ibp2304_10 rio oil gas 2010 rev 6

______________________________ 1 Ph.D., Geomechanics Advisor – Schlumberger Brazil Research and Geoengineering Center

IBP2304_10

DETERMINATIONS OF THE DYNAMIC ELASTIC

CONSTANTS OF A TRANSVERSE ISOTROPIC ROCK BASED

ON BOREHOLE DIPOLE SONIC ANISOTROPY IN DEVIATED

WELLS

Marcelo Frydman1

Copyright 2010, Brazilian Petroleum, Gas and Biofuels Institute - IBP This Technical Paper was prepared for presentation at the Rio Oil & Gas Expo and Conference 2010, held between September, 13-

16, 2010, in Rio de Janeiro. This Technical Paper was selected for presentation by the Technical Committee of the event according to

the information contained in the abstract submitted by the author(s). The contents of the Technical Paper, as presented, were not

reviewed by IBP. The organizers are not supposed to translate or correct the submitted papers. The material as it is presented, does

not necessarily represent Brazilian Petroleum, Gas and Biofuels Institute’ opinion, nor that of its Members or Representatives.

Authors consent to the publication of this Technical Paper in the Rio Oil & Gas Expo and Conference 2010 Proceedings.

Abstract Shale is characterized by thin laminate or parallel layering or bedding and is a major component of sedimentary basins.

Clastic sediments, particularly shales, exhibit transversely isotropic properties and their symmetric axes are

perpendicular to the bedding (TIV). For example, sonic well logs acquired in deviated boreholes will reflect different

velocity data than those acquired in vertical boreholes drilled through the same shale. This situation is common in

deepwater environments, where deviated wells are drilled from the same surface location. As a result, the calculated

elastic constants are inconsistent across the field. These anisotropic rock properties play an important role in all aspects

of the exploitation of hydrocarbon reservoirs. A mathematical formulation is presented that determines the dynamic

elastic constants of a transversely isotropic rock based on borehole dipole sonic anisotropy in deviated boreholes. The

anisotropic elastic constants are used in different case studies and the results are compared with those from the

traditional isotropic formulation.

1. Introduction

Although most sedimentary rocks are anisotropic, traditional models assume rock is isotropic. Isotropic

theories require less data and are easier to implement. However, shales are better described as transversely isotropic

rocks that have the symmetric axes perpendicular to bedding. In laboratory measurements of shale, anisotropy as high as

100% has been widely reported for both static (Amadei, 1996) and dynamic conditions (Wang, 2001). Such a high

magnitude of anisotropy cannot be neglected.

Five independent elastic constants are needed to describe the stiffness matrix for a transversely isotropic

material (Lekhnitskii, 1981; Amadei, 1983). The reasons for the petroleum industry’s use of isotropic simplification are

related to a limitation in measuring the dynamic elastic constants (Mavko and Bandyopadhyay, 2009). For the isotropic

material, only two elastic constants are needed, and we can fully characterize those with vertical P-wave (Vp) and S-

wave (Vs) data from a sonic log and with density data (ρ). Cross-dipole sonic data, yielding two vertically polarizations

of shear (Vs1, Vs2) might allow three constants to be estimated, two vertical shear moduli (C44 and C55) and the vertical

X3-axis (C33). However, for vertical wells with horizontal bedding planes, the two vertical shear moduli are equivalent

(C44 = C55), and only two elastic constants are estimated. The horizontally polarized shear wave (C66) can be estimated

from the Stoneley tube wave velocity (Norris and Sinha, 1993; Walsh et al. 2006), and three of the five can obtained.

For vertical wells with horizontal bedding planes, different approximations exist to estimate the remaining two

parameters. However, for deviated wells the axes of the TIV medium (bedding planes) are not normal to the borehole

axis, and thus wave propagation within the borehole is not axis-symmetric. This lack of symmetry introduces a case in

which the compressional and shear waves measured in the wellbore are a function of both horizontal and vertical

stiffness.

