孔弹性波、电磁波和震电波场的曲线网格有限差分模拟

孔隙介质由固体骨架和孔隙流体构成，是良好的储层介质如含水层、油气层等。孔隙中的流体通常是含有带电离子的电解质溶液，在固体表面电荷的作用下孔隙介质中形成双电层结构。由于双电层的存在，孔隙介质中可以形成动电效应：当地震波产生外力在孔隙内产生流体渗流，带动双电层中的扩散电荷与固定在骨架表面的电荷发生相对运动，进而形成流动电流；当孔隙介质中所处的电场发生变化时，扩散电荷在电场中运动产生传导电流，同时带动扩散层的流体运动。这种固体和流体相对运动产生的动电效应形成了孔隙介质中所特有的震电耦合现象。动电效应会受到孔隙介质参数的影响，研究认为这种效应可以作为探测含水层和油气藏的有效工具。此外，动电效应也是天然地震相关电磁异常现象的一种可能的产生机制。

    Pride将Biot方程和麦克斯韦方程通过耦合系数结合得到了饱和孔隙介质中的震电控制方程。数值模拟是研究震电信号特征、分析震电信号强弱变化的常用工具。起伏地形和界面会影响地震波的传播，同时也会影响震电信号的传播和散射。基于直角网格的有限差分方法在模拟起伏界面时会产生人为的反射和耗散，因此不适合于带起伏界面模型的模拟。为了克服这一问题，本论文提出使用曲线同位网格有限差分方法求解Pride方程，模拟震电波场的传播。首先使用曲线同位网格有限差分方法求解Biot方程，模拟孔弹性波；然后求解麦克斯韦方程模拟电磁波的传播；最后通过震电耦合，实现震电波场的模拟。

       孔隙介质中慢P波传播速度比快P波小得多，同时由于流体黏度影响，具有高衰减的特性，这为孔隙介质中弹性波模拟增加了难度。若要准确计算在界面附近的渗流速度，需要足够细密的网格，同时差分模拟的稳定性进一步限制了时间步，显著地增加了计算量和计算时间。为了兼顾孔隙介质中弹性波模拟的准确性和计算效率，本论文提出采用间断网格和变时间步的方法模拟孔弹性波。通过间断网格的方法将计算区域分块剖分，用非均匀的变时间步格式更新波场，细网格区域和粗网格区域使用不同的时间步，通过泰勒展开保持时间步进中精度的一致性。模型测试中，该算法相比均匀网格方法可以节约80%以上的计算时间，极大地提高了计算效率，实现了孔弹性波的高效模拟。

       传统电磁波有限差分模拟中普遍使用交错网格有限差分法，使用同位网格有限差分模拟电磁波的相关研究很少。为了保持震电模拟算法的一致性，使用同位网格有限差分模拟电磁波。一般交错网格不需要考虑界面条件的问题，但对于同位网格来说，由于所有的电磁场分量均定义在相同的网格点上，界面条件不能再被忽略。因此为了解决同位网格下电磁场的边界连续性问题，本文提出了两种界面处理方法：界面等效方程法和界面虚拟点法，前者通过推导得到了基于等效场的等效方程，后者结合空间差分的特征在界面利用虚拟点进行差分。模拟算例表明，两种方法都能够解决同位网格下的电磁场连续性的求解，实现了高精度的曲线同位网格有限差分电磁波模拟。

  将Biot方程和麦克斯韦方程耦合可以得到饱和流体孔隙介质中的震电电场。通过耦合项将孔弹性波模拟算法和电磁波模拟算法结合，实现了曲线网格有限差分的时变震电波场模拟算法。为了加速计算，本文提出弹性波时间步和电磁波时间步的曲线网格异步更新算法。通过全空间模型的解析解和水平层状模型半解析解对比，验证了算法正确性。通过对比含有水平界面、凸起界面和凹陷界面的三种模型，分析了不规则界面对震电信号的影响。接着讨论了算法的计算效率，本论文提出了放大参数法加速计算局部同震信号，提出了多重网格法实现准静态震电场的快速求解。分析了孔隙流体盐度、孔隙度、地层厚度和震源主频等对震电信号的影响特征。最后通过实测数据的对比表明，本文算法可以有效模拟实测震电信号，用于反演地下结构信息。

[1] Thompson R R. The seismic electric effect [J]. Geophysics, 1936, 1(3): 327-335.
[2] Pride S. Governing equations for the coupled electromagnetics and acoustics of porous media[J]. Physical Review B, 1994, 50(21): 15678-15696.
[3] Sun Y C, Ren H, Zheng X, et al. Numerical simulations to explain the coseismic electromagnetic signals: a case study for a M5.4 aftershock of the 2016 Kumamoto earthquake[J]. Earth, Planets and Space, 2019: 1-24.
