Page 34 - Physical Principles of Sedimentary Basin Analysis

16 Properties of porous media

where ¯u is the average velocity of the fluid in the pipe, r is the radius of the pipe and μ is
the viscosity. The Darcy flux v D , which is the volume rate of fluid per unit time out of a
cross-section, is then
4
Nπr 2 Nπr dp
v D =− ¯ u =− (2.41)
A 8Aμ dx
for a cross-section A with N pipes, when it is assumed that the cross-section A is normal
to the tubes. The permeability is then identified as

Nπr 4
k = . (2.42)
8A
The porosity of the medium of pipes is

Nπr 2
φ = (2.43)
A
which is used to replace the pore radius in the permeability (2.42). Using the porosity (2.43)
in the Darcy flux (2.41) gives that v D = φ ¯u, which is the general relationship (2.30)
between the average fluid velocity in the pores and the Darcy flux. The permeability of the
medium becomes
A 2
k(φ) = φ (2.44)
8π N
when the porosity (2.43) replaces the tube radius in expression (2.42). The permeability
2
decreases as ∼φ with decreasing porosity, and it is proportional to the area of the cross-
section (A), and inversely proportional to the number of pipes (N). The permeability can
also be written as

2 2
φ φ A
0
k(φ) = k 0 · with k 0 = (2.45)
φ 0 8π N
where k = k 0 is the permeability at the reference porosity φ = φ 0 . Notice that permeability
2
has units of length squared (m ). We have already seen that the specific surface of the
3
2
pore space has units of inverse length, (m /m ). It is therefore possible to express the
permeability k with the inversely of the specific surface squared. A porous medium of
tubes has the specific surface area
2π Nr 2 4Nπφ
S = or S = (2.46)
A A
when the tube radius is replaced by the porosity. The permeability as a function of porosity
can then be written
φ 3
k(φ) = . (2.47)
2S 2
This form of permeability function becomes the Kozeny–Carman relationship

φ 3
k(φ) = (2.48)
2
CS (1 − φ) 2
s