Hornby et al. (1999) presented a method to invert anisotropy using well logs from multiple vertical and

deviated wells. Vernik (2008) developed an anisotropy correction based on shale volume (Vsh). However, the full

determination of the five independent elastic constants required remains challenging. It was determined the five dynamic

elastic constants based on the borehole dipole sonic anisotropy, solving an inverse problem. Based on the stiffness

matrix in material coordinates, it is possible to calculate sonic velocities in any direction (Mavko et al., 1998). For the

problem of deviated borehole, there is information on borehole coordinates (Vp , Vs1, Vs2) and the material angles (dip

Rio Oil & Gas Expo and Conference 2010

2

and dip direction). Any approximation (Batugin and Nirenburg, 1972; Schoenberg et al., 1996) is only valid at the

material coordinates. In this paper, the author present the methodology used to solve this inverse problem: the Newton-

Raphson method. This methodology largely concerns layering induced anisotropy and intrinsic anisotropy of shale and

shale-sand laminations; i.e., transversely anisotropic media with bedding-normal symmetry axis.

2. Constitutive Modeling

Many sedimentary rocks have the ability to recover from deformations produced by external forces. The

simplest model for the simulation assumes a linear relationship between the applied forces and the corresponding

deformations. This is called linear elasticity. Under certain conditions, rock behavior follows the linear elastic

assumption, such as when the applied stress is sufficient small and the deformation response is linear. Acoustic wave

propagation in rocks can be studied using the linear elastic assumption.

2.1. Linear Elasticity

The theory of linear elasticity follows the Hooke’s law that states the strain is proportional to the applied

stresses. This can be expressed as

klijklij

klijklij

S

C

σε

εσ

=

= (1)

where Cijkl is known as stiffness tensor, Sijkl is the compliance tensor (Sijkl= Cijkl-1

), and both are fourth-rank tensors

containing 81 components. σ is the stress tensor, while ε is the strain tensor. Since σ and ε are symmetric tensors. Some

other thermodynamic considerations reduce the most general case of Cijkl to 21 independent constants (Carcione 2001).

For most practical applications, it is possibly to simplify the stiffness tensor to consider the symmetry with

respect to some material direction. Figure 1 is a schematic representation of the three types of isotropy commonly

considered in rock mechanics: complete isotropy, transverse isotropy, and orthotropy. Isotropy occurs when the rock has

the same properties in all directions. Orthotropy (orthorhombic symmetry) implies that three orthogonal planes of elastic

symmetry exist and the rock has different properties in the three perpendicular directions. Transverse isotropy describes

a rock that is isotropic within a plane and exhibits different properties perpendicular to this plane.

Figure 1. The three types of isotropy commonly considered in rock mechanics

Using the matrix notation and applying the symmetry assumptions for the orthotropic condition, Equation 1

reduces to


3

=

xy

yz

xz

z

y

x

xy

yz

xz

z

y

x

C

C

C

CCC

CCC

CCC

γ

γ

γ

ε

ε

ε

τ

τ

τ

σ

σ

σ

66

55

44

332313

232212

131211

00000

00000

00000

000

000

000

, (2)

which implies nine independent elasticity constants for the orthotropic condition. One of the most common justifications

for assuming the orthotropic condition in sedimentary sequences is a combination of parallel vertical fractures with

horizontal bedding. If the rock has three perpendicular set of fractures, it will behave as an orthotropic medium. We can

reduce the number of independent elasticity constants in the stiffness matrix further by considering assumptions of

isotropy or transverse isotropy.

2.2. Isotropy

The rock is described as elastic and isotropic if its elastic properties are constant in all directions. For the

isotropic material, only two elastic constants are needed; these can be expressed by

=

xy

yz

xz

z

y

x

xy

yz

xz

z

y

x

C

C

C

CCC

CCC

CCC

γ

γ

γ

ε

ε

ε

τ

τ

τ

σ

σ

σ

66

66

66

111212

121112

121211

00000

00000

00000

000

000

000

, 661112 2CCC −= . (3)

Compare Equations 2 and 3, C11= C22= C33; C12= C13= C23 and C44= C55= C66. The stiffness matrix can be

fully characterized with the vertical P-wave (Vp) and S-wave (Vs) data from the sonic log and with the density log data

(ρ). For the isotropic symmetry, the relationship between the phase velocity of wave propagation and the stiffness

constants is given by

( )22

12

2

66

2

11

2 sp

s

p

VVC

VC

VC

−=

=

=

ρ

ρ

ρ

. (4)

