Modeling phosphorus in the environment - Chapter 2 ppsx

21
2
Modeling Runoff and
Erosion in Phosphorus
Models
Mary Leigh Wolfe
Virginia Polytechnic Institute and State University,
Blacksburg, VA
CONTENTS
2.1 Introduction 22
2.2 Modeling Runoff 22
2.2.1 Runoff Volume 22
2.2.1.1 Curve Number Method 23
2.2.1.2 Curve Number Method Implementation 25
2.2.1.3 Infiltration-Based Approaches 29
2.2.2 Hydrograph Development 32
2.2.2.1 Kinematic Flow Routing 33
2.2.2.2 SCS Unit Hydrograph 34
2.2.2.3 Hydrograph Development Implementation 35
2.2.3 Streamflow, or Channel, Routing 36
2.2.3.1 Hydrologic, or Storage, Routing 37
2.2.3.2 Muskingum Routing Method 37
2.2.3.3 Streamflow, or Channel,
Routing Implementation 39
2.2.4 Peak Rate of Runoff 40
2.2.4.1 Rational Formula 40
2.2.4.2 SCS TR-55 Method 41
2.2.4.3 Peak Runoff Rate Implementation 41
2.3 Modeling Erosion and Sediment Yield 44
2.3.1 USLE-Based Approaches 45
2.3.2 USLE-Based Approach Implementation 48

2.3.3 Process-Based Approaches 49
2.3.4 Process-Based Approach Implementation 52
2.3.5 Channel Erosion 53
2.3.6 Channel Erosion Implementation 54
2.4 Summary 56
References 60
© 2007 by Taylor & Francis Group, LLC
22 Modeling Phosphorus in the Environment
2.1 INTRODUCTION
Runoff and erosion are the overland processes that transport phosphorus. The processes
and equations describing the processes have been described in many references. The
purpose of this chapter is to present the common approaches used for modeling runoff
and erosion processes in models that simulate phosphorus transport and to illustrate
similarities and differences in implementation among selected phosphorus models.
Implementation of the processes varies among the phosphorus models, depending on
model characteristics such as spatial representation of the drainage area (e.g., lumped
or distributed), spatial scale (e.g., field or watershed), purpose of the model (e.g., event
prediction or average annual predictions for management), computational time step
(e.g., daily vs. shorter time steps during rainfall or runoff events), and land uses and
conditions represented (e.g., agricultural, urban, forested land uses, frozen soils).
Examples of implementation from the following models are included in the
chapter: Annualized Agricultural Nonpoint Source (AnnAGNPS) (Cronshey and
Theurer 1998), Areal Nonpoint Source Watershed Environment Response Simulation
2000 (ANSWERS-2000) (Bouraoui 1994), Erosion Productivity Impact Calculator
(EPIC) (Sharpley and Williams 1990), Groundwater Loading Effects of Agricultural
Management Systems (GLEAMS) (Knisel 1993; Leonard et al. 1987), Hydrologic
Simulation Program-Fortran (HSPF) (Bicknell et al. 2001), and Soil and Water
Assessment Tool (SWAT) (Neitsch et al. 2002). Unless otherwise noted, the infor-
mation about the models is from the sources cited in this paragraph. Because of the
variety of equations and variability in how they are implemented in different models,

a mixture of units is used in this chapter. Generally, units are expressed as length,
mass, time (L, M, T, respectively) or in both International System of Units (SI) and
English units for empirical equations or as used in the cited models.
2.2 MODELING RUNOFF
Runoff is a complex, variable process, influenced by many factors such as soil
characteristics, land cover, and topography. Runoff calculations typically include
estimating runoff volume, peak runoff rate, and hydrographs, or the time distribution
of runoff. For some phosphorus models, peak runoff rate is computed only for use
in erosion calculations. For example, in GLEAMS the peak rate is used in the erosion
component for calculating the characteristic discharge rate, sediment transport capa-
city, and shear stress in a concentrated flow. Common approaches used in phosphorus
models for estimating runoff volume, hydrographs, and peak discharge are described
in the following sections.
2.2.1 R
UNOFF
V
OLUME
Runoff volume, often termed rainfall excess, is the total amount of rainfall minus
infiltration and interception. Two general approaches are used to model runoff
volume in phosphorus transport models: (1) the curve number (CN) method and (2)
infiltration methods. The CN method directly calculates runoff volume, whereas the
infiltration methods calculate infiltration first and then estimate runoff as the differ-
ence between rainfall and infiltration. Some phosphorus models include both CN
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 23
and infiltration methods. The CN method is usually used when daily rainfall values
are available; infiltration methods require hourly — or other intervals shorter than
daily — rainfall values.
2.2.1.1 Curve Number Method
The most common method used to estimate runoff volume in phosphorus models is

the U.S. Department of Agriculture (USDA) Soil Conservation Service (SCS) (now
Natural Resources Conservation Service, NRCS) runoff approach. The CN method
correlates runoff with rainfall, antecedent moisture condition (AMC), soil type, and
vegetative cover and cultural practices. Runoff volume is computed using the fol-
lowing relationships (SCS 1972):
(2.1)
, S in mm or , S in in. (2.2)
where Q is direct storm runoff volume (mm or in.), P is storm rainfall depth (mm
or in.), S is the retention parameter or maximum potential difference between rainfall
and runoff at the time the storm begins (mm or in.), and CN is the runoff curve
number, which represents runoff potential of a surface based on land use, soil type,
management, and hydrologic condition. Rainfall depth, P, must be greater than 0.2S
(referred to as the initial abstraction, I
a
) for the equation to be applicable. Values of
CN have been tabulated (Table 2.1) by hydrologic soil group for AMC II, or average
conditions. The CN ranges from 1 to 100, with runoff potential increasing with
increasing CN. Required information to determine a CN value from the table includes
the hydrologic soil group (defined in Table 2.2), the vegetal and cultural practices
of the site, and the AMC (defined in Table 2.3). The CN obtained from Table 2.1
for AMC II can be converted to AMC I (dry) or III (wet) using the values in Table 2.3.
Curve numbers can be determined from rainfall-runoff data for a particular site.
Investigations have been conducted to determine CN values for conditions not
included in Table 2.1 or similar tables. Examples include exposed fractured rock
surfaces (Rasmussen and Evans 1993), animal manure application sites (Edwards
and Daniel 1993), and dryland wheat–sorghum–fallow crop rotation in the semi-arid
western Great Plains (Hauser and Jones 1991).
The CN approach is widely used for estimating runoff volume. Because the CN
is defined in terms of land use treatments, hydrologic condition, AMC, and soil type,
the approach can be applied to ungaged watersheds. Errors in selecting CN values

can result from misclassifying land cover, treatment, hydrologic conditions, or soil
type (Bondelid et al. 1982). The magnitude of the error depends on both the size
of the area misclassified and the type of misclassification. In a sensitivity analysis
of runoff estimates to errors in CN estimates, Bondelid et al. (1982) found that
effects of variations in CN decrease as design rainfall depth increases and confirmed
Hawkins’s (1975) conclusion that errors in CN estimates are especially critical
near the threshold of runoff.
Q
PS
PS
=
−
+
(.)
.
02
08
2
S
CN
=−
25 400
254
,
S
CN
=−
1000
10
© 2007 by Taylor & Francis Group, LLC

24 Modeling Phosphorus in the Environment
TABLE 2.1
Runoff Curve Numbers for Hydrologic Soil-Cover Complexes
Land Use Description/Treatment/Hydrologic Condition Hydrologic Soil Group
Residential:
a
ABCD
Average lot size (ha) Average % impervious
b

