地下黏弹性介质波动方程及波场数值模拟

宋利伟; 石颖; 陈树民; 柯璇; 侯晓慧; 刘志奇

doi:10.7498/aps.70.20210005

摘要

实际介质普遍具有黏弹性, 波在传播过程中常伴有能量的耗散、相位畸变和频带变窄等, 对于含有液体和气体的介质, 衰减现象尤为突出. 由于经典的波动理论未考虑介质的黏弹性效应, 基于完全弹性假设的模拟波场和实际传播特征之间的差异明显, 波动理论在工程技术中的应用效果还有提升的空间. 在岩石物理中, 品质因子Q是量化地震衰减强度的参数, 为了研究波在地下介质中的传播规律, 本文从常Q理论出发, 在黏弹性介质频散关系中, 利用多项式拟合和Taylor展开法将频率的分数阶转化为整数阶, 进而推导了时间域复数形式的地下黏弹性介质波动方程. 该近似处理避免了频散关系经域转换后出现分数阶时间微分项, 能有效地降低计算成本. 最后, 采用有限差分法联合伪谱法对均质模型实现了波场的数值模拟, 验证了方程的有效性.

关键词:

黏弹性介质 /
波动方程 /
数值模拟 /
伪谱法

Abstract

The energy of wavefield is gradually attenuated in all real materials, which is a fundamental feature and more obvious in the media containing liquid and gas. Because the viscosity effect is not considered in the classical wave theory, the actual wavefield is different from the simulated scenario based on the assumption of complete elasticity so that the application of wavefield does not meet the expectations in engineering technology, such as geophysical exploration. In the rock physics field, the well-known constant-Q theory gives a linear description of attenuation and Q is regarded as independent of the frequency. The quality factor Q is a parameter for calculating the phase difference between stress and strain of the media, which, as an index of wavefield attenuation behavior, is inversely proportional to the viscosity. Based on the constant-Q theory, a wave equation can be directly obtained by the Fourier transform of the dispersion relation, in which there is a fractional time differential operator. Therefore, it is difficult to perform the numerical simulation due to memory for all historical wavefields. In this paper, the dispersion relation is approximated by polynomial fitting and Taylor expansion method to eliminate the fractional power of frequency which is uncomfortably treated in the time domain. And then a complex-valued wave equation is derived to characterize the propagation law of wavefield in earth media. Besides the superiority of numerical simulation, the other advantage of this wave equation is that the dispersion and dissipation effects are decoupled. Next, a feasible numerical simulation strategy is proposed. The temporal derivative is solved by the finite-difference approach, moreover, the fractional spatial derivative is calculated in the spatial frequency domain by using the pseudo-spectral method. In the process of numerical simulation, only two-time slices, instead of the full-time wavefields, need to be saved, so the demand for data memory significantly slows down compared with solving the operator of the fractional time differential. Following that, the numerical examples prove that the novel wave equation is capable and efficient for the homogeneous model. The research work contributes to the understanding of complex wavefield phenomena and provides a basis for treating the seismology problems.

Keywords:

viscoelastic media /
wave equation /
numerical simulation /
pseudo-spectral method

作者及机构信息

宋利伟¹,
石颖²,
陈树民³,
柯璇²,
侯晓慧¹,
刘志奇²

1.
东北石油大学物理与电子工程学院, 大庆　163318

2.
东北石油大学地球科学学院, 大庆　163318

3.
大庆油田有限责任公司勘探开发研究院, 大庆　163712

通信作者: 宋利伟, zhidao90@163.com

基金项目: 国家自然科学基金(批准号: 41930431, 41804133)、中央支持地方高校改革发展资金人才培养支持计划(批准号: 140119001)和黑龙江省普通本科高等学校青年创新人才培养计划(批准号: UNPYSCT-2020149)资助的课题

Authors and contacts

Song Li-Wei¹,
Shi Ying²,
Chen Shu-Min³,
Ke Xuan²,
Hou Xiao-Hui¹,
Liu Zhi-Qi²

1.
School of Physics and Electronic Engineering, Northeast Petroleum University, Daqing 163318, China

2.
School of Earth Sciences, Northeast Petroleum University, Daqing 163318, China

3.
Exploration and Development Research Institute of Daqing Oilfield Co Ltd, Daqing 163712, China

Corresponding author: Song Li-Wei, zhidao90@163.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 41930431, 41804133), the Local Universities Reformation and Development Personnel Training Supporting Project from Central Authorities, China (Grant No. 140119001), and the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province, China (Grant No. UNPYSCT-2020149)

