Floods and Scour

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Floods rank high among the natural events that have devastating effects on peoples’ lives and property. Many low-lying areas are flooded frequently, as a result of high tides, especially when assisted by storm surges, or by heavy rainfall. In Amsterdam, for example, you can find tall poles with dates, such as 1953, inscribed at heights which indicate the depth of flooding in those years. To look up at those inscriptions, far above your head, is a strange experience. In other parts of the world, floods affect millions of people.

In 1953 also, over 200 mm of rain fell on Exmoor in a 24 hour period. Soon afterwards, a wall of water fell upon Lynmouth, a small village at the mouth of the steep and narrow valley of the river Lyn. The loss of life and devastation were great, and one of the responses to the disaster was to confine the river to a deep walled channel. In 2004, a similar event occurred in Boscastle, Cornwall.

Like flowing lava, flowing water is exceedingly hard to keep out. At one tonne per cubic metre, it creates hydrostatic pressure and pressure caused by its momentum. And hydrodynamic forces can lift objects with a forces that rise with a high power of the speed. Put another way, the faster the flow, the bigger the objects that are moved. So flooding creates damage both by the presence of the water, pushing structures and ruining, and the speed of the water, damaging structures and vehicles in various ways. Vehicles may be lifted and carried away, and they may be damaged by other objects being carried by the water.

Other pages in this web-site describe some of the bridges over the river Severn. From Gloucester up to Coalbrookdale and beyond, the river Severn causes misery to people who live near it. The residents of Alney Island, near Gloucester, were told in August 2001 that protection from the river Severn would be too expensive to implement.

And among the problems that confront the designer of structures, floods rank high in importance, in areas where they occur.

Scour

In fact, even the normal flow of river water past bridge piers can generate scour which can bring down a bridge. The presence of the piers changes the flow, producing acceleration and turbulence. The lifting and carrying power of a fluid increase as a high power of the speed. The ancient Romans knew about this, and took precautions. Foundations need to penetrate to secure ground, and a pavement around piers can help to protect the bed. You can often see the results of scour around a post or a boulder on a sandy beach, after the tide has gone out. The next diagrams, which are sections at right angles to the flow, show the general effect. The photographs, taken on a beach, show how the presence of objects, even small ones, can affect the flow of water quite strongly, which in turn disturbs the distribution of the sand.

The picture at left shows another pebble on a beach. The red graph shows the brightness variation across the picture, while the white graph shows the calculated brightness for a particular profile of the depression in the sand. The presence of the stone and its shadow cause the profile to be meaningless in that region.

Furthermore, because of turbulence, the pressure on the bridge fluctuates with a wide frequency spectrum; people on a bridge that is nearly submerged report feeling strong vibrations.

In times of spate, the speed of a river can increase greatly, and in times of flood, the width and depth of the river can change dramatically.

We can see from the picture above that this flood is far from being the worst that has happened. This picture shows the line of higher floods in the past. Remembering that the water flows fastest near the surface, we see that the bridge is a very substantial obstacle in time of flood.

The next diagram shows the effect of doubling the water level in valleys of two different depths. The coloured layers represent water, mud and solid rock.

In the shallow valley, the effects on the bridge are more pronounced.

The Loyn bridge spans the river Lune in three arches. The simple design gives a monumental effect. The piers are extended by pointed cutwaters, to steer the water around them. The line of the cutwaters reaches the top of the bridge, where they provide refuges for pedestrians when vehicles cross the bridge.  

Bridges across flood plains often include flood arches, which in normal times carry the route over the dry plain, but in time of flood allow the water to flow through more freely than the river arches alone. Here are some examples.

In the pages about Severn bridges you can find many examples of bridges which have spans which normally cross fields populated with crops or animals, or even parts of river-side towns. These spans play their part when the flood-plain lives up to its name, helping to allow the water to pass under the road or railway.

Unfortunately, even these measures often don’t prevent the occasional flooding of people’s homes and workplaces. One problem is the statistical distribution of magnitudes, whether they be speeds of winds, depths of floods, or magnitudes of earthquakes. Perhaps it would be possible to build a structure to withstand any imaginable event, but the cost would be completely uneconomic. The practicable course is to build for a magnitude of event that is expected on average every N years, where N is determined by some kind of authority, using some kind of economic calculation. For the same reason, it is impossible to purchase and maintain emergency equipment which will cope with any event that will ever happen.

Even this is does not have completely calculable consequences, because random events, however unusual, can happen at any time. If events are truly random, the time between them follows an exponential formula, which means that if the future is divided into uniform periods, the next event is more likely to occur in the next period than in any other. This often leads people to conclude that events are connected when they are not.

These lines represent the results of a calculation using pseudo-random numbers.  These numbers are not "truly random", but they pass a number of tests that are considered to indicate a degree of randomness. The horizontal scale can represent space or time. If you had been asked to draw some randomly spaced lines, would you have drawn something like this?

In the next diagram, the events have random sizes as well as random times.  Bunching of rarer larger events is apparent, though not as close as the bunching of the commoner smaller events. Note also that the larger events show up more on the diagram than the more common smaller ones.

There is no "law of averages" that says that when tossing a coin, a run of heads is more likely to be followed by a tail than by another head. If you are playing a lottery, you might as well choose the numbers 1 2 3 4 5 6 as any other, as all combinations are equally likely.