0.05 or less 65 77 85 90 92
0.10 38 61 75 83 87
0.13 30 57 72 81 86
0.20 25 54 70 80 85
0.40 20 51 68 79 84
Paved parking lots, roofs, driveways, etc.
c
98 98 98 98
Street and roads:

Paved with curbs and storm sewers
c
98 98 98 98
Gravel 76 858991
Dirt 72 82 87 89
Commercial and business areas (85% impervious) 89 92 94 95
Industrial districts (72% impervious) 81 88 91 93
Open spaces, lawns, parks, golf courses, cemeteries, etc.
Good condition: grass cover on 75% or more of the area 39 61 74 80
Fair condition: grass cover on 50 to 75% of the area 49 69 79 84

Fallow Straight row — 77 86 91 94
Row crops Straight row Poor 72 81 88 91
Straight row Good 67 78 85 89
Contoured Poor 70 79 84 88
Contoured Good 65 75 82 86
Contoured and terraced Poor 66 74 80 82
Contoured and terraced Good 62 71 78 81
Small grain Straight row Poor 65 76 84 88
Good 63 75 83 87
Contoured Poor 63 74 82 85
Good 61 73 81 84
Contoured and terraced Poor 61 72 79 82
Good 59 70 78 81
(continued)
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 25
2.2.1.2 Curve Number Method Implementation
The curve number method is used in several phosphorus models to compute runoff
volume. The most common implementation (e.g., AnnAGNPS, GLEAMS, EPIC,
SWAT) includes a modification of the CN to account for daily changes in soil
moisture content (Williams et al. 1990). Typically, the models require the user to
input a value for CN
2
, the curve number for average conditions, or AMC II. Then,
TABLE 2.1 (CONTINUED)
Runoff Curve Numbers for Hydrologic Soil-Cover Complexes
Land Use Description/Treatment/Hydrologic Condition Hydrologic Soil Group
Close–seeded Straight row Poor 66 77 85 89
legumes

d
Straight row Good 58 72 81 85
or Contoured Poor 64 75 83 85
rotation Contoured Good 55 69 78 83
meadow Contoured and terraced Poor 63 73 80 83
Contoured and terraced Good 51 67 76 80
Pasture Poor 68 79 86 89
or range Fair 49 69 79 84
Good 39 61 74 80
Contoured Poor 47 67 81 88
Contoured Fair 25 59 75 83
Contoured Good 6 35 70 79
Meadow Good 30 58 71 78
Woods or Poor 45 66 77 83
forest land Fair 36 60 73 79
Good 25 55 70 77
Farmsteads — 59 74 82 86
Note: Antecedent moisture condition II and I
a
= 0.2S.
a
Curve numbers are computed assuming 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.
b
The remaining pervious areas (lawn) are considered to be in good pasture condition for these
curve numbers.
c
In some warmer climates of the country a curve number of 95 may be used.
d

Close-drilled or broadcast.
Source: SCS. 1972. Hydrology, Section 4: National Engineering Handbook, U.S. Soil Conser-
vation Service, Washington, D.C., Government Printing Office. With permission.
© 2007 by Taylor & Francis Group, LLC
26 Modeling Phosphorus in the Environment
curve numbers corresponding to AMC I (dry), CN
1
, and AMC III (wet), CN
3
, are
computed as a function of CN
2
. The retention parameter, S, also changes due to
fluctuations in soil moisture content. For example, the same relationship is used in
EPIC and SWAT, with the soil water content expressed differently:
in EPIC or in SWAT
(2.3)
where S
1
(L) and S
max
(L) is the value of S associated with CN
1
(computed with
Equation 2.2), FFC is the fraction of field capacity, SW is the soil water content
(L
3
/L
3
), and w

1
and w
2
are shape parameters. FFC is computed in EPIC as
(2.4)
TABLE 2.2
Hydrologic Soil Group Descriptions and Antecedent Rainfall Conditions
for Use with SCS Curve Number Method
Soil Group Description
A Lowest Runoff Potential. Includes deep sands with very little silt and clay, also deep,
rapidly permeable loess.
B Moderately Low Runoff Potential. Mostly sandy soils less deep than A, and loess less
deep or less aggregated than A, but the group as a whole has above-average
infiltration after thorough wetting.
C Moderately High Runoff Potential. Comprises shallow soils and soils containing
considerable clay and colloids, though less than those of group D. The group has
below-average infiltration after presaturation.
D Highest Runoff Potential. Includes mostly clays of high swelling percent, but the
group also includes some shallow soils with nearly impermeable subhorizons near
the surface.
5-Day Antecedent Rainfall
(mm)
Dormant Growing
Condition General Description Season Season
I Optimum soil condition from about lower plastic
limit to wilting point
< 6.4 < 35.6
II Average value for annual floods 6.4 to 27.9 35.6 to 53.3
III Heavy rainfall or light rainfall and low temperatures
within 5 days prior to the given storm

> 27.9 > 53.3
Source: SCS. 1972. Hydrology, Section 4: National Engineering Handbook, U.S. Soil Conservation
Service, Washington, D.C., Government Printing Office. With permission.
SS
FFC
FFC e
wwFFC
=−
+






−
1
1
12
[()]
SS
SW
SW e
wwSW
=−
+







−
max
()
1
12
FFC
SW WP
FC WP
=
−
−
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 27
where SW is the soil water content in the root zone, WP is the wilting point water
content (corresponds to 1500 kPa matric potential for many soils) (L
3
/L
3
), and FC
is the field capacity water content (corresponds to 33 kPa matric potential for many
soils) (L
3
/L
3
). In EPIC, values for w
1
and w
2

are obtained by simultaneous solution
of Equation 2.3 with the assumptions that S = S
2
when FFC = 0.5 and S = S
3
when
FFC = 1.0. In SWAT, w
1
and w
2
are determined by solving Equation 2.3 with the
following assumptions: S = S
1
when SW = WP, S = S
3
when SW = FC, and the soil
has a CN of 99 (S = 2.54) when completely saturated.
The soil water content can be taken as being uniformly distributed through the
root zone or top meter or some other depth of soil, or a nonuniform distribution of
soil water can be considered. If more of the soil water is at the surface than deeper
in the profile, the potential for runoff is greater. Some of the phosphorus models
keep track of soil moisture by layer, so they have the potential to include the soil
water distribution in their runoff calculations. For example, because EPIC estimates
water content of each soil layer daily, the effect of depth distribution on runoff is
expressed by using a depth-weighted FFC value in Equation 2.3:
(2.5)
TABLE 2.3
Conversion Factors for Converting Runoff
Curve Numbers
Curve Number for

Factor to Convert Curve Number
for Condition II to
Condition II Condition I Condition III
10 0.40 2.22
20 0.45 1.85
30 0.50 1.67
40 0.55 1.50
50 0.62 1.40
60 0.67 1.30
70 0.73 1.21
80 0.79 1.14
90 0.87 1.07
100 1.00 1.00
Note:

AMC II to AMC I and III (I
a
= 0.2S).
Source: SCS. 1972. Hydrology, Section 4: National Engineer-
ing Handbook, U.S. Soil Conservation Service, Washington,
D.C., Government Printing Office. With permission.
FFC
FFC
Z
i
M
i
ZZ
Z
i

