Note: Descriptions are shown in the official language in which they were submitted.
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ESTIMATING FORMATION STRESSES USING RADIAL PROFILES OF
THREE SHEAR MODULI
Field of the Invention
[001] The invention is generally related to analysis of subterranean
formations,
and more particularly to estimating formation stresses using radial profiles
of
three shear moduli.
Background of the Invention
[002] Formation stresses can affect geophysical prospecting and development
of oil and gas reservoirs. For example, overburden stress, maximum and
minimum horizontal stresses, pore pressure, wellbore pressure and rock
strength can be used to produce a failure model to aid in well planning,
wellbore stability calculations and reservoir management. It is known that
elastic wave velocities change as a function of prestress in a propagating
medium. For example, sonic velocities in porous rocks change as a function
of effective prestress. However, estimating formation stresses based on
velocity can be problematic because of influences on the horizontal shear
modulus 066. For example, the horizontal shear modulus 066 is reduced in
the presence of horizontal fluid mobility in a porous reservoir. Generally,
the
tube wave slowness decreases by about 2 to 3% in the presence of fluid
mobility, resulting in a decrease in 066 by about 4 to 6%. Conversely, the
horizontal shear modulus 066 is increased in the presence of high clay content
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in a shale interval. Consequently, it is difficult to accurately estimate the
ratio
of vertical to horizontal stress ratios without compensating for the changes
in
C66.
[003] Various devices are known for measuring formation characteristics based
on sonic data. Mechanical disturbances are used to establish elastic waves in
earth formations surrounding a borehole, and properties of the waves are
measured to obtain information about the formations through which the waves
have propagated. For example, compressional, shear and Stoneley wave
information, such as velocity (or its reciprocal, slowness) in the formation
and
in the borehole can help in evaluation and production of hydrocarbon
resources. One example of a sonic logging device is the Sonic Scanner
from Schlumberger. Another example is described in Pistre et al., "A modular
wireline sonic tool for measurements of 3D (azimuthal, radial, and axial)
formation acoustic properties, by Pistre, V., Kinoshita, T., Endo, T.,
Schilling,
K., Pabon, J., Sinha, B., Plona, T., lkegami, T., and Johnson, D.",
Proceedings of the 46th Annual Logging Symposium, Society of Professional
Well Log Analysts, Paper P, 2005. Other tools are also known. These tools
may provide compressional slowness, Atc, shear slowness, Ats, and Stoneley
slowness, Atst, each as a function of depth, z, where slowness is the
reciprocal of velocity and corresponds to the interval transit time typically
measured by sonic logging tools. An acoustic source in a fluid-filled borehole
generates headwaves as well as relatively stronger borehole-guided modes.
A standard sonic measurement system uses a piezoelectric source and
hydrophone receivers situated inside the fluid-filled borehole. The
piezoelectric source is configured as either a monopole or a dipole source.
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The source bandwidth typically ranges from a 0.5 to 20 kHz. A monopole
source primarily generates the lowest-order axisymmetric mode, also referred
to as the Stoneley mode, together with compressional and shear headwaves.
In contrast, a dipole source primarily excites the lowest-order flexural
borehole
mode together with compressional and shear headwaves. The headwaves
are caused by the coupling of the transmitted acoustic energy to plane waves
in the formation that propagate along the borehole axis. An incident
compressional wave in the borehole fluid produces critically refracted
compressional waves in the formation. Those refracted along the borehole
surface are known as compressional headwaves. The critical incidence angle
0, = sin-1(Vi/V,), where Vf is the compressional wave speed in the borehole
fluid; and V, is the compressional wave speed in the formation. As the
compressional headwave travels along the interface, it radiates energy back
into the fluid that can be detected by hydrophone receivers placed in the
fluid-
filled borehole. In fast formations, the shear headwave can be similarly
excited by a compressional wave at the critical incidence angle 0, = sin
1(V1/Vs), where Vs is the shear wave speed in the formation. It is also worth
noting that headwaves are excited only when the wavelength of the incident
wave is smaller than the borehole diameter so that the boundary can be
effectively treated as a planar interface. In a homogeneous and isotropic
model of fast formations, as above noted, compressional and shear
headwaves can be generated by a monopole source placed in a fluid-filled
borehole for determining the formation compressional and shear wave
speeds. It is known that refracted shear headwaves cannot be detected in
slow formations (where the shear wave velocity is less than the borehole-fluid
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compressional velocity) with receivers placed in the borehole fluid. In slow
formations, formation shear velocities are obtained from the low-frequency
asymptote of flexural dispersion. There are standard processing techniques
for the estimation of formation shear velocities in either fast or slow
formations
from an array of recorded dipole waveforms. A different type of tool described
in U.S. Patent No. 6,611,761, issued August 26, 2003 for "Sonic Well Logging
for Radial Profiling" obtains radial profiles of fast and slow shear
slownesses
using the measured dipole dispersions in the two orthogonal directions that
are characterized by the shear moduli c44 and c55 for a borehole parallel to
the
X3-axis in an orthorhombic formation.
Summary of the Invention
[004] In accordance with an embodiment of the invention, a method for
estimating stress in a formation in which a borehole is present comprises:
determining radial profiles of Stoneley, fast dipole shear and slow dipole
shear
slownesses; estimating maximum and minimum horizontal stresses by
inverting differences in far-field shear moduli with difference equations
obtained from radial profiles of dipole shear moduli 044 and 055, and
borehole stresses proximate to the borehole; and producing an indication of
the maximum and minimum horizontal stresses in tangible form.
[005] In accordance with another embodiment of the invention, apparatus
for estimating stress in a formation in which a borehole is present comprises:
at least one acoustic sensor that provides radial profiles of Stoneley, fast
dipole shear and slow dipole shear slownesses; processing circuitry that
estimates maximum and minimum horizontal stresses by inverting differences
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in far-field shear molduli with difference equations obtained from radial
profiles of
dipole shear moduli C44 and C55, and borehole stresses proximate to the
borehole;
and an output that produces an indication of the maximum and minimum
horizontal
stresses in tangible form.
