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|This lab will be conducted on the Mt. Hope River, which borders
Trinity College's field station. Besides learning about stream
discharge, fluvial sediments and flood prediction you will also practice
your spreadsheet skills and will learn simple statistical and fitting
techniques. These techniques will allow us to put our data into
historical context and make predictions about the severity of future
(Image by Scott Smedley)
Stream Discharge Q
a) A very simple channelThe amount of water that flows down a stream per unit time is called discharge and is usually abbreviated by the symbol Q. Before we deal with more realistic looking streams, let's find out how to calculate stream discharge for a rather simple stream geometry. In Figure 1 we assume a stream having a rectangular cross section with a cross-sectional area A = w * d. The water velocity is indicated by the blue arrow and is perpendicular to the cross-sectional area A. It is also assumed to be constant throughout the channel. We then can write for the stream discharge Q:
Q = A * v
or in terms of channel width w and water depth h:
Q = w * d * v
The quantity Q would be called a flux in a physics course, geologists call it discharge, and in both cases Q describes how much water is coming down the stream. In both cases it is important to realize that the cross-sectional area A is perpendicular to the flow of water.
For this simplified channel it would be rather easy to obtain a value for Q:
Measure the width (w) and depth (d) of the channel with a measuring tape, measure the water velocity (v) anywhere in the channel with a flow meter and calculate Q using the relationship Q = w * d * v.
A quick dimensional analysis shows us that discharge has the dimensions of
[Q] = Length * Length * Velocity =
Length * Length * Length / Time = Volume / Time.
The units of discharge are :
[Q] = cubic meters / second, or in standard units
[Q] = cubic feet / second (cfs)
b) A More Realistic Channel GeometryUnfortunately, real life rarely provides rectangular channels with constant water velocities throughout. Real life tends to be a bit more complicated and we will have to take some of these complications into account.
Figure 2a shows a slightly more complicated channel geometry, which already looks rather similar to the real thing. Before we go on let's consider some of the complications that we will face when measuring Q:
How to get the area A.
The channel in Figure 2a has no uniform depth any more, so the product w * d is most likely to give us some bogus number. Sit back for a while and try to decide whether our old equation for area: A = w * d is likely to over- or underestimate the actual cross-sectional area of the stream channel.
Is the velocity really uniform?
Well, most likely not. We will see that velocity changes both across the stream and with depth. We will deal with this problem in a while. Let's focus on the cross-sectional area first.
Approximating the crossectional area A:
As it is shown in Fig. 2.1 we can subdivide our stream channel into a number of rectangular channels similar to the one shown in Fig. 2.1. Each channel has a cross-section of
A(rect. channel) = w * d
and in order to obtain the total cross-sectional area of the stream we add up the areas of all channels as it is shown in Figure 2.3:
A(total) = A(1) + A(2) + ....
Sticklers might agree that the approximation shown in Fig. 2.2 is better than assuming a rectangular channel, but that its accuracy is not good enough. In other words: The area covered by all the boxes does not match exactly the area of the stream. One can avoid this problem by simply making the width w of the rectangular channels smaller. You can easily check for yourself, that a stream that is approximated by twice as many rectangular channels does a better job of getting the crossectional area straight. However, it also means that you have to do twice as many measurements in the field. In practice you will have to decide on a reasonable spacing for your measurements. Otherwise you might want to bring a tent and camp out for a week.
Many of you might have seen this technique before. The mathematicians call this kind of summation a Riemann sum. If you are a stickler and want to be as accurate as it gets you would increase the number of summation terms to arrive at an integral over the cross-sectional area of the stream.
The practice of approximating the stream's cross-sectional area by a number of rectangular channels is equivalent to numerical integration techniques used in computer applications.
c) Getting a Velocity EstimateAs I have mentioned earlier, it is very unlikely that water velocity across our cross-section has a uniform value v. You might have noticed that a stream tends to flow faster near its center and slows down near its banks due to friction between the moving water and the stream bed.
In principle we could use a similar technique as employed above to get a handle on the water velocity, but there are simples ways to arrive at a reasonable estimate of v.
Water velocity is mainly defined by the force of gravity and therefore the slope of the river (water flows downhill, and the steeper the slope the faster it flows), which accelerates the water, and frictional forces, which try to slow it down. As a result the water velocity changes with depth as shown in Fig. 3.1. It is zero at the stream bed, increases as one goes farther away from the bed and decreases again slightly at the water surface.
If you want to know more about this velocity distribution click here.
In real life nobody measures the exact velocity distribution, but approximates it by one mean velocity. This procedure is shown in Figure 3.2. It is possible to estimate the mean velocity by a velocity measurement at a depth 0.6 of the distance from the surface to the bed. Another estimate uses the average of two velocity measurements taken at 0.2 and 0.8 depths.
We will use one velocity measurement, taken at 0.6 depth to approximate the mean velocity of our rectangular channels.
|d) Putting it all
Now we have all the tools to measure discharge for the Mount Hope River. We will stretch a measuring tape across the river and take a series of equally spaced depth and velocity measurements. The equal spacing is not absolutely necessary, but simplifies things because:
A(total) = A(1) + A(2) + ...
= v(1) * w(1) * d(1) + v(2) * w(2) * d(2) + ...
A(total) = v(1) * w * d(1) + v(2) * w * d(2) + ...
