TY - JOUR
AU - Liu,, Yingming
AB - Abstract In the process of dipole-source acoustic far-detection logging, the azimuth of the fracture outside the borehole can be determined with the assumption that the SH–SH wave is stronger than the SV–SV wave. However, in slow formations, the considerable borehole modulation highly complicates the dipole-source radiation of SH and SV waves. A 3D finite-difference time-domain method is used to investigate the responses of the dipole-source reflected shear wave (S–S) in slow formations and explain the relationships between the azimuth characteristics of the S–S wave and the source–receiver offset and the dip angle of the fracture outside the borehole. Results indicate that the SH–SH and SV–SV waves cannot be effectively distinguished by amplitude at some offset ranges under low- and high-fracture dip angle conditions, and the offset ranges are related to formation properties and fracture dip angle. In these cases, the fracture azimuth determined by the amplitude of the S–S wave not only has a |$180^\circ $| uncertainty but may also have a |$90^\circ $| difference from the actual value. Under these situations, the P–P, S–P and S–S waves can be combined to solve the problem of the |$90^\circ $| difference in the azimuth determination of fractures outside the borehole, especially for a low-dip-angle fracture. dipole-source radiation, azimuth characteristics, fracture dip angle, slow formation 1. Introduction Single-well reflection logging, which uses a source in a fluid-filled borehole to radiate energy into the formation and detects the reflected waves from geological structures, can provide information on the development and lateral extension of structures near the borehole. Monopole-source acoustic reflection logging, first introduced by Hornby (1989), has achieved good results in evaluating geological structures outside the borehole (Esmersoy et al.1997; Yamamoto et al.2000). However, the early monopole-source acoustic reflection logging tools, which adopted an axisymmetric source and receiver array, could not determine the reflector azimuth. The sonic scanner logging tool, which arranges receivers at different positions in the circumferential direction, resolves the azimuth insensitivity, but its radial detection depth is shallow due to high excitation frequency (Haldorsen et al.2006). Dipole-source acoustic reflection logging has been proposed to overcome shallow detection depth and azimuth insensitivity (Tang 2004; Tang & Patterson 2009; Wei & Tang 2012). The extraction (Zheng & Tang 2005; Gong et al.2017; Yue 2018) and migration imaging of reflected waves (Li et al.2013, 2014, 2015; Przebindowska et al.2016) has considerably improved in the past decade. However, practical and theoretical problems remain. Previous studies have mainly focused on the optimization of dipole-source excitation frequency (Wei et al.2013a, 2013b; Tan et al.2016), the analysis of dipole-source radiation performance and radiation direction (Tang et al.2014) and the feasibility analysis of dipole-source acoustic far detection under different conditions (Cao et al.2014). The effects of dipole-source polarization direction and source–receiver offset on the amplitude of reflected waves have been investigated in fast and slow formations (Tang & Wei 2012; Wei & Tang 2012; Wei et al.2013c; Li et al.2017). Nonetheless, research on the azimuth characteristics of the reflected shear waves from the dipole source, especially in slow formations, is limited. In the process of data processing and interpretation for dipole-source acoustic far-detection logging, the SH–SH wave is considered much stronger than the SV–SV wave, and a pure SH–SH wave can be generated in the plane normal to the fracture when the dipole-source polarization direction is parallel to the reflector azimuth. Therefore, the reflector azimuth can be determined by the amplitude of the S–S wave. However, the shear wavelength of the dipole-source radiation in slow formation is shorter than that in fast formation. Consequently, borehole modulation becomes more remarkable in the former, therefore complicating the radiation of SH and SV waves further. In addition, acoustic far-detection logging in slow formation requires a longer recording time compared with that in a fast one; thus, the reflected waves are greatly attenuated. Whether the tool can measure effective reflected wave signals under this situation is also uncertain. This study investigates the responses of the dipole-source reflected shear waves in slow formations. We use a 3D finite-difference time-domain (FDTD) algorithm to simulate the dipole-source acoustic field numerically and study the variation characteristics of dipole-source reflected shear waves. Results reveal the relationships between the azimuth characteristics of the S–S wave and the source–receiver offset under different fracture dip angle conditions, thereby providing some guidance to determine the fracture azimuth in slow formations. 2. Theory and method We discuss the numerical modeling of the reflected waves from a dipole source in a fluid-filled borehole in this section. First, we consider a theoretical model for the reflection measurements in the borehole and analyze the radiations of P and S waves from a dipole source. Then, we introduce the 3D FDTD method to solve the elastic wave equation for the dipole-source acoustic field when no analytical solutions are available. 