4 Tsvankin's extended Thomsen parameters for orthorhombic media
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40
Elasticity and Hooke’s law
Unlike in a VTI medium, S-waves propagating along
x3-axis in an orthorhombic
pthe
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
medium can have two different velocities, bx2 ¼ c44 = and bx1 ¼ c55 =, for
waves polarized in the x2 and x1 directions, respectively.
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Either polarization can be
chosen as a reference, though here we take b ¼ c55 = following the definitions of
Tsvankin (1997). Some results shown in later sections will use redefined polarizations in the definition of b.
For the seven constants, we can write
e2ị ẳ
c11 c33
2c33
c13 ỵ c55 ị2 c33 c55 ị2
2c33 c33 c55 ị
c66 c44
ẳ
2c44
e1ị ẳ
c23 ỵ c44 ị2 c33 c44 ị2
2c33 c33 c44 ị
c66 c55
ẳ
2c55
d2ị ẳ
d1ị ẳ
g2ị
g1ị
d3ị ẳ
c22 c33
2c33
c12 ỵ c66 ị2 c11 À c66 Þ2
2c11 ðc11 À c66 Þ
Here, the superscripts (1), (2), and (3) refer to the TI-analog parameters in the
symmetry planes normal to x1, x2, and x3, respectively. These definitions assume that
one of the symmetry planes of the orthorhombic medium is horizontal and that the
vertical symmetry axis is along the x3 direction.
These Thomsen–Tsvankin parameters play a useful role in modeling wave propagation and reflectivity in anisotropic media.
Uses
Tsvankin’s notation for weak elastic anisotropy is useful for conveniently characterizing the elastic constants of an orthorhombic elastic medium.
Assumptions and limitations
The preceding equations are based on the following assumptions:
material is linear, elastic, and has orthorhombic or higher symmetry;
the constants are definitions. They sometimes appear in expressions for anisotropy
of arbitrary strength, but at other times the applications assume that the anisotropy
is weak, so that e, g, d ( 1.
2.5
Third-order nonlinear elasticity
Synopsis
Seismic velocities in crustal rocks are almost always sensitive to stress. Since so much
of geophysics is based on linear elasticity, it is common to extend the familiar linear
elastic terminology and refer to the “stress-dependent linear elastic moduli” – which can
41
2.5 Third-order nonlinear elasticity
have meaning for the local slope of the strain-curves at a given static state of stress. If the
relation between stress and strain has no hysteresis and no dependence on rate, then it is
more accurate to say that the rocks are nonlinearly elastic (e.g., Truesdell, 1965; Helbig,
1998; Rasolofosaon, 1998). Nonlinear elasticity (i.e., stress-dependent velocities) in
rocks is due to the presence of compliant mechanical defects, such as cracks and grain
contacts (e.g., Walsh, 1965; Jaeger and Cook, 1969; Bourbie´ et al., 1987).
In a material with third-order nonlinear elasticity, the strain energy function E
(for arbitrary anisotropy) can be expressed as (Helbig, 1998)
E ¼ 12 cijkl eij ekl ỵ 16 cijklmn eij ekl emn
where cijkl and cijklmn designate the components of the second- and third-order elastic
tensors, respectively, and repeated indices in a term imply summation from 1 to 3.
The components cijkl are the usual elastic constants in Hooke’s law, discussed earlier.
Hence, linear elasticity is often referred to as second-order elasticity, because the
strain energy in a linear elastic material is second order in strain. The linear elastic
tensor (cijkl) is fourth rank, having a minimum of two independent constants for a
material with the highest symmetry (isotropic) and a maximum of 21 independent
constants for a material with the lowest symmetry (triclinic). The additional tensor of
third-order elastic coefficients (cijklmn) is rank six, having a minimum of three
independent constants (isotropic) and a maximum of 56 independent constants
(triclinic) (Rasolofosaon, 1998).