[4] 高玉涛，高永新，何晓，等. 含起伏地形地层中点源激发的震电响应研究[J]. 地球物理学报, 2020, 63(05): 2069-2083.
[5] Wang D, Gao Y, Tong P, et al. Electroseismic and seismoelectric responses at irregular interfaces: Possible application to reservoir exploration [J]. Journal of Petroleum Science and Engineering, 2021: 108513.
[6] Zhang W, Chen X. Traction image method for irregular free surface boundaries in finite difference seismic wave simulation[J]. Geophysical Journal International, 2006, 167(1): 337-353.
[7] Zhang W, Zhang Z, Chen X. Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids: 3-D elastic wave modelling with topography[J]. Geophysical Journal International, 2012, 190(1): 358- 378.
[8] Sun Y C, Ren H, Zheng X Z, et al. 2-D poroelastic wave modelling with a topographic free surface by the curvilinear grid finite-difference method[J]. Geophysical Journal International, 2019, 218(3): 1961-1982.
[9] Sun Y C, Zhang W, Ren H, et al. 3D seismic-wave modeling with a topographic fluid–Solid interface at the sea bottom by the curvilinear-grid finite-difference method[J]. Bulletin of the Seismological Society of America, 2021, 111(5): 2753-2779.
[10] Hu H, Liu J. Simulation of the converted electric field during acoustoelectric logging[C]. SEG Technical Program Expanded Abstracts 2002. Society of Exploration Geophysicists, 2002: 348-351.
[11] Gao Y, Hu H. Seismoelectromagnetic waves radiated by a double couple source in a saturated porous medium[J]. Geophysical Journal International, 2010, 181(2): 873-896.
[12] Ren H, Huang Q, Chen X. A new numerical technique for simulating the coupled seismic and electromagnetic waves in layered porous media[J]. Earthquake Science, 2010, 23(2): 167-176.
[13] 任恒鑫, 黄清华, 陈晓非. 层状孔隙介质震电波场数值模拟高频不稳定性问题的解析处理方法[J]. 地球物理学报, 2010, 53(03): 506-511.
[14] Okubo K, Takeuchi N, Utsugi M, et al. Direct magnetic signals from earthquake rupturing: Iwate-Miyagi earthquake of M 7.2, Japan[J]. Earth and Planetary Science Letters, 2011, 305(1-2): 65-72.
[15] Yamazaki K. Piezomagnetic fields arising from the propagation of teleseismic waves in magnetized crust with finite conductivity: Piezomagnetic fields due to teleseismic waves[J]. Geophysical Journal International, 2011, 184(2): 626-638.
[16] Honkura Y, Ogawa Y, Matsushima M, et al. A model for observed circular polarized electric fields coincident with the passage of large seismic waves[J]. Journal of Geophysical Research, 2009, 114(B10): B10103.
[17] Gao Y, Chen X, Hu H, et al. Induced electromagnetic field by seismic waves in Earth’s magnetic field: Motional induction effect[J]. Journal of Geophysical Research: Solid Earth, 2014, 119(7): 5651-5685.
[18] Gao Y, Jiang P, Xu Y, et al. On magnetic disturbances induced by rotation of coil-type magnetometer driven by seismic waves[J]. Geophysical Journal International, 2021, 226(3): 1948-1974.
[19] Zhu Z, Haartsen M W, Toksöz M N. Experimental studies of seismoelectric conversions in fluid-saturated porous media[J]. Journal of Geophysical Research: Solid Earth, 2000, 105(B12): 28055-28064.
[20] Ivanov A G. Effect of electrization of earth layers by elastic waves passing through them[J]. Doklady Akademii nauk SSSR, 1939, 24(1): 42-45.
[21] Martner S T, Sparks N R. The electroseismic effect [J]. Geophysics, 1959, 24(2): 297-308.
[22] Thompson A H, Gist G A. Geophysical applications of electrokinetic conversion[J]. The Leading Edge, 1993, 12(12): 1169-1173.
[23] Butler K E, Russell R D, Kepic A W, et al. Measurement of the seismoelectric response from a shallow boundary[J]. Geophysics, 1996, 61(6): 1769-1778.
[24] Mikhailov O V, Haartsen M W, Toksöz M N. Electroseismic investigation of the shallow subsurface: Field measurements and numerical modeling[J]. Geophysics, 1997, 62(1): 97-105.
[25] Russell R D, Butler K E, Kepic A W, et al. Seismoelectric exploration[J]. The Leading Edge, 1997, 16(11): 1611-1615.
[26] 严洪瑞，刘洪，李幼铭，等. 震电勘探方法在大庆油田的实验研究[J]. 地球物理学报, 1999, 42(2): 257-267.
[27] Haines S S, Pride S R, Klemperer S L, et al. Development of electroseismic experimental methods[C]. 17th EEGS Symposium on the Application of Geophysics to Engineering and Environmental Problems. Colorado Springs, Colorado, USA, European Association of Geoscientists & Engineers, 2004: 1490-1503.
