DS-5 - Appendix E  The Rational Method

DS-5 - Appendix E.1   General

The Rational Method is widely used for peak flow estimation in small, ungauged catchments. It has limitations on its validity and on the accuracy of results that can be attained. A major limitation of the Rational Method is that the output is a single value for peak flow estimate, with no further detail on the time-varying nature of flood flow. Because of this, use of the Rational Method frequently requires additional analyses and assumptions to be used for design purposes. 

This appendix describes the Rational Method. A set of constraints is provided to assist with decisions on an appropriate analysis technique. An example is also given where the results from a Rational Method based analysis are used to size mitigation works.

Due to the inexact nature of this method, which relies on a degree of engineering judgement being applied, it is recommended that results obtained be tested for sensitivity.

DS-5 - Appendix E.2   Basic Syntax

The rational method formula for peak flow estimation can be stated as follows:

Qp = 2.78 C . i . A

Where: 
Qp = peak flow (L/s)
C = runoff coefficient (dimensionless)
i = average rainfall intensity over the duration equal to catchment time of concentration (mm/hr)
A = catchment area (ha).

The numeric 2.78 in the formula is required for unit conversion and applies only if the units, as indicated above, are used for the inputs. This numeric will change if the units of input data are not as stated above.

DS-5 - Appendix E.3   Method: Peak Flow Estimation

Application of the Rational Method to peak flow estimation can be done using a sequence of steps as outlined below:

1. DS-5 - Appendix E.3.1 Determine Catchment Area upstream of the point of interest in the catchment (i.e. the point at which peak flow estimates are required).
2. DS-5 - Appendix E.3.2 Consider Validity of the Rational Method for the selected location. This will require some knowledge of the catchment area upstream of the point of interest and may rely on engineering judgement.
3. DS-5 - Appendix E.3.3 Determine Time of Concentration for the contributing catchment area to the point of interest.
4. DS-5 - Appendix E.3.4 Rainfall Intensity for the selected Average Recurrence Interval (ARI), look up the average rainfall intensity for the rainfall duration equal to catchment time of concentration.
5. DS-5 - Appendix D.3.5 Determine Runoff Coefficient e.g. look up, derive or develop the Runoff Coefficient applicable to the catchment upstream of the point of interest.
6. DS-5 - Appendix D.3.6 Apply Slope Correction to C that is applicable.
7. DS-5 - Appendix D.3.7 Estimate Peak Flow Using Rational Method Apply the formula, ensuring the units are correct, to obtain the peak flow estimate for the given ARI at the point of interest.
8. DS-5 - Appendix D.3.8 Test Sensitivity. Undertake a sensitivity assessment to develop a likely range of results.

These steps are summarised in Figure 1: Flow Chart for Rational Method.

Figure 1: Flow Chart for Rational Method

DS-5 - Appendix E.3.1   Determine Catchment Area

A catchment plan showing the location of the specific point of interest will be required. An accurate measure of upstream contributing catchment area can be made using this. 

DS-5 - Appendix E.3.2   Consider Validity of the Rational Method

The Rational Method has a specific range of validity, outside of which the results obtained may be unreliable. The limits on validity can be established by following Figure 2: Validity Chart for Rational Method. If site specific information exists then use of this will generally be preferred over the Rational Method results. Hydrological understanding of the catchment is likely to be required.

Figure 2: Validity Chart for Rational Method

DS-5 - Appendix E.3.3   Determine Time of Concentration

The time of concentration for a catchment is the shortest time that it takes for stormwater runoff from the entire catchment to begin to contribute to the flow measured at the point of interest. It is taken as the travel time for runoff to flow from the farthest point in the catchment to the point of interest.
Two alternatives for calculation of time of concentration are given:

1. Urban Catchments: Based on summing the travel and entry times within largely urban networks and over open spaces contained within an urban context. 
2. Rural Catchments.

Time of concentration should be calculated as per Verification Method E1/VM1 (New Zealand Building Code Compliance Document E1, Surface Water), as replicated below:

DS-5 - Appendix E.3.3.1 Time of Concentration for Urban Catchments

Time of concentration TC can be estimated using the following formula:
TC = Te + Tt
Where:
Te = time of entry (travel time for runoff to reach the point of entry)
Tt = travel time.
Time of entry is the time taken for stormwater runoff to reach a dedicated stormwater network element. Travel time is the time taken within the dedicated stormwater network (side channels, pipes, open channels) for flow to reach the point of interest.
Regardless of the outcome of calculations, TC should never be taken as less than 10 minutes in residential and commercial areas and never less than 25 minutes for open space areas (parks, paddocks, etc.).

1. Estimation of time of entry Te for Urban Catchments

For urbanised catchments where the time of entry can be assessed on the assumption of there being a well-defined and regularly-repeating pattern of flow to stormwater sumps or open channels, the following may be applied:

1. Te = 5 minutes. For commercial or industrial areas where the contributing catchment area is greater than 50% impervious.
2. Te = 7 minutes. For residential areas with greater than 50% impervious cover.
3. Te = 10 minutes. For low density residential areas with less than 50% impervious cover.

1. Estimation of Travel Time Tt

Travel time estimation is based on the type of network element (pipe, open channel) that is conveying the flow to the point of interest. 

