In situ calibrated velocity-to-stress transforms using shear sonic radial profiles for time-lapse production analysis

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1 In situ calibrated velocity-to-stress transforms using shear sonic radial profiles for time-lapse production analysis J. A. Donald 1 and R. Prioul 2 Abstract Borehole acoustic waves are affected by near- and far-field stresses within rocks that exhibit stress sensitivity, typically in medium- to high-porosity formations. Nonlinear, or third-order, elastic constants are obtained from the inversion of borehole sonic shear radial profiles with an elastic wellbore stress model. The stress-to-velocity relationship determined from these profiles in the elastic region surrounding the wellbore is used for calibration to compare with empirical laboratory data traditionally used in time-lapse seismic-feasibility studies to assess simulated production. This analysis enables rock physicists to use the wellbore as a laboratory and to examine the stress dependence of the acoustic velocities from in situ field data in their zone of interest. Laboratory experiments on core samples can yield both empirical and mathematical rock-physics models to describe the relationship between stress and velocity to link rock properties to in situ measurements of acoustic data (seismic and sonic). In an example from offshore Malaysia, full-waveform borehole sonic data are processed to produce shear radial profiles in a deepwater environment. The compressional velocities are mainly sensitive to stress in the polarization-propagation direction, and shear velocities are mainly sensitive to stresses in propagation and polarization directions, as expected from nonlinear elasticity. The three compressional and shear velocities vary greatly with vertical stress depending on the stress path because they depend on the three principal stress magnitudes. In contrast, a classical empirical model that depends on porosity, clay content, and effective stress cannot capture differences caused by stress path because it relies on only one stress. Results show that stress sensitivities are significantly stronger with borehole radial profiles than the empirical model for all considered stress paths (K = -0.5, 0, 0.5, and 1). Introduction Time evolution of reservoir-geomechanics properties over the life of producing reservoirs can be characterized from timelapse seismic data and 3D geomechanics models (Herwanger and Koutsabeloulis, 2011). One key ingredient needed to link seismic data and geomechanics models is a relationship among the three principal stresses (as well as pore pressure) and the elastic stiffnesses or velocities. Although the variation of the effective elastic moduli of rocks as a function of compressive stress caused by nonlinear elasticity or the closing of cracks has been reported in the laboratory for more than 40 years (Mavko et al., 1998, section 2.4), the most challenging practical task ever since has been to find a stress-stiffness relationship that captures the representative physics with few parameters to be calibrated in situ. One such model has been based on the theory of acoustoelasticity (Thurston, 1974), also sometimes called nonlinear elasticity, in which it can be shown that an initially isotropic rock described by two so-called second-order elastic constants 1 Schlumberger. 2 Schlumberger-Doll Research Center. that is subjected to three stresses is characterized by only six (instead of nine) effective elastic constants (i.e., it belongs to a special class of orthorhombic media) (Rasolofosaon, 1998) and only three so-called third-order nonlinear elastic constants (acting as stress sensitivity parameters). Early applications of acoustoelasticity to rocks in the laboratory showed that nonlinear stresssensitivity constants could be estimated if nonlinearity remains small to moderate under the application of stress (Johnson and Rasolofosaon, 1996; Winkler and Liu, 1996). One of the important results shown by the acoustoelasticity theory is that compressional velocities are affected mostly by stresses in the direction of propagation-polarization, whereas shear velocities are sensitive to stresses in both propagation and polarization directions; see evidence from the laboratory in Prioul et al. (2004). In boreholes, the local principal stress directions and magnitudes are known to be perturbed by the presence of the circular cavity, which translates into azimuthal and radial velocity variations (Winkler, 1996). The first manifestation of such velocity variations is the observation of shear-wave splitting using dipole sonic-logging tools (Esmersoy et al., 1994; Mueller et al., 1994), which can be used to identify stress-related characteristics (Tang et al., 1999; Tang and Cheng, 2004). Furthermore, the analysis of borehole flexural waves from dipole sonic showed a crossover in flexural dispersions for the radial polarization aligned parallel and normal to the stress direction theoretically (Sinha and Kostek, 1996), experimentally in the laboratory (Winkler et al., 1998), and in situ with log data (Plona et al., 2000; Sinha et al., 2000), which has now become the classical signature of stress-induced anisotropy effects on dipole sonic data. Flexural dispersion curves have been used to estimate radial profiles of shear moduli (Sinha et al., 2006; Tang and Patterson, 2010), which then have been used to estimate in situ nonlinear elastic constants and stress magnitudes (Lei et al., 2012; Donald et al., 2013). Alternative stress-velocity models applied to borehole sonic also have been considered to interpret nonelastic velocity variations in the one-radius region from the borehole wall (Sayers et al., 2007; Fang et al., 2013). In practice, although all rocks have some degree of stress sensitivity, this phenomenon is more likely to be observed within medium- to high-porosity rocks, given the current accuracy of borehole acoustic-logging technology to resolve changes in slowness with stress (Donald et al., 2013). We present here a case study from offshore Malaysia in which we identify clear stress-induced anisotropy signatures and several zones where assumptions of the acoustoelasticity model are satisfied. We recall several key steps of the method to estimate the minimum and maximum horizontal stresses and the nonlinear and reference parameters that fully describe the velocity-to-stress THE LEADING EDGE March 2015 Special Section: Borehole geophysics and sonic logging

2 transforms. Then we show that the in situ calibrated transform can be used to understand stress-path effects on velocities and, as a perspective, could be used for time-lapse seismic and reservoirgeomechanics simulations (Donald et al., 2013). Identifying stress-induced anisotropy In a well from offshore Malaysia, full-waveform sonic logs were acquired (Pistre et al., 2005) to obtain compressionalmonopole, cross-dipole, and Stoneley waveforms. The dipole sources (oriented orthogonally to each other) were processed to determine the fast and slow shear-wave slownesses (slowness = 1/velocity) and the polarization azimuth of the far-field fast shear wave (Esmersoy et al., 1994; Donald et al., 2013), which are shown in Figure 1. At discrete depths, the fast (red) and slow (blue) flexural and Stoneley (cyan) wave-train data were transformed to obtain the slowness-dispersion curves, as shown in Figure 2. The dipole crossover from the dispersion analysis clearly indicates that the dominant mechanism of anisotropy is differential horizontal stress (Donald et al., 2013). This crossover signature is present in all the clean zones throughout the logged section. Also note that polarization of the fast shear wave (or fast shear azimuth) is constant through the interval and is independent of tool rotation. In Figure 2, solid lines represent the theoretical homogeneous isotropic dispersion for each wave, taking into account the borehole fluid bulk modulus and far-field formation moduli (shear and bulk), borehole diameter, and presence of the sonic tool in the wellbore (Donald et al., 2013). The difference between the theoretical model dispersion and the measured dispersions as a function of frequency then were used to obtain a dynamic shear modulus as a function of wavelength and as a function of radii from the borehole wall into the far field as many as seven borehole radii away (so-called shear radial profile) (Sinha et al., 2006; Donald et al., 2013), as shown in Figure 3. We note that the homogenous reference model that is used in the perturbation model to derive the radial profile also requires an estimate of the mud slowness, which often is derived using the high-frequency portion of the leaky-p compressional wave. Alternatively, it is common to calibrate mud slowness within a homogenous and isotropic zone. Figure 1. Cross-dipole anisotropy processing of flexural-wave data to determine fast and slow shear slownesses, along with polarization direction of the fast shear wave. The depth track shows the differences in inline and crossline energies from the fast and slow dipoles. Track 1 shows tool and hole orientation, along with gamma ray and caliper. Track 2 shows the fast shear azimuth. Track 3 shows the slowness anisotropy (DT-based) and time-based anisotropy (differences in average arrival times) and fast and slow shear slownesses. Track 4 shows the processed waveforms at level 7 of the fast and slow dipole firing. Stress characterization and parameter estimation Assuming that one principal stress is vertical, σv, we can define a coordinate system with X 3 pointing to the vertical axis, X1 pointing to the azimuth of maximum horizontal stress σh, and X 2 pointing to the azimuth of minimum horizontal stress σh. When the rock is stress sensitive, sonic velocities change as a function of incremental changes in effective stress above and beyond a reference state. In near-vertical wellbores where there are indications of stress-induced anisotropy from dipole