Page 25 - Physical Principles of Sedimentary Basin Analysis

2.2 The correlation function and specific surface 7

which expresses how likely it is that the porous medium is void at position r 2 , when it is
void at position r 1 . The medium is statistically homogeneous when the correlation function
depends only on the distance r = |r 2 − r 1 | as

C(r 1 , r 2 ) = C(r 2 − r 1 ) = C(r). (2.8)

From the definition (2.7) it follows that the statistically homogeneous correlation function
has the following properties:

2
C(0) = φ and lim C(r) = φ . (2.9)
r→∞

The latter relation assumes that the pore space is uncorrelated between any two positions
separated by a “large” distance. An important reason for introducing the two-point correla-
tion function is that it gives the surface area of the pore space per unit volume – the specific
surface area. It is obtained from the two-point correlation function by the following simple
relation (Berryman, 1987):

dC(r)
S =−4 . (2.10)
dr
r=0
This relation also holds for anisotropic porous media as shown in Note 2.1. The correlation
function can be found experimentally for real porous media or exactly for simple models,
and once it is obtained we will have the porosity from relation (2.9) and the specific surface
area from relation (2.10).

Note 2.1 The derivation of the expression (2.10) follows Berryman (1987). We first
introduce the angular average of the correlation function

1
C a (r) = C rn r (θ, ϕ) sin θ dθ dϕ
4π
1
= f (x) f (x + rn r ) dV sin θ dθ dϕ
4πV V
1
= f (x + rn r ) dV sin θ dθ dϕ (2.11)
4πV
V p

where n r (θ, ϕ) is the unit vector in the direction of r. The integration of ϕ is from 0 to
2π, and the integration of θ is from 0 to π, see Figure 2.5. The last equality holds because