Fluid Dynamics: Darcy's Law
Fluid Dynamics: Darcy's Law
Kevin Shump

1. Introduction
2. Procedure
3. Data
4. Errors
5. Conclusion
6. Bibliography
Introduction

In 1856, a hydraulic engineer named Henri Darcy, from the city of Dijon, France developed the first documented apparatus for observing the movement of water through a porous medium. He found that the water flow through his apparatus was proportional to the difference in the height of the water between the two ends of the apparatus, which, in turn, is inversely proportional to the length of the flow path. Also, Darcy discovered that the quantity of flow depends on the nature of the porous medium. However, before Darcy made these discoveries, there were two other scientists, Gotthilf Hagen (1839) and Jean Louis Poiseulle (1846), who discovered that the rate of flow of water through capillary tubes is proportional to the hydraulic gradient. From his experiment, Darcy was able to derive a formula through which the rate of flow could be calculated. This became known as Darcy’s Law:

Q=KiA

Where
Q = rate of flow (cm^3/s)
K = hydraulic conductivity of the porous medium (cm/sec)
A = area normal to the direction of flow (cm^2)
TOC

Procedure
In this experiment I needed the following materials:
• A sand box model
• Tubing with which to put water in and take water out of the model
• A water reservoir
• A waste water container
• Colored dye
• A syringe
• A stopwatch
• Ample amounts of water
To run this experiment effectively, I first had to set up the experiment so that the water reservoir was at a higher elevation (head) than the sand box, which was at a higher elevation than the waste container. After setting up the experiment, I needed to siphon the water out of the reservoir to get it flowing into the apparatus so the sand box would be fully saturated before I began my testing. It was critical to saturate the model first because the dye would not flow horizontally through dry sand. After the model was fully saturated, I injected colored dye into the model from the syringe into the second well from the left on the model. Before injecting the water however, I needed to set a distance over which I would measure the rate of water flow. I used blue painter’s tape to indicate the starting and finishing points for the water. The distance between tape edges was 10.4 cm. After establishing a field with which to work, it was time to inject the colored water into the sand box. 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.

TOC

Data

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

Next, we must calculate the hydraulic conductivity. To do this we will use the equation vn/i (seepage velocity*porosity/flow gradient).
Table 4

 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.

TOC

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.

TOC

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.

TOC

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)

Ground-Water Hydraulics – U.S. Department of the Interior (U.S. Government Printing Office, Washington D.C. 1972)

TOC

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.

TOC

Updated 6 June, 2006