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

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 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).

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) and 3.6 is a unit conversion factor.

* The equations in section 2:1.3 use the subbasin area rather than the HRU area. Unlike HRUs, subbasins are geographically contiguous areas. Using the subbasin area makes the equations for time of concentration and peak runoff rate easier to conceptualize. In the model, these calculations are performed at the HRU level. Two modifications are made to adapt the equations to HRUs. First, the area of the subbasin is replaced by the area of the HRU. Second, the channel length term, L, used in the channel flow time of concentration calculation is multiplied by the fraction of the subbasin area with the HRU of interest.

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, 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.