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In Saskatchewan we have a man made lake (Lake Diefenbaker) and dam (Gardiner Dam). If Gardiner Dam blew up how long would it take for the lake water to hit Saskatoon (whose center is roughly 120km away via approximate tracing on Google Maps).

My friend who worked for the water authority said that they estimated 30 minutes and that the average speed of the water would be 200km/hr.

To me (uneducated, and likely incorrect) - 200km/hr seems like it's too fast and even with the force of this massive body of water behind it.. that's like free fall speeds, across land.

In general - what's the fastest reasonable speed that water can travel in a situation like this? If the Gardiner Dam were to be removed how long would the people of Saskatoon have to prepare before being wiped out by a massive wall of water?

Asking for a friend.

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    $\begingroup$ The speed of water $u$ through a restriction, is a function of the driving pressure $P$ and a constant $k$ depending on the restriction geometry, $u=k \sqrt P$. In principle, this is only limited by the speed of sound in water, around 1400 m/s. $\endgroup$
    – RC_23
    Oct 6, 2022 at 19:23
  • $\begingroup$ Yes but will it be there in 30 minutes or less :)? $\endgroup$ Oct 6, 2022 at 19:41
  • $\begingroup$ Dam collapses have happened before. It should be possible to do an internet search whereby you can calculate data for water speed vs. average elevation drop for past floods. There may even be a correlation that has already been developed for this problem. $\endgroup$ Oct 6, 2022 at 19:43
  • $\begingroup$ Looking at other dam failures, it seems 50 km/hr is more reasonable. It depends on many factors, but there are cases that seems to range between 30 and 40 mph. $\endgroup$
    – UVphoton
    Oct 6, 2022 at 23:46

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A proper answer to this question requires a detailed expert study, which apparently does exist. According to page 489 of some Saskatchewan Legislature Committee minutes, the flood from a Gardiner Dam failure would take 42 hours to reach Saskatoon and the water levels in the worst-case would peak 15 metres above the city's Broadway Bridge, i.e. roughly 45 m above the river's normal level!

Since this is a physics site, however, it is still fun to make some rough estimates, especially since similar questions have been asked but not answered in detail: "How fast does the water travel down river when the discharge gates of a large dam are opened?" and "Is there a quick way to roughly estimate how quickly a flood will move downstream?".

A flow speed of 200 km/h is high but not completely ridiculous for water at the foot of the Gardiner dam after a sudden complete breach. From Bernoulli's Principle, a crude estimate for the initial flow velocity would be $$ v_{initial}=\sqrt{2gh_{dam}}=128\,km/h$$ where $g=9.81\,m/s^2$ is the acceleration due to gravity and $h_{dam}=64\,m$ is the Gardiner Dam height.

The water should, however, quickly slow down. For dam breaks studied in "A simple method to estimate inundation due to dam break", the fastest arrival time of the flood peak 100 km downstream appears to be about 7 hours. In the 1923 Gleno dam failure, the water took about 45 minutes to sweep down a 21 km long , very steep valley with a vertical drop of 1250 m (i.e. about 30 km/h). Similarly in the 1976 Teton Dam failure, it took about 8 minutes for a "50 to 75 ft wall-of-water" to travel 2.5 miles down the steep walled canyon (i.e. again about 30 km/h). The South Saskatchewan River has a much more gentle slope, dropping only about 30 m as it flows the 120 km from the dam to Saskatoon, so we'd expect the flood to move more slowly. After the Teton Dam flood left the canyon it slowed down and took 22 hours to travel 112 miles, for an average speed of about 8 km/h.

We can also try to roughly estimate the speed ($V$) of the flood waters using the Gauckler–Manning formula for flow in an open channel:

$$V = \frac{k}{n} {R_h}^{2/3} \, S^{1/2}$$

where $k$ is a unit conversion unit equal to one if working in SI units, and $S$ is the slope of the channel. $R_h = A/P$ is the hydraulic radius of a channel with wet cross-sectional area $A$ and wetted perimeter $P$. The Gauckler–Manning Coefficient $n$ parameterizes the flow resistance of the channel, i.e the water flows faster for smaller $n$. Values can range from $n=0.01$ for a smooth straight concrete channel to $n=0.2$ for a floodplain in summer with dense willows or similar bush.

To do this properly would (at the very least) require integrating the Gauckler–Manning formula along the river, taking into account the local geography to continuously evaluate $R_h$ and $n$. Instead, we will make some very crude order-of-magnitude estimates.

The Gardiner Dam is 5000 m wide and 64 m tall, so if it failed completely we'd at first have $A=320000\,\textrm{m}^2$ and $P=5128\,\textrm{m}$. As the water spread and flowed though the downstream valley and floodplains, $A$ and $P$ would constantly change. Looking at the local topography, perhaps a flood width of 15000 m and height of 21 m might be a plausible guess. For $n$, a guess of 0.06 might be reasonable, e.g "light brush and trees, in summer", but this could be off by a factor of two in either direction.

These values give a flow velocity of $V=7\,\textrm{km/h}$, so the flood would arrive in Saskatoon after about 16 hours. Given our very large uncertainties, this is consistent with (possibly optimistic) official estimate of 42 hours. The first response of the river, however, might happen sooner.

When the dam initially breaks, the initial burst of water should generate a wave in the river. This can travel faster than the water flow, but since it is a perturbation of the water already in the river, it can't be very high. A shallow-water gravity wave has a velocity $v=\sqrt{gd}$, where $d$ is the depth of the river. The monitored depth of the South Saskatchewan River at Saskatoon is 2-3 m, which seems consistent with the 11 foot height of the Saskatoon Weir. If we assume $d=3\,\textrm{m}$ as the average pre-flood depth of the river from the dam to Saskatoon, this wave will propagate at 20 km/h and will arrive in 6 hours.

It is important to emphasize that all these estimates are very crude with very large uncertainties. They should not be trusted for making risk assessments of life-threatening floods. If I were you, I'd want to track down that expert report.

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