具有离散裂缝空间分布的二维固体中地震波传播的有限差分模拟
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摘要
结合有限差分方法和等效介质理论,模拟了离散分布裂缝介质中地震波的传播.基于等效介质理论,利用二维有限差分实现封闭裂缝的离散分布;裂缝可以处理成固体岩石中的高度柔性界面,并可以用线性滑动或者位移间断模型进行裂缝的物理描述.对于含有多组裂隙的破裂固体,其有效柔度可以认为是固体骨架背景柔度和裂缝附加柔度之和.在一阶近似条件下,固体骨架和裂缝参数可以通过有效各向异性系数联系起来,有效各向异性系数决定了各向异性(裂缝效应)对于地震波传播的影响.通过与射线理论方法的对比检验,说明本文提出的模拟方法的有效性,并通过几个数值算例说明本方法可有效模拟不同的裂缝分布效应.结果表明,即使在裂缝密度很小的情况下,具有相同裂缝密度的不同的空间分布可以产生不同的波场特征.同时,也验证了不同裂缝尺度对波长的不同影响,以及裂缝尺度具有幂率分布(分形)时,尺度对波场的影响.最后得出结论:在运用建立在等效介质理论基础上的地震各向异性概念来描述裂缝固体的特征时,要倍加小心,等效介质理论中尚未合理处理的裂缝尺度和空间分布对波的传播特征具有重要的影响.
We present an effective technique to model seismic wave propagation in media with discrete distributions of fractures based on the combination of the finite difference and equivalent medium theories. The distribution of discrete fractures with vanishing apertures in the 2_D finite difference grids is implemented using the equivalent medium theory. Fractures are treated as highly compliant interfaces inside a solid rock mass. For the physical representation of the fractures the concept of linear slip or displacement discontinuity model is used. The effective compliance of a fractured solid with multiple fracture sets can be found as the sum of the compliances of the matrix solid (background) and extra compliance introduced as a result of the presence of fractures. To the first order, the matrix solid and fracture parameters can be related to the effective anisotropic coefficients, which govern the influence of anisotropy (a result from fractures) on seismic wave propagation. We test the validity of the method by comparison with theoretical ray method. Several numerical examples are presented to demonstrate the effectiveness of our method showing the effects of different fracture distributions. We show that different spatial distributions with the same fracture density can produce significant different wave_field characteristics even when fracture density is small. We also examine the effects of variation of fracture scale_length (size) as compared to the wavelength. In the case of fractures having a power_law (fractal) distribution of sizes, we show how the variation of scale_length affects the wave_field. We conclude that characterization of fractured solids based on the concept of seismic anisotropy using effective medium theories must be used with caution. Scale_length and the spatial distributions of fractures, which are not properly treated in equivalent medium theories, have a strong influence on the characteristics of wave propagations.
We present an effective technique to model seismic wave propagation in media with discrete distributions of fractures based on the combination of the finite difference and equivalent medium theories. The distribution of discrete fractures with vanishing apertures in the 2_D finite difference grids is implemented using the equivalent medium theory. Fractures are treated as highly compliant interfaces inside a solid rock mass. For the physical representation of the fractures the concept of linear slip or displacement discontinuity model is used. The effective compliance of a fractured solid with multiple fracture sets can be found as the sum of the compliances of the matrix solid (background) and extra compliance introduced as a result of the presence of fractures. To the first order, the matrix solid and fracture parameters can be related to the effective anisotropic coefficients, which govern the influence of anisotropy (a result from fractures) on seismic wave propagation. We test the validity of the method by comparison with theoretical ray method. Several numerical examples are presented to demonstrate the effectiveness of our method showing the effects of different fracture distributions. We show that different spatial distributions with the same fracture density can produce significant different wave_field characteristics even when fracture density is small. We also examine the effects of variation of fracture scale_length (size) as compared to the wavelength. In the case of fractures having a power_law (fractal) distribution of sizes, we show how the variation of scale_length affects the wave_field. We conclude that characterization of fractured solids based on the concept of seismic anisotropy using effective medium theories must be used with caution. Scale_length and the spatial distributions of fractures, which are not properly treated in equivalent medium theories, have a strong influence on the characteristics of wave propagations.
引文
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[3]Zhang Z J,Wang G J,Harris J M.Multi_component wavefield simulationin viscous extensively dilatancy anisotropic media.Physics ofthe Earth and PlanetaryInteriors,1999,114:25~38
[4]Yang D H,Liu E R,Zhang Z J,Teng J W.Finite_difference modelingintwo_dimensional anisotropic media using a flux_corrected transport technique.Geophys.J.Int.,2002,148(2):320~328
[5]Yang D H,Wang S,Zhang Z J,Teng J W.n_times absorbing boundary conditions for compact finite difference modeling of acoustic and elastic wave propagation in the2D TI medium.Bulletin oftheSeismological SocietyofAmerica,2003,93(6):2389~2401
[6]Zhang ZJ,He Q D,Xu Z X.The absorping of artificial boundary reflecting in2_D transversely isotropic media with FDM elastic wavefield modeling.Chinese J.Geophys.,1993,36(4):519~527
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[10]Coates RT,Schoenberg M.Finite_difference modelling of faults and fractures.Geophysics,1995,60(5):1514~1526
[11]Hsu CJ,Schoenberg M.Elastic waves through a simulatedfractured medium.Geophysics,1993,58(7):964~977
[12]Liu E R,Zhang Z J.Numerical study of elastic wave scattering by cracks or inclusions usingthe boundaryintegral equation method.J.Comp.Acoust,2001,9(3):1039~1054
[13]Liu E R,Crampin S,Hudson J A.Diffraction of seismic waves with applicationto hydraulicfracturing.Geophysics,1997,62(1):253~265
[14]Liu E R,Hudson J A,Pointer T.Equivalent mediumrepresentation of fractured rock.J.Geophys.Res.,2000,105(B2):2981~3000
[15]Muir F,Dellinger D,Etgen J,Nichols D.Modelling elastic wave_fields across irregular boundaries.Geophysics,1992,57(9):1189~1193.