Third-order elasticity is sometimes used to describe the stress-sensitivity of seismic
velocities and apparent elastic constants in rocks. The apparent fourth-rank stiffness
tensor, c~eff , which determines the speeds of infinitesimal-amplitude waves in a rock
under applied static stress can be written as
c~eff
ijkl ẳ cijkl ỵ cijklmn emn
where emn are the principal strains associated with the applied static stress.
Approximate expressions, in Voigt notation, for the effective elastic constants of a
stressed VTI (transversely isotropic with a vertical axis of symmetry) solid can be
written as (Rasolofosaon, 1998; Sarkar et al., 2003; Prioul et al., 2004)
0
ceff
11 % c11 ỵ c111 e11 ỵ c112 e22 ỵ e33 ị
0
ceff
22 % c11 þ c111 e22 þ c112 ðe11 þ e33 Þ
0
ceff
33 % c33 ỵ c111 e33 ỵ c112 e11 ỵ e22 ị
0
ceff
12 % c12 ỵ c112 e11 ỵ e22 ị ỵ c123 e33
0
ceff
13 % c13 ỵ c112 e11 ỵ e33 ị ỵ c123 e22
0
ceff
23 % c13 ỵ c112 e22 ỵ e33 ị þ c123 e11
0
ceff
66 % c66 þ c144 e33 þ c155 e11 ỵ e22 ị
0
ceff
55 % c44 ỵ c144 e22 ỵ c155 e11 ỵ e33 ị
0
ceff
44 % c44 ỵ c144 e11 þ c155 ðe22 þ e33 Þ
42
Elasticity and Hooke’s law
Table 2.5.1 Experimentally determined third-order elastic constants c111, c112,
and c123 and derived constants c144, c155, and c456, determined by Prioul and
Lebrat (2004), using laboratory data from Wang (2002). Six different sandstone
and six different shale samples are shown.
c111 (GPa)
c112 (GPa)
c123 (GPa)
À10 245
À9 482
À6 288
À8 580
À8 460
À12 440
À966
À1745
À1744
À527
À1162
À3469
À966
À1745
À1744
À527
À1162
À3094
À6 903
À4 329
À7 034
À4 160
1 294
À1 203
À976
À2122
À2147
À2013
À510
À637
À976
À1019
296
À940
À119
À354
c144 (GPa)
c155 (GPa)
c456 (GPa)
0
0
0
0
0
À188
À2320
À1934
À1136
À2013
À1825
À2243
À1160
À967
À568
À1006
À912
À1027
0
À552
À1222
À536
À196
À141
À1482
À552
À1222
À536
À196
À141
À741
0
0
0
0
0
Sandstones
Shales
where the constants c011 , c033 , c013 , c044 , c066 are the VTI elastic constants at the
unstressed reference state, with c012 ¼ c011 À 2c066 . e11, e22, and e33 are the principal
strains, computed from the applied stress using the conventional Hooke’s law,
eij ¼ sijkl kl . For these expressions, it is assumed that the direction of the applied
principal stress is aligned with the VTI symmetry (x3-) axis. Furthermore, for
these expressions it is assumed that the stress-sensitive third-order tensor is
isotropic, defined by the three independent constants, c111, c112, and c123 with
c144 ẳ c112 c123 ị=2, c155 ẳ c111 c112 ị=4, and c456 ẳ c111 3c112 ỵ 2c123 ị=8.
It is generally observed (Prioul et al., 2004) that c111 < c112 < c123, c155 < c144,
c155 < 0, and c456 < 0. A sample of experimentally determined values from
Prioul and Lebrat (2004) using laboratory data from Wang (2002) are shown in the
Table 2.5.1.
The third-order elasticity as used in most of geophysics and rock physics (Bakulin
and Bakulin, 1999; Prioul et al., 2004) is called the Murnaghan (1951) formulation
of finite deformations, and the third-order constants are also called the Murnaghan
constants.
