Page 102 - Physical Principles of Sedimentary Basin Analysis

84 Compressibility of rocks and sediments

reciprocal theorem can be expressed in the same way as above: if a linearly elastic struc-
ture is subjected to two different stress–strain systems, the work done by the stress of the
first system with the strain of the second system is equal to the work done by the stress of
the second system with the strain of the first system.

4.2 More compressibilities
More compressibilities than those introduced by definitions (4.1)–(4.4) are measured and
used. It is common to measure compressibilities as a function of the effective pressure,
which is the difference between the bulk pressure and the fluid pressure, instead of the
bulk pressure. The bulk volume and the pore volume are now the following functions:

V b = V b (p s , p f ) and V p = V p (p s , p f ) (4.24)
where the effective pressure is
p s = p b − p f . (4.25)

The compressibilities of the bulk volume and the pore volume with respect to effective
pressure and fluid pressure are
1 1 ∂V b
=− (4.26)
K V b ∂ p s
p f
1 1 ∂V b
=− (4.27)
K V b ∂ p f
s p s
1 1 ∂V p
=− (4.28)
K p V p ∂ p s
p f
1 1 ∂V p
=− (4.29)
K φ V p ∂ p f
p s
where the notation for the moduli follow Wang (2000). The bulk pressure is here taken to
be the same as the confining pressure in Wang (2000). These compressibilities have the

following names: 1/K is the drained bulk compressibility,1/K is the unjacketed bulk
s
compressibility,1/K p is the drained pore compressibility and 1/K φ is the unjacketed pore
compressibility. The drained compressibilities are defined by a volume change at constant
(pore) fluid pressure in response to an effective pressure change. The pore fluid pressure
can only be kept constant at changing bulk pressure by allowing fluid to move out of
or into (by draining) the pore space, because the pore volume is changing with changing
bulk pressure. The unjacketed compressibilities are defined by volume changes in response
to changes in the pore fluid pressure, when the effective pressure is constant. The effec-
tive pressure is kept constant by letting the bulk pressure follow the pore pressure, which
is achieved in experiments by measuring the volume changes in a porous sample that is
immersed in the fluid. The (confining) fluid pressure on the walls of the sample are then