Modeling phosphorus in the environment - Chapter 2 ppsx

For some phosphorus models, peak runoff rate is computed only for use in erosion calculations.. The CN method is usually used when daily rainfall valuesare available; infiltration method

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

2.1 INTRODUCTION

Runoff and erosion are the overland processes that transport phosphorus The processesand equations describing the processes have been described in many references Thepurpose of this chapter is to present the common approaches used for modeling runoffand erosion processes in models that simulate phosphorus transport and to illustratesimilarities and differences in implementation among selected phosphorus models.Implementation of the processes varies among the phosphorus models, depending onmodel 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., eventprediction or average annual predictions for management), computational time step(e.g., daily vs shorter time steps during rainfall or runoff events), and land uses andconditions represented (e.g., agricultural, urban, forested land uses, frozen soils) Examples of implementation from the following models are included in thechapter: Annualized Agricultural Nonpoint Source (AnnAGNPS) (Cronshey andTheurer 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 AgriculturalManagement Systems (GLEAMS) (Knisel 1993; Leonard et al 1987), HydrologicSimulation Program-Fortran (HSPF) (Bicknell et al 2001), and Soil and WaterAssessment 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 thevariety 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) andEnglish 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 soilcharacteristics, land cover, and topography Runoff calculations typically includeestimating 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 erosioncomponent for calculating the characteristic discharge rate, sediment transport capa-city, and shear stress in a concentrated flow Common approaches used in phosphorusmodels 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 minusinfiltration and interception Two general approaches are used to model runoffvolume in phosphorus transport models: (1) the curve number (CN) method and (2)infiltration methods The CN method directly calculates runoff volume, whereas theinfiltration methods calculate infiltration first and then estimate runoff as the differ-ence between rainfall and infiltration Some phosphorus models include both CN

and infiltration methods The CN method is usually used when daily rainfall valuesare available; infiltration methods require hourly — or other intervals shorter thandaily — rainfall values.

2.2.1.1 Curve Number Method

The most common method used to estimate runoff volume in phosphorus models isthe U.S Department of Agriculture (USDA) Soil Conservation Service (SCS) (nowNatural Resources Conservation Service, NRCS) runoff approach The CN methodcorrelates runoff with rainfall, antecedent moisture condition (AMC), soil type, andvegetative 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 curvenumber, 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, Ia) for the equation to be applicable Values of

CN have been tabulated (Table 2.1) by hydrologic soil group for AMC II, or averageconditions The CN ranges from 1 to 100, with runoff potential increasing withincreasing CN Required information to determine a CN value from the table includesthe 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 2.1for 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 notincluded in Table 2.1 or similar tables Examples include exposed fractured rocksurfaces (Rasmussen and Evans 1993), animal manure application sites (Edwardsand Daniel 1993), and dryland wheat–sorghum–fallow crop rotation in the semi-aridwestern 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 valuescan result from misclassifying land cover, treatment, hydrologic conditions, or soiltype (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 thateffects of variations in CN decrease as design rainfall depth increases and confirmedHawkins’s (1975) conclusion that errors in CN estimates are especially criticalnear the threshold of runoff

TABLE 2.1

Runoff Curve Numbers for Hydrologic Soil-Cover Complexes

Land Use Description/Treatment/Hydrologic Condition Hydrologic Soil Group

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

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)

2.2.1.2 Curve Number Method Implementation

The curve number method is used in several phosphorus models to compute runoffvolume The most common implementation (e.g., AnnAGNPS, GLEAMS, EPIC,SWAT) includes a modification of the CN to account for daily changes in soilmoisture content (Williams et al 1990) Typically, the models require the user toinput a value for CN2, 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

meadow Contoured and terraced Poor 63 73 80 83

Contoured and terraced Good 51 67 76 80

Good 39 61 74 80 Contoured Poor 47 67 81 88 Contoured Fair 25 59 75 83 Contoured Good 6 35 70 79

Note: Antecedent moisture condition II and Ia= 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.

curve numbers corresponding to AMC I (dry), CN1, and AMC III (wet), CN3, arecomputed as a function of CN2 The retention parameter, S, also changes due to

fluctuations in soil moisture content For example, the same relationship is used inEPIC and SWAT, with the soil water content expressed differently:

(2.3)

where S1 (L) and Smax (L) is the value of S associated with CN1 (computed with

Equation 2.2), FFC is the fraction of field capacity, SW is the soil water content

(L3/L3), and w1 and w2 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

Source: SCS 1972 Hydrology, Section 4: National Engineering Handbook, U.S Soil Conservation

Service, Washington, D.C., Government Printing Office With permission.

