BR112012001198B1 - method and apparatus for estimating stress, methods and apparatus for estimating stress relationship in a formation - Google Patents

Description

METHOD AND APPARATUS TO ESTIMATE TENSION, METHODS AND APPARATUS TO ESTIMATE THE TENSION RELATIONSHIP IN A FORMATION

Field of invention

The invention is generally related to the analysis of underground formations and, more particularly, to the estimation of stresses in formation using radial profiles of three shear modules.

Background of the Invention

Formation stresses can affect the prospecting and geophysical development of oil and gas reservoirs. For example, overload stress, maximum and minimum horizontal stresses, pore pressure, well pressure and rock resistance can be used to produce a failure model, to assist with well planning, well stability calculations and reservoir management . It is known that the speeds of elastic waves change due to the pre-tension in a propagation medium. For example, sonic speeds in porous rocks change depending on the effective pre-tension. However, the estimation of stresses in formation based on speed can be problematic, because of the influences on the horizontal shear modulus C66 · For example, the horizontal shear modulus C6 is reduced in the presence of horizontal fluid mobility in a reservoir porous.

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Generally, the slowness of the tube wave decreases by about 2 to 3% in the presence of fluid mobility, resulting in a decrease in Cô6 of about 4 to 6%. On the other hand, the Cô6 horizontal shear modulus is increased in the presence of high clay content in a shale gap. Consequently, it is difficult to accurately estimate the relationship between vertical and horizontal stresses, without compensating for changes in C66.

Several devices are known to measure formation characteristics, based on sound data.

Mechanical disturbances are used to establish elastic waves in terrestrial formations around a well, and wave properties are measured to obtain information about the formations, where the waves propagate. For example, information about compression, shear and Stoneley waves, such as speed (or its inverse, slowness) in the formation and in the well, can help in the assessment and production of hydrocarbon resources. An example of a sound recording device is Schlumberger's Sonic Scanner®. Another example is described by

Pistre and others. A modular fixed network sonic tool for 3D measurements (azimuth, radial and axial) of the formation's acoustic properties, by Pistre, V.,

Kinoshita, T., Endo, T., Schilling, K., Pabon, J., Sinha,

B., Plona, T., Ikegami, T., and Johnson, D., Annals of 46 °

Annual Profiling Symposium, Society of Professional Well

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Log Analysts, Article P, 2005. Other tools are also known. These tools can provide compression slowness, At _c , shear slowness, At _s , and Stoneley slowness, At _s t, each as a function of depth, z, where slowness is the inverse of speed and corresponds to the time interval of transit usually measured by sonic profiling tools. An acoustic source in a well filled with fluid generates frontal waves, as well as relatively stronger modes guided by the well. A standard sound measurement system uses a piezoelectric source and hydrophone receivers located inside the fluid-filled well. The piezoelectric source is configured as a monopolar, or bipolar, source. The source's bandwidth generally ranges from 0.5 to 20 kHz. A monopolar source mainly generates the lowest-order asymmetric mode, also known as the Stoneley mode, along with frontal compression and shear waves. In contrast, a bipolar source excites mainly the lowest order bending mode in the well, along with frontal compression and shear waves. The frontal waves are caused by the coupling of the acoustic energy transmitted to plane waves in the formation, which propagate along the shaft of the well. A compression wave incident on the well fluid produces critically refracted compression waves in the formation. Those waves refracted along the surface of the well are known as frontal compression waves. O

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4/36 critical angle of incidence θί = sin ^-1 (Vf / V _c ), where Vf is the velocity of the compression wave in the well fluid; and V _c is the velocity of the compression wave in the formation. As the frontal compression wave moves along the interface, it radiates energy back into the fluid, which can be detected by hydrophone receivers placed in the fluid-filled well. In fast formations, the frontal shear wave can also be excited by a compression wave at the critical angle of incidence θί = sin ^-1 (Vf / V _c ), where V _s is the speed of the shear wave in the formation. It is also important to note that frontal waves are only excited when the wavelength of the incident wave is less than the diameter of the well, so that the contour can be treated effectively as a planar interface. In a homogeneous and isotropic model of fast formations, as mentioned above, frontal compression and shear waves can be generated by a monopolar source placed in a well filled with fluid, to determine the speed of the compression and shear wave of the formation. It is known that refracted frontal shear waves cannot be detected in slow formations (where the speed of the shear wave is less than the speed of fluid / well compression) with receptors placed in the well fluid. In slow formations, shear velocities of the formation are obtained from the low frequency dispersion asymptote at

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5/36 flexion. There are standard processing techniques for estimating the shear velocities of the formation in fast or slow formations, from a variety of recorded bipolar waveforms. A different type of tool described in U.S. Patent No. 6,611.7 61,

published on August 26 2003 to Sonic record in well for radial profiling get profiles radial in slowness shear fast and slow, using at dispersions bipolar measures at two directions orthogonal,

which are characterized by the C44 shear modules and

C55 for a well parallel to the X3 axis in an orthorhombic formation.

Summary of the Invention

According to an embodiment of the invention, a method for estimating stress in a formation, in which a well is present, comprises: determining radial Stoneley profiles, slow bipolar shear and slow bipolar shear; estimate maximum and minimum horizontal stresses by inversion of differences in remote field shear modules with difference equations obtained from radial profiles of bipolar shear modules C44 and C55, and stresses close to the well, and produce an indication of maximum horizontal stresses and minimally, tangibly.

