Page 105 - Physical Principles of Sedimentary Basin Analysis

4.3 Compressibility of porosity and the solid volume 87

The relative change of the solid volume with respect to changes in the bulk volume and
pore volume is

V s 1 V b φ V p
= − . (4.48)
V s (1 − φ) V b (1 − φ) V p

The compressibility of the solid volume and the bulk volume are equal at zero porosity.
Inserting relationships (4.5) and (4.6) for the bulk and the pore volume, we get

1
V s
= (φα pc − α bc ) p b + (α bp − φα pp ) p f . (4.49)
V s (1 − φ)
The porosity and the solid volume could have been used instead of the bulk volume and
the pore volume as a basis for the definitions of the four basic compressibilities. How the
compressibilities of the porosity and the solid volume are related to the already introduced
compressibilities are summarized as follows:

1 ∂φ
= α bc − α pc (4.50)
φ ∂ p b
p f
1 ∂φ
= α pp − α bp (4.51)
φ ∂ p f
p b
1 ∂V s
= α bc − φα pc (4.52)
V s ∂ p b
p f
1 ∂V s
= φα pp − α bp . (4.53)
V s ∂ p f
p b
We have already shown by equation (4.19) that the compressibility of a pure solid is α s =
α bc − φα pc . It turns out that later in the formulation of pressure equations we need the
porosity change (4.47) and solid volume change (4.49).

Exercise 4.1 Derive relationship (4.46) for the change φ in the porosity.
Solution: We have φ = V p /V b and therefore

V p − V b V p + V b V p
φ = = 2 (4.54)
V b V
b

V p V b V p V p
=− + (4.55)
V b V b V b V p
which is (4.46).

Exercise 4.2 Derive relationship (4.48) for the change V s /V s in the solid volume.