基于改进BISQ模型弹性波数值模拟的交错网格有限差分方法
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摘要
地震波勘探方法是目前进行能源资源勘探开发中最为广泛应用的一种方法。在现实中,主要能源资源大多数都埋藏在具有复杂几何构造和复杂物理属性的地球岩石圈中,在工程勘探中,往往把含资源地层中的复杂地质构造简化成较为简单便于数值模拟的模型,简化的过程中难免产生误差使数值模拟得到的结果不准确,与实际的地下情况不符。所以,模拟出更符合实际地质结构的数值模拟图是快速解决当今地震勘探领域中问题的关键。
在地震勘探中,勘探层往往表现为双相介质。所以,对双相介质波动理论的研究可以解决勘探中的难题,本文基于改进BISQ模型利用交错网格有限差分方法进行数值模拟。改进BISQ模型是结合Biot理论机制与喷射流理论机制,对理论计算公式进行简化的双相介质模型,能够更全面、更具体地反应了地下介质的真实情况的同时,计算量相对较少。
交错网格有限差分方法被广泛的应用在双相介质地震波数值模拟中。论文基于改进BISQ模型,对二维( x , z )情况下一阶弹性波动方程利用交错网格有限差分方法进行差分离散,采用一网格点处的变量取相邻两网格点变量之和求平均的差分离散格式。利用PML完全匹配层进行吸收边界反射,对加入阻尼因子之后的波动方程同样利用上述差分格式进行离散,通过对固相波场进行数值模拟,观测到PML完全匹配层有效的吸收了人工边界反射。对用交错网格有限差分方法离散波动方程过程中产生的数值频散现象产生的原因及影响进行了分析,并采用FCT(通量校正传输)方法进行压制数值频散,讨论FCT方法中漫射系数的选定标准,通过固相波场数值模拟观测压制数值频散的效果。
最后,对单层、双层地质模型进行数值模拟。通过观测数值模拟得到的波场快照及合成地震记录图,分析、讨论波场特征,对固相、流相中的快纵波、慢纵波、S波的特征进行分析讨论。同时观测FCT方法压制数值频散与PML完全匹配层吸收边界反射的效果。
Seismic wave exploration is the most widely used method for exploration and development of energy resources. Most of the major mineral and energy resources are buried in the Earth's lithosphere,which is of complicated geometry structures and complex physical properties.In practical engineering exploration, the complicated structure in the stratum which contains natural resources is usually simplified to a model in order to apply numerical simulation. It is hard to avoid errors during the simplification, which is because it will cause the inaccuracy in numerical simulation and thus will not match with the actual underground conditions. Therefore, establishing a more realistic numerical simulation of geological structure is the key of quickly resolving the issue in the field of seismic exploration nowadays.
In seismic exploration, the two-phase medium wave theory can be used into practice to solve the problem of exploration. Seismic waves can be regarded as elastic waves spreading in the bed stratum, and elastic theory is the foundation of the elastic waves study. The paper is based on the numerical simulation with finite difference method by improving BISQ theory, and Improving BISQ theory is based on Biot assumption. Solid skeleton and interconnected pores form the fluid saturated porous media, and connected pores is filled with viscous fluid which is flow and can be compressed. The media formed by the solid skeleton and the fluid media is the basis of the theoretical research, which is closer to exploration and the actual situation
Staggered-grid finite difference method is widely used in the numerical simulation of the two-phase media in the Seismic wave. Based on the improved BISQ model, we transform the elastic wave equation with two-dimensional (x, z) case into difference discrete situation with the method of staggered-grid finite difference, then use perfectly matched layer PML to absorb boundary reflection. By improving the PML difference scheme with the method of assigning the average of its two adjacent grid points to the variable in semi-grid point, we can eliminate the effect of artificial boundary reflection effectively. Then, by using FCT (Flux-Corrected Transport) simulation to suppress the numerical dispersion, we finally achieve numerical simulation for single and double-phase geological model. After getting wave field maps and synthetic seismograms, we further analysised and discussed the characteristics of the fast primary wave, low primary wave and S wave. At the same time, the impact of the selection of the different diffusion factors to the numerical dispersion effect in FCT method is also being analysised. The numerical simulation then explains the feasibility of the improved format discrete PML.
Finally, the single and double numerical simulation of the geological model. Numerical simulation by observing the snapshots and synthetic seismograms, wave field analysis and discussion of the characteristics of the solid phase, fluid phase of the fast P wave, slow wave, wave characteristics discussed. Both methods suppress the observed numerical dispersion and the perfectly matched layer absorbing boundary reflection effect.
