Study on the Numerical Simulation of Array Sonic Logging Responses and Their Velocity Dispersion Characteristics in Fractured Formation
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摘要:
为分析裂缝性储层中裂缝发育带对阵列声波测井响应的影响，本文采用交错网格有限差分算法实现孔弹介质阵列声波测井波列响应的数值模拟。通过对模拟波形速度频散信息的提取，总结裂缝发育带与扩径井段不同模式波的吸收衰减与速度频散响应特征的差异性。首先基于Biot孔弹介质模型导出柱坐标下弹性波的速度-应力方程；其次，用交错网格有限差分方法对孔弹波动方程进行离散化，为提高模拟算法效率与精度，使用高阶差分格式与NPML；最后，基于含倾斜裂缝带模型与井壁坍塌地层模型的响应模拟，用前后向振幅相位估计法分析阵列声波的波形幅度相位响应特征与速度频散的差异性。结果表明，低速裂缝带纵横波波场时差会增大，且衰减幅度随裂缝带倾角变化；井壁塌陷在径向与轴向上扩大时，会增强斯通利波与伪瑞利波衰减。声波纵横波时差、速度频散与幅度衰减均可反映一定的裂缝带特征，这些影响对评价和开发裂缝性储层具有重要意义。
Abstract:To understand the effects of fractured formation on the responses of array sonic logging, the wave propagation of array acoustic logging was simulated based on the staggered grid finite difference method and poroelastic theory. Methods of velocity dispersion analysis and attenuation estimation were carried out on simulated sonic waveforms, and the differences of velocity dispersion and wave amplitude attenuation among these wave modes were summarized. First, the velocity-stress equation in cylindrical coordinates was deduced for the Biot poroelastic model. Next, the velocity-stress wave equation was discretized by using the staggered grid finite difference method. Furthermore, the higher order finite difference format and nearly perfect match layer (NPML) were adopted to improve the efficiency and accuracy of the finite difference method. Finally, simulated sonic waveforms in formation models with a tilted fracture zone and collapsed interval were applied to the velocity dispersion analysis by using forward and back amplitudes and the phase estimation method. The results show that the time difference of the P and S wave field in the low-velocity fracture zone increases and that the attenuation amplitude changes with the dip angle of the fracture zone. The attenuation of Stoneley waves and pseudo-Rayleigh waves is enhanced when the shaft collapse expands in the radial and axial directions. The time difference of acoustic waves, velocity dispersion, and amplitude attenuation can reflect the characteristics of the fracture zone, which is of great significance to the evaluation and development of fractured reservoirs.
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图 2 轴向对称性地层的配置（a）与非轴对称空间域离散化的交错网格剖分示意图（b）
Figure 2. Scheme of formation with (a) axial symmetric and (b) asymmetric staggered grids in a three-dimensional discretized space domain
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图 7 有限差分数值解与经典波数积分解结果对比
Figure 7. Comparisons between FD numerical simulation results and those of a classic wave number integral
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图 9 （a）基质地层，（b）30°，（c）50°，（d）70°倾斜裂缝带地层模型的波场模拟结果
Figure 9. (a) No slow formation, (b) inclined slow formation (dip = 30), (c) inclined slow formation (dip = 50), (d) inclined slow formation (dip = 70)
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图 10 源处于5 m处时7.59 m处第1个接收器在4个模型中的全波波形
Figure 10. Waveforms of the first receiver when the source is at 5.0 and the receiver is at 7.59 m in four models
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图 11 含50°倾角裂缝带与基质地层在7.59 m处的频率-慢度图
Figure 11. Dispersion plot velocity slowness at 7.59 m, (a) formation with an inclined fracture zone （dip=50） and （b） matrix formation
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图 12 8个不同倾角裂缝带地层模型的模拟波形用STC方法提取横波速度谷值点随倾角与深度的变化
注：（a）速度随裂缝带倾角与深度的变化，（b）速度谷值点随裂缝带倾角与深度的变化。
Figure 12. STC analysis of simulated scenarios with different inclined sediments
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图 13 8个不同倾角裂缝带地层模型的模拟波形用STC方法提取纵波速度谷值点随倾角与深度的变化
注：（a）速度随裂缝带倾角与深度的变化，（b）速度谷值点随裂缝带倾角与深度的变化。
Figure 13. STC analysis of simulated scenarios with different inclined sediments
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图 14 含径向（a）和纵向（b）扩径模拟地层配置示意图
Figure 14. Scheme of simulated formation configurations with radial borehole enlargement (a) and vertical borehole enlargement (b)
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图 15 （a）无扩径，（b）0.2 m扩径，（c）0.4 m扩径，（d）0.6 m扩径全波列数据
Figure 15. Full waveform data of (a) no enlargement, and with enlargement of (b) 0.2 m, (c) 0.4 m and (d) 0.6 m
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图 17 （a）无扩径，（b）扩径范围0.4 m，（c）扩径范围0.8 m，（d）扩径范围1.2 m全波列数据
Figure 17. Full waveform data with (a) no enlargement and enlargement of (b) 0.4 m, (c) 0.8 m and (d) 1.2 m
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图 16 （a）0.2 m扩径，（b）0.6 m扩径在8.59 m处的FBAPES结果
Figure 16. FBAPES results at 8.59 m of (a) 0.2 m and (b) 0.6 m
表 1 含倾斜慢速地层模型物性参数
Table 1 Simulated parameters of inclined slow formation
分区介质 分层位置 固相参数 流相参数 耦合参数 耗散参数 A11 A13 A44 ρ11 R ρ22 Q ρ12 b 井中流体 0.1 R/m 2.25 2.25 0 1.00 0 1.00 0 0 0 基质地层 3.0 R/m 43.39 13.62 14.89 2.27 0.30 0.16 1 -0.06 1 慢速地层 -6.7 Z/m 8.00 2.24 2.88 2.08 0.30 0.16 1 -0.06 1 
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表 2 阵列声波测井模拟参数
Table 2 Simulated schematic for array sonic logging
参数 取值 起始深度/m -11 结束深度/m -6 深度间隔/m 0.2 源类型 单极子源 子波类型 Ricker子波 采样点数 12000 采样间隔/μs 0.47 首接收器与源的距离/m 2.59 接收器间隔/m 0.152 接收器数量 8
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表 3 阵列声波测井模拟参数
Table 3 Simulated parameters of inclined slow formation
分区介质 分层位置 固相参数 流相参数 耦合参数 耗散参数 A11 A13 A44 ρ11 R ρ22 Q ρ12 b 井中流体 0.1 R/m 2.25 2.25 0 1.00 0 1.00 0 0 0 快速地层 3.0 R/m 43.39 13.62 14.89 2.27 0.30 0.16 1 -0.06 1 坍塌填充 -7.0 Z/m 2.25 2.25 0 1.00 0 1.00 0 0 0
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