Page 103 - Physical Principles of Sedimentary Basin Analysis

4.2 More compressibilities 85

the same as the pore fluid pressure. Since the rock sample is immersed in the fluid it wears
no “jacket,” and thus the name unjacketed. Changes in the effective pressure and the fluid
pressure lead to the following volume changes:

V b 1 1
=− p s − p f (4.30)
V b K K s
V p 1 1
=− p s − p f . (4.31)
V p K p K φ
The compressibilities defined in terms of effective pressure and fluid pressure are not inde-
pendent of the compressibilities defined in terms of bulk pressure and fluid pressure. These
new volume compressibilities are related to the previous compressibilities by the chain rule
of differentiation. The following relationships are derived in Note 4.2:

1
α bc = (4.32)
K
1 1
α bp = − (4.33)
K K
s
1
α pc = (4.34)
K p
1 1
α pp = − . (4.35)
K p K φ
The inverse relationships, which are straightforward to obtain, are given for the sake of
completeness:

1
= α bc (4.36)
K
1
= α bc − α bp (4.37)
K s
1
= α pc (4.38)
K p
1
= α pc − α pp . (4.39)
K φ
Note 4.2 The relationship between the two sets of compressibilities follows from the rela-
tionship between the pressures, where p s = p b − p f .Let V be either the bulk volume or
the pore volume, and let the pair of compressibilities with respect to the pressures p b and
p f be C 1 and C 2 , respectively. A volume change is then related to the pressure change as

V
=−C 1 p b + C 2 p f
V
=−C 1 p s + (−C 1 + C 2 ) p f . (4.40)