To ensure that the colored water was going to flow in the correct direction, I maintained the water level in the main reservoir such that water continued to flow through the sand box from left to right. For that to occur, the water level on the left needed to be higher than that of the right. The difference in these heights, divided by the distance between them, is referred to as the hydraulic gradient. For this experiment, the independent variable is the hydraulic gradient, and the dependent variable is time. The purpose of the experiment is to analyze how changes in the hydraulic gradient affect the time required for water to flow a set distance. Moving right along, after injecting the colored dye into the well, I had to wait for the water to flow over the starting line so I could begin the stopwatch. Once it did cross the line, I measured the time it took to move from the starting line 10.4 cm to the right to the finish line. After it crossed the finish line, I injected clean water into the same well in which I had previously injected colored water to clear the dye out and make way for more colored water. I ran the test in the same fashion with different hydraulic gradients seven additional times.
Moving right along, after injecting the colored dye into the well, I had to wait for the water to flow over the starting line so I could begin the stop watch. Once it did cross the line, I measured the time it took to move from the starting line 10.4 cm to the right to the finish line. After it crossed the finish line, I injected clean water into the same well in which I had previously injected colored water to clear the dye out and make way for more colored water. I ran the test in the same fashion with different hydraulic gradients seven additional times.
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Data
My data were as follows (organized from least head difference [Change Head] to most head difference):
Table 1
Trial | Head Up (mm) | Head Down (mm) | Change Head (mm) | Elapsed Time (min) | |
3 | 28 | 55 | 27 | 17.15 | |
2 | 28 | 57 | 29 | 13.34 | |
4 | 30 | 79 | 49 | 10.47 | |
5 | 32 | 104 | 72 | 8.96 | |
6 | 32 | 126 | 94 | 8.13 | |
7 | 33 | 159 | 126 | 6.87 | |
8 | 33 | 184 | 151 | 6.1 | |
1 | 33 | 190 | 157 | 5.33 |
The head up is the water level from the top of the apparatus on the left hand side, and the head down is the water level on the right hand side of the box (think upstream and downstream). From that I could calculate the Change in Head (|head up – head down|). The Change in Head divided by the distance between the upgradient and downgrdient measureing points equals the hydraulic gradient. As described above, I recorded the time it took for the water to flow from the starting line to the finish. Since Darcy’s Law is Q=KiA, we can divide both sides of the equation by A to get Q/A=Ki because A (area) in my experiment is a constant. Ki, (cm/s) is often referred to as the Darcy Velocity. Darcy velocity is the rate at which water will flow through a medium. The problem with this, however, is that it does not take into account that water cannot actually flow through a grain of sand, it must flow around it. For example, in a river, if there is a large rock sitting in the middle, the water will move around the rock rather than flow through it. This is the same principle on a different scale. Because of this, Darcy Velocity is not an accurate measurement of seepage velocity. It must be converted. In order to check that our hydraulic conductivity from the experiment is correct, we must convert the equation so it is calculates the actual velocity, as opposed to the Darcy Velocity. To do this, we need to divide both sides of the equation by the porosity. Porosity is the volume of all the open space (pores) in a material divided by the volume of the whole. A typical porosity for clean sand (which is the medium in question) is 25%, which is usually signified as n. So, now we must divide the right side of the equation by n to get the seepage velocity which ends up being v=(Ki)/n.
The seepage velocity is as follows:
Table 2
Trial | Seepage Velocity (cm/s) |
3 | 0.0404 |
2 | 0.052 |
4 | 0.0662 |
5 | 0.0774 |
6 | 0.0853 |
7 | 0.1009 |
8 | 0.1137 |
1 | 0.1301 |
0 | 0.1137 |
As a check on the accuracy of my experiment, I decided to convert Darcy’s law to calculate the hydraulic conductivity (K) of the sand in the sand box. To do this, we can multiply both sides of the equation by n, so we have vn=Ki. Now, in order to isolate K, the variable which we are trying to check, we need to divide both sides of the equation by i, so (vn)/i=K.
