Garden with Insight v1.0 Help: Hydrology - Surface Runoff - Peak Runoff Rate

There are two options for estimating the peak runoff rate -- the modified Rational formula and the SCS TR-55 method (USDA-SCS, 1986). A stochastic element is included in the Rational equation to allow realistic simulation of peak runoff rates, given only daily rainfall and monthly rainfall intensity.

Rational Equation Method

The rational equation can be written in the form

q(rho) = rho * r * A / 360 (Equation 13)

where q(rho) is the peak runoff rate in meters cubed per second, rho is a runoff coefficient expressing the watershed infiltration characteristics, r is the rainfall intensity in mm per hour for the watershed's time of concentration, and A is the drainage area in ha.

The runoff coefficient can be calculated for each storm if the amount of rainfall and runoff are known in this equation

tho = Q / R (Equation 14)

Since R is input and Q is computed with [Equation 1], rho can be calculated directly.

Rainfall intensity r can be expressed with the relationship

r = R(tc) / t(c) (Equation 15)

where R(tc) is the amount of rainfall in mm during the watershed's time of concentration, t(c) in h.

The value of R(tc) can be estimated by developing a relationship with total R. The Weather Service's TP- 40 (Hershfield 1961) provides accumulated rainfall amounts for various durations and frequencies. Generally, R(tc) and R(24) (24-h duration is appropriate for the daily time step model) are proportional for various frequencies. Thus,

R(tc) = alpha * R(24) (Equation 16)

where alpha is a dimensionless parameter that expresses the proportion of total rainfall the occurs during t(c).

The peak runoff equation [Equation 17] is obtained by substituting equations 14, 15 and 16 into equation 13.

Equation 17:

q(p) = alpha * Q * A / (360 * t(c))
Code:
q(p) = alpha * Q / t(c)
Variables:
q(p) = PeakRunoffRateByRationalEquation_m3Psec
alpha = propRainInTimeOfConc_frn
Q = runoffVolume_mm
A = watershedArea_ha
t(c) = timeOfConc_hr

The time of concentration can be estimated by adding the surface and channel flow times [Equation 18] where t(cc) is the time of concentration for channel flow and t(cs) is the time of concentration for surface flow in h.

Equation 18:

t(c) = t(cc) + t(cs)
Code:
same
Variables:
t(c) = TimeOfConc_hr
t(cc) = timeOfConcForChannelFlow_hr
t(cs) = timeOfConcForSurfaceFlow_hr

The t(cc) can be computed using the equation

t(cc) = L(c)/v(c) (Equation 19)

where L(c) is the average channel flow length for the watershed in km and v(c) is the average channel velocity in meters per second.

The average channel flow length can be estimated by using the equation

L(c) = sqrt(L * L(ca)) (Equation 20)

where L is the channel length from the most distant point to the watershed outlet in km and L(ca) is the distance from the outlet along the channel to the watershed centroid in km.

Average velocity can be estimated by using Manning's equation and assuming a trapezoidal channel with 2:1 side slopes and a 10:1 bottom width/depth ratio. Substitution of these estimated and assumed values gives

t(cc) = (sqrt(L * L(ca)) * n^0.75) / 0.489 * q(c)^0.25 * sigma^0.375 (Equation 21)

where n is Manning's n, q(c) is the average flow rate in meters cubed per second, and sigma is the average channel slope in meters/meter.

Assuming that L(ca) = 0.5L and converting units (L from m to km, t(cc) from sec to hr, and q(c) from meters cubed per second to mm/hr) gives the equation

t(cc) = (1.75 * L* * n^0.75) / (q*(c) * A)^0.25 * sigma^0.375 (Equation 22)

where L* is the channel length in km and q*(c) is the average flow rate in mm/hr.

The average flow rate is obtained from the estimated average flow rate from a unit source in the watershed (one hectare area) and the relationship

q*(c) = q*(o) * A^-0.5 (Equation 23)

where q*(o) is the average flow rate from a one hectare area in mm/hr.

The unit source flow rate is estimated with the equation [Equation 24] where DUR, the rainfall duration in hours, is calculated using the equation [Equation 25] where alpha(0.5) is computed with equation 16 using R(0.5) instead of R(tc). Equation 25 is derived assuming rainfall intensity is exponentially distributed. (Details of the procedure for estimating alpha(0.5) and DUR are given in the water erosion section of this section.)

