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
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摘要
本文以基于改进BISQ模型的二维双相各向同性介质一阶速度-应力方程为基础,推导出了曲线坐标系下对应的方程,然后采用低频散、低耗散的同位网格MacCormack有限差分法来离散方程,并采用紧致的单边MacCormack差分格式结合牵引力镜像法来施加自由地表边界条件,实现了地震波场数值模拟.曲线网格有限差分法采用贴体网格来描述自由表面,地表的网格线紧贴地形,避免了台阶近似造成的数值散射.数值模拟结果表明,在双相介质起伏自由地表和分界面处,各类波型复杂的反射透射规律可以清晰展现,曲线网格有限差分法可以精确地解决地震波在含起伏地表的双相各向同性介质中的传播问题.
In this article,the corresponding equations in curvilinear coordinate system based on the first-order velocity stress equation of two-phase isotropic media based on the improved BISQ model are derived,then the equations are numerically solved by an optimized high-order nonstaggered finite difference scheme,that is,DRP/opt MacCormack scheme.To implement the undulating free-surface topography,we derive the analytical relationship between derivatives of velocity components and use the compact finite-difference scheme and traction-image method.The curvilinear grid finite difference method uses body-conforming grid to describe the free surface,thus avoids the numerical approximation caused by scattering.In the undulating free surface andthe undulating interface of two-phase medium,the complex reflection wave and transmission wave can be clearly demonstrated by the numerical simulation results.The simulation results show that the curvilinear grid finite-difference method can accurately solve the propagation problem of seismic waves in a two-phase isotropic medium with undulating surface.
In this article,the corresponding equations in curvilinear coordinate system based on the first-order velocity stress equation of two-phase isotropic media based on the improved BISQ model are derived,then the equations are numerically solved by an optimized high-order nonstaggered finite difference scheme,that is,DRP/opt MacCormack scheme.To implement the undulating free-surface topography,we derive the analytical relationship between derivatives of velocity components and use the compact finite-difference scheme and traction-image method.The curvilinear grid finite difference method uses body-conforming grid to describe the free surface,thus avoids the numerical approximation caused by scattering.In the undulating free surface andthe undulating interface of two-phase medium,the complex reflection wave and transmission wave can be clearly demonstrated by the numerical simulation results.The simulation results show that the curvilinear grid finite-difference method can accurately solve the propagation problem of seismic waves in a two-phase isotropic medium with undulating surface.
引文
Biot M A.1956a.Theory of propagation of elastic waves in a fluidsaturated porous solid.I.Low-frequency range.J.Acoust.Soc.Am.,28(2):168-178.
Biot M A.1956b.Theory of propagation of elastic waves in a fluid-saturated porous solid.II.Higher frequency range.J.Acoust.Soc.Am.,28(2):179-191.
Diallo M S,Appel E.2000.Acoustic wave propagation in saturated porous media:reformulation of the Biot/Squirt flow theory.J.Appl.Geophys.,44(4):313-325.
DialloM S,Prasad M,Appel E.2003.Comparison between experimental results and theoretical predictions for P-wave velocity and attenuation at ultrasonic frequency.Wave Motion,37(1):1-16.
Dvorkin J,Nur A.1993.Dynamic poroelasticity:a unified model with the squirt and the Biot mechanisms.Geophysics,58(4):524-533.
Gist G A.1994.Interpreting laboratory velocity measurements in partially gas-saturated rocks.Geophysics,59(7):1100-1109.
Gurevich B,Lopatnikov S L.1995.Velocity and attenuation of elastic waves in finely layered porous rocks.Geophys.J.Int.,121(3):933-947.
Han C,Chen X F,Jiang D,et al.2007.Application of the improved BISQ model to seismic numerical simulation of two-phase media.Natural Gas Industry(in Chinese),27(10):49-52.