[28] Pride S R, Haartsen M W. Electroseismic wave properties[J]. The Journal of the Acoustical Society of America, 1996, 100(3): 1301-1315.
[29] Zhu Z, Haartsen M W, Toksöz M N. Experimental studies of electrokinetic conversions in fluid‐saturated borehole models[J]. Geophysics, 1999, 64(5): 1349-1356.
[30] Zhu Z, Toksöz M N. Crosshole seismoelectric measurements in borehole models with fractures[J]. Geophysics, 2003, 68(5): 1519-1524.
[31] 胡恒山，李长文，王克协，等. 声电效应测井模型实验研究[J]. 测井技术, 2001, 25(2): 89-95.
[32] 陈本池，牟永光，狄帮让，等. 井中震电勘探模型实验研究[J]. 石油物探, 2003, 42(01): 35-38.
[33] 王军，胡恒山，徐小蓉，等. 基于动电效应的岩芯渗透率实验测量[J]. 地球物理学报, 2010, 53(8): 1953-1960.
[34] 彭蓉. 含流体孔隙介质中震电响应机制及实验研究[D]. 中国石油大学, 2019.
[35] Peng R, Di B, Glover P W J, et al. Seismo-electric conversion in shale: experiment and analytical modelling[J]. Geophysical Journal International, 2020, 223(2): 725-745.
[36] 彭蓉, 王佳新, 刘子淳, 等. 薄互层地层震电波场响应特征[J]. 地球物理学报, 2022, 65(2): 785-795.
[37] Wang J, Zhu Z, Gao Y, et al. Measurements of the seismoelectric responses in a synthetic porous rock[J]. Geophysical Journal International, 2020, 222(1): 436-448.
[38] Fujinawa Y, Takahashi K. Electromagnetic radiations associated with major earthquakes[J]. Physics of the Earth and Planetary Interiors, 1998, 105 (3-4): 249-259.
[39] Fujinawa Y, Takahashi K, Noda Y, et al. Remote detection of the electric field change induced at the seismic wave front from the start of fault rupturing[J]. International Journal of Geophysics, 2011, 2011: 1-11.
[40] 杨皓，詹艳，赵凌强，等. 2013年甘肃岷县漳县Ms6.6地震震电磁信号和同震信号观测[J]. 震灾防御技术, 2015(B10): 785-793.
[41] 汤吉，詹艳，王立凤，等. 汶川地震强余震的电磁同震效应[J]. 地球物理学报, 2010, 53(3): 526-534.
[42] Gao Y, Harris J M, Wen J, et al. Modeling of the coseismic electromagnetic fields observed during the 2004 Mw 6.0 Parkfield earthquake[J]. Geophysical Research Letters, 2016, 43(2): 620-627.
[43] Gao Y, Zhao G, Chong J, et al. Coseismic electric and magnetic signals observed during 2017 Jiuzhaigou Mw 6.5 earthquake and explained by elec- trokinetics and magnetometer rotation[J]. Geophysical Journal International, 2020, 223(2): 1130-1143.
[44] Dzieran L, Thorwart M, Rabbel W. Seismoelectric monitoring of aquifers using local seismicity a feasibility study[J]. Geophysical Journal International, 2020, 222(2): 874-892.
[45] Kasdi A S, Bouzid A, Hamoudi M, et al. Numerical modeling of the co-seismic electromagnetic signals observed during the 2014 Mw 4.9 Hammam Melouane earthquake, Northern Algeria[J]. Journal of Applied Geophysics, 2022, 206: 104840.
[46] Andre R. The Seismoelectric Method[M]. 2015.
[47] Frenkel J. On the theory of seismic and seismoelectric phenomena in a moist soil[J]. Journal of Engineering Mechanics, 2005, 131(9): 879-887.
[48] Biot M A. Theory of propagation of elastic waves in a fluid‐saturated porous solid. I. low‐frequency range[J]. The Journal of the Acoustical Society of America, 1956, 28(2): 168-178.
[49] Biot M A. Theory of propagation of elastic waves in a fluid‐saturated porous solid. II. higher frequency range[J]. The Journal of the Acoustical Society of America, 1956, 28(2): 179-191.
[50] Biot M A. Mechanics of deformation and acoustic propagation in porous media[J]. Journal of Applied Physics, 1962, 33(4): 1482-1498.
[51] Neev J, Yeatts F R. Electrokinetic effects in fluid-saturated poroelastic media[J]. Physical Review B, 1989, 40(13): 9135-9141.
[52] Fujinawa Y, Noda Y. Characteristics of seismoelectric wave fields associated with natural microcracks[J]. Pure and Applied Geophysics, 2016, 173(1): 255-268.
[53] Revil A, Mahardika H. Coupled hydromechanical and electromagnetic disturbances in unsaturated porous materials[J]. Water Resources Research, 2013, 49(2): 744-766.
