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Double-porosity poroelastic model takes into account the effect of mesoscopic flow induced by rock heterogeneity on dispersion and attenuation of elastic waves, and has obtained good application results in the quantitative explanation of seismic data in heterogeneous reservoirs. Wavefield simulation based on double-porosity model not only helps visualize the propagation characteristics of the elastic waves but also lays the foundation for seismic imaging. In this work, we perform wavefield simulation and analysis based on the Santos-Rayleigh model which incorporates mesoscopic and global flow in a partially-saturated double-porosity medium. Specifically, the mesoscopic flow mechanism is represented with a Zener viscoelastic model. The comparison shows that the Zener model can accurately capture the propagation characteristics of fast P-wave, but fails to describe the attenuation characteristics of slow P3 wave in the low-frequency band. It implies that Zener viscoelastic model and slow wave modes follow different mechanisms. Then the staggered grid finite-difference method is used to simulate wave propagation in a double-porosity medium, and the stiff problem is solved with a time-splitting algorithm, which can significantly improve computational efficiency. Based on the above methods, the correctness of our algorithm is verified with derived analytical solution for a P-wave source in a uniform partially saturated poroelastic medium. Analytical and numerical solutions are in good agreement and mean error is 0.33%. We provide some examples of wavefield snapshots and seismograms in homogeneous and layered heterogeneous media at seismic and ultrasonic frequencies. The simulation results demonstrate the strong attenuation of fast P-wave and no change of S-wave in the seismic band due to mesoscopic flow mechanism, which is consistent with the theoretical prediction of double-porosity model. Moreover, the energy of fast P-wave is concentrated in solid phase while slow waves are stronger in fluid phase. This work contributes to the understanding of broadband elastic wave propagation in a heterogeneous partially saturated porous medium and can be applied to the reservoir imaging with broadband geophysical data.
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Keywords:
- double-porosity media /
- finite difference /
- dispersion and attenuation
[1] Biot M A 1956 J. Acoust. Soc. Am. 28 179
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[2] Santos J E, Corberó J M, Douglas J 1990 J. Acoust. Soc. Am. 87 1428
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Google Scholar
[4] Berryman J G, Wang H F 2000 Int. J. Rock Mech. Min. 37 63
Google Scholar
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图 1 Santos-Rayleigh模型和黏弹性Santos模型预测的快纵波频散(a)和衰减(b)
Figure 1. Comparison of P1 wave velocities (a) and attenuations (b) as the function of frequencies predicted by Santos-Rayleigh and viscoelastic Santos models.
图 2 Santos-Rayleigh模型和黏弹性Santos模型预测的慢纵波频散(a)和衰减(b)
Figure 2. Comparison of P2 and P3 wave velocities (a) and attenuations (b) as the function of frequencies predicted by Santos-Rayleigh and viscoelastic Santos models.
图 3 各场分量在交错网格的相对位置
Figure 3. Relative position of field components on the staggered grids.
图 4 S2 = 0.9, ${\eta }_{\mathrm{{f}}}^{(1)}={\eta }_{\mathrm{{f}}}^{(2)}=0 $时数值解与解析解比较 (a)固相; (b)非润湿相; (c)润湿相
Figure 4. Comparison of analytical and numerical solutions when S2 = 0.9, ${\eta }_{\mathrm{{f}}}^{(1)}={\eta }_{\mathrm{{f}}}^{(2)}=0 $: (a) Solid phase; (b) non-wetting phase; (c) wetting-phase.
图 5 fp = 20 Hz, 0.18 s固相垂直速度分量的波场快照 (a) 无介观流影响; (b) 有介观流影响
Figure 5. Snapshots of the vertical components of solid particle velocities at 0.18 s with fp = 20 Hz: (a) Without mesoscopic attenuation; (b) with mesoscopic attenuation.
图 6 fp = 20 Hz, (270, 270) m处有无介观流两种情况下固相接收的波形对比
Figure 6. Comparison of seismograms of the solid phase at (270, 270) m with fp = 20 Hz.
图 7 fp = 30 kHz, 0.15 ms时刻固相垂直速度分量的波场快照 (a) 无介观流影响; (b) 有介观流影响
Figure 7. Snapshots of the vertical components of solid particle velocities at 0.15 ms with fp = 30 kHz: (a) Without mesoscopic attenuation; (b) with mesoscopic attenuation.
图 8 fp = 30 kHz, (0.1, 0.3) m处有无介观流两种情况下$V_z^{(1)} $的波形对比
Figure 8. Comparison of seismograms of the $V_z^{(1)} $ at (0.1, 0.3) m with fp = 30 kHz.
图 9 fp = 30 kHz, 0.15 ms非润湿相流体的垂向速度分量波场快照
Figure 9. Snapshots of the vertical components of non-wetting particle velocities at 0.15 ms with fp = 30 kHz.
图 10 fp = 20 Hz, 0.2 s固相垂向速度分量波场快照 (a) 无介观流影响; (b) 有介观流影响
Figure 10. Snapshots of the vertical components of solid velocities at 0.2 s with fp = 20 Hz: (a) Without mesoscopic attenuation; (b) with mesoscopic attenuation.