在地震勘探中,勘探层往往表现为双相介质。所以,对双相介质波动理论的研究可以解决勘探中的难题,本文基于改进BISQ模型利用交错网格有限差分方法进行数值模拟。改进BISQ模型是结合Biot理论机制与喷射流理论机制,对理论计算公式进行简化的双相介质模型,能够更全面、更具体地反应了地下介质的真实情况的同时,计算量相对较少。
交错网格有限差分方法被广泛的应用在双相介质地震波数值模拟中。论文基于改进BISQ模型,对二维( x , z )情况下一阶弹性波动方程利用交错网格有限差分方法进行差分离散,采用一网格点处的变量取相邻两网格点变量之和求平均的差分离散格式。利用PML完全匹配层进行吸收边界反射,对加入阻尼因子之后的波动方程同样利用上述差分格式进行离散,通过对固相波场进行数值模拟,观测到PML完全匹配层有效的吸收了人工边界反射。对用交错网格有限差分方法离散波动方程过程中产生的数值频散现象产生的原因及影响进行了分析,并采用FCT(通量校正传输)方法进行压制数值频散,讨论FCT方法中漫射系数的选定标准,通过固相波场数值模拟观测压制数值频散的效果。
最后,对单层、双层地质模型进行数值模拟。通过观测数值模拟得到的波场快照及合成地震记录图,分析、讨论波场特征,对固相、流相中的快纵波、慢纵波、S波的特征进行分析讨论。同时观测FCT方法压制数值频散与PML完全匹配层吸收边界反射的效果。
Seismic wave exploration is the most widely used method for exploration and development of energy resources. Most of the major mineral and energy resources are buried in the Earth's lithosphere,which is of complicated geometry structures and complex physical properties.In practical engineering exploration, the complicated structure in the stratum which contains natural resources is usually simplified to a model in order to apply numerical simulation. It is hard to avoid errors during the simplification, which is because it will cause the inaccuracy in numerical simulation and thus will not match with the actual underground conditions. Therefore, establishing a more realistic numerical simulation of geological structure is the key of quickly resolving the issue in the field of seismic exploration nowadays.
In seismic exploration, the two-phase medium wave theory can be used into practice to solve the problem of exploration. Seismic waves can be regarded as elastic waves spreading in the bed stratum, and elastic theory is the foundation of the elastic waves study. The paper is based on the numerical simulation with finite difference method by improving BISQ theory, and Improving BISQ theory is based on Biot assumption. Solid skeleton and interconnected pores form the fluid saturated porous media, and connected pores is filled with viscous fluid which is flow and can be compressed. The media formed by the solid skeleton and the fluid media is the basis of the theoretical research, which is closer to exploration and the actual situation
Staggered-grid finite difference method is widely used in the numerical simulation of the two-phase media in the Seismic wave. Based on the improved BISQ model, we transform the elastic wave equation with two-dimensional (x, z) case into difference discrete situation with the method of staggered-grid finite difference, then use perfectly matched layer PML to absorb boundary reflection. By improving the PML difference scheme with the method of assigning the average of its two adjacent grid points to the variable in semi-grid point, we can eliminate the effect of artificial boundary reflection effectively. Then, by using FCT (Flux-Corrected Transport) simulation to suppress the numerical dispersion, we finally achieve numerical simulation for single and double-phase geological model. After getting wave field maps and synthetic seismograms, we further analysised and discussed the characteristics of the fast primary wave, low primary wave and S wave. At the same time, the impact of the selection of the different diffusion factors to the numerical dispersion effect in FCT method is also being analysised. The numerical simulation then explains the feasibility of the improved format discrete PML.
Finally, the single and double numerical simulation of the geological model. Numerical simulation by observing the snapshots and synthetic seismograms, wave field analysis and discussion of the characteristics of the solid phase, fluid phase of the fast P wave, slow wave, wave characteristics discussed. Both methods suppress the observed numerical dispersion and the perfectly matched layer absorbing boundary reflection effect.
引文
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[6] Dvorkin J,Nur A. Dynamic poroelasticity: a unified model with the squirt and the Biot mechanisms[J]. Geop hysics,1993,58(4):52-533.
[7] Mamadou Sanou. Acoustic wave propagation in saturated porous media:reformulation of the Biot/Squirt Flow Theory[J]. Journal of Apolied Geophysics,2000,44:313-325.
[8]董良国,马在田,等.一阶弹性波方程交错网格高阶差分解法[J].地球物理学报,2000,43(3):631-419.
[9]杨宽德,杨顶辉,王书强.基于BISQ高频极限方程的交错网格法数值模拟[J].石油地球物理勘探,2002,37(5):463-468.
[10]韩翀,李正文.基于改性BISQ模型双相介质波场数值模拟[J].内蒙古石油化工,2007,5(2):56-59.