The next picture shows a typical frequency distribution of 10000 events, with magnitude plotted horizontally. The largest event, marked with a red arrow, is far bigger than the mean, marked with a blue arrow.

If we take another 10000 events, we will find that the mean is very similar, but the maximum will probably be different. If we take many sets of 10000 events, we will find that the maxima produce a wide distribution. This is very unfortunate in many practical situations. Very often we are not interested in the average: we want to know the extreme. The rolling, yawing and pitching ship may make us sea-sick, but the real quantity of interest is the probability that the ship will turn turtle. The level of a river may excite interest from the local newspaper, but the real quantity of interest is the probability that it will overtop the banks. The general accuracy of aircraft landing systems is of course important, but the real quantity of interest is the probability that it will reach the ground on the runway, and not before it or off to one side. It is so often the extreme that we care about, even though it is so hard to predict.

This is a very serious problem when designing components, because the buyer almost always wants to know the lowest possible strength that mechanical parts will have, or the longest rise-time of integrated circuits, and so on. Yet these are precisely the things that are difficult to measure, whether the distribution is exponential or normal (Gaussian). The extrema of distributions are not "good" parameters. The "good" parameters are quantities such as mean and the the standard deviation from the mean.

The next pictures show the previous type of plot, but with a logarithmic vertical scale, showing that the more events we observe, the larger the biggest event becomes. We have take many sets of two events, many sets of ten events, many sets of a hundred events, and so on. The black histograms are the distributions of the actual data, while the red ones record the distributions of the maxima. Of course, for two events and ten events, there isn’t much of a black histogram. At the other extreme, 5000 sets of 100000 events and 1000000 events would have taken too long to compute.

What we see is that however many data we take, the spread of the maxima never becomes significantly smaller. What does change is that with more events, the average of the extremes increases. The maxima are shown by red arrows in the left-hand plots and by red histograms in the right-hand plots.

So even if nothing changes, the deepest recorded flood will on average become bigger over the years, though there may be long periods when no new record is set. The extreme value of a distribution is in fact a poor guide to the trend. To see properly what is going on we should look at the mean or the standard deviation, or some other function that uses all the data, rather than one value. Note how the range of the histogram does not decrease much as we add more events. The mean and standard deviation are known more and more accurately as we add more data, but the the extremes never become good variables, unless the distribution is strictly bounded. An example of a bounded distribution would be the timing of the first goal in a football match, which can never happen before the start of the game.

For a normal distribution also, the range is a poor measure of the size of the distribution. The reason that the maximum is ill defined is found by considering 1000 sets of 1000 events, and the same events in one group of a million. Clearly, one of the groups of 1000 must include the biggest event in the million. If you look carefully at the fifth histogram you will see signs of artefacts, always a danger with rare events in statistical calculations.

Because of the weak but real dependence of the extremes on the number of data, people cannot build flood defences that are never breached. They can only build for an estimated probability per year, per decade, or per century. The cost of building has to be somehow balanced against the cost and distress of a flood. And even if you make the "50-year flood" defence, you might still get a "once-per-100-year flood" the very next year. Not very likely, but possible, because all things being equal, any year is as likely as any other. Looking at it another way, if you build fifty structures that take care of fifty different phenomena, and you build for the fifty year probability, you are not unlikely to get a problem with one of them in any one year. The unlikely can happen, and does happen. So blaming the authorities or the designers, while it may be understandable, may not actually be justified.

The problem with distributions arises in the testing of integrated circuits for computers. Fast logic pulses are not the tidy rectangles of the text book: they are in some cases almost sinusoids, and the timing of the decision between 1 and 0 is crucial. The ICs are built to a specification which includes the mean timing and a maximum or a minimum, which must never be breached. The problem is exactly as described earlier – it’s easy enough to find the mean and standard deviation, but hard to find the extremes. If the IC has to run for several years without error, how can you test it? If you test a new design for several years before releasing it, your competitors may get ahead. You can, of course, assume that the distribution is normal, and calculate the range for a given number of data from the mean and standard deviation. The problem there is that the value you need to measure is extremely sensitive to the shape of the tails of the distribution.

The pictures below were taken during flooding of the river Severn near Gloucester.  Some older pictures are included for comparison with more normal conditions. The first two show the road between Westgate and Maisemore. One picture shows the camber of the road: the second includes the depth indicating posts, showing over two feet/0.6 m of water on the road. It also shows clear standing waves as the water flows off the road at left. The third and fourth, taken a few days later, show about eighteen inches less water, and a cyclist making bow waves. Strangely, the river Severn, the proximate cause of the problem, is on the left, yet the flood water is coming from flooded fields on the right. They received the water from the river upstream.

Here are some pictures of bridges and their flood spans in the Severn flood plain.  They all have to cope with flooding.

This isn’t a flood, but the picture shows that when a channel widens, making the flow slower, flotsam is likely to be deposited. Once this has happened, more rubbish is likely to be stopped. Why do you think the channel was made wider at the bridge, thus increasing the cost of the bridge? Much of the rubbish that is seen in this type of stream in a town has been thrown in by people. Objects as large as supermarket trolleys may sometimes be seen at intervals of only a few hundred metres.

Links

Statistics of distributions of events

Report on scour

Transitory waves

Flood folklore

Waldo-Hancock bridge scour

Bridge failures including scour effect

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