Various representations of the third-order constants that can be found in the
literature (Rasolofosaon, 1998) include the crystallographic set (c111, c112, c123)
presented here, the Murnaghan (1951) constants (l, m, n), and Landau’s set (A, B, C)
(Landau and Lifschitz, 1959). The relations among these are (Rasolofosaon, 1998)
43
2.6 Effective stress properties of rocks
c111 ẳ 2A ỵ 6B ỵ 2C ẳ 2l þ 4m
c112 ¼ 2B þ 2C ¼ 2l
c123 ¼ 2C ẳ 2l 2m ỵ n
Chelam (1961) and Krishnamurty (1963) looked at fourth-order elastic coefficients, based on an extension of Murnaghan’s theory. It turns out from group theory
that there will only be n independent nth-order coefficients for isotropic solids, and
n2 2n ỵ 3 independent nth-order constants for cubic systems. Triclinic solids have
126 fourth-order elastic constants.
Uses
Third-order elasticity provides a way to parameterize the stress dependence of
seismic velocities. It also allows for a compact description of stress-induced anisotropy, which is discussed later.
Assumptions and limitations
2.6
The above equations assume that the material is hyperelastic, i.e., there is no
hysteresis or rate dependence in the relation between stress and strain, and there
exists a unique strain energy function.
This formalism assumes that strains are infinitesimal. When strains become finite,
an additional source of nonlinearity, called geometrical or kinetic nonlinearity,
appears, related to the difference between Lagrangian and Eulerian descriptions of
motion (Zarembo and Krasil’nikov, 1971; Johnson and Rasolofosaon, 1996).
Third-order elasticity is often not general enough to describe the shapes of real
stress–strain curves over large ranges of stress and strain. Third-order elasticity is
most useful when describing stress–strain within a small range around a reference
state of stress and strain.
Effective stress properties of rocks
Synopsis
Because rocks are deformable, many rock properties are sensitive to applied stresses
and pore pressure. Stress-sensitive properties include porosity, permeability, electrical resistivity, sample volume, pore-space volume, and elastic moduli. Empirical
evidence (Hicks and Berry, 1956; Wyllie et al., 1958; Todd and Simmons, 1972;
Christensen and Wang, 1985; Prasad and Manghnani, 1997; Siggins and Dewhurst,
2003, Hoffman et al., 2005) and theory (Brandt, 1955; Nur and Byerlee, 1971;
Zimmerman, 1991a; Gangi and Carlson, 1996; Berryman, 1992a, 1993; Gurevich,
2004) suggest that the pressure dependence of each of these rock properties, X, can be
44
Elasticity and Hooke’s law
represented as a function, fX, of a linear combination of the hydrostatic confining
stress, PC, and the pore pressure, PP:
X ¼ fX ðPC nPP ị;
n
1
The combination Peff ẳ PC nPP is called the effective pressure, or more generally,
C
the tensor eff
ij ¼ ij À nPP dij is the effective stress. The parameter n is called the
“effective-stress coefficient,” which can itself be a function of stress. The negative
sign on the pore pressure indicates that the pore pressure approximately counteracts
the effect of the confining pressure. An expression such as X ¼ fX ðPC À nPP Þ is
sometimes called the effective-stress law for the property X. It is important to point
out that each rock property might have a different function fi and a different value of
ni (Zimmerman, 1991a; Berryman, 1992a, 1993; Gurevich, 2004). Extensive discussions on the effective-stress behavior of elastic moduli, permeability, resistivity, and
thermoelastic properties can be found in Berryman (1992a, 1993). Zimmerman
(1991a) gives a comprehensive discussion of effective-stress behavior for strain and
elastic constants.
