In the previous chapter Darcy’s law for the flow of a fluid through a porous medium has been formulated, in its simplest form, as
This means that the hydraulic conductivity k can be determined if the specific discharge q can be measured in a test in which the gradient dh/ds is known. An example of a test setup is shown in Figure 7.1. It consists of a glass tube, filled with soil. The two ends are connected to small reservoirs of water, the height of which can be adjusted. In these reservoirs a constant water level can be maintained. Under the influence of a difference in head Deltah between the two reservoirs, water will flow through the soil. The total discharge Q can be measured by collecting the volume of water in a certain time interval. If the area of the tube is A, and the length of the soil sample is DeltaL, then Darcy’s law gives
Because Q = qA this formula is in agreement with (7.1). Darcy performed tests as shown in Figure 7.1 to verify his formula (7.2). For this purpose he performed tests with various values of Deltah, and indeed found a linear relation between Q and h. The same test is still used very often to determine the hydraulic conductivity (coefficient of permeability) k. For sand normal values of the hydraulic conductivity k range from 10^-6 m/s to 10^−3 m/s. For clay the hydraulic conductivity usually is several orders of magnitude smaller, for instance k = 10^−9 m/s,or even smaller. This is because the permeability is approximately proportional to the square of the grain size of the material, and the particles of clay are about 100 or 1000 times smaller than those of sand. An indication of the hydraulic conductivity of various soils is given in Table 7.1.
As mentioned before, the permeability also depends upon properties of the fluid. Water will flow more easily through the soil than a thick oil.
This is expressed in the formula (6.11),
where μ is the dynamic viscosity of the fluid. The quantity K (the intrinsic permeability) depends upon the geometry of the grain skeleton only.
A useful relation is given by the formula of Kozeny-Carman,
Here d is a measure for the grain size, and c is a coefficient, that now only depends upon the tortuosity of the pore system, as determined by the shape of the particles. Its value is about 1/200 or 1/100. Equation (7.4) is of little value for the actual determination of the value of the permeability , because the value of the coefficient c is still unknown, and because the hydraulic conductivity can easily be determined directly from a permeability test. The Kozeny-Carman formula (7.4) is of great value, however, because it indicates the dependence of the permeability on the grain size and on the porosity. The dependence on d^2 indicates, for instance, that two soils for which the grain size differs by a factor 1000 (sand and clay) may have a difference in permeability of a factor 106. Such differences are indeed realistic. The large variability of the permeability indicates that this may be a very important parameter. In constructing a large dam, for instance, the dam is often built from highly permeable material, with a core of clay. This clay core has the purpose to restrict water losses from the reservoir behind the dam. If the core is not very homogeneous, and contains thin layers of sand, or if the clay core is not well encased into the rock bottom, the function of the clay core is disturbed to a high degree, and large amounts of water may be leaking through the dam. Severe accidents of this type have happened, see for instance Figure 7.2, which shows the collapse of the Teton Dam, in Idaho, USA, in 1976, photographed by Mrs. Eunice Olson.
