabhaneni et al., 2007).
Time
Figure 7-1 Components of wet-weather wastewater flow.
Quantitative estimates of RDII are almost always derived from actual wastewater flow records as
opposed to attempting to model the distributed set of small scale physical processes directly
responsible for RDII. Methods for modeling RDII are reviewed by Bennet et al. (1999) and Lai
(2008). SWMM uses the RTK unit hydrograph approach, which is among the most flexible and
widely used RDII methods (Vallabhaneni et al., 2007). (The initials RTK stand for the three
parameters that characterize the unit hydrographs used by the method.)
The RTK unit hydrograph method was first developed by CDM-Smith consultants in an RDII
study for the East Bay Municipal Utility District in Oakland, CA (Giguere and Riek, 1983). It
represents the response of a sewershed to a rainfall event through a series of up to three triangular
197
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unit hydrographs. These unit hydrographs can be applied to any particular storm event to produce
a resulting time history of RDII flow rates.
Figure 7-2 shows a single triangular unit hydrograph assumed to represent the RDII flow induced
by one unit of rainfall over a unit of time. This unit hydrograph is characterized by the following
parameters:
R: the fraction of rainfall volume that enters the sewer system and equals the volume under
the hydrograph
T: the time from the onset of rainfall to the peak of the unit hydrograph
K: the ratio of time to recession of the unit hydrograph to the time to peak
Qpeak'. peak flow (per unit area) on the unit hydrograph.
!
Q
TIME
T + KT
Figure 7-2 Example of an RDII triangular unit hydrograph.
198
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Figure 7-3 shows how this single unit hydrograph would be applied to a storm that consists of
three time periods of varying rainfall volume. The original unit hydrograph is replicated for each
rainfall time period, with its origin offset by the time period and its ordinates multiplied by the
rainfall volume for that period. The overall response to the storm is the hydrograph obtained by
summing the ordinates of the volume-adjusted hydrographs at each time point. The volumetric
RDII inflow into the conveyance system is the ordinate of the composite hydrograph multiplied
by the contributing area of the affected sewershed. This process of adding together the rainfall-
adjusted, time-shifted hydrographs is known as convolution (Chow et al, 1988) and is expressed
mathematically as:
(7-1)
where:
7 = 1
Qt = RDII flow per unit area during time period t,
Ut = ordinate of the unit hydrograph for time period^,
Pj = depth of rainfall for time period/
PI P2 P3
fT
!
Q
Resultant RDII flow per unit area
TIME
Figure 7-3 Application of a unit hydrograph to a storm event.
199
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The ordinate value Uj for time periody is determined from the shape parameters R, T, and K of the
unit hydrograph as follows. One can write:
Uj = fjQpeak (7-2)
where fj is the fraction of the rising limb (or falling limb) that corresponds to time period/ Because
the area under the unit hydrograph is R, the value of Qpeak is:
2R
Vpeak T . IV-T \'~-))
.
Thus Uj can be expressed as
U, = -- (7-4)
1 T + KT
By convention, the time TJ on the unit hydrograph base corresponding to time period 7 is taken as
the midpoint between either ends of the time interval:
TJ = (j- 0.5)Ar (7-5)
where A ris the time interval over which precipitation is recorded. The fraction^ is then determined
as:
fj = j for TJ < = T (7-6)
fj = l~~KT~ forT T+KT (7-8)
Because actual RDII hydrographs have complex shapes, three different hydrographs of increasing
durations are typically used to represent the overall RDII unit response (Vallabhaneni et al.,
2007). The first hydrograph models the most rapidly responding inflow component usually caused
by direct sources of inflow, and has a time to peak T of one to three hours. The second includes
both rainfall-derived inflow and infiltration, and has a longer T value. The third represents
infiltration that may continue long after the storm event has ended and has the longest T value.
Figure 7-4 depicts how the three unit hydrographs are summed together to produce a total RDII
hydrograph in response to a unit of rainfall over one unit of time. Equation 7-1 is still used to
compute the overall RDII hydrograph to any given storm event, with a separate Qt computed for
200
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each of the three unit hydrographs. These are then added together to produce the total flow per unit
area for time period t.
TOTAL RDII HYDROGRAPH
RESULTING FROM
RAINFALL, P
R..T,. ANDK.
SCOM3 UNIT
HYDKOQLATH CS1
.. ANDK.
THIRD UNIT
HYDKQQSAFHOF
K,, T.. AND K.
0 Ti Tz Ti+Ki'Ti T3 T2+K:'T: TTMI
Figure 7-4 Use of three unit hydrographs to represent RDII (Vallabhaneni et al., 2007).