[005a] In accordance with another embodiment of the invention, there is
provided a method for estimating stress in a formation in which a borehole is
present
comprising: determining radial profiles of Stoneley slowness, fast dipole
shear
slowness, and slow dipole shear slowness; calculating dipole shear moduli C44
and
C55 using the fast dipole shear slowness and the slow dipole shear slowness;
calculating modulus C66 using the Stoneley slowness; estimating maximum and
minimum horizontal stresses by inverting (i) difference equations obtained
from
far-field shear moduli C44, C55, and C66, (ii) difference equations obtained
from
radial profiles of dipole shear moduli C44 and C55 at two or more different
radial
positions, and (iii) a relationship between stresses within the formation
proximate to
the borehole and stresses in the far-field of the formation; and producing an
indication
of the maximum and minimum horizontal stresses in tangible form.
[005b] In accordance with another embodiment of the invention, there
is
provided an apparatus for estimating stress in a formation in which a borehole
is
present comprising: at least one acoustic sensor that provides radial profiles
of
Stoneley, fast dipole shear and slow dipole shear slownesses; processing
circuitry
that estimates maximum and minimum horizontal stresses by inverting (i)
differences
in far-field shear moduli C44, C55, and C66, (ii) difference equations
obtained from
radial profiles of dipole shear moduli C44 and C55 at two or more different
radial
positions, and (iii) a relationship between stresses within the formation
proximate to
the borehole and stresses in the far-field of the formation; and an output
that
produces an indication of the maximum and minimum horizontal stresses in
tangible
form.
[005c] In accordance with another embodiment of the invention, there
is
provided a method for estimating stress ratio in a formation in which a
borehole is
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present comprising: determining radial profiles of Stoneley slowness for the
formation
and dipole shear slownesses for the formation; calculating at least one of
dipole
shear moduli C44 and C55 using the dipole shear slownesses; calculating
modulus
C66 using the Stoneley slowness; estimating horizontal to overburden stress
ratio by
inverting (i) a difference equation obtained from far-field shear modulus C66
and at
least one of moduli C44 and C55, (ii) a difference equation obtained from a
radial
profile of at least one of dipole shear moduli C44 and C55 at two or more
different
radial positions within the formation, and (iii) a relationship between
stresses within
the formation proximate to the borehole and stresses in the far-field of the
formation;
and producing an indication of the horizontal to overburden stress ratio in
tangible
form.
[005d] In accordance with another embodiment of the invention, there
is
provided apparatus for estimating stress ratio in a formation in which a
borehole is
present comprising: at least one acoustic sensor that provides radial profiles
of
Stoneley slowness for the formation and dipole shear slownesses for the
formation;
processing circuitry that estimates horizontal to overburden stress ratio by
inverting
(i) a difference equation obtained from far-field shear modulus C66 and at
least one
of moduli C44 and C55, (ii) a difference equation obtained from a radial
profile of at
least one of dipole shear moduli C44 and C55 at two or more different radial
positions
within the formation, and (iii) a relationship between stresses within the
formation
proximate to the borehole and stresses in the far-field of the formation; and
an output
that produces an indication of the horizontal to overburden stress ratio in
tangible
form.
[005e] In accordance with another embodiment of the invention, there
is
provided a method for estimating stress ratio in a formation in which a
borehole is
present comprising: determining compressional slowness for the formation;
determining radial profiles of Stoneley slowness for the formation and dipole
shear
slownesses for the formation; calculating compressional moduli C33 in a far-
field of
the formation using compressional slowness; calculating at least one of dipole
shear
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moduli C44 and C55 in the far-field of the formation using the dipole shear
slownesses; calculating modulus C66 in the far-field of the formation using
the
Stoneley slowness; estimating horizontal to overburden stress ratio by
inverting (i) a
difference equation obtained from C33 in the far-field of the formation at two
different
depths within an interval of the formation, (ii) a difference equation
obtained from at
least one of dipole shear moduli C44 and C55 in the far-field of the formation
at two
different depths within the interval of the formation, and (iii) a difference
equation
obtained from modulus C66 in the far-field of the formation at two different
depths
within the interval of the formation; and producing an indication of the
horizontal to
overburden stress ratio in tangible form.
[005f] In accordance with another embodiment of the invention, there
is
provided apparatus for estimating stress ratio in a formation in which a
borehole is
present comprising: at least one acoustic sensor that provides (i)
compressional
slowness for the formation and (ii) radial profiles of Stoneley slowness and
dipole
shear slownesses for the formation; processing circuitry that estimates
horizontal to
overburden stress ratio by inverting (i) a difference equation obtained from
C33 in the
far-field of the formation at two different depths within an interval of the
formation,
(ii) a difference equation obtained from at least one of dipole shear moduli
C44 and
C55 in the far-field of the formation at two different depths within the
interval of the
formation, and (iii) a difference equation obtained from modulus C66 in the
far-field of
the formation at two different depths within the interval of the formation;
and an output
that produces an indication of the horizontal to overburden stress ratio in
tangible
form.
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Brief Description of the Figures
[006] Figure 1 is a schematic diagram of a borehole in a formation subject to
the far-field principal stresses.
[007] Figure 2 is a schematic diagram of a borehole of radius "a" subject to
formation stresses in a poroelastic formation with pore pressure Pp and
wellbore pressure Pw.
[008] Figure 3 illustrates a logging tool for estimating formation stresses
using
radial profiles of three shear moduli.
[009] Figure 4 is a flow diagram illustrating steps of a method in accordance
with an embodiment of the invention.
[0010] Figure 5 illustrates radial variation of axial (GZZ), hoop (GOO), and
radial
(Grr) effective stresses at an azimuth parallel to the maximum horizontal
stress direction at a given depth.