Zimmerman (1991a) points out the distinction between the effective-stress behavior for finite pressure changes versus the effective-stress behavior for infinitesimal
increments of pressure. For example, increments of the bulk-volume strain, eb, and
the pore-volume strain, ep, can be written as
1 @VT
eb ðPC ; PP ị ẳ Cbc PC mb PP ịdPC nb dPP ị; Cbc ẳ
VT @PC PP
1 @VP
ep PC ; PP ị ẳ Cpc PC mp PP ịdPC np dPP ị; Cpc ẳ
VP @PC PP
where the compressibilities Cbc and Cpc are functions of PC À mPP . The coefficients
mb and mp govern the way that the compressibilities vary with PC and PP. In contrast,
the coefficients nb and np describe the relative increments of additional strain
resulting from pressure increments dPC and dPP. For example, in a laboratory experiment, ultrasonic velocities will depend on the values of Cpc, the local slope of the
stress–strain curve at the static values of PC and PP. On the other hand, the sample
length change monitored within the pressure vessel is the total strain, obtained by
integrating the strain over the entire stress path.
The existence of an effective-stress law, i.e., that a rock property depends only on
the state of stress, requires that the rock be elastic – possibly nonlinearly elastic. The
deformation of an elastic material depends only on the state of stress, and is independent of the stress history and the rate of loading. Furthermore, the existence of an
effective-stress law requires that there is no hysteresis in stress–strain cycles. Since
no rock is perfectly elastic, all effective-stress laws for rocks are approximations. In
fact, deviation from elasticity makes estimating the effective-stress coefficient from
laboratory data sometimes ambiguous. Another condition required for the existence
of an effective-pressure law is that the pore pressure is well defined and uniform
throughout the pore space. Todd and Simmons (1972) show that the effect of pore
45
2.6 Effective stress properties of rocks
pressure on velocities varies with the rate of pore-pressure change and whether the
pore pressure has enough time to equilibrate in thin cracks and poorly connected
pores. Slow changes in pore pressure yield more stable results, describable with an
effective-stress law, and with a larger value of n for velocity.
Much discussion focuses on the value of the effective-stress constants, n (and m).
Biot and Willis (1957) predicted theoretically that the pressure-induced volume
increment, dVT, of a sample of linear poroelastic material depends on pressure
increments ðdPC À ndPP Þ. For this special case, n ¼ a ¼ 1 À K=KS , where a is
known as the Biot coefficient or Biot–Willis coefficient. K is the dry-rock (drained)
bulk modulus and KS is the mineral bulk modulus (or some appropriate average of the
moduli if there is mixed mineralogy), defined below. Explicitly,
!
dVT
1
K
dPP
¼À
dPC À 1 À
VT
K
KS
where dVT, dPC, and dPP signify increments relative to a reference state.
Pitfall
A common error is to assume that the Biot–Willis effective-stress coefficient a for
volume change also applies to other deformation-related rock properties. For example,
although rock elastic moduli vary with crack and pore deformation, there is no
theoretical justification for extrapolating a to elastic moduli and seismic velocities.
Other factors determining the apparent effective-stress coefficient observed in the
laboratory include the rate of change of pore pressure, the connectivity of the pore
space, the presence or absence of hysteresis, heterogeneity of the rock mineralogy,
and variation of pore-fluid compressibility with pore pressure.
Table 2.6.1, compiled from Zimmerman (1984), Berryman (1992a, 1993), and H. F.
Wang (2000), summarizes the theoretically expected effective incremental stress laws
for a variety of rock properties. These depend on four defined rock moduli:
1
1 @VT
; K ¼ modulus of the drained porous frame;
¼À
K
VT @Pd PP
1
1 @VT
¼À
; KS ¼ unjacketed modulus; if monomineralic;
KS
VT @PP Pd
KS ¼ Kmineral ; otherwise KS is a poorly understood
average of the mixed mineral moduli
1
1 @V
¼À
; if monomineralic; KS ¼ K ¼ Kmineral
K
V @PP Pd
1
1 @V
1 1
1
¼À
¼
À
KP
V @Pd PP K KS
46
Elasticity and Hooke’s law
Theoretically predicted effective-stress laws for incremental
changes in confining and pore pressures (from Berryman, 1992a).