Not all storms will result in measurable inflow/infiltration. Just as with ordinary runoff, a certain
initial volume of rainfall will be captured by surface ponding, interception by flat roofs and
vegetation, and surface wetting and will not contribute to RDII. This phenomenon is represented
in SWMM by three user-supplied "initial abstraction" (IA) parameters that accompany each RDII
unit hydrograph. IAmax (in or mm) is the maximum depth of initial abstraction capacity available
for the sewershed. IAo (in or mm) is the amount of that capacity already used up at the start of the
simulation. IAr (in/day or mm/day) is the rate at which capacity becomes available again during
periods of no rainfall. During storm events, the volume of rainfall applied to the unit hydrograph
convolution formula, Equation 7-1, is reduced by the amount of initial abstraction capacity
remaining. During dry periods, this capacity is regenerated based on the user-supplied recovery
rate.
201
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7.3 Computational Scheme
SWMM generates RDII inflows for specific nodes of a sewer system. Recall from Section 1.2 that
SWMM uses a network of links and nodes to represent the conveyance portion of a drainage area.
For RDII applications this network would be the sewer system (either sanitary or combined), the
links are the sewer pipes and the nodes are points where pipes connect to one another (e.g.,
manholes or pipe fittings).
It should be noted once again that RDII is computed independently from any surface runoff or
groundwater flow generated from the subcatchments contained in a SWMM model. The sewershed
that produces RDII flow for a specific sewer system node is not represented explicitly in SWMM
and need not correspond to any of the runoff subcatchments defined for the study area. In fact it is
perfectly acceptable (and quite common for sanitary sewer systems) to conduct an RDII analysis
without including any subcatchments in the model. In this case the model would consist of a set
of Rain Gage objects (and their data sources), the node and link objects that make up the sewer
network and sets of user-supplied time series that describe groundwater (GWI) and sanitary (BSF)
flows.
SWMM computes all RDII inflow time series prior to the start of a simulation and saves these
inflow values to an interface file. Each line of the file contains, in chronological order, a node ID
name, a date, a time of day, and the RDII inflow value for that node. Dates with no RDII inflows
are not recorded. To compute the entries of this file the following quantities are assumed known
for each node of the conveyance system node that receives RDII inflows:
• the area (^4) of the sewershed that contributes RDII to the node,
• the R-T-K parameters for each of three RDII unit hydrographs,
• the initial abstraction parameters (IAmax, IAo, and IAr) associated with each RDII unit
hydrograph,
• the time series of rain volumes that fall on the sewershed and their recording interval AT
(sec) as provided by a SWMM Rain Gage object.
The steps used to process a precipitation record against a set of unit hydrographs to produce a
record of RDII inflows for a specific conveyance node are described in the sidebar shown
below.
202
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Computational Scheme for RDII
First define the following variables:
Ttot = total elapsed time (sec)
Tgage = total time elapsed time for the rainfall record (sec)
Tbase = time base of a unit hydrograph, = T + KT (sec)
Tdry = time since the last rainfall (sec)
AtwET = wet time step used for runoff computations (sec) (see Section 3.6)
P = vector of past rainfall volumes (ft)
Then do the following for each conveyance system node designated to receive RDII flow:
1 . Initialize the following quantities:
Ttot = 0
Tgage = 0
i dry i base ' 1
IA = lAmax - IAo
P = 0
2. Repeat the following sub-steps until Tgage > Ttot.
a. retrieve the rain volume v over the rain gage recording interval at time Tgage
b. if there is any rainfall, reduce it by any available initial abstraction; otherwise
recover initial abstraction over the time step AT
c. if there is still rainfall excess and Tdry > Tbase then begin a new RDII event by
setting all entries in P to 0 and set Tdry = 0; otherwise add AT to Tdry
d. save the rain volume in the next available entry in P
e. add AT to
3 . If Tdry < Tbase then apply convolution to the vector of past rainfall volumes and the
unit hydrograph ordinates to compute an RDII flow per unit of sewershed area.
4. If the RDII flow is non-zero, multiply it by the node's sewershed area and save the
current date at Ttot and the RDII flow value to the interface file.
5. Add AtwET to Ttot and return to Step 2 if Ttot is less than the total duration.
(Continued on next page)
203
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Note that RDII flows are computed for each runoff wet time step but that precipitation
records and the RDII convolution are processed at the rain gage recording interval time step.