= w (v(1) * d(1) + v(2) * d(2) + ... )
The calculations and sums are best done in Excel, though you could do them by hand as well.
e) A Note on Significant Figures
When reporting your results make sure you only report significant figures. Just think for a second (or two) about all the assumptions and uncertainties that go into your result. We approximated the crossectional area by a sum of rectangles and guesstimeted the mean velocity from one (or two) measurements).
On top of that you might have substantial measurement errors: People splashing around, holding the meter stick crookedly...
With all those uncertainties, how many significant figures make sense?
|We are lucky enough that the United States
Geological Survey has been measuring streamflow on the Mt. Hope River for
over 60 years and that the data can be downloaded from the web. The
site that has all the relevant information can be found at:|
USGS Real Time Water data
Go to this site and find the station homepage for the Mt. Hope River. Note, every dot on this map is a gauging station operated by the USGS, but a click on Connecticut lets you choose from a list of, maybe, 30 stations, so it shouldn't be that hard. The station you are looking for is 01121000. By default he site displays the discharge measured for the last 7 days or so. You should see something like the image on the right.
The blue line shows the hourly mean discharge at the station. In the image to the right you can see the effect of a rainstorm on stream discharge. Discharge increases rapidly at the onset of the storm, but takes a few days to decline back to base level.
What could explain this particular shape of the discharge curve?
To download discharge data into Excel set the output format to Tab-separated and click on the get data button again. You will get a simple text file (ASCII) that you can save on your computer and import into Excel (select File > Save As and choose an appropriate file name and location).
The graph also shows the long term (30-year) daily discharge average. This are the red triangles in the graph. How could you decide whether stream flow on the Mt. Hope River is abnormally high / low? For your downloaded dataset you will calculate the weekly discharge average and compare it with these long term daily averages.
|Large numbers of people live on floodplains and can
be affected by flooding such as shown in the images to the right.
The satellite image shows a false color image of the Missouri,
Mississippi and Illinois rivers at St. Louis, MO. If you ever
wondered why Budweiser tastes the way it tastes, this might be one
explanation. The images below gives you an idea what the flood looks
on the ground and makes a pretty good point for flood control and
One way to predict the likelihood of flooding uses a statistical technique called the flood recurrence interval. In this part of the lab exercise you will download long term flood data for the Mt. Hood River, calculate the gage height of a 100-year flood event and determine the portion of our field station that would be affected by such a flood.
To calculate recurrence intervals we have to download the necessary data first. Go back to the USGS website and select peak streamflow from the dropdown menu. You will get a graph (or table) that shows the highest discharge values for any given water year since 1938 (A water year goes from October until September the next year).
To calculate flood recurrence intervals the flood events are ranked by their maximum discharge and the recurrence interval R is determined by the following equation:
R = (N + 1) / M
Where M is the rank of the flooding event (largest flood M = 1, smallest flood M = N) and N is the number of years that are on record. To make that equation a bit clearer, consider the largest flood that occurred during a 50 year long recording period. For this flood event M = 1 and N = 50 (that is constant for all events in that record), and we obtain:
R = (50 + 1) / 1 = 51 years.
A flood recurrence interval of 51 years means that, on average, a flood of that size is to occur approximately twice a century. The figure below shows the flood record for the Housatonic as recorded at Stevenson, CT. Note that the record is only 80 years or so long. How can you estimate a 100-year flood if your record is only 80 years long?
The answer lies in an extrapolation procedure that is explained below. After downloading the necessary data and calculating flood occurrence intervals for each flood, the discharge value for each flood is plotted against the recurrence interval R. This is shown for the Housatonic in the figure below.
Please note that the data are plotted on a semi-logarithmic scale. The recurrence interval is plotted on a logarithmic scale, while discharge is plotted on a linear scale. When plotted that way the discharge data can often be approximated by a straight line. It is then possible to fit the data with a linear fit and extrapolate it all the way to a recurrence interval of 100 years as it is done for the Housatonic data (solid red line). However, the fit is not perfect, and several other extrapolations are possible as indicated by the dashed red lines, which represent some case of worst or best case scenario. This uncertainty allows us only to predict a range of discharges that might occur in a 100-year flood. this range is indicated by a thick red bar. For the Housatonic we estimate that a 100-year flood might have a peak discharge ranging from 80,000 to 100,000 cfs (cubic feet per second).
We can also link discharge to gage height, or how high the water level will rise during a flood. The figure below shows a graph of discharge vs. gage height for the Stevenson station. Again, it is possible to extrapolate from the historical data all the way to peak discharges of 100,000 cfs (red line). Then it is possible to read the expected gage height from the y-axis of the graph. For the Housatonic at Stevenson a 100-year flood would lead to a gage height of approximately 25 - 28 ft. This means that land that is less than 28 ft above the Housatonic river will be affected by a 100 year flood.
Figures 3.1 and 3.2 are from L.B. Leopold, Wolman M.G. and Miller J.P., 1964, Fluvial Processes in Geomorphology, reprinted by Dover Publications, New York
All other images are either originals or linked to their original sources.
Streamflow data is from USGS real time data site indicated above.
This laboratory exercised is based on an idea presented by Jim Welsh and Carolyn Dobler, Gustavus Adolphus College, at the Quantskills Workshop, Northfield 2002.