Table 2.6.1
Property
General mineralogy
Sample volumea
dVT
VT
dV
V
Pore volumeb
Porosityc
Solid volumed
Permeabilitye
Velocity/elastic modulif
¼ K1 dPC adPP ị
ẳ K1P dPC bdPP ị
d
a
ẳ K dPC dPP ị
1
ẳ 1ịK
dPC dPP ị
S
h
i
a
dk
2
k ẳ h K ỵ 3K dPC kdPP ị
dVS
VS
dVP
VP
ẳ f dPC yPP ị
Notes:
VT is the total volume. a ẳ 1 K=KS , Biot coefficient; usually in the range a 1;
if monomineralic, a ¼ 1 À K=Kmineral . b V ¼ VT , pore volume. b ¼ 1 À KP =K , usually,
b 1, but it is possible that b > 1. c ¼ bÀ
aÀ a; if monomineralic, ẳ 1.
21aị
d
ẳ a a
1:
a b. e k ẳ 1 3haịỵ2
1a a ị; if monomineralic, ẳ .
a
S ị=@PC
h % 2 ỵ m % 4, where m is Archie’s cementation exponent. f y ¼ 1 @1=K
@1=Kị=@PC ; if monomineralic,
y ẳ 1.
where Pd ẳ PC À PP is the differential pressure, VT is the sample bulk (i.e., total)
volume, and Vf is the pore volume. The negative sign for each of these rock
properties follows from defining pressures as being positive in compression and
volumes positive in expansion.
There is still a need to reconcile theoretical predictions of effective stress with
certain laboratory data. For example, simple, yet rigorous, theoretical considerations
(Zimmerman, 1991a; Berryman, 1992a; Gurevich, 2004) predict that nvelocity ¼ 1 for
monomineralic, elastic rocks. Experimentally observed values for nvelocity are sometimes close to 1, and sometimes less than one. Speculations for the variations
have included mineral heterogeneity, poorly connected pore space, pressure-related
changes in pore-fluid properties, incomplete correction for fluid-related velocity
dispersion in ultrasonic measurements, poorly equilibrated or characterized pore
pressure, and inelastic deformation.
Uses
Characterization of the stress sensitivity of rock properties makes it possible to invert
for rock-property changes from changes in seismic or electrical measurements. It also
provides a means of understanding how rock properties might change in response to
tectonic stresses or pressure changes resulting from reservoir or aquifer production.
47
2.7 Stress-induced anisotropy in rocks
Assumptions and limitations
2.7
The existence of effective-pressure laws assumes that the rocks are hyperelastic, i.e.,
there is no hysteresis or rate dependence in the relation between stress and strain.
Rocks are extremely variable, so effective-pressure behavior can likewise be variable.
Stress-induced anisotropy in rocks
Synopsis
The closing of cracks under compressive stress (or, equivalently, the stiffening of
compliant grain contacts) tends to increase the effective elastic moduli of rocks (see
also Section 2.5 on third-order elasticity).
When the crack population is anisotropic, either in the original unstressed condition or as a result of the stress field, then this condition can impact the overall elastic
anisotropy of the rock. Laboratory demonstrations of stress-induced anisotropy have
been reported by numerous authors (Nur and Simmons, 1969a; Lockner et al., 1977;
Zamora and Poirier, 1990; Sayers et al., 1990; Yin, 1992; Cruts et al., 1995).
The simplest case to understand is a rock with a random (isotropic) distribution of
cracks embedded in an isotropic mineral matrix. In the initial unstressed state, the
rock is elastically isotropic. If a hydrostatic compressive stress is applied, cracks in
all directions respond similarly, and the rock remains isotropic but becomes stiffer.
However, if a uniaxial compressive stress is applied, cracks with normals parallel or
nearly parallel to the applied-stress axis will tend to close preferentially, and the rock
will take on an axial or transversely isotropic symmetry.
