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## Replicating the rational method in SWMM |

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.

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

Where:

Q | = | Runoff rate (m^{3}/s|ft^{3}/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:

Where:

t_{c} |
= | time of concentration, minutes |

L | = | catchment or watershed length, m |

S_{w} |
= | 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.

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.

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.

From there the Rational equation is solved:

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:

The peak runoff generated for subcatchment S1 model was 13.24 m^{3}/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 m^{3}/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 ;;-------------- ---------------- ----------------

Do you have a simple model you've used to test SWMM or illustrate a SWMM concept?

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