基于保角变换法的地震裂缝波场研究
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摘要
针对实际地层中的裂缝形状,提出了两种裂缝模型,一种是广义四边形裂缝模型,另一种是角形裂缝模型。基于保角变换,对单个裂缝的波场进行了研究。首先,对物理域中的裂缝区域施加保角变换,将其变换为计算域中的上半平面,对物理域中的裂缝边界条件进行保角变换为计算域中的边界条件;其次,在计算域中求出裂缝边界及其附近的波场;最后,通过保角变换的反变换得到物理域中裂缝边界波场变化规律。提出的保角变换法对研究单个和多个任意形状的裂缝波场之间的定量关系以及勘探地球物理中的非均质问题提供了新的思路。
In view of the actual shape of fracture formation,two fracture models are presented in this paper.One is general quadrilateral fracture model,and the other is the horny fracture model.The wave fields of single fracture were studied based on the conformal mapping method.First of all,the conformal mapping on the fracture area in physical domain was transformed to the upper half plane in the computational domain and the boundary conditions in physical domain to the boundary conditions in the computational domain;secondly,the wave fields near the boundary of the fracture in the computational domain were calculated;fInally,the wave fields variation curve in the boundary of physical domain through the inverse conformal mapping transformation was obtained.The conformal mapping method are used in our research on the quantitative relationship of the wave fields of a single and several any shape fracture provides some new ideas about the heterogeneity problem in the exploration geophysical.
In view of the actual shape of fracture formation,two fracture models are presented in this paper.One is general quadrilateral fracture model,and the other is the horny fracture model.The wave fields of single fracture were studied based on the conformal mapping method.First of all,the conformal mapping on the fracture area in physical domain was transformed to the upper half plane in the computational domain and the boundary conditions in physical domain to the boundary conditions in the computational domain;secondly,the wave fields near the boundary of the fracture in the computational domain were calculated;fInally,the wave fields variation curve in the boundary of physical domain through the inverse conformal mapping transformation was obtained.The conformal mapping method are used in our research on the quantitative relationship of the wave fields of a single and several any shape fracture provides some new ideas about the heterogeneity problem in the exploration geophysical.
引文
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[12]Xianfeng Gu,Yalin Wang,Chan T F,et al.Genus ze-ro surface conformal mapping and its application to brainsurface mapping[J].IEEE Transactions on Medical Imag-ing,2004,23(8):949–958.
[13]Bowie O L.Solutions of plane fracture problems by map-ping techniques,in methods of analysis and solutions offracture problems[M].Leiden:Noordhoof Int.,1973:81–106.
[14]Driscoll T,Trefethen L N.Schwarz-Christoffel map-ping[M].Cambridge Univ.Press,2002:204–130.
[15]Samek M,Slean C,Weghorst H.Texture mapping anddistortions in digital graphics[J].The Visual Computer,1986,2(5):313–320.
[16]Floater M.Parametrization and smooth approximation ofsurface triangulations[J].Computer Aided Geometric De-sign,1997,14(3):231–250.
[17]Gu X,Yau S.Computing conformal structures of sur-faces[J].Commun.Inform.Syst.,2002,2(2):121–146.
[2]沈建中,应崇福,李明轩.固体中带状裂缝对脉冲超声波散射的理论分析—普适解[J].声学学报,1985,10(1):13–22.Shen Jianzhong,Ying Chongfu,Li Mingxuan.Scatter-ing of an ultrasonic pulse by a two-dimensional crackin solid-general theoretical analysis[J].Acta Acoustica,1985,10(1):13–22.
[3]盖秉政.论弹性波绕射与动应力集中问题[J].力学进展,1987,17(4):447–465.Gai Bingzheng.On the diffraction of elastic wave and dy-namic stress concentration[J].Advances in Mechanics,1987,17(4):447–465.
[4]刘殿魁.各向异性介质中III型裂缝的动力分析[J].地震工程与工程振动,1991,11(2):29–38.Liu Diankui.Analysis of dynamic stresses of mode IIIcrack in anisotropic materials[J].Earthquake Engineeringand Engineering Vibration,1991,11(2):29-38.
[5]刘殿魁,刘宏伟.孔边裂纹对SH波的散射及其动应力强度因子[J].力学学报,1999,31(3):292–299.Liu Diankui,Liu Hongwei.Scattering of SH-wave byfractures originating at a circular hole edge and dynamicstress intensity factor[J].Acta Mechanica Sinica,1999,31(3):292–299.
[6]刘殿魁,陈志刚.椭圆孔边裂纹对SH波的散射及其动应力强度因子[J].应用数学和力学,2004,25(9):958–966.Liu Diankui,Chen Zhigang.Scattering of SH-wave bycracks originating at an elliptic hole and dynamic stressintensity factor[J].Applied Mathematics and Mechanics,2004,25(9):958–966.
[7]杨秀夫,陈勉,刘希圣,等.保角变换法在水压裂缝缝内流场数值模拟中的应用研究[J].中国海上油气(工程),2002,14(5):28–30.Yang Xiufu,Chen Mian,Liu Xisheng,et al.Applicationresearch of conformal conversion to the numerical simu-lation of fluid field inner hydraulic fracture[J].China SeaOi(l Engineering),2002,14(5):28–30
[8]钟伟芳,聂国华.弹性波的散射理论[M].武汉:华中理工大学出版社,1997:174–189.
[9]Michele G,Francesco B,Paolo C,et al.A generalconformal-mapping approach to the optimum electrodedesign of coplanar waveguides with arbitrary cross sec-tion[J].IEEE Transactions on Microwave Theory andTechniques,2001,49(9):1573–1580.
[10]FranssensGR,LagassePE.Scatteringofelasticwavesbya cylindrical obstacle embedded in a multilayered medi-um[J].J Acoust Soc Am,1984,76(5):1535–1542.
[11]Angel Y C,Bogy D B.Scattering of rayleigh wave bya plate attached to a half space through a viscous cou-plant[J].ASME J Appl Mech,1981,48(4):881–888.
[12]Xianfeng Gu,Yalin Wang,Chan T F,et al.Genus ze-ro surface conformal mapping and its application to brainsurface mapping[J].IEEE Transactions on Medical Imag-ing,2004,23(8):949–958.
[13]Bowie O L.Solutions of plane fracture problems by map-ping techniques,in methods of analysis and solutions offracture problems[M].Leiden:Noordhoof Int.,1973:81–106.
[14]Driscoll T,Trefethen L N.Schwarz-Christoffel map-ping[M].Cambridge Univ.Press,2002:204–130.
[15]Samek M,Slean C,Weghorst H.Texture mapping anddistortions in digital graphics[J].The Visual Computer,1986,2(5):313–320.
[16]Floater M.Parametrization and smooth approximation ofsurface triangulations[J].Computer Aided Geometric De-sign,1997,14(3):231–250.
[17]Gu X,Yau S.Computing conformal structures of sur-faces[J].Commun.Inform.Syst.,2002,2(2):121–146.
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