Equation 24:

q*(o) = Q / DUR
Code:
if (Q > DUR) q*(o) = Q / DUR else q*(o) = 1.0
Variables:
q*(o) = AvgFlowRateForOneHectare_mmPhr
Q = runoffVolume_mm
DUR = rainfallDuration_hr
alpha(0.5) = meanPropTotalRainFallsInFirstHalfHourForMonth_frn

Equation 25:
DUR = 4.605 / (-2.0 * ln(1.0 - alpha(0.5)))
Code:
same except if R <= 0 DUR = 24
also added upper bound check at 24 if R > 0
Variables:
DUR = RainfallDuration_hr
alpha(0.5) = propRainInFirstHalfHourModBySnow
R = rainfallWithoutSnowmeltForDay_mm

Substituting equation 24 into equation 22 gives the final equation for t(cc), [Equation 26].

Equation 26:

t(cc) = 1.75 * L* * pow(n, 0.75) / (pow(q*(o), 0.25) * pow(A, 0.125) * pow(sigma, 0.375))
Code:
same, broken into two equations, one calculated at input (unchanging), one daily (changing)
at input:
tccf = 1.75 * L* * pow(n, 0.75) / (pow(A, 0.125) * pow(sigma, 0.375))
daily:
t(cc) = tccf / pow(q*(o), 0.25)
Variables:
t(cc) = TimeOfConcForChannelFlow_hr
tccf = TimeOfConcForChannelFlowFactor
L* = channelLength_km
n = manningsChannelRoughnessCoeff
q*(o) = avgFlowRateForOneHectare_mmPhr
A = watershedArea_ha
sigma = channelSlope_mPm

A similar approach is used to estimate t(cs):

t(cs) = lambda / v(s) (Equation 27)

where lambda is the surface slope length in m and v(s) is the surface flow velocity in m/sec.

Applying Manning's equation to a strip 1 m wide down the slope length, assuming flow is concentrated into a small trapezoidal channel with 1:1 side slopes and 5:1 bottom width-depth ratio gives the velocity equation

v(s) = (0.8375 * d^2/3 * s^0.5) / n (Equation 28)

where v(s) is the flow velocity in meters cubed per sec, d is flow depth in m, s is the land surface slope in m/m and n is Manning's roughness Coeff for the surface.

The average flow depth, d, can be calculated from Manning's equation as a class function of flow rate

d = ((q(o) * n) / (5.025 * s^0.5))^0.375 (Equation 29)

where q(o) is the average flow rate in meters cubed per sec.

Substituting equations 28 and 29 into equation 27 gives [Equation 30].

Equation 30:

t(cs) = 0.0216 * pow(lambda * n, 0.75) / (pow(q*(o), 0.25) * pow(s, 0.375))
Code:
same, broken into two equations, one calculated at input (unchanging), one daily (changing)
at input:
tcsf = 0.0216 * pow(lambda * n, 0.75) / pow(s, 0.375)
daily:
t(cs) = tcsf / pow(q*(o), 0.25)
Variables:
t(cs) = TimeOfConcForSurfaceFlow_hr
tcsf = TimeOfConcForSurfaceFlowFactor
lambda = slopeLength_m
n = manningsSurfaceRoughnessCoeff
q*(o) = avgFlowRateForOneHectare_mmPhr
s = slopeSteepness_mPm

To properly evaluate alpha, variation in rainfall patterns must be considered. For some short duration storms, most or all the rain occurs during t(c) causing alpha to approach its upper limit of 1.0. Other storms of uniform intensity cause alpha to approach a minimum value. All other patterns cause higher alpha values than the uniform pattern, because r(tc) is greater than r(24) for all patterns except the uniform. By substituting the products of intensity and time into Equation 16, an expression for the minimum value of alpha, alpha(mn), is obtained [Equation 31].

Equation 31:

alpha = triangular distribution with t(c)/24 < alpha < 1.0
Code:
the code does not use a triangular distribution. It uses this equation:
alpha = 1 - exp(-maxFractionOfRainfallInTimeOfConcParam * t(c) * r(p) / R)
Variables:
alpha = PropRainInTimeOfConc_frn
t(c) = timeOfConc_hr
r(p) = peakRainfallRate_mmPhr
R = rainfallForDay_mm

Thus, alpha ranges within the limits of t(c)/24 <= alpha <= 1.0. Although confined between limits, the value of alpha is assigned with considerable uncertainty when only daily rainfall and simulated runoff amounts are given. Thus, alpha is generated from a triangular distribution with the base ranging from t(c)/24 to 1.0.

The peak of the alpha distribution changes monthly because of seasonal differences in rainfall intensities. The Weather Service (US Department of Commerce, 1979) provides information on monthly maximum rainfall intensities that can be used to estimate the peak alpha for each month. (Peak alpha is an input for each month. If not input, it is estimated.)