Hayashi K,Bums D R,Toks9z M N.2001.Discontinuous-grid finite-difference seismic modeling including surface topography.Bull.Seismol.Soc.Am.,91(6):1750-1764.
Hestholm S O,Ruud B O,Husebye E S.1999.3-D versus 2-D finitedifference seismic synthetics including real surface topography.Phys.Earth Planet.Int.,113(1-4):339-354.
Hestholm S O,Ruud B O.2000.2Dfinite-difference viscoelastic wave modelling including surface topography.Geophys.Prosp.,48(2):341-373.
Hestholm S O,Ruud B O.1994.2Dfinite-difference elastic wavemodelling including surface topography.Geophys.Prosp.,42(5):371-390.
Hixon R.1997.Evaluation of a high-accuracy MacCormack-type scheme using benchmark problems.J.Comput.Acous.,6(3):291-305.
Hixon R,Turkel E.2000.Compact implicit MacCormack-type schemes with high accuracy.J.Comput.Phys.,158(1):51-70.
Jiang L L,Sun J G.2008.Source terms of elliptic system in grid generation.Global Geology(in Chinese),27(3):298-305.
Jiang L L.2010.Body-fitted grid generation for the geological conditions[Ph.D.thesis].Changchun:Jilin University.
Jih R,McLaughlin K L,Der Zoltan A.1988.Free-boundary conditions of arbitrary polygonal topography in a two-dimensional explicit elastic finite-difference scheme.Geophysics,53(8):1045-1055.
Jones T D.1986.Pore fluids and frequency-dependent wave propagation in rocks.Geophysics,56(10):1939-1953.
Lan H T,Liu C,Guo Z Q.2014.Modelling of seismic wave propagation in two-phase medium based on reformulated BISQmodel using time-splitting staggered pseudospectral method.Global Geology(in Chinese),33(1):190-199.
Li N.2012.Finite difference seismoelectric wave modeling in 2Dporous media with surface topography[Ph.D.thesis](in Chinese).Hefei:University of Science and Technology of China.
Liu C,Lan H T,Guo Z Q et al.2003.Pseudo-spectral modeling and feature analysis of wave propagation in two-phase HTImedium based on reformulated BISQ mechanism.Chinese J.Geophys.(in Chinese),56(10):3461-3473,doi:10.6038/cjg20131021.
Mavko G M,Jizba D.1991.Estimating grain-scale fluid effects on velocity dispersion in rocks.Geophysics,56(12):1940-1949.
Opr2al I,Zahradník J.1999.Elastic finite-difference method for irregular grids.Geophysics,64(1):240-250.
Peng C Z,Li C M,Wang M C.2007.Seismic wave-field modeling based on the modefied BISQ model.Computing Techniques for Geophysical&Geochemical Exploration(in Chinese),29(1):15-18.
Sun Y C,Zhang W,Chen X F.2016.Seismic-wave modeling in the presence of surface topography in 2Dgeneral anisotropic media by a curvilinear grid finite-difference method.Bull.Seism.Soc.Am.,106(3):1036-1054.
Tam C K W,Webb J C.1993.Dispersion-relation-preserving finitedifference schemes for computational acoustics.J.Comput.Phys.,107(2):262-281.
Tarrass I,Giraud L,Thore P.2011.New curvilinear scheme for elastic wave propagation in presence of curved topography.Geophys.Prosp.,59(5):889-906.
Tessmer E,Kosloff D,Behle A.1992.Elastic wave propagation simulation in the presence of surface topography.Geophys.J.Int.,108(2):621-632.
Thompson J F,Warsi Z U A,Mastin C W.1985.Numerical Grid Generation:Foundations and Applications.New York:Elsevier North-Holland,Inc.
Winkler K W.1985.Dispersion analysis of velocity and attenuation in Berea sandstone.Journal of Geophysical Research,90(B8):6793-6800.
Yang D H,Zhang Z J,Teng J W,et al.2000.The study of twophase anisotropy,questions and applied prospects.Progress in Geophysics(in Chinese),15(2):7-21.