An initially isotropic rock with arbitrary stress applied will have at least orthorhombic symmetry (Nur, 1971; Rasolofosaon, 1998), provided that the stress-induced
changes in moduli are small relative to the absolute moduli.
Figure 2.7.1 illustrates the effects of stress-induced crack alignment on seismicvelocity anisotropy discovered in the laboratory by Nur and Simmons (1969a). The
crack porosity of the dry granite sample is essentially isotropic at low stress.
As uniaxial stress is applied, crack anisotropy is induced. The velocities (compressional and two polarizations of shear) clearly vary with direction relative to the stressinduced crack alignment. Table 2.7.1 summarizes the elastic symmetries that result
when various applied-stress fields interact with various initial crack symmetries
(Paterson and Weiss, 1961; Nur, 1971).
A rule of thumb is that a wave is most sensitive to cracks when its direction of
propagation or direction of polarization is perpendicular (or nearly so) to the crack faces.
The most common approach to modeling the stress-induced anisotropy is to assume
angular distributions of idealized penny-shaped cracks (Nur, 1971; Sayers, 1988a, b;
Gibson and Toksoăz, 1990). The stress dependence is introduced by assuming or
inferring distributions or spectra of crack aspect ratios with various orientations.
48
Elasticity and Hooke’s law
5.0
Stress (bars)
300
4.6 200
VP (km/s)
4.8
4.4
100
4.2
4.0
0
3.8
3.6
0
20
40
60
80
Angle from stress axis (deg)
3.2
3.2
300
2.9
200
2.8
100
2.7
Vs (SV) (km/s)
Vs (SH) (km/s)
400
3.0
2.6
Stress (bars)
3.1 400
Stress (bars)
3.1
200
2.9
2.8 100
2.7
0
0
300
3.0
20
40
60
80
Angle from stress axis (deg)
2.6
0
0
20
40
60
80
Angle from stress axis (deg)
Figure 2.7.1 The effects of stress-induced crack alignment on seismic-velocity anisotropy measured
in the laboratory (Nur and Simmons, 1969a).
The assumption is that a crack will close when the component of applied compressive stress normal to the crack faces causes a normal displacement of the crack faces
equal to the original half-width of the crack. This allows us to estimate the crack
closing stress as follows:
close ¼
3pð1 À 2nị
p
aK0 ẳ
am
41 n2 ị
21 nị 0
where a is the aspect ratio of the crack, and n, m0, and K0 are the Poisson ratio, shear
modulus, and bulk modulus of the mineral, respectively (see Section 2.9 on the
deformation of inclusions and cavities in elastic solids). Hence, the thinnest cracks
will close first, followed by thicker ones. This allows one to estimate, for a given
aspect-ratio distribution, how many cracks remain open in each direction for any
applied stress field. These inferred crack distributions and their orientations can be
put into one of the popular crack models (e.g., Hudson, 1981) to estimate the resulting
effective elastic moduli of the rock. Although these penny-shaped crack models have
been relatively successful and provide a useful physical interpretation, they are
limited to low crack concentrations and may not effectively represent a broad range
of crack geometries (see Section 4.10 on Hudson’s model for cracked media).
49
2.7 Stress-induced anisotropy in rocks
Dependence of symmetry of induced velocity anisotropy on initial crack
distribution and applied stress and its orientation.
