Physics for Geologists, Second edition

water above a vertical thickness h of water-saturated sand:
Fluids and fluid flow 1 19
The depth of water does, of course, contribute to the total stress, S.
Reynolds numbers
Osborne Reynolds (1842-1912) carried out some beautiful experiments in
the 1880s on the flow of water in pipes, and was interested in the conditions
that separated the stable from the unstable, the laminar from the turbulent.
He found that it was a function of the internal diameter of the pipe and the
velocity and viscosity of the water in it, but argued that since units are arbi-
trary, there must be a dimensionless relationship. Noting that the dimensions
of kinematic viscosity (v = qlp) are those of a length multiplied by a veloc-
ity, he took as his dimensionless number dV/v, where d is the diameter of
the pipe and V is the velocity of the liquid (volume per unit area normal to
flow, L~T-' LP2 = LT-l). This is now called a Reynolds number. It con-
sists of a characteristic length multiplied by a characteristic velocity divided
by the kinematic viscosity of the fluid. It has the indefinite article because
there are different ways of defining the components. (Reynolds actually took
the inverse of this but it was changed to the form given, presumably to give
a large number rather than a small fraction, and that is how it is now defined.)
In pipe and channel flow, the characteristic length is the hydraulic radius
(which we shall come to shortly). What should it be in porous rocks? It is
common practice to take the mean grain size, but this is a very indirect quan-
tity because it is precisely the part of the rock that fluids do not flow through.
We shall return to this problem when we consider water flow through porous
rocks.
Hydraulic radius
With pipe flow, there is no difficulty in measuring the radius. But what if the
pipe were square or elliptical? or if the pipe were not quite full?
It has been found that the unifying parameter is obtained by taking the
volume of water and dividing it by its wetted surface (if the dimensions
remain constant, you can take the cross-sectional area divided by the wetted
perimeter). This is called the hydraulic radius, and it has the dimension of
length. The hydraulic radius of a pipe of circular cross-section and constant
internal diameter d is
If we never used anything but pipes that were full, this would not be neces-
sary. But pipes are not always full, and water can flow in channels as well
as pipes.
Copyright 2002 by Richard E. Chapman