According to another embodiment of the invention, an apparatus for estimating stress in a formation,

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6/36 in which a well is present, comprises: at least one acoustic sensor, which provides radial Stoneley profiles, fast bipolar and slow bipolar shear slowness; processing circuit, which estimates maximum and minimum horizontal stresses by inversion of differences in remote field shear modules with difference equations obtained from radial profiles of bipolar shear modules C44 and C55, and stresses close to the well; and an output, which produces an indication of the maximum and minimum horizontal stresses, in a tangible form.

Brief Description of the Figures

Figure 1 is a schematic diagram of a well in a formation subject to major remote field stresses.

Figure 2 is a schematic diagram of a ray well subject to formation stresses in a poroelastic formation, with pore pressure Pp and well pressure P _w .

Figure 3 illustrates a profiling tool to estimate formation stresses using radial profiles of three shear modules.

Figure 4 is a flow chart, showing the steps of a method, according to an embodiment of the invention.

Figure 5 illustrates the radial variation of the effective axial (σΖΖ), circumferent (σθθ), and radial (orr) stresses in an azimuth parallel to the direction of maximum horizontal stress at a given depth.

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Figure 6 illustrates a radial variation of the effective axial (oZZ), circumferent (σθθ), and radial (orr) stresses at an azimuth perpendicular to the direction of maximum horizontal stress.

Figure 7 illustrates dispersion curves of

shear fast and slow, that exhibit a signature transversal together with the dispersal asymmetric in Stoneley, from order more low. Figure 8 illustrates curves dispersal in shear fast and slow, not feature an

transverse signature, and the fast and slow bipolar dispersions essentially overlap, implying that SHmax = Shmin.

Figure 9 illustrates an algorithm to solve the unknowns Ae and oH / oV, which allows the determination of SH, and an elastic acoustic coefficient.

Detailed Description

Embodiments of the invention will be described with reference to main stresses illustrated in Figure 1, and triaxial stresses Txx, Tyy, and Tzz, together with pressure well Pw, and pressure in the pores Pp, as illustrated in Figure 2. The well radius is denoted by a. As discussed earlier, sonic velocities and, therefore, slowness, are sensitive to effective stresses in the propagation medium. Effective stress Oij = Tij - õijPp, where Tij is the applied stress, õij is the Kronecker delta, and Pp is the pore pressure.

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Figure 3 illustrates an example of a profiling tool (106) used to acquire and analyze data, according to an embodiment of the invention. The tool has a plurality of receivers and transmitters.

The profiling tool illustrated (106) also includes multipolar transmitters, such as crossed bipolar transmitters (120, 122) (only one dipole terminal (120) being visible in Figure 1) and monopolar transmitters (109) (close) and (124 ) (remote), capable of exciting shear, Stoneley, and bending waves.

compression,

The profiling tool (106) also includes receivers (126), which are spaced at a distance from the transmitters. Each receiver can include several hydrophones mounted azimuthal at regular intervals around the circumference of the tool.

Other settings, such as a tool

Digital Sonic

Imaging (DSI), with four receivers in each of the eight receiver stations, or incorporating other multipolar sources, such as quadrupole, is also possible. The use of a plurality of receivers and transmitters results in better signal quality and adequate extraction of signals from several holes over a wide range of frequencies. However, the distances, number and types of receivers and transmitters shown in this embodiment are just a configuration

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9/36 possible, and should not be construed to limit the invention.

The subsurface formation (102) is crossed by a well (104), which can be filled with drilling fluid or mud. The profiling tool (106) is suspended by a shielded cable (108), and can have optional centralizers (not shown). The cable (108) extends from the well (104) on a pulley wheel (110) in a tower (112) to a winch, forming part of the surface equipment, which may include an analyzer unit (114). Known depth measurement equipment (not shown) can be provided to measure the displacement of the cable over the pulley wheel (110). The tool (106) can include any of the many devices known to produce a signal indicating orientation of the tool. Processing and interface circuits within the tool (106) amplify, sample and digitize information signals from the tool for transmission, and communicate them to the analyzer unit (114), via the cable (108). Electric power and control signals to coordinate the operation of the tool (106) can be generated by the analyzer unit (114) or some other device, and communicated, via the cable (108), to the circuits inside the tool (106) . Surface equipment includes a processor subsystem (116) (which may include a microprocessor, memory, clock and calendar, and functions

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10/36 input / output not shown separately), standard peripheral equipment (not shown separately), and a recorder (118).

Figure 4 is a flowchart, which illustrates a method for estimating formation stresses, using radial profiles or three shear modules. If possible, a depth range characterized by relatively uniform lithology is selected for stress assessment. Transverse monopoly and bipolar sound data are then obtained with the tool (106, Figure 3) over the selected range, as indicated by step (400). Transverse monopolar and bipolar data, density (pf), remote field velocity (Vf), and well radius (a) are used to determine Stoneley radial profiles, fast bipolar shear slowness and slow bipolar shear slowness, as indicated by step (402). Mass density of the formation (bp) obtained in step (404) is then used to calculate radial profiles of three shear modules (C44, C55, C66), as shown in step (406). The next step is selected based on the result of calculating the shear modules.