The application of the initial abstraction at Step 2b of this process proceeds as follows:
1. Ifv>0then:
a. if IA > v then IA = IA - v and v = 0;
b. else if IA > 0 then v = v - IA and IA = 0.
2. If v = 0 then IA = min(IAmax, IA + IArAi)
Calculation of the RDII flow at Step 3 is carried out by adding together the products UjPi for
each of the unit hydrographs as the hydrograph indexy is incremented from 1 to the number
of hydrograph intervals (equal to Tbase/ AT} while the rainfall index /' is decreased from the
current period back an equal number of time intervals. Equations 7-2 through 7-7 are used
to compute Uj for each of the three unit hydrographs.
7.4 Parameter Estimates
To use SWMM's RDII option a user must supply estimates of the three parameters (R, T, and K)
that define each of three unit hydrographs for each node where RDII enters the sewer system. Each
unit hydrograph can also have a set of initial abstraction parameters (lao, Iamax, and Iar). SWMM
also allows one to specify different sets of unit hydrographs and initial abstraction parameters for
different months of the year. In addition, the area of the RDII contributing sewershed must also be
specified.
R-T-K parameters are derived from site-specific flow monitoring data. There are no general values
that can be applied in the absence of actual field data. All of these parameters require that a
continuous flow monitoring program be implemented at strategic points in the sewer system. As
described in Vallabhaneni et al., 2007, estimating the RDII unit hydrograph parameters for a
sewershed involves the following activities:
1. Identify the sewershed areas that are tributary to the flow monitor (see Figure 7-5).
2. Extract the RDII portion of the recorded flow at the monitoring station during a wet weather
event (see Figure 7-6).
204
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3. Estimate the R-T-K values for each of three unit hydrographs whose resultant hydrograph best
matches the RDII flow extracted from the flow record (see Figure 7-7).
Sewer Service Area
Undeveloped Area
Flow Monitor
Model Extent
Unmodeted Sewer
Figure 7-5 Sewershed delineation (Vallabhaneni et al., 2007).
205
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Time
Figure 7-6 Extracting RDII flow from a continuous flow monitor (Vallabhaneni et al.,
2007).
Calculated RDII
(yellow), curve fitted
with observed RDII
Flow (red)
^^^^
Figure 7-7 Fitting unit hydrographs to an RDII flow record (Vallabhaneni et al., 2007).
206
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7.5 Numerical Example
A simple example illustrates how SWMM constructs an RDII interface file for use within a
hydraulic simulation. Assume there is a single rain gage whose rainfall time series is shown in
Table 7-1. Note that the recording interval is 1 hour, and that there are two events separated by 22
hours. SWMM will use data from this gage to construct a time series of RDII flows for a node
named Nl in the conveyance system that services an area of 10 acres. There is a single group of 3
unit hydrographs used to derive RDII from the rain gage data. The shapes and parameters of the
unit hydrographs (UH1, UH2, and UH3) are shown in Figure 7-8. Note that the R-values of this
set of unit hydrographs sum to 0.36, implying that 36 percent of total rainfall volume winds up as
RDII. To keep things simple, initial abstraction is not considered in this example.
Table 7-1 Rainfall time series for the illustrative RDII example
Hour
0:00
1:00
2:00
3:00
4:00
5:00
6:00
27:00
28:00
29:00
30:00
Rainfall (inches)
0.0
0.25
0.5
0.8
0.4
0.1
0.0
0.0
0.4
0.2
0.0
207
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UH1
UH2 UH3
•UH-Total
10
Hour
12
14
16
18
20
Figure 7-8 Unit hydrographs used for the illustrative RDII example.
The resulting RDII flows are depicted in Figure 7-9. SWMM places these flows into an RDII
interface file, a portion of which is displayed in Figure 7-10. This file is accessed during the flow
routing portion of a SWMM run to add RDII inflow into node Nl at each time step of the routing
process.
D RDII D Rainfall
15
22
29
36
43
Hour
Figure 7-9 Time series of RDII flows for the illustrative RDII example.
208
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SWMM5 Interface File
900 - reporting time step in sec
1 - number of constituents as listed
FLOW CFS
1 - number of nodes as listed below:
Nl
Node
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Nl
Year
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
002
002
002
002
002
002
002
002
002
002
002
002
002
002
002
002
002
002
002
002
Mon
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
Day Hr
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
01
01
01
02
02
02
02
03
03
03
03
04
04
04
04
05
05
05
05
06
below:
Min Sec FLOW
15
30
45
00
15
30
45
00
15
30
45
00
15
30
45
00
15
30
45
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
0
0
0
0
0
0
0
0
1.