2.3. Transverse Isotropy (TI)

Most rocks are anisotropic. The petroleum industry uses isotropic simplification because of a limitation in

measuring the dynamic elastic constants (Mavko and Bandyopadhyay, 2009). Transverse isotropy in sedimentary

sequences can be related to the following modes (Figure 2): a) intrinsic anisotropy that is due to the constituent plate-

shaped clay particles oriented parallel to each other (Sayers, 2005), b) horizontally or titled layered sedimentary rocks

(each layer can be isotropic in small scale), and c) a system of parallel fractures or microcracks within the rock.


4

a) SEM image of shale (Fjaer et al, 1991).

b) Horizontally layered rock.

c) System of parallel set of fractures

Figure 2. Examples of transverse isotropy in sedimentary sequences

The stiffness matrix for the transverse isotropic condition is expressed as

=

xy

yz

xz

z

y

x

xy

yz

xz

z

y

x

C

C

C

CCC

CCC

CCC

γ

γ

γ

ε

ε

ε

τ

τ

τ

σ

σ

σ

66

44

44

331313

131112

131211

00000

00000

00000

000

000

000

, 1211662 CCC −= , (5)

which implies five independent elasticity constants. Comparing Equations 2 and 5, C11= C22≠ C33, C12≠ C13= C23, and

C44= C55≠ C66. The compliance matrix is expressed as

−−

−−

−−

=

xy

yz

xz

z

y

x

xy

yz

xz

z

y

x

EEE

EEE

EEE

τ

τ

τ

σ

σ

σ

µ

µ

µ

νν

νν

νν

γ

γ

γ

ε

ε

ε

00000

0´0000

00´000

000´1´´´´

000´´1

000´´1

. (6)

The two values for Young’s modulus (E´ and E), Poisson’s ratio (ν´ and ν), and shear modulus (µ´ and µ) can

be expressed as a function of the stiffness constants


5

( )( )

( )( )

.12

;´ ; ;´

2 ;2´

66442

133311

2

131233

1211

13

2

133311

3312

2

1333111211

1211

2

1333

νµµνν

+===

−

−=

+=

−

+−−=

+−=

ECC

CCC

CCC

CC

C

CCC

CCCCCCCE

CC

CCE

(7)

where E, ν and µ define properties in the plane of isotropy, and E’, ν´ and µ’ are properties in a plane containing the

normal to the plane of isotropy.

3. Wave Propagation in a TI Medium

Considering the wave propagation in a plane containing the symmetry axis (z-axis) of a transversely isotropic

medium, the velocities of the three modes are given by

( ) ( )[ ] ( ) ,2sincossin)(

cossin)(

)(cossin)2()(

)(cossin)2()(

22

4413

22

4433

2

4411

2

44

2

66

44

2

33

2

11

21

44

2

33

2

11

21

θθθθ

ρ

θθθ

θθθρθ

θθθρθ

CCCCCCK

CCV

KCCCV

KCCCV

SH

SV

p

++−−−=

+=

−++=

+++=

−

−

(8)

where θ is the angle between the wave vector and the axis of symmetry. Figure 3 shows the longitudinal-wave (Vp) and

shear-wave (VSV and VSH) modes.

Figure 3. Longitudinal and shear-wave modes (Hudson and Harrison, 1997)

Stoneley tube wave velocity is related to the stiffness coefficients (Norris and Sinha, 1993) by

( ) θθ

θθθθθ

θρ 2

66

42

44

4

33

4

13

4

1122

22

cos2

sincos1sinsin2sin

8

1

)(

)(CCCCC

VV

VV

Tf

fT

f +

−−++−=

−, (9)

where VT is the Stoneley wave velocity, Vf is the drilling fluid velocity, and ρf is the drilling fluid density. It is important

to note that the use of Stoneley waves to estimate shear modulus requires an in-gauge hole and no fractures.

