The bulk modulus (K) is defined as the ratio
between the increase in equal-all-round stress and the
resulting volumetric compression (compaction):
K ¼ Δσ=ε vol
(11.12)
For an isotropic material the bulk modulus is
K ¼ E=3ð1 À 2vÞ. The shear modulus (G) is the ratio
between the increase in shear stress and the resulting
shear strain (angular change due to deformation). For
an isotropic material it may be shown that
G ¼ E=2ð1 þ vÞ.
For a uniaxial strain compaction situation with
strain only in the vertical direction and no lateral strain
allowed, the compaction (compression) modulus (M)
may be expressed as M ¼ Eð1 À vÞ=ð1 þ vÞð1 À 2vÞ.
As stated in Sect. 11.3.1, the compaction process in a
fairly homogeneous and wide sedimentary basin may
be considered uniaxial. This is not the case in a narrow
basin or where there are abrupt changes in depth of
basin, lithology and material compressibility, for
instance due to faulting and block rotations.
The strains induced by the transmission of seismic
signals are so small that linear elastic behaviour may
be assumed. However, anisotropic deformation
characteristics should be allowed for when the seismic
signals are used to derive deformation characteristics
of the sedimentary rocks. A special type of anisotropy,
which is a useful extension of the isotropic material
theory, assumes the sedimentary rock to be transversely isotropic. This implies that the elastic
properties are equal for all directions within a plane,
but different in the other directions. Transverse isotropy may be considered to be a representative symmetry for horizontally layered sedimentary rocks. The
properties are assumed isotropic, but different, in the
vertical and horizontal planes. Five elastic constants
fully define all the deformation characteristics for such
a material.
In general, sedimentary rocks cannot be treated as
linearly elastic materials because the normal and shear
strains are not recovered when the element is
unloaded. However, the modulus concepts from elasticity theory, as outlined above, are very useful even
when analysing the more realistic non-linear
behaviour of such rocks, including permanent (plastic)
deformation.
It should be pointed out that for a linearly elastic
ideal material like the one described above, there is no
coupling between the effects of normal stresses and
shear stresses and between normal strains and shear
strains. For instance, if an element is only subjected to
shear stresses, there will not be any normal strains. As
discussed below, the behaviour of sediments and sedimentary rocks is not that simple, and there can be
significant volume expansion (dilatancy) or contraction
even if only shear stresses are applied. This phenomenon is also related to the degree of overconsolidation, as
the higher the overconsolidation ratio, the more significant is the degree of shear dilatancy.
11.4.2 Non-linear, Inelastic Behaviour in
Uniaxial Strain Compression
The stress–strain results from a saturated sediment
tested in uniaxial strain compression are shown in
Fig. 11.5a. The starting point (I) for the curve
represents the initial state of stress, and it is assumed
that initially the specimen is normally consolidated.
The uniaxial compression modulus increases with the
level of applied vertical effective stress. We therefore
use the term tangent modulus (M t ) and/or secant modulus (M s ) to represent the behaviour. The tangent
modulus gives the slope of the curve at any specified
stress level, while the secant modulus gives the slope
of the secant between two stress levels, commonly
between the initial point I and point A. At stress
level A the specimen is unloaded to the initial stress
level (point B). As may be seen from the figure, a
significant irrecoverable strain has been accumulated,
given by the horizontal distance IB. The specimen is
then reloaded back to A and up to a higher level, point
C. It is found that the curve from A to C is a natural
elongation of the curve from I to A. From I to A and A
to C the specimen is normally consolidated, while
during the unloading/reloading sequence it is
overconsolidated. It should be noted that linearly elastic loading and unloading behaviour would be
represented by only one common straight line in this
diagram, from I to C.