Before describing the selection of the next stage, the effects of the presence of the well in a formation will be explained with reference to Figures 5 and 6. The presence of a well causes stress distributions close to the well,

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11/36 that can be described by the following equations (Jaeger and

Cook, 1969) _{} +} Í £ i ^ _ £ i2 ₍ | _ _{+) cos} 2ô_, r * 2 r- r? (1)

(2)

0) (4) effective radial, circumferent, radial shear, and respectively, the effective remote field overload, horizontal maximum, horizontal minimum; a denotes the radius of the well, and azimuth measured from the direction of maximum horizontal stress. In the absence of a well, remote field voltages are given by:

2 2 15) * 2 2 (6) 2Θ. and (7) ÍTjj = tT,.,. (3)

of tensions

Subtracting the remote fields from the stresses close to the well produces incremental stresses, when the surface of the well is approximated, as follows:

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Z r 2 r r r ~ (1U ¢ 12) where Pw denotes well pressure at a certain depth.

In the case of an isotropically loaded reference state, shear modules in the three orthogonal planes are the same, that is, C44

C55

C6 μ ·

When this type of formation is subject to incremental anisotropic stresses, changes in shear modules can be expressed as (13) where AC55 is obtained from the rapid bipolar shear slowness from sound data acquired by a bipolar transmitter aligned parallel to the direction Xi and parallel to the well in direction X3; the quantities C55, μ, and v are the linear elastic modules, while C144 and C155 are

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13/36 the non-linear formation constants in the chosen reference state; and Δθ33, Δσιι, and Δθ22 denote, respectively, the effective overload stresses (parallel to the X3 direction), maximum horizontal (parallel to the Xi direction) and minimum horizontal (parallel to the X2 direction) at a chosen depth of interest,

(14) where AC44 is obtained from the slow bipolar shear slowness, from sound data acquired by a bipolar transmitter aligned parallel to the X2 direction and parallel to the well in the X3 direction;

CI = CA * + v) * C 4 - C _jaj v + (1 - v) _JLA C j ^(Atfj + + [- (] + 2r - 2vC "] ^AcTj3 (15) which is obtained from ACô6 dispersion of the Stoneley shear slowness from the sound data acquired by a monopolar transmitter at a chosen depth of interest.

Referring to Figures 4 and 7, in case the fast and slow shear dispersion curves have a transverse signature, step (408) is used. Step (408) includes two component calculations, which are done

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14/36 with the aid of the data of the overload voltage (Sv), well pressure (P _w ) _r pore pressure (Pp), Biot coefficient a, and well radius (a), obtained in step (410) . A first component calculation is to form two difference equations, using the remote field shear modules 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 axes Xi, X2, and X3 are, respectively, parallel to the maximum horizontal (oh), minimum horizontal (Oh) and vertical (σ _ν ) stresses, equations (13) - (15) produce difference equations in the modules effective shear, in terms of differences in the principal stress magnitudes, through an acoustic elastic coefficient defined in terms of nonlinear formation constants, which refer to a chosen reference state and for a given formation lithology. The following three equations relate to changes in shear modules to correspond to changes in effective principal stresses in a homogeneously stressed formation, as would be the case in the remote field, sufficiently far from the well surface:

AC ' -AQ ^ A ^ -AíTJ, (16) ac ₃₅ - -Δ ^ = ^. (Δσ _$ -Δσ ₂₁ ), (17} -ÚC _4I = Α _£ (Δσ _ιι -Δσ _Η ), (18)

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15/36 where Δθ33 _Λ Δσιι, and Δθ22 denote, respectively, changes in the effective overload stresses, maximum horizontal, and minimum horizontal; and

Λ, = 2 ₊ ^.

A (19) is the acoustic elastic coefficient, C55 and C44 denote, respectively, the shear modules for slow and fast shear waves; C456 = (C155-C144) / 2, is a non-linear parameter of the formation, which defines the acoustic elastic coefficient; and μ 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 transversal plane of the well causes the splitting of the bipolar shear wave, and the observed shear slope anisotropy can be used to calculate the acoustic elastic coefficient Ae from equation (18), provided that estimates of the three main stresses depending on depth are available. Note that bipolar shear waves are largely unaffected by fluid mobility. The stress-induced change in the Stoneley Cô6 shear modulus is then calculated using equations (16) and (17), and the magnitudes of the effective stress Δσ _ν , Δοη, and Aoh at a given depth are obtained.

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The next step (412) is to use an algorithm to solve the unknowns. Combining the two difference equations (16) and (17) in terms of the remote field shear modules produces four independent equations to solve the four unknowns Δοη, Aoh, C144 and C155. Equations (13), (14), and (15) can be expressed in terms of the main stress parameters Δοη, Δσ ^, and Δσ _ν , as follows:

t [2a <1 + r) + C „- k q. + (I - v> c, J.

(20)

AC = [- (1 + + C _J44 - 2r

4- [2μ (1 4-μ) + Γ ^ -νΓ _ω 4- (1 -v) í? _ls J, (21)

ΔΓ _Κ = [«(I + 1 ') + C _M - l' C _t4í + (1 - l ') C _13i ] ^<Agfl * ^Ag,) 2p (1 + i) (22)

Defining the following dimensionless parameters _ δ & η _ Δσ _; , _

2μ (Ι + μ) '' ^Λ 2χΐ + ν) '' ^r 2 ^ (1+ ν) ¹ (23} e

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A- “• 'Ç + rKI-É), (24) it is possible to express the non-linear constants C144 and C155 in terms of Ai and A2, and the acoustic elastic coefficient Ae can also be expressed in terms of Ai and A2, as shown below:

_{c =} IvA, + (1 - | /) A, _{c =} A, + vA _t ^{l +} '' (1-21 /) (1 + μ) '(l-2i /) (l + v) * (25)

(26) (27) ~ = Â _£ (Δσ _μ -), (28)

Δ6 ' _Β -ΔΓ ₍₆ = Α _Α (Δσ _μ -Δσ _ή ).