TR-55 Method

The SCS TR-55 method for estimating peak runoff rate has been described in detail (USDA-SCS, 1986). Only the essential equations are presented here. The peak runoff rate is dependent on the rainfall distribution and amount, the runoff curve number, and the time of concentration as expressed in the equation [Equation 32].

Equation 32:

q(p) = q*(p) * R
Code:
same (no calculation of q*(p) given in publication)
Variables:
q(p) = PeakRunoffRateBySCSTR55_m3Psec
q*(p) = peakRunoffRatePerUnitRainfall_mmPhrmm
R = rainfallForDay_mm

A set of curves is available (USDA-SCA, 1986) for estimating q*(p) given rainfall distribution (SCS Type I, IA, II, or III), the runoff curve number, and the watershed time of concentration. A map of the U.S. is given (USDA-SCS, 1986) to determine appropriate storm type. The curves relating q*(p), t(c), and CN have been fitted with a seventh degree polynomial for use in computer solutions.

Time of concentration is estimated with the equation [Equation 33] where t(c) is the watershed time of concentration in hr and t(cc), t(cs) and t(sf) are the t(c) components attributed to channel flow, surface flow, and shallow channel flow.

Equation 33:

t(c) = t(cc) + t(cs) + t(sf)
Code:
same
Variables:
t(c) = TimeOfConcTR55_hr
t(cc) = timeOfConcForChannelFlow_hr
t(cs) = timeOfConcForSurfaceFlow_hr
t(sf) = timeOfConcForShallowChannelFlow_hr

The channel component is computed with the equation [Equation 34] where t(cc) is the channel time of concentration in hr, L is the channel lengh from the most distant point to the watershed outlet in km, lambda is the surface slope length in km, L(sf) is the shallow flow length in km, n is Manning's roughness Coeff, d is the average channel flow depth in m, and sigma is the channel slope in m/m.

Equation 34:

t(cc) = (L - lambda - L(sf)) * n / (3.6 * pow(d, 0.67) * pow(sigma, 0.5))
Code:
same except first group (L - lambda - L(sf)) cannot be below zero
Variables:
t(cc) = TimeOfConcForChannelFlow_hr
L = channelLength_km
lambda = slopeLength_km
L(sf) = shallowFlowLength_km
n = manningsChannelRoughnessCoeff
d = channelDepth_m
sigma = channelSlope_mPm

The shallow flow component of t(c) is estimated with the equation [Equation 35] where V(sf), the average shallow flow velocity in km/h, is estimated with [Equation 36] where S is surface flow in m/m.

Equation 35:

t(sf) = L(sf) / V(sf)
Code:
same
Variables:
t(sf) = TimeOfConcForShallowChannelFlow_hr
L(sf) = shallowFlowLength_km
V(sf) = avgShallowFlowVelocity_mPhr
S = slopeSteepness_mPm

Equation 36:

v(sf) = 17.7 * sqrt(S), <= 2.19 km/h
Code:
same except this is in m/hr
Variables:
v(sf) = AvgShallowFlowVelocity_mPhr
S = slopeSteepness_mPm

The length of shallow flow, L(sf), is estimated with [Equation 37]. Note that t(sf) is constrained to less than 0.1 hr even with S = 0.001. Thus, t(sf) is generally insignificant.

Equation 37:

if L > 0.1 km L(sf) = 0.05
if 0.05 < L < 0.1 km L(sf) = L - 0.05
if L < 0.05 km L(sf) = 0.0
Code:
same except in middle case L(sf) is multiplied by 0.1, and aResult is in m, not km
Variables:
L(sf) = ShallowFlowLength_m
L = channelLength_km

The surface flow component of t(c) is estimated with the equation [Equation 38].

Equation 38:

t(cs) = 0.0913 * pow(lambda * n, 0.8) / (pow(S, 0.4) * pow(R, 0.5))
Code:
same, broken into two equations, one calculated at input (unchanging), one daily (changing)
at input:
tcsf = 0.0913 * pow(lambda * n, 0.8) / pow(S, 0.4)
daily:
t(cs) = tcsf / pow(R, 0.5)
Variables:
t(cs) = TimeOfConcForSurfaceFlowTR55_hr
tcsf = TimeOfConcForSurfaceFlowTR55Factor
lambda = slopeLength_m
n = manningsSurfaceRoughnessCoeff
S = slopeSteepness_mPm
R = rainfallForDay_mm