Research progress on seismic reflection and transmission based on solid acoustoelasticity theory
-
摘要: 多期沉积埋深及成岩过程使地下储层介质具有复杂初始地应力,应力范围跨度大且分布方向复杂. 初始地应力对储层介质岩石物理与地震波传播响应具有显著影响. 地震反射透射系数方程可以量化储层性质与地球物理观测数据的关系,构建考虑初始地应力作用的地震反射透射系数方程有利于更好理解深部储层界面处地震波传播特征,为深层油气勘探奠定理论基础,正逐渐受到国内外学者的关注. 当前已提出的相关理论模型主要从固体声弹理论出发,考虑初始应力与地震波扰动引起的微小固体应变,聚焦岩石物理与地震反射透射参数化的理论方法研究,在应力诱导各向异性参数反演、地应力预测、油气储层识别等方面开展了初步实际应用探索. 本文首先介绍固体声弹理论核心内容,然后介绍了基于声弹理论的地震反射透射模型、基本假设及其阶段性应用进展,讨论了方程的局限性以及在地震勘探领域的发展方向.Abstract: Multi-stage sedimentation and diagenetic processes lead to complex in situ stresses in subsurface hydrocarbon reservoirs. The in situ stresses have a broad magnitude range and their distribution directions are complicated, significantly impacting the rock physics properties and seismic propagation response in reservoirs. Seismic reflection and transmission (R/T) coefficient equations can quantify the relationship between reservoir properties and observed geophysical data. Therefore, seismic R/T coefficient equations considering the effects of in situ stresses can help to better understand the characteristics of wave propagation on deep-strata stratigraphic interfaces, laying a theoretical foundation for exploring deep-strata hydrocarbon. This research topic is attracting the attention of international scholars. Most reported stress-dependent wave R/T coefficient equations consider the infinitesimal deformations induced by the in situ stress and wave perturbation based on the theory of acoustoelasticity; these approaches have been preliminarily applied in several theoretical and practical scenarios such as seismic inversion for stress-induced anisotropy parameters, in situ stress detection, and discrimination of oil and gas reservoirs. In this paper, we introduce the main body of solid acoustoelasticity theory and the stress-dependent seismic R/T coefficient equations based on acoustoelasticity theory, along with their basic assumptions, preliminary applications, limitations, and prospects in seismic exploration.
-
图 1 介质颗粒所处状态示意图. 图中
${u^{\rm{i}}}$ 、${u^{\rm{f}}}$ 和$u$ 分别表示从自然状态到初始状态的质点位移、自然状态到最终状态的位移、初始状态到最终状态的位移.${x_1}$ 、${x_2}$ 和${x_3}$ 为主坐标轴Figure 1. Sketch of the states of medium particle, where
${u^{\rm{i}}}$ ,${u^{\rm{f}}}$ and$u$ represent particle displacements from natural state to initial state, natural state to final state, and initial state to final state, respectively.${x_1}$ ,${x_2}$ and${x_3}$ are principal coordinate axes图 2 实测干燥与饱和Berea砂岩
$\rho {W^2}$ 的斜率(左图实心圆)与用右图所示3oECs预测的结果(左图空心圆)对比(修改自Winkler and McGowan, 2004). 图中$\rho $ 和$W$ 分别为密度和速度. PH和SH为静水压力下砂岩纵、横波速度,P2、S21和S32为单轴应力作用下纵、横波速度. A、B和C是三个独立的3oECsFigure 2. Comparison between the measured
$\rho {W^2}$ (solid circles in left panels) of Berea sandstone and the predicted results (open circles in left panels) with 3oECs shown in the right hand panels (modified from Winkler and McGowan, 2004), where$\rho $ and$W$ are the density and wave velocity of rock, respectively. PH and SH represent the P- and S-wave velocities in sandstone under hydrostatic pressure. P2, S21 and S32 represent one P-wave and two S-wave velocities in sandstone under the uniaxial stress. A, B and C are three independent 3oECs图 3 单轴应力作用(a)与孔隙压力作用下(b)实测岩样速度(几何图形)与孔隙声弹模型预测结果(实线与虚线)对比图(修改自Liu et al., 2021). 图中VP1为纵波速度,V31和V32为两个不同方向的横波速度
Figure 3. Comparisons between measured (geometries) and predicted (lines) velocities of rock sample under uniaxial stress (a) or pore pressure (b) (modified from Liu et al., 2021), where VP1 is P-wave velocity, V31 and V32 are S-wave velocities along two different directions.