¢ 29)

An expression for the stress relationship is obtained from equations (16) and (17) ₌ ] _ lÍAÇ '- ^, W _Δ ^ -Δς, J

Δ0, (δ ^ -δ £; _1? / Μ

30)

Subtracting equation (28) from (29), and replacing Ae from equation (27), produces

Δί% - AC = [2ü (I + v) + A - Λ,] ^{A <y} «~ ^{A <r} * * ¹ * 2 // (1 + 1 /) (31)

This results in one of the two equations relating Ai and A2

Λ, -A, = 2 // (1 + v)

ΔΓ ,, - ΔΓ,., _T .Δσ ,, - Δσ * (32)

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Different values of Ai and A2 can be obtained for different Δοη / Δ o _v , as follows = ^ + 4) (57 (^ = 6) -07 (^ = 2)] + (- / 41 + 2r) 4- A) [57 (r / ü = 6) -57 (r / tf = 2)] + ^ + 4) (^ (^ = 6) - ^ 7 (^ = 2)) (33)

AÇ '(r / ú = 6) -AÍ7' (rM = 2)

ΔίΤ „(/ z + 4) 5. + ^ - 21 o ;. '<ξ. c \, (- / 41 + 2ι /) + α) ^ 51 + ^ + ^ 21 (7 _V <7 _V + ^ + 4 ^ - ^ + ^ σ _ν (34) where

177 17

2592 ⁺ 32

JJTJ

105

2392

J) _ ’

4 ⁺ 4

.32 <35) _ 7 ¹ 2592 32

4

4

36 ^ 2 ’(T

£

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Equation (34) can be rewritten in a compact way, shown below

Jí, = (/ t + Λ,) (\ + Λ \) -I- (- // (1 + 2v) 4- Λ,) .S ‘, í-36) where

X _f = ACg ⁽ 'lr fu = 6) - AC £ ° 07 tf = 2) i

(37) dff j; _vv Cf σ _ν σ _ν

Similarly,

^ ^Ac,: (r ia = 6) - = 2)

Ά + A Jbr (r / «= 6; - (r / a = 2)] + (- / Al + 2r) + A,) [^ (f / ü = 6) - X / '(ría = 2) ] + Uf + A,) | .S '™ (r / «= 6) - S /' (r /« = 2)] (38) âCtr / fl = 6) - àC ™ (rfa = 2)

(39) where

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i

Equation (39) can be rewritten in a compact way, as follows:

= (μ + Λ) (. S ' ₄ 4- SJ 4 - (- // (I + 2r) 4- A> 5 _S „(41) where

Resolving

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21/36 (44)

- = ₊ 4 XS, + i -,) + (-X41 + 2> Z) + A) S,

Δσ _Γ ^f = (// + + 5 (,) + (—A <1 + 2i ^z ) + A.) 5j

Δσ _{| (} for Ai and A2, produces _A 5Ζλ-5 / ή ¹ S ₂ (S ₄ + 5 _fi ) - S _f (5] + Sp ' _Λ , (5, + ^) 6, - (5, + 5 ,) 6, ij (5, + \) - S ₅ (5, + S, J where ¢ = - ^ + (- // CS + 5j) + // (1+ 2v) 5j,

Λ rT _T.

6, = - + [- // (5, + 5J + // (1 + 2v) 5,]

X

SHmax, Shmin, and an acoustic elastic coefficient Ae are calculated in step (414). The acoustic elastic parameter Ae is calculated as a function of the oh / ov stress ratio, using equation (28). Oh / ov is calculated in terms of oh / ov, using equation (30). The real Ae is calculated in terms of C55, C44, oh and Oh, using equation (18). A1 and A2 are calculated from equations (45) and (46). Ae is again calculated from equation (27). The error e = abs [(Ae - Ae Real) / Ae Real] is minimized as a function of oh / ov. An estimated value of oh / ov is obtained when the error is minimal. Oh / ov can be calculated using equation (30). The pore pressure is then added to the estimated effective stresses to obtain the absolute magnitude of the magnitudes

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22/36 formation stress in the selected depth range.

Returning to step (406) and Figures 4 and 8, in case the fast and slow shear dispersion curves do not have a transversal signature, and the fast and slow bipolar dispersions essentially overlap, step (416) is used. When the fast and slow bipolar dispersions overlap, implying that there is no division of the shear wave for propagation along the shaft of the well, this suggests that the maximum and minimum horizontal stresses (SHmax = Shmin) are practically the same. In these circumstances, it is possible to estimate the relationship between the overload voltage Sv and the horizontal voltage SH (SH = SHmax = Shmin) in a chosen depth interval with reasonably uniform lithology, using sonic speed gradients (or slowness) in remote field . Step (416) includes two-component calculations for this particular case: forming a difference equation, using the remote field shear modules C44 (= C55), and C66; and form another differential equation, using [C55 (r / a = 6) C55 (r / a = 2)]. An algorithm is then used to solve the unknowns Ae and the effective tension ratio oh / ov, as indicated by step (418), which allows the determination of the horizontal stress magnitude SH, and an acoustic elastic coefficient Ae, as indicated by step (420).