dispersions, the slow, fast, and Stoneley shear provide estimates of the three shear moduli (Donald et al., 2013), c44, c55, and c66, where cij = [1/ (shear slowness)]2ρb. The vertically propagating compressionalwave velocity yields the compressional modulus c 33. Then the three P-wave moduli and three S-wave moduli can be expressed in terms of the diagonal elements of elastic stiffness tensor as follows (Donald et al., 2013): Special Section: Borehole geophysics and sonic logging Figure 2. Slowness-dispersion analysis indicating stress-induced anisotropy with classic crossover behavior between fast and slow dipole firings. Monopole compressional and shear head waves are also evident at high frequencies, whereas the dispersive Stoneley waves can be seen at lower frequencies. March 2015 THE LEADING EDGE 287

3 c 11 V 112, c 22 V 222, c 33 V 332, c 44 V 322, V 312, c 66 V 322, (1) where ρ b denotes the formation bulk density and V ij (i, j = 1, 2, 3) denotes the velocity of a wave traveling along axis X i and polarized along X j (Donald et al., 2013). Following Lei et al. (2012), stress-sensitivity coefficients for the compressional moduli rely on M ref, μ ref, c 111, and c 112, whereas the stress-sensitivity coefficients for the shear moduli rely on M ref, μ ref, c 144, and c 155. M ref and μ ref are the two independent second-order elastic constants in a hydrostatically loaded reference stress (the rock is assumed to be isotropic in the unstressed or hydrostatically loaded state). There are three independent thirdorder elastic constants (Donald et al., 2013), c 111, c 112, and c 123, with c 144 = (c 112 c 123 )/2 and c 155 = (c 111 c 112 )/4. As shown by Pistre et al. (2009) and by Sun and Prioul (2010), the stress regime, or Q factor, can be related to the relative ranking of the shear moduli (Donald et al., 2013), as defined in Table 1. For a given zone that shows stress-induced anisotropy for a normal faulting regime, the ratio of shear moduli to the corresponding formation stresses yields (Donald et al., 2013) c 66 D = σ =, (2) V σ h σ H σ h where the acoustoelastic parameter D = 3/2 + (c 155 c 144 )/2μ. With measurements of the three shear moduli, the overburden stress, and the minimum horizontal stress directly measured from extended leak-off tests (XLOT) or minifracs, equation 2 can be rearranged to solve for the maximum horizontal stress directly as (Donald et al., 2013) σ H = [ ] (σ V σ h ) + σ h. (3) c 66 If both the minimum and maximum horizontal stresses are unknown, then the D parameter must be solved independently. Subsequent work by Lei et al. (2012) shows a method of obtaining D independently from the dipole radial profiles (Donald et al., 2013) combined with a borehole stress model. Inversion of the measured dipole dispersions yields a radial profile of the shear modulus from the sand face into the far field. In conjunction with the measurements, an equivalent isotropic model of the simulated dipole dispersions for each direction (maximum and minimum horizontal stress directions) can be generated in an anisotropic stress environment. The dipole measurements are affected by a combination of the near-wellbore stresses (axial, radial, and tangential) and the far-field stresses (vertical, maximum horizontal, and minimum horizontal) (Donald et al., 2013). By combining the elastic solution from Kirsch (1898) with the effect of the dipole measurements in the near and far fields by Sinha and Kostek (1996), we obtain the following relationships (Lei et al., 2012): a 2 3 a 4 (r, ϕ) ϕ=0 = m 1 + ( ) +, (4) r 2 2 r 4 Figure 3. Shear radial profiles from fast and slow dipoles and from the Stoneley wave at a single depth. The change in shear slowness is different for each of the three orthogonal shear measurements. Table 1. Shear moduli, stress regimes, and Q factor. Shear moduli ranking Stress regime Q factor > c 44 > c 66 Normal faults 0 1 c 66 > c 66 > c 44 Strike-slip faults + c 66 2c 44 1 c 2 55 c 66 > > c 44 Thrust faults 3c 66 2c 44 2 c 3 66 a 2 3 a 4 c 44 (r, ϕ) ϕ=π/2 = m 2 ( ) + c 44, (5) r 2 2 r 4 where a is the distance from the wellbore wall; r is the radius of the wellbore; and m 1 and m 2 are functions of c 144, c 155, and the reference moduli. The full derivation is shown in Lei et al. (2012) for the m 1 and m 2 terms. The model radial profiles from equations 4 and 5 are compared with the measured radial profiles from the fast (related to ) and slow (related to c 44 ) dipoles, respectively (Donald et al., 2013). A least-squares regression is performed between the model and measured profiles, as shown in Figure 4. The region in which the model and the measurements diverge represents the area that is not related to elastic behavior, and thus the data from that point to the wellbore wall are excluded. This computation is done at the same sampling rate as the sonic data are processed, and as a result, the nonlinear elastic constants are obtained at 15- cm intervals. Once the nonlinear elastic constants are known, the minimum and maximum horizontal stress magnitudes can be obtained independently. The remaining independent third-order elastic constant c 111 can be determined by changes in the compressional modulus between depths z 1 and z 2 over a reasonably uniform lithology layer. 288 THE LEADING EDGE March 2015 Special Section: Borehole geophysics and sonic logging

