The SCS runoff equation is an empirical model that came into common use in the 1950s. It was the product of more than 20 years of studies involving rainfall-runoff relationships from small rural watersheds across the U.S. The model was developed to provide a consistent basis for estimating the amounts of runoff under varying land use and soil types (Rallison and Miller, 1981).

The SCS curve number equation is (SCS, 1972):

$Q_{surf}=\frac{(R_{day}-I_a)^2}{(R_{day}-I_a+S)}$ 2:1.1.1

where $Q_{surf}$ is the accumulated runoff or rainfall excess (mm H$_2$O), $R_{day}$ is the rainfall depth for the day (mm H$_2$O), $I_a$ is the initial abstractions which includes surface storage, interception and infiltration prior to runoff (mm H$_2$O), and $S$ is the retention parameter (mm H$_2$O). The retention parameter varies spatially due to changes in soils, land use, management and slope and temporally due to changes in soil water content. The retention parameter is defined as:

2:1.1.2

where is the curve number for the day. The initial abstractions, , is commonly approximated as and equation 2:1.1.1 becomes

2:1.1.3

Runoff will only occur when . A graphical solution of equation 2:1.1.3 for different curve number values is presented in Figure 2:1-1.

The time of concentration is the amount of time from the beginning of a rainfall event until the entire subbasin area is contributing to flow at the outlet. In other words, the time of concentration is the time for a drop of water to flow from the remotest point in the subbasin to the subbasin outlet. The time of concentration is calculated by summing the overland flow time (the time it takes for flow from the remotest point in the subbasin to reach the channel) and the channel flow time (the time it takes for flow in the upstream channels to reach the outlet):

$t_{conc}=t_{ov}+t_{ch}$ 2:1.3.2

where $t_{conc}$ is the time of concentration for a subbasin (hr), $t_{ov}$ is the time of concentration for overland flow (hr), and $t_{ch}$ is the time of concentration for channel flow (hr).

SCS defines three antecedent moisture conditions:

I—dry (wilting point), II—average moisture, and III—wet (field capacity). The moisture condition I curve number is the lowest value the daily curve number can assume in dry conditions. The curve numbers for moisture conditions I and III are calculated with the equations:

$CN_1=CN_2-\frac{20*(100-CN_2)}{(100-CN_2+exp[2.533-0.0636*(100-CN_2)])}$ 2:1.1.4

$CN_3=CN_2*exp[0.00673*(100-CN_2)]$ 2:1.1.5

where $CN_1$ is the moisture condition I curve number, $CN_2$ is the moisture condition II curve number, and $CN_3$ is the moisture condition III curve number.

The peak runoff rate is the maximum runoff flow rate that occurs with a given rainfall event. The peak runoff rate is an indicator of the erosive power of a storm and is used to predict sediment loss. SWAT+ calculates the peak runoff rate with a modified rational method.

The rational method is widely used in the design of ditches, channels and storm water control systems. The rational method is based on the assumption that if a rainfall of intensity $i$ begins at time $t=0$ and continues indefinitely, the rate of runoff will increase until the time of concentration, $t=t_{conc}$, when the entire subbasin area* is contributing to flow at the outlet. The rational formula is:

$q_{peak}=\frac{C*i*Area}{3.6}$ 2:1.3.1

where $q_{peak}$ is the peak runoff rate ($m^3 s^{-1}$), $C$ is the runoff coefficient, $i$ is the rainfall intensity (mm/hr), Area is the subbasin area (km$^2$) and 3.6 is a unit conversion factor.

The runoff coefficient is the ratio of the inflow rate, $i*Area$, to the peak discharge rate, $q_{peak}$. The coefficient will vary from storm to storm and is calculated with the equation:

$C=\frac{Q_{surf}}{R_{day}}$ 2:1.3.15

where $Q_{surf}$ is the surface runoff (mm H$_2$O) and $R_{day}$ is the rainfall for the day (mm H$_2$O).

The overland flow time of concentration, tov, can be computed using the equation

$t_{ov}=\frac{L_{slp}}{3600*v_{ov}}$ 2:1.3.3

where $L_{slp}$ is the subbasin slope length (m), $v_{ov}$ is the overland flow velocity (m s$^{-1}$) and 3600 is a unit conversion factor.