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Figure 9 shows a workflow to implement the necessary corrections in the C66 shear module, in the presence of fluid mobility in a reservoir, or structural anisotropy in shales. The estimation of the stress magnitudes of the formation (910) using the three shear modules requires compensation (912) for the purpose of fluid mobility in a reservoir, and the effects of structural anisotropy in overload or underload shales, as calculated in step ( 900). When fluid mobility estimates are known using any of the known techniques

in art, as measurements in a sample in a testimony, analysis of transient pressure, or data in resonance magnetic nuclear is possible calculate The decrease

induced by mobility in the Stoneley C66 shear module, using a standard Biot model. In this situation, the workflow increases the C6 measured by the same amount, before inverting the three shear modules for forming stresses. In contrast, the effects of structural anisotropy on shales make the Stoneley Cô6 shear modulus larger than the C44 or C55 bipolar shear modulus. One way to compensate for the effects of structural or intrinsic anisotropy, as indicated in step (902), is to measure shear velocities with parallel propagation and perpendicular to the layer in a core sample.

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24/36 subjected to confining pressure, and estimate the parameter γ of Thomsen's anisotropy. As shale anisotropy increases the magnitude of C66 _f, we must decrease Cô6 by the same value in step (904), before inverting the three shear modules in step (906) for stress magnitudes of the formation.

One option (Algorithm I) is to estimate the relationship between horizontal and overload stress over a reasonably uniform depth range, using differences in compression and shear slowness at two depths within the chosen range. Another option (Algorithm II) is to estimate the relationship between horizontal and overload stresses, using the two remote field shear modules (C44 and Côô) _r and the radial profile of the bipolar shear module (C44). This algorithm can be applied at depth intervals, which are axially heterogeneous, as the difference equations use shear modules at the same chosen depth.

The algorithm I will now be described. There are two sets of quantities, which need to be estimated from the inversion of changes in sonic velocities, caused mainly by changes in the formation voltage.

The first set involves the determination of the three nonlinear constants, and the second set comprises the estimation of the stress magnitudes. The elastic equations

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25/36 acoustics related to changes in sonic velocities caused by changes in voltages in the propagation medium contain the product of unknown quantities. When the horizontal stresses SHmax and Shmin are approximately the same, they can be approximately obtained from the Poisson's coefficient in the vertical plane X2-X3 in terms of changes in the effective vertical stress Δον or Δθ33. The steps described below can be used.

Given (49) where Δσιι AND Δθ22 denote the maximum and minimum horizontal stresses, an incremental change in effective stiffness ΔΟ55 can be expressed in terms of changes in the effective vertical tension between two depths over a reasonably uniform lithology range

(50) where C155 is one of the three nonlinear constants, which refers to the chosen depth range. An incremental change in effective stiffness Δϋδδ can be expressed in terms of changes in vertical effective tension between two depths over a reasonably uniform lithology range

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where C144 is another nonlinear constant, what if refers to interval in depth chosen An change increment at effective stiffness AC33 can be expressed in

changes in effective vertical tension between two depths over a reasonably uniform lithology interval

Γ (1-κ-2γ ² ) 3 r) A k

(52) where Cm is the third non-linear constant, which refers to the chosen depth range. Equations (50), (51) and (52) are used to calculate the three non-linear constants (Cm, C144, C155) in the chosen depth range. These non-linear constants, together with linear constants, allow the determination of the stress coefficients of velocities in the chosen depth range.

A second step is to use general 3D equations, relating changes in the effective stresses to corresponding changes in the effective elastic modules, and estimate the unknown stress magnitudes SHmax and Shmin in the chosen depth range, where the three nonlinear constants were estimated. These equations are as shown below:

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27/36 + (^ - () + 2+) 0.-2 + 0,, 1 ^^ + (^ 0 ++) + 0, - + 0 ,, + (1- +) 0 ,,, ( , (53) where AC55 is obtained from the rapid bipolar shear slowness and volumetric density of the formation at the top and bottom of the range.

AÍC = [- (1 d + C _JUJ - 2r '[- ^ G + j ⁺ Cj * η- ti —1 <) <?, „] ——— 2p CJ + v) + (2 ^ (1 + + ) +0 .- + 0 ,, + (1- +) 0,, 1,

Ά * U Τ ’F J (54) where AC44 is obtained from the slow bipolar shear slowness and volumetric density of the formation at the top and bottom of the range.

ΔΡ = Líí (I +>') + C - k C + p - c jc. ] ^(Ασ _ί ι ^{+ Ασ} ^ ^{) w} * “ ^lji 2μ (14- i ') + [- 0 + 2 +) 0. +0 ,. - i + Cj,

2jü (I + v) (55) where ACô6 is obtained from the dispersion of the Stoneley shear slowness and the volumetric density of the formation in the upper and lower part of the interval.

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ΔίΖ = ((1-2 ^ (7 ,,,] (Δσ „+ Δσ _& + Δσ _Β )

2Α (I + ν) (Δσ ,, + Δσ · ^ λίΛΙ + ν) + [- (l + 2r) C _B -4 (| -v) C _1W ] + hí / (i _{+ r) +} (., ₊₂ , _{) cü + skc} . _sl] _ ^ _ (56) where ΔΟ33 is obtained from the slow compression and the volumetric density of the formation in the upper and lower parts of the interval. Note that the modulus of elasticity

Y = 2μ (1 + v);

and (57) (53) ^ C _i5 = ftVA.i.-AV ^ (59) (SO) where indices A and B denote the upper and lower parts of the chosen depth range.