[11]彭传正,李才明.基于改进BISQ模型的地震波场数值模拟[J].物探化探计算技术,2007,29(1):15-20.
[12]韩翀,陈雪菲,等.改进BISQ模型在双相介质地震波数值模拟中的应用[J].天然气工业,2007,27(10):49-52.
[13]李红星,刘财,陶春辉.基于BISQ的各向同性孔隙介质弹性波三维交错网格高阶有限差分数值模拟[J].世界地质,2007,26(4):501-508.
[14]赵海波,王秀明,王东.完全匹配层吸收边界在孔隙介质弹性波动模拟中的应用[J].地球物理学报,2007,50(2):581-591.
[15]周清强,刘志远.基于改进的BISQ地震波储层参数反演[J].内蒙古石油化工,2008,(14):60-62.
[16]董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法稳定性研究[J].地球物理学报,2000,43(6):856- 864.
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[22] Alterman Z,Karal F C Jr.Propagation of elastic waves in layered media by finite difference methods[J]. Bull.Seism.Soc. Am.,1968,58(1):367-396.
[23] Boore D M. Finite difference methods for seismic wave propagation in heterogeneous materials[M]. In:Method in Bolt B A Ed.Computational physics.Academic Press. Inc.1972.
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[25] Kelly K R,Ward R W,Treitel S et al. Synthetic seismograms,a finite difference approach.[J]Geophysics,1976,41(1)2-27.
[26] Aki K , Richards P G. Quantitative Seismology[M]W.H . Freeman and Company, 1980.
[27] Virieux J. SH-wave propagation in heterogeneous media: velocity-stress finite-difference method[J]. Geophysics,1984,49(11):1933-1957.
[28] Virieux J. P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method[J]. Geophysics,1986,51(4):889-901.
[29] 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.
[30] O’Connell R J Budiansky B. Viscodastic properties of fluid-saturated crackedsolids[J]. J Geophys,Res.1977,82(36):5719-5736
[31] Murphy W FⅢWinkler K W,Kleinberg R B. Acoustic relaxation in sedimentary rocks:dependence on grain contacts fluid saturation[J]. Gephysicy 1986,51(23):757-766.
[32] Smith W D. A non-reflecting plane boundary for wave propagation problems[J]. J Comput. Phys,1974,15:492-503.
[33] Cerjan C,Kosloff D,Reshef M. A nonreflecting boundary condition for discrete acoustic and elastic wave equation[J]. Geophysics,1985,50(4):705-708.
[34] Burns D R. Acoustic and elastic scattering from seamounts in three dimensions-numerical modeling study[J]. J. Acoust. Soc. Amer.,1992,92:2784-2791.
[35] Kosloff R,Kosloff D. Absorbing boundaries for wave propagation problems[J]. J Comput. Phys.,1986,63(2):363-376.
[36] Clayton R,Engquist B. Aborbing boundary condition for acoustic and elastic wave equations[J]. Bull. Seism. Soc. Am.,1977,67(6):1529-1540.
[37] Liao Z P,Wong H L,Yang B P,et al. A transmitting boundary for transient wave analyses[J]. Sci.Ainica Ser. A,1984,27(10):1063-1076.
[38] Keys R G.. Absorbing boundary condition for acoustic media.[J]. Geophysics,1985,50(6):892-902.
[39] Higdon R L. Absorbing boundary conditions for difference approximations to the multidimensional wave equation.[J] Mathematics of Computation,1986,47(176):437-459
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[58] Francis Collino,Chrysoula Tsogka. Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogenenous media[J]. Geophysicds,2001,66(1):294-307
[2] Biot M A. Theory of Propagation of Elastic Waves in a Fluid—Saturated Porous Solid:I. Low—Frequency Range[J]. The Journal of the Acoustic Society of America,1956,28(2):168-178.
[3] Biot M A. Theory of Propagation of Elastic Waves in a Fluid Saturated Porous Solid:Ⅱ.Higher Frequency Range[J]. The Journal of the Acoustic Society of America,1956,28(2):179-191.
[4] Biot M .A. Mechanics of deformations and acoustic propation in porous media[J]. J. Appl. Phys,1962a,33(4):1482-1498.
[5] Biot M A. Generalized theory of Acoustic Propagation in porous dissipative media[J]. The Journal of Acoustic Society of America,1962b,34(5):1254-1264.
[6] Dvorkin J,Nur A. Dynamic poroelasticity: a unified model with the squirt and the Biot mechanisms[J]. Geop hysics,1993,58(4):52-533.
[7] Mamadou Sanou. Acoustic wave propagation in saturated porous media:reformulation of the Biot/Squirt Flow Theory[J]. Journal of Apolied Geophysics,2000,44:313-325.