[54] Warden S, Garambois S, Jouniaux L, et al. Seismoelectric wave propagation numerical modelling in partially saturated materials[J]. Geophysical Journal International, 2013, 194(3): 1498-1513.
[55] Grobbe N. Seismoelectric exploration - Geophysical monograph series[M]. 2020.
[56] Dupuis J C, Kepic A W, Butler K E. Design of field instrumentation and noise removal techniques for seismoelectric measurements[M]. Seismoelectric Exploration. American Geophysical Union (AGU), 2020: 319-345.
[57] Kepic A W, Maxwell M, Russell R D. Field trials of a seismoelectric method for detecting massive sulfides[J]. Geophysics, 1995, 60(2): 365-373.
[58] Djuraev U, Jufar S R, Mahardika H, et al. Numerical simulation of seismoelectric effect for monitoring foam propagation through a reservoir[J]. Journal of Petroleum Science and Engineering, 2018, 171: 618-635.
[59] Kulessa B, Garambois S, Dietrich M, et al. Seismoelectric characterization of ice sheets and glaciers[M]. Seismoelectric Exploration. American Geophysical Union (AGU), 2020: 383-399.
[60] Rabbel W, Iwanowski-Strahser K, Strahser M, et al. Seismoelectric field measurements in unconsolidated sediments in compare- son with other methods of near-surface prospecting[M]. Seismoelectric Exploration. American Geophysical Union (AGU), 2020: 347-363.
[61] Dzieran L, Rabbel W, Thorwart M, et al. Seismoelectric ground response to local and regional earthquakes[M]. Seismoelectric Exploration. American Geophysical Union (AGU), 2020: 401-413.
[62] Slob E, Mulder M. Seismoelectromagnetic homogeneous space Green’s functions[J]. Geophysics, 2016, 81(4): F27-F40.
[63] Haartsen M W, Pride S R. Electroseismic waves from point sources in layered media[J]. Journal of Geophysical Research: Solid Earth, 1997, 102(B11): 24745-24769.
[64] Hu H, Gao Y. Electromagnetic field generated by a finite fault due to electrokinetic effect[J]. Journal of Geophysical Research: Solid Earth, 2011, 116(B8): B08302.
[65] Garambois S, Dietrich M. Full waveform numerical simulations of seismoelectromagnetic wave conversions in fluid-saturated stratified porous media[J]. Journal of Geophysical Research: Solid Earth, 2002, 107(B7): ESE- 5.
[66] Kennett B L N, Kerry N J. Seismic waves in a stratified half space[J]. Geophysical Journal International, 1979, 57(3): 557-583.
[67] Grobbe N, Slob E C, Thorbecke J W. Comparison of eigenvectors for coupled seismo-electromagnetic layered-Earth modelling[J]. Geophysical Journal International, 2016, 206(1): 152-190.
[68] Ren H, Huang Q, Chen X. Existence of evanescent electromagnetic waves resulting from seismoelectric conversion at a solid–porous interface[J]. Geophysical Journal International, 2016, 204(1): 147-166.
[69] Ren H, Huang Q, Chen X. Numerical simulation of seismo-electro- magnetic fields associated with a fault in a porous medium[J]. Geophysical Journal International, 2016, 206(1): 205-220.
[70] Zheng X, Ren H, Butler K E, et al. Seismoelectric and electroseismic modeling in stratified porous media with a shallow or ground surface source[J]. Journal of Geophysical Research: Solid Earth, 2021, 126(9): e2021JB021950.
[71] Monachesi L B, Zyserman F I, Jouniaux L. SH seismoelectric response of a glacier: an analytic study[J]. Journal of Geophysical Research: Earth Surface, 2018, 123(9): 2135-2156.
[72] Cheng Q, Gao Y, Zhou G, et al. Seismoelectric waves generated by a point source in horizontally stratified vertical transversely isotropic porous media[J]. Geophysics, 2022: 1-91.
[73] 段韵达, 胡恒山. 流体-孔隙介质界面震电反射和透射系数的简化表达式[J]. 地球物理学报, 2022, 65(11): 4513-4525.
[74] 胡恒山. 声电效应测井的理论、数值与实验研究[D]. 哈尔滨工业大学, 2000.
[75] 崔志文. 多孔介质声学模型与多极源声电效应测井和多极随钻声测井的理论与数值研究[D]. 吉林大学, 2004.
[76] Han Q, Wang Z. Time-domain simulation of SH-wave–induced electro- magnetic field in heterogeneous porous media: A fast finite-element algorithm[J]. Geophysics, 2001, 66(2): 448-461.
[77] Kröger B, Yaramanci U, Kemna A. Numerical analysis of seismo- electric wave propagation in spatially confined geological units: Seismo- electric wave propagation in geological units[J]. Geophysical Prospecting, 2014, 62(1): 133-147.