Yang D H.2002.Finite Element method of the elastic wave equation and wavefield simulation in two-phase anisotropic media.Chinese J.Geophys.(in Chinese),45(4):575-583.
Zhang W,Chen X F.2006.Traction image method for irregular free surface boundaries in finite difference seismic wave simulation.Geophys.J.Int.,167(1):337-353.
Zhang W,Zhang Z G,Chen X F.2012.Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids.Geophys.J.Int.,190(1):358-378.
Zhang Y,Xu Y X,Xia J H,et al.2015.Characteristics and application of surface wave propagation in fluid-filled porous media.Chinese J.Geophys.(in Chinese),58(8):2759-2778,doi:10.6038/cjg20150812.
Zhou Z S,Tang L.2012.Seismic modeling and numerical dispersion correcting in double-phase medium based on reformulated BISQmodel.Journal of Central South University:Science and Technology(in Chinese),43(4):1411-1418.
Zhu H J,Zhang W,Chen X F.2009.Two-dimensional seismic wave simulation in anisotropic media by non-staggered finite difference method.Chinese Journal of Geophysics(in Chinese),52(6):1536-1546,doi:10.3969/j.issn.0001-5733.2009.06.015.
韩翀,陈雪菲,蒋东等.2007.改进BISQ模型在双相介质地震波数值模拟中的应用.天然气工业,2007,27(10):49-52.
蒋丽丽,孙建国.2008.基于Poisson方程的曲网格生成技术.世界地质,27(3):298-305.
蒋丽丽.2010.面向地质条件的贴体网格生成技术[博士论文].长春:吉林大学.
兰慧田,刘财,郭智奇.2014.利用时间分裂的错格伪谱法模拟地震波在基于改进BISQ模型的双相介质中的传播.世界地质,33(1):190-199.
李娜.2012.含起伏地表的二维双相介质中地震波及电磁波传播有限差分算法[博士论文].合肥:中国科学技术大学.
刘财,兰慧田,郭智奇等.2013.基于改进BISQ机制的双相HTI介质波传播伪谱法模拟与特征分析.地球物理学报,56(10):3461-3473,doi:10.6038/cjg20131021.
彭传正,李才明,王明春.2007.基于改进BISQ模型的地震波场数值模拟.物探化探计算技术,29(1):15-18.
杨顶辉,张中杰,滕吉文等.2000.双相各向异性研究、问题与应用前景.地球物理学进展,15(2):7-21.
杨顶辉.2002.双相各向异性介质中弹性波方程的有限元解法及波场模拟.地球物理学报,45(4):575-583.
张煜,徐义贤,夏江海等.2015.含流体孔隙介质中面波的传播特性及应用.地球物理学报,58(8):2759-2778,doi:10.6038/cjg20150812.
周竹生,唐磊.2012.改进BISQ模型的双相介质地震波场数值模拟及频散校正.中南大学学报:自然科学版,43(4):1411-1418.
祝贺君,张伟,陈晓非.2009.二维各向异性介质中地震波场的高阶同位网格有限差分模拟.地球物理学报,52(6):1536-1546,doi:10.3969/j.issn.0001-5733.2009.06.015.
Biot M A.1956b.Theory of propagation of elastic waves in a fluid-saturated porous solid.II.Higher frequency range.J.Acoust.Soc.Am.,28(2):179-191.
Diallo M S,Appel E.2000.Acoustic wave propagation in saturated porous media:reformulation of the Biot/Squirt flow theory.J.Appl.Geophys.,44(4):313-325.
DialloM S,Prasad M,Appel E.2003.Comparison between experimental results and theoretical predictions for P-wave velocity and attenuation at ultrasonic frequency.Wave Motion,37(1):1-16.
Dvorkin J,Nur A.1993.Dynamic poroelasticity:a unified model with the squirt and the Biot mechanisms.Geophysics,58(4):524-533.