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) (L3/L3), and FC

is the field capacity water content (corresponds to 33 kPa matric potential for manysoils) (L3/L3) In EPIC, values for w1 and w2 are obtained by simultaneous solution

of Equation 2.3 with the assumptions that S = S2 when FFC = 0.5 and S = S3 when

FFC = 1.0 In SWAT, w1 and w2 are determined by solving Equation 2.3 with the

following assumptions: S = S1 when SW = WP, S = S3 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 theroot zone or top meter or some other depth of soil, or a nonuniform distribution ofsoil 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 modelskeep track of soil moisture by layer, so they have the potential to include the soilwater distribution in their runoff calculations For example, because EPIC estimateswater content of each soil layer daily, the effect of depth distribution on runoff isexpressed 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

Note:AMC II to AMC I and III (Ia= 0.2S).

Source: SCS 1972 Hydrology, Section 4: National ing Handbook, U.S Soil Conservation Service, Washington,

Engineer-D.C., Government Printing Office With permission.

FFC

FFC

Z

i M i

Z Z Z i

M Z Z Z

i

i i i

i i i

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 FFCi is divided by Zi 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 Wi is the weighting factor, SMi 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 CN2 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 CN2s is the handbook CN2 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 ofthe 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 soilgroup is assigned to each series However, a series name can include different soiltextures, which would have different runoff potentials but would still be in the samehydrologic soil group The developers of GLEAMS expanded Table 2.1 to give arange of curve numbers for each combination in the table to allow users to distinguishbetween similar soils within a series (Table 2.4) For example, CN2 for row cropswith straight rows in good hydrologic condition could be 78 for a Cecil sandy loamand 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

UL

i i i i

D RD

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 toincreased 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 — tomatch measured runoff values — decreased with increasing drainage area for 18semi-arid watersheds in southeastern Arizona Some phosphorus models divide largewatersheds into subwatersheds or other smaller hydrologic response units It isreasonable to apply the CN to the smaller response units and then to determine howthe runoff from the individual units contributes to streamflow, through routing orother 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 timeand asymptotically approaches a final infiltration rate The final infiltration rate is

approximately equal to the saturated hydraulic conductivity, Ks, of the soil The

amount and rate of infiltration depend on infiltration capacity of the soil and theavailability 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 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 equalthe infiltration capacity If the rainfall rate is less than the saturated hydraulic

infil-TABLE 2.4

Excerpt of Expanded Curve Number Table for GLEAMS Model

Land Use Treatment

or Practice

Hydrologic Condition

Hydrologic Soil Group

Note: SR = straight row; CT = conservation tillage; CNT = contoured.

Source: Knisel, W.G., GLEAMS: Groundwater Loading Effects of Agricultural Management tems, Version 2.10, University of Georgia, Coastal Plain Experiment Station, Biological and Agri-

Sys-cultural Engineering Department, Publication 5, p 130, 1993 With permission.

conductivity of the soil, the infiltration rate will equal the rainfall rate, and pondingwill not occur.