Extensive laboratory testing of different types of
sediments has shown that the tangent modulus may be
determined by the following general expression:
M t ¼ m p 0
σ
0
v
p 0
1Àa
(11.13)
11 Introduction to Geomechanics: Stress and Strain in Sedimentary Basins
309
between the increase in equal-all-round stress and the
resulting volumetric compression (compaction):
K ¼ Δσ=ε vol
(11.12)
For an isotropic material the bulk modulus is
K ¼ E=3ð1 À 2vÞ. The shear modulus (G) is the ratio
between the increase in shear stress and the resulting
shear strain (angular change due to deformation). For
an isotropic material it may be shown that
G ¼ E=2ð1 þ vÞ.
For a uniaxial strain compaction situation with
strain only in the vertical direction and no lateral strain
allowed, the compaction (compression) modulus (M)
may be expressed as M ¼ Eð1 À vÞ=ð1 þ vÞð1 À 2vÞ.
As stated in Sect. 11.3.1, the compaction process in a
fairly homogeneous and wide sedimentary basin may
be considered uniaxial. This is not the case in a narrow
basin or where there are abrupt changes in depth of
basin, lithology and material compressibility, for
instance due to faulting and block rotations.
The strains induced by the transmission of seismic
signals are so small that linear elastic behaviour may
be assumed. However, anisotropic deformation
characteristics should be allowed for when the seismic
signals are used to derive deformation characteristics
of the sedimentary rocks. A special type of anisotropy,
which is a useful extension of the isotropic material
theory, assumes the sedimentary rock to be transversely isotropic. This implies that the elastic
properties are equal for all directions within a plane,
but different in the other directions. Transverse isotropy may be considered to be a representative symmetry for horizontally layered sedimentary rocks. The
properties are assumed isotropic, but different, in the
vertical and horizontal planes. Five elastic constants
fully define all the deformation characteristics for such
a material.
In general, sedimentary rocks cannot be treated as
linearly elastic materials because the normal and shear
strains are not recovered when the element is
unloaded. However, the modulus concepts from elasticity theory, as outlined above, are very useful even
when analysing the more realistic non-linear
behaviour of such rocks, including permanent (plastic)
deformation.
It should be pointed out that for a linearly elastic
ideal material like the one described above, there is no
coupling between the effects of normal stresses and
shear stresses and between normal strains and shear
strains. For instance, if an element is only subjected to
shear stresses, there will not be any normal strains. As
discussed below, the behaviour of sediments and sedimentary rocks is not that simple, and there can be
significant volume expansion (dilatancy) or contraction
even if only shear stresses are applied. This phenomenon is also related to the degree of overconsolidation, as
the higher the overconsolidation ratio, the more significant is the degree of shear dilatancy.
11.4.2 Non-linear, Inelastic Behaviour in
Uniaxial Strain Compression
The stress–strain results from a saturated sediment
tested in uniaxial strain compression are shown in
Fig. 11.5a. The starting point (I) for the curve
represents the initial state of stress, and it is assumed
that initially the specimen is normally consolidated.
The uniaxial compression modulus increases with the
level of applied vertical effective stress. We therefore
use the term tangent modulus (M t ) and/or secant modulus (M s ) to represent the behaviour. The tangent
modulus gives the slope of the curve at any specified
stress level, while the secant modulus gives the slope
of the secant between two stress levels, commonly
between the initial point I and point A. At stress
level A the specimen is unloaded to the initial stress
level (point B). As may be seen from the figure, a
significant irrecoverable strain has been accumulated,
given by the horizontal distance IB. The specimen is
then reloaded back to A and up to a higher level, point
C. It is found that the curve from A to C is a natural
elongation of the curve from I to A. From I to A and A
to C the specimen is normally consolidated, while
during the unloading/reloading sequence it is
overconsolidated. It should be noted that linearly elastic loading and unloading behaviour would be
represented by only one common straight line in this
diagram, from I to C.
Extensive laboratory testing of different types of
sediments has shown that the tangent modulus may be
determined by the following general expression:
M t ¼ m p 0
σ
0
v
p 0
1Àa
(11.13)
11 Introduction to Geomechanics: Stress and Strain in Sedimentary Basins
309