The overland flow velocity can be estimated from Manning’s equation by considering a strip 1 meter wide down the sloping surface:

$v_{ov}=\frac{q_{ov}^{0.4}*slp^{0.3}}{n^{0.6}}$ 2:1.3.4

where is the average overland flow rate (), is the average slope in the subbasin (m ), and is Manning’s roughness coefficient for the subbasin. Assuming an average flow rate of 6.35 mm/hr and converting units

2:1.3.5

Substituting equation 2:1.3.5 into equation 2:1.3.3 gives

2:1.3.6

With SWAT+, users are allowed to select between two methods for calculating the retention parameter. The traditional method is to allow the retention parameter to vary with soil profile water content. An alternative added in SWAT+ allows the retention parameter to vary with accumulated plant evapotranspiration. Calculation of the daily CN value as a function of plant evapotranspiration was added because the soil moisture method was predicting too much runoff in shallow soils. By calculating daily CN as a function of plant evapotranspiration, the value is less dependent on soil storage and more dependent on antecedent climate.

When the retention parameter varies with soil profile water content, the following equation is used:

$S=S_{max}*(1-\frac{SW}{[SW+exp(w_1-w_2*SW)]})$ 2:1.1.6

where $S$ is the retention parameter for a given day (mm), $S_{max}$ is the maximum value the retention parameter can achieve on any given day (mm), $SW$ is the soil water content of the entire profile excluding the amount of water held in the profile at wilting point (mm H$_2$O), and $w_1$ and $w_2$ are shape coefficients. The maximum retention parameter value, , is calculated by solving equation 2:1.1.2 using .

The shape coefficients are determined by solving equation 2:1.1.6 assuming that

1. the retention parameter for moisture condition I curve number corresponds to wilting point soil profile water content,

2. the retention parameter for moisture condition III curve number corresponds to field capacity soil profile water content, and

3. the soil has a curve number of 99 (S = 2.54) when completely saturated.

2.1.1.7

2.1.1.8

where is the first shape coefficient, is the second shape coefficient, is the amount of water in the soil profile at field capacity (mm HO), is the retention parameter for the moisture condition III curve number, is the retention parameter for the moisture condition I curve number, is the amount of water in the soil profile when completely saturated (mm HO), and 2.54 is the retention parameter value for a curve number of 99.

When the retention parameter varies with plant evapotranspiration, the following equation is used to update the retention parameter at the end of every day:

2:1.1.9

where is the retention parameter for a given day (mm), is the retention parameter for the previous day (mm), is the potential evapotranspiration for the day (mm d), is the weighting coefficient used to calculate the retention coefficient for daily curve number calculations dependent on plant evapotranspiration, is the maximum value the retention parameter can achieve on any given day (mm), Rday is the rainfall depth for the day (mm HO), and is the surface runoff (mm HO). The initial value of the retention parameter is defined as

When the top layer of the soil is frozen, the retention parameter is modified using the following equation:

2:1.1.10

where is the retention parameter adjusted for frozen conditions (mm), is the maximum value the retention parameter can achieve on any given day (mm), and is the retention parameter for a given moisture content calculated with equation 2:1.1.6 (mm).

The daily curve number value adjusted for moisture content is calculated by rearranging equation 2:1.1.2 and inserting the retention parameter calculated for that moisture content:

2:1.1.11

where is the curve number on a given day and is the retention parameter calculated for the moisture content of the soil on that day.

The moisture condition II curve numbers provided in the tables are assumed to be appropriate for 5% slopes. Williams (1995) developed an equation to adjust the curve number to a different slope:

$CN_{2s}=\frac{(CN_3-CN_2)}{3}*[1-2*exp(-13.86*slp)]+CN_2$ 2:1.1.12

where $CN_{2s}$ is the moisture condition II curve number adjusted for slope, $CN_3$ is the moisture condition III curve number for the default 5% slope, $CN_2$ is the moisture condition II curve number for the default 5% slope, and $slp$ is the average fraction slope of the subbasin. SWAT+ does not adjust curve numbers for slope. If the user wishes to adjust the curve numbers for slope effects, the adjustment must be done prior to entering the curve numbers in the management input file.

Table 2:1-1: SWAT+ input variables that pertain to surface runoff calculated with the SCS curve number method.

In large subbasins with a time of concentration greater than 1 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.

