ibp2304_10 rio oil gas 2010 rev 6
TRANSCRIPT
______________________________ 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
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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
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=
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.
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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
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( )( )
( )( )
.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):
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)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:
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( ) ( )( ) 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.
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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
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,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
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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.
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