[78] Zyserman F, Gauzellino P, Jouniaux L. Finite element modeling of electroseismics and seismoelectrics[J]. Seismoelectric Exploration: Theory, Experiments, and Applications, 2020: 245-267.
[79] Pain C C, Saunders J H, Worthington M H, et al. A mixed finite-element method for solving the poroelastic Biot equations with electro- kinetic coupling[J]. Geophysical Journal International, 2005, 160(2): 592- 608.
[80] Haines S S, Pride S R. Seismoelectric numerical modeling on a grid[J]. Geophysics, 2006, 71(6): N57-N65.
[81] Singarimbun A, Mahardika H, Srigutomo W, et al. A preliminary result of seismoelectric responses study on shallow fluid-saturated layer: numerical modeling using transfer function approach[J]. Indonesian Journal of Physics, 2016, 19(3): 61-68.
[82] Garambois S, Dietrich M. Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[J]. Geophysics, 2001, 66(5): 1417-1430.
[83] Gao Y, Wang D, Yao C, et al. Simulation of seismoelectric waves using finite-difference frequency-domain method: 2D SHTE mode[J]. Geophysical Journal International, 2018, 216(1): 414-438.
[84] Tohti M, Wang Y, Slob E, et al. Seismoelectric numerical simulation in 2D vertical transverse isotropic poroelastic medium[J]. Geophysical Prospecting, 2020, 68(6): 1927-1943.
[85] 穆纳尔丁∙托合提，王一博，肖文交，等. 三维正交各向异性孔隙介质震电波场数值模拟研究[J]. 地球物理学报, 2022, 65(11): 4471-4484.
[86] Ji Y, Han L, Huang X, et al. A high order finite-difference scheme for time-domain modelling of time-varying seismoelectric waves[J]. Geophysics, 2021: 1-49.
[87] Han L, Ji Y, Ye W, et al. Seismoelectric wave propagation simulation by combining poro-viscoelastic anisotropic model with Cole-Cole depression model[J]. IEEE Transactions on Geoscience and Remote Sensing, 2022: 1-1.
[88] Guan W, Hu H. 2D seismoelectric log simulation using a finite‐difference method[C]. SEG Technical Program Expanded Abstracts 2009. Society of Exploration Geophysicists, 2009: 2567-2571.
[89] 黄兴兴. 震电效应的有限元波场模拟及实验研究[D]. 中国石油大学, 2021.
[90] Zhang W, Shen Y, Zhao L. Three-dimensional anisotropic seismic wave modelling in spherical coordinates by a collocated-grid finite-difference method: 3-D synthetics in spherical coordinates[J]. Geophysical Journal Inter- national, 2012, 188(3): 1359-1381.
[91] Hassanzadeh S. Acoustic modeling in fluid‐saturated porous media[J]. Geophysics, 1991, 56(4): 424-435.
[92] Zhu X, Mcmechan G A. Numerical simulation of seismic responses of poroelastic reservoirs using Biot theory[J]. Geophysics, 1991, 56(3): 328-339.
[93] Dai N, Vafidis A, Kanasewich E R. Wave propagation in heterogeneous, porous media: A velocity‐stress, finite‐difference method[J]. Geophysics, 1995, 60(2): 327-340.
[94] Zhang J. Quadrangle-grid velocity–stress finite difference method for poroelastic wave equations[J]. Geophysical Journal International, 1999, 139(1): 171-182.
[95] Zeng Y Q, He J Q, Liu Q H. The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media[J]. Geophysics, 2001, 66(4): 1258-1266.
[96] 王秀明, 张海澜, 王东. 利用高阶交错网格有限差分法模拟地震波在非均匀孔隙介质中的传播[J]. 地球物理学报, 2003, 46(6): 1206-1217.
[97] Saenger E H, Bohlen T. Finite‐difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid[J]. Geophysics, 2004, 69(2): 583-591.
[98] Sheen D H, Tuncay K, Baag C E, et al. Parallel implementation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation[J]. Computers & Geosciences, 2006, 32(8): 1182-1191.
[99] Masson Y J, Pride S R, Nihei K T. Finite difference modeling of Biot’s poroelastic equations at seismic frequencies[J]. Journal of Geophysical Research, 2006, 111(B10): B10305.
[100] Masson Y J, Pride S R. Poroelastic finite difference modeling of seismic attenuation and dispersion due to mesoscopic-scale heterogeneity[J]. Journal of Geophysical Research, 2007, 112(B3): B03204.
[101] Mueller M T, Krzikalla F. Simulation of wave propagation in poroelastic structures using a modified finite-difference scheme[C]. 69th EAGE Conference and Exhibition incorporating SPE EUROPEC 2007. European Association of Geoscientists & Engineers, 2007: cp-27.
[102] Wenzlau F, Müller T M. Finite-difference modeling of wave propagation and diffusion in poroelastic media[J]. Geophysics, 2009, 74(4): T55-T66.
