Simulation of the effect of stress-induced anisotropy on borehole compressional wave propagation

Formation elastic properties near a borehole may be altered from their original state due to the stress concentration around the borehole. This can lead to an incorrect estimation of formation elastic properties measured from sonic logs. Previous work has focused on estimating the elastic properties of the formation surrounding a borehole under anisotropic stress loading. We studied the effect of borehole stress concentration on sonic logging in a moderately consolidated Berea sandstone using a two-step approach. First, we used an iterative approach, which combines a rock-physics model and a finite-element method, to calculate the stress-dependent elastic properties of the rock around a borehole subjected to an anisotropic stress loading. Second, we used the anisotropic elastic model obtained from the first step and a finite-difference method to simulate the acoustic response of the borehole. Although we neglected the effects of rock failure and stress-induced crack opening, our modeling results provided important insights into the characteristics of borehole P-wave propagation when anisotropic in situ stresses are present. Our simulation results were consistent with the published laboratory measurements, which indicate that azimuthal variation of the P-wave velocity around a borehole subjected to uniaxial loading is not a simple cosine function. However, on field scale, the azimuthal variation in P-wave velocity might not be apparent at conventional logging frequencies. We found that the low-velocity region along the wellbore acts as an acoustic focusing zone that substantially enhances the P-wave amplitude, whereas the high-velocity region caused by the stress concentration near the borehole results in a significantly reduced P-wave amplitude. This results in strong azimuthal variation of P-wave amplitude, which may be used to infer the in situ stress state. INTRODUCTION Borehole acoustic-logging data provide important information about formation elasticity (Mao, 1987; Sinha and Kostek, 1995). Monopole and crossdipole measurements are widely used for determining the formation of P-wave velocity and S-wave anisotropy (Sinha and Kostek, 1995, 1996; Winkler et al., 1998; Tang et al., 1999, 2002). Most conventional unfractured reservoir rocks, such as sands, sandstones, and carbonates, show very little intrinsic anisotropy in an unstressed state (Wang, 2002). However, stress-induced anisotropy caused by the opening or closing of the compliant and crack-like parts of the pore space due to tectonic stresses can significantly affect the elastic properties of rocks. Drilling a borehole in a formation strongly alters the local stress distribution. When the in situ stresses are anisotropic, drilling causes the closure or opening of cracks in the formation around a borehole and leads to an additional stress-induced anisotropy. Winkler (1996) experimentally measures the azimuthal variation of the P-wave velocity in a direction parallel to a borehole that was subjected to a uniaxial stress loading and showed that the borehole stress concentration has a strong impact on the velocity measurements. To fully understand the effect of borehole stress concentration on borehole sonic logging, a thorough analysis of the propagation of waves in a 3D borehole embedded in a medium with stress-dependent elastic properties needs to be conducted. The stiffness tensor of the formation around a borehole is governed by the constitutive relation between the stress field applied around the borehole and the elasticity of the rock with microcracks embedded in the matrix. Several approaches (Sinha and Kostek, 1996; Winkler et al., 1998; Tang et al., 1999; Brown and Cheng, Manuscript received by the Editor 14 May 2013; revised manuscript received 23 January 2014; published online 27 May 2014. Massachusetts Institute of Technology, Department of Earth, Atmospheric and Planetary Sciences, Cambridge, Massachusetts, USA. E-mail: xinfang@mit .edu; [email protected]. Halliburton, Houston, Texas, USA. E-mail: [email protected]. © 2014 Society of Exploration Geophysicists. All rights reserved. D205 GEOPHYSICS, VOL. 79, NO. 4 (JULY-AUGUST 2014); P. D205–D216, 20 FIGS., 1 TABLE. 10.1190/GEO2013-0186.1 D ow nl oa de d 10 /3 0/ 15 to 1 8. 