The estimation of C144 and C155 over a uniform lithology range will now be described. Assuming that transverse bipolar dispersions overlap, implying that there is no division of the shear wave,

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29/36 then SHmax (ou) = Shmin (022), and in equations (52) and (53), on = 022. The objective is to invert the data of Stoneley and bipolar slowness to estimate the Aov / Aoh ratio in a given depth interval with a reasonably uniform lithology. Above a given depth range, <T (61) * 2 ^{11 J} [2 (1 + 1 /) 2 (1 + 1 /) ” ^{Α | Τ;} ---- ------- (Λ (Γμ +)

2 (l + r) 2 / l + i / J ^IJ _ ν (I - r 1

--A <r ,. + ------ (AP .. +, '(1 + 1 /)' ^s 2 (1 + 1 /) ¹¹ (62)

It is now possible to solve the nonlinear parameters C144 and

C155, using the following matrix equation:

A, ₂

A, ¢ 63} where

Petition 870180143351, of 10/22/2018, p. 36/54

30/36

----- ¹¹ --- (Δσ ,, + Δ ¢ 7)

Α „= 2 (Ι + Ρ) 2 (1 + 1-) (64)

(66)

(Ê7)

V (I + F) ^Δί Ά (I - r)

2 --- ΗΔσ _Ν + Δσ „)

2 (1+ κ) (68)

(69)

Having estimated ο acoustic elastic coefficient Ae in terms of C144 and C155, as described by equation (68),

At „= 2-i- ^{C |} :, ^J ~ ^{C | LJ} (68) it is possible to estimate the relationship between the vertical and horizontal effective stresses from the equation below:

Petition 870180143351, of 10/22/2018, p. 37/54

31/36 _A _AC _M -AC * ^£ Δσ _μ - Δσ _k '(69) where all quantities in the RHS are estimated at a certain depth. Therefore, the relationship between the effective vertical and horizontal stresses can now be calculated from the equation given below:

Apr. _{1 |} AC _to -AC _l ,

Αί7 _ή (70) or equivalent, | ACÇ - ACÇ,

B.C. Δ <η, A _K (71) algorithm II is a preferred technique in environments, where the formation presents azimuthal isotropy in transverse bipolar dispersions, implying that horizontal stresses are practically the same in all azimuths. In addition, this technique is more suitable for formations, where lithology changes rapidly with depth, than algorithm I. The presence of a well causes stress distributions close to the well, which can be described by equations (1) to ( 4). In the absence of a well, remote field voltages are given by equations (5) to (8). Subtracting remote field stresses from stresses close to the well produces incremental stresses,

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32/36 as we approach the surface of the well, and are given by equations (9) to (12).

When the formation lithology changes rapidly with depth, a better way to invert the sound data from the well for the relationship between horizontal and overload stress may be the use of radial profiles of the three shear modules in conjunction with close stress distributions to the well in terms of remote field stresses, as described by Kirsch's equations (Jaeger and Cook, 1969). This procedure includes the steps described below. The effective radial and circumferential stresses as a function of the radial position in polar coordinates can be expressed in terms of the horizontal remote field voltage and the APw = (Pw P _p ) overpressure, in the well fluid, by means of:

^ (7 = 2 ^ = 0 ^) = - 1 (^ + ^.), (72) ^ / + = 2 ^ = (7) = - + ^ + ΛΡ), (73) where Θ = 0 ^o , corresponds to the azimuth parallel to the direction of the maximum horizontal tension. Likewise, the radial and circumferential stresses at r = 5a can be obtained from equations (9) to (12) and they take the form

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33/36 σ _ΓΓ (Γ = 5 _ί ; ^ = 0 °) = - ^ (σ _{/ Ρ} + ΔΛ).

(74) a _w (r = 5 _fl ^ = 0 ') = ± (^ _f + lPJ, (75)

Changes in the bipolar shear modulus C55 at the radial positions r = 2a and r = 5a, caused by the presence of local stresses, can be expressed as

AC _S5 (λ = 2a) = ^ (2C _3S + C _IS5 - C, J (1 + v) (σ „+ Δ / ζ.).

¢ 76)

AC _S (/ · = 5α) + C »- C _w ) ('+ ν) (σ„ + Δ / ;.).

(77)

Form a difference equation

AC _ai (t- = 2ü) - = 5ΰ) = ς (2C _JS + C _Jdí - C _m ) (1 + t>) (σ „+ Δ / *.), (7β) where ζ = 1/4 - 1/25 = 0.21.

Use the following equation to calculate the left side of equation (78) (/ = 2u) - dC _s _ (r = 5a) = p [V / (r = 2a) - V / (r = 5 «)].

(79) where V _s denotes the shear velocity at different radial positions obtained from Bipolar Profiling

Shear speed (DRP). The acoustic elastic coefficient Ae can be written as

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34/36

(80) where μ is ο remote field shear module in the assumed reference state. Replacing (C155 - C144) in terms of the remote field shear module C55 and Cô6 of equation (80) produces, from equation (78), p [Vf (r = - V / 0 · = 5d)] = [ 2C ₃₅ - Ίμ - 2)] (1 + υ) (σ „+ Δ / ζ.) Ζ.

(61)

Since the effective overload voltage is generally known, it is possible to resolve the relationship between the horizontal and overload voltages, from the following equation

P ÍV / (r = 2ú) - v; (r = 5ü) 1 = [2C „- 2μ & ^C “ -2)] (I + u) (χσ _ν +) ζ.