[8]董良国,马在田,等.一阶弹性波方程交错网格高阶差分解法[J].地球物理学报,2000,43(3):631-419.
[9]杨宽德,杨顶辉,王书强.基于BISQ高频极限方程的交错网格法数值模拟[J].石油地球物理勘探,2002,37(5):463-468.
[10]韩翀,李正文.基于改性BISQ模型双相介质波场数值模拟[J].内蒙古石油化工,2007,5(2):56-59.
[11]彭传正,李才明.基于改进BISQ模型的地震波场数值模拟[J].物探化探计算技术,2007,29(1):15-20.
[12]韩翀,陈雪菲,等.改进BISQ模型在双相介质地震波数值模拟中的应用[J].天然气工业,2007,27(10):49-52.
[13]李红星,刘财,陶春辉.基于BISQ的各向同性孔隙介质弹性波三维交错网格高阶有限差分数值模拟[J].世界地质,2007,26(4):501-508.
[14]赵海波,王秀明,王东.完全匹配层吸收边界在孔隙介质弹性波动模拟中的应用[J].地球物理学报,2007,50(2):581-591.
[15]周清强,刘志远.基于改进的BISQ地震波储层参数反演[J].内蒙古石油化工,2008,(14):60-62.
[16]董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法稳定性研究[J].地球物理学报,2000,43(6):856- 864.
[17] Thomson W T. Transmission of elastic waves through a stratified solid[J]. Journal of Applide physics,1950,21(2):89-93.
[18] Alterman Z,Karal F C Jr. Propagation of elastic waves in layered media by finite difference methods[J]. Bull.Seism. Soc.Am.,1968,58(1):367-398.
[19] Smith W D. The application of finite-element analysis to body wave propagation problems[J] Geophysics J.Roy Astr. Soc.,1975,42(2):747-768.
[20] Hong T L,Helmberger D V. Glorified optics and wave propagation in nonplanar structure[J]. Bull.Seism. Soc. Am.,1978,68(5):1313-1330.
[21] Ber P,et al. A spectral method for seismic wave propagation in elastic media[J]. Wave Motion,1990,12(5):415-427.
[22] Alterman Z,Karal F C Jr.Propagation of elastic waves in layered media by finite difference methods[J]. Bull.Seism.Soc. Am.,1968,58(1):367-396.
[23] Boore D M. Finite difference methods for seismic wave propagation in heterogeneous materials[M]. In:Method in Bolt B A Ed.Computational physics.Academic Press. Inc.1972.
[24] Alford R M,Kelly K R,Boore D M. Accuracy of finite-difference modeling of the acoustic wave equation[J]. Geophysics,1974,39(6):834-842.
[25] Kelly K R,Ward R W,Treitel S et al. Synthetic seismograms,a finite difference approach.[J]Geophysics,1976,41(1)2-27.
[26] Aki K , Richards P G. Quantitative Seismology[M]W.H . Freeman and Company, 1980.
[27] Virieux J. SH-wave propagation in heterogeneous media: velocity-stress finite-difference method[J]. Geophysics,1984,49(11):1933-1957.
[28] Virieux J. P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method[J]. Geophysics,1986,51(4):889-901.
[29] 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.
[30] O’Connell R J Budiansky B. Viscodastic properties of fluid-saturated crackedsolids[J]. J Geophys,Res.1977,82(36):5719-5736
[31] Murphy W FⅢWinkler K W,Kleinberg R B. Acoustic relaxation in sedimentary rocks:dependence on grain contacts fluid saturation[J]. Gephysicy 1986,51(23):757-766.
[32] Smith W D. A non-reflecting plane boundary for wave propagation problems[J]. J Comput. Phys,1974,15:492-503.
[33] Cerjan C,Kosloff D,Reshef M. A nonreflecting boundary condition for discrete acoustic and elastic wave equation[J]. Geophysics,1985,50(4):705-708.
[34] Burns D R. Acoustic and elastic scattering from seamounts in three dimensions-numerical modeling study[J]. J. Acoust. Soc. Amer.,1992,92:2784-2791.
[35] Kosloff R,Kosloff D. Absorbing boundaries for wave propagation problems[J]. J Comput. Phys.,1986,63(2):363-376.
[36] Clayton R,Engquist B. Aborbing boundary condition for acoustic and elastic wave equations[J]. Bull. Seism. Soc. Am.,1977,67(6):1529-1540.
[37] Liao Z P,Wong H L,Yang B P,et al. A transmitting boundary for transient wave analyses[J]. Sci.Ainica Ser. A,1984,27(10):1063-1076.
[38] Keys R G.. Absorbing boundary condition for acoustic media.[J]. Geophysics,1985,50(6):892-902.
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