2.1. Basic principles of dipole-source acoustic reflection logging The coordinate system shown in figure 1 is used to discuss the source radiation, reflector reflection and receiver reception in a dipole-source acoustic logging survey. The wave incident plane, which is perpendicular to the plane of the reflector, is selected as the reference plane. When the dipole-source polarization direction is in the incident plane, the compressional and shear waves vibrating in this plane are defined as P and SV waves, respectively. When the dipole-source polarization direction is parallel to the reflector plane, the shear wave vibrating along the polarization direction is defined as the SH wave. For a dipole-source exciting in the borehole in the case of far detection, the reflected waves received by the receivers in the borehole are multivariable functions of the source far-field radiation, the well-side reflector reflection, the borehole receiving modulation and the inelastic attenuation of the formation, and can be written as follows (Tang & Patterson 2009): $$\begin{equation}\left\{ {\begin{array}{@{}*{1}{l}@{}} {{u_P} = \displaystyle\frac{{S(\omega )}}{{4\pi \rho V_P^2}}\displaystyle\frac{{{e^{i\omega D/{V_P}}}}}{D}{e^{ - \omega D/2{V_P}{Q{_P}}}}R{F_P}\cos {\theta _1}\cos {\theta _2}\sin \varphi = P\sin \varphi }\\ {{u_{SH}} = \displaystyle\frac{{S(\omega )}}{{4\pi \rho V_S^2}}\displaystyle\frac{{{e^{i\omega D/{V_S}}}}}{D}{e^{ - \omega D/2{V_S}{Q{_S}}}}R{F_{SH}}\cos \varphi = SH\cos \varphi }\\ {{u_{SV}} = \displaystyle\frac{{S(\omega )}}{{4\pi \rho V_S^2}}\displaystyle\frac{{{e^{i\omega D/{V_S}}}}}{D}{e^{ - \omega D/2{V_S}{Q{_S}}}}R{F_{SV}}\cos {\theta _1}\cos {\theta _2}\sin \varphi = SV\sin \varphi } \end{array}} \right.,\end{equation}$$ (1) where |$\rho $|⁠, |${V_P}$| and |${V_S}$| are the formation density, P- velocity and S wave velocity, respectively; |$S(\omega )$| is the source spectrum; |$\omega $| is the circular frequency; D is the total distance from the source to the radiation field and from the radiation field to the receiver; |$R{F_P}$|⁠, |$R{F_{SH}}$| and |$R{F_{SV}}$| are the reflection coefficients of the P, SH and SV waves, respectively; |${\theta _1}$| and |${\theta _2}$| are the emergent and incident angles, respectively; |$\varphi $| is the angle between the radiation direction of the dipole source and the polarization direction; and |${Q{_P}}$| and |${Q{_S}}$| are the quality factors of the P and S waves, respectively. Figure 1. Open in new tabDownload slide Coordinate system for the analysis of the radiation, reflection and reception of a dipole-source acoustic logging system in a fluid-filled borehole. Figure 1. Open in new tabDownload slide Coordinate system for the analysis of the radiation, reflection and reception of a dipole-source acoustic logging system in a fluid-filled borehole. We can use two orthogonal sets of dipole emission and reception systems in the borehole to receive these reflected waves. One set is along the x-direction, and the other is along the y-direction. Projecting the incident displacement vectors into the x- and y-directions can yield the following four components: $$\begin{equation} \left\{ {\begin{array}{@{}*{1}{l}@{}} {x{x_P} = P\,{{\sin }^2}\varphi }\\ {x{y_P} = P\sin \varphi \cos \varphi }\\ {y{y_P} = P\,{{\cos }^2}\varphi }\\ {y{x_P} = P\sin \varphi \cos \varphi } \end{array},} \right.\end{equation}$$ (2) $$\begin{equation}\left\{ {\begin{array}{@{}*{1}{l}@{}} {x{x_s} = SH\,{{\cos }^2}\varphi + SV{{\sin }^2}\varphi }\\ {x{y_s} = - (SH - SV)\sin \varphi \cos \varphi }\\ {y{y_s} = SH\,{{\sin }^2}\varphi + SV{{\cos }^2}\varphi }\\ {y{x_s} = - (SH - SV)\sin \varphi \cos \varphi } \end{array}} \right..\end{equation}$$ (3) For the P–P wave, we can determine the maximum reflected wave only with two components (⁠|$x{x_P}$|⁠, |$y{y_P}$|⁠), namely, $$\begin{equation}P = x{x_P} + y{y_P}.\end{equation}$$ (4) The azimuth of the reflector outside the borehole can be determined by comparing the amplitudes of |$x{x_P}$| and |$y{y_P}$|⁠. However, for S–S waves, we must combine the four components to acquire the SH–SH and SV–SV waves, i.e. $$\begin{equation}\left\{ {\begin{array}{@{}*{1}{l}@{}} {SH = x{x_s}\,{{\cos }^2}\varphi - (x{y_s} + y{x_s})\sin \varphi \cos \varphi + y{y_s}\,{{\sin }^2}\varphi }\\ {SV = x{x_s}\,{{\sin }^2}\varphi + (x{y_s} + y{x_s})\sin \varphi \cos \varphi + y{y_s}\,{{\cos }^2}\varphi } \end{array}} \right..\end{equation}$$ (5) When the reflector is approximately parallel to the borehole or the angle between the borehole and the reflector is relatively small, the SV–SV wave can be negligible. The size and azimuth of the reflector can then be determined only with the SH–SH wave in this case, which is similar to the P–P wave condition, i.e. $$\begin{equation} \left\{ {\begin{array}{@{}*{1}{l}@{}} {x{x_s} = SH\,{{\cos }^2}\varphi }\\ {y{y_s} = SH\,{{\sin }^2}\varphi \quad (SV \sim 0)}\\ {x{x_s} + y{y_s} = SH} \end{array}} \right.. \end{equation}$$ (6) The preceding discussion indicates that the results remain the same when |$\varphi $| is replaced with |$\varphi + 180^\circ $| in equations (2–6). This condition means that the dipole-source acoustic far-detection logging has a |$180^\circ $| uncertainty. In other words, the dipole-source method can only determine the direction of the reflector but not its inclination. Determining the direction of geological structures is also important in many situations, such as fracture detection. 