Page 30 - Physical Principles of Sedimentary Basin Analysis

12 Properties of porous media

2.4 Darcy’s law
Darcy’s law is an expression for the flux of fluid that is flowing through a porous medium
in response to a pressure difference. Figure 2.8 shows a cylindrical core with length l where
the fluid is driven through by a pressure difference p f = p 2 − p 1 . The flux v D of the fluid
is measured as volume per cross-section area and per time, and it is expressed as follows
by Darcy’s law:

k p
v D =− . (2.28)
μ l

The minus sign is introduced because the flow is positive in the direction of the pressure
drop. The Darcy flux is, for a given core and a given fluid, proportional to the pressure
difference across the core and inversely proportional to the length of the core. There are
two parameters in Darcy’s law (2.28) and they are the permeability k and the fluid viscosity
μ. The permeability is a rock property characterizing the rocks’ ability to conduct fluid,
while the viscosity is a fluid property expressing the fluids’ resistance to deformations. The
unit often used for permeability is the darcy (D) or millidarcy (mD). A permeability of 1 D
gives the flux 1 cm/s for a fluid with viscosity 1 cP (centipoise) when the pressure gradient
is 1 atm/cm. One darcy becomes

(1cm/s)(1cP)(1cm) (10 −2 m/s)(10 −3 Pa s)(10 −2 m)
1D = =
5
1atm 1.01325 · 10 Pa
= 0.986923 · 10 −12 m 2 (2.29)
5
where we have that 1 atm = 1.01325 · 10 Pa, and that 1 cP = 10 −3 Pa s. SI units for
2
2
permeability are m , and without much loss of accuracy we let 1 D = 10 −12 m and
1mD = 10 −15 in applications.
The permeability has been measured for a large range of sediments and rocks, and it is a
property that spans several orders of magnitude. Figure 2.9 shows the permeability range
for some common rocks. Notice how difficult it is to constrain the permeability within one
rock type, for instance sandstone.

Area = A

length = dl
p
p 2 1
Figure 2.8. The Darcy flow through a core plug is proportional to the pressure difference p =
p 2 − p 1 and inversely proportional to the length of the core l.