M
ZZ
Z
i
ii
i
ii
i
*
,=
∑
(
)
∑
≤
=
−
=
−
−
−
1
1
1
1
110. m
© 2007 by Taylor & Francis Group, LLC
28 Modeling Phosphorus in the Environment
where FFC
*

is the depth-weighted FFC value for use in Equation 2.3, Z is the depth
(m) to the bottom of soil layer i, and M is the number of soil layers. Equation 2.5
reduces the influence of lower layers because FFC
i
is divided by Z
i
and gives proper
weight to thick layers relative to thin layers because FFC is multiplied by the layer
thickness.
GLEAMS also computes a depth-weighted retention parameter:
(2.6)
where W
i
is the weighting factor, SM
i
is the water content in soil layer i (L), and
UL
i
is the upper limit of water storage in layer i (L). The weighting factors decrease
with depth according to the equation:
(2.7)
where D
i
is the depth to the bottom of layer i (L) and RD is the root zone depth (L).
The sum of the weighting factors equals one.
Assuming that the CN
2
value in Table 2.1 (SCS 1972) is appropriate for a 5%
slope, Williams et al. (1990) developed an equation to adjust that value for other slopes:
(2.8)

where CN
2s
is the handbook CN
2
value adjusted for slope and s is the average slope
of the watershed (L/L). This adjustment is included in EPIC but not in SWAT.
EPIC also accounts for uncertainty in the retention parameter, or CN, by gen-
erating the final curve number estimate from a triangular distribution. The mean of
the distribution is the best estimate of CN based on using Equations 2.2 through
2.5, and 2.8 and an equation to adjust S for frozen ground. The extremes of the
distribution are ±5 curve numbers from the mean.
Another example of a modification in implementation of the CN method is seen
in GLEAMS. In the U.S., soils are grouped by series name, and a hydrologic soil
group is assigned to each series. However, a series name can include different soil
textures, which would have different runoff potentials but would still be in the same
hydrologic soil group. The developers of GLEAMS expanded Table 2.1 to give a
range of curve numbers for each combination in the table to allow users to distinguish
between similar soils within a series (Table 2.4). For example, CN
2
for row crops
with straight rows in good hydrologic condition could be 78 for a Cecil sandy loam
and 82 for a Cecil clay loam.
Care must be taken in utilization of the CN method in different scale models.
The CN method was developed based on data from small watersheds, so it should
SS W
SM
UL
i
i
i

i
N
=−














=
∑
1
1
10.
We e
i
D
RD
D
RD
ii
=−

−






−






−
1 016
416 416
1
.









CN CN CN e CN
s

s
232
13 86
2
1
3
12=−− +
−
()[]
.
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 29
not be applied to a whole watershed larger than that. A larger watershed is subject
to spatial variability in rainfall amounts and increased transmission losses due to
increased flow path lengths, changing the CN value from that of a smaller watershed.
For example, Simanton et al. (1996) found that the optimum curve number — to
match measured runoff values — decreased with increasing drainage area for 18
semi-arid watersheds in southeastern Arizona. Some phosphorus models divide large
watersheds into subwatersheds or other smaller hydrologic response units. It is
reasonable to apply the CN to the smaller response units and then to determine how
the runoff from the individual units contributes to streamflow, through routing or
other methods, as described in the following sections (2.2.3).
2.2.1.3 Infiltration-Based Approaches
Infiltration is defined as the entry of water from the surface into the soil profile.
From a ponded surface or a rainfall situation, infiltration rate decreases over time
and asymptotically approaches a final infiltration rate. The final infiltration rate is
approximately equal to the saturated hydraulic conductivity, K
s
, of the soil. The
amount and rate of infiltration depend on infiltration capacity of the soil and the

availability of water to infiltrate. Infiltration capacity is influenced by soil properties,
soil texture, initial soil moisture content, surface conditions, and availability of water
to be infiltrated, i.e., precipitation or ponded water. Rainfall intensity affects infil-
tration rate. If the infiltration capacity of the soil is exceeded by the rainfall intensity
(L/T), then water will pond on the soil surface, and the infiltration rate will equal
the infiltration capacity. If the rainfall rate is less than the saturated hydraulic
TABLE 2.4
Excerpt of Expanded Curve Number Table for GLEAMS Model
Land Use Treatment
or Practice
Hydrologic
Condition
Hydrologic Soil Group
ABCD
Row Crops
SR Poor 65 72 77 78 81 85 86 87 88 90 91 92
SR Good 60 67 73 74 78 82 83 85 87 88 89 90
SR + CT Poor 66 71 75 76 79 83 84 86 87 88 89 90
SR + CT Good 57 64 70 71 75 79 80 82 83 84 85 86
CNT Poor 64 70 75 76 79 81 82 84 86 87 88 89
CNT Good 59 65 70 71 75 79 80 82 84 85 86 87
Note: SR = straight row; CT = conservation tillage; CNT = contoured.
Source: Knisel, W.G., GLEAMS: Groundwater Loading Effects of Agricultural Management Sys-
tems, Version 2.10, University of Georgia, Coastal Plain Experiment Station, Biological and Agri-
cultural Engineering Department, Publication 5, p. 130, 1993. With permission.
© 2007 by Taylor & Francis Group, LLC
30 Modeling Phosphorus in the Environment
conductivity of the soil, the infiltration rate will equal the rainfall rate, and ponding
will not occur.
A number of infiltration equations have been developed, ranging from solving the

Richards (1931) equation to empirical equations. The Richards equation, the generalized
equation for flow in porous media, is a partial differential equation derived from conser-
vation of mass and Darcy’s equation describing flux. To simulate infiltration, the Richards
equation is solved subject to appropriate boundary and initial conditions. Empirical
infiltration equations typically include coefficients or exponents to represent soil prop-
erties and to generate the relationship of decreasing infiltration rate with time. Some
infiltration equations that have been used in phosphorus models include the Holtan (1961)
equation, which was used in the original ANSWERS event-based model (Beasley et al.
1982); the Philip (1957) equation, which is the basis of the infiltration calculations in
HSPF; and the Green and Ampt (1911) equation.
2.2.1.3.1 Green and Ampt Approach Description
In phosphorus models that include infiltration simulation (e.g., ANSWERS-2000,
SWAT), the Green and Ampt (1911) equation as modified by Mein and Larson (1973)
is the most common approach used to estimate infiltration. This approach is phys-
ically based, and its parameters can be determined from readily available soil and
vegetal cover information. The approach has been tested for a variety of conditions
and has successfully simulated the effects of different management practices on
infiltration.
The original Green and Ampt (1911) equation was derived using Darcy’s law
for infiltration from a ponded surface into a deep, homogeneous soil profile with
uniform initial water content. Water is assumed to enter the soil as slug flow resulting
in a sharply defined wetting front that separates a zone that has been wetted from
an unwetted zone. Infiltration rate is expressed as
(2.9)
where f is infiltration rate (L/T), K
s
is saturated hydraulic conductivity (L/T), M is
the difference between final and initial moisture content (the difference in moisture
content across the wetting front) (L
3

/L
3
), S
av
is average wetting front suction (L),
and F is cumulative infiltration (L). Substituting into Equation 2.9 and
integrating with F = 0 at time (t) = 0 yields
(2.10)
Mein and Larson (1973) extended the Green and Ampt equation to rainfall
conditions by first determining cumulative infiltration at the time of surface ponding:
(2.11)
fK
MS
F
s
av
=+