[16]Nihei K T,Nakagawa S,Myer L R.VSP fracture imaging with elastic reverse_time migration.In:70th Ann.Internat.Mtg.Soc.of Expl.Geophys.,2000.1784~1751
[17]Pointer T,Liu E,HudsonJ A.Numerical modellingof seismic waves scattered by hydrofractures:Application of the indirect boundary element method.Geophys.J.Int.,1998,135(1):289~303
[18]Pyrak_Nolte LJ,Myer L R,Cook N G W.Transmission of seismic waves accross single natural fractures.J.Geophys.Res.,1990,95(B5):8617~8638
[19]Saenger E H,Shapiro S A.Effective velocitiesinfractured media:a numerical study using the rotated staggered finite_difference grid.Geophys.Prosp.,2002,50(2):183~194
[20]van Baren G B,Mulder W A,Herman G C.Finite_difference modelling of scalar_wave propagation in cracked media.Geophysics,2001,66(1):267~276
[21]Vlastos S,Liu E R,Main I G,Li X Y.Numerical simulation of wave propagation in media with discrete distributions of fractures:effects of fracture sizes and spatial distributions.Geophys.J.Int.,2003,152:649~668
[2]Zhang Z J,He Q D,Teng J W,et al.Forward modeling of3_component seismic records in2_D transversely isotropic media with finite difference method.Can.J.Exp.Geophys.,1993,29(1):51~58
[3]Zhang Z J,Wang G J,Harris J M.Multi_component wavefield simulationin viscous extensively dilatancy anisotropic media.Physics ofthe Earth and PlanetaryInteriors,1999,114:25~38
[4]Yang D H,Liu E R,Zhang Z J,Teng J W.Finite_difference modelingintwo_dimensional anisotropic media using a flux_corrected transport technique.Geophys.J.Int.,2002,148(2):320~328
[5]Yang D H,Wang S,Zhang Z J,Teng J W.n_times absorbing boundary conditions for compact finite difference modeling of acoustic and elastic wave propagation in the2D TI medium.Bulletin oftheSeismological SocietyofAmerica,2003,93(6):2389~2401
[6]Zhang ZJ,He Q D,Xu Z X.The absorping of artificial boundary reflecting in2_D transversely isotropic media with FDM elastic wavefield modeling.Chinese J.Geophys.,1993,36(4):519~527
[7]何樵登,张中杰.横向各向同性介质中地震波及其数值模拟.长春:吉林大学出版社,1996He QD,Zhang ZJ.Seismic Wavesin TransverselyIsotropic Medium and Numerical Modelling(in Chinese).Changchun:JiLin University Press,1996
[8]Benites R,Aki K,Yomogida K.Multiple scattering of SHwaves in2D media with many cavities.Pure appl.Geophys.,1992,138(3):353~390
[9]Bonnet E,Bour O,Odling N E,Davy P,Main I,Cowie P,Berkowitz B.Scaling of fracture systems in geological media.Rev.Geophys.,2001,39(3):347~383
[10]Coates RT,Schoenberg M.Finite_difference modelling of faults and fractures.Geophysics,1995,60(5):1514~1526
[11]Hsu CJ,Schoenberg M.Elastic waves through a simulatedfractured medium.Geophysics,1993,58(7):964~977
[12]Liu E R,Zhang Z J.Numerical study of elastic wave scattering by cracks or inclusions usingthe boundaryintegral equation method.J.Comp.Acoust,2001,9(3):1039~1054
[13]Liu E R,Crampin S,Hudson J A.Diffraction of seismic waves with applicationto hydraulicfracturing.Geophysics,1997,62(1):253~265
[14]Liu E R,Hudson J A,Pointer T.Equivalent mediumrepresentation of fractured rock.J.Geophys.Res.,2000,105(B2):2981~3000
[15]Muir F,Dellinger D,Etgen J,Nichols D.Modelling elastic wave_fields across irregular boundaries.Geophysics,1992,57(9):1189~1193.
[16]Nihei K T,Nakagawa S,Myer L R.VSP fracture imaging with elastic reverse_time migration.In:70th Ann.Internat.Mtg.Soc.of Expl.Geophys.,2000.1784~1751
[17]Pointer T,Liu E,HudsonJ A.Numerical modellingof seismic waves scattered by hydrofractures:Application of the indirect boundary element method.Geophys.J.Int.,1998,135(1):289~303
[18]Pyrak_Nolte LJ,Myer L R,Cook N G W.Transmission of seismic waves accross single natural fractures.J.Geophys.Res.,1990,95(B5):8617~8638
[19]Saenger E H,Shapiro S A.Effective velocitiesinfractured media:a numerical study using the rotated staggered finite_difference grid.Geophys.Prosp.,2002,50(2):183~194
[20]van Baren G B,Mulder W A,Herman G C.Finite_difference modelling of scalar_wave propagation in cracked media.Geophysics,2001,66(1):267~276
[21]Vlastos S,Liu E R,Main I G,Li X Y.Numerical simulation of wave propagation in media with discrete distributions of fractures:effects of fracture sizes and spatial distributions.Geophys.J.Int.,2003,152:649~668
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