图 11 fp = 30 kHz, 0.15 ms时, 流体垂向速度分量波场快照 (a)固相; (b)非润湿相; (c)润湿相
Figure 11. Snapshots of the vertical components of velocities with fp = 30 kHz, 0.15 ms: (a) Solid phase; (b) non-wetting phase; (c) wetting-phase.
表 1 部分饱和双重孔隙介质物性参数表
Table 1. Physical parameters of partially saturated double-porosity media.
符号 参数 层1 层2 Ks 基质体积模量/GPa 36 40 ρs 基质密度/(kg·m–3) 2650 2800 Km 骨架体积模量/GPa 6.21 9.5 μm 骨架剪切模量/GPa 4.55 6.2 ϕ 孔隙度 0.33 0.2 κ 渗透率/m2 4.93×10–12 2.96×10–12 ${K}_{\mathrm{{f}}}^{(1)} $ 润湿相流体体积模量/GPa 2.223 $ {\rho}_{\mathrm{{f}}}^{(1)}$ 润湿相流体密度/(kg·m–3) 1000 ${\eta}_{\mathrm{{f}}}^{(1)} $ 润湿相流体黏度/(Pa·s) 0.001 ${K}_{\mathrm{{f}}}^{(2)} $ 非润湿相流体体积模量/GPa 0.022 ${\rho}_{\mathrm{{f}}}^{(2)} $ 非润湿相流体密度/(kg·m–3) 100 ${\eta}_{\mathrm{{f}}}^{(2)} $ 非润湿相流体黏度/(Pa·s) 1.5×10–5 
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[1] Biot M A 1956 J. Acoust. Soc. Am. 28 179
Google Scholar
[2] Santos J E, Corberó J M, Douglas J 1990 J. Acoust. Soc. Am. 87 1428
Google Scholar
[3] Liu L, Zhang X, Wang X 2022 J. Theor. Comp. Acout. 30 2150002
Google Scholar
[4] Berryman J G, Wang H F 2000 Int. J. Rock Mech. Min. 37 63
Google Scholar
[5] Huang J D, Yang D H, He X J, Chang Y F 2023 Geophysics 88 T121
Google Scholar
[6] Pride S R, Berryman J G, Harris J M 2004 J. Geophys. Res. 109 B01201
Google Scholar
[7] Zheng P, Ding B Y, Sun X T 2017 Int. J. Rock Mech. Min. 91 104
Google Scholar
[8] Ba J, Carcione J M, Nie J X 2011 J. Geophys. Res. 116 B06202
Google Scholar
[9] Sun W T, Ba J, Carcione J M 2016 Geophys. J. Int. 205 22
Google Scholar
[10] Ba J, Xu W H, Fu L Y, Carcione J M, Zhang L 2017 J. Geophys. Res. Solid Earth 122 1949
Google Scholar
[11] 石志奇, 何晓, 刘琳, 陈德华, 王秀明 2023 72 069101
Google Scholar
Shi Z Q, He X, Liu L, Chen D H, Wang X M 2023 Acta Phys. Sin. 72 069101
Google Scholar
[12] Shi Z Q, He X, Chen D H, Wang X M 2024 Geophys. J. Int. 236 1172
Google Scholar
[13] Ba J, Nie J X, Cao H, Yang H Z 2008 Geophys. Res. Lett. 35 L04303
Google Scholar
[14] Liu X, Greenhalgh S, Zhou B 2009 Geophys. J. Int. 178 375
Google Scholar
[15] Liu X, Greenhalgh S 2019 Geophysics 84 WA59
Google Scholar
[16] Wang E, Carcione J M, Ba J 2019 Geophysics 84 WA11
Google Scholar
[17] Carcione J M 2015 Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media (Amsterdam Boston: Elsevier
[18] Jiang Y C, Gao Y X, Cheng Q L, Song Y J 2023 Geophys. J. Int. 235 970
Google Scholar
[19] Virieux J 1986 Geophysics. 51 889
Google Scholar
[20] Wenzlau F, Müller T M 2009 Geophysics 74 T55
Google Scholar
[21] Guan W, Hu H 2011 Commun. Comput. Phys. 10 695
Google Scholar
[22] 孔丽云, 王一博, 杨慧珠 2013 62 139101
Google Scholar
Kong L Y, Wang Y B, Yang H Z 2013 Acta Phys. Sin. 62 139101
Google Scholar
[23] 刘财, 罗玉钦 2023 地球 66 3840
Google Scholar
Liu C, Luo Y Q 2023 Chin. J. Geophys. 66 3840
Google Scholar
[24] Zhao H, Wang X 2008 Sci. China Ser. G-Phys. Mech. Astron. 51 723
Google Scholar
[25] Deng W B, Fu L Y, Wang Z W, Hou W T, Han T C 2023 Geophys. J. Int. 235 1218
Google Scholar
[26] Carcione J M, Quiroga-Goode G 1995 J. Comput. Acoust. 3 261
Google Scholar
[27] Komatitsch D, Martin R 2007 Geophysics 72 SM155
Google Scholar
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