2.2. Finite-difference method If fractures in the formation are obliquely intersecting the borehole, the borehole acoustic field has no analytical solutions. Hence, a 3D FDTD algorithm is used in this study to solve this problem. In the Cartesian coordinate system, the motion and constitutive equations expressed in terms of stress and velocity in an isotropic medium can be expressed as (Wang & Qiao 2015) $$\begin{equation} \begin{array}{@{}l@{}} \begin{array}{@{}*{1}{l}@{}} {\rho \displaystyle\frac{{\partial {v_x}}}{{\partial t}} = \displaystyle\frac{{\partial {\sigma _{xx}}}}{{\partial x}} + \displaystyle\frac{{\partial {\sigma _{xy}}}}{{\partial y}} + \displaystyle\frac{{\partial {\sigma _{xz}}}}{{\partial z}} + {f_x}}\\ {\rho \displaystyle\frac{{\partial {v_y}}}{{\partial t}} = \displaystyle\frac{{\partial {\sigma _{xy}}}}{{\partial x}} + \displaystyle\frac{{\partial {\sigma _{yy}}}}{{\partial y}} + \displaystyle\frac{{\partial {\sigma _{yz}}}}{{\partial z}} + {f_y}}\\ {\rho \displaystyle\frac{{\partial {v_z}}}{{\partial t}} = \displaystyle\frac{{\partial {\sigma _{xz}}}}{{\partial x}} + \displaystyle\frac{{\partial {\sigma _{yz}}}}{{\partial y}} + \displaystyle\frac{{\partial {\sigma _{zz}}}}{{\partial z}} + {f_z}} \end{array},\\ \end{array}\end{equation}$$ (7) $$\begin{equation} \left( {\begin{array}{@{}*{1}{l}@{}} {\frac{{\partial {\sigma _{xx}}}}{{\partial t}}}\\ {\frac{{\partial {\sigma _{yy}}}}{{\partial t}}}\\ {\frac{{\partial {\sigma _{zz}}}}{{\partial t}}}\\ {\frac{{\partial {\sigma _{yz}}}}{{\partial t}}}\\ {\frac{{\partial {\sigma _{xz}}}}{{\partial t}}}\\ {\frac{{\partial {\sigma _{xy}}}}{{\partial t}}} \end{array}} \right) = \left( {\begin{array}{@{}*{6}{l}@{}} {\lambda + 2\mu }&\quad \lambda &\quad \lambda &\quad 0&\quad 0&\quad 0\\ \quad \lambda &{\lambda + 2\mu }&\quad \lambda &\quad 0&\quad 0&\quad 0\\ \quad \lambda &\quad \lambda &{\lambda + 2\mu }&\quad 0&\quad 0&\quad 0\\ \quad 0&\quad 0&\quad 0&\quad \mu &\quad 0&\quad 0\\ \quad 0&\quad 0&\quad 0&\quad 0&\quad \mu &\quad 0\\ \quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad \mu \end{array}} \right) \cdot \left( {\begin{array}{@{}*{1}{c}@{}} {\frac{{\partial {v_x}}}{{\partial x}}}\\ {\frac{{\partial {v_y}}}{{\partial y}}}\\ {\frac{{\partial {v_z}}}{{\partial z}}}\\ {\frac{{\partial {v_z}}}{{\partial y}} + \frac{{\partial {v_y}}}{{\partial z}}}\\ {\frac{{\partial {v_x}}}{{\partial z}} + \frac{{\partial {v_z}}}{{\partial x}}}\\ {\frac{{\partial {v_x}}}{{\partial y}} + \frac{{\partial {v_y}}}{{\partial x}}} \end{array}} \right) + \left( {\begin{array}{@{}*{1}{c}@{}} {{g_{xx}}}\\ {{g_{yy}}}\\ {{g_{zz}}}\\ {{g_{yz}}}\\ {{g_{xz}}}\\ {{g_{xy}}} \end{array}} \right),\end{equation}$$ (8) where|${v_x}$|⁠, |${v_y}$| and |${v_z}$| are the velocity components; |${\sigma _{xx}}$|⁠, |${\sigma _{yy}}$| and |${\sigma _{zz}}$| are the positive stress components; |${\sigma _{xy}}$|⁠, |${\sigma _{yz}}$| and |${\sigma _{xz}}$| are the shear stress components; |$\rho $| is the density; |$\lambda $| and |$\mu $| are the Lame coefficients; and |${f_{(x,y,z)}}$| and |${g_{(x,y,z)}}$| are combined to simulate the sources. The complex frequency-shifted method is applied to absorb the boundary reflections. Two point sources with close distance and opposite phases are used to simulate the actual dipole source. A Ricker wavelet is used as the source and can be expressed as $$\begin{equation}S(t) = \left[ {1 - 2{{(\pi {f_0}t - \pi {f_0}{t_0})}^2}} \right]{e^{ - {{\left[ {\pi {f_0}(t - {t_0})} \right]}^2}}}.\end{equation}$$ (9) In the Cartesian coordinate system, the stability conditions of the finite-difference method are $$\begin{equation}\left\{ {\begin{array}{@{}*{1}{l}@{}} {\displaystyle \frac{{\Delta t{v_{\max }}}}{{\Delta x}} \le {{\left(\sqrt 3 \sum\limits_{m = 0}^{N - 1} {\left| {{a_m}} \right|} \right)}^{ - 1}}}\\ {\sqrt {{{(\Delta x)}^2} + {{(\Delta y)}^2} + {{(\Delta z)}^2}} < \displaystyle\frac{{{\lambda _{\min }}}}{2}} \end{array}} \right.,\end{equation}$$ (10) where |${v_{\max }}$| is the maximum velocity of the mode waves; |${a_m}$| is the differential coefficient; N is half of the amount of the space order; |$\Delta x$|⁠, |$\Delta y$| and |$\Delta z$| are the space steps; and |${\lambda _{\min }}$| is the smallest wavelength of the mode waves. 3. Numerical simulation results We mainly study dipole-source reflected shear waves in a slow formation. The near-borehole fracture model has a size of 12 m in the x-direction, 5 m in the y-direction and 10 m in the z-direction and is illustrated in figure 2. The model physical parameters are listed in Table 1. The fracture length is 8 m, the width is 2 cm and the dip angle is changing from |$45^\circ $| to |$90^\circ $|⁠. The fracture extension depth is 3 m, the distance between the fracture and borehole is 6 m, and the fracture azimuth is along the y-axis. The dipole source is located at |$x = 0.6\,\,m$|⁠, |$y = 1.5\,\,m$| and |$z = 1\,\,m$|⁠. Four sets of receiver arrays with equal intervals in the circumferential direction (shown in figure 3) are deployed along the borehole axis to receive the waveforms. Each receiver array contains 75 receivers. The R1 and R3 arrays are in the y-direction, whereas the R2 and R4 arrays are in the x-direction. The distance between adjacent receivers is 0.1 m, with a 1 m distance from the first receiver to the source. The main frequency of the dipole source is 3 kHz. Figure 2. Open in new tabDownload slide Configuration of the