Mavko G., Mukerji T., Dvorkin J. The Rock Physics Handbook

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e 



ðÞV



ðÞ



ðÞ

Therefore, e best describes what is usually called the “P-wave anisotropy.”

Similarly, for weak anisotropy the constant g can be seen to describe the fractional

difference between the SH-wave velocities parallel and orthogonal to the symmetry axis,

which is equivalent to the difference between the velocities of S-waves polarized parallel

and normal to the symmetry axis, both propagating normal to the symmetry axis:

g 



ðÞV



ðÞ



ðÞ



ðÞV



ðÞ



ðÞ

The small-offset normal moveout velocity is affected by VTI (transversely isotropic

with vertical symmetry axis) anisotropy. In terms of the Thomsen parameters, NMO

(normal moveout) velocities, V

NMO;P

, V

NMO;SV

, and V

NMO;SH

for P-, SV-, and SH-

modes are (Tsvankin, 2001):

NMO;P

¼ a

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ 2d

NMO;SV

¼ b

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ 2

;¼



ðe  dÞ

NMO;SH

¼ b

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ 2g

An additional anellipticity parameter, , was introduced by Alkhalifah and Tsvankin

(1995):

 ¼

e  d

1 þ 2d

 is important in quantifying the effects of anisotropy on nonhyperbolic moveout

and P-wave time-processing steps (Tsvankin, 2001), including NMO, DMO (dip

moveout), and time migration.

The Thomsen parameters can be inverted for the elastic constants as follows:

¼ a

; c

¼ b

¼ c

1 þ 2eðÞ; c

¼ c

1 þ 2gðÞ

¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

 c

ðÞ

d þ c

 c

ðÞ

 c

Note the nonuniqueness in c

that results from uncertainty in the sign of (c

þ c

Tsvankin (2001, p. 19) argues that for most cases, it can be assumed that (c

þ c

) > 0,

and therefore, the þsign in the equation for c

is usually appropriate.

Tsvankin (2001) summarizes some bounds on the values of the Thomsen

parameters:



the lower bound for d: d 1  b



=2;



an approximate upper bound for d: d  2= a

 1



; and



in VTI materials resulting from thin layering, e > d and g > 0.

37 2.3 Thomsen’s notation for weak elastic anisotropy

Transversely isotropic media consisting of thin isotropic layers always have

Thomsen (1986) parameters, such that e – d  0andg  0 (Tsvankin, 2001). Berryman

et al. (1999) find the additional conditions summarized in Table 2.3.1, based on

Backus (1962) average analysis and Monte Carlo simulations of thinly laminated

media. Although all of the cases in the table have d  0, Berryman et al. find that

d can be positive if the layers have significantly varying and positively correlated

shear modulus, m, and Poisson ratio, n.

Bakulin et al. (2000) studied the Thomsen parameters for anisotropic HTI (trans-

versely isotropic with horizontal symmetry axis) media resulting from aligned vertical

fractures with crack normals along the horizontal x

-axis in an isotropic background.

For example, when the Hudson (1980) penny-shaped crack model (Section 4.10)

is used to estimate weak anisotropy resulting from crack density, e, the dry-rock

Thomsen parameters in the vertical plane containing the x

-axis can be approximated as

ðVÞ

dry

 c

¼

e  0;

ðVÞ

dry

ðc

þ c

ðc

 c

ðc

 c

¼

e 1 þ

gð1  2gÞ

ð3  2gÞð1  gÞ



 0;

ðVÞ

dry

 c

¼

3ð3  2gÞ

 0;



ðVÞ

dry

ðVÞ

dry

 d

ðVÞ

dry

1 þ 2d

ðVÞ

dry

gð1  2gÞ

ð3  2gÞð1  gÞ



 0

where g ¼ V

is a property of the unfractured background rock. In the case of

fluid-saturated penny-shaped cracks, such that the crack aspect, a, is much less than

the ratio of the fluid bulk modulus to the mineral bulk modulus, K

fluid

mineral

, then

the weak-anisotropy Thomsen parameters can be approximated as

ðVÞ

saturated

 c

¼ 0

ðVÞ

saturated

ðc

þ c

ðc

 c

ðc

 c

¼

32ge

3ð3  2gÞ

 0

Table 2.3.1 Ranges of Thomsen parameters expected for thin laminations of

isotropic materials (Berryman et al., 1999).

m ¼ const

l ¼ const

m 6¼ const

lþm ¼ const

m 6¼ const

lþ2m ¼ const

m 6¼ const

n ¼ const

l, m 6¼ const

e 0 00 0 0

d 0 0 0 00

g 0 0 0  0 0

e – d 0 0 0 0 0

38 Elasticity and Hooke’s law

ðVÞ

saturated

 c

¼

3ð3  2gÞ

 0



ðVÞ

saturated

¼d

ðVÞ

saturated

32ge

3ð3  2gÞ

 0

In intrinsically anisotropic shales, Sayers (2004) finds that d can be positive or

negative, depending on the contact stiffness between microscopic clay domains and

on the distribution of clay domain orientations. He shows, via modeling, that distri-

butions of clays domains, each having d  0, can yield a composite with d > 0 overall

if the domain orientations vary significantly from parallel.