1
fdFdt=
Kt F S M
F
MS
sav
av

=− +






ln 1
F
SM
R
K
p
av
s
=
−1
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 31
where F
p
is cumulative infiltration at time of ponding (L) and R is rainfall rate (L/T).
Before ponding occurs, the infiltration rate is equal to the rainfall rate. After ponding
occurs, the infiltration rate is a function of the infiltration capacity of the soil. The
Green-Ampt-Mein-Larson (GAML) model for infiltration rate is a two-stage model
with f = R for t ≤ t
p
, and f is computed with Equation 2.9 after ponding. If R < K
s
,

surface ponding will not occur, providing the profile is deep and homogeneous as was
assumed in the derivation of the equations and f = R. By recognizing that f = dF/dt
and accounting for the preponding stage, Mein and Larson (1971) developed an
equation for cumulative infiltration over time:
(2.12)
where t′
p
is the equivalent time (T) to infiltrate F
p
under initially ponded conditions.
Since Equation 2.12 is implicit in F, it might be desirable to increment F and to
solve directly for time, t, and then for f from Equation 2.9.
A number of studies have been conducted related to estimating the values of
the GAML parameters K
s
, M, and S
av
. Skaggs and Khaleel (1982) reported that
Bouwer (1966) and Bouwer and Asce (1969) showed that the hydraulic conductivity
parameter should be less than the saturated value because of entrapped air. When
measured values are not available, Bouwer (1966) suggested that an effective hydrau-
lic conductivity of 0.5 K
s
be used in place of K
s
. Other adjustments to K
s
have also
been recommended to account for the impact of surface conditions on infiltration,
resulting in the concept of an effective hydraulic conductivity, K

e
. Similarly, it has
been suggested that the final moisture content included in the determination of M
be something less than saturation due to air entrapment in the field.
The most difficult Green-Ampt parameter to estimate is the suction term, pre-
sented as effective suction at the wetting front by Green and Ampt and then as
average suction at the wetting front by Mein and Larson. One accepted estimation
method (Mein and Larson 1973) is using the unsaturated hydraulic conductivity as
a weighting factor and defining S
av
as
(2.13a)
where
ψ
is the soil water suction (negative of the matric potential) (L) and K
r
is the
relative hydraulic conductivity = K(
ψ
)/K
s
. Neuman (1976) obtained a theoretical
expression for S
av
, similar to Equation 2.13a, by relating it to the physical charac-
teristics of the soil:
(2.13b)
Kt t t F MS
F
MS

spp av
av
() ln−+
′
=− +






1
SdK
av r
=
∫
ψ
0
1
SKd
av r
i
=
∫
ψ
ψ
0
© 2007 by Taylor & Francis Group, LLC
32 Modeling Phosphorus in the Environment
where

ψ
i
is the suction at the initial water content. Morel-Seytoux and Khanji (1974)
found that for most cases the value of S
av
given by Equation 2.13a or 2.13b is a
reasonable approximation for an effective matric drive, which is dependent on the
relative conductivities of both air and water (Skaggs and Khaleel 1982). Because
Equation 2.13 requires the K(
ψ
) relationship, researchers have developed predictive
equations for K(
ψ
) or S
av
, and K
e
as well, based on readily available soil characteristics
(e.g., Brakensiek and Rawls 1983; Rawls and Brakensiek 1986; Rawls et al. 1989).
2.2.1.3.2 Green-Ampt Approach Implementation
The Green-Ampt approach is implemented in several phosphorus models. In
ANSWERS-2000, Equation 2.10 is solved using Newton’s iteration technique to
determine cumulative infiltration depth, F. The infiltration rate is then computed
using Equation 2.9. ANSWERS-2000 replaces K
s
with K
e
, which is computed as the
weighted sum of K
e

under canopy cover and K
e
for the area outside the canopy. The
values are computed as functions of soil parameters, (i.e., K
s
, effective porosity, bulk
density, residual soil water, sand, silt, and clay fractions) and vegetation parameters
(i.e., percent area outside the canopy, percent bare area under canopy, percent canopy
area) using equations primarily from Rawls et al. (1989). The available porosity, M,
is computed as the difference between porosity corrected for rocks and the antecedent
soil moisture content. S
av
is computed using an empirical equation developed by
Rawls and Brakensiek (1985), in which S
av
is a function of total porosity, sand
fraction, and clay fraction.
For implementation in the SWAT model, Equation 2.10 applied at time (t – ∆t)
was subtracted from Equation 2.10 applied at time t to yield the following expression
for F at time t:
(2.14)
SWAT uses a successive substitution technique to solve Equation 2.14. For each
time step, SWAT calculates the amount of water entering the soil. The water that
does not infiltrate into the soil becomes surface runoff. SWAT uses effective hydraulic
conductivity, K
e
, in place of K
s
; K
e

is computed as a function of K
s
and CN, thus
incorporating land-cover impacts into the hydraulic conductivity value (Nearing et al.
1996). Similar to ANSWERS-2000, SWAT uses the expression developed by Rawls
and Brakensiek (1985) to calculate S
av
.
2.2.2 HYDROGRAPH DEVELOPMENT
A hydrograph is the relationship between flow rate and time. In some phosphorus
models, only the runoff hydrograph as it enters a receiving stream is computed. In
others, the spatial variability of the hydrograph along the slope is simulated as well.
Some phosphorus models do not directly compute a hydrograph; instead, all of the
runoff is assumed to reach the receiving water in a certain time frame or the runoff
is lagged in some way to determine when it arrives at the receiving water.
FF KtSM
FSM
FSM
tt s av
tav
tav
=+ +
+
+







−
−
1
1
∆ ln
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 33
Runoff, or overland flow, can be visualized as sheet-type flow — as opposed to
channel flow — with small depths of flow and slow velocities (less than 0.3 m/sec).
Considerable volumes of water can move through overland flow. Overland flow is
spatially varied and usually unsteady and nonuniform — that is, the velocity and
flow depth vary in both time and space. Input (rainfall) to the flow is distributed
over the flow surface. Methods for determining hydrographs range from direct
calculations of the basic hydrograph descriptors — peak flow, time to peak, and
base time — to overland flow routing, which yields the relationship of runoff rate
with space and time. Examples of the methods used in some phosphorus models are
described in the following sections.
2.2.2.1 Kinematic Flow Routing
The theoretical hydrodynamic equations attributed to St. Venant are based on the
fundamental laws of conservation of mass (continuity) and conservation of
momentum applied to a control volume or fixed section of channel with the
assumptions of one-dimensional flow, a straight channel, and a gradual slope (Hug-
gins and Burney 1982). With these assumptions, a uniform velocity distribution and
a hydrostatic pressure distribution can be assumed, resulting in quasi-linear partial
differential equations. Because these equations are not typically implemented in
phosphorus models, they are not given here. Detailed derivations of continuity and
momentum equations as they apply to unsteady, nonuniform flow can be found in
Strelkoff (1969).
Lighthill and Whitham (1955), cited by Huggins and Burney (1982), proposed
that the dynamic terms in the momentum equation had negligible influence for cases

in which backwater effects were absent. Neglecting these terms yields a quasi-steady
approach, known as the kinematic wave approximation. The kinematic approxima-
tion is composed of the continuity equation
(2.15)
and a flow (depth-discharge) equation of the general form
(2.16)
where y is local depth of flow (L), q is discharge per unit width (L
3
/T/L), I is lateral
inflow per unit length and width of the flow plane (L
3
/T/L
2
), f is lateral outflow per
unit length and width of the flow plane (L
3
/T/L
2
), t is time (T), x is the flow direction
axis (L), and
α
and m are parameters.
The flow equation can be one describing laminar or turbulent channel flow, with
the overland flow plane represented by a wide channel. Overton (1972) analyzed
200 hydrographs for relatively long, impermeable planes and found that flow was
turbulent or transitional. Foster et al. (1968) concluded that both Manning’s and
Darcy-Weisbach flow equations were satisfactory for describing overland flow on
short erodible slopes.
∂
∂