near-borehole fracture model with a dipole-source acoustic logging system in the borehole. Figure 2. Open in new tabDownload slide Configuration of the near-borehole fracture model with a dipole-source acoustic logging system in the borehole. Figure 3. Open in new tabDownload slide Circumferential deployment of receiver arrays. Figure 3. Open in new tabDownload slide Circumferential deployment of receiver arrays. Table 1. Parameter values for model calculation. P velocity (m s−1) S velocity (m s−1) Density (kg cm−3) Radius (m) Fluid 1500 0 1000 0.1 Formation 1 2400 1000 2050 – Formation 2 2800 1200 2150 – Fracture Filled 1950 600 1750 – P velocity (m s−1) S velocity (m s−1) Density (kg cm−3) Radius (m) Fluid 1500 0 1000 0.1 Formation 1 2400 1000 2050 – Formation 2 2800 1200 2150 – Fracture Filled 1950 600 1750 – Open in new tab Table 1. Parameter values for model calculation. P velocity (m s−1) S velocity (m s−1) Density (kg cm−3) Radius (m) Fluid 1500 0 1000 0.1 Formation 1 2400 1000 2050 – Formation 2 2800 1200 2150 – Fracture Filled 1950 600 1750 – P velocity (m s−1) S velocity (m s−1) Density (kg cm−3) Radius (m) Fluid 1500 0 1000 0.1 Formation 1 2400 1000 2050 – Formation 2 2800 1200 2150 – Fracture Filled 1950 600 1750 – Open in new tab We first investigate the full waveforms received in the borehole in Formation 1 when the dipole source is excited along the x- and y-directions. Figure 4 shows the full waveforms recorded by R2 when the source is excited along the x-direction (shown in figure 4a) and those recorded by R1 (shown in figure 4b) when the source is excited along the y-direction. The fracture dip angle is |$75^\circ $|⁠. Obvious leakage P and flexural waves can be observed on R1 and R2. The P–P wave is mixed with the flexural waves. An S–S wave can also be observed on R1. Figure 4. Open in new tabDownload slide Full waveforms received by the receiver arrays when the fracture dip angle is |$75^\circ $|⁠. (a) Full waveforms received by R2 when the dipole source is excited along the x-direction. (b) Full waveforms received by R1 when the dipole source is excited along the y-direction. Figure 4. Open in new tabDownload slide Full waveforms received by the receiver arrays when the fracture dip angle is |$75^\circ $|⁠. (a) Full waveforms received by R2 when the dipole source is excited along the x-direction. (b) Full waveforms received by R1 when the dipole source is excited along the y-direction. Next, we examine the reflected waveforms from the dipole source. The low-fracture dip angle conditions (dip angle |$\le\! {\rm{60}}^\circ $| in this study) are analyzed, as shown in figure 5. As the source is excited along the y-direction, a pure SH–SH wave is generated in the XOZ plane. No reflected waves can be observed at the long offset range due to the geometry of the model in this study. The SH–SH wave amplitude decreases with an increase in the source–receiver offset. When the source is excited along the x-direction, P–P, P–S, S–P and pure SV–SV waves are all generated (shown in figures 5a and c). The amplitude of the SV–SV wave is relatively large at the short offset range and decreases continuously with an increase in the offset. The arrivals of P–S and S–P waves largely differ and can be clearly distinguished. Figure 5. Open in new tabDownload slide (a) Reflected waveforms received by R2 when the dipole source is excited in the x-direction and (b) reflected waveforms received by R1 when the dipole source is excited in the y-direction. The fracture dip angle is |$45^\circ $|⁠. (c) Reflected waveforms received by R2 when the dipole source is excited in the x-direction and (d) reflected waveforms received by R1 when the dipole source is excited in the y-direction. The fracture dip angle is |$60^\circ $|⁠. Figure 5. Open in new tabDownload slide (a) Reflected waveforms received by R2 when the dipole source is excited in the x-direction and (b) reflected waveforms received by R1 when the dipole source is excited in the y-direction. The fracture dip angle is |$45^\circ $|⁠. (c) Reflected waveforms received by R2 when the dipole source is excited in the x-direction and (d) reflected waveforms received by R1 when the dipole source is excited in the y-direction. The fracture dip angle is |$60^\circ $|⁠. In high-fracture dip angle cases (⁠|${\rm{dip\ angle\ >\!60}}^\circ $| in this study), the SH–SH wave amplitude decreases with an increase in the source–receiver offset (shown in figures 6b and d). The SV–SV wave is evident at the long offset range, and its amplitude increases with an increase in the offset (shown in figures 6a and c). The P–S and S–P waves begin to mix. Figure 6. Open in new tabDownload slide (a) Reflected waveforms received by R2 when the dipole source is excited in the x-direction and (b) reflected waveforms received by R1 when the dipole source is excited in the y-direction. The fracture dip angle is |$75^\circ $|⁠. (c) Reflected waveforms received by R2 when the dipole source is excited in the x-direction and (d) reflected waveforms received by R1 when the dipole source is excited in the y-direction. The fracture dip angle is |$90^\circ $|⁠. Figure 6. Open in new tabDownload slide (a) Reflected waveforms received by R2 when the dipole source is excited in the x-direction and (b) reflected waveforms received by R1 when the dipole source is excited in the y-direction. The fracture dip angle is |$75^\circ $|⁠. (c) Reflected waveforms received by R2 when