1
1
1
1
1
1
1
0
0
0
0.
.204
.204
.204
.204
.554
.554
.554
.554
195
195
195
195
604
604
604
604
021479
.021
.021
.021
.001
.001
.001
.001
479
479
479
312
312
312
312
.703842
.703842
.703842
7038
42
Figure 7-10 Excerpt from the RDII interface file for the illustrative RDII example.
209
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Glossary
Aquifer - as defined in SWMM, it is the underground water bearing layer below a land surface,
containing both an upper unsaturated zone and a lower saturated zone.
Areal Depletion - the process by which the land area covered by snow decreases as the total
volume of snow decreases due to melting.
Capillary Suction Head - the soil water tension at the interface between a fully saturated and
partly saturated soil.
Climate Data Online - an interactive web based data retrieval service operated by NOAA's
National Climatologic Data Center for retrieving historical rainfall and climate data.
Cold Content - the difference between the heat content of a frozen snow pack and its base melt
temperature.
Continuous Simulation - refers to a simulation run that extends over more than just a single
rainfall event.
Curve Number - a factor, dependent on land cover, used to compute a soil's maximum moisture
storage capacity.
Curve Number Method - a method that uses a soil's maximum moisture storage capacity as
derived from its curve number to determine how cumulative infiltration changes with cumulative
rainfall during a rainfall event. Not to be confused with the NRCS (formerly SCS) Curve Number
runoff method as embodied in TR-55.
D
Darcy's Law - states that flow velocity of water through a porous media equals the hydraulic
conductivity of the media times the gradient of the hydraulic head it experiences.
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Depression Storage - the volume over a surface that must be filled prior to the occurrence of
runoff. It represents such initial abstractions as surface ponding, interception by flat roofs and
vegetation, and surface wetting.
Design Storm - a rainfall hyetograph of a specific duration whose total depth corresponds to a
particular return period (or recurrence interval), usually chosen from an IDF curve.
Directly Connected Impervious Area - impervious area whose runoff flows directly into the
collection system without the opportunity to run onto pervious areas such as lawns.
Dividing Temperature - the temperature below which precipitation falls in the form of snow.
Field Capacity - the amount of water a well-drained soil holds after free water has drained off, or
the maximum soil moisture held against gravity. Usually defined as the moisture content at a
tension of 1/3 atmospheres.
Global Historical Climatology Network - a data base administered by NOAA's National
Climatic Data Center that archives daily climate observations from approximately 30 different
sources for about 30,000 stations across the globe.
Green-Ampt Method - a method for computing infiltration of rainfall into soil that is based on
Darcy's Law and assumes there is a sharp wetting front that moves downward from the surface,
separating saturated soil above from drier soil below.
H
Hargreaves Method - an empirical formula for estimating daily evaporation that depends on air
temperature and solar radiation.
Horton Curve - an empirical curve that describes the exponential decrease in infiltration rate with
time during a rainfall event.
Horton Method - a method for computing infiltration of rainfall into soil that uses the Horton
Curve to relate infiltration rate to time, with modifications made to consider times where the
rainfall rate is less than the curve's infiltration rate.
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Hydraulic Conductivity - the rate of water movement through soil under a unit gradient of
hydraulic head. Its value increases with increasing soil moisture, up to a maximum for a completely
saturated soil (known as the saturated hydraulic conductivity or Ksat).
Hydrograph - a plot that shows how runoff flow varies with time.
Hydrologic Soil Group - a classification that indicates a soil's ability to infiltrate water.
Hyetograph - a plot that shows how rainfall rate varies with time.
I
IDF Curves - a series of curves that determine the average rainfall intensity (I) for a given duration
of storm (D) that occurs at a specific annual frequency (F), e.g., the intensity of a 6-hour storm that
occurs once every 10 years.
Impervious Surface - a surface that does not allow infiltration of rain water, such as a roof,
roadway or parking lot.
Infiltration - the process by which rainfall penetrates the ground surface and fills the pores of the
underlying soil.
Infiltrometer - a device used to measure the rate of water infiltration into soil or other porous
media.
Initial Abstraction - precipitation that is captured on vegetative cover or within surface
depressions that is not available to become runoff and is removed by either infiltration or
evaporation.
LID Control - a low impact development practice that provides detention storage, enhanced
infiltration and evapotranspiration of runoff from localized surrounding areas. Examples include
rain gardens, rain barrels, green roofs, vegetative swales, and bio-retention cells.
Link - a connection between two nodes of a SWMM conveyance network that transports water.
Channels, pipes, pumps, and regulators (weirs and orifices) are all represented as links in a SWMM
model.
