SWMM Knowledge Base
https://www.openswmm.org/Topic/9896

Replicating the rational method in SWMM

Kat Finney

The rational equation is a simple method used to determine the peak discharge from a drainage basin for a known design storm.  The following example illustrates how the rational method can be used to compute the peak drainage for a watershed with a known area, slope, flow length, and imperviousness.  

Using the rational method parameters, an equivalent EPA SWMM5 model was created and compared to the rational method calculation.  

Equations

The relationship between the runoff rate and rainfall intensity is expressed in the rational formula:

Syntax error.

Where:

Q = Runoff rate (m3/s|ft3/s)
C = Runoff coefficient
i = Rainfall intensity (mm/h|in/h)
A = Subcatchment area (ha|ac)
k = conversion const. (1.0 for US units and 0.0028 for SI units)

For simplicity, in this example it will be assumed that the study area is 100% impervious.  For an area that is 100% impervious, a runoff coefficient of 0.95 is considered appropriate (https://www.waterboards.ca.gov/water_issues/programs/swamp/docs/cwt/guidance/513.pdf).

The time of concentration measures the time it takes a drop of rain to travel the longest flow path in a catchment area.  The equation used to compute the time of concentration changes based on the runoff coefficient of the catchment.  For example, the Bransby William Formula is used if the Rational Method runoff coefficient is greater than 0.4 and is as follows:

Syntax error.

Where:

tc = time of concentration, minutes
L = catchment or watershed length, m
Sw = catchment or watershed slope, %
A = catchment area,  ha

(www.mto.gov.on.ca/english/publications/drainage/hydrology/section10.shtml)

In this exercise, because the runoff coefficient is 0.95, the Bransby William Formula will be used to calculate the time of concentration.  

Hand calculations

The following illustrates one example of how the rational method might be used to determine the peak rainfall for a catchment.  

For the comparison, the below assumptions are made:

Parameter                Symbol     US units    SI units
--------------------------------------------------------
Runoff coefficient       C         0.95         0.95
Percent Impervious      Imperv.    95           95
Subcatchment Slope       Sw        0.99%        0.99% 
Flow Length              L         6561.7 ft    2000 m
Area of subcachment      A         263.2 ac     106.5 ha
Duration                           100-year     100-year

We'll also assumed this set of IDF curves is applicable:

Duration 5-min (mm/hr) 10-min (mm/hr) 15-min (mm/hr) 30-min (mm/hr) 1-hr (mm/hr) 2-hr (mm/hr) 6-hr (mm/hr) 12-hr (mm/hr) 24-hr (mm/hr)
2-year 84.00 61.20 50.40 33.60 20.60 12.70 5.95 3.61 2.06
5-year 117.60 85.20 68.80 47.00 28.80 17.65 7.75 4.65 2.69
10-year 140.40 100.80 80.80 56.00 34.20 20.90 8.95 5.34 3.11
25-year 168.00 120.60 96.00 67.40 41.00 25.05 10.45 6.23 3.64
50-year 189.60 135.00 107.60 75.80 46.10 28.15 11.58 6.88 4.03
100-year 210.00 149.40 118.80 84.00 51.10 31.20 12.68 7.52 4.43

Since the runoff coefficient is greater than 0.4, the Bransby William Formula will be used to compute a time of concentration of 72 minutes. 

Syntax error.

Linearly interpolating the peak rainfall intensity using 100-year duration row from the IDF table above, a peak intensity of 47.12 mm/hr is calculated. 

Syntax error.

From there the Rational equation is solved:

Syntax error.

Input Files

The input file (SI units) below provides one attempt at a SWMM5 representation of the rational method calculations presented above.  

This example consists of two subcatchments, both assigned the same area, slope and flow length as the rational method example above.  

Since SWMM5 uses a non-linear reservoir model to compute runoff, it does not require a runoff coefficient.  Instead, SWMM5 uses a percent imperviousness value, which in this case, will be set to 95%. Infiltration for the remaining 5% is modeled using the Green and Ampt method using infiltration parameters representative for sand.  