the dipole source is excited in the x-direction and (d) reflected waveforms received by R1 when the dipole source is excited in the y-direction. The fracture dip angle is |$90^\circ $|⁠. The preceding discussion implies that the SH–SH wave amplitude decreases with an increase in the source–receiver offset under low- and high-fracture dip angle conditions. However, the relationships between the SV–SV wave and the source–receiver offset are highly complex. Therefore, the amplitudes of the SH–SH and SV–SV waves may have different magnitude relationships at different offsets. In the models shown in this paper, the S–P wave has a comparable amplitude with that of the P–P wave and even greater than that of the P–P wave under several conditions. The S–P wave is also directional, and it can be considered a supplement to the dipole-source acoustic far detection. To determine the azimuth of the fracture outside the borehole in practice, we need to synthesize the S–S wave at different dipole-source polarization directions with the cross-dipole data. The fracture azimuth can then be determined in accordance with the amplitude of the S–S wave. The amplitude of the SH–SH wave is generally considered much larger than that of the SV–SV wave. Therefore, the direction at which the S–S wave is the strongest is the fracture azimuth because the S–S wave is the pure SH–SH wave in this case. However, figures 5 and 6 show that the amplitude of the SV–SV wave is comparable with that of the SH–SH wave at the short offset range under low-fracture dip angle conditions and at the long offset range under high-fracture dip angle conditions. Figures 7 and 8 are the acoustic field snapshots of the dipole source when the fracture dip angles are |$45^\circ $| and |$90^\circ $|⁠, respectively. The figures present that the SH wave has better radiation coverage, and the energy of the SH wave radiated into the formation from the dipole source is greater than that of the SV wave. Although the radiation energy of the SH wave is greater than that of the SV wave, the amplitude of the SV–SV wave reflected by the low-dip-angle fracture is almost equivalent to that of the SH–SH wave, as shown in the red oval in figures 7b and 7d. The SV–SV wave received by the receivers in the borehole may be stronger than the SH–SH wave at the short offset range after borehole receiving modulation. When the fracture angle is |$90^\circ $|⁠, the SH–SH wave amplitude in the lower part of the snapshots is significantly greater than the SV–SV wave, as depicted in figures 8c and 8d. Nevertheless, the amplitude of the SV–SV wave in the upper part of the snapshots increases, and the amplitude of the SV–SV wave received by the receivers in the borehole may be greater than the SH–SH wave in the long offset range after borehole receiving modulation. Figure 7. Open in new tabDownload slide (a) and (b) Acoustic field snapshots of the XOZ plane at different times when the dipole source is excited in the x-direction. (c) and (d) Acoustic field snapshots of the XOZ plane at different times when the dipole source is excited in the y-direction. The fracture dip angle is |$45^\circ $|⁠. Figure 7. Open in new tabDownload slide (a) and (b) Acoustic field snapshots of the XOZ plane at different times when the dipole source is excited in the x-direction. (c) and (d) Acoustic field snapshots of the XOZ plane at different times when the dipole source is excited in the y-direction. The fracture dip angle is |$45^\circ $|⁠. Figure 8. Open in new tabDownload slide (a) and (b) Acoustic field snapshots of the XOZ plane at different times when the dipole source is excited in the x-direction. (c) and (d) Acoustic field snapshots of the XOZ plane at different times when the dipole source is excited in the y-direction. The fracture dip angle is |$90^\circ $|⁠. Figure 8. Open in new tabDownload slide (a) and (b) Acoustic field snapshots of the XOZ plane at different times when the dipole source is excited in the x-direction. (c) and (d) Acoustic field snapshots of the XOZ plane at different times when the dipole source is excited in the y-direction. The fracture dip angle is |$90^\circ $|⁠. From the discussions above, we conclude that the magnitude relationships between the amplitudes of the SH–SH and SV–SV waves are complicated at the short offset range under low-fracture dip angle conditions and at the long offset range under high-fracture dip angle conditions. We next discuss the relationships between the S–S wave and the dipole-source polarization direction to analyze the azimuth characteristics of the S–S wave in these cases. We mainly focus on the dip angle of |$45^\circ $| for the low-fracture dip angle conditions and the dip angle of |$90^\circ $| for the high-fracture dip angle conditions to simplify the discussion. The reflected waves received by R1 and R3 at different dipole-source polarization directions are calculated by equation 3 when the fracture dip angle is |$45^\circ $|⁠, as shown in figure 9. |$\varphi $| is the angle between the dipole-source polarization direction and Y-axis (when the dipole-source polarization direction is along the y-axis, |$\varphi = 0^\circ $|⁠; when the dipole-source polarization direction is perpendicular to the y-axis, |$\varphi = 90^\circ $|⁠). We mainly analyze the S–S wave at short offsets (⁠|${\rm{offset\ }} \le\! {\rm{4}}{\rm{.5}}\ \ {\rm{m}}$| in this study), considering the S–S wave is very weak at long offsets. The