Darcy’s Law

Feb 29, 2016

Environmental Enlightenment #144

When water flows downhill inside an aquifer, from a higher elevation (h1) to a lower elevation (h2) along a distance (d), we know that the a larger difference in the elevations (h1 – h2) will results in a faster flow.

On the other hand, an increase in the horizontal distance between the (h1)elevation to the (h2elevation will interspace more aquifer material (sand, stone, clay or silt) between the start and end points, thus more resistance, resulting in a slower flow.

We represent the flow by a letter “Q” expressed, for example, in gallons per minute running through every square foot of cross section of the aquifer.

We say that the flow Q is directly proportional to the difference in elevation, but is inversely proportional to the distance and we express these relationships in a mathematical formula:

Q  = [ K• (h1 – h2) ] / d   

The dot (•) in the formula indicates multiplication, the slash (/) indicates division and the “K” is a factor that determines the proportionality. The K magnitude is determined by the materials of which the interspacing material is made. We experiment with different materials and find their K value.

We call K, “hydraulic conductivity.” Hydraulic means water and conductivity is the ability to transmit.  Thus, K represents the ability of a material to transmit water

We immediately see that the K will be large for a highly permeable material like sand and gravel. It will be small for dense clay or silt.

In 1856, Henry Darcy, the Engineer of the town of Dijon, in Southern France, investigated the flow of water in a sand filter in connection with the city’s fountains. From his experiments, he developed the above formula which will forever now memorialize his investigative spirit.

When groundwater moves down-hill through the pore space of the aquifer, we name it “porous flow.”  Darcy’s law is operative in porous flow environments.

We need to know the K to enable us calculate groundwater flow. Estimates can be made from knowledge of the geologic material, or from aquifer hydraulic tests.

Here are comparative values of K (expressed in units of foot per day)

Clay                         K =  0.00005

Silty Sand                 K =  0.3

Well Sorted Sands      K =  30

Well Sorted Gravels     K =  3000

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