文章全文 : translate this paragraph

参考文献

[1]	Yang P, Brossier R, Metivier L 2018 SIAM J. Sci. Comput. 40 B1101 Google Scholar
[2]	Chen H, Zhou H, Yao Y 2020 Geophysics 85 S169 Google Scholar
[3]	Keating S, Innanen K 2020 Geophysics 85 R397 Google Scholar
[4]	Zhang, W, Shi Y 2019 Geophysics 84 S95 Google Scholar
[5]	Liu H P, Anderson D L, Kanamori H 1976 Geophys. J. Int. 47 41 Google Scholar
[6]	Emmerich H, Korn M 1987 Geophysics 52 1252 Google Scholar
[7]	Zhu T, Carcione J M, Harris J M 2013 Geophys. Prospect. 61 931 Google Scholar
[8]	Kjartansson E 1979 Geophys. Prospect. 84 4737
[9]	Carcione J M 2008 Geophysics 74 T1
[10]	Carcione J M, Cavallini F, Mainardi F, Hanyga A 2002 Pure Appl. Geophys. 159 1719 Google Scholar
[11]	Lu J F, Hanyga A 2004 Geophys. J. Int. 159 688 Google Scholar
[12]	Podlubny I 1999 Fractional Differential Equations (California: Academic Press) pp270−217
[13]	Yang J, Zhu H 2018 Geophys. J. Int. 215 1064 Google Scholar
[14]	Chen X W, Zhang R C, Mei F X 2000 Acta Mech. Sin. 16 282 Google Scholar
[15]	Dorodnitsyn V, Kozlov R 2010 J. Eng. Math. 66 253 Google Scholar
[16]	方刚, 张斌 2013 物理学报 62 154502 Google Scholar Fang G, Zhang B 2013 Acta Phys. Sin. 62 154502 Google Scholar
[17]	Li H X, Tao C H, Liu C, Huang G N, Yao Z A 2020 Chin. Phys. B 29 064301 Google Scholar
[18]	周聪, 王庆良 2015 物理学报 64 239101 Google Scholar Zhou C, Wang Q 2015 Acta Phys. Sin. 64 239101 Google Scholar
[19]	Zhang Z J, Wang G J, Harris J M 1999 Phys. Earth Planet. Inter. 114 25 Google Scholar
[20]	董良国, 马在田, 曹景忠 2000 地球物理学报 43 856 Google Scholar Dong L G, Ma Z T, Cao J Z 2000 Chin. J. Geophys 43 856 Google Scholar
[21]	孟路稳, 程广利, 张明敏, 尚建华 2017 海军工程大学学报 29 57 Meng L W, Cheng G L, Zhang M M, Shang J H 2017 J. Naval Univ. Eng. 29 57
[22]	杜启振, 刘莲莲, 孙晶波 2007 物理学报 56 6143 Google Scholar Du Q, Liu L, Sun J 2007 Acta Phys. Sin. 56 6143 Google Scholar
[23]	唐春安 1997 岩石力学与工程学报 4 75 Tang C A 1997 Chin. J. Rock Mech. Eng. 4 75
[24]	Carcione J M 2014 Wave Fields in Real Media (Amsterdam: Elsevier Science) p75
[25]	Chen W, Holm S 2004 J. Acoust. Soc. Am. 115 1424 Google Scholar
[26]	Carcione J M 2010 Geophysics 75 A53 Google Scholar

施引文献

图 1 拟合曲线 (Q = 30)

Fig. 1. Fitting curve (Q = 30).

下载: 全尺寸图片幻灯片

图 2 数值模拟模型

Fig. 2. Numerical simulation model.

下载: 全尺寸图片幻灯片

图 3 600 ms波场快照

Fig. 3. Wavefield snapshot at 600 ms.

下载: 全尺寸图片幻灯片

图 4 不同时刻的1维波场

Fig. 4. One dimensional wavefield at different times.

下载: 全尺寸图片幻灯片

图 5 波场记录

Fig. 5. Wavefield record.

下载: 全尺寸图片幻灯片

图 6 记录频谱

Fig. 6. Record spectrum.

下载: 全尺寸图片幻灯片

图 7 组合波场

Fig. 7. Combined wavefield.