Rasolofosaon (1998) shows that under the assumptions of third-order elasticity an

isotropic rock obtains ellipsoidal symmetry with respect to the propagation of qP

waves. Hence e ¼ d in the symmetry planes (see Section 2.4) on Tsvankin’s extended

Thomsen parameters.

Uses

Thomsen’s notation for weak elastic anisotropy is useful for conveniently character-

izing the elastic constants of a transversely isotropic linear elastic medium.

Assumptions and limitations

The preceding equations are based on the following assumptions:



material is linear, elastic, and transversely isotropic;



anisotropy is weak, so that e, g, d  1.

2.4 Tsvankin’s extended Thomsen parameters for orthorhombic media

The well-known Thomsen (1986) parameters for weak anisotropy are well suited for

transversely isotropic media (see Section 2.3). They allow the five independent elastic

constants c

, c

, and c

to be expressed in terms of the more intuitive

P-wave velocity, a, and S-wave velocity, b, along the symmetry axis, plus additional

constants e, g , and d. For orthorhombic media, requiring nine independent elastic

constants, the conventional Thomsen parameters are insufficient.

Analogs of the Thomsen parameters suitable for orthorhombic media can be

defined (Tsvankin, 1997), recognizing that wave propagation in the x

–x

and x

–x

symmetry planes (pseudo-P and pseudo-S polarized in each plane and SH polarized

normal to each plane) is analogous to propagation in two different VTI media. We

once again define vertically propagating (along the x

-axis) P- and S-wave velocities,

a and b, respectively:

a ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

=

; b ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

=

39 2.4 Tsvankin’s extended Thomsen parameters

Unlike in a VTI medium, S-waves propagating along the x

-axis in an orthorhombic

medium can have two different velocities, b

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

=

and b

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

=

, for

waves polarized in the x

and x

directions, respectively. Either polarization can be

chosen as a reference, though here we take b ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

=

following the definitions of

Tsvankin (1997). Some results shown in later sections will use redefined polariza-

tions in the definition of b.

For the seven constants, we can write

ð2Þ

 c

ð2Þ

ðc

þ c

ðc

 c

ðc

 c

ð2Þ

 c

ð1Þ

 c

ð1Þ

ðc

þ c

ðc

 c

ðc

 c

ð1Þ

 c

ð3Þ

ðc

þ c

ðc

 c

ðc

 c

Here, the superscripts (1), (2), and (3) refer to the TI-analog parameters in the

symmetry planes normal to x

, x

, and x

, 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 x

direction.

These Thomsen–Tsvankin parameters play a useful role in modeling wave propa-

gation and reflectivity in anisotropic media.

Uses

Tsvankin’s notation for weak elastic anisotropy is useful for conveniently character-

izing 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

40 Elasticity and Hooke’s law

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

ijkl

ijklmn

where c

ijkl

and c

ijklmn

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 c

ijkl

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 (c

ijkl

) 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 (c

ijklmn

) 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,

eff

, which determines the speeds of infinitesimal-amplitude waves in a rock

under applied static stress can be written as

eff

ijkl

¼ c

ijkl

þ c

ijklmn

where e

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)

eff

 c

þ c

111

þ c

112

ðe

þ e

eff

 c

þ c

111

þ c

112

ðe

þ e

eff

 c

þ c

111

þ c

112

ðe

þ e

eff

 c

þ c

112

ðe

þ e

Þþc

123

eff

 c

þ c

112

ðe

þ e

Þþc

123

eff

 c

þ c

112

ðe

þ e

Þþc

123

eff

 c

þ c

144

þ c

155

ðe

þ e

eff

 c

þ c

144

þ c

155

ðe

þ e

eff

 c

þ c

144

þ c

155

ðe

þ e

41 2.5 Third-order nonlinear elasticity

where the constants c

, c

are the VTI elastic constants at the

unstressed reference state, with c

¼ c

 2c

. e

, e

, and e

are the principal

strains, computed from the applied stress using the conventional Hooke’s law,

¼ s

ijkl



. For these expressions, it is assumed that the direction of the applied

principal stress is aligned with the VTI symmetry (x

-) axis. Furthermore, for

these expressions it is assumed that the stress-sensitive third-order tensor is

isotropic, defined by the three independent constants, c

111

, c

112

, and c

123

with

144

¼ c

112

 c

123

ðÞ=2, c

155

¼ c

111

 c

112

ðÞ=4, and c

456

¼ c

111

 3c

112

þ 2c

123

ðÞ=8.

It is generally observed (Prioul et al., 2004) that c

111

< c

112

< c

123

, c

155

< c

144

155

< 0, and c

456

< 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 (c

111

, c

112

, c

123

)

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)

Table 2.5.1 Experimentally determined third-order elastic constants c

111

, c

112

and c

123

and derived constants c

144

, c

155

, and c

456

, determined by Prioul and

Lebrat (2004), using laboratory data from Wang (2002). Six different sandstone

and six different shale samples are shown.

