San Diego Modified Rational Formula in InfoSWMM and InfoSWMM SA

San Diego Modified Rational Formula in InfoSWMM and InfoSWMM SA

The Rational method is a widely used technique for estimation of peak flows from urban and rural drainage basins (Maidment 1993; Mays 2001). Mathematically, the Rational formula is expressed as:

where Q = peak runoff rate (flow unit)

C = runoff coefficient (unitless)

I = rainfall intensity (intensity unit)

A = watershed area (area unit)

Stormwater modeling applications such as the design of detention basins require knowledge of total inflow volume obtained from runoff hydrographs. For these applications peak flow information alone may not be sufficient. The San Diego modified rational formula is a technique adopted by the San Diego County to generate runoff hydrograph by extending the traditional rational formula.

The San Diego County uses a 6-hour storm event for many stormwater design applications. Accordingly, for designs that are dependent on total storm volume a hydrograph has to be generated by creating a rainfall distribution consisting of blocks of rain, creating an incremental hydrograph for each block of rain using the modified rational formula, and adding the hydrographs from each block of rain where each rainfall block lasts for a duration of time of concentration. The rainfall blocks are distributed across the 6-hour duration using the “2/3, 1/3” distribution in which the peak rainfall block is centered at the 4-hour time within the 6-hour rainfall duration, and the additional blocks are distributed in a sequence alternating two blocks to the left and one block to the right of the 4-hour time. The actual rainfall amount for each rainfall block is calculated as:

where PN = actual rainfall amount for each rainfall block (inch or mm)

P6-hr = six-hour storm depth (inch or mm)

N = an integer representing position of a given block number of rainfall. N is 1 for the peak rainfall block, and is assigned according to the “2/3, 1/3” distribution rule for other rainfall blocks.

Tc = time of concentration (minutes)

Once rainfall distribution is created, triangular hydrograph is generated for each rainfall block. Peak flow for the triangular hydrograph is computed according to the rational formula. Finally, the overall hydrograph for the 6-hr storm event is determined by adding all the triangular hydrographs from each block of rain. The final hydrograph has its peak at 4 hours plus ½ of the Tc. The total volume under the hydrograph is the product of runoff coefficient, the six-hour precipitation depth, and area of the subcatchment.

Stormwater Runoff and the Rational Method from Innovyze H2OCalc for Reference

For storm sewer loading, the focus shifts to hydrologic analysis of excess precipitation and associated runoff. Common techniques for analysis include the rational method and unit hydrograph methods, as well as the use of more advanced hydrologic models.

For small drainage areas, peak runoff is commonly estimated by the rational method. This method is based on the principle that the maximum rate of runoff from a drainage basin occurs when all parts of the watershed contribute to flow and that rainfall is distributed uniformly over the catchment area. Since it neglects temporal and spatial variability in rainfall, and ignores flow routing in the watershed, collection system, and any storage facilities, the rational method should be used with caution only for applications where the assumptions of rational method are valid.

Rational Method from Innovyze H2OCalc for Reference

The rational formula is expressed as

where Qp = peak runoff rate (m3/s, ft3/s)

C = dimensionless runoff coefficient (see Table 3-9)

I = average rainfall intensity (mm/hr, in/hr) for a duration of the time of concentration (tc)

A = drainage area (km2, acres)

K = conversion constant (0.28 in SI, 1 in English)

The time of concentration tc used in the rational method is the time associated with the peak runoff from the watershed to the point of interest. Runoff from a watershed usually reaches a peak at the time when the entire watershed is contributing; in this case, the time of concentration is the time for a drop of water to flow from the remotest point in the watershed to the point of interest. Time of concentration, tc (min), for the basin area can be computed using one of the formulas listed in Table 3-10.

