Simulation of borehole sonic waveforms in dipping, anisotropic, and invaded formations

Transcription

1 GEOPHYSICS, VOL. 76, NO. 4 (JULY-AUGUST 2011); P. E127 E139, 17 FIGS., 2 TABLES / Simulation of borehole sonic waveforms in dipping, anisropic, and invaded formations Robert K. Mallan 1, Carlos Torres-Verdín 2, and Jun Ma 2 ABSTRACT A numerical simulation study has been made of borehole sonic measurements that examined shoulder-bed, anisropy, and mud-filtrate invasion effects on frequency-dispersion curves of flexural and Stoneley waves. Numerical simulations were considered for a range of models for fast and slow formations. Computations are performed with a Cartesian 3D finitedifference time-domain code. Simulations show that presence of transverse isropy (TI) alters the dispersion of flexural and Stoneley waves. In slow formations, the flexural wave becomes less dispersive when the shear modulus (c 44 )governingwave propagation parallel to the TI symmetry axis is lower than the shear modulus (c 66 ) governing wave propagation normal to the TI symmetry axis; conversely, the flexural wave becomes more dispersive when c 44 > c 66. Dispersion decreases by as much as 30% at higher frequencies for the considered case where c 44 < c 66. Dispersion of Stoneley waves, on the her hand, increases in TI formations when c 44 > c 66 and decreases when c 44 < c 66. Dispersion increases by more than a factor of 2.5 at higher frequencies for the considered case where c 44 < c 66.Simulations also indicate that the impact of invasion on flexural and Stoneley dispersions can be altered by the presence of TI. For the case of a slow formation and TI, where c 44 decreases from the isropic value, separation between dispersion curves for cases with and without the presence of a fast invasion zone increases by as much as 33% for the flexural wave and by as much as a factor of 1.4 for the Stoneley wave. Lastly, presence of a shoulder bed intersecting the sonic tool at high dip angles can alter flexural dispersion significantly at low frequencies. For the considered case of a shoulder bed dipping at 80, ambiguity in the flexural cutoff frequency might lead to shear-wave velocity errors of 8% 10%. INTRODUCTION Although the analysis of sonic wave propagation in a fluid-filled borehole has been documented extensively, relatively little attention has been given to studying the combined effects of dipping bed boundaries, anisropy, and invasion. Studies examine the effects of isropic, dipping layered formations on sonic waveforms (Yoon and McMechan, 1992; Cheng et al., 1995; Liu et al., 1996), effects of invasion on Stoneley dispersion in isropic formations penetrated by a vertical well (Baker, 1984), effects of a dipping transversely isropic (TI) formation on dipole and monopole waveforms (Wang et al., 2002) and flexural and Stoneley dispersions (Leslie and Randall, 1992; Sinha et al., 2006a), and the effects of azimuthal shear-wave anisropy (Tang and Patterson, 2001) and orthorhombic anisropy (Cheng et al., 1995) on flexural dispersion. Peyret and Torres-Verdín (2006) study the combined effects of shoulder beds and invasion on dipole and monopole waveforms and slowness-time-coherence pls for an isropic formation penetrated by a vertical well. This paper considers a range of 3D models to quantify the relative influence of layer dip angle, invasion, bed boundary, and transverse isropy (TI) on borehole sonic measurements. Our objective is to characterize the nature of these effects and quantify their relative impact on borehole sonic measurements. We focus our analysis on the impact of these effects on slowness-frequency dispersion curves that are processed from the simulated time-domain waveforms. Flexural and Stoneley dispersion data are used commonly to infer formation properties. Furthermore, these data are sensitive to radial variations of properties extending several borehole radii away from the well (Sinha et al., 2006b). Models considered in our study include Manuscript received by the Editor 2 March 2010; revised manuscript received 17 February 2011; published online 16 June Formerly at University of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin, Texas, U.S.A.; presently at Chevron Energy Technology Co., Houston, Texas, U.S.A. rkmallan@chevron.com. 2 University of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin, Texas, U.S.A. cverdin@uts.cc.utexas.edu; junma@mail.utexas.edu. VC 2011 Society of Exploration Geophysicists. All rights reserved. E127

