Page 33 - Physical Principles of Sedimentary Basin Analysis

2.6 Permeability as a function of porosity 15

where the hydrostatic pressure
z
p h,0 = f gdz (2.37)
z 0
is relative to the reference level z 0 . A natural choice for a reference level could for instance
be the sea level. The reason for introducing the pseudo-potential is that it becomes a true
potential when the fluid density is constant. In that case we can write

k
v D =− ∇ (2.38)
μ
and the Darcy flow becomes proportional to the gradient of , where is the potential.
Flow that is proportional to the gradient of a quantity is potential flow, and the quantity
is denoted the potential. The unit for the potential is the same as for pressure (Pa in SI
units).
It is common to express Darcy’s law in terms of the gradient of the hydraulic head
instead of the fluid flow potential. The hydraulic head h is the fluid flow potential divided
by f g, and Darcy’s law becomes

k f g
v D =− ∇h. (2.39)
μ
The hydraulic head measures the fluid pressure in terms of the corresponding height of
a fluid column. The factor k f g/μ is called the hydraulic conductivity, and it has SI
units m/s. The hydraulic conductivity becomes the Darcy flux driven by a gradient of the
hydraulic head equal to 1.

2.6 Permeability as a function of porosity

Fluid flow on a pore scale is from one pore through a pore throat to a neighbor pore. The
permeability of the rock is determined by two properties – the size of the pore throats
and how well the pores are connected. We will now look at a simple model that accounts
for the size of the pore throats, which is a porous medium made of parallel tubes. It is
possible to obtain a relationship between the permeability and the porosity for this model,
and it leads to the often used Kozeny–Carman relationship for permeability. It turns out that
this permeability model applies for porous media of well sorted grains, like for instance
well sorted sands and sandstones. Equal grain size corresponds to equal pore throats and a
constant tube radius. The starting point for Darcy flow in such a medium is laminar flow in
a single tube, which is written as
2
r dp
¯ u =− (2.40)
8μ dx