Table 2.7.1
Symmetry of initial
crack distribution
Applied stress
Random
Hydrostatic
Uniaxial
Triaxiala
Axial
Hydrostatic
Uniaxial
Uniaxial
Uniaxial
Triaxiala
Triaxiala
Orthorhombic
Hydrostatic
Uniaxial
Uniaxial
Uniaxial
Triaxiala
Triaxiala
Triaxiala
Orientation of
applied stress
Parallel to axis
of symmetry
Normal to axis
of symmetry
Inclined
Parallel to axis
of symmetry
Inclined
Parallel to axis
of symmetry
Inclined in plane
of symmetry
Inclined
Parallel to axis
of symmetry
Inclined in plane
of symmetry
Inclined
Symmetry of induced
velocity anisotropy
Number of
elastic
constants
Isotropic
Axial
Orthorhombic
2
5
9
Axial
Axial
5
5
Orthorhombic
9
Monoclinic
Orthorhombic
13
9
Monoclinic
13
Orthorhombic
Orthorhombic
9
9
Monoclinic
13
Triclinic
Orthorhombic
21
9
Monoclinic
13
Triclinic
21
Note:
a
Three generally unequal principal stresses.
As an alternative, Mavko et al. (1995) presented a simple recipe for estimating
stress-induced velocity anisotropy directly from measured values of isotropic VP and
VS versus hydrostatic pressure. This method differs from the inclusion models,
because it is relatively independent of any assumed crack geometry and is not limited
to small crack densities. To invert for a particular crack distribution, one needs to
assume crack shapes and aspect-ratio spectra. However, if rather than inverting for a
crack distribution, we instead directly transform hydrostatic velocity–pressure data to
stress-induced velocity anisotropy, we can avoid the need for parameterization in
terms of ellipsoidal cracks and the resulting limitations to low crack densities. In this
sense, the method of Mavko et al. (1995) provides not only a simpler but also a more
general solution to this problem, for ellipsoidal cracks are just one particular case of
the general formulation.
50
Elasticity and Hooke’s law
The procedure is to estimate the generalized pore-space compliance from the
measurements of isotropic VP and VS. The physical assumption that the compliant
part of the pore space is crack-like means that the pressure dependence of the
generalized compliances is governed primarily by normal tractions resolved across
cracks and defects. These defects can include grain boundaries and contact regions
between clay platelets (Sayers, 1995). This assumption allows the measured
pressure dependence to be mapped from the hydrostatic stress state to any applied
nonhydrostatic stress.
The method applies to rocks that are approximately isotropic under hydrostatic
stress and in which the anisotropy is caused by crack closure under stress. Sayers
(1988b) also found evidence for some stress-induced opening of cracks, which is
ignored in this method. The potentially important problem of stress–strain hysteresis
is also ignored.
The anisotropic elastic compliance tensor Sijkl(s) at any given stress state s may be
expressed as
ÁSijkl ị ẳ Sijkl ị S0ijkl
Z p=2 Z 2p
0
0
^ T mÞ
^ À 4W2323
^ T mÞ
^ mi mj mk ml sin y dy d
ẳ
W3333 m
m
ẳ0
yẳ0
Z
ỵ
p=2
yẳ0
Z
0
^ T mị
^ dik mj ml þ dil mj mk þ djk mi ml
W2323
ðm
¼0
Ã
þdjl mi mk sin y dy d
2p
where
1
Siso pị
2p jjkk
1
0
iso
W2323
pị ẳ ẵSiso
abab pị Saabb pị
8p
0
W3333
pị ẳ
The tensor S0ijkl denotes the reference compliance at some very large confining
hydrostatic pressure when all of the compliant parts of the pore space are closed. The
iso
0
expression ÁSiso
ijkl ðpÞ ¼ Sijkl ðpÞ À Sijkl describes the difference between the compliance under a hydrostatic effective pressure p and the reference compliance at high
pressure. These are determined from measured P- and S-wave velocities versus the
0
0
hydrostatic pressure. The tensor elements W3333
and W2323
are the measured normal
and shear crack compliances and include all interactions with neighboring cracks and
pores. These could be approximated, for example, by the compliances of idealized
ellipsoidal cracks, interacting or not, but this would immediately reduce the general^ ðsin y cos ; sin y sin ; cos yÞT denotes the unit normal to the
ity. The expression m
crack face, where y and f are the polar and azimuthal angles in a spherical coordinate
system, respectively.