2 E128 Mallan et al. bh soft and hard formations, and the elastic properties are consistent with a gas-bearing layer invaded with water-based mud filtrate. The sensitivity analysis is performed with a 3D finite-difference time-domain (FDTD) code. This second-order explicit algorithm solves the first-order, coupled velocity-stress elasticwave equations with staggered-grid central differencing in bh space and time. In addition, we implement perfectly matchedlayer (PML) absorbing boundary conditions to truncate the simulation domain, thereby significantly reducing computation time. Validation of the simulation algorithm is performed against 1D and 2D computations that include presence of the borehole and a wide range of soft and hard formations and source frequencies. Presence of a rigid sonic tool is n included in this study; the simulation code assumes point sources and receivers with source-receiver array dimensions adapted from a commercial array sonic tool (Pistre et al., 2005). In the following sections, we first describe the method used to perform the numerical simulations. This includes details of the 3D FDTD algorithm used in the study, along with a description of the source-receiver configuration and the formation models examined. Next, we compare the 3D FDTD results to an independent 1D code to benchmark the accuracy of the simulations. Finally, we present results that examine the effects of vertical and dipping TI, invasion, and presence of a shoulder bed intersecting the tool at a high angle. Effects on simulated sonic waveforms are illustrated through frequency-slowness dispersion curves of flexural and Stoneley waves. SIMULATION METHOD Borehole sonic measurements are numerically simulated using a Cartesian, 3D FDTD algorithm that solves the coupled velocity-stress differential equations (following Leslie and Randall, 1992) q ov ¼rs; (1) Figure 1. Stress and velocity components in the staggered grid. Stress component s aa represents the normal stress components s xx, s yy, and s zz. and os ¼ C e; (2) where q is density, t is time, v is the velocity vector T; v ¼ v x ; v y ; v z (3) and s and e are the stress and strain tensors, respectively, expressed in vector form as T s ¼ s xx ; s yy ; s zz ; s yz ; s xz ; s xy (4) and h i e ¼ ov x ox ; ov y oy ; ov z ; ov z oy þ ov y ; ov x þ ov z ox ; ov y T: ox þ ov x oy (5) Here, the superscript T denes transpose, and C is the fourthrank stiffness tensor describing a transversely isropic medium, namely, 0 1 c 11 c 12 c c 12 c 11 c c C ¼ 13 c 13 c c : (6) B c 44 0 A c 66 Appendix A describes the ration of the stiffness tensor and the expanded differential equations for tilted TI media. Equations 1 and 2 (or equations A-1 and A-2) are discretized using staggered-grid, second-order central finite differences in bh space and time. Figure 1 shows the locations of the stress and velocity components on the staggered grid. The coupled system of velocity-stress FD equations are solved explicitly, where time-stepping is performed in a leapfrog fashion in time intervals 1=2Dt. Stability of the pffiffifd scheme is ensured by taking time steps Dt < D min =ðv max 3 Þ, where Dmin is the smallest grid spacing, and v max is the maximum wave velocity in the model. Grid dispersion is mitigated by maintaining a maximum grid spacing D max v min =ð2:5f c NÞ, where v min is the minimum wave velocity in the model, f c is the central frequency of the source wavelet, and N ¼ 20. The FD grid used in the simulations consists of cells in the z-, x-, and y-directions, respectively; grid dimensions in the x- and y-directions are identical with the exception that, because of model symmetry in the y-direction (for the simulation examples described in this paper), the grid in the y-direction is truncated at y ¼ 0, and it is designated with Neumann boundary conditions. Cell size is Dz ¼ 0:4 cm across the source-receiver array, and Dx ¼ Dy ¼ 0:4 cm near the well, incrementally increasing outward from the well to extend the grid to approximately 1 m. Simulations define the circular cross section of the borehole with a staircase approximation; therefore, small cell sizes across the borehole in the x-y plane are necessary to attain accurate simulation results. A nonsplitting perfectly matched layer (PML) is implemented at the boundaries, following that described by Wang and Tang (2003), to mitigate spurious reflections. This implementation decreases the grid size and shortens the computation time.

