Page 99 - Physical Principles of Sedimentary Basin Analysis

4.1 Rock compressibility 81

Notice that the compressibilities α bp and α pp with respect to the pore fluid are written with
a plus sign. That is because a positive fluid pressure increment is a tensile pressure that
expands the rock. On the other hand, a positive bulk pressure increment is a compressive
pressure that reduces the rock volume.
These compressibilities are not independent of each other or the compressibility of a
(pure) solid matrix material, α s . The following relations between the compressibilities fol-
low from the work of Zimmerman et al. (1986). The bulk volume and the pore volume are
both functions of the bulk pressure and the pore fluid pressure. These volumes are therefore
written as
V b = V b (p b , p f ) and V p = V p (p b , p f ). (4.7)
A change in the pressure state is now given by the pressure pair (dp b , dp f ). Assuming that
the rock behaves linearly with respect to pressure changes we can write that
dV b (dp b , dp f ) = dV b (dp b , 0) + dV b (0, dp f ). (4.8)

The volume change caused by a pressure change (dp b , dp f ) is decomposed into a vol-
ume change from (dp b , 0) and (0, dp f ), respectively, using the assumption of linearity.
Although the assumption about linearity is not in general valid for rocks, it is at least valid
for “small” changes to a given pressure state. The next step is the observation that a vol-
ume change caused by a uniform stress increment (dp, dp) turns out to be dV b =−α s dp.
The reason for this is that a volume change caused by equally large pressure changes in
the pore fluid as in the bulk is the same as if the pores were filled with matrix material.
Equation (4.8) can therefore be written as
(−α s dp) = (−α bc dp) + (α bp dp) (4.9)

where the definition of the compressibilities α bc and α bp from equations (4.1) and (4.2),
respectively, are used. This yields the first relation between the compressibilities,

α bp = α bc − α s . (4.10)
An analogous argument can now be applied to the pore volume. The pressure change
(dp, dp) is applied to V p , and the resulting pore volume change dV p (dp, dp) is decom-
posed into the pore volume change from the bulk pressure dV p (dp, 0) and the pore volume
change from the pore pressure dV p (0, dp),
dV p (dp, dp) = dV p (dp, 0) + dV p (0, dp). (4.11)

Using again that the volume change from a uniform pressure change (dp, dp) is the same as
if the rock had the pore space filled up by matrix material, we can write equation (4.11)as

(−α s dp) = (−α pc dp) + (α pp dp) (4.12)
where the pore space compressibilities, given by definitions (4.3) and (4.4), are used. This
leads to the second relationship between the compressibilities:
α pp = α pc − α s . (4.13)