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Contact 52 North

Montija catchment In this case study the peak runoff of a small watershed in Southern Spain is determined using the Modified Rational Formula method, and the SCS Dimensionless Unit Hydrograph for an excess rainfall of 1 cm.

Theoretical Background

An important formula for determining the peak runoff rate is the Rational Formula. It is characterised by:

  • consideration of the entire drainage area as a single unit,
  • estimation of flow atthe most downstream point only,
  • the assumption that rainfall is uniformly distributed over the drainage area.

The Rational Formula reads:

Qp= 0.28*C*I*A



= Peak runoff rate [m/sec]
C =Runoff coefficient [-]
I = Rainfall intensity [mm/hr]
A = Drainage area [km]

The Rational Formula follows the assumption that:

  • the predicted peak discharge has the same probability of occurence (return period) as the used rainfall intensity (I),
  • the runoff coefficient (C) is constant during the rain storm, and
  • the recession time is equal to the time of rise.

In the modified version of the Rational Formula, a storage coefficient is included to account for a recession time larger than the time the hydrograph takes to rise. The Modified Rational Formula reads:

Qp = 0.28 * Cs * C * I * A

Cs = Storage coefficient [-]

The maximum runoff rate in a catchment is reached when all parts of the watershed are contributing to the outflow. This happens when the time of concentration, the time after which the runoff rate equals the excess rainfall rate, is reached. In this exercise, the Kirpich/Ramser formula is used to calculate the time of concentration:

tc = 0.0195 * L0.77 * S-0.385

tc = Time of concentration [min]
L = Length of main river [m]
S = Distance weighted channel slope [m/m]

Another important empirical formula for determining the quantity of runoff is the SCS Dimensionless Unit Hydrograph. In order to use this method, the time to peak and the peak discharge are estimated. The method assumes that:

  • the duration of excess rainfall is less or equal to 0.133 x the time of concentration, and that
  • the rainfall duration is not too long (D<0.2 time to peak)

If these conditions are met, the following formulas can be used:

Formula 6

qpeak = Peak runoff rate [m/s]


= Storm runoff or excess rainfall volume [cm]
A = Watershed area [km]
tpeak = Time to peak [hr]

Formula 7

tpeak = Time to peak [hr]
D = Duration of excess rainfall [hr]
tlag = The lag time of the watershed [hr]

Formula 8

tc = Time of concentration [hr]


Formula 9

tlag = Lag time [hr]
L = Hydraulic watershed length [m]
CN = Hydrologic area-weighted curve number [-]
H = Average watershed land slope [%]

L= 110 A0.6

A = Watershed area [ha]

Determination of peak runoff rate using the Modified Rational Formula

Determine time of concentration

In order to determine the time of concentration using the Kirpich/Ramser formula you need to calculate the drainage length (L) and the distance weighted channel slope (S). The river length is calculated by performing a weighted Distance operation on the rasterized river map using the most downstream point of the river map as starting point.

To evaluate the weighted channel slope, a DEM of the catchment must be created by means of Contour Interpolation. This DEM is needed for investigating the height in the drainage area. Once you have found the height for two points along the river and the distance between them, it is possible to compute the weighted channel slope and the time of concentration with some mapcalc and tabcalc statements.

Determine the area weighted runoff coefficient

Soil group & Land cover complexes

The rational formula uses C, the runoff coefficient. This coefficient is related to the different land covers and hydrologic soil groups. Within the catchment, more than one land cover type and soil group exists. In order to find a representative runoff coefficient, an overall catchment runoff coefficient has to be determined using the areas of the different land cover/hydrologic soil group complexes as weighting factor.

First, the soils map is reclassified into a map showing the Hydrologic Soil Groups. Also the Land Use map is reclassified. The reclassification of this map is based on a newly created attribute table that correlates information about land cover with a certain type of crop. To determine the area occupied by the different land cover/hydrologic soil group complexes, both reclassified maps are crossed.

Runoff coefficients differ per soil group. To get coefficients for the different soil units, a conversion column is added to the cross table. This cross table is joined with a table that contains runoff values for different crops and hydrological conditions. Multiplying both columns with a tabcalc statement results in runoff coefficients for the various land cover/hydrologic soil group complexes. Aggregating this output column, using the areas of the complexes as weighting factors, gives the overall runoff coefficient.

Determine the 10-year return period peak runoff rate

When determining the peak runoff rate according to the Modified Rational Formula, a rainfall intensity has to be selected with a certain return period. The duration of the storm has to be equal to the time of the concentration of the catchment. In this exercise the 10-year return period peak runoff rate is calculated with a tabcalc formula using a rainfall intensity of 100 mm/h.

Determination of peak runoff rate using the SCS Dimensionless Unit Hydrograph

Determine the area weighted curve number

The SCS method uses curve numbers. These numbers are related to the different land cover types, soil properties and antecedent moisture conditions. Within the catchment more than one land cover type and soil type exist. In order to find a representative curve number, an overall catchment curve number has to be determined using the areas of the different land cover and soil types as weighting factor. In this study, the overall curve number is computed according to the same procedure as the overall runoff coefficient C is calculated.

Determine the lag time

For the calculation of the lag time, a number of watershed parameters have to be determined:

  • Hydraulic watershed length (L).
    This parameter is calculated with a tabcalc statement using the total catchment area in hectares as input.
  • Average watershed slope (H).
    To determine the average watershed slope, first a slope map has to be created.
    The slope map is calculated by filtering the DEM of the catchment area in x and y direction, using the mapcalc statement
    Slope = ((HYP(dx,dy)) / 29.5)*100 and masking the output map so that areas out of the catchment
    are eliminated.
    Secondly, a histogram of the slope map is created.
    Finally, the average slope is calculated via the aggregate operation using theaverage function.
  • Overall Curve Number.
    This value is already calculated in the previous step.

Finally, the formula to get the lag time can be applied in the cross table with a tabcalc expression.

Determine the unit duration of excess rainfall

The application of the SCS Dimensionless Unit Hydrograph is only valid for storms with a duration less than 0.133 x time of concentration. The duration D of excess rainfall is calculated in this excise using the formula D = 0.133*tlag/0.6.

Determine the peak runoff rate for 1 cm of excess rainfall

When calculating the peak runoff of any unit hydrograph, the amount of excess rainfall equals a unit depth, i.e. 1 mm or 1 cm. To determine the peak runoff for 1 cm of excess rainfall, first the time to peak should be determined before the SCS formula can be entered as a tabcalc expression in the cross table. Based on the unit hydrograph, the peak flow rate for other storms with a different rainfall excess amount can be calculated.


  • Schwab, G.O., Fangmeier, D.D., Elliot, W.J. and Freveret, R.K. (1993). Soil and water conservation engineering. J. Wiley and sons. New York. 507 pp.
  • Sherman, L.K. (1932). Stream-flow from rainfall by the unit-graph method. Eng. News-Rec. 108: 501-505.
  • U.S. Soil Conservation Service. (1964). Hydrology. Section 4, SCS National Engineering Handbook. Washington, D.C.