[0011] Figure 6 illustrates radial variation of axial (GZZ), hoop (cr9e), and
radial
(Gm) effective stresses at an azimuth perpendicular to the maximum horizontal
stress direction.
[0012] Figure 7 illustrates fast and slow shear dispersion curves which
exhibit a
crossing signature together with the lowest-order axisymmetric Stoneley
dispersion
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[0013] Figure 8 illustrates fast and slow shear dispersion curves which do not
exhibit a crossing signature, and the fast and slow dipole dispersions
essentially overlay implying that SHmax = Shmin.
[0014] Figure 9 illustrates an algorithm for solving for the unknowns Ae and
aH/
a V, which enables determination of SH, and an acoustoelastic coefficient Ae.
Detailed Description
[0015] Embodiments of the invention will be described with reference to
principal
stresses illustrated in Figure 1, and triaxial stresses T)oc, Tyy, and Tzz,
together with wellbore
pressure Pw, and pore pressure Pp as illustrated in Figure 2. Also source Sr
and receivers
Re as illustrated in Figures 1 and 2 and liquid Li as illustrated in Figure 2.
The borehole radius
is denoted by "a". As previously discussed, sonic velocities, and therefore
slownesses, are
sensitive to effective stresses in the propagating medium. Effective stress
aij =1-4- ,5õ Pp,
where Ty is the applied stress, 154 is the Kronecker delta, and Pp is the pore
pressure.
[0016] Figure 3 illustrates one example of a logging tool (106) used to
acquire
and analyze data in accordance with an embodiment of the invention. The
tool has a plurality of receivers and transmitters. The illustrated logging
tool
(106) also includes multi-pole transmitters such as crossed dipole
transmitters
(120, 122) (only one end of dipole (120) is visible in Figure 1) and monopole
transmitters (109) (close) and (124) (far) capable of exciting compressional,
shear, Stoneley, and flexural waves. The logging tool (106) also includes
receivers (126), which are spaced apart some distance from the transmitters.
Each receiver may include multiple hydrophones mounted azimuthally at
regular intervals around the circumference of the tool. Other configurations,
such as a Digital Sonic Imaging (DSI) tool with four receivers at each of
eight
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receiver stations, or incorporating other multi-pole sources such as
quadrupole, are also possible. The use of a plurality of receivers and
transmitters results in improved signal quality and adequate extraction of the
various borehole signals over a wide frequency band However, the
distances, number and types of receivers and transmitters shown in this
embodiment are merely one possible configuration, and should not be
construed as limiting the invention.
[0017] The subsurface formation (102) is traversed by a borehole (104) which
may be filled with drilling fluid or mud. The logging tool (106) is suspended
from an armored cable (108) and may have optional centralizers (not shown).
The cable (108) extends from the borehole (104) over a sheave wheel (110)
on a derrick (112) to a winch forming part of surface equipment, which may
include an analyzer unit (114). Well known depth gauging equipment (not
shown) may be provided to measure cable displacement over the sheave
wheel (110). The tool (106) may include any of many well known devices to
produce a signal indicating tool orientation. Processing and interface
circuitry
within the tool (106) amplifies, samples and digitizes the tool's information
signals for transmission and communicates them to the analyzer unit (114) via
the cable (108). Electrical power and control signals for coordinating
operation of the tool (106) may be generated by the analyzer unit (114) or
some other device, and communicated via the cable (108) to circuitry
provided within the tool (106). The surface equipment includes a processor
subsystem (116) (which may include a microprocessor, memory, clock and
timing, and input/output functions--not separately shown), standard peripheral
equipment (not separately shown), and a recorder (118).
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[0018] Figure 4 is a flow diagram that illustrates a method of estimating
formation stresses using radial profiles of three shear moduli. If practical,
a
depth interval characterized by relatively uniform lithology is selected for
evaluation of stresses. Monopole and cross-dipole sonic data is then
obtained with the tool (106, Figure 3) over the selected interval, as
indicated
by step (400). The monopole and cross-dipole data, density (pt,) far field
velocity (Vo, and borehole radius (a) are used to determine radial profiles of
Stoneley, fast dipole shear slowness and slow dipole shear slowness as
indicated by step (402). Formation bulk density (Pb) obtained in step (404) is
then used to calculate radial profiles of the three shear moduli (044, 055,
066) as shown in step (406). The next step is selected based on the result of
the shear moduli calculation.
[0019] Prior to describing selection of the next step, the effects of borehole
presence in a formation will be explained with reference to Figures 5 and 6.
The presence of a borehole causes near-wellbore stress distributions that can
be described by the following equations (Jaeger and Cook, 1969)
igh) a2 (aH ¨ ah) 4a2 3a4 a 2
IfYõ = _________ (l_
(l_ ) + __________________________ (1 + ¨4) COS
20 ¨ Pw ¨, (1)
2 r 2 r 2
r r2
(a H + a h) a2 (0-H ¨ ah ) 3a4 a 2
ago = (1 2 ) _______ (1+ ) cos 28+ Pw , (2)
2 r 2 r r
(aa H ¨ a h) 2a2 3a4 =
re (1+ ¨2 ¨ ---)sin 20, (3)
2 r
r
(:7 H ¨ a h) 2a2
a zz = a v V 2 cos 20, (4)
2 r
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where an, 099, aro, and azz denote the effective radial, hoop, radial-shear,
and axial stresses, respectively; aH,av, and a, represent the effective
far-
field overburden, maximum horizontal, minimum horizontal stresses,
respectively; and "a" denotes the borehole radius, and 0 is the azimuth
measured from the maximum horizontal stress direction. In the absence of a
borehole, the far-field stresses are given by:
a = (cH + ah ) + (aH ¨ ah ) cos 20, (5)
IT
2 2
=
(cH + ah ) (aH ¨ ah) cos 20,
coo (6)
2 2
a
(cH ¨a)h si
= 2 n 20,
re and (7)
azz = av = (8)
Subtracting the far-field stresses from the near-wellbore stresses yields
incremental stresses as the borehole surface is approached as follows:
17 = 2
(aH ah ) a2 (aH ¨ ch )(4a2 3a4 ) cos 28¨ /3 , (9)
a 2
A a ( r 2 ) 2 ' r2
r 4
r
(CH + CO a2 (CH ¨CO 3a4 a 2
2 2 r r
(10)
= (aH ¨ 0-3 (2a2 3a4 )sin 28,
A 0-re
r 4
(1 1 )
2a 2
Aazz = ¨v(aH ¨ a 3 (. ) c 0 s 20,
r
(12)
where 13, denotes the wellbore pressure at a given depth.