3 Simulation of 3D sonic waveforms E129 Borehole sonic measurements are numerically simulated for variants of the model illustrated in Figure 2, which depicts a fluid-filled borehole penetrating a sand formation layer that is shouldered by shale. The borehole radius is 11.1 cm, with the borehole fluid assumed to have a density of 1000 kg=m 3 and a compressional velocity of 1500 m=s. For cases with presence of invasion, the radius of the invaded zone is 26.2 cm, where the invasion zone exhibits a circular, piston-shaped front. Velocities and densities are assigned to the sand layer, representing fast and slow rock formations (Table 1). A formation is described as fast or slow when the shear-wave velocity of the formation is greater or less, respectively, than the borehole fluid velocity. Formation properties listed in Table 1 are chosen to be consistent with 30% (slow-formation) and 10% (fast-formation) porosity, gas-bearing sands invaded with water-base mud filtrate. We assume water saturation close to 100% for the invaded zone, which yields higher compressional velocities in the invaded zone relative to the uninvaded zone. When an appreciable amount of trapped gas remains in the invaded zone, then compressional velocities in the invaded zone can be slower than in the uninvaded zone when calculated using Gassmann s fluidsubstitution model (Gassmann, 1951). However, assuming patchy saturation for the invaded zone, velocities calculated with the patchy fluid-substitution model (Hill, 1963; Dvorkin and Nur, 1998) still would yield higher compressional velocities in the invaded zone than in the uninvaded zone. Transverse isropy (TI) is assigned to the formation layer, with the axis of symmetry normal to layer bedding. A degree of TI is assumed, where wave velocities perpendicular to layer bedding, V \, are lower with respect to wave velocities parallel to layer bedding, V jj (the isropic velocities listed in Table 1), such that V \ = V jj ¼ 0.8. Elements of the stiffness tensor are calculated from the assumed compressional and shear-wave velocities, V P and V S, respectively, using: and c 11 ¼ qv 2 Pjj ; c 33 ¼ qv 2 P? ; c 44 ¼ qv 2 S? c 66 ¼ qv 2 Sjj c 12 ¼ c 11 2c 66 : Even though a value of v P in the direction of 45 with respect to the TI symmetry axis is necessary to calculate a true value for c 13, we use the approximation (Schoenberg et al., 1996) c 13 ¼ c 33 2c 44 : Table 2 summarizes the calculated elastic constants. Presence of a rigid sonic tool is n considered in the simulations; instead, we assume point sources and receivers. Simulations assume a near source-receiver offset of m, and a receiver array length of m, with receiver spacings of 15.2 cm (Figure 3). This source-receiver array configuration is adapted from a commercially available borehole sonic-logging tool (Pistre et al., 2005). Table 1. Formation properties assumed in the numerical simulations considered in this paper. Borehole fluid density and velocity are 1000 kg=m 3 and 1500 m=s, respectively. For cases where the formation layer is transversely isropic (TI), velocities parallel to the TI symmetry axis are decreased by a factor of 1.25 relative to tabulated velocities. Model Uninvaded q=q f v P (m=s) v S (m=s) Invaded q=q f v P (m=s) v S (m=s) Slow Fast Table 2. Summary of the formation elastic constants calculated from the properties listed in Table 1. Model c 11 (GPa) c 12 (GPa) c 13 (GPa) c 33 (GPa) c 44 (GPa) Figure 2. Layered velocity model with borehole and invasion. Radius of the borehole and invasion are 11.1 and 26.2 cm, respectively. Borehole dip angle is located in the x-z Cartesian plane and is measured from the normal direction to the layers. Tables 1 and 2 summarize the assumed elastic properties of the formation layer. Slow uninvaded Slow invaded Fast uninvaded Fast invaded

4 E130 Mallan et al. The simulated sonic source is a Ricker wavelet with a center frequency of 3 khz. Simulated time-domain waveforms are converted to frequency-domain phase velocities using the extended Prony method described by Donghong et al., (2008). The 3D FDTD code is validated against an independent, analytical 1D sonic code (Chi and Torres-Verdín, 2004). Simulations are compared to those of a vertical well in an infinitely Figure 3. Sonic tool configuration assumed in the numerical simulations. The near source-receiver offset is m, and the receiver array length is m, with 15.2-cm receiver spacings. thick, anisropic layer with presence of invasion. Figure 4a shows waveforms simulated with a monopole source in the slow formation model. Displayed waveform data are pressures. Figure 4b displays Stoneley dispersion curves processed from these waveforms. Figure 5a shows waveforms simulated with a dipole source in the slow formation model. Displayed waveform data are the pressure gradient. Flexural dispersion curves processed from these waveforms are shown in Figure 5b. Similarly, comparisons between 3D and 1D code results are examined for the fast formation model. Figure 6a shows synthetic waveforms simulated with a monopole source, and Figure 6b shows processed Stoneley dispersions. Figure 7a shows synthetic waveforms simulated with a dipole source, and Figure 7b shows processed flexural dispersions. For additional reference, we display the corresponding analytical dispersion curves, which are generated directly by solving the dispersion equation described by Tang and Cheng (2004) for isropic or TI, radially layered Figure 4. (a) Synthetic waveforms simulated for the case of a monopole source in the slow formation. Results are displayed for 3D and 1D codes. (b) Stoneley dispersion processed from waveforms shown in (a). The analytical dispersion solution is also shown for reference. Simulations consider the case of a vertical well penetrating an infinitely thick, anisropic, slow formation with presence of invasion. Figure 5. (a) Synthetic waveforms simulated for the case of a dipole source in the slow formation. Results are displayed for 3D and 1D codes. (b) Flexural dispersion processed from waveforms shown in (a). The analytical dispersion solution also is shown for reference. Simulations consider the case of a vertical well penetrating an infinitely thick, anisropic, slow formation with presence of invasion.