下载: 全尺寸图片幻灯片

图 8 部分波场

Fig. 8. Part wavefield.

下载: 全尺寸图片幻灯片

图 9 复杂模型波场模拟　(a) 速度模型; (b) 0.3 s波场; (c) 0.4 s波场; (d) 0.5 s波场

Fig. 9. Wavefield simulation for the complex model: (a) Velocity; (b) wavefield at 0.3 s; (c) wavefield at 0.4 s; (a) wavefield at 0.5 s.

下载: 全尺寸图片幻灯片

表 1 不同Q值拟合系数

Table 1. Fitting coefficients of different Q values.

Q 5 10 20 30 60 100 5000

a₁ 0.8144 0.9081 0.9545 0.9698 0.9850 0.991 0.9998
a₂ 58.452 30.728 15.662 10.499 5.276 3.1716 0.0636
a₃ –2078.2 –1133.1 –587.58 –396.08 –200.14 –120.57 –2.4255

下载: 导出CSV

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[1]	Yang P, Brossier R, Metivier L 2018 SIAM J. Sci. Comput. 40 B1101 Google Scholar
[2]	Chen H, Zhou H, Yao Y 2020 Geophysics 85 S169 Google Scholar
[3]	Keating S, Innanen K 2020 Geophysics 85 R397 Google Scholar
[4]	Zhang, W, Shi Y 2019 Geophysics 84 S95 Google Scholar
[5]	Liu H P, Anderson D L, Kanamori H 1976 Geophys. J. Int. 47 41 Google Scholar
[6]	Emmerich H, Korn M 1987 Geophysics 52 1252 Google Scholar
[7]	Zhu T, Carcione J M, Harris J M 2013 Geophys. Prospect. 61 931 Google Scholar
[8]	Kjartansson E 1979 Geophys. Prospect. 84 4737
[9]	Carcione J M 2008 Geophysics 74 T1
[10]	Carcione J M, Cavallini F, Mainardi F, Hanyga A 2002 Pure Appl. Geophys. 159 1719 Google Scholar
[11]	Lu J F, Hanyga A 2004 Geophys. J. Int. 159 688 Google Scholar
[12]	Podlubny I 1999 Fractional Differential Equations (California: Academic Press) pp270−217
[13]	Yang J, Zhu H 2018 Geophys. J. Int. 215 1064 Google Scholar
[14]	Chen X W, Zhang R C, Mei F X 2000 Acta Mech. Sin. 16 282 Google Scholar
[15]	Dorodnitsyn V, Kozlov R 2010 J. Eng. Math. 66 253 Google Scholar
[16]	方刚, 张斌 2013 物理学报 62 154502 Google Scholar Fang G, Zhang B 2013 Acta Phys. Sin. 62 154502 Google Scholar
[17]	Li H X, Tao C H, Liu C, Huang G N, Yao Z A 2020 Chin. Phys. B 29 064301 Google Scholar
[18]	周聪, 王庆良 2015 物理学报 64 239101 Google Scholar Zhou C, Wang Q 2015 Acta Phys. Sin. 64 239101 Google Scholar
[19]	Zhang Z J, Wang G J, Harris J M 1999 Phys. Earth Planet. Inter. 114 25 Google Scholar
[20]	董良国, 马在田, 曹景忠 2000 地球物理学报 43 856 Google Scholar Dong L G, Ma Z T, Cao J Z 2000 Chin. J. Geophys 43 856 Google Scholar
[21]	孟路稳, 程广利, 张明敏, 尚建华 2017 海军工程大学学报 29 57 Meng L W, Cheng G L, Zhang M M, Shang J H 2017 J. Naval Univ. Eng. 29 57
[22]	杜启振, 刘莲莲, 孙晶波 2007 物理学报 56 6143 Google Scholar Du Q, Liu L, Sun J 2007 Acta Phys. Sin. 56 6143 Google Scholar
[23]	唐春安 1997 岩石力学与工程学报 4 75 Tang C A 1997 Chin. J. Rock Mech. Eng. 4 75
[24]	Carcione J M 2014 Wave Fields in Real Media (Amsterdam: Elsevier Science) p75
[25]	Chen W, Holm S 2004 J. Acoust. Soc. Am. 115 1424 Google Scholar
[26]	Carcione J M 2010 Geophysics 75 A53 Google Scholar

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地下黏弹性介质波动方程及波场数值模拟