图 4 岩石纵(a)、横(b)波速度随孔隙压力改变图(修改自Fu and Fu, 2018). 图中虚线和实线分别为经典孔隙声弹理论和考虑软孔作用的新孔隙声弹理论预测的速度,空心圆为实测值
Figure 4. P-wave (a) and S-wave (b) velocities of a rock sample variation with pore pressure (modified from Fu and Fu, 2018), where the dashed and solid lines represent the velocities provided by classic poro-acoustoelasticity theory and the new poro-acoustoelasticity theory with compliant pore, respectively. Circles represent the measured velocities
图 5 不同垂直应力T作用下连续界面纵波反射系数(a)、透射系数(b)与横波反射系数(c)、透射系数(d)随入射角变化图(修改自Chen et al., 2021)
Figure 5. P-wave reflection (a) and transmission (b) coefficients and S-wave reflection (c) and transmission (d) coefficients on the welded interface versus incidence angle under the different vertical in situ stresses (modified from Chen et al., 2021)
图 6 30°入射角时不同频率及垂直应力作用下非连续界面纵波反射系数(a)、透射系数(b)与横波反射系数(c)、透射系数(d)随法向界面柔度变化图(修改自Chen et al., 2022b). 图中实线、虚线与点线分别表示20 Hz、40 Hz与60 Hz时的反射透射系数
Figure 6. P-wave reflection (a) and transmission (b) coefficients and S-wave reflection (c) and transmission (d) coefficients on the nonwelded interface versus normal compliance under in situ stresses at different frequencies at an incidence angle of 30°(modified from Chen et al., 2022b), where the solid, dashed and dotted lines represent the reflection/transmission coefficients at frequencies of 20 Hz, 40 Hz and 60 Hz, respectively
图 7 随机噪声(a)与常噪声(b)条件下水平应力预测结果(修改自Chen et al., 2022c)
Figure 7. Prediction results of horizontal uniaxial stress under random noise (a) and constant noise (b) (modified from Chen et al., 2022c)
图 8 应力诱导各向异性因子反演结果(修改自Chen and Zong, 2022). 其中图(a)、(b)分别为
$100\times \left|\varPsi \right|$ 和$ 100\times \left|Y\right| $ 反演结果. SIAF:应力诱导各向异性因子Figure 8. Inverted results of two stress-induced anisotropy factors (modified from Chen and Zong, 2022). (a) For
$100\times \left|\varPsi \right|$ and (b) for$ 100\times \left|Y\right| $ 图 9 应力诱导各向异性参数反演结果(修改自Pan and Zhao, 2023)
Figure 9. Inverted results of stress-induced anisotropy parameters (modified from Pan and Zhao, 2023)
-
[1] Abdideh M, Moghimzadeh M. 2019. Geomechanical study of gas reservoir rock using vertical seismic profile and petrophysical data (continental shelf in southern Iran)[J]. Geomechanics and Geoengineering, 14: 118-135. doi: 10.1080/17486025.2019.1573322 [2] Abiza Z, Destrade M, Ogden R W. 2012. Large acoustoelastic effect[J]. Wave Motion, 49: 364-374. doi: 10.1016/j.wavemoti.2011.12.002 [3] Aki K, Richards P G. 2002. Quantitative Seismology, 2nd ed[M]. University Science Book. [4] Ba J, Carcione J M, Cao H, et al. 2013. Poro-acoustoelasticity of fluid-saturated rocks[J]. Geophysical Prospecting, 61: 599-612. doi: 10.1111/j.1365-2478.2012.01091.x [5] Berjamin H, Pascalis R D. 2022. Acoustoelastic analysis of soft viscoelastic solids with application to pre-stressed phononic crystals[J]. International Journal of Solids and Structures, 241: 111529. doi: 10.1016/j.ijsolstr.2022.111529 [6] Biot M A. 1956a. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range[J]. Journal of the Acoustical Society of America, 28: 179-191. doi: 10.1121/1.1908241 [7] Biot M A. 1956b. Thermoelasticity and irreversible thermodynamics[J]. Journal of Applied Physics, 27: 240-253. doi: 10.1063/1.1722351 [8] Borgomano J V, Pimienta L X, Fortin J, et al. 2019. Seismic dispersion and attenuation in fluid-saturated carbonate rocks: Effect of microstructure and pressure[J]. Journal of Geophysical Research: Solid Earth, 124: 12498-12522. doi: 10.1029/2019JB018434 [9] Buland A, More H. 2003. Bayesian linearized AVO inversion[J]. Geophysics, 68(1): 185–198. doi: 10.1190/1.1543206 [10] Carcione J M, Cavallini F, Wang E, et al. 2019. Physics and simulation of wave propagation in linear thermoporoelastic media[J]. Journal of Geophysical Research: Solid Earth, 124: 8147–8166. doi: 10.1029/2019JB017851 [11] Chaisri S, Krebes E S. 2000. Exact and approximate formulas for P-SV reflection and transmission coefficients for a nonwelded contact interface[J]. Journal of Geophysical Research: Solid Earth, 105: 28045-28054. doi: 10.1029/2000JB900296 [12] Chattopadhyay A, Bose S, Chakraborty M. 1982. Reflection of elastic waves under initial stress at a free surface: P and SV motion[J]. Journal of the Acoustical Society of America, 72: 255-263. doi: 10.1121/1.387987 [13] Chen F B, Zong Z Y. 2021. P-wave reflectivity parameterization for HTI medium with the effect of horizontal stress[C]// First International Meeting for Applied Geoscience and Energy (IMAGE), Denver, Colorado, 241-254. [14] Chen F B, Zong Z Y, Jiang M. 2021. Seismic reflectivity and transmissivity parameterization with the effect of normal in-situ stress[J]. Geophysical Journal International, 229(1): 311-327. doi: 10.1093/gji/ggab475 [15] Chen F B, Zong Z Y. 2022. PP wave reflection coefficient in stress-induced anisotropic media and amplitude variation with incident angle and azimuth inversion[J]. Geophysics, 87: C155–C172. doi: 10.1190/geo2021-0706.1 [16] Chen F B, Zong Z Y, Yin X Y. 2022a. Acoustothermoelasticity for joint effects of stress and thermal fields on wave dispersion and attenuation[J]. Journal of Geophysical Research: Solid Earth, 127: e2021JB023671. [17] Chen F B, Zong Z Y, Yin X Y, et al. 2022b. Accurate formulae for P-wave reflectivity and transmissivity for a non-welded contact interface with the effect of in situ vertical stress[J]. Geophysical Journal International, 229(1): 311-327. [18] Chen F B, Zong Z Y, Yin X Y. 2022c. Horizontal stress inversion based on azimuthal difference in P-wave reflectivity[C]// Second International Meeting for Applied Geoscience and Energy (IMAGE), Houston, Texas, 218-222. [19] Chen F B, Zong Z Y, Yin X Y. 2022d. Monitoring the change in horizontal stress with multi-wave time-lapse seismic response based on nonlinear elasticity theory[J]. Petroleum Science, 20: 815-826. [20] Chen F B, Zong Z Y, Alexey S, et al. 2023. Wave reflection and transmission coefficients for layered transversely isotropic media with vertical symmetry axis under initial stress[J]. Geophysical Journal International, 223(3): 1580-1595. [21] Degtyar A D, Rokhlin S I. 1998. Stress effect on boundary conditions and elastic wave propagation through an interface between anisotropic media[J]. Journal of the Acoustical Society of America, 104: 1992-2003. doi: 10.1121/1.423765 [22] Dey S, Addy S K. 1977. Reflection of plane waves under initial stresses at a free surface[J]. International Journal of Non-Linear Mechanics, 12: 371-381. doi: 10.1016/0020-7462(77)90038-5 [23] Fatti J L. 1994. Detection of gas in sandstone reservoirs using AVO analysis: A 3-D seismic case history using the Geostack technique[J]. Geophysics, 59: 1362-1376. doi: 10.1190/1.1443695 [24] Fu B Y, Fu L Y. 2017. Poro-acoustoelastic constants based on Pade approximation[J]. Journal of the Acoustical Society of America, 142(5): 2890-2904. doi: 10.1121/1.5009459 [25] Fu B Y, Fu L Y. 2018. Poro-acoustoelasticity with compliant pores for fluid-saturated rocks[J]. Geophysics, 83(3): WC1–WC14. doi: 10.1190/geo2017-0423.1 [26] Fu L Y, Fu B Y, Sun W J, et al. 2020. Elastic wave propagation and scattering in prestressed porous rocks[J]. Science China Earth Sciences, 63: 