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[0020] In the case of an isotropically loaded reference state, shear moduli in
the
three orthogonal planes are the same, i.e., 044= 055 = 066 = . When this type
of formation is subject to anisotropic incremental stresses, changes in the
shear moduli can be expressed as
i Ao-ii
AC55 = [C55 ¨ v C144 (1¨ V) C155 ] __
2,u (1+ v)
[C144 ¨ (1 2v )C55 ¨ 2v C155] Aa22
2,u (1+ v)
i Ao-õ
+ [2,u (1 + v ) + C55 ¨ V C144 + (1 ¨ V )C155 i __ ,
2,u (1+ v)
(13)
where AC55 is obtained from the fast-dipole shear slowness from sonic data
acquired by a dipole transmitter aligned parallel to the X1-direction and
borehole parallel to the X3-direction; the quantities 055, [I., and v are the
linear elastic moduli, whereas 0144 and 0155 are the formation nonlinear
constants in the chosen reference state; and 4,533, Ac, and 4,522,
respectively, denote the effective overburden (parallel to the X3-direction),
maximum horizontal (parallel to the X1-direction), and minimum horizontal
(parallel to the X2-direction) stresses at a chosen depth of interest,
AC44 = [¨ (1+ 2v)C44 + C144 ¨ 2v C155] __ Au"
2,u (1+ v)
+[¨ v C144 C44 (1¨ v) c155] A(722
2,u (1+ v)
i Ao-õ
¨v C144 + (1 ¨ V )C155 i _______________________ ,
2,u (1+ v)
(14)
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where AC44 is obtained from the slow-dipole shear slowness from sonic data
acquired by a dipole transmitter aligned parallel to the X2-direction and
borehole parallel to the X3-direction;
(Aa
AC66 = [u(1 + V ) + C66 -v C144 +(1 ¨ V ) C155 1 J õ + Aa22)
2,u (1+ v)
i
+ [¨ (1+ 2v ) C66 + C144 ¨ 2v C155 i __ ,
2,u (1 + v )
(15)
where AC66 is obtained from the Stoneley shear slowness dispersion from
sonic data acquired by a monopole transmitter at a chosen depth of interest.
Referring to Figures 4 and 7, in the case where the fast and slow shear
dispersion curves exhibit a crossing signature, step (408) is used. Step (408)
includes two component calculations that are made with the aid of overburden
stress (Sv), wellbore pressure (Pw), pore pressure (Pp), Biot coefficient a,
and
borehole radius (a) data, obtained in step (410). A first component
calculation
is to form two difference equations using the far-field shear moduli C44, C55,
and C66. A second component calculation is to form two difference equations
using [C55(r/a=6) ¨ C55(r/a=2)] and [C44(r/a=6) ¨ C55(r/a=2)]. Assuming
that the X1-, X2-, and X3-axes, respectively, are parallel to the maximum
horizontal (aH), minimum horizontal (cm), and vertical (av) stresses,
equations
(13)-(15) yield difference equations in the effective shear moduli in terms of
differences in the principal stress magnitudes through an acoustoelastic
coefficient defined in terms of formation nonlinear constants referred to a
chosen reference state and for a given formation lithology. The following
three equations relate changes in the shear moduli to corresponding changes
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in the effective principal stresses in a homogeneously stressed formation as
would be the case in the far-field, sufficiently away from the borehole
surface:
AC44 ¨ AC66 = AE(Acrõ ¨ Acrii),
(16)
ACõ ¨ AC66 = AE(Acrõ ¨
(17)
ACõ ¨ AC44 = AE(Acrii ¨
(18)
where Aa33, Am 1, and Aa22 denote changes in the effective overburden,
maximum horizontal, and minimum horizontal stresses, respectively; and
C
AE = 2 456 ,
Al
(19)
is the acoustoelastic coefficient, 055 and 044 denote the shear moduli for the
fast and slow shear waves, respectively; C456=(C155-C144)12, is a formation
nonlinear parameter that defines the acoustoelastic coefficient; and p
represents the shear modulus in a chosen reference state. However, only two
of the three difference equations (16), (17), and (18) are independent. The
presence of differential stress in the cross-sectional plane of the borehole
causes dipole shear wave splitting and the observed shear slowness
anisotropy can be used to calculate the acoustoelastic coefficient AE from
equation (18), provided estimates of the three principal stresses as a
function
of depth are available. Note that the dipole shear waves are largely
unaffected by the fluid mobility. The stress-induced change in the Stoneley
shear modulus 066 is then calculated using equations (16) and (17), and the
effective stress magnitudes Aav, AaH, and Aah at a given depth are obtained.
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[0021] The next step (412) is to use an algorithm to solve for the
unknowns.
