# Time of concentration: Bransby Williams formula

The Bransby Williams formula for the time of concentration dates from 1922 when George Bransby Williams published a paper “Flood discharge and dimensions of spillways in India“.  This refers back to an earlier paper by “Mr Chamier” who noted that the:

The maximum flood will generally be produced by the greatest possible rainfall of a duration corresponding to the… time taken for the water to reach the point of discharge from the most distance point of the watershed.

Bransby Williams provides a formula for this “time taken…”, what we would now call the time of concentration.  In the original paper this is provided for a circular catchment as:

$a = \sqrt{t_c^5h} \qquad \qquad \mathrm{(1)}$

Where:

• $a$ is the catchment area in square miles
• $t_c$ is the time of concentration
• $h$ is the average number of feet fall per 100 ft from the edge of the watershed to the outfall.

For a non-circular catchment, $t_c$ is increased by:

$\frac{l}{d} \qquad \qquad \mathrm{(2)}$

Where:

• $l$ is the greatest distance from the edge of the watershed to the outfall
• $d$ is the diameter of a circular catchment of the same area.

It is possible to combine equations (1) and (2) to derive the the usual Bransby Williams formula as presented in, for example, the Handbook of Hydrology (Equation 9.4.2, page 9.16).

For a circular catchment, area and diameter are related:

$a = \frac{1}{4} \pi d^2 \qquad \qquad \mathrm{(3)}$

Therefore:

$d = \frac{2}{\sqrt{\pi}} a^{0.5} \qquad \qquad \mathrm{(4)}$

Now, in equation (1), it is more common to use $S$, slope (ft/ft, or m/m) rather than $h$.

$h = 100 S\qquad \qquad \mathrm{(5)}$

From (1)

$a = \sqrt{t_c^5 100 S}\qquad \qquad \mathrm{(6)}$

$a^2 = 100St_c^5\qquad \qquad \mathrm{(7)}$

$t_c = 100^{-0.2}S^{-0.2}a^{0.4} \qquad \qquad \mathrm{(8)}$

For a non-circular catchment, multiply equation (8) by equation (2) and use equation (4) for $d$.

$t_c = \frac{\sqrt{\pi}}{2 \times 100^{0.2} } l S^{-0.2}a^{-0.5}a^{0.4}\qquad \qquad \mathrm{(9)}$

$t_c = \frac{\sqrt{\pi}}{2 \times 100^{0.2} } l S^{-0.2}a^{-0.1}\qquad \qquad \mathrm{(10)}$

$t_c = 0.3528 \; l S^{-0.2}a^{-0.1} \qquad \mathrm{(hours)} \qquad \qquad \mathrm{(11)}$

$t_c = 21.169 \; l S^{-0.2}a^{-0.1} \qquad \mathrm{(minutes)} \qquad \qquad \mathrm{(12)}$

In equation (12), area, a, is in square-miles and stream length, l, is in miles.

If area is in square-km, stream length in is km and slope in m/m then the formula is:

$t_c = 14.467 \; l S^{-0.2}a^{-0.1} \qquad \mathrm{(minutes)} \qquad \qquad \mathrm{(13)}$

There are Bransby Williams calculators on the Internet e.g. here.

Bransby Williams doesn’t say how this formula was derived and doesn’t present any data.  He does note that: “This formula gives a somewhat more rapid concentration than actually takes place in most instances”.  He also states that his interest is Eastern India where: “a mean annual rainfall of less than 30 in. does not come into consideration”.

Despite the lack of supporting data and narrow geographic focus, the Bransby Williams formula has been widely used.  In the 1977 version of Australian Rainfall and Runoff it was recommended for the whole of Australia “until a better formula is developed” (equation 7.2, page 73).

The 1987 version of ARR (Book IV, Section 1.2) acknowledges that there is no: “universal method…applicable for the whole of Australia” but does describe the Bransby Williams formula as an “arbitrary but reasonable approach” where these is no specific formula available for a particular region. This recommendation was based on the work of French et al. (1974) who tested eight formulae for estimating the time of concentration and showed the Bransby Williams formula was least biased and straightforward to calculate.

The formula provided in ARR 1987 is:

$t_c = \frac{58L}{A^0.1 S_e^{0.2}}$

Where:

• $S_e$ is the equal area slope of the main stream projected to the catchment divide (m/km)
• $L$ is the main stream length measured to the catchment divide (km)
• $A$ is the area of the catchment (km2)

The equal area slope is: “the slope of a line drawn on a profile of a stream such that the line passes through the outlet and has the same area under it as the stream profile”.  (I’ve added a post on the calculation of the equal area slope).

ARR 1987 acknowledges that Bransby Williams used the average slope rather than the equal area slope but preferred the equal area slope and noted the reasonable fit to data in French et al. (1974), (although French et al. used the average slope in their analysis).   Other work suggested that the Bransby Williams formula wasn’t satisfactory as a measure of the characteristic response time of a catchment because it gave inconsistent values particularly when used with the equal area slope (McDermott and Pilgrim, 1982, p 33; Pilgrim and McDermott, 1982). The accuracy of the Bransby Williams formula was also questioned in the UK (Berans, 1979).  Despite this, the Bransby Williams formula was recommended, in ARR1987, for the Northern Territory and northern and western semi-arid areas.

As of 2017, it is 30 years since ARR 1987 and close to 100 years since the publication of the Bransby Williams formula, yet the method is still occasionally used.  With the latest revision of the Australian Rainfall and Runoff, a new regional tool (the RFFE) has been developed which should provide better estimates of design floods without the need to calculate the time of concentration.

### References

Berans, M. (1979) The Bransby Williams formula an evaluation.  Procedings of the Institution of Civil Engineers. 66(2):293-299 (https://doi.org/10.1680/iicep.1979.2355).  Also see the discussion (https://doi.org/10.1680/iicep.1980.2512).

Bransby Williams, G. (1922). Flood discharge and the dimensions of spillways in India.  The Engineer (London) 121: 321-322. (link)

Chamier (1891) Minutes of Proceedings of the Institution of Civil Engineers Vol. cxxxiv., page 313 (as cited by Bransby Williams).

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.  Research Project No. 78/104.  Australian Water Resources Council Technical Paper No. 73.  Department of National Development and Energy.  Australian Government Publishing Services (link)

Pilgrim, D. H. and Cordery, I. (1993) Flood runoff. In: Maidment, D. (ed) Handbook of Hydrology. McGraw-Hill.

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

Pattison, A. (1977) Australian Rainfall and Runoff: flood design and analysis.  The Institution of Engineers, Australia.  ISBN 0 85825 077 2