amplitudes of P–P, P–S and S–P waves all increase with an increase in |$\varphi $| at different offsets, whereas the amplitude variation of the S–S wave is more complicated. Figure 9. Open in new tabDownload slide (a) Reflected waves at different dipole-source polarization directions at an offset of 1.5 m, (b) reflected waves at different dipole-source polarization directions at an offset of 2.5 m, (c) reflected waves at different dipole-source polarization directions at an offset of 3.5 m and (d) reflected waves at different dipole-source polarization directions at an offset of 4.5 m. The fracture dip angle is |$45^\circ $|⁠. Figure 9. Open in new tabDownload slide (a) Reflected waves at different dipole-source polarization directions at an offset of 1.5 m, (b) reflected waves at different dipole-source polarization directions at an offset of 2.5 m, (c) reflected waves at different dipole-source polarization directions at an offset of 3.5 m and (d) reflected waves at different dipole-source polarization directions at an offset of 4.5 m. The fracture dip angle is |$45^\circ $|⁠. The S–S wave amplitude is calculated at different offsets to analyze the relationships between the S–S wave amplitude and the dipole-source polarization direction, as shown in figure 10. The horizontal axis shows the angle between the dipole-source polarization direction and the y-axis. The amplitude in the figure has been normalized. The S–S wave amplitude increases with an increase in |$\varphi $| at short offsets and reaches the maximum value at |$\varphi = 90^\circ $|⁠. In other words, the S–S wave is the pure SH–SH wave when the dipole-source polarization direction is perpendicular to the y-axis; hence, the determined fracture azimuth is perpendicular to the y-axis. However, the pure SH–SH wave is generated when the dipole-source polarization direction is parallel to the y-axis. At this time, the SH–SH and SV–SV waves cannot be effectively distinguished by the amplitude, and the fracture azimuth determined by the short offset data has a |$90^\circ $| difference from the actual value. Figure 10. Open in new tabDownload slide Amplitude variation of the S–S wave with dipole-source polarization direction at different offsets when the fracture dip angle is 45º. Figure 10. Open in new tabDownload slide Amplitude variation of the S–S wave with dipole-source polarization direction at different offsets when the fracture dip angle is 45º. The reflected waveforms received by R1 and R3 shown in figure 9 imply that the S–S waves received by R1 and R3 are almost the same at different offsets when the S–S wave is dominated by the SV–SV wave (⁠|$\varphi = 90^\circ $|⁠). As |$\varphi $| changes from |$90^\circ $| to |$0^\circ $|⁠, the S–S wave changes from an SV–SV wave-dominant mode to an SH–SH wave-dominant mode. The S–S waves received by R1 and R3 then begin to have a phase difference. Therefore, the SH–SH and SV–SV waves can be effectively distinguished by the phase information of the diagonal receiver arrays, and the azimuth of the fracture can be exactly determined. If we consider the P–P and S–P waves, then we can observe that the amplitudes of the P–P and S–P waves reach the maximum values when the dipole-source polarization direction is perpendicular to the y-axis. This condition means that the dipole-source polarization direction is perpendicular to the fracture azimuth at |$\varphi = 90^\circ $|⁠, and the azimuth at which the S–S wave is the pure SH–SH wave can also be determined exactly. When the fracture dip angle is |$90^\circ $|⁠, the SH–SH wave is much stronger than the SV–SV wave at the short offset range. Therefore, we only analyze the reflected waves of different dipole-source polarization directions at long offsets (⁠|${\rm{offset\ > \! 4}}{\rm{.5}}\ \ {\rm{m}}$| in this study). The reflected waves received by R1 and R3 at different dipole-source polarization directions are shown in figure 11. The variations in P–P, P–S and S–P waves are the same as those in figure 9. When the offset is relatively short, the S–S wave amplitude decreases with an increase in |$\varphi $|⁠. At an offset of 8.5 m, the amplitude of the SV–SV wave is evidently greater than that of the SH–SH wave and the S–S wave amplitude increases with an increase in |$\varphi $|⁠. Figure 11. Open in new tabDownload slide (a) Reflected waves at different dipole-source polarization directions at an offset of 5.5 m, (b) reflected waves at different dipole-source polarization directions at an offset of 6.5 m, (c) reflected waves at different dipole-source polarization directions at an offset of 7.5 m and (d) reflected waves at different dipole-source polarization directions at an offset of 8.5 m. The fracture dip angle is |$90^\circ $|⁠. Figure 11. Open in new tabDownload slide (a) Reflected waves at different dipole-source polarization directions at an offset of 5.5 m, (b) reflected waves at different dipole-source polarization directions at an offset of 6.5 m, (c) reflected waves at different dipole-source polarization directions at an offset of 7.5 m and (d) reflected waves at different dipole-source polarization directions at an offset of 8.5 m. The fracture dip angle is |$90^\circ $|⁠. Figure 12 shows the relationships between the S–S wave amplitude and the dipole-source polarization direction at different offsets when the fracture dip angle is |$90^\circ $|⁠. The amplitude of the S–S wave reaches the maximum value at |$\varphi = 0^\circ $| at the short offset range and at |$\varphi = 90^\circ $| at the long offset range. At this time, the fracture azimuth determined by the long offset data also has a |$90^\circ $| difference from the actual value. However, if we examine the phase difference of the reflected waveforms on R1 and R3 and consider the P–P and S–P waves, then the azimuth of the fracture can be effectively determined, as discussed in the low-fracture dip angle case. Figure 12. Open in new tabDownload slide Amplitude variation of the S–S wave with dipole-source polarization direction at different offsets when the fracture dip angle is |$90^\circ $|⁠. Figure 12. Open in new tabDownload slide Amplitude variation of the S–S wave with dipole-source polarization direction at different offsets when the fracture dip angle is |$90^\circ $|⁠. The discussions above present that the S–S wave amplitude reaches the maximum value when the dipole-source polarization direction is perpendicular to the fracture azimuth at some offset ranges under low- and high-fracture dip angle conditions. The SH–SH and SV–SV waves cannot be effectively distinguished by the amplitude under these situations, and this limitation may cause problems in determining the fracture azimuth. In these cases, the phase difference of the reflected waves received by the diagonal receiver arrays can be used, and the P–P, S–P and S–S waves can be combined to determine the fracture azimuth. The dipole-source acoustic field in Formation 2 is also simulated to analyze further the azimuth characteristics of the S–S wave in slow formations. The geometric parameters of the model remain unchanged, and the physical parameters are shown in Table 1. Figures 13 and 14 show the amplitude variation of the S–S wave with dipole-source polarization direction and source–receiver offset in Formations 1 and 2 under different fracture dip angle conditions. The offset range at which the S–S wave is dominated by the SV–SV wave is marked by the red rectangle. In low-fracture dip angle cases (⁠|${\rm{dip\ angle\ }} \le\! {\rm{60}}^\circ $| in this study), the S–S wave is dominated by the SV–SV wave at the short offset range, and the amplitude of the S–S wave reaches the maximum value at |$\varphi = 90^\circ $|⁠, as shown in figure 13. The offset range at which the SV–SV wave plays a dominant role extends as the formation slows down (shown in the red rectangle in figures 13a and b). The offset range at which the SV–SV wave plays a dominant role in Formations 1 and 2 also increases as the fracture dip angle decreases (shown in the red rectangle in figures 13a and c). Under high-fracture dip angle conditions (⁠|${\rm{dip\ angle\ >\!60}}^\circ $| in this study), the S–S wave is dominated by the SV–SV wave at the long offset range, and the amplitude of the S–S wave reaches the maximum value at |$\varphi = 90^\circ $|⁠. The offset range increases as the formation slows down (shown in the red rectangle in figures 14a and b). The offset range also increases as the fracture dip angle increases in Formations 1 and 2 (shown in the red rectangle in figures 14b and d). In these cases, the fracture azimuth determined by the S–S wave at long offsets has a |$90^\circ $| difference from the actual value. Figure 13. Open in new tabDownload slide (a) Amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 1 and (b) amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 2. The fracture dip angle is |$45^\circ $|⁠. (c) Amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 1 and (d) amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 2. The fracture dip angle is |$60^\circ $|⁠. Figure 13. Open in new tabDownload slide (a) Amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 1 and (b) amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 2. The fracture dip angle is |$45^\circ $|⁠. (c) Amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 1 and (d) amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 2. The fracture dip angle is |$60^\circ $|⁠. Figure 14. Open in new tabDownload slide (a) Amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 1 and (b) amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 2. The fracture dip angle is |$75^\circ $|⁠. (c) Amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 1 and (d) amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 2. The fracture dip angle is |$90^\circ $|⁠. Figure 14. Open in new tabDownload slide (a) Amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 1 and (b) amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 2. The fracture dip angle is |$75^\circ $|⁠. (c) Amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 1 and (d) amplitude variation of the S–S wave with offset and dipole-source polarization direction in Formation 2. The fracture dip angle is |$90^\circ $|⁠. In summary, the offsets at which the SV–SV wave plays a dominant role are at the short offset range, and increase as the formation slows down and the fracture dip angle decreases under low-fracture dip angle conditions in the model shown in this paper. The offsets at which the SV–SV plays a dominated role are at the long offset range and shorten as the formation slows down and the fracture dip angle increases under high-fracture dip angle conditions. The offsets at which the SV–SV play a dominated role can be within the offset range of the existing dipole-source acoustic logging tool, especially under low-fracture dip angle