228
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M
Manning Equation - the equation that relates flow rate to the slope of the hydraulic grade line
for gravity flow in open channels.
Manning Roughness - a coefficient that accounts for friction losses in the Manning flow equation.
Modified Horton Method - a modified form of the Horton infiltration method that tracks
cumulative infiltration volume instead of time along the Horton curve to determine how infiltration
rate changes with time during a rainfall event.
Moisture Deficit - the difference between a soil's current moisture content and its moisture
content at saturation.
N
Newton-Raphson Method - a commonly used iterative numerical method for solving nonlinear
equations that makes use of the derivative of the equation with respect to the unknown variable.
Node - a point in a runoff conveyance system that receives runoff and other inflows, that connects
conveyance links together, or that discharges water out of the system. Nodes can be simple
junctions, flow dividers, storage units, or outfalls. Every conveyance system link is attached to
both an upstream and downstream node.
Nonlinear Reservoir Model - a simple conceptual model of a storage reservoir where the change
in volume with respect to time equals the difference between a known inflow rate and an outflow
rate that is a nonlinear function of the current stored volume.
O
Overland Flow Path - the path that runoff follows as it flows over the surface of a catchment area
until it reaches a collection channel or storm drain.
Pervious Surface - a surface that allows water to infiltrate into the soil below it, such as a natural
undeveloped area, a lawn or a gravel roadway.
Pollutograph - a plot of the concentration of a pollutant in runoff versus time.
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Porosity - the fraction of void (or air) space in a volume of soil.
R
Rainfall File - an external text file that contains rainfall data for a single rain gage in one of the
several different formats that SWMM can recognize.
Rainfall Interface File - a binary file generated by SWMM that contains the rainfall time series
used in a simulation for all of the rain gages in the project. This file can be used to input rainfall
in subsequent simulation runs.
Rain Gage - a SWMM object that provides precipitation data, either as an internal time series or
through an external data file, to one or more subcatchment areas in a SWMM model.
RDII - rainfall dependent inflow and infiltration are stormwater flows that enter sanitary or
combined sewers due to "inflow" from direct connections of downspouts, sump pumps, foundation
drains, etc. as well as "infiltration" of subsurface water through cracked pipes, leaky joints, poor
manhole connections, etc.
Richards Equation - the nonlinear partial differential equation that describes the physics of water
flow in unsaturated soil as a function of moisture content and moisture tension.
Routing Interface File - a text file that contains the time history of external flow and water quality
inflow to different locations of the conveyance network of a SWMM model. It can be generated
from a previous SWMM run or can serve as a replacement for SWMM's runoff calculations.
RTK Unit Hydrograph - a triangular unit hydrograph that represents the time pattern of rainfall
entering a sewer system as RDII. R is the fraction of total rainfall entering the system (i.e., the area
under the hydrograph), T is the time at the hydrograph peak, and K is the ratio of the length of the
receding limb of the hydrograph to the time to peak.
Runge-Kutta Method - a numerical method for solving systems of ordinary differential equations
over a series of sequential time steps.
Runoff Coefficient - the ratio of total runoff to total rainfall over a study area.
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Shape Factor - the ratio of a watershed's area to the length of its main drainage channel squared.
It is used to estimate the runoff width of a catchment area.
Snow Catch Factor - a multiplier used to correct for inaccurate snowfall measurements due to
wind blowing snow away from the precipitation gage.
Snow Pack - the accumulation of snow cover that blankets an area. Snow pack depth increases as
new snow falls and decreases as snow melts.
Subcatchment - a sub-area of a larger catchment area whose runoff flows into a single drainage
pipe or channel (or onto another subcatchment).
Subcatchment Discretization - the process of dividing a study area into subcatchments that
properly characterize the spatial variability in overland drainage pathways, surface properties and
connections into drainage pipes and channels.
Two-Zone Groundwater Model - a conceptual model that represents the subsurface region
beneath a subcatchment as consisting of an unsaturated upper zone that lies above a lower saturated
zone. The extent of each zone and the moisture content of the upper zone can change in response
to variations in surface infiltration, evapotranspiration, and groundwater outflow.
U
Unit Hydrograph - represents the unit response of a watershed (in terms of runoff volume and
timing) to a unit input of rainfall. Unit hydrographs are specific to particular catchments and
typically have either a triangular or bell curve shape.
W
Wilting Point - the soil moisture content at which plants can no longer extract moisture to meet
their transpiration requirements. Usually defined as the moisture content at a tension of 15
atmospheres.
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