3.1. Wave Vector Parallel to the Transversely Isotropic Axis of Symmetry

For vertical boreholes within horizontal bedding planes, the wave vector is perpendicular to the bedding and θ

= 00. Cross-dipole sonic data yielding two vertical polarizations of shear (Vs1, Vs2) might allow three constants to be

estimated: two vertical shear moduli (C44 and C55) and the vertical X3-axis (C33):


6

)0(

)0(

)0(

2

55

2

44

2

33

°=

°=

°=

SV

SH

p

VC

VC

VC

ρ

ρ

ρ

(10)

However, for vertical boreholes within horizontal bedding planes, the two vertical shear moduli are equivalent

(C44 = C55) and only two elastic constants are estimated. The horizontally polarized shear wave (C66) can be estimated

from the Stoneley tube wave velocity (Equation 9, with θ = 0°):

)0(

)0(22

22

66 o

o

Tf

fT

fVV

VVC

−= ρ , (11)

where VT is the Stoneley wave velocity, Vf is the drilling fluid velocity, and ρf is the drilling fluid density.

For vertical wells drilled through horizontal bedding planes, in the best scenario, three of the five independent

elasticity constants can be obtained. Different approximations exist (Batugin and Nirenburg, 1972; Schoenberg et al.,

1996) to estimate the remaining two parameters.

Considering that intrinsic anisotropy is not the dominant factor for the transversely isotropic elasticity (see

Figure 2 for more on modes for transverse isotropy in sedimentary sequences), it is possible to calculate the equivalent

continuous material representative of the rock mass. The compliance matrix (Equation 1) that includes the effect of a set

of fractures must include the additional compliance due to the presence of fractures. If we assume the background rock

is isotropic, the stiffness matrix for the transversely isotropic condition reduces to four independent elasticity constants,

where C12= C13 = C23. This is one assumption of the ANNIE approximation from Schoenberg et al. (1996).

A second approximation is necessary. The ANNIE approximation assumes the Thomsen’s anisotropy

parameter δ (Thomsen, 1986) is 0, which gives

66443311 22 CCCC +−= . (12)

An alternative to δ = 0 is the relationship developed by Batugin and Nirenburg (1972) based on different rocks:

´´)21(

´´ 44

EE

EEC

νµ

++== . (13)

3.2. Wave Vector Deviated to Transversely Isotropic Axis of Symmetry

In deviated wells, the axis of the isotropy planes (bedding planes) are not normal to the borehole axis, and thus

wave propagation within the borehole is not axis-symmetric. This lack of symmetry introduces a case in which the

compressional and shear waves measured in the wellbore are a function of both horizontal and vertical stiffness.

The first ANNIE assumption (Schoenberg et al., 1996), in which C12= C13 = C23=C11-2 C66, was used to reduce

to four independent elastic constants (C11, C33; C44 and C66). The borehole axis is titled through an angle θ with respect

to the TI axis of symmetry. Equations 8 and 9 can be combined as

( )

( ) ( )[ ] ( ) .2sin2cossin)(

0cos2

sin

2

sincos1

8

sin

0cossin

02)(cossin

02)(cossin

22

664411

22

4433

2

4411

22

22

24

66

42

441133

4

4

22

44

2

663

2

44

2

33

2

112

2

44

2

33

2

111

θθθθ

ρθθθ

θθ

ρθθ

ρθθθ

ρθθθ

CCCCCCCK

VV

VVCCCCf

VCCf

VKCCCf

VKCCCf

Tf

fT

f

SH

SV

p

−++−−−=

=−

−

++

−−+−=

=−+=

=−−++=

=−+++=

(14)

Equation 14 can be expressed in the compact form F(X)= 0, where F represents the set of nonlinear equations

and X is the vector of unknowns (C11, C33; C44 and C66). The Newton-Raphson method was used to solve the set of

nonlinear equations, which resulted in an iterative process with the following recurrence formula:


7

( ) ( )( ) 01 =−+ + iiii XXXJXF , (15)

where

=

66

44

33

11

C

C

C

C

X,

=

4

3

2

1

f

f

f

f

F, and J(X

i) is a matrix of first-order partial derivatives of F(X) with respect to X,

termed the Jacobian, evaluated at Xi.