Combining the two difference equations (16) and (17) in terms of the far-field
shear moduli yields four independent equations to solve for the four unknowns
--- AaH, Aah , C144 and C155. Equations (13), (14), and (15) can be expressed
in terms of the principal stress parameters AaH, Aah, and Aav as follows:
AC55 = [C55 - v C144 + (1 - v) C155 1 j
AaH
- 2,u (1+ v)
h
+ [C144 - (1+ 2v )C55 - 2v C155 ]
2,u (1+ v)
1 Acv
+ [2,u (1 + v ) + C55 - V C144 + (1 - v )C,55 j ,
2,u (1+ v)
(20)
1
AC44 = [¨ (1+ 2v)C44 + C144 ¨ 2v c155]
AaH
- 2,u (1+ v)
h
+ [- v C144 C44 (1 - 11) C155 1 j Aa
2,u (1+ v)
1 Aav
+ [2,u (1 + v ) + C44 - 11 C144 (1 - 11 ) C155 j ,
2,u (1+ v)
(21)
AC66 = [p(1+ v ) + C66 - 11 C144 + (1 - 11 )C155 1 (Aall + Ach )
2,u(l+v)
1 Acv
+ [- (1 + 2v )C66 + C144 - 2v C155 i ,
2,u (1+ v)
(22)
Defining the following non-dimensional parameters
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ACTH Auh Au,
SH = , S=h , S y = ________
2/41+ v) 2/41+ v) 2/41+ v) '
(23)
and
4 = -v c;44+0 -0 cis, 4 = C144 -2v C155 ) (24)
it is possible to express the nonlinear constants 0144 and 0155 in terms of A1
and A2, and the acoustoelastic coefficient AE can also be expressed in terms
of A1 and A2 as shown below:
2vA1 + (1¨ v)A2 A1 + vA2
C144 = _________ , C155
(1 - 2v)(1+ v) (1- 2v)(1+ v) '
(25)
C155 - C144 = Al ¨ A2
1+
(26)
V'
AE = 2 +-1(C155 -C144)=2+ A1-A2 (27)
2,u 2,u(1+ v) '
AC44 - AC66 = AE (Ao-v - Au,), (28)
AC55 ¨ AC66 = AE (Ao-v h).-Au
(29)
An expression for the stress ratio is obtained from equations (16) and (17)
Au\Au AC -AC (AC -AC \Au
h=i 55 66+ 55 66 H 44 55 + 55 66 H
&Tv AC44 -AC66 0E44 -AC66 j Ao-v AC44 -AC66 0C44 - AC66) Aav '
30)
Subtracting equation (28) from (29), and substituting for AE from equation
(27), yields
l -uh
AC55 - AC44 = [214(1 + V) + A1- AdAu H
2,u(1A+ v) '
(31)
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This results in one of the two equations relating A1 and A2
AC55 - 44
A1¨A2 = 2,u(1+ v) ___ AC .
H Auh
(32)
Different A1 and A2 values can be obtained for different Acri, /Acry as
follows
AC57 (rla= 6) ¨AC59'5 )(rIa= 2) = + AASH"(rla = ¨ SH6"(rla=2)1
+(¨ ,u(1+ 2v) + A)[S h6"(r1 a = 6) ¨ S h6" (r I a =2)]
+(u+ A,)[S,6"(r I a = 6) ¨ S,6"(r I a= 2)]
(33)
AC57 (rla= 6) ¨ AC57 (rla= 2)
(ii+A) 11111 - +1111 ' +1111117
Ao-v 0-v
+ (¨ itt(1+ 2v) + A2) h 2"
_ V
(34)
where
177 17
1HH
2592 32
105 9
2HH
2592 32
1 1
j3HH_
36 4
h 33 1
_
2592 32
(35)
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hh= 39 7
2 2592 32 '
,t hh 1 1
36 4
2 1
"1 =¨ ¨36 ,
2 1
J "2 =T6¨ =
Equation (34) can be rewritten in a compact form shown below
X
= CU + Ai)(Si + S3) + (¨,u(1+ 2v) + A2)S2 ,
Au,
(36)
where
Xc ¨ AC5e5 (r 1 a 6) AC5e5 (r 1 a 2)
S1 = " 11
Ji ¨ail+ j2HH ah + j311 PW
(IV aV aV
S2= Jihh-c7H+ j2hh (3-h + j3hh PW
0-v 0-v
(37)
q3 = jivv CH + j2vv C717
UV UV
Similarly,
AC:94=90 (rla= 6) ¨ AC57 (rla= 2) = (eu + AAS:=9 (rla = 6) ¨ S He' (r1 a =
2)1
+ (¨ ,u(1+ 2v) + A2 )[S :=9 (r I a = 6)¨ S he= (r I a = 2)]
+ (it + AAS ve=9 (r I a = 6) ¨ Svelr/ a = 2)]
(38)
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AC484=9 (r I a =6) - AC57 (r I a =2)
(I1 + 4 )[V1hil 1 + V2hil ' + V3hil 2'713 1
Aav av av av
+ (-,u(1+ 2v) + A2 )Hillh 1+ V211h ' + V311h 2171:1 1
CV CV CV
+ (Al + A1V1vv 1+ K H
CV CV
(39)
where
yi hH = 105 17
2592 32 '
177 9
yi h2 H_
,
2592 32
H 1 1
36 4
yf Hh= 39 1
2592 32 '
yi Hh= 33 7
2592 32 '
(40)
wHh 1
_ 1 _
36 4 '
36 2 '
1 .
36 2
Equation (39) can be rewritten in a compact form as follows:
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Yc
_______________________________________________________ = Cu + Ai)(S 4 + S 6)
+ (¨,u(1+ 2v) + A2)S 5 , (41)
Au,
where
Yc = AC 449=9 (r I a =6) - AC57 (r I a =2) ,
S4 = VhH 0-H+ hH 0-h hH Pw
I ¨V12
a-v a (7-
v v
0h u P
S5 = v iHh H + v 2H h + v311h
[ W1 ,
0 0 0
V V V
(42)
vv 1 + V122.v 1
s6= Vi
Civ
Solving
Xc ___________________________________________________ = (II + Ai )(Si + S3) +
(-11(1 + 2v) + A2)S2 , (43)
Ao-1,
Yc = cu+Aixs4 + so + (-,u(i+ 2v) + A2)S5 , (44)
Au,
for A1 and A2, yields
S2G2 S5Gi
Al= (45)
S2(S4 + S6)¨ S5(S1+ S3) '
A (S4 + S6)G1 (Si+ S3)G2
112 = (46)
S2 (S4 + S6 ) ¨ S5 (Si+ S3) '
where
Xr , ,
G1 = c + L itil,-)1 + S3) + ,u(1 + 2v)S21 , (47)
Auv
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Y
G2= ______________________ c + [ 11(S4 + S6) + /(l + 2V)S51 . (48)
Ao-v
[0022] SHmax, Shmin, and an acoustoelastic coefficient Ae are calculated in
step (414). The acoustoelastic parameter Ae is calculated as a function the
stress ratio aH/av using equation (28). ah/av is calculated in terms of aidhav
using equation (30). True Ae is calculated in terms of 055, 044, GH, and Gil
using equation (18). Al and A2 are calculated from equations (45) and (46).