[103] Masson Y J, Pride S R. Finite-difference modeling of Biot’s poroelastic equations across all frequencies[J]. Geophysics, 2010, 75(2): N33-N41.
[104] Chiavassa G, Lombard B, Piraux J. Numerical modeling of 1D transient poroelastic waves in the low-frequency range[J]. Journal of Computational and Applied Mathematics, 2010, 234(6): 1757-1765.
[105] Chiavassa G, Lombard B. Time domain numerical modeling of wave propagation in 2D heterogeneous porous media[J]. Journal of Computational Physics, 2011, 230(13): 5288-5309.
[106] Blanc E. Time-domain numerical modeling of poroelastic waves: the Biot-JKD model with fractional derivatives[J]. 2014: 158.
[107] Yang Q, Mao W. Simulation of seismic wave propagation in 2-D poroelastic media using weighted-averaging finite difference stencils in the frequency–space domain[J]. Geophysical Journal International, 2017, 208(1): 148-161.
[108] 杨尚倍, 白超英, 周兵. 起伏地表下基于改进BISQ模型双相介质中曲线网格有限差分法波场模拟[J]. 地球物理学报, 2018, 61(08): 3356-3373.
[109] Moczo P, Gregor D, Kristek J, et al. A discrete representation of material heterogeneity for the finite-difference modelling of seismic wave propagation in a poroelastic medium[J]. Geophysical Journal International, 2019, 216(2): 1072-1099.
[110] Gregor D, Moczo P, Kristek J, et al. Seismic waves in medium with poroelastic/elastic interfaces: a two-dimensional P-SV finite-difference modelling[J]. Geophysical Journal International, 2021, 228(1): 551-588.
[111] Wolf S, Galis M, Uphoff C, et al. An efficient ADER-DG local time stepping scheme for 3D HPC simulation of seismic waves in poroelastic media[J]. Journal of Computational Physics, 2021: 110886.
[112] Wang E, Carcione J M, Ba J. Wave simulation in partially saturated porothermoelastic media[J]. IEEE Transactions on geoscience and remote sensing, 2022, 60: 1-14.
[113] Yee K. Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media[J]. IEEE Transactions on Antennas and Propagation, 1966, 14(3): 302-307.
[114] Bergmann T, Robertsson J O A, Holliger K. Numerical properties of staggered finite-difference solutions of Maxwell’s equations for ground-penetrating radar modeling[J]. Geophysical Research Letters, 1996, 23(1): 45-48.
[115] Xu T, Mcmechan G A. GPR attenuation and its numerical simulation in 2.5 dimensions[J]. Geophysics, 1997, 62(2): 403-414.
[116] Bo Yang, Rappaport C. Response of realistic soil for GPR applications with 2-D FDTD[J]. IEEE Transactions on Geoscience and Remote Sensing, 2001, 39(6): 1198-1205.
[117] 李静，曾昭发，吴丰收，等. 探地雷达三维高阶时域有限差分法模拟研究[J]. 地球物理学报, 2010, 53(04): 974-981.
[118] Liu Q, Zhao G. Review of PSTD methods for transient electromagnetics[J]. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 2004, 17(3): 299-323.
[119] Lee Jin-Fa, Lee R, Cangellaris A. Time-domain finite-element methods[J]. IEEE Transactions on Antennas and Propagation, 1997, 45(3): 430-442.
[120] Morency C. Electromagnetic wave propagation based upon spectral- element methodology in Dispersive and Attenuating Media[J]. Geophysical Journal International, 2019, 220(2): 951-966.
[121] Zhan Q, Fang Y, Zhuang M, et al. Stabilized DG-PSTD Method With Nonconformal Meshes for Electromagnetic Waves[J]. IEEE Transactions on Antennas and Propagation, 2020, 68(6): 4714-4726.
[122] Xie Z, Chan C H, Zhang B. An Explicit Fourth-Order Orthogonal Curvilinear Staggered-Grid FDTD Method for Maxwell’s Equations[J]. Journal of Computational Physics, 2002, 175(2): 739-763.
[123] Taflove A, Hagness S C. Computational electrodynamics: the finite-difference time-domain method[M]. 3rd ed. Boston: Artech House, 2005.
[124] Tajdini M M, Gonzalez-Valdes B, Martinez-Lorenzo J A, et al. Real-time modeling of forward-looking synthetic aperture ground Penetrating radar scattering from rough terrain[J]. IEEE Transactions on Geoscience and Remote Sensing, 2019, 57(5): 2754-2765.
[125] Janaswamy R, Liu Y. An unstaggered colocated finite-difference scheme for solving time-domain Maxwell’s equations in curvilinear coordinates[J]. IEEE Transactions on Antennas and Propagation, 1997, 45(11): 1584-1591.