To represent the equivalent rainfall intensity in SWMM5, a time series with a constant rainfall intensity of 47.12 mm/hr was assigned to the subcatchment S1. This was the value calculated above using the IDF cuve for the 100-year design storm.  As a comparison, S2 was assigned a representative Chicago storm distribution using the IDF curve to calculate the abc value using the following equation:

  i equal fraction numerator a over denominator left parenthesis t subscript D plus b right parenthesis to the power of c end fraction

The peak runoff generated for subcatchment S1 model was 13.24 m3/s and the runoff coefficient for this model was calculated to be 0.95.  This differed for subcatchment S2 which had a peak runoff of 13.94 m3/s and a runoff coefficient of 0.96.   In the case of S2, it was observed that when a larger rainfall time step was used, the peak runoff decreased.  

A copy of the input file for the SWMM5 model can be viewed and downloaded below.

[TITLE]

;File: "Rational Method vs SWMM5.inp"


[OPTIONS]

;;Options            Value

;;------------------ ------------

FLOW_UNITS          CMS

INFILTRATION        GREEN_AMPT

FLOW_ROUTING        DYNWAVE

START_DATE          11/17/2014

START_TIME          00:00:00

REPORT_START_DATE    11/17/2014

REPORT_START_TIME    00:00:00

END_DATE            11/17/2014

END_TIME            06:00:00

SWEEP_START          01/01

SWEEP_END            12/31

DRY_DAYS            0

REPORT_STEP          00:01:00

WET_STEP            00:05:00

DRY_STEP            00:05:00

ROUTING_STEP        5

ALLOW_PONDING        YES

INERTIAL_DAMPING    PARTIAL

VARIABLE_STEP        0.75

LENGTHENING_STEP    0

MIN_SURFAREA        0

NORMAL_FLOW_LIMITED BOTH

SKIP_STEADY_STATE    NO

FORCE_MAIN_EQUATION H-W

LINK_OFFSETS        DEPTH

MIN_SLOPE            0

MAX_TRIALS          8

HEAD_TOLERANCE      0.0015

SYS_FLOW_TOL        5

LAT_FLOW_TOL        5

MINIMUM_STEP        0.5

THREADS              4


[EVAPORATION]

;;Type          Parameters

;;------------- ----------

CONSTANT    0.0

DRY_ONLY     NO


[RAINGAGES]

;;              Rain      Time  Snow  Data      

;;Name          Type      Intrvl Catch Source    

;;-------------- --------- ------ ------ ----------

Chicago_3h      INTENSITY 0:10  1.0    TIMESERIES Chicago_3h      

Constant        INTENSITY 0:30  1.0    TIMESERIES Constant        


[SUBCATCHMENTS]

;;                                              Total    Pcnt.            Pcnt.    Curb    Snow    

;;Name          Raingage        Outlet          Area    Imperv  Width    Slope    Length  Pack    

;;-------------- ---------------- ---------------- -------- -------- -------- -------- -------- --------

S1              Chicago_3h      OF2              106.5    95      532.5    0.99    0                        

S2              Constant        OF1              106.5    95      532.5    0.99    0                        


[SUBAREAS]

;;Subcatchment  N-Imperv  N-Perv    S-Imperv  S-Perv    PctZero    RouteTo    PctRouted 

;;-------------- ---------- ---------- ---------- ---------- ---------- ---------- ----------

S1              0.001      0.01      0          0          0          OUTLET    

S2              0.001      0.01      0          0          0          OUTLET    


[INFILTRATION]

;;Subcatchment  Suction    HydCon    IMDmax    

;;-------------- ---------- ---------- ----------

S1              49.02      120.34    0.413     

S2              49.02      120.34    0.413     


[OUTFALLS]

;;              Invert    Outfall      Stage/Table      Tide

;;Name          Elev.      Type        Time Series      Gate Route To        

;;-------------- ---------- ------------ ---------------- ---- ----------------

OF1              0          FREE                        NO                  

OF2              0          FREE                        NO                  


[TIMESERIES]

;;Name          Date      Time      Value     

;;-------------- ---------- ---------- ----------

;Rainfall (mm/hr)