Gist G A.1994.Interpreting laboratory velocity measurements in partially gas-saturated rocks.Geophysics,59(7):1100-1109.
Gurevich B,Lopatnikov S L.1995.Velocity and attenuation of elastic waves in finely layered porous rocks.Geophys.J.Int.,121(3):933-947.
Han C,Chen X F,Jiang D,et al.2007.Application of the improved BISQ model to seismic numerical simulation of two-phase media.Natural Gas Industry(in Chinese),27(10):49-52.
Hayashi K,Bums D R,Toks9z M N.2001.Discontinuous-grid finite-difference seismic modeling including surface topography.Bull.Seismol.Soc.Am.,91(6):1750-1764.
Hestholm S O,Ruud B O,Husebye E S.1999.3-D versus 2-D finitedifference seismic synthetics including real surface topography.Phys.Earth Planet.Int.,113(1-4):339-354.
Hestholm S O,Ruud B O.2000.2Dfinite-difference viscoelastic wave modelling including surface topography.Geophys.Prosp.,48(2):341-373.
Hestholm S O,Ruud B O.1994.2Dfinite-difference elastic wavemodelling including surface topography.Geophys.Prosp.,42(5):371-390.
Hixon R.1997.Evaluation of a high-accuracy MacCormack-type scheme using benchmark problems.J.Comput.Acous.,6(3):291-305.
Hixon R,Turkel E.2000.Compact implicit MacCormack-type schemes with high accuracy.J.Comput.Phys.,158(1):51-70.
Jiang L L,Sun J G.2008.Source terms of elliptic system in grid generation.Global Geology(in Chinese),27(3):298-305.
Jiang L L.2010.Body-fitted grid generation for the geological conditions[Ph.D.thesis].Changchun:Jilin University.
Jih R,McLaughlin K L,Der Zoltan A.1988.Free-boundary conditions of arbitrary polygonal topography in a two-dimensional explicit elastic finite-difference scheme.Geophysics,53(8):1045-1055.
Jones T D.1986.Pore fluids and frequency-dependent wave propagation in rocks.Geophysics,56(10):1939-1953.
Lan H T,Liu C,Guo Z Q.2014.Modelling of seismic wave propagation in two-phase medium based on reformulated BISQmodel using time-splitting staggered pseudospectral method.Global Geology(in Chinese),33(1):190-199.
Li N.2012.Finite difference seismoelectric wave modeling in 2Dporous media with surface topography[Ph.D.thesis](in Chinese).Hefei:University of Science and Technology of China.
Liu C,Lan H T,Guo Z Q et al.2003.Pseudo-spectral modeling and feature analysis of wave propagation in two-phase HTImedium based on reformulated BISQ mechanism.Chinese J.Geophys.(in Chinese),56(10):3461-3473,doi:10.6038/cjg20131021.
Mavko G M,Jizba D.1991.Estimating grain-scale fluid effects on velocity dispersion in rocks.Geophysics,56(12):1940-1949.
Opr2al I,Zahradník J.1999.Elastic finite-difference method for irregular grids.Geophysics,64(1):240-250.
Peng C Z,Li C M,Wang M C.2007.Seismic wave-field modeling based on the modefied BISQ model.Computing Techniques for Geophysical&Geochemical Exploration(in Chinese),29(1):15-18.
Sun Y C,Zhang W,Chen X F.2016.Seismic-wave modeling in the presence of surface topography in 2Dgeneral anisotropic media by a curvilinear grid finite-difference method.Bull.Seism.Soc.Am.,106(3):1036-1054.
Tam C K W,Webb J C.1993.Dispersion-relation-preserving finitedifference schemes for computational acoustics.J.Comput.Phys.,107(2):262-281.
Tarrass I,Giraud L,Thore P.2011.New curvilinear scheme for elastic wave propagation in presence of curved topography.Geophys.Prosp.,59(5):889-906.