Ae is again calculated from equation (27). The error c = abs [( Ae ¨ True Ae)
/ True Ae] is minimized as a function of aH/av. An estimated value of aH/sav
is
obtained when the error c is minimum. ah/av can then be calculated using
equation (30). Pore pressure is then added to the estimated effective
stresses to obtain absolute magnitude of formation stress magnitudes at the
selected depth interval.
[0023] Referring back to step (406) and to Figures 4 and 8, in the case where
the
fast and slow shear dispersion curves do not exhibit a crossing signature, and
the fast and slow dipole dispersions essentially overlay, step (416) is used.
When the fast and slow dipole dispersions overlay, implying that there is no
shear wave splitting for propagation along the borehole axis, it suggests that
the maximum and minimum horizontal stresses (SHmax = Shmin) are nearly
the same. Under these circumstances it is possible to estimate the ratio of
the overburden stress Sv and the horizontal stress SH (SH= SHmax = Shmin)
in a chosen depth interval with reasonably uniform lithology using the far-
field
sonic velocity (or slowness) gradients. Step (416) includes two component
calculations for this particular case: Forming a difference equation using the
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far-field shear moduli C44(=C55), and 066; and forming another difference
equation using [C55(r/a=6) ¨ C55(r/a=2)]. An algorithm is then used to solve
for the unknowns Ae and the effective stress ratio o-H/ o-v, as indicated by
step (418), which enables determination of the horizontal stress magnitude
SH, and an acoustoelastic coefficient Ae, as indicated by step (420).
[0024] Figure 9 shows a workflow for implementing necessary corrections in the
shear modulus 066 in the presence of either fluid mobility in a reservoir or
structural anisotropy in shales. Estimation of formation stress magnitudes
(910) using the three shear moduli requires compensation (912) for fluid
mobility effects in a reservoir and structural anisotropy effects in
overburden
or underburden shales as calculated in step (900).
When fluid mobility estimates are known from any of the techniques known in
the art, such as measurements on a core sample, pressure transient analysis,
or nuclear magnetic resonance data, it is possible to calculate mobility-
induced decrease in the Stoneley shear modulus 066 using a standard Biot
model. Under this situation, the workflow increases the measured 066 by the
same amount before inverting the three shear moduli for formation stresses.
In contrast, structural anisotropy effects in shales cause the Stoneley shear
modulus 066 to be larger than the dipole shear moduli 044 or 055. One way to
compensate for the structural or intrinsic anisotropy effects, as indicated in
step (902), is to measure shear velocities propagating parallel and
perpendicular to the layer in a core sample subjected to a confining pressure
and estimate the Thomsen anisotropy parameter 7. Insofar as the shale
anisotropy increases the magnitude of 066, we must decrease the measured
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066 by the same amount in step (904) before inverting the three shear moduli
in step (906) for formation stress magnitudes.
[0025] One option ("Algorithm I"), is to estimate the horizontal to overburden
stress ratio in a reasonably uniform depth interval using differences in the
compressional and shear slownesses at two depths within the chosen
interval. Another option ("Algorithm II") is to estimate the horizontal to
overburden stress ratio using the two far-field shear moduli (044 and 066),
and
radial profile of the dipole shear modulus (044). This algorithm can be
applied
in depth intervals that are axially heterogeneous insofar as the difference
equations use shear moduli at the same chosen depth.
[0026] Algorithm I will now be described. There are two sets of quantities
that
need to be estimated from the inversion of changes in sonic velocities caused
primarily by formation stress changes. The first set involves determination of
the three nonlinear constants, and the second set comprises the estimation of
stress magnitudes. The acoustoelastic equations relating changes in the sonic
velocities caused by changes in stresses in the propagating medium contain
product of unknown quantities. When the horizontal stresses SHmax and
Shmin are nearly the same, and they can be approximately obtained from the
Poisson's ratio in the vertical X2-X3 plane in terms of changes in the
vertical
effective stress Aav or Aa33. The steps described below can be utilized.
[0027] Given
r
v
Ault = Au = __________
22 0 -11/Au" ,
(49)
where Ami and Aa22 denote the maximum and minimum horizontal stresses,
an incremental change in the effective stiffness AC55 can be expressed in
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terms of changes in the effective vertical stress between two depths in a
reasonably uniform lithology interval
AC5, = [1+ (1¨ v ¨ 2v2)11 Cõ
Acrõ ,
2(1¨ v2 1+) eti j (50)
where 0155 is one of the three nonlinear constants referred to the chosen
depth interval. An incremental change in the effective stiffness AC66 can be
expressed in terms of changes in the effective vertical stress between two
depths in a reasonably uniform lithology interval
[
v (1¨ v ¨ 2v2)r C144 1\ A u33 ,
AC66 = +
1 - v 2(1¨ v2) /1 1 1 (51)
where 0144 is another nonlinear constants referred to the chosen depth
interval. An incremental change in the effective stiffness AC33 can be
expressed in terms of changes in the effective vertical stress between two
depths in a reasonably uniform lithology interval
AC33 =[1+ (1¨ v ¨ 2v2) r Cõ, 3 C33 \1 A 0_33 ,
2(1¨ v2) ,u /1 J (52)
where Ciii is the third nonlinear constant referred to the chosen depth
interval. Equations (50), (51), and (52) are used to calculate the three
nonlinear constants (C111, 0144, C155) in the chosen depth interval. These
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nonlinear constants together with linear constants enable determination of
the stress coefficients of velocities in the chosen depth interval.