1309–1329. doi: 10.1007/s11430-019-9615-3 [27] Fuck R F, Tsvankin I. 2009. Analysis of the symmetry of a stressed medium using nonlinear elasticity[J]. Geophysics, 74(5): WB79–WB87. doi: 10.1190/1.3157251 [28] Goldberg Z A. 1961. Interaction of plane longitudinal and transverse elastic waves[J]. Soviet Physical Acoustics, 6: 306–310. [29] Grana D, Russell B, Mukerji T. 2022. Petrophysical inversion based on f-s-r amplitude-variation-with-offset linearization and canonical correlation analysis[J]. Geophysics, 87(6): M247–M258. doi: 10.1190/geo2021-0747.1 [30] Grinfeld M A, Norris A N. 1996. Acoustoelasticity theory and applications for fluid saturated porous media[J]. Journal of the Acoustical Society of America, 100(3): 1368-1374. doi: 10.1121/1.415983 [31] Gurevich B, Makarynska D, Pervukhina M. 2009. Ultrasonic moduli for fluid-saturated rocks: Mavko-Jizba relations rederived and generalized[J]. Geophysics, 74: N25–N30. doi: 10.1190/1.3123802 [32] Han T C, Gurevich B, Pervukhina M, et al. 2016. Linking the pressure dependency of elastic and electrical properties of porous rocks by a dual porosity model[J]. Geophysical Journal International, 205: 378-388. doi: 10.1093/gji/ggw019 [33] Hughes D S, Kelly J L. 1953. Second-order elastic deformation of solids[J]. Physics Review, 92: 1145–1149. doi: 10.1103/PhysRev.92.1145 [34] 贾承造, 庞雄奇. 2015. 深层油气地质理论研究进展与主要发展方向[J]. 石油学报, 36(12): 1457-1469Jia C Z, Pang X Q. 2015. Research processes and main development directions of deep hydrocarbon geological theories[J]. Acta Petrolei Sinica, 36(12): 1457-1469 (in Chinese). [35] Johnson P A, Rasolofosaon P. 1996. Nonlinear elasticity and stress-induced anisotropy in rock[J]. Journal of Geophysical Research, B2: 3113-3124. doi: 10.1029/95JB02880 [36] Jones G L, Kobett D. 1963. Interaction of elastic waves in an isotropic solid[J]. Journal of the Acoustical Society of America, 35: 5–10. doi: 10.1121/1.1918405 [37] Li W Q, Hu H S. 2023. Reflection and transmission of plane waves in stressed media with an imperfectly bonded interface[J]. Geophysical Journal International, 233: 2232-2252. doi: https://doi.org/10.1016/B978-0-08-099999-9.00006-6 [38] 李阳, 薛兆杰, 程喆, 等. 2020. 中国深层油气勘探开发进展与发展方向[J]. 中国石油勘探, 25(1): 45-57Li Y, Xue Z J, Cheng Z, et al. 2020. Progress and development directions of deep oil and gas exploration and development in China[J]. China Petroleum Exploration, 25(1): 45-57 (in Chinese). [39] Ling W C, Ba J, Carcione J M, et al. 2021. Poroacoustoelasticity for rocks with a dual-pore structure[J]. Geophysics, 86: MR17–MR25. doi: 10.1190/geo2020-0314.1 [40] Liu J X, Cui Z W, Wang K X. 2009. The relationships between uniaxial stress and reflection coefficients[J]. Geophysical Journal International, 179: 1584-1592. doi: 10.1111/j.1365-246X.2009.04353.x [41] Liu J X, Cui Z W, Wang K X. 2012. Effect of stress on reflection and refraction of plane wave at the interface between fluid and stressed rock[J]. Soil Dynamics and Earthquake Engineering, 42: 47-55. doi: 10.1016/j.soildyn.2012.05.022 [42] 刘金霞, 崔志文, 王克协. 2016. 水平单轴应力与横波反射系数[J]. 