+
∂
∂
=−
y
t
q
x
If
qy
m
=
α
© 2007 by Taylor & Francis Group, LLC
34 Modeling Phosphorus in the Environment
The most commonly used flow equation for overland flow is Manning’s equation,
which can be written for overland flow as
(2.17)
where q is discharge (m
3
/s/m of width), n is the roughness coefficient, y is flow
depth (m), and S is the slope of energy gradeline, usually taken as surface slope
(m/m). Values of Manning’s n factor vary from 0.02 for smooth pavement to 0.40
for average grass cover. Manning’s n values are tabulated in a variety of sources
(e.g., Linsley et al. 1988; Novotny and Olem 1994).
Woolhiser and Liggett (1967) developed an accuracy parameter to assess the
effect of neglecting dynamic terms in the momentum equation:
(2.18)
where k is a dimensionless parameter, S
o

is the bed slope (L/L), L is the length of
bed slope (L), H is the equilibrium flow depth at the outlet (L), and F is the
equilibrium Froude number for flow at the outlet (dimensionless). For values of k
greater than 10, very little advantage in accuracy is gained by using the momentum
equation in place of a depth–discharge relationship. Since k is usually much greater
than 10 in virtually all overland flow conditions, the kinematic wave equations
generally provide an adequate representation of the overland flow hydrograph (Huggins
and Burney 1982).
2.2.2.2 SCS Unit Hydrograph
The SCS unit hydrograph method generates a triangular-shaped hydrograph. The
relationship among the time parameters of the hydrograph is described as
(2.19)
where T
p
is time to peak (T), D is duration of excess rainfall (T), T
L
is lag, or mean
travel time (T), and T
c
is the time of concentration (T), or travel time from the most
hydraulically remote point in the watershed to the watershed outlet. Lag time (h) is
computed as
(2.20)
where L is the maximum length of flow (m), CN is the runoff curve number, and S
g
is the average watershed gradient (m/m). The peak discharge is a function of amount
of runoff computed with Equation 2.1:
(2.21)
q
n

yS=
1
53 12//
k
SL
HF
=
o
2
T
D
T
D
T
p
L
c
=+=+
22
0.6
L
g
T
=L
CN
S
08
07
05
1000

9
4407
.
.
.
()
−






p
u
q
=q AQ
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 35
where q
p
is peak runoff rate (m
3
/s), q
u
is unit peak flow rate (m
3
/s per ha/mm of
runoff), Q is the runoff volume (mm), and A is watershed area (ha). Unit peak flow
rate, q

u
, can be obtained from charts (e.g., SCS 1986) or computed from Haan et al.
(1994):
(2.22)
where the Cs are from a table as a function of I
a
/P, with I
a
generally taken as 0.2
S, with S computed from Equation 2.2.
2.2.2.3 Hydrograph Development Implementation
Phosphorus models use different methods to develop runoff hydrographs or route
overland flow. For example, ANSWERS-2000 uses kinematic routing, HSPF
includes a flow equation with an empirical relationship, SWAT computes a runoff
lag time for delivery to channels, and AnnAGNPS applies the SCS unit hydrograph.
ANSWERS-2000 applies the kinematic wave equations to route overland flow
with an explicit, backward difference solution of the continuity equation combined
with Manning’s stage-discharge equation. The hydraulic radius in Manning’s equa-
tion is assumed equal to the average detention depth in the cell — ANSWERS-2000
represents a watershed as a grid of cells. Detention depth is calculated as the total
volume of surface water in a cell minus the retention volume divided by the area of
the cell. This implies that the entire specified retention volume of an element must
be filled before any water becomes available for surface detention and runoff.
ANSWERS-2000 uses a surface detention model developed by Huggins and Monke
(1966) to describe the surface storage potential of a surface as a function of the
water depth on the soil surface. Each cell acts as an overland flow plane with a user-
specified slope steepness and direction. Flow is routed from one cell to another until
it enters a channel element, which then routes the runoff to the watershed outlet.
In HSPF, overland flow is treated as a turbulent flow process. It is simulated
using the Chezy-Manning equation and an empirical expression that relates outflow

depth to detention storage. The rate of overland flow discharge is determined by the
equations
, for S
d
< S
e
(2.23a)
, for S
d
≥ S
e
(2.23b)
where O is surface outflow (in./interval), ∆t is the length of the time interval, C
r
is
the routing variable = , S
d
is the mean surface detention storage over the
time interval (in.), S
e
is the equilibrium surface detention storage (in.) for current
log( ) log (log )qCCtC t
ucc
=+ +
01 2
2
OtCS
S
S
rd

d
e
=+



















∆ 10 06
3
167

.






OtS S
rd
=
()
∆ 16
167
.
.
1020 snL/
© 2007 by Taylor & Francis Group, LLC
36 Modeling Phosphorus in the Environment
supply rate, s is slope of the overland flow plane (ft/ft), n is Manning’s roughness
coefficient, and L is length of the overland flow plane (ft). Equation 2.23a applies
when the overland flow rate is increasing, whereas Equation 2.23b applies when the
overland flow rate is at equilibrium or receding.
SWAT determines when overland flow reaches a channel based on the time of
concentration for the overland flow area. If t
c
is less than one day, SWAT adds the
runoff generated on that day to the channel on that day. If t
c
is greater than one day,
only a portion of the surface runoff will reach the main channel on the day it is
generated. SWAT incorporates a surface runoff storage feature to lag a portion of
the surface runoff release to the main channel. After surface runoff is calculated, the
amount of surface runoff released to the main channel is calculated as
(2.24)
where Q

surf
is the amount of surface runoff discharged to the main channel on a
given day (mm), Q′
surf
is the amount of surface runoff generated on a given day
(mm), Q
stor, i-1
is the surface runoff stored or lagged from the previous day (mm),
surlag is the surface runoff lag coefficient, and t
c
is the time of concentration for
the sub-basin (h). The expression [1 − exp(-surlag/t
c
)] in Equation 2.24 represents
the fraction of the total available water that will be allowed to enter the reach on
any one day. For a given t
c
, as surlag decreases more water is held in storage. The
delay in release of surface runoff smooths the streamflow hydrograph simulated in
the reach.
2.2.3 STREAMFLOW, OR CHANNEL, ROUTING
Many phosphorus models include channel processes in addition to upland processes.
Routing streamflow entails computing the effect of channel storage on the shape
and movement of a hydrograph, or flood wave. When a flood wave advances into a
reach segment, inflow exceeds outflow and a wedge of storage is produced. As the
flood wave recedes, outflow exceeds inflow in the reach segment, and a negative
wedge is produced. In addition to the wedge storage, the reach segment contains a
prism of storage formed by a volume of constant cross-section along the reach length.
Streamflow routing involves relationships among inflow, outflow, and storage in
each reach or segment of the stream. The continuity equation for unsteady flow

relating inflow, outflow, and storage in a reach is
(2.25)
where I is inflow (L
3
/T), O is outflow (L
3
/T), S is storage (L
3
), and t is time (T).
Flow routing procedures typically assume that the average of flows at the beginning
QQQ e
surf surf stor i
surlag
t
c
=
′
+−
−
−