conditions. These cases will cause problems in determining the fracture azimuth. The above discussions indicate that the SV–SV wave can be stronger than the SH–SH wave under low- and high-dip-angle conditions. Next, we examine whether the S–S wave can be effectively recorded in these cases with the actual dipole-source acoustic logging tools. During the acquisition of acoustic signals, the signals are quantized in accordance with the maximum amplitude of the recorded signals. When the dipole source is excited in a fluid-filled borehole, the full waveform received in the borehole is dominated by the flexural wave. Therefore, we need to consider the relative magnitude between the amplitudes of the flexural and S–S waves to record the S–S wave effectively. The ratio of the flexural wave amplitude to the S–S wave amplitude is calculated by $$\begin{equation}\begin{array}{@{}*{1}{c}@{}} {{A_{ref}} = {{\left\| {{s_{ref}}(t)} \right\|}_2}/\sqrt N }\\ {{A_{flex}} = {{\left\| {{s_{flex}}(t)} \right\|}_2}/\sqrt N }\\ {RA = 20\log {A_{flex}}/A{}_{ref}} \end{array},\end{equation}$$ (11) where |${A_{ref}}$| and |${A_{flex}}$| are the amplitudes of the S–S and flexural waves, respectively; |${s_{ref}}(t)$||${\rm{\ }}$|and |${s_{flex}}(t)$| are the signals of the S–S and flexural waves, respectively; |${\| {{s_{ref}}(t)} \|_2}$| and |${\| {{s_{flex}}(t)} \|_2}$| are the two norm numbers of the S–S and flexural wave signals, respectively; N is the length of the signal and |$RA$|is the ratio of flexural wave amplitude to S–S wave amplitude. Figure 15 shows the variations in the ratio with the source–receiver offset in the two formations. The change rules of the ratio with the offset are basically the same in the two formations. Taking 40 dB as the upper limit of the current dipole-source acoustic logging tool, this means that the ratio of flexural wave amplitude to S–S wave amplitude should be less than 40 dB. The entire offset range can receive effective S–S wave signals in the high-fracture dip angle cases. However, the range is small when the fracture dip angle is low, but it is still within the offset range of the existing logging tools. Figure 15. Open in new tabDownload slide (a) Variation in the ratio of flexural wave amplitude to S–S wave amplitude with source–receiver offset in Formation 1 and (b) variation in the ratio of flexural wave amplitude to S–S wave amplitude with source–receiver offset in Formation 2. Figure 15. Open in new tabDownload slide (a) Variation in the ratio of flexural wave amplitude to S–S wave amplitude with source–receiver offset in Formation 1 and (b) variation in the ratio of flexural wave amplitude to S–S wave amplitude with source–receiver offset in Formation 2. The discussions above indicate that in the cases in which the SV–SV wave dominates the S–S wave, the actual tools can measure effective S–S wave signals. Therefore, we need to pay attention to the problem of the |$90^\circ $| difference from the actual value when the S–S wave is used to determine the azimuth of the fracture outside the borehole in the slow formation, especially the azimuth of a low-dip-angle fracture. 4. Conclusions The 3D FDTD method is used to simulate the dipole-source acoustic field in a slow formation with a fracture outside the borehole. The dip angle of the fracture changes from |$45^\circ $| to |$90^\circ $|⁠. In the models discussed in this study, the SV–SV wave can be stronger than the SH–SH wave at some offset ranges under low- and high-fracture dip angle conditions. This condition leads to the complex relationships between the S–S wave and the dipole-source polarization direction. The offset ranges are related to formation properties and the fracture dip angle. They may be within the offset range of the existing dipole-source acoustic logging tool, especially in low-fracture dip angle cases, under which conditions the tool can still measure effective reflected wave signals. In such situations, the SH–SH and SV–SV waves cannot be effectively distinguished by the amplitude. Therefore, the azimuth of the fracture determined by the S–S wave not only has a |$180^\circ $| uncertainty but also has a |$90^\circ $| difference from the actual value. As the dipole-source polarization direction is not perpendicular to the azimuth of the fracture, the reflected waves received by the receiver arrays that are |$180^\circ $| apart from one another at the circumference direction have a phase difference. The phase difference can help distinguish whether the S–S wave is dominated by the SH–SH wave or by the SV–SV wave, and can be adopted in the data acquisition process to solve the problem of the |$90^\circ $| difference from the actual fracture azimuth. The dipole source excited in a slow formation not only generates large amplitudes of the P–P and S–S waves but also that of the S–P wave. The S–P wave, sometimes even stronger than the P–P wave, is also directional. It can be used as a complement to the dipole-source acoustic far-detection survey and help solve the problem of the |$90^\circ $| difference. 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TI - Responses of dipole-source reflected shear waves in acoustically slow formations
JF - Journal of Geophysics and Engineering
DO - 10.1093/jge/gxz078
DA - 2020-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/responses-of-dipole-source-reflected-shear-waves-in-acoustically-slow-0PIrN4bKdC
SP - 1
VL - 17
IS - 1
DP - DeepDyve
ER -