111

(GPa) c

112

(GPa) c

123

(GPa) c

144

(GPa) c

155

(GPa) c

456

(GPa)

Sandstones

10 245 966 966 0 2320 1160

9 482 1745 1745 0 1934 967

6 288 1744 1744 0 1136 568

8 580 527 527 0 2013 1006

8 460 1162 1162 0 1825 912

12 440 3469 3094 188 2243 1027

Shales

6 903 976 976 0 1482 741

4 329 2122 1019 552 552 0

7 034 2147 296  1222 1222 0

4 160 2013 940 536 536

1 294 510 119 196 196 0

1

203 637 354 141 141 0

42 Elasticity and Hooke’s law

111

¼ 2A þ 6B þ 2C ¼ 2l þ 4m

112

¼ 2B þ 2C ¼ 2l

123

¼ 2C ¼ 2l 2m þ n

Chelam (1961) and Krishnamurty (1963) looked at fourth-order elastic coeffi-

cients, 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

–2n þ 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 aniso-

tropy, which is discussed later.

Assumptions and limitations



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.

2.6 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, elec-

trical 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

43 2.6 Effective stress properties of rocks

represented as a function, f

, of a linear combination of the hydrostatic confining

stress, P

, and the pore pressure, P

X ¼ f

ðP

 nP

Þ; n  1

The combination P

eff

¼ P

 nP

is called the effective pressure, or more generally,

the tensor 

eff

¼ 

 nP

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 f

ðP

 nP

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 f

and a different value of

(Zimmerman, 1991a; Berryman, 1992a, 1993; Gurevich, 2004). Extensive discus-

sions 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 behav-

ior for finite pressure changes versus the effective-stress behavior for infinitesimal

increments of pressure. For example, increments of the bulk-volume strain, e

, and

the pore-volume strain, e

, can be written as

ðP

; P

Þ¼C

ðP

 m

ÞðdP

 n

Þ; C

¼



ðP

; P

Þ¼C

ðP

 m

ÞðdP

 n

Þ; C

¼



where the compressibilities C

and C

are functions of P

 mP

. The coefficients

and m

govern the way that the compressibilities vary with P

and P

. In contrast,

the coefficients n

and n

describe the relative increments of additional strain

resulting from pressure increments dP

and dP

. For example, in a laboratory experi-

ment, ultrasonic velocities will depend on the values of C

, the local slope of the

stress–strain curve at the static values of P

and P

. 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 indepen-

dent 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

44 Elasticity and Hooke’s law

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, dV

, of a sample of linear poroelastic material depends on pressure

increments ðdP

 ndP

Þ. For this special case, n ¼ a ¼ 1  K=K

, where a is

known as the Biot coefficient or Biot–Willis coefficient. K is the dry-rock (drained)

bulk modulus and K

is the mineral bulk modulus (or some appropriate average of the

moduli if there is mixed mineralogy), defined below. Explicitly,

¼

 1 





where dV

, dP

, and dP

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:

¼



; K ¼ modulus of the drained porous frame;

¼



; K

¼ unjacketed modulus; if monomineralic;

¼ K

mineral

; otherwise K

is a poorly understood

average of the mixed mineral moduli



¼





; if monomineralic; K

¼ K



¼ K

mineral

¼











45 2.6 Effective stress properties of rocks

where P

¼ P

 P

is the differential pressure, V

is the sample bulk (i.e., total)

volume, and V

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 n

velocity

¼ 1 for

monomineralic, elastic rocks. Experimentally observed values for n

velocity

are some-

times 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.

Table 2.6.1 Theoretically predicted effective-stress laws for incremental

changes in confining and pore pressures (from Berryman, 1992a).

Property General mineralogy

Sample volume

¼

ðdP

 adP

Pore volume



¼

ðdP

 bdP

Porosity

d



¼

a

K



ðdP

 dP

Solid volume

¼

ð1ÞK

ðdP

 dP

Permeability

¼h

a

K



ðdP

 kdP

Velocity/elastic moduli

¼ f ðdP

 yP

Notes:

is the total volume. a ¼ 1  K=K

, Biot coefficient; usually in the range   a  1;

if monomineralic, a ¼ 1  K=K

mineral



¼ V

, pore volume. b ¼ 1  K



, usually,

  b  1, but it is possible that b > 1.

 ¼

b

a



a; if monomineralic,  ¼ 1.

 ¼ a 

a

1a



ða  Þ; if monomineralic,  ¼ .   a  b.

k ¼ 1 

2ð1aÞ

3hðaÞþ2

 1:

h  2 þ m  4, where m is Archie’s cementation exponent.

y ¼ 1 

@ð1=K

Þ=@P

@ð1=KÞ=@P

; if monomineralic,

y ¼ 1.

46 Elasticity and Hooke’s law