()
,1
1


()
IO
dS
dt
−=
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 37
and end of a short time period ∆t (routing period) equals the average flow during
the period. Expressing Equation 2.25 for a finite time interval yields
(2.26)
where the subscripts 1 and 2 refer to the beginning and end of the time period,
respectively. The routing period should be selected to be sufficiently short to ensure
that this assumption is not seriously violated. If ∆t is too long (> travel time through
reach), the wave crest could pass through the reach during the routing period.
Generally, ∆t is 1/2 to 1/3 the travel time through the reach.
Most storage routing methods are based on Equation 2.26, which includes two
unknowns: the storage and outflow at the end of the time interval. A second rela-
tionship is required to determine the unknowns. Different methods define the second
relationship in different ways.
2.2.3.1 Hydrologic, or Storage, Routing
Hydrologic, or storage, routing is the simplest form of routing and is based on the
continuity equation (Equation 2.26) (Haan et al. 1994). The storage in a channel
reach depends on the channel geometry and depth of flow. The flow rate can be
related to the depth or cross-sectional area of flow, assuming steady, uniform flow
using Manning’s equation for each cross-section:
(2.27)
where q is the flow rate (m
3
/s); R is the hydraulic radius (m), which is equal to the

cross-sectional area of flow divided by the wetted perimeter; S is the slope of the
channel (m/m); and A is the cross-sectional area of flow (m
2
). Manning’s equation
was previously presented for overland flow (Equation 2.17), in which an overland
flow plane is assumed to be a very wide, shallow channel — which leads to R being
approximated by y — and discharge is computed per unit width. A simple method
for computing the storage in a reach is to multiply the length of the reach by the
average cross-sectional area of the reach at a given flow rate.
Flow routed down the channel as the outflow from one reach becomes the inflow
to the next reach. Additional inflows from overland flow, tributaries, and ground
water can be added to the inflow or outflow of each reach as well.
2.2.3.2 Muskingum Routing Method
The Muskingum flow routing method is described in many references. The following
description is adapted from Linsley et al. (1988).
The Muskingum routing method models the storage volume in a channel length
as a combination of wedge and prism storages. As defined by Manning’s equation,
II
t
OO
tS S
12 1 2
21
22
+
−
+
=−∆∆
q
n

RSA=
1
2
3
1
2
/
/
© 2007 by Taylor & Francis Group, LLC
38 Modeling Phosphorus in the Environment
the cross-sectional area of flow is assumed to be directly proportional to the discharge
for a given reach segment. The Muskingum flow routing method is based on the
continuity equation (Equation 2.25). The second relationship between storage and
outflow is based on the following expression for storage in a reach of a stream:
(2.28)
where a and n are constants from the mean stage–discharge relation for the reach,
q = ay
n
, and b and m are constants from the mean stage–storage relation for the
reach, S = by
m
. In a uniform rectangular channel, storage would vary with the first
power of stage (m = 1), and discharge would vary as the 5/3 power of stage
(Manning’s formula). In a natural channel with overbank floodplains, the exponent
n may approach or become less than unity. The constant X expresses the relative
importance of inflow and outflow in determining storage. For a simple reservoir,
X = 0 (storage has no effect), whereas if inflow and outflow were equally effective,
X would be 0.5. For most streams, X is between 0 and 0.3, with a mean value near 0.2.
The Muskingum method assumes that m/n = 1 and lets b/a = K, yielding
S = K[XI + (1 – X)O] (2.29)

The constant K, known as the storage constant, is the ratio of storage to discharge
and has the dimension of time. It is approximately equal to the travel time through
the reach and in the absence of better data is sometimes estimated in this way. If
flow data on previous floods are available, K and X are determined by plotting S vs.
[XI + (1 – X)O] for various values of X. The best value of X is that which causes
the data to plot most nearly as a single-valued curve. The Muskingum method
assumes that this curve is a straight line with slope K. The units of K depend on the
units of flow and storage.
Substituting Equation 2.29 for S in Equation 2.26 and collecting like terms yields
O
2
= c
0
I
2
+ c
1
I
1
+ c
2
O
1
(2.30)
where
(2.30a)
(2.30b)
(2.30c)
c
0

+ c
1
+ c
2
= 1 (2.30d)
S
b
a
XI X O
mn mn
=+−[()]
//
1
c
KX t
KKX t
0
05
05
=−
−
−+
.
.
∆
∆
c
KX t
KKX t
1

05
05
=
+
−+
.
.
∆
∆
c
KKX t
KKX t
2
05
05
=
−−
−+
.
.
∆
∆
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 39
The routing period ∆t is in the same units as K. With K, X, and ∆t established,
the values of c
0
, c
1
, and c

2
can be computed. The routing operation then consists of
solving Equation 2.30 with the O
2
of one routing period becoming the O
1
of the
succeeding period. To maintain numerical stability and to avoid the computation of
negative outflows, the following condition must be met:
(2.31)
2.2.3.3 Streamflow, or Channel, Routing Implementation
The SWAT model utilizes either the variable storage routing method developed by
Williams (1969) and used in the hydrologic model (HYMO) (Williams and Hann 1973)
and routing outputs to outlets (ROTO) (Arnold et al. 1995) models or the Muskingum
method. The variable storage-routing method is based on the continuity equation. Travel
time is computed by dividing the volume of water in the channel by the flow rate.
Substituting the relationship for travel time into the continuity equation (Equation 2.26)
and simplifying yields the expression for outflow from the reach segment:
(2.32)
where TT is travel time.
In the implementation of the Muskingum routing procedure in SWAT, the value
for the weighting factor, X, is input by the user. As just noted, for most streams X
is between 0 and 0.3 with a mean value near 0.2. The user can use this information
or site-specific knowledge to assign a value for X. The value for K, the storage time
constant, is estimated as
(2.33)
where K is the storage time constant for the reach segment, coef
1
and coef
2

are
weighting coefficients input by the user, K
bnkfull
is the storage time constant calculated
for the reach segment with bankfull flows, and K
0.1bnkfull
is the storage time constant
calculated for the reach segment with one tenth of the bankfull flows. An equation
developed by Cunge (1969) is used to calculate K
bnkfull
and K
0.1bnkfull
.
Routing in HSPF is also based on the continuity equation. The second relation-
ship is based on the assumption that outflows are functions of volume or time, or a
combination of the two. If the outflow is a function of volume, the user defines the
depth-surface area-volume-discharge relationship in an input table (an FTABLE in
HSPF terms). This is one of four optional methods for computing overland flow.
Another method is a simple power function method.
Some phosphorus models also include transmission, or leaching, losses from
channels during periods when a stream receives no groundwater contributions.
221KX t K X<< −∆ ()
O
t
TT t
II
S
2
12
1