(8Ξ) where oh = x ov, and it is possible to solve this question from equation (82), which is a quadratic equation in x. Alternatively, it is possible to build a cost function e, which needs to be minimized due to the unknown parameter x. This cost function takes the form l £ l (Ê3) where

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35/36 ç, = = M>

(84) and

ξ. = [2C _U -Ίμ & ~ ^Ctà - 2)] (1 + υ + ^) ζ.

(I - .ϊ) (35)

Since the Cô6 shear module is also affected by the mobility of the reservoir fluid and the anisotropy of the overhead shale, there is a need to compensate for these effects on the estimation of the stress magnitudes, using sound data. One approach is to estimate the relationship between horizontal and overload voltages as a function of the parameter γ = C'66 / C66. If fluid mobility causes a 10% decrease in the measured C6, then the measured C6 can be increased by 10% (referred to as the corrected C'66). Therefore, it is possible to estimate stress magnitudes, using this algorithm for γ = 1.1, and this estimate is effectively compensated, due to the trend of fluid mobility in the measured C6.

In the same way as the approach described above, in order to estimate stress magnitudes in the overhead shale, which generally exhibits transversely isotropic anisotropy (TI), one approach is to determine the magnitude of TI anisotropy in terms of the C66 / C44 ratio from testimony data. If this ratio is 1.3, then the measured C6 is reduced by 30% in order to remove the effects of

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36/36 structural anisotropy from the magnitude stress estimation algorithm. Consequently, the estimation of the stress magnitudes in this shale interval corresponds to the values obtained for γ = 0.7. Therefore, parameter x is estimated for an interval of C'66 = γθ66. Finally, the relationship between horizontal and overload voltages is given by:

Sn / Sv = (λ Ον η · a Pp) / (oy + ct Pp).

¢ 86)

Although the invention is described through the exemplary embodiments above, it should be understood by those of ordinary skill in the art, that modifications and variations of the illustrated embodiments can be made, without departing from the creative concepts disclosed herein. In addition, although preferred embodiments are described in connection with various illustration structures, one skilled in the art will recognize that the system can be incorporated, using a variety of specific structures. Thus, the invention should not be seen as limited, except for the scope and spirit of the attached claims.

Claims (19)

