# Time of concentration: Pilgrim McDermott formula

There are many formulas for the time of concentration.  A previous post discussed the Bransby Williams approach. Here I look at the Pilgrim McDermott formula, which is another method commonly used in Australia and relates time of concentration to catchment area (A):

$t_c = 0.76A^{0.38}$    (hours)                                                     (equation 1)

where A is measured in km2.

This formula is a component of the Probabilistic Rational Method as discussed in Australian Rainfall and Runoff 1987 (ARR1987) Book IV and is recommended for use in:

• Eastern New South Wales
• Victoria (as developed by Adams, 1987)
• Western Australia – wheatbelt region

McDermott and Pilgrim (1982) needed a formula for the time of concentration to develop their probabilistic rational method approach which was ultimately adopted in ARR1987.  They make the point that, for their statistical method, it is not necessary that the time of concentration closely matches the time for water to traverse a catchment, rather a characteristic time is required for a catchment to determine the duration of the design rainfall.  This characteristic time must be able to be determined directly by designers and lead to consistent values of the runoff coefficient and design flood values.

The basic formula for the probabilistic rational method is:

$Q_y = C_y I_{(y,t_c)} A$                                                                  (equation 2)

Where:

• $Q_y$ is the flood of $y$ years average recurrence interval.
• $C_y$ is the runoff coefficient for a particular average recurrence interval.
• $I$ is the rainfall intensity which is a function of $t_c$ (time of concentration) and $y$.
• $A$ is the catchment area.

For a catchment with a stream gauge, where flood frequency analysis can be undertaken, this will provide the $Q_y$ values on the left hand side of equation 2. We also know the catchment area ($A$). If $t_c$ can be estimated via a time of concentration formula, then the rainfall intensity can be looked up in an IFD table for the location and the only unknown is $C_y$.

$C_y = \frac{Q_y}{I_{(y,t_c)} A}$                                                              (equation 3)

This was the approach used in ARR1987. A large number of gauges were selected and $C_y$ values calculated. Ultimately $C_{10}$ values were mapped in Volume 2 of Australian Rainfall and Runoff.   For floods other than those with a 10 year average recurrence interval, frequency factors were provided to calculate the required runoff coefficient values.  This meant design floods could be estimated for ungauged catchments given information on design rainfall intensity which is available everywhere in Australia.

For this approach to work, some relationship is required between $t_c$ and catchment characteristics i.e. we need a time of concentration formula. McDermott and Pilgrim (1982) began their development of such a formula by testing the Bransby Williams approach because that had been shown to be the best of 8 methods examined by French et al. (1974). McDermott and Pilgrim found that Bransby Williams wasn’t suitable for their purposes because it often resulted in runoff coefficients greater than 1 and they thought the use of such large values would be resisted by practising engineers. Equation 2 doesn’t preclude runoff coefficient values greater than 1 but the intuitive definition of $C$ as being “the proportion of rainfall that runs off” requires it.

An alternative time of concentration formula was developed by considering the ‘minimum time of rise of the flood hydrograph’ which McDermott and Pilgrim collected or collated for 96 catchments. This is the time from when storm rainfall starts until stream discharge begins to increase. McDermott and Pilgrim adopted this as their definition of the time of concentration.

The measured times of concentration were regressed against catchment characteristics that included:

• Catchment area
• Main stream length
• Main stream equal area slope
• Main stream average slope
• Catchment shape factor
• Stream slope non-uniformity index
• Vegetation cover
• Median annual rainfall
• Soil type.

Three formulas provided a similar fit to the data with the simple relationship with catchment area ultimately adopted (equation 1).

One of the important implications of the probabilistic rational method approach is that the time of concentration used for design must be calculated using the same formula that was used in the derivation of the runoff coefficients (equation 3).   So, in Victoria (and Eastern NSW and the Wheatbelt of WA), when using the probabilistic rational method to estimate floods in ungauged catchments, it is important to adopt the Pilgrim McDermott formula for the time of concentration and not use any of the many other approaches.

### References

Adams, C. A. (1987) Design flood estimation for ungauged rural catchments in Victoria.  Road Construction Authority, Victoria. (link)

French, R., Pilgrim, D. H. and Laurenson, E. M. (1974) Experimental examination of the rational method for small rural catchments. Civil Engineering Transactions CE16: 95-102.

McDermott, G. E. and Pilgrim, D. H. (1982) Design flood estimation for small catchments in New South Wales.  Department of National Development and Energy.  Australian Water Resources Council Technical Paper No. 73, pp. 233. (link)

Pilgrim, D. H. and McDermott, G. E. (1982) Design floods for small rural catchments in eastern New South Wales. Civil Engineering Transactions.  Institution of Engineers CE24:226-234.