Tessmer E,Kosloff D,Behle A.1992.Elastic wave propagation simulation in the presence of surface topography.Geophys.J.Int.,108(2):621-632.
Thompson J F,Warsi Z U A,Mastin C W.1985.Numerical Grid Generation:Foundations and Applications.New York:Elsevier North-Holland,Inc.
Winkler K W.1985.Dispersion analysis of velocity and attenuation in Berea sandstone.Journal of Geophysical Research,90(B8):6793-6800.
Yang D H,Zhang Z J,Teng J W,et al.2000.The study of twophase anisotropy,questions and applied prospects.Progress in Geophysics(in Chinese),15(2):7-21.
Yang D H.2002.Finite Element method of the elastic wave equation and wavefield simulation in two-phase anisotropic media.Chinese J.Geophys.(in Chinese),45(4):575-583.
Zhang W,Chen X F.2006.Traction image method for irregular free surface boundaries in finite difference seismic wave simulation.Geophys.J.Int.,167(1):337-353.
Zhang W,Zhang Z G,Chen X F.2012.Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids.Geophys.J.Int.,190(1):358-378.
Zhang Y,Xu Y X,Xia J H,et al.2015.Characteristics and application of surface wave propagation in fluid-filled porous media.Chinese J.Geophys.(in Chinese),58(8):2759-2778,doi:10.6038/cjg20150812.
Zhou Z S,Tang L.2012.Seismic modeling and numerical dispersion correcting in double-phase medium based on reformulated BISQmodel.Journal of Central South University:Science and Technology(in Chinese),43(4):1411-1418.
Zhu H J,Zhang W,Chen X F.2009.Two-dimensional seismic wave simulation in anisotropic media by non-staggered finite difference method.Chinese Journal of Geophysics(in Chinese),52(6):1536-1546,doi:10.3969/j.issn.0001-5733.2009.06.015.
韩翀,陈雪菲,蒋东等.2007.改进BISQ模型在双相介质地震波数值模拟中的应用.天然气工业,2007,27(10):49-52.
蒋丽丽,孙建国.2008.基于Poisson方程的曲网格生成技术.世界地质,27(3):298-305.
蒋丽丽.2010.面向地质条件的贴体网格生成技术[博士论文].长春:吉林大学.
兰慧田,刘财,郭智奇.2014.利用时间分裂的错格伪谱法模拟地震波在基于改进BISQ模型的双相介质中的传播.世界地质,33(1):190-199.
李娜.2012.含起伏地表的二维双相介质中地震波及电磁波传播有限差分算法[博士论文].合肥:中国科学技术大学.
刘财,兰慧田,郭智奇等.2013.基于改进BISQ机制的双相HTI介质波传播伪谱法模拟与特征分析.地球物理学报,56(10):3461-3473,doi:10.6038/cjg20131021.
彭传正,李才明,王明春.2007.基于改进BISQ模型的地震波场数值模拟.物探化探计算技术,29(1):15-18.
杨顶辉,张中杰,滕吉文等.2000.双相各向异性研究、问题与应用前景.地球物理学进展,15(2):7-21.
杨顶辉.2002.双相各向异性介质中弹性波方程的有限元解法及波场模拟.地球物理学报,45(4):575-583.
张煜,徐义贤,夏江海等.2015.含流体孔隙介质中面波的传播特性及应用.地球物理学报,58(8):2759-2778,doi:10.6038/cjg20150812.
周竹生,唐磊.2012.改进BISQ模型的双相介质地震波场数值模拟及频散校正.中南大学学报:自然科学版,43(4):1411-1418.
祝贺君,张伟,陈晓非.2009.二维各向异性介质中地震波场的高阶同位网格有限差分模拟.地球物理学报,52(6):1536-1546,doi:10.3969/j.issn.0001-5733.2009.06.015.