Once surface runoff is calculated with the Curve Number or Green & Ampt method, the amount of surface runoff released to the main channel is calculated:

$Q_{surf}=(Q'_{surf}+Q_{stor,i-1})*(1-exp[\frac{-surlag}{t_{conc}}])$ 2:1.4.1

where $Q_{surf}$ is the amount of surface runoff discharged to the main channel on a given day (mm H$_2$O), is the amount of surface runoff generated in the subbasin on a given day (mm H$_2$O), $Q_{stor,i-1}$ is the surface runoff stored or lagged from the previous day (mm H$_2$O), $surlag$ is the surface runoff lag coefficient, and $t_{conc}$ is the time of concentration for the subbasin (hrs).

The expression in equation 2:1.4.1 represents the fraction of the total available water that will be allowed to enter the reach on any one day. Figure 2:1-3 plots values for this expression at different values for and .

Figure 2:1-3: Influence of and on fraction of surface runoff released.

Note that for a given time of concentration, as decreases in value more water is held in storage. The delay in release of surface runoff will smooth the streamflow hydrograph simulated in the reach.

Table 2:1-6: SWAT+ input variables that pertain to surface runoff lag calculations.

The rainfall intensity is the average rainfall rate during the time of concentration. Based on this definition, it can be calculated with the equation:

$i=\frac{R_{tc}}{t_{conc}}$ 2:1.3.16

where $i$ is the rainfall intensity (mm/hr), $R_{tc}$ is the amount of rain falling during the time of concentration (mm H$_2$O), and $t_{conc}$ is the time of concentration for the subbasin (hr).

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 was proportional to the amount of rain falling during the 24-hr period.

$R_{tc}=\alpha_{tc}*R_{day}$ 2:1.3.17

where is the amount of rain falling during the time of concentration (mm HO), is the fraction of daily rainfall that occurs during the time of concentration, and is the amount of rain falling during the day (mm HO).

For short duration storms, all or most of the rain will fall during the time of concentration, causing to approach its upper limit of 1.0. The minimum value of would be seen in storms of uniform intensity (). This minimum value can be defined by substituting the products of time and rainfall intensity into equation 2:1.3.17

2:1.3.18

Thus, falls in the range

SWAT+ estimates the fraction of rain falling in the time of concentration as a function of the fraction of daily rain falling in the half-hour of highest intensity rainfall.

2:1.3.19

where is the fraction of daily rain falling in the half-hour highest intensity rainfall, and is the time of concentration for the subbasin (hr). The determination of a value for is discussed in Chapters 1:2 and 1:3.

The modified rational formula used to estimate peak flow rate is obtained by substituting equations 2:1.3.15, 2:1.3.16, and 2:1.3.17 into equation 2:1.3.1

$q_{peak}=\frac{\alpha_{tc}*Q_{surf}*Area}{3.6*t_{conc}}$ 2:1.3.20

where $q_{peak}$ is the peak runoff rate ($m^3 s^{-1}$), $\alpha_{tc}$ is the fraction of daily rainfall that occurs during the time of concentration, $Q_{surf}$ is the surface runoff (mm H$_2$O), $Area$ is the subbasin area (km$^2$), $t_{conc}$ is the time of concentration for the subbasin (hr) and 3.6 is a unit conversion factor.

Table 2:1-5: SWAT+ input variables that pertain to peak rate calculations.

The channel flow time of concentration, $t_{ch}$, can be computed using the equation

$t_{ch}=\frac{L_c}{3.6*v_c}$ 2:1.3.7

where $L_c$ is the average flow channel length for the subbasin (km), $v_c$ is the average channel velocity (m s$^{-1}$), and 3.6 is a unit conversion factor.

The average channel flow length can be estimated using the equation

$L_c=\sqrt{L*L_{cen}}$ 2:1.3.8

where is the channel length from the most distant point to the subbasin outlet (km), and is the distance along the channel to the subbasin centroid (km). Assuming , the average channel flow length is

2:1.3.9

The average velocity can be estimated from Manning’s equation assuming a trapezoidal channel with 2:1 side slopes and a 10:1 bottom width-depth ratio.