[126] Pinho P, Domingues M O, Ferreira P J S G, et al. Interpolating Wavelets and Adaptive Finite Difference Schemes for Solving Maxwell’s Equations: The Effects of Gridding[J]. IEEE Transactions on Magnetics, 2007, 43(3): 1013-1022.
[127] Sheu T W H, Chung Y W, LI J H, et al. Development of an explicit non-staggered scheme for solving three-dimensional Maxwell’s equations[J]. Computer Physics Communications, 2016, 207: 258-273.
[128] Layek M K, Sengupta P. Forward modeling of GPR data by Unstaggered finite difference frequency domain (FDFD) method: an approach towards an appropriate numerical scheme[J]. Journal of Environmental and Engineering Geophysics, 2019, 24(3): 487-496.
[129] Gao Y, Chen X, Hu H, et al. Early electromagnetic waves from earthquake rupturing: II. validation and numerical experiments[J]. Geophysical Journal International, 2013, 192(3): 1308-1323.
[130] Jiang L. Rotation-induced magnetic field in a coil magnetometer generated by seismic waves[J]. Geophysical Journal International, 2018, 212(2): 743-759.
[131] Ren H, Huang Q, Chen X. Quantitative understanding on the amplitude decay characteristic of the evanescent electromagnetic waves generated by seismoelectric conversion[J]. Pure and Applied Geophysics, 2018, 175(8): 2853-2879.
[132] Butler K E, Kulessa B, Pugin A J M. Multimode seismoelectric phenomena generated using explosive and vibroseis sources[J]. Geophysical Journal International, 2018, 213(2): 836-850.
[133] Dupuis J C, Butler K E, Kepic A W. Seismoelectric imaging of the vadose zone of a sand aquifer[J]. Geophysics, 2007, 72(6): A81-A85.
[134] Dean T, C. Dupuis J. The vibroelectric method - a new tool for near-surface characterisation and improved seismic data quality[C]. 73rd EAGE Conference and Exhibition incorporating SPE EUROPEC 2011. Vienna, Austria, 2011: cp-238.
[135] Haines S S. Seismoelectric imaging of shallow targets[D]. Stanford University, 2005.
[136] Markov M, Markov A, Levin V, et al. Electromagnetic field generated by acoustic wave scattering at a poroelastic inclusion located in a fluid[J]. International Journal of Engineering Science, 2022, 181: 103766.
[137] Frenkel J. On the theory of seismic and seismoelectric phenomena in a moist soil[J]. Journal of Physics, 1944, 8: 230-241.
[138] Johnson D L, Koplik J, Dashen R. Theory of dynamic permeability and tortuosity in fluid-saturated porous media[J]. Journal of Fluid Mechanics, 1987, 176(1): 379-402.
[139] 张伟. 含起伏地形的三维非均匀介质中地震波传播的有限差分算法及其在强地面震动模拟中的应用[D]. 北京大学, 2006.
[140] Thompson J F, Warsi Z U A, Mastin C W. Numerical grid generation: foundations and applications[M]. New York: Elsevier North-Holland, 1985.
[141] Tarrass I, Giraud L, Thore P. New curvilinear scheme for elastic wave propagation in presence of curved topography [J]. Geophysical Prospecting, 2011, 59(5): 889-906.
[142] Hixon R. On increasing the accuracy of MacCormack schemes for aero- acoustic applications[C]. 3rd AIAA/CEAS Aeroacoustics Conference, 1997: 1586.
[143] Suli E, Mayers D F. An Introduction to Numerical Analysis[M]. Cambridge university press, 2003.
[144] Carcione J M, Quiroga-Goode G. Some aspects of the physics and numerical modeling of Biot compressional waves[J]. Journal of Computational Acoustics, 1995, 03(04): 261-280.
[145] Alkhimenkov Y, Räss L, Khakimova L, et al. Resolving wave propagation in anisotropic poroelastic media using graphical processing units (GPUs)[J]. Journal of Geophysical Research: Solid Earth, 2021, 126(7): e2020JB021175.
[146] Picotti S, Carcione J M, Germán Rubino J, et al. P-wave seismic attenuation by slow-wave diffusion: Numerical experiments in partially saturated rocks[J]. Geophysics, 2007, 72(4): N11-N21.
[147] Pride S R, Garambois S. The role of Biot slow waves in electroseismic wave phenomena[J]. The Journal of the Acoustical Society of America, 2002, 111(2): 697-706.
[148] Kristek J, Moczo P, Galis M. Stable discontinuous staggered grid in the finite-difference modelling of seismic motion: Stable discontinuous staggered grid[J]. Geophysical Journal International, 2010, 183(3): 1401-1407.
[149] Zhang Z, Zhang W, Li H, et al. Stable discontinuous grid implementation for collocated-grid finite-difference seismic wave modelling[J]. Geophysical Journal International, 2013, 192(3): 1179-1188.