5 Simulation of 3D sonic waveforms E131 media. Results indicate that the 3D FDTD simulations match within 1% in phase slowness the results obtained with the 1D sonic code. Although the 3D simulations compare extremely well to the 1D simulations, discrepancies are observed near the flexural cutoff frequency between the flexural dispersion curves processed from simulated waveforms (from the 3D and 1D codes) versus the analytic flexural dispersion curve (Figure 5b and Figure 7b). We believe this behavior arises from the fact that the analytical solution does n consider the finite dimension of a source-receiver array and considers the ideal case of a single mode. By contrast, dispersion curves obtained from processing waveforms collected across a finite receiver array in the proximity of the source must cope with diminished resolution at low frequencies (longer wavelengths) and resolving the presence of additional modes. We ran simulations, using the analytical 1D code, for the cases with an extended receiver array. Resultant flexural dispersion curves processed from 50 receivers, with the nearest receiver at approximately 10 m from the source, compare (with significant improvement) to the analytic dispersion curves. Vertical well SIMULATION RESULTS We first investigate the relative impact of the presence of anisropy and invasion on borehole sonic simulations in a vertical well. The models consider a circular, piston-shaped invasion front. Simulations are compared for the cases of slow and fast, infinitely thick formations, where the formation is either isropic or transversely isropic, with and without the presence of invasion. Figure 8 displays flexural and Stoneley dispersions, processed from numerically simulated waveforms, for the slow formation involving the different cases. These results indicate that presence of TI alters the degree of dispersion observed in the Figure 6. (a) Synthetic waveforms simulated for the case of a monopole source in the fast formation. Results are displayed for 3D and 1D codes. (b) Stoneley dispersion processed from waveforms shown in (a). The analytical dispersion solution also is shown for reference. Simulations consider the case of a vertical well penetrating an infinitely thick, anisropic, fast formation with presence of invasion. Figure 7. (a) Synthetic waveforms simulated for the case of a dipole source in the fast formation. Results are displayed for 3D and 1D codes. (b) Flexural dispersion processed from waveforms shown in (a). The analytical dispersion solution is also shown for reference. Simulations consider the case of a vertical well penetrating an infinitely thick, anisropic, fast formation with presence of invasion.

6 E132 Mallan et al. flexural and Stoneley waves, relative to the cases of an isropic formation. The amount of dispersion in the flexural wave is decreased by as much as 30%, whereas Stoneley-wave dispersion is increased by more than a factor of 2.5. Flexural and Stoneley dispersions exhibit sensitivity to the presence of invasion; furthermore, this sensitivity appears enhanced when the formation is transversely isropic. Separation observed between dispersion curves for cases with and without the presence of invasion increases with respect to cases in isropic formations, by as much as 33% in the flexural wave and by as much as a factor of 1.4 in the Stoneley wave. Figure 9 displays flexural and Stoneley dispersions, processed from simulated waveforms, for the fast formation involving the different cases. Results indicate that the presence of TI, as with the slow formation, alters the degree of dispersion observed in the flexural and Stoneley waves, relative to the isropic formation. Similar to slow formation models, the amount of dispersion decreases in the flexural wave. However, in contrast to slow formation models, Stoneley-wave dispersion decreases significantly, to be closely null. Flexural and Stoneley dispersions show sensitivity to presence of invasion; however, unlike for the cases of soft formation models, the sensitivity to invasion appears unchanged when the formation is transversely isropic. In addition to the Stoneley mode, a pseudo-rayleigh mode is present in the isropic case, whereas a leaky compressional mode is present in the anisropic case. The results in Figure 8 and Figure 9 consider the case where the TI is introduced by reducing c 44 relative to the isropic value, such that c 44 < c 66. To further elucidate the nature of the dispersion in response to presence of TI and invasion, we simulate the case where c 44 is increased from the isropic value (c 44 > c 66 ) by a factor of 1.25, and we simulate the anisropic cases where c 44 is fixed and c 66 is increased (c 66 > c 44 ) and decreased (c 66 < c 44 ) by a factor of Figure 10 displays analytic dispersions calculated for these anisropic cases applied to Figure 8. (a) Flexural and (b) Stoneley dispersions processed from simulated waveforms. The model is a vertical well in an infinitely thick, slow formation layer. Simulations consider the cases of isropic and anisropic layers, with and without presence of invasion. Analytical dispersion solutions (red line) are also shown for reference. Figure 9. (a) Flexural and (b) Stoneley dispersions processed from simulated waveforms. The model is a vertical well in an infinitely thick, fast formation layer. Simulations consider the cases of isropic and anisropic layers, with and without presence of invasion. Analytic dispersion solutions (red line) are also shown for reference.

