Physics for Geologists, Second edition

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Fluids and fluid flow 127
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Flow in litreslrninute
Figure 12.9 Darcy's results. These are plotted as the difference of head in metres
due to flow at the rates given in litres per minute through filters of
different sands and different lengths (marked). The straight lines
are the regression lines constrained to pass through the origin - no
difference of head when there is no flow.
of a velocity and is not therefore a true constant but a material constant that
varies with both the properties of the fluid and the pore characteristics.
The coefficient of permeability, K, clearly takes several factors into
account. Darcy found different values for K for different sands. Had he used
a liquid other than water, with a different viscosity, no doubt K would have
had different values again. And the hydraulic gradient is clearly an energy
term for unit weight of the liquid, lacking the factor g. So K must include
the acceleration due to gravity, g, and the mass density of the liquid, p.
Dimensional considerations lead to
where q is the coefficient of viscosity, and k is the permeability attributable
to the permeable material alone, called the intrinsic permeability. But k has
the dimensions of an area, so it too is a material constant (as you would
expect), and must embrace various attributes of the sand.
The size of the pores clearly affects water flow through them, so we
would expect the hydraulic radius [L] to play a part in intrinsic perme-
ability. Hydraulic radius, as we have seen, is a measure of both size and
shape of the passages for the water flow. It is the characteristic dimension of
the pores, but must not be regarded as an average dimension: it is just the
volume of pore space available for water flow divided by the bounding area.
The hydraulic radius is the same (with but statistical variation) for 1 m3 as
for 10 m3. Dimensions indicate that intrinsic permeability is proportional to
the square of the hydraulic radius.
Copyright 2002 by Richard E. Chapman