( ) ( )[ ] ( ) ( )

( ) ( )[ ] ( ) ( )

( ) ( )[ ]( ) ( )[ ]( ) ( ) ( )θCCCθθθ-CCθ--CC(θK

θθ-CCθ--CC(θK

θCCC+θ-CCθ--CCθK

θCCCCCCCK

θθ

θθ

θθ

θθ

K(θ

θ(CCC

K(θ

θK

K(θ

θKθ

K(θ

θKθ

K(θ

θ(CCC

K(θ

θK

K(θ

θKθ

K(θ

θKθ

2sin22cossincossin2=)

coscossin)

2sin22cossinsin2)(

2sin2cossin)(

2

sincos

2

sincos1

8

sin

8

sinsincos00

)

)2sin)2(2

)2

)(1

)

)(cos

)2

)(sin

)

)2sin)2(2

)2

)(1

)

)(cos

)2

)(sin

2

664411

222

4433

2

44113

22

4433

2

44112

2

664411

2

4433

2

4411

2

1

22

664411

22

4433

2

4411

42

42

44

22

2

66441132212

2

66441132212

−+++−

=

−+=

−++−−−=

+−−−

−+−+−

−+−+−+

=

θ

θθθ

J

4. Results of Investigation

The anisotropic elastic constants were used in different case studies and the results were compared with those

from the traditional isotropic formulation. Initially, the dynamic elastic constants were calculated for a transversely

isotropic rock in an inclined borehole following the methodology described previously.

4.1. Effect of Sonic Velocity Anisotropy on Estimate of Pore Pressure

An accurate knowledge of pore pressure is necessary for wellbore-stability modeling and appropriate well

design. Pore pressure can be determined using several methods, each typically relating velocity to the pressure signal in

the formation that is due to undercompaction. Predrill pore pressure predictions are often obtained from sonic log

velocities using an empirical velocity–to–pore pressure transform (Gutierrez et al., 2006).

Compaction disequilibrium is one of the main causes of overpressure. This concept is associated with a

timescale that is short for the sedimentation compared with that for the fluid expulsion from the porous space and

consequently the pore pressure increase. When considering one-dimensional compaction (vertical consolidation), it is

usually assumed that the elastic wave velocity is a function only of the vertical effective stress. Since compaction is

predominately an inelastic process, Terzaghi's theory of one-dimensional consolidation is used for the effective stress

definition. Examples of the use of the vertical effective stress (σV’) to predict pore pressures include the methods of

Bowers (1994) and Eaton (1975). The method of Bowers (1994) is based on the following assumed empirical relation

between the vertical effective stress and the velocity:

B

p

VVA

vvp

/1

0'

−=−= σσ , (16)

where vp is the P-wave sonic velocity, v0 is the velocity of the unconsolidated fluid-saturated sediments (taken in this

paper to be 5,000 ft/s), and A and B describe the variation in velocity with increasing vertical effective stress (where A =

9.185 and B = 0.765). Note that in Equation 16 a velocity value for an anisotropic medium is not defined. However,

the same justification that relates the one-dimensional compaction to vertical effective stress suggests the vertical P-

wave velocity should be used.


8

This experiment investigated the effect of sonic velocity anisotropy on estimates of pore pressure. Some

properties of the modeling material used and stresses are described in Table 1.

Table 1. Properties of the modeling material and stresses

Property Value

TVD 3500 m

Water depth 1000 m

Overburden stress 9163 psi

P-wave sonic velocity (Vertical θ = 00) 2566 m/s

S-wave velocity (Vertical θ = 00) 1040 m/s

Bulk density 2.24 g/cm3

Thomsen’s anisotropy parameter ε 0.25

Thomsen’s anisotropy parameter γ 0.25

The stiffness elastic constants C33 and C44 are calculated based on Equation 10. Thomsen’s anisotropy

parameters ε and γ are used to calculate C11 and C66 based on Equation 17 (Thomsen, 1986), and it was considered the

first assumption of the ANNIE approximation (Schoenberg et al., 1996), where C12= C13 = C11-2 C66:

33

3366

33

3311

2 ;

2 C

CC

C

CC −=

−= γε (17)

The estimated pore pressure through Bowers’ method is shown is Figure 4. The P-wave velocity through an

angle θ with respect to the TI axis of symmetry is represented in blue (Equation 8). The figure shows an increase of

longitudinal velocity occurs when the well is drilled parallel to bedding (θ = 90°). The calibrated pore pressure is

represented in green at 6866 psi, and the equivalent circulation density at 11.5 lbm/gal. The estimated pore pressure

obtained through Bowers’ method without any anisotropic consideration (Equation 16) is shown in red. It

underestimates the pore pressure by 1787 psi, or 3 lbm/gal.

