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Darcy's Law

In 1856 Henri Darcy published in an appendix to his book Italiass Les Fontaines Publiques de la Ville de Dijon, his results of an experiment he conducted for the city of Dijon, France. From the generalized results of his experiments we have the empiricism known as Darcy's Law which describes the rate of flow through filter sands and can be expressed as

where

q     = volume of water crossing a unit cross sectional area per unit time  [LT]

K = factor of proportionality or

(Appendix A)[LT]

= water heights above a reference level measured by manometers

which terminate above and below the sand column [L]

L = length of the flow path [L]

Using an apparatus similar to the one in Figure 1, Darcy filled the circular tube with sand with an input tube at the top and a matching output tube at the bottom.

  
Figure 1: Experimental apparatus to illustrate Darcy's Law

Manometers, tubes used to determine the elevation of water level, or the measurement of the hydraulic head are fitted into the cylinder at elevations and . The elevations of the fluid inside the manometers are and with their difference in elevation equal to and the length between them . Water is directed into the input tube until all the pores are filled with water and consequently the inflow rate Q is equal to the outflow rate. If we let A be a cross section of the cylinder with dimensions and since the flow rate Q has dimensions , then v, the specific discharge throughout the cylinder defined as has dimensions . Note that we are calculating the specific discharge on a macroscopic scale which is easily measured as opposed to the microscopic view of the almost impossible measurement of the individual velocities between the grains of the porous medium. This type of ``sand-tank'' model is still being used today since Darcy's Law is still based on empirical evidence although numerous attempts have been made to derive his law from known physical laws.

Darcy's experiments showed conclusive evidence that the flow of water is directly related to , the slope of the hydraulic gradient. K, although a constant of proportionality actually depends on the permeability of the soil as well as the fluid flowing through the soil. Darcy was able to show that v is directly proportional to when is held constant and inversely proportional to when is held constant. [8]
If we define and with ,
we now have

The negative in the equation implies that the groundwater flows in the direction of head loss. Although Darcy's original experiments were conducted with the tube in a vertical position such that the flow was perpendicular to the variable z = 0, the law holds for groundwater flow in any direction even if the flow is forced upward from the datum z = 0 against gravitational forces. [2]

Although there are some theoretical and practical limitations to the use of Darcy's Law, it ``provides an accurate description of the flow of groundwater in almost all hydrogeological environments. In general, Darcy's Law holds for

  1. for saturated flow and for unsaturated flow
  2. for steady-state flow and for transient flow
  3. for flow in aquifers and for flow in aquitards
  4. for flow in homogeneous systems in anisotropic media
  5. flow in both rocks and granular media. [2]




next up previous contents
Next: Reynolds Number Up: No Title Previous: Introduction



Rhonda Macleod
Mon Apr 17 16:32:58 EDT 1995