Page 164 - Fundamentals of Reservoir Engineering

DARCY'S LAW AND APPLICATIONS 103

frictionless process, to transport a unit mass of fluid from a state of atmospheric
pressure and zero elevation to the point in question, thus

p dp
Φ= + gz (4.5)
1atm ρ
−
Although defined in this way, fluid potentials are not always measured with respect to
atmospheric pressure and zero elevation, but rather, with respect to any arbitrary base
pressure and elevation (p b, z b ) which modifies equ. (4.5) to

p dp
−
Φ= + g(z z ) (4.6)
b
p b ρ
The reason for this is that fluid flow between two points A and B is governed by the
difference in potential between the points, not the absolute potentials, i.e.

p A dp p B dp p A dp
B
Φ− Φ = + g(z − z ) − + g(z − z ) = + g(z − z )
b
B
b
A
B
A
A
p b ρ p b ρ p B ρ
It is therefore conventional, in reservoir engineering to select an arbitrary, convenient
datum plane, relative to the reservoir, and express all potentials with respect to this
plane. Furthermore, if it is assumed that the reservoir fluid is incompressible (ρ
independent of pressure) then equ. (4.5) can be expressed as
p
Φ= + gz (4.7)
ρ

which is precisely the term appearing in equ. (4.4). It can therefore be seen that the h
term in Darcy's equation is directly proportional to the difference in fluid potential
between the ends of the sand pack.

The constant K/g is only applicable for the flow of water, which was the liquid used
exclusively in Darcy's experiments. Experiments performed with a variety of different
liquids revealed that the law can be generalised as

kdΦ
ρ
u = (4.8)
µ dl
in which the dependence of flow velocity on fluid density ρ and viscosity µ is fairly
obvious. The new constant k has therefore been isolated as being solely dependent on
the nature of the sand and is described as the permeability. It is, in fact, the absolute
permeability of the sand, provided the latter is completely saturated with a fluid and,
because of the manner of derivation, will have the same value irrespective of the
nature of the fluid.

This latter statement is largely true, under normal reservoir pressures and flow
conditions, the exception being for certain circumstances encountered in real gas flow.
At very low pressures there is a slippage between the gas molecules and the walls of