According to the Ground Water Manual published by the Department of the Interior, the hydraulic conductivity or clean sand should fall somewhere within the realm of 50 – 1500 feet per day. Unfortunately, the units used by Henri Darcy and the Department of the Interior are in a bit of a conflict with each other so some unit conversion will need to occur. Basically, we will need to convert cm/s into ft/day:
(cm/s)(1in/2.54cm)(1ft/12in)(60s/min)(60min/hr)(24hr/day) = ((60)(60)(24))/((2.54)(12))
So, from that equation, we get 2834.646 feet per day for each centimeter per second of flow.
Now that we have the rate of flow in feet per day, we will need to find the flow gradient, which is Change in Head/(head distance). The flow gradients have been calculated below with the head distance (between measuring points) equal to 52.2cm:
Table 3
Trial | Gradient |
3 | 0.052 |
2 | 0.056 |
4 | 0.094 |
5 | 0.138 |
6 | 0.18 |
7 | 0.241 |
8 | 0.289 |
1 | 0.301 |
0 | 0.289 |
Trial | Hydraulic Conductivity (K) (cm/s) |
3 | 0.1954 |
2 | 0.234 |
4 | 0.1764 |
5 | 0.1403 |
6 | 0.1185 |
7 | 0.1045 |
8 | 0.0983 |
1 | 0.1082 |
0 | 0.0983 |
In order to get one value for the hydraulic conductivity, we will next average the measurements for the hydraulic conductivity which comes out to be 0.141526cm/s. Finally, we can find the hydraulic conductivity of the entire experiment in feet per day by multiplying 0.141526 by 2834.646 which comes out to be 401.1749. This is near the middle of the range of hydraulic conductivity values provided by the Department of Interior. This suggests my measurements were reasonably accurate.
This graph compares the flow velocity (Darcy Velocity) versus the hydraulic gradient. As is seen in the graph, the higher the difference in the hydraulic gradient, the more quickly the water will flow. Also, when plotted on a graph, the data forms almost a perfect line (with the exception of a couple unruly data points) which is what Darcy’s Law formula suggests we should end up with.
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Errors
During the lab, I encountered some difficulties that could’ve contributed to the gathering of less accurate data. One major difficulty was keeping the hydraulic gradient at a constant measurement. Ideally, to keep the head gradients constant, we would need a water source to fill the water reservoir at the same rate that water was flowing out of the reservoir. Unfortunately, that equipment was not available so I had to just try to keep the water level somewhat constant throughout the experiment.
Another difficulty was that it was not easy to determine exactly when the water crossed the starting and finish lines. There were some judgment errors there because, as one could imagine, water tends to move rather slowly through the sand, so it was difficult to tell exactly when to start and to stop the stop watch.
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Conclusion
After running the experiment several times, and after several tedious calculations, we can conclude that the experiment was a success. The results mirrored those of Henri Darcy and the Department of the Interior, so, assuming they were correct to begin with, I can safely say that our experiment and calculations were done correctly. Simply looking at Darcy’s Law, you can see a direct relationship with the hydraulic gradient and the flow rate, so one could quickly see that as the hydraulic gradient increased, the flow rate would increase assuming the area and hydraulic conductivity stayed constant. The experiment was a success.
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Bibliography
Ground Water Manual – U.S. Department of the Interior (United States Government Printing office, Denver CO, 1981)
Applied Hydrogeology – C.W. Fetter (Merril Publishing Co., Columbus OH, 1988)
Groundwater – R. Allan Freeze/John A. Cherry (Prentice-Hall Inc., Englewood Cliffs NJ, 1979)
http://www.haestad.com/library/books/awdm/online/wwhelp/wwhimpl/java/html/wwhelp.htm
Ground-Water Hydraulics – U.S. Department of the Interior (U.S. Government Printing Office, Washington D.C. 1972)
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Massachusetts Institute of Technology - A large database on how water flows in general.
Absolute Astronomy - An encyclopedia entry on Darcy's Law and how it works.
PSI Gate - A resource with different links to various websites that have a huge quantity of information.
Groundwater and Geologic Process(PDF) - Very helpful in understanding some of the more basic principles of how groundwater behaves in different situations.
Oklahoma State UniversityA well put together site that gives abundant information on Darcy's law with graphics and formulas.
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