2:1.3.10

where is the average channel velocity (m s), is the average channel flow rate (), is the channel slope (m m), and is Manning’s roughness coefficient for the channel. To express the average channel flow rate in units of mm/hr, the following expression is used

2.1.3.11

where is the average channel flow rate (mm hr), is the subbasin area (km), and 3.6 is a unit conversion factor. The average channel flow rate is related to the unit source area flow rate (unit source area = 1 ha)

2:1.3.12

where is the unit source area flow rate (mm hr), is the subbasin area (km), and 100 is a unit conversion factor. Assuming the unit source area flow rate is 6.35 mm/hr and substituting equations 2:1.3.11 and 2:1.3.12 into 2:1.3.10 gives

2:1.3.13

Substituting equations 2:1.3.9 and 2:1.3.13 into 2:1.3.7 gives

2:1.3.14

where is the time of concentration for channel flow (hr), is the channel length from the most distant point to the subbasin outlet (km), n is Manning’s roughness coefficient for the channel, Area is the subbasin area (km), and is the channel slope (m m).

Although some of the assumptions used in developing equations 2:1.3.6 and 2:1.3.14 may appear liberal, the time of concentration values obtained generally give satisfactory results for homogeneous subbasins. Since equations 2:1.3.6 and 2:1.3.14 are based on hydraulic considerations, they are more reliable than purely empirical equations.

Many semiarid and arid watersheds have ephemeral channels that abstract large quantities of streamflow (Lane, 1982). The abstractions, or transmission losses, reduce runoff volume as the flood wave travels downstream. Chapter 19 of the SCS Hydrology Handbook (Lane, 1983) describes a procedure for estimating transmission losses for ephemeral streams which has been incorporated into SWAT+. This method was developed to estimate transmission losses in the absence of observed inflow-outflow data and assumes no lateral inflow or out-of-bank flow contributions to runoff.

The prediction equation for runoff volume after transmission losses is

2:1.5.1

where is the volume of runoff after transmission losses (), is the regression intercept for a channel of length and width (), is the regression slope for a channel of length and width

Surface runoff occurs whenever the rate of water application to the ground surface exceeds the rate of infiltration. When water is initially applied to a dry soil, the infiltration rate is usually very high. However, it will decrease as the soil becomes wetter. When the application rate is higher than the infiltration rate, surface depressions begin to fill. If the application rate continues to be higher than the infiltration rate once all surface depressions have filled, surface runoff will commence.

SWAT+ provides two methods for estimating surface runoff: the SCS curve number procedure (SCS, 1972) and the Green & Ampt infiltration method (1911).

The Green & Ampt equation was developed to predict infiltration assuming excess water at the surface at all times (Green & Ampt, 1911). The equation assumes that the soil profile is homogenous and antecedent moisture is uniformly distributed in the profile. As water infiltrates into the soil, the model assumes the soil above the wetting front is completely saturated and there is a sharp break in moisture content at the wetting front. Figure 2:1-2 graphically illustrates the difference between the moisture distribution with depth modeled by the Green & Ampt equation and what occurs in reality.

Mein and Larson (1973) developed a methodology for determining ponding time with infiltration using the Green & Ampt equation. The Green-Ampt Mein-Larson excess rainfall method was incorporated into SWAT+ to provide an alternative option for determining surface runoff. This method requires sub-daily precipitation data supplied by the user.

The Green-Ampt Mein-Larson infiltration rate is defined as:

2:1.2.1

The U.S. Natural Resource Conservation Service (NRCS) classifies soils into four hydrologic groups based on infiltration characteristics of the soils. NRCS Soil Survey Staff (1996) defines a hydrologic group as a group of soils having similar runoff potential under similar storm and cover conditions. Soil properties that influence runoff potential are those that impact the minimum rate of infiltration for a bare soil after prolonged wetting and when not frozen.

These properties are depth to seasonally high water table, saturated hydraulic conductivity, and depth to a very slowly permeable layer.

Soil may be placed in one of four groups, A, B, C, and D, or three dual classes, A/D, B/D, and C/D.

Definitions of the classes are:

A: (Low runoff potential). The soils have a high infiltration rate even when thoroughly wetted. They chiefly consist of deep, well drained to excessively drained sands or gravels. They have a high rate of water transmission.

The SCS curve number is a function of the soil’s permeability, land use and antecedent soil water conditions. Typical curve numbers for moisture condition II are listed in tables 2:1-1, 2:1-2 and 2:1-3 for various land covers and soil types (SCS Engineering Division, 1986). These values are appropriate for a 5% slope.

Table 2:1-1: Runoff curve numbers for cultivated agricultural lands (from SCS Engineering Division, 1986)

[1] Poor: < 50% ground cover or heavily grazed with no mulch; Fair: 50 to 75% ground cover and not heavily grazed;