Chicago_3h      11/17/2014 00:00:00  0.315     

Chicago_3h      11/17/2014 00:10:00  0.418     

Chicago_3h      11/17/2014 00:20:00  0.599     

Chicago_3h      11/17/2014 00:30:00  0.973     

Chicago_3h      11/17/2014 00:40:00  2.028     

Chicago_3h      11/17/2014 00:50:00  8.862     

Chicago_3h      11/17/2014 01:00:00  62.238    

Chicago_3h      11/17/2014 01:10:00  10.454    

Chicago_3h      11/17/2014 01:20:00  3.863     

Chicago_3h      11/17/2014 01:30:00  2.093     

Chicago_3h      11/17/2014 01:40:00  1.347     

Chicago_3h      11/17/2014 01:50:00  0.958     

Chicago_3h      11/17/2014 02:00:00  0.726     

Chicago_3h      11/17/2014 02:10:00  0.575     

Chicago_3h      11/17/2014 02:20:00  0.472     

Chicago_3h      11/17/2014 02:30:00  0.396     

Chicago_3h      11/17/2014 02:40:00  0.34      

Chicago_3h      11/17/2014 02:50:00  0.296     

Chicago_3h      11/17/2014 03:00:00  0         


Constant                  0:00      47.12     

Constant                  0:30      47.12     

Constant                  1:00      47.12     

Constant                  1:30      47.12     

Constant                  2:00      47.12     

Constant                  2:30      47.12     

Constant                  3:00      47.12     

Constant                  3:30      47.12     

Constant                  4:00      47.12     

Constant                  4:30      47.12     

Constant                  5:00      47.12     

Constant                  5:30      47.12     

Constant                  6:00      47.12     

Constant                  6:30      47.12     

Constant                  7:00      47.12     

Constant                  7:30      47.12     

Constant                  8:00      47.12     

Constant                  8:30      47.12     

Constant                  9:00      47.12     

Constant                  9:30      47.12     

Constant                  10:00      47.12     

Constant                  10:30      47.12     

Constant                  11:00      47.12     

Constant                  11:30      47.12     

Constant                  12:00      47.12     

Constant                  12:30      47.12     

Constant                  13:00      47.12     

Constant                  13:30      47.12     

Constant                  14:00      47.12     

Constant                  14:30      47.12     

Constant                  15:00      47.12     

Constant                  15:30      47.12     

Constant                  16:00      47.12     

Constant                  16:30      47.12     

Constant                  17:00      47.12     

Constant                  17:30      47.12     

Constant                  18:00      47.12     

Constant                  18:30      47.12     

Constant                  19:00      47.12     

Constant                  19:30      47.12     

Constant                  20:00      47.12     

Constant                  20:30      47.12     

Constant                  21:00      47.12     

Constant                  21:30      47.12     

Constant                  22:00      47.12     

Constant                  22:30      47.12     

Constant                  23:00      47.12     

Constant                  23:30      47.12     

Constant                  24:00:00  47.12     


[REPORT]

INPUT      YES

CONTROLS  NO

SUBCATCHMENTS ALL

NODES ALL

LINKS ALL


[TAGS]


[MAP]

DIMENSIONS      452976.7498      5141115.2564    453008.4342      5141131.8576    

UNITS            Meters


[COORDINATES]

;;Node          X-Coord          Y-Coord         

;;-------------- ---------------- ----------------

OF1              452983.275      5141116.186     

OF2              453001.994      5141116.103     


[VERTICES]

;;Link          X-Coord          Y-Coord         

;;-------------- ---------------- ----------------


[POLYGONS]

;;Subcatchment  X-Coord          Y-Coord         

;;-------------- ---------------- ----------------

S1              453001.994      5141116.103     

S1              452996.994      5141121.103     

S1              452996.994      5141131.103     

S1              453006.994      5141131.103     

S1              453006.994      5141121.103     

S1              453001.994      5141116.103     

S2              452983.19        5141116.011     

S2              452978.19        5141121.011     

S2              452978.19        5141131.011     

S2              452988.19        5141131.011     

S2              452988.19        5141121.011     

S2              452983.19        5141116.011     


[SYMBOLS]

;;Gage          X-Coord          Y-Coord         

;;-------------- ---------------- ----------------
 

Robert Dickinson

Thanks, a lot of good tips here, Karen.

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