Physics for Geologists, Second edition

126 Fluids and fluid flow
Figure 12.8 Diagram of Darcy's apparatus (in section). The sand is packed in
the cylinder over the length I, and the water energy lost to friction by
flow through the sand is proportional to the difference in elevation
of the water surfaces in the manometers. If the flow were reversed,
the levels in the manometers would be reversed, but the value of Ah
would be the same. If the apparatus were to be inclined, the results
would still be the same.
where Q is the volumetric rate of flow, A is the internal cross-sectional area
of the cylinder, Ah is the difference of the water levels in the manometer
(total head), 1 is the macroscopic length of flow in the sand, and K is the
proportionality constant called the coefficient of permeability or hydraulic
conductivity.
Darcy was not measuring the difference of pressure across the sand; he
was measuring the difference of energy as measured by the elevations in the
manometers (see Bernoulli's theorem, page 123). This is the difference of
total head, h.
We now write Equation 12.10
where q is the specific discharge (or approach velocity, because it is the real
velocity of the water in the cylinder just above the sand), and it has the
dimensions of a velocity [LT-'1. The term, Ahll, called the hydraulic gra-
dient, is dimensionless; so the coefficient of permeability has the dimensions
Copyright 2002 by Richard E. Chapman