2500

2600

2700

2800

2900

3000

3100

3200

3300

3400

3500

5000

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

0 10 20 30 40 50 60 70 80 90

P-w

av

e v

elo

city

(m/s

)

Po

re p

ress

ure

(psi

)

Brorehole angle with respect to the TI axis of symmetry (degrees)

Estimated Pp (psi)

Calibrated Pp (psi)

Vp (m/s)

Figure 4. Effect of sonic velocity anisotropy on estimate of pore pressure

4.2. Effect of Sonic Velocity Anisotropy on Estimate of Horizontal Stress Profile

In the second case study, the stress profile distribution was calculated for on a transversely isotropic rock.

Thiercelin and Plumb (1991) developed the elastic equations that relate the effective horizontal stress with gravity

loading and horizontal strain as


9

,11

'1

'

''

22 HhVh

EE

E

Eε

ν

νε

νσ

ν

νσ

−+

−+

−= (18)

where εh and εH are respectively the tectonic strain in the minimum horizontal stress direction and the maximum

horizontal stress direction. Leak-off test, minifracturing, microfracturing, and hydraulic fracturing are data points used to

calibrate for a more complete stress profile. There are other constraints that can be used to help build a horizontal stress

profile. These include breakouts and drilling-induced fractures identified from image logs and drilling events (caving,

losses, etc.) in near-by wells. Frydman and Ramirez (2006) detailed the process of horizontal stress profile calibration

through geomechanical modeling.

Figure 5 presents the minimum horizontal stress profile calculated based on isotropic (blue) and TIV conditions

(green). The red curve is the clay volume (VCL). There is not much difference in this example between the isotropic and

TIV conditions for the sandstones, but the shale barriers are clearly anisotropic.

0

0.5

1

1.5

2

2.5

0

1000

2000

3000

4000

5000

6000

7000

8000

6240 6260 6280 6300 6320 6340 6360 6380 6400 6420

VC

L (V

/V)

σσ σσh

(psi

)

TVD (ft)

Sh_Isotropic

Sh_TIV

VCL

Sandstone

Shale

Figure 5. Effect of sonic velocity anisotropy on estimate of horizontal stress

This stress profile is further used to calculate the propagation of a hydraulic fracture during stimulation through

a pseudo-3D fracturing simulator. Figure 5 compares the results of the fracture geometry of the TIV model (left) and the

isotropic assumption (right). The isotropic assumption underestimates the stresses in the shales, which would result in a

fracture propagating into the barriers. The fracture is more confined in the TIV model, which is more consistent with

field experience. An accurate horizontal stress profile reduces the potential for inappropriate treatments that will result

in early screen-outs or in failure to cover the entire pay zone. At the same time, it adds confidence that fracture growth

will be limited inside barriers, thereby avoiding gas or water zones.

Figure 6. Comparison of fracture geometry from TIV model (left) and isotropic assumption (right)

5. Conclusions


10

Although most sedimentary rocks are anisotropic, the majority of current geomechanics models use simple

isotropic assumptions. This paper discusses one of the required steps to move forward to a more realistic anisotropic

(TIV) model. The elastic stiffnesses for these anisotropic rocks play an important role in all aspects of hydrocarbon

exploitation. During geophysical investigations, elastic properties are used to obtain reliable time-to-depth conversion,

pore pressure from seismic data, and seismic imaging in sedimentary basins. Drilling problems associated with wellbore

instability are normally aggravated across shale layers. During a hydraulic fracture operation, both orientation and

propagation of the fractures are strongly influenced by the stress distribution (magnitude and direction) in the reservoir

and surrounding rocks (mainly shales). During production, the pressure depletion during withdrawal or the pressure

increase during injection will affect the state of stress inside the reservoir and its overburden (mainly shales). These

stress changes may cause reservoir compaction, changes in reservoir performance, movement of the overburden,

reactivation of faults, and other compromises to the well integrity.