2
22
=
+






+
+






∆
∆
KcoefK coef K
bnkfull bnkfull
=+
1201.
© 2007 by Taylor & Francis Group, LLC
40 Modeling Phosphorus in the Environment
For example, SWAT uses Lane’s method described in Chapter 19 of the SCS Hydrol-
ogy Handbook (SCS 1983) to estimate transmission losses from intermittent or ephem-
eral channels. Water losses from the channel are a function of effective hydraulic
conductivity of the channel alluvium, flow travel time, wetted perimeter, and channel

length. Both volume and peak rate are adjusted when transmission losses occur in
tributary channels.
2.2.4 PEAK RATE OF RUNOFF
2.2.4.1 Rational Formula
The rational method is the most common method used for peak flow estimation in
practice. The method is presented in many references. The following description is
based on Haan et al. (1994).
The rational equation is
(2.34)
where q
p
is the peak rate of runoff (cfs), C is a dimensionless runoff coefficient,
i is the rainfall intensity (in./h) for a duration equal to the time of concentration, t
c
,
and A is the drainage area (ac). To be dimensionally correct, a conversion factor of
1.008 should be included to convert acre-inches per hour to cubic feet per second;
however, this factor is generally neglected. Time of concentration, t
c
, is defined as
the travel time from the most hydraulically remote point in the watershed to the
watershed outlet and is typically computed as a function of the length of flow and
the slope of the watershed.
The basic concept of the rational equation is as follows. If a steady rainfall
occurs on a watershed, the runoff rate will increase until the entire watershed is
contributing runoff. If a rainfall of duration less than t
c
occurs, the entire basin will
not be contributing, so the resulting runoff rate will be less than from a rainfall with
a duration equal to t

c
. If a rainfall of duration greater than t
c
occurs, the relationship
between average rainfall intensity and duration for a given frequency shows that the
average intensity will be less than if the duration was equal to t
c
. Thus, it is reasoned
that a rainfall of duration t
c
produces the maximum flow rate.
The rational equation is based on several assumptions:
The rainfall occurs uniformly over the drainage area.
The peak rate of runoff can be reflected by the rainfall intensity averaged
over a time period equal to the time of concentration for the drainage area.
The frequency of runoff is the same as the frequency of the rainfall used in
the equation.
The runoff coefficient, C, is the most difficult factor to accurately determine since
it represents the impact of many factors — such as interception, infiltration, surface
detention, and antecedent moisture conditions — on the peak runoff. Various tables
qCiA
p
=
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 41
of C values are available in many sources. Some provide ranges of values for different
conditions such as land use, rainfall intensity, and soil texture. Others provide single
values for combinations of factors.
Haan et al. (1994) note that in spite of the recognized shortcomings of the rational
method, it continues to be widely used because of its simplicity, entrenchment in

practice, coverage in texts, and lack of a comparable alternative.
2.2.4.2 SCS TR-55 Method
SCS (1986) presented the Graphical method for computing peak discharge from
rural and urban areas. The Graphical method was developed from hydrograph anal-
yses using TR-20, “Computer Program for Project Formulation — Hydrology” (SCS
1983).
In the Graphical method, the peak discharge equation is computed as
q
p
= q
u
AQF
p
(2.35)
where q
p
is the peak discharge (cfs), q
u
is the unit peak discharge (cfs/mi
2
/in. runoff),
A is the drainage area (mi
2
), Q is the runoff (in.), and F
p
is a pond and swamp
adjustment factor. The unit peak discharge, q
u
, is a function of t
c

and I
a
/P, the ratio
of initial abstraction to rainfall amount. SCS (1986) provided tables and figures for
determining q
u
based on known t
c
, CN, and P. Q is computed from the curve number
equation, Equation 2.1, based on a 24-h P with a return period of the peak flow. F
p
is based on the percentage of ponds and swampy areas, assumed to be distributed
throughout the watershed. The value of F
p
ranges from 1.0 for 0% pond and swamp
areas to 0.72 for 5% pond and swamp areas.
2.2.4.3 Peak Runoff Rate Implementation
Some phosphorus models do not include calculations of peak runoff rate. In some
models, peak discharge can be determined from overland flow routing. In other
models, peak runoff rate is computed for use in erosion calculations but not in
hydrologic calculations. EPIC and SWAT compute peak runoff rate using a modified
rational formula, whereas AnnAGNPS computes peak discharge similar to the TR-55
method, and GLEAMS uses a different empirical relationship.
In EPIC and SWAT, the peak runoff rate is predicted based on a modification
of the rational formula, with different units used in the two models:
in EPIC in SWAT (2.36)
where q
p
is peak runoff (m
3

/s),
α
is a dimensionless parameter expressing the
proportion of total rainfall that occurs during t
c
, Q is runoff volume (mm), A is
drainage area (ha), t
c
is time of concentration (h),
α
tc
is the fraction of daily rainfall
q
QA
t
p
c
=
α
360
q
QArea
t
p
t
c
c
=
α
36.

© 2007 by Taylor & Francis Group, LLC
42 Modeling Phosphorus in the Environment
that occurs during t
c
, Area is drainage area (km
2
), and 360 and 3.6 are unit conver-
sions. Equation 2.36 results from relationships for C and i, described in the following
paragraphs, substituted into Equation 2.34 with changes in units.
The runoff coefficient, C, is calculated for each storm as the ratio of the runoff
volume and the amount of rainfall. The runoff volume is computed in the model,
and rainfall is an input to the model. This modification eliminates the need to estimate
C from published tables.
The rainfall intensity, i, is determined as the average rainfall rate during the time
of concentration. An analysis of rainfall data collected by Hershfield (1961) for
different durations and frequencies showed that the amount of rain falling during
the time of concentration, R
tc
, was proportional to the amount of rain falling during
the 24-h period, or where
α
tc
is the fraction of daily rainfall that occurs
during the time of concentration. For short-duration storms, all or most of the rain
will fall during the time of concentration, causing
α
tc
to approach its upper limit of
1.0. The minimum value of
α

tc
would be seen in storms of uniform intensity (i
24
=
i). Thus,
α
tc
falls in the range t
c
/24 ≤
α
tc
≤ 1.0.
When the value of
α
tc
is assigned based on only daily rainfall and simulated
runoff, there is considerable uncertainty. Thus, in EPIC
α
is generated from a gamma
function with the base ranging from t
c
/24 to 1.0.
SWAT estimates
α
tc
as a function of the fraction of daily rain falling in the half
hour of highest intensity rainfall:
(2.37)
where

α
0.5
is the fraction of daily rain falling in the half hour of highest-intensity
rainfall. If subdaily precipitation data are used in the model, SWAT calculates the
maximum half hour rainfall fraction directly from the precipitation data. If daily
precipitation data are used, SWAT generates a value for
α
0.5
from a triangular
distribution, which requires four inputs: (1) average monthly half hour rainfall frac-
tion, (2) maximum value for half hour rainfall fraction allowed in month, (3) min-
imum value for half hour rainfall fraction allowed in month, and (4) a random number
between 0.0 and 1.0.
The time of concentration is computed in EPIC and SWAT as the sum of surface
and channel flow times. The time of concentration for channel flow is computed as
the average channel flow length for the watershed divided by the average channel
velocity. Average channel flow length is estimated as
(2.38)
where L is the channel length from the most distant point to the watershed outlet
(km) and L
ca
is the distance along the channel to the watershed centroid (km).
Average velocity is estimated using Manning’s equation, assuming a trapezoidal
channel with 2:1 side slopes and a 10:1 bottom width-to-depth ratio. Further,
RR
ttc
c
=
α
24

,
α
α
t
t
c
c
e=−
−
1
21
05
[ln( )]
.
LLL
cca
= ()( )
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 43
assuming that L
ca
= 0.5 L and that the average flow rate is about 6.35 mm/h and is
a function of the square root of drainage area yields the final equation for the time
of concentration for channel flow, t
cc
:
(2.39)
where n is Manning’s roughness coefficient and
σ
is the average channel slope (m/m).