4 Figure 4. Fast and slow radial profiles compared with wellbore elastic model. Note the divergence of the model regressions and the field data at the plastic yielding point. In stress-sensitive formations, it is common to observe that the moduli increase with depth, and the hydrostatic and overburden stresses increase accordingly. Equation 6 describes this evaluation using the same inputs as described above (Lei et al., 2012, equation 78): c111 = Mref z=z Mref z=z 8 4 c + μ + K ref z=z, ref z=z σref z=z σref z=z (6) 2 where Kref = Mref 4μref /3. Figure 5 shows the results of the stress-magnitude inversion. The requirements for choosing each zone were a positive indication for dispersion crossover, intrinsic formation isotropy (no layering) and consistent formation properties over a minimum length of 3 m, and a minimum of 2% shear slowness anisotropy between fast and slow dipole sources. The results of the three shear moduli indicate that this section has a normal stress regime, where c55 > c44 > c66, and the stress Q factor is computed to be 0.66 (Donald et al., 2013). A result of a leak-off test is plotted for the upper part of the interval, along with the equivalent mud weight used to drill the well. Both values are consistent with the results of the minimum horizontal stress values. The difference between the minimum and maximum horizontal stresses ranges from 2.5 to 3.5, or 8% to 12%. The sediments are relatively high in porosity, and the change in the three shear moduli with depth is very evident (Donald et al., 2013). Pore pressure (Pp ) is measured in this section using a wireline formation tester, and Biot s alpha (α) is assumed to be As an example, we report the complete stress determination in Table 2 for a depth of 2504 m. It should be noted that the dual-axis caliper measurements show no ovality over the logged section. The difference in the shear moduli is evidence that the principal formation stresses are different (Donald et al., 2013). Special Section: Borehole geophysics and sonic logging Figure 5. Stress magnitude and nonlinear elastic constant analysis from offshore Malaysia. Gamma ray is displayed in track 1. Bulk density and oriented calipers are shown in track 2. Three shear moduli from fast and slow dipoles and Stoneley are in track 3. Stress magnitudes with pore pressure and overburden stress gradients and calibration points (leak-off test, mud weight, and pore-pressure tests) are shown in track 4. Nonlinear elastic constants from the dipole radial profiling are shown in track 5. Table 2. In situ stress magnitudes for a well offshore Malaysia at a depth of 2504 m. SV 37.4 SH Sh 35.1 Pp α Table 3. Reference moduli and stress-sensitive constants used for stress determination at 2504 m. Mref 12.2 µref 3.4 σref 34.5 c c c111 24,669 Velocity variations under different stress paths Once all three stress magnitudes and the stress-sensitivity coefficients of the zone of interest have been determined, we have in situ calibrated velocity-to-stress transforms that can be used readily for time-lapse seismic reservoir geomechanics (Donald et al., 2013). Table 3 shows the coefficients for the case study. During primary depletion, vertical effective stress and horizontal effective stress increase within the reservoir because of a decrease in pore pressure, whereas in the caprock, vertical March 2015 THE LEADING EDGE 289

5 effective stress decreases and horizontal stresses can increase (Herwanger and Horne, 2009). The stress path K is a convenient way to characterize tensor stress changes from an initial stress state by a single parameter. It is defined as the ratio between the change in minimum horizontal effective stress and the change in vertical effective stress (Donald et al., 2013): σ h K =. (7) σ V We assume here that the maximum and minimum horizontal stresses are changing in the same way ( σ H = σ h ). We consider several modes of deformation, i.e., different values of K, to illustrate how seismic velocities change as a function of different stress paths. For example, under hydrostatic stress changes such as porepressure changes, the horizontal and vertical stresses are increased simultaneously by equal amounts, i.e., K = 1 (Donald et al., 2013). If deformation of the reservoir is constrained by a no-lateraldeformation boundary condition (such as in uniaxial strain experiments), elasticity theory tells us that vertical stress changes are associated with horizontal stress changes as 0 < K = (ν/1 ν) < 1, where ν is Poisson s ratio. For laterally unconstrained compression using only vertical force (i.e., horizontal stress changes as Δσ H = Δσ h = 0 σ H = σ h = 