1. Method for estimating stress in a formation (102) in which a well (104) is present, characterized by the fact that it comprises: determination (402) of radial profiles of Stoneley slowness, fast bipolar shear slowness and slow bipolar shear slowness; calculation (406) of bipolar shear modules C44 and C55 using fast bipolar shear slowness and slow bipolar shear slowness; calculation (406) of C6 module using Stoneley slowness; estimate of maximum and minimum horizontal stresses, by inversion of the difference equations obtained from the remote field shear modules C44, C55 and C66, of the difference equations obtained from radial profiles of bipolar shear modules C44 and C55 in two or more different radial positions, and a relationship between stresses within the formation (102) near the well (104) and stresses in the remote field of the formation (102); and production of an indication of the maximum and minimum horizontal stresses, in a tangible form. 2. Method, according to claim 1, characterized by the fact that it still comprises the determination that fast and slow shear dispersion curves of the formation (102) exhibit a transversal signature. Petition 870180143351, of 10/22/2018, p. 44/54 2/9 3. Method, according to claim 1, characterized by the fact that it still comprises the estimate of a reduction in the Stoneley shear modulus (Cô6) _r caused by fluid mobility in a reservoir, and an increase in Cô6 measured by the same amount, to obtain the magnitudes of the maximum and minimum horizontal stresses, which are compensated for the purpose of fluid mobility. 4. Apparatus for estimating stress in a formation (102), in which a well (104) is present, characterized by the fact that it comprises: at least one acoustic sensor, which provides radial Stoneley profiles, slow bipolar shear and slow bipolar shear; processing circuit, which estimates maximum and minimum horizontal stresses by reversing the differences in remote field shear modules C44, C55 and C66, of the difference equations obtained from radial profiles of bipolar shear modules C44 and C55, in two or more different radial positions, and a relationship between stresses within the formation (102) near the well (104) and stresses in the remote field of the formation (102); and emission that produces an indication of the maximum and minimum horizontal stresses, in a tangible form. 5. Apparatus, according to claim 4, characterized by the fact that the processing circuit is operative to use the functions of determination and Petition 870180143351, of 10/22/2018, p. 45/54 3/9 estimate, only when fast and slow shear dispersion curves of the formation exhibit a transverse signature. 6. Apparatus, according to claim 4, characterized by the fact that the processing circuit estimates a reduction in the Stoneley shear module (C66), caused by fluid mobility in a reservoir, and increases the C66 measured by the same amount to obtain the magnitudes of the maximum and minimum horizontal stresses, which are compensated for the purpose of fluid mobility. 7. Method for estimating the stress ratio in a formation (102), in which a well (104) is present, characterized by the fact that it comprises: locating a well tool in the well (104) of the formation (102), cited well tool having at least one acoustic sensor (126), and using the well tool to provide slowness of Stoneley radial profiles for the formation (102 ) and bipolar shear slows for formation (102); use of electronic circuit for: calculate at least one bipolar shear module C44 and C55 using bipolar shear slowness; calculate module C66 using Stoneley slowness; Petition 870180143351, of 10/22/2018, p. 46/54 4/9 estimate the relationship between horizontal stress and overload by inversion of a difference equation obtained from the remote field shear module C66 and at least one of the modules C4 4 and C55, from a difference equation obtained from a radial profile of at least one of the bipolar shear modules C44 and C55 in two or more different radial positions within the formation (102), and a relationship in between tensions within the formation (102) next to the well (104) and voltages in the remote field of formation (102); and production of a indication of the relationship between tension horizontal and overload, tangibly. 8. Method, according to claim 7, characterized by the fact that it still comprises the determination that the curves of the fast and slow shear dispersion of the formation overlap, and do not have a transversal signature. 9. Method, according to claim 7, characterized by the fact that it still comprises the determination that the radial profiles of the bipolar shear modules C44 and C55 overlap essentially, implying that C44 = C55. 10. Method, according to claim 7, characterized by the fact that it still comprises an estimate of an increase in the Stoneley shear modulus (C66), caused by structural anisotropy in a formation Petition 870180143351, of 10/22/2018, p. 47/54 5/9 containing shales determined from measurements on core samples subjected to hydrostatic pressures, and, before inversion, the decrease in C66 measured by the same amount, to obtain the relationship between horizontal and overload tension, which is compensated with views to intrinsic anisotropy in a formation containing shales. 11. Apparatus for estimating the stress ratio in a formation (102), in which a well (104) is present, characterized by the fact that it comprises: at least one acoustic sensor, which provides radial profiles of Stoneley slowness for formation (102) and fast bipolar and slow bipolar shear for formation (102); processing circuit, which estimates the relationship between horizontal and overload stresses by inversion of the difference equations obtained from the remote field shear module C66 and at least one of the modules C44 and C55 for the formation (102), of an equation difference obtained from a radial profile of at least one of the bipolar shear modules C44 and C55 in two or more different radial positions within the formation (102) and a relationship between stresses within the formation (102) near the well ( 104) and tensions in the remote field of formation (102); and emission that produces a tangible indication of the relationship between horizontal and overload voltages. Petition 870180143351, of 10/22/2018, p. 48/54 6/9 Apparatus according to claim 11, characterized in that the processing circuit is operative to determine that the fast and slow shear dispersion curves of the formation do not exhibit a transverse signature. 13. Apparatus according to claim 11, characterized in that the processing circuit is operative to determine that the radial profiles of the modules in bipolar shear C44 and C55 if overlap, implying in that C44 = C55. 14. Apparatus, in a deal with The claim 11, characterized by fact of circuit in processing be operative to estimate an increase in the Stoneley shear modulus (C66), caused by structural anisotropy in a formation containing shales, as determined from measurements in core samples subjected to hydrostatic pressures, and, before inversion, decrease the measured C66 by the same amount to obtain the relationship between horizontal and overload stresses, which is compensated for intrinsic anisotropy in a formation containing shales. 15. Method for estimating the stress ratio in a formation (102), in which a well (104) is present, characterized by the fact that it comprises: location of a well tool in the well (104) of the formation (102), mentioned well tool having at least one acoustic sensor, and use of the tool Petition 870180143351, of 10/22/2018, p. 49/54 Ί / S well for provide slowness compression for The formation, is for to provide profiles slow radials in Stoneley for The formation and shear slowness bipolar for training (102); use of electronic circuit for: calculate compression module C33 in a remote field of formation (102) using slow compression; calculate at least one bipolar shear module C44 and C55 in the remote field of the formation using bipolar shear slowness; calculate module C66 in the remote field using Stoneley's slowness; estimate the relationship between horizontal voltage and overload by inversion of a difference equation obtained from C33 in the remote field of the formation (102) at two different depths within a range of the formation (102), of a difference equation obtained at from at least one of the bipolar shear modules C44 and C55 in the remote field of the formation (102) at two different depths within the range of the formation (102), and from a difference equation obtained from module C66 in the remote field of the formation (102) at two different depths within the formation range (102); and producing a tangible indication of the relationship between horizontal voltage and overload voltage. Petition 870180143351, of 10/22/2018, p. 50/54 8/9 16. Method, according to claim 15, characterized by the fact that it still comprises the determination that the fast and slow shear dispersion curves of the formation overlap, and do not have a transversal signature. 17. Method, according to claim 15, characterized by the fact that it still comprises the determination that the radial profiles of the bipolar shear modules C44 and C55 overlap essentially, implying that C44 = C55. 18. Apparatus for estimating the stress ratio in a formation (102), in which a well (104) is present, characterized by the fact that it comprises: at least one acoustic sensor, which provides compression slows for formation (102) and radial Stoneley slowness profiles and bipolar shear slits for formation (102); processing circuit, which estimates the relationship between horizontal and overload stresses by inversion of the difference equations obtained from C33 in the remote field of the formation (102) at two different depths within a range of the formation (102), of an equation of difference obtained from at least one of the bipolar shear modules C44 and C55 in the remote field of the formation (102) at two different depths within the range of the formation (102), and a difference equation obtained at Petition 870180143351, of 10/22/2018, p. 51/54 9/9 from module C66 in the remote field of the formation (102) at two different depths within the range of the formation (102), and emission that produces a tangible indication of the relationship between horizontal and overload stresses. 19. Apparatus according to claim 18, characterized in that the processing circuit is operative to determine that the fast and slow shear dispersion curves of the formation do not exhibit a transverse signature. 20. Apparatus according to claim 18, characterized in that the processing circuit is operative to determine that the radial profiles of the C44 and C55 bipolar shear modules overlap, implying that C44 = C55.