The obtained results contribute to the understanding of the material properties and stress development in a

transversely isotropic rock. The methodology developed has allowed the full determination of the dynamic elastic

constants of a transversely isotropic rock based on borehole dipole sonic anisotropy in deviated boreholes.

6. References

AMADEI, B. Rock anisotropy and theory of stress measurements. In: Brebbia, Orszag Eds, Lecture Notes in

Engineering, Springer Verlag, 1983.

AMADEI, B. Importance of anisotropy when estimating and measuring in situ stress in rock, Int. J. Rock Mech. Min.

Sci. & Geomech. Abstr. v. 33, n. 3, 1996.

BATUGIN, S.A., NIRENBURG, R.K. Approximate relation between the elastic constants of anisotropic rocks and the

anisotropy parameters, translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznikh Iskopaemykh, n. 1, p.

7–12, Jan.–Feb. 1972.

BOWERS, G. L. Pore pressure estimation from velocity data: Accounting for overpressure mechanisms besides

undercompaction. Paper SPE 27488, 1994.

CARCIONE, J. M. Wave fields in real media: wave propagation in anisotropic, anelastic and porous media. Seismic

Exploration Volume 31, PERGAMON An Imprint of Elsevier Science, 2001.

EATON, B. A. The equation for geopressure prediction from well logs. Paper SPE 5544, 1975.

FJAER, E., HOLT, R. M., HORSRUD, P., RAAEN, A. M., RISNES, R. Petroleum Related Rock Mechanics, Elsevier

Science Publishers, 1991.

FRYDMAN, M., RAMIREZ, H.A. Using breakouts for in situ stress estimation in tectonically active areas. In:

GoldenRocks 41st U.S. Rock Mechanics Symposium, A. ARMA/USRMS 06-985, Denver, June 17–21 2006.

GUTIERREZ, M. A., BRAUNSDORF, N. R., COUZENS, B. A. Calibration and ranking of pore-pressure prediction

models, The Leading Edge, v. 25, n. 12, p. 1516–1523, 2006.

HORNBY, B. E., HOWIE, J. M., INCE, D. W. Anisotropy correction for deviated well sonic logs: Application to

seismic well tie. In: SEG Extended Abstracts for 69th

Annual International Meeting, p. 112–115, 1999.

HUDSON, J. A., HARRISON, J. P. Engineering rock mechanics—An introduction to the principles. Elsevier Science

Ltd, 1997.

LEKHNITSKII, S G. Theory of elasticity of an anisotropic body, Mir Publishers, 1981.

MAVKO, G., MUKERJI, T., DVORKIN, J. The rock physics handbook, Cambridge University Press, 1998.

MAVKO, G., BANDYOPADHYAY, K. Approximate fluid substitution for vertical velocities in weakly anisotropic

VTI rocks, Geophysics, v. 74, n. 1, 2009.

NORRIS, A., SINHA, B. Weak elastic anisotropy and the tube wave. Geophysics, 58, p 1091–1098, 1993.

SAYERS, C. M. Seismic anisotropy of shales, Geophysical Prospecting, v. 53, p. 667–676, 2005.

SCHOENBERG, M., MUIR, F., SAYERS, C. Introducing Annie: A Simple Three-Parameter Anisotropic Velocity

Model for Shales, Journal of Seismic Exploration, 5, 35-49, 1996.

THIERCELIN, M. J., PLUMB, R. A. A Core-Based Prediction of Lithologic Stress Contrasts in East Texas Formations,

SPE Formation Evaluation, 1991.

THOMSEN, L. Weak elastic anisotropy, Geophysics, vol. 51, n. 10, p. 1954-1966, 1986.

VERNIK, L. Anisotropic correction of sonic logs in wells with large relative dip, Geophysics, v.73, n.1,[need page

numbers] 2008.

WANG, Z. Z. Fundamentals of seismic rock physics, Geophysics, v. 66, n. 2, 2001.

WALSH, J., SINHA, B, DONALD, D. Formation anisotropy parameters using borehole sonic data. In the 47th Annual

Logging Symposium Transactions, Society of Petrophysicists and Well Log Analysts, 2006.

ibp2304_10 rio oil gas 2010 rev 6

Documents