A number of infiltration equations have been developed, ranging from solving theRichards (1931) equation to empirical equations The Richards equation, the generalizedequation 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 Richardsequation is solved subject to appropriate boundary and initial conditions Empiricalinfiltration equations typically include coefficients or exponents to represent soil prop-erties and to generate the relationship of decreasing infiltration rate with time Someinfiltration 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 inHSPF; 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 ically based, and its parameters can be determined from readily available soil andvegetal cover information The approach has been tested for a variety of conditionsand has successfully simulated the effects of different management practices oninfiltration

phys-The original Green and Ampt (1911) equation was derived using Darcy’s lawfor infiltration from a ponded surface into a deep, homogeneous soil profile withuniform 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), Ks is saturated hydraulic conductivity (L/T), M is

the difference between final and initial moisture content (the difference in moisturecontent across the wetting front) (L3/L3), Sav 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

p av

s

=

−1

where Fp 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 pondingoccurs, the infiltration rate is a function of the infiltration capacity of the soil TheGreen-Ampt-Mein-Larson (GAML) model for infiltration rate is a two-stage model

with f = R for t ≤ tp , 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 anequation for cumulative infiltration over time:

(2.12)

where t′p is the equivalent time (T) to infiltrate Fp 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 Ks, M, and Sav Skaggs and Khaleel (1982) reported that

Bouwer (1966) and Bouwer and Asce (1969) showed that the hydraulic conductivityparameter should be less than the saturated value because of entrapped air Whenmeasured values are not available, Bouwer (1966) suggested that an effective hydrau-

lic conductivity of 0.5 Ks be used in place of Ks Other adjustments to Ks have also

been recommended to account for the impact of surface conditions on infiltration,

resulting in the concept of an effective hydraulic conductivity, Ke 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, sented as effective suction at the wetting front by Green and Ampt and then asaverage suction at the wetting front by Mein and Larson One accepted estimationmethod (Mein and Larson 1973) is using the unsaturated hydraulic conductivity as

pre-a weighting fpre-actor pre-and defining Spre-av pre-as

(2.13a)

where ψ is the soil water suction (negative of the matric potential) (L) and Kr is the

relative hydraulic conductivity = K(ψ)/Ks Neuman (1976) obtained a theoretical expression for Sav, similar to Equation 2.13a, by relating it to the physical charac-

teristics of the soil:

where ψi is the suction at the initial water content Morel-Seytoux and Khanji (1974) found that for most cases the value of Sav given by Equation 2.13a or 2.13b is a

reasonable approximation for an effective matric drive, which is dependent on therelative 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 Sav, and Ke 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 InANSWERS-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 Ks with Ke, which is computed as the weighted sum of Ke under canopy cover and Ke for the area outside the canopy The values are computed as functions of soil parameters, (i.e., Ks, 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 Sav is computed using an empirical equation developed by Rawls and Brakensiek (1985), in which Sav 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 eachtime step, SWAT calculates the amount of water entering the soil The water thatdoes not infiltrate into the soil becomes surface runoff SWAT uses effective hydraulic

conductivity, Ke, in place of Ks; Ke is computed as a function of Ks 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 Sav.

2.2.2 H YDROGRAPH D EVELOPMENT

A hydrograph is the relationship between flow rate and time In some phosphorusmodels, only the runoff hydrograph as it enters a receiving stream is computed Inothers, 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 therunoff 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

1

Runoff, or overland flow, can be visualized as sheet-type flow — as opposed tochannel 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 isspatially varied and usually unsteady and nonuniform — that is, the velocity andflow depth vary in both time and space Input (rainfall) to the flow is distributedover the flow surface Methods for determining hydrographs range from directcalculations of the basic hydrograph descriptors — peak flow, time to peak, andbase time — to overland flow routing, which yields the relationship of runoff ratewith space and time Examples of the methods used in some phosphorus models aredescribed in the following sections

2.2.2.1 Kinematic Flow Routing

The theoretical hydrodynamic equations attributed to St Venant are based on thefundamental laws of conservation of mass (continuity) and conservation ofmomentum applied to a control volume or fixed section of channel with theassumptions 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 partialdifferential equations Because these equations are not typically implemented inphosphorus models, they are not given here Detailed derivations of continuity andmomentum equations as they apply to unsteady, nonuniform flow can be found inStrelkoff (1969)

