Page 27 - Physical Principles of Sedimentary Basin Analysis

2.3 The penetrable grain model 9

2.3 The penetrable grain model
The porosity and the characteristic function are not exactly known for other than some
simple porous media. One example of such a porous medium is N randomly placed spher-
ical grains of equal radius in a volume V . This model is called the penetrable grain model
because the grains are allowed to overlap. A porous medium of penetrable spheres is shown
in Figure 2.6a. The inverse of the penetrable sphere model, where solid and void are inter-
changed, is shown to the right. The inverse model is sometimes called a “Swiss cheese”
model, because the pores are now overlapping spheres.
The porosity of the penetrable grain model is equal to the probability that a given point
inside V is not overlapped by any of the N grains of volume V g ,

N
V g
φ = 1 − . (2.14)
V
The probability that a point in V is overlapped by a single grain is V g /V , when it is
assumed that the grains are uniformly distributed. We can replace the volume V by the
grain density = N/V , which is the number of grains per unit volume. The porosity
is then
N

V g
φ = 1 − (2.15)
N
which becomes
φ = exp(− V g ) (2.16)
N
x
in the limit N →∞. (We have that (1 + x/N) → e when N →∞.) A porous medium
of penetrable spheres of radius a has
V g = (4/3)πa 3 (2.17)

(a) (b)
Figure 2.6. (a) Porous medium formed by overlapping spheres. (b) The inverse porous medium of the
overlapping sphere model where the solid and the void are interchanged.