7 Simulation of 3D sonic waveforms E133 the slow formation, with and without the presence of invasion. Although the low-frequency slowness of the flexural mode is controlled primarily by c 44, the amount of dispersion with increasing frequency is governed by c 66. Relative to the isropic case, when c 66 is decreased (c 66 < c 44 ), the amount of dispersion increases, whereas when c 66 is increased (c 66 > c 44 ), dispersion decreases. In contrast, the low-frequency slowness (approaching the tube wave at 0 Hz) of the Stoneley mode is controlled by c 66, and c 44 governs the degree of dispersion with increasing frequency. Specifically, dispersion of the Stoneley mode increases when c 44 decreases (c 44 < c 66 ), and dispersion decreases when c 44 increases (c 44 > c 66 ). These results are consistent with flexural and Stoneley modes sensitivities to V \ = V jj presented by Tang and Cheng, Regarding presence of invasion, the cases show that presence of TI can alter the impact to the dispersion curve relative to the impact observed in the isropic case. Deviated well Next, we examine the effects of dipping TI on flexural and Stoneley dispersion curves, and we assess the impact on the invasion effect. Inline dipole (x-dipole), crossline dipole (y-dipole), and monopole configurations are simulated for a borehole dip angle of 60 measured from the normal to the layer. For cases of TI dipping in the x-z plane, the y-dipole produces a pure shear wave (SH) with particle mion perpendicular to the x-z plane, and the x-dipole produces a quasi-shear wave (qsv) with particle mion in the x-z plane. Figure 11 displays dispersions processed from simulated waveforms for the case of the slow formation, with and without the presence of invasion. Although the SH and SV flexural dispersion curves are roughly parallel, the SH flexural wave appears slightly more dispersive than the SV flexural wave. Figure 10. (a) Flexural and (b) Stoneley dispersions calculated analytically. The model is a vertical well in an infinitely thick, slow formation layer. Simulations consider the cases of isropic and anisropic layers, with and without presence of invasion. In addition to the anisropic cases considered to produce the dispersion curves in Figure 8, where c 44 is reduced from the isropic value (c 44 < c 66iso ), dispersions are shown for the case where c 44 is increased from the isropic value (c 44 < c 66iso ) and the anisropic cases where c 44 is fixed and c 66 is increased (c 66 > c 44iso ) and decreased (c 66 < c 44iso ). Figure 11. (a) The x- and y-dipole flexural dispersions for cases with and without presence of invasion, and (b) Stoneley dispersions for cases with and without presence of invasion, processed from simulated waveforms. The model is a well dipping at 60 in an infinitely thick, slow formation layer exhibiting transverse isropy.

8 E134 Mallan et al. Regarding the impact of the fast invasion zone, separation between dispersion curves appears less discernable than for the case of the vertical TI. Regarding Stoneley dispersion, the amount of dispersion observed is between that observed for the cases of a vertical well in isropic and anisropic formation layers. The effect of invasion is ambiguous, as dispersion curves tend to overlay one anher. Figure 12 displays the corresponding results for the case of the fast formation layer, with and without the presence of invasion. Similar to the slow-formation cases, the SH flexural wave appears slightly more dispersive than the SV flexural wave. However, the effect of invasion appears greater than for cases of a vertical TI, as a larger separation at higher frequencies arises between the dispersion curves for cases with and without the presence of invasion. Regarding the Stoneley dispersion, the amount of dispersion observed is similar to that observed for the cases of a vertical well in the isropic formation layer. The effect of invasion is also similar to that observed for the cases of a vertical well in an isropic formation layer. At low frequencies, simulated flexural dispersion curves approach the theoretical ray slowness (or group slowness in the anisropic sense) described by Thomsen (1986). We believe this is a result of the strongly TI formation, which produces a strongly anelliptic wave surface (Thomsen anisropy parameters e ¼ 0.28 and c ¼ 0.16 for bh soft and hard formations considered in the analysis). This effect also is discussed by Hornby et al. (2003). Figure 12. (a) The x- and y-dipole flexural dispersions for cases with and without presence of invasion, and (b) Stoneley dispersions for cases with and without presence of invasion, processed from simulated waveforms. The model is a well, dipping at 60 in an infinitely thick, fast formation layer that exhibits transverse isropy. Figure 13. Half-space formation models dipping at (a) 0, (b) 60, (c) 80, and (d) 90. The upper half-space has elastic properties equal to those of the formation layer (listed in Table 1 and Table 2), whereas the lower half-space (shoulder bed) has the properties shown in Figure 2. The half-space boundary intersects the borehole at z ¼ 0 cm. Source location (indicated with a white *) is z ¼ cm, and the array of 13 receivers (indicated by white x marks), uniformly spaced 15.2 cm apart, is located cm above the source. Shoulder bed Finally, we examine the impact of the presence of a layer boundary (shoulder bed) on borehole sonic measurements. Simulations are performed for cases of the slow and fast formation layers overlying the shale shoulder bed, such that the layer boundary intersects the sonic tool between the source and the receiver array. The source is located in the shale layer, cm below the layer boundary, and the receiver array is located in the formation layer. Simulations consider cases of the layer boundary intersecting the borehole at 0, 60, 80, and 90. Figure 13 displays the half-space models and the locations of the source and receiver array.