Lighthill and Whitham (1955), cited by Huggins and Burney (1982), proposedthat the dynamic terms in the momentum equation had negligible influence for cases

in which backwater effects were absent Neglecting these terms yields a quasi-steadyapproach, 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 (L3/T/L), I is lateral

inflow per unit length and width of the flow plane (L3/T/L2), f is lateral outflow per

unit length and width of the flow plane (L3/T/L2), 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, withthe overland flow plane represented by a wide channel Overton (1972) analyzed

200 hydrographs for relatively long, impermeable planes and found that flow wasturbulent or transitional Foster et al (1968) concluded that both Manning’s andDarcy-Weisbach flow equations were satisfactory for describing overland flow onshort erodible slopes

∂

y t

q

x I f

q=αy m

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 (m3/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 theeffect of neglecting dynamic terms in the momentum equation:

(2.18)

where k is a dimensionless parameter, So 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 equationsgenerally provide an adequate representation of the overland flow hydrograph (Hugginsand Burney 1982)

where L is the maximum length of flow (m), CN is the runoff curve number, and Sg

is the average watershed gradient (m/m) The peak discharge is a function of amount

of runoff computed with Equation 2.1:

where qp is peak runoff rate (m3/s), qu is unit peak flow rate (m3/s per ha/mm of

runoff), Q is the runoff volume (mm), and A is watershed area (ha) Unit peak flow rate, qu, 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 Ia/P, with Ia 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 routeoverland flow For example, ANSWERS-2000 uses kinematic routing, HSPFincludes a flow equation with an empirical relationship, SWAT computes a runofflag time for delivery to channels, and AnnAGNPS applies the SCS unit hydrograph.ANSWERS-2000 applies the kinematic wave equations to route overland flowwith an explicit, backward difference solution of the continuity equation combinedwith 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-2000represents a watershed as a grid of cells Detention depth is calculated as the totalvolume of surface water in a cell minus the retention volume divided by the area ofthe 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 thewater 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 simulatedusing the Chezy-Manning equation and an empirical expression that relates outflowdepth to detention storage The rate of overland flow discharge is determined by theequations

, for S d < S e (2.23a)

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.), Se is the equilibrium surface detention storage (in.) for current

log( )q u =C0+C1logt c+C2(log )t c 2

S

r d

d e

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 theoverland 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 tc is less than one day, SWAT adds the runoff generated on that day to the channel on that day If tc is greater than one day,

only a portion of the surface runoff will reach the main channel on the day it isgenerated SWAT incorporates a surface runoff storage feature to lag a portion ofthe surface runoff release to the main channel After surface runoff is calculated, theamount 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 representsthe 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 inthe reach

2.2.3 S TREAMFLOW , OR C HANNEL , R OUTING

Many phosphorus models include channel processes in addition to upland processes.Routing streamflow entails computing the effect of channel storage on the shapeand movement of a hydrograph, or flood wave When a flood wave advances into areach segment, inflow exceeds outflow and a wedge of storage is produced As theflood wave recedes, outflow exceeds inflow in the reach segment, and a negativewedge is produced In addition to the wedge storage, the reach segment contains aprism of storage formed by a volume of constant cross-section along the reach length Streamflow routing involves relationships among inflow, outflow, and storage ineach reach or segment of the stream The continuity equation for unsteady flowrelating inflow, outflow, and storage in a reach is

(2.25)

where I is inflow (L3/T), O is outflow (L3/T), S is storage (L3), and t is time (T).

Flow routing procedures typically assume that the average of flows at the beginning

Q surf Q surf Q stor i e

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 ensurethat 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 twounknowns: 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 secondrelationship 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 thecontinuity equation (Equation 2.26) (Haan et al 1994) The storage in a channelreach depends on the channel geometry and depth of flow The flow rate can berelated to the depth or cross-sectional area of flow, assuming steady, uniform flowusing Manning’s equation for each cross-section:

(2.27)

where q is the flow rate (m3/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 (m2) Manning’s equationwas 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 theaverage 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 groundwater 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 followingdescription 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,

the cross-sectional area of flow is assumed to be directly proportional to the dischargefor a given reach segment The Muskingum flow routing method is based on thecontinuity equation (Equation 2.25) The second relationship between storage andoutflow 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

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 throughthe 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

