Page 104 - Physical Principles of Sedimentary Basin Analysis

86 Compressibility of rocks and sediments

The compressibilities with respect to the pressures p s and p f are 1/K 1 and 1/K 2 ,
respectively, and the volume change is written

V 1 1
=− p s − p f (4.41)
V K 1 K 2
which then gives
1 1
= C 1 and = C 1 − C 2 (4.42)
K 1 K 2
or
1 1 1
C 1 = and C 2 = − . (4.43)
K 1 K 1 K 2
These are precisely the relationships (4.32) and (4.33) for the bulk volume, and relation-
ships (4.34) and (4.35) for the pore volume.

Note 4.3 The relationships between the two different set of compressibilities follow from
the chain rule of differentiation. Let V (x, y) be a volume with respect to the pressures x
and y, and let V (s, t) be the same volume with respect to a new pressure pair s and t.In
the case of effective pressure the two alternative pressure pairs are related as s = x − y
and t = y. The inverse relationship between the pressures are x = s + t and y = t.
Differentiation then leads to
∂V ∂V ∂x ∂V ∂y ∂V
= + = (4.44)
∂s ∂x ∂s ∂y ∂s ∂x
∂V ∂V ∂x ∂V ∂y ∂V ∂V
= + = + . (4.45)
∂t ∂x ∂t ∂y ∂t ∂x ∂y
These partial derivative can be applied to the definitions of the compressibilities, which
then give the relationship between them.

4.3 Compressibility of porosity and the solid volume
This section shows how the porosity and the volume of the solid matrix depend on (con-
fining) bulk pressure and (pore) fluid pressure in a porous rock. Both relationships are
based on the compressibilities for the bulk volume and the pore volume of Section 4.2.
The porosity is treated first, where a change in the porosity is

V p V p V b
φ = = φ − (4.46)
V b V p V b
where the porosity is φ = V p /V b (the pore volume over the bulk volume). It then follows
from (4.5) and (4.6) for changes in the bulk and pore volume that

φ
= (α bc − α pc ) p b + (α pp − α bp ) p f . (4.47)
φ