9 Simulation of 3D sonic waveforms E135 Figure 14. (a) The x-dipole flexural, (b) y-dipole flexural, and (c) monopole Stoneley dispersions processed from simulated waveforms. Simulations consider the cases of half-space formation models dipping at 0,60,80, and 90 (Figure 13). The formation layer is slow and isropic, and it does n include invasion. Figure 15. (a) The x-dipole flexural, (b) y-dipole flexural, and (c) monopole Stoneley dispersions processed from simulated waveforms. Simulations consider the cases of half-space formation models dipping at 0,60,80, and 90 (Figure 13). The formation layer is fast and isropic, and it does n include invasion.

10 E136 Mallan et al. Figure 14 displays x- and y-dipole flexural and monopole Stoneley dispersion curves processed from simulated waveforms. Simulations consider the case of the slow formation layer, isropic and uninvaded, overlying the shale layer. As a reference, the analytic dispersion curves corresponding to the formation layer (solid line) and the shoulder bed (dashed line) overlay the pl. No appreciable effect is observed in the flexural and Stoneley dispersion curves for cases of 0,60, and 80 dip. However, the compressional wave slowness, observed in the monopole dispersion pl, is altered for the cases of 60 and 80 dip, such that the observed slownesses are an average of the formation layer and shoulder bed compressional wave slownesses. For the case of 90 dip, dispersion curves are affected significantly. At the lower frequencies (<5 to 6 khz), the flexural and Stoneley dispersion curves tend toward an average between the dispersion curves representing, respectively, homogeneous models of the formation layer and the shoulder bed. At the higher frequencies, dispersion processing discerns two modes, representative of the respective homogeneous cases. In similar fashion, Figure 15 displays x- and y-dipole flexural and monopole Stoneley dispersion curves processed from simulated waveforms for the case of the fast formation layer, isropic and uninvaded, overlying the shale layer. At 0 dip, the flexural and Stoneley dispersion curves are unaffected by the presence of the layer boundary between the source and the receiver array. The Stoneley dispersion curve remains unaltered for the cases of 60 and 80 dip. Conversely, at 60 and 80 dip, flexural dispersion curves become irregular at lower frequencies, with the effect becoming more severe with increasing dip angle. At higher frequencies, dispersion curves are unaffected by the presence of the shoulder bed. For the extreme case of 90 dip, the Stoneley dispersion curve is altered and exhibits the character of a slow formation, where the slowness increases with frequency. The x- and y-dipole dispersion curves appear to exhibit two flexural modes, characteristic of the two formations comprising the half-space model. A closer examination of the case of the 80 dipping halfspace model is shown in Figure 16. Although the simulated Figure 16. The x-dipole (a) waveforms, (b) amplitude spectra, (c) phase spectra, and (d) dispersion. Simulations consider the case of the half-space formation model dipping at 80 (Figure 13c). The formation layer is fast and isropic, and it does n include invasion.

11 Simulation of 3D sonic waveforms E137 waveforms exhibit no apparent effect from the dipping shoulder bed, the modulation pattern of the amplitude spectrum (Figure 16b) clearly indicates mode interference. This behavior results in the irregular nature of the dispersion (Figure 16d), where the low frequency part of the dispersion is n well determined. It is worth ning that locations (in frequency) of the sharp inflections seen in the dispersion pl at roughly 2, 3, and 4.5 khz correspond to inflection points in the phase spectrum (Figure 16c). Dispersion results processed from the simulated waveforms were confirmed independently by Xiao-Ming Tang (personal communication, 2009). We do n believe that dipping shoulder-bed effects toward the dispersion curves (Figure 15) are the result of refracted energy along the shoulder bed formation layer interface (as prescribed by Snell s law), because the distorting effect is n observed in monopole dispersion results. Such an effect is because of the lack of azimuthal symmetry in elastic properties across the borehole. For the case of the 90 dipping half-space model, the monopole Stoneley and dipole flexural modes each appear to separate (or decouple) into two separate modes respective of either half space, especially at higher frequencies, where the wavelength becomes small with respect to borehole radius, thereby allowing the mode to decouple. For the cases of 60 and 80 dip, the region along the borehole which has nonsymmetric properties