Table: Runoff Coefficients for 2 to 10 Year Return Periods

Description of drainage area Runoff coefficient
Business Downtown 0.70-0.95
Neighborhood 0.50-0.70
Residential Single-family 0.30-0.50
Multi-unit detached 0.40-0.60
Multi-unit attached 0.60-0.75
Suburban 0.25-0.40
Apartment dwelling 0.50-0.70
Industrial Light 0.50-0.80
Heavy 0.60-0.90
Parks and cemeteries 0.10-0.25
Railroad yards 0.20-0.35
Unimproved areas 0.10-0.30
Pavement Asphalt 0.70-0.95
Concrete 0.80-0.95
Brick 0.75-0.85
Roofs 0.75-0.95
Lawns Sandy soils Flat (2%) 0.05-0.10
Average (2-7%) 0.10-0.15
Steep (≥7%) 0.15-0.20
Heavy soils Flat (2%) 0.13-0.17
Average (2-7%) 0.18-0.22
Steep (≥7%) 0.25-0.35

Source: Nicklow et al. (2006)

Table 3-10: Formulas for Computing Time of Concentration

Method Formula
Kirpich (1940)

L = length of channel (ft)

S = average watershed slope (ft/ft)

California Culverts Practice (1942)

L = length of the longest channel (mi)

H = elevation difference between divide and outlet (ft)

Izzard (1946)

i = rainfall intensity (in/h)

c = Retardance coefficient

Retardance factor, c, ranges from 0.007 for smooth pavement to 0.012 for concrete and to 0.06 for dense turf; product i times L should be < 500
Federal Aviation Administration (1970)

C = rational method runoff coefficient (see Table 3.9)

Kinematic wave

n = Manning’s roughness coefficient

SCS lag equation

CN = SCS runoff curve number (see Table 3.11)

SCS average velocity charts

V = average velocity (ft/s)

Yen and Chow (1983)

KY = Coefficient

N = Overland texture factor

(see Table 3.13)

KY ranges from 1.5 for light rain (i<0.8) to 1.1 for moderate rain (0.8<i<1.2), and to 0.7 for heavy rain (i>1.2)

Source: Nicklow et al. (2004)

Table 3-11: Runoff Curve Numbers for Urban Land Uses

Land use description Soil Group
Lawns, open spaces, parks, golf courses:
Good condition: grass cover on 75% or more area 39 61 74 80
Fair condition: grass cover on 50% to 75% of area 49 69 79 84
Poor condition: grass cover on 50% or less of area 68 79 86 89
Paved parking lots, roofs, driveways, etc 98 98 98 98
Streets and roads:
Paved with curbs and storm sewers 98 98 98 98
Gravel 76 85 89 91
Dirt 72 82 87 89
Paved with open ditches 83 89 92 93
Commercial and business areas (85% impervious) 89 92 94 95
Industrial districts (72% impervious) 81 88 91 93
Row houses, town houses and residential with lot sizes of 1/8 ac or less (65% impervious) 77 85 90 92
Residential average lot size:
1/4 ac (38% impervious) 61 75 83 87
1/3 ac (30% impervious) 57 72 81 86
1/2 ac (25% impervious) 54 70 80 85
1 ac (20% impervious) 51 68 79 84
2 ac (12% impervious) 46 65 77 82
Developing urban area (newly graded; no vegetation) 77 86 91 94

Adapted from SCS (1985)

Table 3-12: Description of NRCS Soil Classifications

Group Description Min. infiltration (in/hr)
A Deep sand; deep losses; aggregated silts 0.30-0.45
B Shallow loess; sandy loam 0.15-0.30
C Clay loams; shallows sandy loam; soils low in organic content; soils usually high in clay 0.05-0.15
D Soils that swell significantly 0-0.05

Adapted from SCS (1985)

Table 3-13: Overland Texture Factor N

Overland flow surface Low Medium High
Smooth asphalt pavement 0.010 0.012 0.015
Smooth impervious surface 0.011 0.013 0.015
Tar and sand pavement 0.012 0.014 0.016
Concrete pavement 0.014 0.017 0.020
Rough impervious surface 0.015 0.019 0.023
Smooth bare packed soil 0.017 0.021 0.025
Moderate bare packed soil 0.025 0.030 0.035
Rough bare packed soil 0.032 0.038 0.045
Gravel soil 0.025 0.032 0.045
Mowed poor grass 0.030 0.038 0.045
Average grass, closely clipped sod 0.040 0.055 0.070
Pasture 0.040 0.055 0.070
Timberland 0.060 0.090 0.120
Dense grass 0.060 0.090 0.120
Shrubs and bushes 0.080 0.120 0.180
Land use Low Medium High
Business 0.014 0.022 0.35
Semi-business 0.022 0.035 0.050
Industrial 0.020 0.035 0.050
Dense residential 0.025 0.040 0.060
Suburban residential 0.030 0.055 0.080
Parks and lawns 0.040 0.075 0.120

Adapted from Yen and Chow (1983)

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