地球物理学报, 59(4): 1469-1476 doi: 10.6038/cjg20160427Liu J X, Cui Z W, Wang K X. 2016. Relationship between uniaxial stress and S-wave reflection coefficients[J]. Chinese Journal of Geophysics, 59(4): 1469-1476 (in Chinese). doi: 10.6038/cjg20160427 [43] Liu J X, Cui Z W, Sevostianov I. 2021. Effect of stresses on wave propagation in fluid-saturated porous media[J]. International Journal of Engineering Science, 167: 103519. doi: 10.1016/j.ijengsci.2021.103519 [44] Lu J, Wang Y, Chen J Y, et al. 2018. Joint anisotropic amplitude variation with offset inversion of PP and PS seismic data[J]. Geophysics, 83(2): N31–N50. doi: 10.1190/geo2016-0516.1 [45] Murnaghan F D. 1937. Finite deformations of an elastic solid[J]. American Journal of Mathematics, 59: 235–260. doi: 10.2307/2371405 [46] Murnaghan F D. 1951. Finite Deformation of an Elastic Solid[M]. New York: John Wiley & Sons, Inc. [47] Norris A N. 1995. The speed of a wave along a fluid/solid interface in the presence of anisotropy and pre-stress[J]. Journal of the Acoustical Society of America, 98: 1147-1154. doi: 10.1121/1.413613 [48] Pan X P, Zhao Z Z. 2023. A decoupled fracture- and stress-induced PP-wave refection coefficient approximation for azimuthal seismic inversion in sressed horizontal transversely isotropic media[J]. Surveys in Geophysics, https://doi.org/10.1007/s10712-023-09791-y. [49] Pan X P, Zhao Z Z, Zhang D Z. 2023. Characteristics of azimuthal seismic refection response in horizontal transversely isotropic media under horizontal in situ stress[J]. Surveys in Geophysics, 44: 387–423. doi: 10.1007/s10712-022-09739-8 [50] Pao Y H, Sachse W, Fukuoka H. 1984. Acoustoelasticity and Ultrasonic Measurement of Residual Stress. Physical Acoustics[M]. London: Academic Press, Inc. London Ltd. [51] Pyrak-Nolte L J, Myer L R, Cook N G W. 1990. Transmission of seismic waves across single natural fractures[J]. Journal of Geophysical Research, 95: 8617-8638. doi: 10.1029/JB095iB06p08617 [52] Ruger A. 1997. P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry[J]. Geophysics, 62: 713-722. doi: 10.1190/1.1444181 [53] Ruger A. 1998. Variation of P-wave reflectivity with offset and azimuth in anisotropic media[J]. Geophysics, 3: 935-947. doi: 10.1190/1.1444405 [54] Sarkar D, Bakulin A, Kranz R L. 2003. Anisotropic inversion of seismic data for stressed media: Theory and a physical modeling study on Berea Sandstone[J]. Geophysics, 68: 1-15. doi: 10.1190/1.1567240 [55] Schmitt D R, Currie C A, Zhang L. 2012. Crustal stress determination from boreholes and rock cores: Fundamental principles[J]. Tectonophysics, 580: 1-26. doi: 10.1016/j.tecto.2012.08.029 [56] Schoenberg M. 1980. Elastic wave behavior across linear slip interfaces[J]. Journal of the Acoustical Society of America, 68: 1516–1521. doi: 10.1121/1.385077 [57] Shapiro S A. 2003. Elastic piezosensitivity of porous and fractured rocks[J]. Geophysics, 68: 482–486. doi: 10.1190/1.1567215 [58] Shapiro S A. 2017. Stress impact on elastic anisotropy of triclinic porous and fractured rocks[J]. Journal of Geophysical Research: Solid Earth, 122: 2034-2053. doi: 10.1002/2016JB013378 [59] Sharma M D. 2007. Effect of initial stress on reflection at the free surface of anisotropic elastic medium[J]. Journal of Earth System Science, 116: 537-551. doi: 10.1007/s12040-007-0049-8 [60] Shaw R K, Sen M K. 2004. Born integral, stationary phase and linearized reflection coefficients in weak anisotropic media[J]. Geophysical Journal International, 158: 225–238. doi: 10.1111/j.1365-246X.2004.02283.x [61] Shuey T R. 1985. A simplification of the Zoeppritz equations[J]. Geophysics, 50: 609-614. doi: 10.1190/1.1441936 [62] Sripanich Y, Vasconcelos I, Tromp J, et al. 2021. Stress-dependent elasticity and wave propagation-New insights and connections[J]. Geophysics, 86: W47-W64. doi: 10.1190/geo2020-0252.1 [63] Thurston R N, Brugger K. 1964. Third-order elastic constants and the velocity of small amplitude elastic waves in homogeneously stressed media[J]. Physical Review, 133: A1604–A1610. doi: 10.1103/PhysRev.133.A1604 [64] 田家勇, 王恩福. 2006. 基于声弹理论的地应力超声测量方法[J]. 