Oakland county, where I’m running for drain commissioner, has ‘flashy rivers’ and river erosion, a pair of problems I hope to solve using weir dams. Currently, the rivers go from being dry, or near dry, to flooding and over-running roads whenever it rains significantly. The current drain commissioner has no engineering background; he has attempted to solve the flood problem by dredging (needed every year) and by getting rid of the few weir dams we have. The result is we have no natural fish — they’re washed out in the rain — and we have lots of erosion and pollution.

Weirs have been used for flood control for centuries. They are put in ditches and rivers as a restriction so the water does not immediately flood the lowest spots, but flows down at a controlled rate. Done right, fish have no problem swimming upstream.

To show how effective a weir is to keep water where you wanted it, some weeks ago, I posed the following math/ engineering problem: show that, if a 2 to 3 foot dam is used to double the depth of water in a river or drain, this has the effect of increasing the residence time by approximately 2.8 times and reducing the speed of the flow by the same factor, 2.8. Below is my solution.

Lets assume the shape of the bottom of the drain is a parabola, e.g. y = x^{2} (it works with all parabolas). Now use integral calculus to calculate how the volume of water in a length of drain is affected by water height: V =2XY- ∫idx = 2XY- 2/3X^{3 }= 4/3 Y√Y. Here, capital Y is the height of water in the drain, and capital X is the distance of the water edge from the drain centerline; X=√Y. If you double the height Y, in parabolic drain, you find you increase the volume of water, V by 2√2, or about 2.8.

To find how this affects residence time and velocity, note that the dam does not significantly affect the volumetric rate of water flow. Let’s call this volumetric flow Q, and measure it in gallons/hr. Let’s now measure the volume per mile of drain, V above, in gallons. From conservation of mass, we find that the residence time in the drain (hours/mile of drain) is V/Q and that the speed (miles per hour) is Q/V. Increasing V by 2.8, is thus found to increase the residence time by 2.8 and decrease the average speed of flow by the same factor, 2.8.

Why is this important? The slower the flow, the less soil erosion. Also, the more residence time and the more oxygenation. The turbulence of water flowing over the dam tens to provide oxygenation, and this is good for fish and __for__ bio-remediation. The use of a weir instead of a solid dam causes the lakes to lower somewhat between rains. This is good for the lake, and makes rivers less flashy as the lakes now have capacity to fill when there is a rain. Between rains, the lakes get rid of their sludge and the continuous flow provides a better waterway for fish. During major rains, the whole dam overflow, but the dams still help by diverting the floodwaters where you want them instead of having them run too quickly downstream. Adding to the efficacy, I’d like to add nearby wetlands perhaps covered by low bicycle paths and floodible picnic areas just upstream of the dams. This is an aspect of beautiful flood control that I’ve seen as a professor at Michigan State: there is a dam like this right in the middle of campus. The fish do fine, the water does fine, and it’s a rather attractive flood solution, besides.

Dr. Robert E. Buxbaum, May 11-Sept 20, 2016. Elect me water commissioner. Here are some related thoughts. And, for those who’d like a challenge, here’s another dam flow problem.

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