Page 29 - Physical Principles of Sedimentary Basin Analysis

2.3 The penetrable grain model 11

The expressions (2.23) and (2.24) apply for the penetrable sphere model. These expres-
sions can easily be converted to the inverse model, the “Swiss cheese model,” by replacing
φ by 1 − φ.

Exercise 2.6 Find the number density as a function of the porosity (2.18), and use this
relation for the number density to show that the pair correlation function (2.19) becomes

r
3 r 1
3

⎧

⎪ exp lnφ 1 + − , r < 2a
⎪
4 a 16 a
⎨
C(r) = (2.26)
⎪
⎪
φ , r ≥ 2a
⎩ 2
when (2.20) and (2.21)givethe volume V 2 . Check that the correlation function fulfills both
relations (2.9).
Exercise 2.7 Show that volume of two overlapping spheres separated by a distance r <
2a is given by expression (2.20). Use that the volume of the carlot of a sphere is V k =
2
2
(πh/6)(3r + h ), where h is the carlot height and r k is the carlot radius, see Figure 2.7b.
k
Solution: We have that V 2 = 2(V g − V k ) where V g is the volume of a sphere and V k is the
volume of one of the two carlots formed by the overlapping spheres, see Figure 2.7a. The
2
2
2
2
2
height of the carlot is h = a −r/2, and the carlot radius is r = a −(a −h) = a −r /4.
k
The volume of the carlot is then
π
r 2 3r 2
r 2 π 2 2 r 3
V k = a − 3a − + a − = 4a − 3a r + . (2.27)
6 2 4 2 6 2
3
We check that V k (r = 0) = 4πa /6, which is a half sphere. Another check is that two
spheres only touching each other have V k (r = 2a) = 0. The volume of the two overlapping
spheres V 2 = 2(V g −V k ) is then found by inserting the volume of a sphere V g = (4/3)πa 3
and the carlot volume V k from (2.27).
Exercise 2.8 (a) Derive expression (2.23). (b) Derive expression (2.24). (c) Derive
expression (2.25).

a r k
h
r

(a) (b)
Figure 2.7. (a) Two overlapping spheres of radius a that are separated by a distance r. (b) The
volume of the carlot is given by the height h and the carlot radius r k .