51 .1 .3 . R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry .s eg .o rg / 2007; Fang et al., 2013) have been proposed to describe the stressdependent response of the elastic properties of the rock around a borehole when it is subjected to anisotropic stress loading. Liu and Sinha (2000, 2003) study the influence of borehole stress concentration on monopole and dipole dispersion using finite-difference methods to solve the elastic wave equation and a nonlinear constitutive relation that accounts for the finite deformation caused by tectonic stresses. In this paper, we use the method of Fang et al. (2013) that combines the rock-physics model of Mavko et al. (1995) and a finite-element method (FEM), to calculate the stiffness tensor of the formation around a borehole. Then, we use a finite-difference method to simulate the wave propagation along the borehole and study the effect of stresses on borehole P-wave propagation. All approaches (Sinha and Kostek, 1996; Winkler et al., 1998; Tang et al., 1999; Brown and Cheng, 2007; Fang et al., 2013) that have been used for calculating the stress-induced formation stiffness changes around a borehole are based on the data measured from compression experiments to determine the mechanical behavior of a rock under stress. The effect of tensile stress on rock stiffness is either neglected or determined by extrapolating the data from the compressive to the tensile regime. This extrapolation has no physical basis (Fang et al., 2013). Although crack opening under tension can be studied through uniaxial or triaxial experiments (Stanchits et al., 2006), further research is needed to study how to quantitatively determine the effect of crack opening from laboratory data in the calculation of borehole stress-induced anisotropy. Thus, the effect of stress-induced crack opening is not considered in this study. Moreover, the inelastic effect due to irreversible mechanical damage is also neglected because the rock-physics model of Mavko et al. (1995), which is used by Fang et al. (2013), is purely elastic. Instead of focusing on rock-physics modeling, our objective here is to discuss the importance of wave propagation simulation in the study of the effect of stress on borehole sonic logging. We first give a brief review of the method of Fang et al. (2013) for calculating the borehole stress-induced anisotropy, and then compare the numerical simulation results with the laboratory measurements of Winkler (1996) for a moderately consolidated Berea sandstone. We then simulate the effect of stress on sonic logs on the field scale. The work presented in this paper gives us a new understanding of the characteristics of sonic-wave propagation in a borehole. BRIEF REVIEW OF THE METHOD FOR BOREHOLE STRESS-INDUCED ANISOTROPY CALCULATION In the approach of Fang et al. (2013), the stress-induced anisotropy around a borehole is obtained through an iterative process (Figure 1) that combines the method of Mavko et al. (1995) and an FEM. First, we measure the Pand S-wave velocities versus hydrostatic pressure for a given rock sample. These data are used to calculate the stress dependent crack compliance of the rock. Second, we apply the workflow shown in Figure 1 to calculate the stiffness tensor of the rock around a borehole subjected to a given stress loading. In the workflow, we first use Mavko’s model to calculate the stiffness of an intact rock under a given stress loading and use it as an initial model. Second, we insert a borehole into the initial model and use an FEM to calculate the stress field around the borehole. We then iteratively use Mavko’s model to calculate the stiffness tensor of each element in the model based on the local stress tensor and replace the old stiffness tensor with the updated one. After the first iteration, the model becomes heterogeneous due to the spatially varying stress field. We use the finite element to calculate the stress distribution in the updated model and iterate over those steps inside the loop shown in Figure 1, until the model stiffness converges to a stable value. The output from the iteration is the stiffness tensor of the model as a function of space and applied stress. Validation of the finite-element program is presented in Appendix A. FINITE-DIFFERENCE MODELING We use the Pand S-wave velocities versus hydrostatic pressure data (shown in Figure 4 of Fang et al., 2013) measured from a moderately consolidated Berea sandstone sample to construct a model for our wave propagation simulation. Table 1 lists the properties of the rock sample (i.e., sample 1). To validate the applicability of our modeling to simulate stress effect on sonic logs, we first compare our numerical simulations with the laboratory measurements of Winkler (1996), in which the P-wave velocity versus azimuth Table 1. Parameters of the Berea sandstone samples in an unstressed state. Samples 1 and 2 are, respectively, the rocks used in our simulation and in the experiment of Winkler (1996). VP (km∕s) VS (km∕s) ρ (kg∕m) Porosity (%) Sample 1 (Fang et al., 2013) 2.83 1.75 2198 17.7 Sample 2 (Winkler, 1996) 2.54 N/A N/A 22 H I Figure 1. Workflow for computation of stress-induced anisotropy around a borehole (from Fang et al., 2013). FEM and M represent finite-element method and the method of Mavko et al. (1995), respectively. See text for explanation. D206 Fang et al. D ow nl oa de d 10 /3 0/ 15 to 1 8. 