Page 100 - Physical Principles of Sedimentary Basin Analysis

82 Compressibility of rocks and sediments

A third relation between the compressibilities is based on a theorem in linear elastic
mechanics called Betti’s reciprocal theorem. It applies when a linearly elastic structure
is subjected to two separate force systems, and it states that the work done by the force of
the first system during deformations of the second system is equal to the work done by the
force of second system during deformations of the first system. (This theorem is shown in
Note 4.1.) The pressure dp b and the corresponding deformation dV b (dp b , 0) is one force
system and the pressure dp f and the corresponding deformation dV p (dp b , 0) is the second
force system. Betti’s reciprocal theorem implies that

− dV b (dp b , 0) · dp f = dV p (0, dp f ) · dp b . (4.14)

The minus sign is needed because the pressure increment dp f is tensile while the increment
dp b is compressive. Equation (4.14) is rewritten as

(4.15)
− (−V b α bc dp b ) dp f = (V p α pp dp f ) dp b
in terms of compressibilities. Then we get the third relation between the compressibilities:

α bc = φα pp (4.16)
where it is used that V p = φV b . The three relationships (4.10), (4.13) and (4.16) can be
used to express any three of the four compressibilities α bc , α bp , α pc and α pp in terms of φ,
α s and any remaining fourth compressibility. The bulk compressibility α bc is considered
to be the “fundamental” compressibility, because it is easier to measure than the other
compressibilities. The other three compressibilities are in terms of φ, α s and α bc :

α bp = α bc − α s (4.17)

α pp = α bc − (1 + φ)α s /φ (4.18)
α pc = (α bc − α s )/φ. (4.19)

Figure 4.1 gives an idea of what these compressibilities (4.17)–(4.19) might be. We notice
that the compressibilities α pp and α pc depend on the porosity. The numbers used are
α s = 2.7 · 10 −11 Pa −1 for pure quartz and α bc = 4 · 10 −11 Pa −1 for the Fontainebleau
sandstone, where these parameters are based on Song and Renner (2008). The compress-
ibilities derived above are constant (for a given porosity) assuming a homogeneous rock
without any specific texture or composition. It turns out that most rocks are heterogeneous
on all length scales, from the grain scale to the basin scale, with variations in porosity,
mineralogy and texture. The compressibilities are also to some degree dependent on effec-
tive pressure, as for instance seen in the study by Song and Renner (2008). It is therefore
difficult to assign precise numbers for these compressibilities.

Note 4.1 Betti’s reciprocal theorem is shown in 1D by stretching a bar. We recall that the
force needed to keep a bar stretched a distance l is F = AE l/l, and that the work done
2
stretching the bar a distance l is W = (1/2)AE l /l, where A is the cross-section of