地下黏弹性介质波动方程及波场数值模拟  

Wave equation for underground viscoelastic media and wavefield numerical simulation

作　　者：宋利伟[1] 石颖[2] 陈树民[3] 柯璇[2] 侯晓慧 刘志奇 Song Li-Wei;Shi Ying;Chen Shu-Min;Ke Xuan;Hou Xiao-Hui;Liu Zhi-Qi(School of Physics and Electronic Engineering,Northeast Petroleum University,Daqing 163318,China;School of Earth Sciences,Northeast Petroleum University,Daqing 163318,China;Exploration and Development Research Institute of Daqing Oilfield Co Ltd,Daqing 163712,China)

机构地区：[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

出　　处：《物理学报》2021年第14期392-398,共7页Acta Physica Sinica

基　　金：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);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.

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.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 fracti

关 键 词：黏弹性介质 波动方程 数值模拟 伪谱法 

分 类 号：P631.4[天文地球—地质矿产勘探]