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4 Tsvankin's extended Thomsen parameters for orthorhombic media

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



2c44



e1ị ẳ



c23 ỵ c44 ị2 c33 c44 ị2

2c33 c33 c44 ị

c66 c55



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



W3333 m

m

ẳ0



yẳ0



Z







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.



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4 Tsvankin's extended Thomsen parameters for orthorhombic media

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