岩石力学与工程学报, 25(2): 3719-3724Tian J Y, Wang E F. 2006. Ultrasonic method for measuring in-situ stress based on acoustoelasticity theory[J]. Chinese Journal of Rock Mechanics and Engineering, 25(2): 3719-3724 (in Chinese). [65] Toupin P A, Bernstein B. 1961. Sound waves in deformed perfectly elastic materials. Acoustoelastic effect[J]. Journal of the Acoustical Society of America, 33: 216–225. doi: 10.1121/1.1908623 [66] Tromp J, Trampert J. 2018. Effects of induced stress on seismic forward modelling and inversion[J]. Geophysical Journal International, 213: 851–867. doi: 10.1093/gji/ggy020 [67] Tromp J, Marcondes M, Wentzcovitch R. 2019. Effects of induced stress on seismic waves: Validation based on ab initio calculations[J]. Journal of Geophysical Research: Solid Earth, 124: 729–741. doi: 10.1029/2018JB016778 [68] Wang H Q, Tian J Y. 2014. Acoustoelastic theory for fluid-saturated porous media[J]. Acta Mechanica Solida Sinica, 27(1): 41-53. doi: 10.1016/S0894-9166(14)60015-X [69] 王密, 田家勇. 2019. 基于岩石声弹理论的波速-静水围压关系耦合模型[J]. 地球物理学进展, 34(2): 0462-0468Wang M, Tian J Y. 2019. Coupled model for velocity change in rocks subjected to hydrostatic confining pressure based on rock acoustoelasticity[J]. Progress in Geophysics, 34(2): 0462-0468 (in Chinese). [70] 王志伟, 符力耘, 韩同城, 等. 2021. 岩石热弹性理论及其在地球物理中的应用[J]. 地球与行星物理论评, 52(6): 623-63. doi: 10.19975/j.dqyxx.2021-009Wang Z W, Fu L Y, Han T C, et al. 2021. Review of thermoelasticity theory in rocks and its applications in geophysics[J]. Reviews of Geophysics and Planetary Physics, 52(6): 623-633 (in Chinese). doi: 10.19975/j.dqyxx.2021-009 [71] Wei Y, Ba J, Carcione J M. 2022. Stress effects on wave velocities of rocks: Contribution of crack closure, squirt flow and acoustoelasticity[J]. Journal of Geophysical Research: Solid Earth, 127: e2022JB025253. doi: 10.1029/2022JB025253 [72] Winkler K W, Liu X. 1996. Measurements of third-order elastic constants in rocks[J]. Journal of the Acoustical Society of America, 100: 1392–1398. doi: 10.1121/1.415986 [73] Winkler K W, McGowan L. 2004. Nonlinear acoustoelastic constants of dry and saturated rocks[J]. Journal of Geophysical Research: Solid Earth, 109: B10204. doi: 10.1029/2004JB003262 [74] 印兴耀, 邓炜, 宗兆云. 2016. 基于逆算子估计的AVO反演方法研究[J]. 地球物理学报, 59(4): 1457-1468 doi: 10.6038/cjg20160426Yin X Y, Deng W, Zong Z Y. 2016. AVO inversion based on inverse operator estimation[J]. Chinese Journal of Geophysics, 59(4): 1457-1468 (in Chinese). doi: 10.6038/cjg20160426 [75] Zoback M, Zoback M, Mount V, et al. 1987. New evidence on the state of stress of the San Andreas fault system[J]. Science, 238: 1105-1111. doi: 10.1126/science.238.4830.1105 [76] 宗兆云, 印兴耀, 张峰, 等. 2012. 杨氏模量和泊松比反射系数近似方程及叠前地震反演[J]. 地球物理学报, 55(11): 3786-3794 doi: 10.6038/j.issn.0001-5733.2012.11.025Zong Z Y, Yin X Y, Zhang F, et al. 2012. Reflection coefficient equation and pre-stack seismic inversion with Young’s modulus and Poisson ratio[J]. Chinese Journal of Geophysics, 55(11): 3786-3794 (in Chinese). doi: 10.6038/j.issn.0001-5733.2012.11.025 [77] Zong Z Y, Yin X Y, Wu G C. 2015. Geofluid discrimination incorporating poroelasticity and seismic reflection inversion[J]. Surveys in Geophysics, 36: 659-681. doi: 10.1007/s10712-015-9330-6 [78] Zong Z Y, Chen F B, Yin X Y, Li K. 2022. Effect of stress on wave propagation in fluid-saturated porous thermoelastic media[J]. Surveys in Geophysics, 44: 425-462. [79] Zuo P, Liu Y, Fan Z. 2021. Modeling of acoustoelastic borehole waves subjected to tectonic stress with formation anisotropy and borehole deviation[J]. Geophysics, 87: D1-D19. doi: 10.1190/geo2020-0859.1 -