Page 163 - Fundamentals of Reservoir Engineering

DARCY'S LAW AND APPLICATIONS 102

∆h

water
manometers

q cc / sec
h
l

z
+ z

datum plane; z = 0, p = 1 atm.

Fig. 4.2 Orientation of Darcy's apparatus with respect to the Earth's gravitational field

It is worthwhile considering the significance of the ∆h term appearing in Darcy's law.
The pressure at any point in the flow path, fig. 4.2, which has an elevation z, relative to
the datum plane, can be expressed in absolute units as

p = ρg(h-z)

with respect to the prevailing atmospheric pressure. In this equation h is the liquid
elevation of the upper manometer, again, with respect to z = 0 and ρ is the liquid
(water) density. The equation can be alternatively expressed as

p
hg = ( + gz) (4.2)
ρ

If equ. (4.1) is written in differential form as

dh
u = K (4.3)
dl

then differentiating equ. (4.2) and substituting in equ. (4.3) gives

Kd p K d(hg)

u = + gz = (4.4)
gdl ρ g dl

p
The term ( + gz), in this latter equation, has the same units as hg which are:
ρ
distance × force per unit mass, that is, potential energy per unit mass. This fluid
potential is usually given the symbol Φ and defined as the work required, by a