Page 28 - Physical Principles of Sedimentary Basin Analysis

10 Properties of porous media

and the porosity is
4 3

φ = exp − πa . (2.18)
3
The pair correlation function C(r) can be obtained in a similar way as the porosity. It is
equal to the probability that two points separated by a distance r inside the volume V is
not occupied by a pair of spheres. Using the number density and going to the limit of a
large number of spheres gives

C(r) = exp − V 2 (r) (2.19)

where V 2 is the volume of two overlapping spheres separated by a distance r.Thevolume
V 2 is
r
4πa 3 3 r 1
3

V 2 = 1 + − , r < 2a (2.20)
3 4 a 16 a
where the sphere radius is a. When r ≥ 2a the spheres do not overlap and we have that V 2
is the sum of the volumes of the two spheres,
8πa 3
V 2 = , r ≥ 2a. (2.21)
3
Relation (2.10) for the specific surface of the porous medium gives

dC(r) 2
S =−4 = 4πa φ (2.22)
dr r=0
where φ is the porosity given by (2.18).
We will later need an expression for the specific surface as a function of the porosity.
This is for applications where the porosity is lost due to precipitation of minerals, and
where the precipitation process is controlled by the available specific surface. Before we
derive an expression for the specific surface as a function of the porosity, we will first
find an expression for the volume of a grain as a function of the porosity. We have from
(2.16) that
lnφ
V g (φ) = V 0 (2.23)
lnφ 0
where the initial porosity φ 0 and the initial grain volume V 0 are used to eliminate the
number density. From the volume of a grain (2.17) we obtain the following expression for
the radius of a sphere as a function of the porosity:

1/3
lnφ
a(φ) = a 0 . (2.24)
lnφ 0
Finally, the specific surface (2.22) as a function of the porosity becomes

2/3
3φlnφ 0 lnφ
S(φ) =− . (2.25)
a 0 lnφ 0