51 .1 .3 . R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry .s eg .o rg / around a borehole in sandstone samples with and without applied uniaxial stress was measured. After the comparison with Winkler’s high-frequency experiment, we upscale our model to the field scale and study the simulation results at frequencies suitable for field sonic logging. Comparison with Winkler’s laboratory measurements Winkler (1996) conducts acoustic experiments on a Berea sandstone sample and a Hanson sandstone sample to measure the P-wave velocity versus azimuth around a borehole. We cannot directly simulate the laboratory experiments because the Pand S-wave velocities of the rock samples versus hydrostatic pressure, which are the necessary input to determine the stress dependent crack compliance in our model construction, are not available. Instead, we use the measurements made by Fang et al. (2013) and scale the results to those of Winkler (1996). Our comparison is limited to the Berea sandstone because we only have velocity versus pressure data for a Berea sandstone sample (sample 1 in Table 1). As shown in Table 1, the Berea sandstone sample used in Winkler’s experiment has similar properties to the sample used in our simulation, so we can expect that they have similar responses under stress. In Winkler’s experiment, the Berea sandstone sample having dimensions of 15 × 15 × 13 cm and with a 2.86 cm (1.125 in) diameter borehole parallel to the short dimension was placed in a water tank for conducting acoustic measurements. The P-wave velocity at each azimuth was measured parallel to the borehole axis using a directional transducer and two receivers, which were 7 and 10 cm, respectively, away from the transducer. The average P-wave velocity before stress was applied to the sample was approximately 2.54 km∕s and there was little variation with azimuth. When Winkler’s model is scaled to a 20 cm (8 in) borehole, the corresponding frequency of the received acoustic signals is 30 kHz. We built a borehole model with the same geometry of the experiment configuration of Winkler (1996), so that the numerical results are comparable to the laboratory measurements. Figure 2 shows the geometry of our borehole model. The formation is Berea sandstone (sample 1 in Table 1) and the borehole is water saturated. A 2.86 cm (1.125 in) borehole is at the center of the model along the z-direction. A uniaxial stress is applied normal to the borehole in the x-direction. The direction of the applied uniaxial stress is defined as 0°. A 0.64 cm (1∕4 in) diameter piston source, which mimics the 1∕4 in diameter directional transducer in Winkler’s experiment, is used in the simulation. A schematic of the piston source is shown in Figure 3. The source amplitude is tapered from the center to the edges using a Hanning window that is shown as the dashed curve in Figure 3. Source time function is a Ricker wavelet with a 213 kHz center frequency, the corresponding frequency is 30 kHz in a 20 cm (8 in) borehole. Figure 4 shows a snapshot at 0.011 ms of the pressure field in the borehole excited by a piston source pointing at 30°. We can see that the wavefield excited by the piston source has good directionality and is antisymmetric with respect to the source plane. Receivers, which are shown as the blue circles in Figure 2, are placed in water and are 0.7 cm away from the borehole axis along the source direction. Perfectly match layer is used at all model boundaries to avoid boundary reflection. The boundary effect is not considered in the comparison because Winkler (1996) only measures the time of the first arriving P-wave and the distance between the model boundary and the wellbore is almost six times the P-wave wavelength. The elastic model obtained from the method of Fang et al. (2013) contains 21 independent elastic constants, which are functions of the applied stress and position. For 10 MPa stress loading, we calculate the average value of the stiffness tensor of the formation around the borehole and plot it in Figure 5. Color intensity of each box in Figure 5 represents the average value of the corresponding component in the stiffness tensor (6 × 6 matrix notation). The nine Figure 2. Borehole model geometry. A piston source (red circle) is located at the borehole center. Receivers (blue circles) are 0.7 cm away from the borehole center along the source direction that is indicated by the red arrow. A uniaxial stress is applied in the x-direction. Borehole diameter is 2.86 cm. x y