[0028] A second step is to use general 3D-equations relating changes in the
effective stresses to corresponding changes in the effective elastic moduli
and
estimate the unknown stress magnitudes SHmax and Shmin in the chosen
depth interval where the three nonlinear constants have been estimated.
These equations are as shown below:
i Au.
AC55 = [C55 ¨ V C1õ (1 ¨ 1/) C155 i ____
2,u (1+ v)
A0-22
+ [C,õ ¨ (1+ 2v )C55 ¨ 2v C155 1 j
2,u(l+v)
i Ao-õ
+ [2,u (1+ v) + C55 ¨ V C1õ (1¨ V )C155 i _______ ,
2,u (1+ v)
(53)
where AC55 is obtained from the fast-dipole shear slowness and formation
bulk density at the top and bottom of the interval.
i Ao-õ
ACõ = [¨(1 + 2v)Cõ + C144 ¨ 2v C155 i _______
2,u (1+ v)
1 A0-22
2,u (1+ v)
i Ao-õ
+ [2,u (1 + v ) + Cõ -V C144 + (1 ¨ v ) C155 i ____ ,
2,u (1+ v)
(54)
where AC44 is obtained from the slow-dipole shear slowness and formation
bulk density at the top and bottom of the interval.
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AC66 = LU (1+ v) C66 ¨ V C144 (1¨ V ) C155 i (Aaõ A0-22)
2,u (1+ v)
i Ao-õ
+ [¨ (1+ 2v ) C66 C1õ ¨ 2v C155 i ________ ,
2,u (1+ v)
(55)
where AC66 is obtained from the Stoneley shear slowness dispersion and
formation bulk density at the top and bottom of the interval.
i(Ao- + Ao-22 + Ao-õ)
ACõ = [(1 ¨ 2v )Cõ, j ____ "
2,u (1+ v)
+ [¨(1 + 2v )Cõ ¨ 4(1 ¨ v ) C155 1 (A (7" A(722 )
2,u (1+ v)
Ao-õ
+ [2,u (1 + v ) + (3 + 2v )Cõ +8v C155] ________ ,
2,u (1+ v)
(56)
where AC33 is obtained from the compressional slowness and formation
bulk density at the top and bottom of the interval. Note that the Young's
modulus
Y=2,u(1+v);
and
AC33 = p,Vc2,,¨ PA VA ,
(57)
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ACõ = p,V,2,A,- pAV,A
,
(58)
AC55 = p,V,¨ PA VA ,
(59)
AC66 = p,V,2,,¨ pAVH2,,, ,
(60)
where the index A and B denote the top and bottom of the chosen depth
interval.
[0029] Estimation of 0144 and 0155 in a uniform lithology interval will now be
described. Assuming cross-dipole dispersions overlay implying that there is
no shear wave splitting, then SHmax (ail) = Shmin (622), and in equations
(52) and (53), G11=G22. The objective is to invert the borehole Stoneley and
dipole slowness data to estimate the ratio of Aav / Aah in a given depth
interval with a reasonably uniform lithology. Over a given depth interval,
AC44 = A crõ + [ (1 + 2v ) A aõ + (A cr22 + , Au )1 C44
2(1 + v ) 2(1 + v ) Al
[ _______________________
A aõ v
+ _______________________________________________ (Au, + Acrõ )1C
¨144
2(1+v) 2(1+v) Al
v (1 - v )
(A o-22 + A aõ )1C
,
[ (1+v) 2(1+v) Al
(61)
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1 (Ao-ll +Ao-22) (1+ 2v ) Ao_33
22 ¨c66
AC66 = -(Ao-11 +Ao- )
2 2(1+v) 2(1+v) g
Ao-33v
+ _____________________________________________ (Ao-ll +Ao-,) ¨C144
2(1+v) 2(1+v) _ g
v (1-v)
+ _________________________ Ao-33 (Ao-ll +Ao-22) C'15, ,
(1+v) 2(1+v ) g
(62)
It is now possible to solve for the nonlinear parameters 0144 and 0155 from
the following matrix equation:
C144
[An A121 1 id } 131 }
A21 A22 Cõ5 B2 '
il
(63)
where
Ault v kL-1t A
(722 + A(733 )
An = 2(1 + v ) 2(1 + v )
,
(64)
v (1 ¨ v ) t A
Al2 = A Cri 1
kL-1(722 + A(733)
(1+v) 2(1+v)
,
(65)
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B1 = AC, ¨ Acyõ + + 2v) (A0-22 + A0-õ)] C44
+ v ) + v )
' ,
(66)
Acrõ
A( 0-11 A 0-22)
A21= 2(1+v) 2(1+v)
(67)
(1¨ v )
A22 = (1 + v) Acrõ + 2(1 + v ) (A0-11 A0-22)
(68)
(A
AC
1 (A ull + Ao-22) (1+ 2v )A0_
331 C66
B2 ¨ ACT22 ) [
2 2(1+ v) 2(1+ v) ,u
(69)
Having estimated the acoustoelastic coefficient AE in terms of 0144 and 0155
as described by equation (68),
A, = 2 + C144 ,
2/./
(68)
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t is possible to estimate the ratio of the vertical to horizontal effective
stresses from the equation given below:
A, = ACõ ¨ AC66
___________________ ,
Aav ¨ Au h
(69)
where all quantities on the RHS are estimated at a given depth. Therefore,
the ratio of the vertical to horizontal effective stresses can now be
calculated from the equation given below:
Aav = 1+ ______________
, ACõ ¨ AC66
,
Aah Aah AE
(70)
or equivalently,
Aah =1 ACõ ¨ AC66
,
Aav Ao-v A,
(71)
[0030] Algorithm II is a preferred technique in environments where the
formation
exhibits azimuthal isotropy in cross-dipole dispersions implying that
horizontal
stresses are nearly the same at all azimuths. Further, this technique is more
suitable for formations where lithology is rapidly changing with depth than
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Algorithm I. The presence of a borehole causes near-wellbore stress
distributions that can be described by equations (1) through (4).