A similar approach is used to estimate time of concentration for surface flow,
t
cs
, assuming that t
cs
is the surface slope length divided by the surface flow
velocity. Manning’s equation was applied to a strip 1 m wide and the average
flow rate was assumed to be 6.35 mm/h. The resulting equation for estimating
t
cs
in hours is
(2.40)
where
λ
is the surface slope length (m) and S is the land surface slope (m/m).
Williams et al. (1990) noted that although some of the assumptions used in
developing the equations for t
cc
and t
cs
may appear liberal, they generally give
satisfactory results from small homogeneous watersheds. The authors also stated
that since the equations are based on hydraulic considerations, they are more reliable
than purely empirical equations.
For implementation in AnnAGNPS, the TR-55 Graphical method was extended
to apply to drainage areas with t
c
greater than 10 h and to I
a
/P ratios from 0 to 1. The

equation is implemented in AnnAGNPS as
(2.41)
where q
p
is peak discharge (m
3
/s), P
24
is 24-h effective rainfall (mm), D
a
is drainage
area (ha), t
c
is time of concentration (h), and a, b, c, d, e, and f are regression
coefficients for a given I
a
/P and rainfall distribution type. The regression coefficients
eliminate the need for look-up tables in the program.
In GLEAMS, peak flow rate for daily runoff is computed with an empirical
relationship:
(2.42)
where q
p
is the peak runoff rate (ft
3
/s), DA is the drainage area (mi
2
), S is the hydraulic
slope (ft/mi), Q is the daily runoff volume (in.), and LW is the length-to-width ratio
of the watershed. The hydraulic slope of a field is defined as the slope of the longest

flow path. The longest flow path is the flow line from the most remote point on the
t
Ln
A
cc
=
11
075
0 125 0 375
.
.

σ
t
n
S
cs
=
()
.
.
λ
06
03
18
qPD
act et
bt dt
pa
cc

cc
=⋅
++
++
−
2 77777778 10
1
3
24
2
2
.
++






ft
c
3
qDASQ LW
p
DA
=
−
200
07 0159 0917 0 187
0 0166

() ( )
. .
.
© 2007 by Taylor & Francis Group, LLC
44 Modeling Phosphorus in the Environment
field (drainage) boundary to the outlet of the field. This length and difference in
elevation from the most remote point to the outlet are the same as those used in
estimating a time of concentration of a drainage area. As the length-to-width ratio
increases, the peak rate of runoff decreases — that is, runoff is attenuated, and the
runoff peak occurs later in the runoff period.
2.3 MODELING EROSION AND SEDIMENT YIELD
Erosion can be divided into three processes: detachment, transport, and deposition.
Detachment is the dislodging of soil particles (sediment) by raindrop impact or
runoff over the soil surface. Detachment occurs when the erosive forces of raindrop
impact or flowing water exceed the soil’s resistance to detachment or when the
kinetic energy of water exceeds the shear strength (cohesiveness) of the soil. After
particles are detached, they are available for transport. Transport is movement of
sediment from its original location (source area) to downslope locations. Some
transport is caused by splash resulting from the impact of raindrops. The transport
capacity of the flowing water is a function of the kinetic energy of the water and
sediment characteristics — particle size, density, and shape. Raindrops impacting
shallow flow will enhance the transport capacity of runoff. The third process, dep-
osition or sedimentation, refers to the settling of sediment particles out of the flowing
water. The transport capacity of flowing water varies for different particle types.
Deposition occurs when the sediment load of a given particle type exceeds its
corresponding transport capacity. Transport capacity varies as the velocity of the
flowing water varies due to changes in surface conditions, such as vegetative cover,
slope steepness, and, in the case of shallow rain-impacted flow, raindrop size,
velocity, and frequency. Sediment is typically deposited on the toe of slopes and on
flood plains.

Because phosphorus is adsorbed more to finer soil particles than to coarser
particles, the distribution of sediment particle sizes is also important. Because dep-
osition of sands occurs first, followed by large aggregates and then small aggregates,
the distribution of particles in the sediment leaving a field or entering a water body
is usually different than the residual soil in the field. This increase in the proportion
of fine soil particles is referred to as enrichment. One measure is the enrichment
ratio; that is, the ratio of the specific surface area of the sediment to the specific
surface area of the residual soil.
The term sheet erosion describes removal of soil in thin layers from sloping
land. Very little soil might move in this manner because water does not typically
flow as a sheet down a slope, except on slopes with low gradients. Often, a flow
pattern of small well-defined channels, or rills, forms, and erosion occurs in the rills
and the areas between the rills, commonly termed rill and interrill erosion. Rills are
generally 3 to 300 mm in depth. Through interrill erosion, soil is moved by raindrop
splash and overland flow to rills. Flowing water in rills detaches soil particles from
the rills and transports sediment originating in both rills and interrill areas. Rill
erosion is a function of soil critical shear stress; cover conditions; soil slope; and
surface runoff depth, velocity, and energy. Gully erosion is an advanced form of rill
© 2007 by Taylor & Francis Group, LLC
Modeling Runoff and Erosion in Phosphorus Models 45
erosion. Gullies can be described as rills, or channels, that are greater than 300 mm
in depth and that cannot be removed by ordinary tillage operations. Erosion also
occurs in stream channels. Soil can be removed from stream banks (undercut banks)
or the bed. Channel erosion is similar to rill and gully erosion except it occurs in
perennial streams (continuously flowing) as opposed to rill and gully erosion, which
are ephemeral in nature.
Approaches used in phosphorus models to simulate upland erosion and sediment
yield can be classified into two general categories: approaches based on the Universal
Soil Loss Equation (USLE) and process-based approaches. Some of the newer, more
physically based erosion modeling approaches such as Water Erosion Prediction

Project (WEPP) (Laflen et al. 1997) are not described here because such detailed
modeling is generally not included in phosphorus models.
2.3.1 USLE-BASED APPROACHES
The USLE was developed by Wischmeier and Smith (1965, 1978) to predict long-
term average annual erosion (rill and interrill combined) for conservation planning
— that is, for determining appropriate combinations of cropping systems and man-
agement practices for specific fields to meet soil conservation goals. Site-specific
characteristics represented in the equation include soil type, rainfall patterns, topog-
raphy, and cropping and management practices. The USLE does not predict depo-
sition or sediment yield and does not predict gully or channel erosion. The USLE
(Wischmeier and Smith, 1978) is expressed as
A = RKLSCP (2.43)
where A is soil loss per unit area (M/L
2
/T), R is the rainfall and runoff factor (EI
units/T, where T is usually annual), K is the soil erodibility factor (M/L
2
/EI unit
from the unit plot), L is the slope length factor (dimensionless), S is the slope
steepness factor (dimensionless), C is the cover and management factor (dimension-
less), and P is the support practice factor (dimensionless).
The numerical value for R in the soil loss equation must quantify the raindrop
impact effect and must also provide relative information on the amount and rate
of runoff likely to be associated with the rain (Wischmeier and Smith 1978). The
rainfall erosion index derived by Wischmeier (1959) was selected as best in
meeting those requirements. Values for average annual R were presented on a map
of the U.S. (Wischmeier and Smith 1978). Further discussion of the R factor by
Wischmeier and Smith (1978) explained the concept of the EI rainstorm parameter.
EI for a given rainstorm is defined as the product of total storm energy (E) and
the maximum 30-min intensity (I

30
), where E is in hundreds of ft-tons per acre
and I
30
is in in./h. The sum of the storm EI values for a given period is a numerical
measure of the erosive potential of the rainfall within that period. The average
annual total of the storm EI values in a particular locality is the rainfall erosion
index for that locality. Because of apparent cyclical patterns in rainfall data, the
published rainfall erosion index values were generally based on station rainfall
records exceeding 20 years.
© 2007 by Taylor & Francis Group, LLC