[150] Kang T S. Finite-Difference Seismic Simulation Combining Discontinuous Grids with Locally Variable Timesteps[J]. Bulletin of the Seismological Society of America, 2004, 94(1): 207-219.
[151] Liu L, Li X, Hu F Q. Non-Uniform Time-Step Runge-Kutta Discontinuous Galerkin Method for Computational Aeroacoustics[J]. Journal of Computa- tional Physics, 2010, 229(19): 6874-6897.
[152] Liu L, Li X, Hu F Q. Nonuniform-time-step explicit Runge–Kutta scheme for high-order finite difference method[J]. Computers & Fluids, 2014, 105: 166-178.
[153] Li H, Zhang W, Zhang Z, et al. Elastic wave finite-difference simulation using discontinuous curvilinear grid with non-uniform time step: two- dimensional case[J]. Geophysical Journal International, 2015, 202(1): 102- 118.
[154] Cerjan C, Kosloff D, Kosloff R, et al. A nonreflecting boundary condition for discrete acoustic and elastic wave equations[J]. Geophysics, 1985, 50(4): 705-708.
[155] Berenger J P. A perfectly matched layer for the absorption of electromagnetic waves[J]. Journal of Computational Physics, 1994, 114(2): 185-200.
[156] Gedney S D, Zhao B. An Auxiliary Differential Equation Formulation for the Complex-Frequency Shifted PML[J]. IEEE Transactions on Antennas and Propagation, 2010, 58(3): 838-847.
[157] Zhang W, Shen Y. Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling[J]. Geophysics, 2010, 75(4): T141-T154.
[158] Kuzuoglu M, Mittra R. Frequency dependence of the constitutive para- meters of causal perfectly matched anisotropic absorbers[J]. IEEE Microwave and Guided Wave Letters, 1996, 6(12): 447-449.
[159] Roden J A, Gedney S D. Convolution PML (CPML): An efficient FDTD implementation of the CFS–PML for arbitrary media[J]. Microwave and optical technology letters, 2000, 27(5): 334-339.
[160] Chen X. A systematic and efficient method of computing normal modes for multilayered half-space[J]. Geophysical Journal International, 1993, 115(2): 391-409.
[161] Ren H, Chen X, Huang Q. Numerical simulation of coseismic electromagnetic fields associated with seismic waves due to finite faulting in porous media: Numerical simulation of coseismic EM fields[J]. Geophysical Journal International, 2012, 188(3): 925-944.
[162] Kristeková M, Kristek J, Moczo P. Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals[J]. Geophysical Journal International, 2009, 178(2): 813-825.
[163] 孙耀充. 曲线网格有限差分模拟地震波在各向异性介质和固-液耦合介质中的传播[D]. 中国科学技术大学, 2017.
[164] Moczo P. 3D Heterogeneous Staggered-Grid Finite-Difference Modeling of Seismic Motion with Volume Harmonic and Arithmetic Averaging of Elastic Moduli and Densities[J]. Bulletin of the Seismological Society of America, 2002, 92(8): 3042-3066.
[165] Robertsson J O A, Blanch J O, Nihei K, et al. Numerical Modeling of Seismic Wave Propagation: Gridded Two-way Wave-equation Methods[M]. Society of Exploration Geophysicists, 2012.
[166] Ward S H, Hohmann G W. Electromagnetic Theory for Geophysical Applications[M]. Electromagnetic Methods in Applied Geophysics: Volume 1, Theory. Society of Exploration Geophysicists, 1988.
[167] Warren C, Giannopoulos A, Giannakis I. gprMax: Open source software to simulate electromagnetic wave propagation for Ground Penetrating Radar[J]. Computer Physics Communications, 2016, 209: 163-170.
[168] Guan W, Hu H. Finite-difference modeling of the electroseismic logging in a fluid-saturated porous formation[J]. Journal of Computational Physics, 2008, 227(11): 5633-5648.
[169] 关威. 孔隙介质弹性波-电磁场耦合效应测井的波场模拟研究[D]. 哈尔滨工业大学, 2009.
[170] Pride S R, Morgan F D. Electrokinetic dissipation induced by seismic waves[J]. Geophysics, 1991, 56(7): 914-925.
[171] Cai J, Wei W, Hu X, et al. Electrical conductivity models in saturated porous media: A review[J]. Earth-Science Reviews, 2017, 171: 419-433.
[172] Wu Y, Han J, Liu W, et al. Coseismic electromagnetic wave signals associated with Zhuolu (China) Ms4.3 earthquake on Sept. 6th, 2014[J]. Journal of Seismology, 2021, 25(4): 1161-1170.
[173] Ma X, Liu Y, Yin C, et al. Estimation of fluid salinity using coseismic electric signal generated by an earthquake[J]. Geophysical Journal International, 2022, 233(1): 127-144.
[174] 段韵达, 胡恒山, 关威. 矿化度界面对震电测井波场的影响[J]. 地球物理学报, 2020, 63(02): 778-787.