Page 31 - Physical Principles of Sedimentary Basin Analysis

2.4 Darcy’s law 13

μD mD D kD

Shale

Marine clay

Limestone
Sandstone
Silty sand

Clean sand

10 −20 10 −18 10 −16 10 −14 10 −12 10 −10 10 −8 m 2
2
Figure 2.9. The permeability for common sediments and sedimentary rocks in units log (m ). Notice
10
how unconstrained the permeability really is. Even the permeability of one rock type will normally
span several orders of magnitude.

The Darcy flux is related to the average fluid velocity v f by multiplication with the
porosity φ

(2.30)
v D = φv f
assuming that the fluid is only flowing through the fraction φ of the area of a cross-section.
An important point concerning compacting and deforming porous media is that the Darcy
flux is measured relative to the solid skeleton. The Darcy flux in terms of velocities is
therefore

v D = φ · (v f − v s ) (2.31)

where v s is the velocity of the solid skeleton.
Darcy’s law (2.28) can also be rewritten as the force balance

μ
sum forces = pA + v D Adl = 0 (2.32)
k
where the first term is the pressure force and the second term is the viscous friction force.
Newton’s second law of motion gives that the momentum is conserved when the forces
add up to zero, and Darcy’s law can therefore be viewed as an expression for conservation
of momentum.

Exercise 2.9 How large a pressure difference is needed across a core plug of length 30 cm,
2
cross-section 100 cm in order for 1 liter per hour to flow through? The core plug has the
permeability 1 mD and the viscosity of the fluid is 10 −3 Pa s. (Answer: 8.33 MPa)