1. For travel time within piped networks, an appropriate pipe flow formula will yield flow velocity for the specific pipes involved (taking into account pipe diameter, material and slope). Alternatively Figure 3: Nomograph for Piped Network Flow Velocity Estimation may be used to estimate pipe flow velocity. Pipe length can be divided by travel velocity to give travel time.
2. For travel time within open channels, much like piped networks, an appropriate hydraulic formula will yield average cross sectional flow velocity (based on hydraulic parameters), but an approximation can also be made as follows:
   1. Relatively flat channel, average slope 1-4% 0.3 – 1.0 m/s
   2. Undulating terrain, average slope 2-8% 0.6 – 2.0 m/s
   3. Hilly area, steep channel, average slope 6-15% 1.5 – 3.0 m/s.
   4. Channel length can be divided by travel velocity to give travel time.
3. For travel time within rural catchments, Te should be calculated using the overland flow and the road channel flow methods outlined below. The time of entry is taken as the sum of the overland flow time of entry and the road channel time of entry. This applies to catchments that do not have a well-defined and regularly-repeating pattern of flow to a stormwater network.

Te = 100 . n . Lo 0.33 + TRC
   S 0.2
Where:
n = surface roughness coefficient (dimensionless)
Lo = length of overland flow (m)
S = slope in percent
TRC = time of entry for flow in road channel.
The first term in the above formula has been reproduced in Figure 4: Nomograph for Rural Catchments.
For flow in road channels, Figure 5: Side Channel Time of Entry Estimation Nomograph may be used to derive an estimate of TRC.

Figure 3: Nomograph for Piped Network Flow Velocity Estimation

Figure 4: Nomograph for Rural Catchments

Figure 5: Side Channel Time of Entry Estimation Nomograph

DS-5 - Appendix E.3.3.2 Time of Concentration for Rural Catchments

For rural catchments without formal stormwater collection and conveyance networks, time of concentration (in minutes) may be estimated from the formula below:
T_C=0.0195 (〖L_c〗^3/H)^0.385
Where:
Lc = length of catchment along flow path (m)
H = rise from point of interest to top of catchment (m).

DS-5 - Appendix E.3.4   Rainfall Intensity

Once time of concentration TC is known, the rainfall intensity of duration equal to this time of concentration for the ARI selected can be read from Table B: Climate-Adjusted Design Rainfall Intensity Estimates (2055) in mm/hour located in DS-5 - Appendix B.1 General. If the values in the table are to be interpolated, this should not be done linearly.
Note that for a detailed hydrological modelling assessment, the above figures within Table B are used to develop a centrally-weighted nested rainfall profile.

DS-5 - Appendix E.3.5   Determine Runoff Coefficient

A comprehensive list of runoff coefficients for use in urban and rural catchments has been developed by the New Zealand Institute of Engineers (NZIE 1980) as shown below: 

For catchments having a mixture of different types, the run-off coefficient shall be determined by averaging the value for individual parts of the catchment by using the formula: 
C=  (∑▒〖C_(i ) A_i 〗)/A
Where: 
C = the run-off coefficient for the catchment. 
Ci = the run-off coefficient for a particular land use. 
Ai = the area of land to which Ci  applies. 
A = the total catchment area.

DS-5 - Appendix E.3.6   Apply Slope Correction

The runoff coefficients listed have been calibrated for slopes of 5% to 10%. If the slope is outside of this range, an adjustment needs to be applied to the runoff coefficient as follows:

DS-5 - Appendix E.3.7   Estimate Peak Flow Using Rational Method

Peak flow can be estimated using data derived as above in the formula as follows:
Q_p=2.78 C.i.A
Where:
Qp = peak flow (L/s)
C = runoff coefficient (dimensionless)
i  = average rainfall intensity over the duration equal to catchment time of concentration (mm/hr)
A = catchment area (ha).

DS-5 - Appendix E.3.8   Test Sensitivity

The process outlined in DS-5 - Appendix D.3 Method: Peak Flow Estimation should be repeated, taking the upper and lower bounds of input parameters to test the sensitivity in the final result (as an outcome of the judgements made in the application of the formula).

If the results are found to be notably sensitive i.e. greater than an acceptable margin of error acceptable for the particular situation, then an alternative method shall be used and the results from the Rational Method shall be discarded.

DS-5 - Appendix E.4   Method: Pre-Development and Post-Development Comparison

DS-5 - Appendix E.3 Method: Peak Flow Estimation gives an indication of how to successfully use the Rational Method for peak flow estimation in small, ungauged catchments. A situation for which this method is frequently used is in assessment of the change in response from an undeveloped catchment to a fully developed catchment.

Using the method described in DS-5 - Appendix E.3 Method: Peak Flow Estimation, the peak discharge from a selected catchment can be assessed for both the pre-development and the post-development cases. A typical finding would be that the peak flow for post-development exceeds that for pre-development and the time of concentration post development is shorter than it was for the pre-development case. It is a requirement that the post-development peak discharge should be reduced, via detention for example, to be no greater than it was for the pre-development land use in response to a rainfall event of given Average Recurrence Interval.

Sizing a detention method to achieve the required mitigation can be done using the estimates of peak flow from the Rational Method, but also requires some additional approximations to be made. Given that the Rational Method does not result in any time-varying description of the flow i.e. a hydrograph is not produced, the total runoff volume cannot be inferred from the method.

Bay of Plenty Regional Council has published a method to achieve this that is based on fitting a triangular hydrograph shape to the peak discharge estimates from the Rational Method as shown in Figure 6: BoPRC Fitted Hydrograph (BoPRC H&H Guidelines).

Figure 6: BoPRC Fitted Hydrograph (BoPRC H&H Guidelines)

Qp = peak discharge estimated using Rational Method (m³/s)
tC = time of concentration (s)
D = rainfall duration (s)

This gives the total runoff estimation as:

Vtot = 4/3 . tC . Qp

Using this fitted hydrograph, for both pre-development and post-development land uses, for a target catchment, the required detention volume to mitigate the effects of development can be estimated. This is done by assessment of the maximum difference in accumulated runoff volume between the two land use scenarios.

Figure 7: Fitted Triangular Hydrographs