O2= c0I2+ c1I1+ c2O1 (2.30)where

∆

∆

The routing period ∆t is in the same units as K With K, X, and ∆t established,

the values of c0, c1, and c2 can be computed The routing operation then consists of

solving Equation 2.30 with the O2 of one routing period becoming the O1 of thesucceeding period To maintain numerical stability and to avoid the computation ofnegative 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 byWilliams (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 Muskingummethod The variable storage-routing method is based on the continuity equation Traveltime 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 T T 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, coef1 and coef2 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 constantcalculated 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 ship is based on the assumption that outflows are functions of volume or time, or acombination of the two If the outflow is a function of volume, the user defines thedepth-surface area-volume-discharge relationship in an input table (an FTABLE inHSPF terms) This is one of four optional methods for computing overland flow.Another method is a simple power function method

relation-Some phosphorus models also include transmission, or leaching, losses fromchannels during periods when a stream receives no groundwater contributions

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 hydraulicconductivity of the channel alluvium, flow travel time, wetted perimeter, and channellength Both volume and peak rate are adjusted when transmission losses occur intributary channels

2.2.4 P EAK R ATE OF R UNOFF

2.2.4.1 Rational Formula

The rational method is the most common method used for peak flow estimation inpractice The method is presented in many references The following description isbased 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 asthe travel time from the most hydraulically remote point in the watershed to thewatershed outlet and is typically computed as a function of the length of flow andthe slope of the watershed

The basic concept of the rational equation is as follows If a steady rainfalloccurs 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 willnot 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 relationshipbetween 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 averagedover 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 inthe 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, surfacedetention, and antecedent moisture conditions — on the peak runoff Various tables

q p=CiA

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 singlevalues for combinations of factors

Haan et al (1994) note that in spite of the recognized shortcomings of the rationalmethod, it continues to be widely used because of its simplicity, entrenchment inpractice, 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 fromrural and urban areas The Graphical method was developed from hydrograph anal-yses using TR-20, “Computer Program for Project Formulation — Hydrology” (SCS1983)

In the Graphical method, the peak discharge equation is computed as

where q p is the peak discharge (cfs), q u is the unit peak discharge (cfs/mi2/in runoff),

A is the drainage area (mi2), 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 swampareas 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 somemodels, peak discharge can be determined from overland flow routing In othermodels, peak runoff rate is computed for use in erosion calculations but not inhydrologic calculations EPIC and SWAT compute peak runoff rate using a modifiedrational formula, whereas AnnAGNPS computes peak discharge similar to the TR-55method, 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:

where q p is peak runoff (m3/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), αtc is the fraction of daily rainfall

p t c

c

=α

3 6

that occurs during tc, Area is drainage area (km2), 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) fordifferent durations and frequencies showed that the amount of rain falling during

the time of concentration, Rtc, was proportional to the amount of rain falling during

the 24-h period, or where αtc is the fraction of daily rainfall that occurs

during the time of concentration For short-duration storms, all or most of the rainwill fall during the time of concentration, causing αtc to approach its upper limit of

1.0 The minimum value of αtc would be seen in storms of uniform intensity (i24=

i) Thus, αtc falls in the range t c/24 ≤αtc≤ 1.0

When the value of αtc is assigned based on only daily rainfall and simulatedrunoff, there is considerable uncertainty Thus, in EPIC α is generated from a gamma

function with the base ranging from tc/24 to 1.0.

SWAT estimates αtc as a function of the fraction of daily rain falling in the half

hour of highest intensity rainfall:

(2.37)

where α0.5 is the fraction of daily rain falling in the half hour of highest-intensityrainfall If subdaily precipitation data are used in the model, SWAT calculates themaximum half hour rainfall fraction directly from the precipitation data If dailyprecipitation data are used, SWAT generates a value for α0.5 from a triangulardistribution, 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 numberbetween 0.0 and 1.0

The time of concentration is computed in EPIC and SWAT as the sum of surfaceand channel flow times The time of concentration for channel flow is computed asthe average channel flow length for the watershed divided by the average channelvelocity Average channel flow length is estimated as