0), the stress path is K = 0. Negative stress paths are predicted for overburden stretching (K < 0) (Donald et al., 2013). Figure 6 shows the variations of the three compressional velocities as a function of small perturbations of effective vertical stress for the stress conditions of depth 2504 m and for the different stress path K = 0.5, 0, 0.5, and 1. Figure 7 shows the same for the three shear velocities. Because the nonlinear model was calibrated near a reference stress, we analyze only perturbations within 6 of the vertical stress of the considered depth (Donald et al., 2013). For comparison of the plots, we show the results from the classical empirical Eberhardt-Phillips et al. (1989) model that depend on porosity (26%) and clay volume V clay (5%) as well as an effective stress P e (velocities in kilometers per second and stress in kilobars) (Donald et al., 2013): V P = φ 1.73 V clay (P e e 16.7P e), (8) V S = φ 1.57 V clay (P e e 16.7P e). (9) For a visual comparison of stress-sensitivity effects, we arbitrarily shifted V P and V S at the reference vertical stress (Donald et al., 2013). We make several observations (Donald et al., 2013): The compressional velocities are sensitive mainly to the stress in the polarization-propagation direction (e.g., V P33, V P11, V P22 depend, respectively, mainly on S V, S H, and S h ) with a slight dependence to stresses in orthogonal directions, as expected from nonlinear elasticity. The shear velocities are sensitive mainly to the stresses in both the propagation and polarization directions (e.g., V S31 to S V and S H, V S32 to S V and S h, and V S12 to S H and S h ), as expected from nonlinear elasticity. The three compressional and shear velocities vary greatly with vertical stress depending on the stress path because they depend on the three principal stress magnitudes. The empirical model cannot capture differences caused by stress path because it relies on only one stress. The stress sensitivities are significantly stronger than the empirical V P and V S for all considered stress paths (K = 0.5, 0, 0.5, and 1). The model is calibrated for in situ conditions for three independent stresses and orthorhombic elastic media (nine independent constants), whereas the empirical model had to be calibrated artificially to the in situ conditions. Figure 6. Variations of vertical (V P33, green) and horizontal (V P11 and V P22, red and blue) compressional velocities as a function of small perturbations of the effective vertical stress for the different stress paths K = 0.5, 0, 0.5, and 1 for the Malaysia case study at a depth of 2504 m. The empirical model V P for 26% porosity and 5% clay volume also is reported as a function of effective vertical stress and is shifted arbitrarily for each reference velocity. Figure 7. Variations of vertical (V S31 and V S32, green and red) and horizontal (V S12, blue) shear velocities as a function of small perturbations of the effective vertical stress for the different stress paths K = 0.5, 0, 0.5, and 1 for the Malaysia case study at a depth of 2504 m. The empirical model V S for 26% porosity and 5% clay volume also is reported as a function of effective vertical stress and is shifted arbitrarily for each reference velocity. 290 THE LEADING EDGE March 2015 Special Section: Borehole geophysics and sonic logging

6 When sensitivities to compressional stress are known, the full elastic stiffness-to-stress transforms are known, and fluid substitution on the anisotropic orthorhombic medium could be pursued easily for advanced time-lapse seismic scenario analysis (Donald et al., 2013). Conclusions Horizontal stress magnitudes and third-order elastic constants were determined using full-waveform borehole acoustic waves along with the effective vertical stress and an acoustoelastic model based on nonlinear elasticity. An example from Malaysia was presented in which rock formations exhibited measurable stress sensitivity to acoustic waves, and this technique provided estimates of stress magnitudes consistent with field observations. When all three stress magnitudes and the stress-sensitivity coefficients of the zone of interest are known, the in situ calibrated velocity-to-stress transforms can be used to understand the effects of the stress path on velocities and could be used for time-lapse seismic and reservoir-geomechanics simulations. These observations illustrate that borehole sonic measurements can be used with coupled geomechanical models to resolve orthotropic stress states and their respective stress paths beyond the traditional empirical relationship to effective pressure. Acknowledgments We thank Schlumberger for the time and permission to publish this article, and we thank an anonymous company for providing the data. Corresponding author: References Donald, J. A., R. Prioul, T. Lei, and B. Sinha, 2013, Stress characterization in deep boreholes using acoustoelasticity, in X.-T. Feng, J. A. Hudson, and F. 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