is finite, at least in regard to the radial length of sensitivity of wave modes. Because the dipole flexural mode has greater radial sensitivity (and is seemingly more susceptible to decoupling because of its asymmetric nature compared to the axisymmetric nature of the Stoneley mode), we believe that the mode is partially impacted by this decoupling effect. Regarding the slow-formation dispersion curves, the impedance contrast between the slow formation and the shoulder bed is too small to have an appreciable impact on waveforms, especially at longer wavelengths. At higher frequencies, the difference in flexural-mode phase velocity between the two layers increases (as a result of the dispersive nature of the flexural mode). This behavior, together with the shorter wavelength, enables the dispersion processing to resolve the two modes. CONCLUSION Numerical simulations performed with the 3D FDTD sonic code show an excellent agreement with simulations of radially 1D, isropic, and transversely isropic formation models. Benchmarks against frequency-slowness dispersions indicate that the accuracy of the 3D code is within 1% in phase slowness. Simulations for the cases of a vertical well in an infinitely thick layer show that the presence of transverse isropy alters the amount of dispersion observed in flexural and Stoneley waves. The slowness at the low-frequency endpoint of the flexural dispersion curve is fixed by c 44 (the shear modulus governing wave propagation parallel to the TI symmetry axis), whereas the change in slowness with increasing frequency is influenced by c 66 (the shear modulus governing wave propagation normal to the TI symmetry axis). For the case of the soft formation, the flexural wave is more dispersive when c 66 < c 44 and less dispersive when c 66 > c 44. In contrast, the low-frequency endpoint of the Stoneley dispersion curve is fixed by c 66, and the change in slowness with increasing frequency is influenced by c 44, such that the Stoneley wave becomes more dispersive when c 44 < c 66 and less dispersive when c 44 > c 66. Furthermore, these simulations suggest that the presence of TI can alter the impact of invasion effects on dispersion curves. For the considered case of the soft formation, the effect of the fast invasion zone on flexural and Stoneley dispersions appears enhanced in the TI (c 44 < c 66 ) formation with respect to the isropic formation. We ne that the impact of invasion observed in this study applies to the modeled circular, piston-like invasion front. An invasion profile with a considerable transition zone could impact borehole sonic measurements in a different manner. Concerning simulations for cases of a layer boundary crossing the tool between the source and receiver array, we found that the Stoneley dispersion was unaffected by the presence of the layer boundary for dip angles below 90. On the her hand, flexural dispersions exhibited significant distortions at lower frequencies for cases of high dip angle. This distortion affects the low-frequency asympte of the flexural mode, thereby biasing the estimation of shear-wave velocity. A modulation pattern in the amplitude frequency-spectrum of the simulated waveforms indicates interference of multiple modes in half-space models dipping at high angles. At low frequencies, dispersion processing is unable to discern individual modes clearly. Moreover, at low frequencies, the increased wavelength diminishes the resolution of the receiver array, making it difficult to distinguish multiple modes that do n differ significantly in phase slowness. ACKNOWLEDGMENTS The work reported in this paper was funded by The University of Texas at Austin s Research Consortium on Formation Evaluation, jointly sponsored by Anadarko, Aramco, Baker Hughes, BG, BHP Billiton, BP, ConocoPhillips, Chevron, ENI, ExxonMobil, Halliburton, Hess, Marathon, Mexican Institute for Petroleum, Nexen, Petrobras, Schlumberger, Statoil, Tal, and Weatherford. We are grateful to Xiao-Ming Tang for his analysis of numerical simulation results. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing high performance computing resources used for the simulation of sonic waveforms. A ne of special gratitude goes to Gilles Guerin and two anonymous reviewers for their technical and editorial comments which improved the original version of the paper. APPENDIX A Ration of the stiffness tensor (Equation 6) through dip angle h about the y-axis yields (following Auld, 1990) 0 c 0 11 c 0 12 c c c 0 21 c 0 22 c c C 0 ¼ M C M T c 0 31 c 0 32 c c ¼ c c 0 ; 46 B c 0 51 c 0 52 c c A c c 0 66 where the transformation matrix M is given by

