Page 26 - Physical Principles of Sedimentary Basin Analysis

8 Properties of porous media

z
θ

r
y

ϕ
x
Figure 2.5. A position in space is given by the spherical coordinates (r,θ,ϕ). The angles, ϕ from 0
to 2π and θ from 0 to π, parameterize the surface of sphere with radius r.

f (x) = 1for x in the pore space V p and otherwise 0. The derivative of the angular average
of the correlation function is
dC a (r) 1 ∂ f (x + rn r )
= dV sin θ dθ dϕ
dr 4πV ∂r
V p
1
= n r ·∇ f (x + rn r ) dV sin θ dθ dϕ
4πV
V p
1
= ∇· n r f (x + rn r ) dV sin θ dθ dϕ
4πV
V p
1
= n · n r f (x + rn r ) dA sin θ dθ dϕ. (2.12)
4πV
A p
The volume integral is converted to a surface integral by means of the divergence theorem,
and n is the outward unit vector of the surface A p of the pore space. The coordinate system
is now centered at x with n = n z , which gives that n · n r = cos θ. The outward normal
vector n of the surface of the pore space is pointing upwards, which means that the surface
is locally in the xy-plane around x. The solid is locally above the xy-plane and the pore
space is locally below the xy-plane. The integral

2π π
I = sin θ n · n r f (x + rn r ) dθ dϕ
0 0
π
= 2π sin θ cos θ f (x + rn r ) dθ
0
π
= 2π sin θ cos θ dθ
π/2
=−π (2.13)
in the limit r → 0. The function f (x + rn r ) is then 0 for n r (θ, ϕ) pointing into the solid,
with angles θ from 0 to π/2. Inserting the integral I into expression (2.12) gives that
dC/dr =−A p /4V , where A p is the surface area of the pore space and the ratio A p /V is
the specific surface.