In the absence of a borehole, the far-field stresses are given by equations
(5) through (8).
Subtracting the far-field stresses from the near-wellbore stresses yields
incremental stresses as we approach the borehole surface and are given
by equations (9) through (12).
[0031] When the formation lithology is rapidly changing with depth, a more
suitable way to invert borehole sonic data for the horizontal to overburden
stress ratio may be to use radial profiles of the three shear moduli together
with near-wellbore stress distributions in terms of the far-field stresses as
described by Kirsch's equations (Jaeger and Cook, 1969). This procedure
includes the steps described below. The effective radial and hoop stresses as
a function of radial position in polar coordinates can be expressed in terms
of
the far-field horizontal stress and over- or under-pressure APw = (Pw ¨ Pp),
in
the wellbore fluid using :
cr,(r =2a,60=0 )=--1(a, + AP, ) ,
4
(72)
+ AP, ) ,
4
(73)
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where 0 = 00, corresponds to the azimuth parallel to the maximum
horizontal stress direction. Similarly, the radial and hoop stresses at r=5a,
can be obtained from equations (9) through (12) and they take the form
arr(r = 5a,9 =0 )=--1(a, + APO ,
(74)
ath,(r = 5a,9 =0 ) = ¨1(a, + APO ,
(75)
Changes in the fast dipole shear modulus C55 at radial positions r = 2a and
r = 5a, caused by the presence of local stresses can be expressed as
AC55 (r =2a)= ¨1(2C55 +C155 - C144 ) (1 V )(0-H + AP,) ,
4
(76)
AC55 (r = 5a)= ¨1(2C55 + C155 - C144) (1+ V )(0-H + AP,) .
(77)
Form a difference equation
AC55(r =2a)¨ AC55(r = 5a)= c (2C55 +C155 -Cl) (1 + V )(0-H + APw ) ,
(78)
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where = 1/4 -1/25 = 0.21.
Use the following equation to calculate the left-hand-side of equation (78)
AC55 (r = 2a) ¨ AC 55(r = 5a) = p [V; (r = 2a) ¨V; (r = 5a)] ,
(79)
where Vs denotes the shear velocity at different radial positions obtained
from the Dipole Radial Profiling (DRP) of shear velocity. The
acoustoelastic coefficient AE can be written as
A ¨ E = C55 C66 = 2 C155 ¨C144
,
a v(1 ¨ x) 2,u
(80)
where is the far-field shear modulus in the assumed reference state.
Replacing (0155 ¨ 0144) in terms of the far-field shear modulus 055 and 066
from equation (80) yields, from equation (78),
p [V 5,2 (r = 2a) ¨V; (r = 5a)] = [2C 55 ¨ 2,u (C 55 ¨ C66 2)] (1+ v)(cr, +
AP, )4-,
av ¨ aH
(81)
Since the effective overburden stress is generally known, it is possible
solve for the horizontal to overburden stress ratio from the following
equation
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-
p [V; (r = 2a) ¨17,2 (r =5a)] = [2C 55 ¨2p (C55 C66 2)] (1+ v)(xo-v
(Tv (1¨ x)
(82)
where oH = X 6V, and it is possible to solve for this unknown from equation
(82), which is a quadratic equation in "x". Alternatively, it is possible to
construct a cost function c that needs to be minimized as a function of the
unknown parameter "x". This cost function c takes the form
E =1 i - 2( XA
11 ,
(83)
where
=p [V; (r = 2a) ¨17,2 (r =5a)] ,
(84)
and
¨
2 = [2 C55 - 2,u ( C55 C66
cry (1¨ x)
(85)
Since the shear modulus 066 is also affected by reservoir fluid mobility and
overburden shale anisotropy, there is a need to compensate for these
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effects on the estimation of stress magnitudes using sonic data. One
approach is to estimate the horizontal to overburden stress ratio as a
function of parameter 7 = 066/066. If the fluid mobility causes a decrease in
the measured 066 by 10%, then the measured 066 can be increased by
10% (referred to as the corrected 066). Hence, it is possible to estimate
stress magnitudes using this algorithm for 7=1.1, and this estimate is
effectively compensated for fluid mobility bias on measured 066.
[0032] Similarly to the approach described above, in order to estimate stress
magnitudes in the overburden shale that generally exhibits transversely-
isotropic (TI) anisotropy, one approach is to determine the magnitude of TI-
anisotropy in terms of the ratio of 066/044 from core data. If this ratio is
1.3,
then the measured 066 is decreased by 30% in order to remove the structural
anisotropy effects from the stress magnitude estimation algorithm.
Consequently, the estimate of stress magnitudes in this shale interval
corresponds to the values obtained for 7=0.7. Therefore, the parameter x is
estimated for a range of 066 = 7 066. Finally, the ratio of the horizontal to
overburden stress is given by:
SH/Sv = (x av + a Pp) / (av + a Pp).
(86)
[0033] While the invention is described through the above exemplary
embodiments, it will be understood by those of ordinary skill in the art that
modification to and variation of the illustrated embodiments may be made
without departing from the inventive concepts herein disclosed. Moreover,
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= .
69897-147
while the preferred embodiments are described in connection with various
illustrative structures, one skilled in the art will recognize that the system
may
be embodied using a variety of specific structures. Accordingly, the invention
should not be viewed as limited except by the scope of the
appended claims.
34