12 E138 Mallan et al. 0 b 2 11 b 2 12 b b 12 b 13 2b 13 b 11 2b 11 b 12 b 2 21 b 2 22 b b 22 b 23 2b 23 b 21 2b 21 b 22 b 2 31 b 2 32 b b M ¼ 32 b 33 2b 33 b 31 2b 31 b 32 : b 21 b 31 b 22 b 32 b 23 b 33 b 22 b 33 þ b 23 b 32 b 21 b 33 þ b 23 b 31 b 22 b 31 þ b 21 b 32 B b 31 b 11 b 32 b 12 b 33 b 13 b 12 b 33 þ b 13 b 32 b 11 b 33 þ b 13 b 31 b 11 b 32 þ b 12 b 31 A b 11 b 21 b 12 b 22 b 13 b 23 b 22 b 13 þ b 12 b 23 b 11 b 23 þ b 13 b 21 b 22 b 11 þ b 12 b 21 Here, b ij constitute the elements of the ration matrix 0 1 cos h cos / cos h sin / sin h b sin / cos / 0 A; sin h cos / sin h sin / cos h where h is the dip angle (in the x-z plane) about the y-axis, and / ¼ 0 (for the simulations considered in this paper) is the strike angle (in the x-y plane) about the z-axis. Expansion of Equations 1 and 2 by inserting Equations 3 through 5 and C 0 yields the coupled velocity-stress differential equations for a TI media rated about the y-axis, given by and os xx os yy os zz os yz os xz os xy q ov x ¼ os xx ox þ os xy oy þ os xz ; q ov y ¼ os xy ox þ os yy oy þ os yz ; q ov z ¼ os xz ox þ os yz oy þ os zz ; ¼ c 0 ov x 11 ox þ ov y c0 12 oy þ ov z c0 13 þ ov z c0 15 ox þ ov x ¼ c 0 ov x 12 ox þ ov y c0 22 oy þ ov z c0 23 þ ov z c0 25 ox þ ov x ¼ c 0 ov x 13 ox þ ov y c0 23 oy þ ov z c0 33 þ ov z c0 35 ox þ ov x ¼ c 0 44 ov z oy þ ov y ¼ c 0 ov x 15 ox þ c0 25 ¼ c 0 46 ov z oy þ ov y þ c 0 46 ov x oy þ ov y ox ov y oy þ ov z c0 35 þ c0 55 þ c 0 66 ov x oy þ ov y ox ; ov z ox þ ov x : (A-1) ; ; ; ; (A-2) Elements of the stiffness tensor associated with calculations of shear stress are averaged over the finite-difference cells neighboring the respective shear stress component using a weighted harmonic average. For example, in calculating osyz is averaged accordingly: c 0 44 ¼ Dy j þ Dy j 1, c0 44 ð Dzk þ Dz k 1 Þ Dy j Dz k c þ Dy jdz k 1 0 c þ Dy j 1Dz k 0 c 0 44ði;j;kÞ 44ði;j;k 1Þ 44ði;j 1;kÞ þ Dy j 1Dz k 1 c 0 44ði;j 1;k 1Þ In the finite-differencing of velocities at nodes for which the respective velocity component does n exist, a weighted, linearly interpolated velocity is used. For example, Figure A-1 shows the v x velocities interpolated to find ~v x, where o~vx is used in calculating the stress field s xx (see equation A-2). : Figure A-1. Finite-difference stencil showing v x velocities interpolated to find ~v x, where o~vx is used in calculating the stress field s xx (see equation A-2). REFERENCES Auld, B. A., 1990, Acoustic fields and waves in solids, 2nd ed.: Robert E. Krieger Publishing Co. Baker, L. J., 1984, The effect of the invaded zone on full wavetrain acoustic logging: Geophysics, 49, Cheng, N., C. H. Cheng, and M. N. Toksöz, 1995, Borehole wave propagation in three dimensions: The Journal of the Acoustical Society of America, 97, Chi, S., and C. Torres-Verdín, 2004, Synthesis of multipole acoustic logging measurements using the generalized reflection=transmission matrices method: 74th Annual International Meeting, SEG, Expanded Abstracts, Donghong, L., H. Wenlong, and C. Zhijie, 2008, SVD-TLS extended Prony algorithm for extracting UWB radar target feature: Journal of Systems Engineering and Electronics, 19, Dvorkin, J. and A. Nur, 1998, Acoustic signatures of patchy saturation: International Journal of Solids Structures, 35, Gassmann, F., 1951, Elasticity of porous media (Uber die elastizität poröser medien): Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 96, Hill, R., 1963, Elastic properties of reinforced solids: Some theoretical principles: Journal of the Mechanics and Physics of Solids, 11, Hornby, B., X. Wang, and K. Dodds, 2003, Do we measure phase or group velocities with dipole sonic tools?: 65th Conference and Exhibition, EAGE, F29. Leslie, H. D., and C. J. Randall, 1992, Multipole sources in boreholes penetrating anisropic formations: Numerical and experimental results: The Journal of the Acoustical Society of America, 91, Liu, Q., E. Schoen, F. Daube, C. Randall, H. Liu, and P. Lee, 1996, A three-dimensional finite difference simulation of sonic logging: The Journal of the Acoustical Society of America, 100, Peyret, A., and C. Torres-Verdín, 2006, Assessment of shoulder-bed, invasion, and lamination effects on borehole sonic logs: A numerical sensitivity study: 47th Annual Logging Symposium, SPWLA, Transactions, Paper PP. Pistre, V., T. Kinoshita, T. Endo, K. Schilling, J. Pabon, B. Sinha, T. Plona, T. Ikegami, and D. Johnson, 2005, A modular wireline sonic tool for measurements of 3D (azimuthal, radial, and axial) formation acoustic properties: 46th Annual Logging Symposium, SPWLA, Transactions, Paper P. Schoenberg, M., F. Muir, and C. M. Sayers, 1996, Introducing ANNIE: A simple three-parameter anisropic velocity model for shales: Journal of Seismic Exploration, 5,

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