双相介质P波波动方程小波域多尺度模拟
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摘要
相比于单相介质理论而言,双相介质理论更接近实际地层的真实情况,因此在地球物理勘探、地震工程和岩土动力学等领域有着广泛的应用。传统的波动方程数值解法由于本身固有的不足不利于求解诸如双相介质波动方程等复杂的非线性和不规则性问题;而小波方法则由于自身良好的特性可以用来构建解决此类问题的自适应性算法。本文详细推导了双相介质P波波动方程的有限差分矩阵表示形式,利用小波变换将其转移到小波域,设置阈值形成更为稀疏的迭代矩阵以构建自适应算法,从而达到减少计算量,增加地震波场数值模拟灵活性和准确性的目的。地球物理勘探的数值模拟实例验证了方法的有效性。
Compared with the single media theory,the two-phase media theory considers the porous elastic solid filled with compressible viscous fluid such as water,oil and etc.So,it describes the actual earth stratum more precisely and can be used widely in many fields,such as geophysical prospecting,earthquake engineering and rock & soil dynamics.The traditional numerical methods have some trouble in solving the nonlinear problem because of its inherent shortcoming.However,based on many good properties,wavelet method could solve the problem better.In this paper,combining the wavelet analysis method with the finite difference method,the multi-scale wavelet finite difference method is introduced to the numerical simulation of P-wave wave equation in the two-phase media.The finite difference matrix form of the P-wave wave equation in the two-phase media is deduced and then is transferred to the wavelet field using the wavelet transformation.The sparser iterative matrix can be obtained and the adaptive algorithm is formed by using the wavelet thresholding.The wavelet field multiscale simulation of seismic wave can reduce the cost of computation and increase the flexibility and the accusacy of the seismic wave numerical simulation,especially for the large scale problem.Two numerical simulation results prove the validity of the method.
Compared with the single media theory,the two-phase media theory considers the porous elastic solid filled with compressible viscous fluid such as water,oil and etc.So,it describes the actual earth stratum more precisely and can be used widely in many fields,such as geophysical prospecting,earthquake engineering and rock & soil dynamics.The traditional numerical methods have some trouble in solving the nonlinear problem because of its inherent shortcoming.However,based on many good properties,wavelet method could solve the problem better.In this paper,combining the wavelet analysis method with the finite difference method,the multi-scale wavelet finite difference method is introduced to the numerical simulation of P-wave wave equation in the two-phase media.The finite difference matrix form of the P-wave wave equation in the two-phase media is deduced and then is transferred to the wavelet field using the wavelet transformation.The sparser iterative matrix can be obtained and the adaptive algorithm is formed by using the wavelet thresholding.The wavelet field multiscale simulation of seismic wave can reduce the cost of computation and increase the flexibility and the accusacy of the seismic wave numerical simulation,especially for the large scale problem.Two numerical simulation results prove the validity of the method.
引文
[1]BIOT M A.Theory of propagation of elastic waves ina fluid-saturated porous solid:low frequency range[J].J Acoust Soc Amer,1956a,28:168-178.
[2]BIOT M A.Theory of propagation of elastic waves ina fluid-saturated porous solid:higher frequence range[J].J Acoust Soc Amer1956b,28:179-191.
[3]PLONA T J.Observation of a second bulk compres-sional wave in porous media at frequences[J].ApplPhys,1980,36:259-261.
[4]AMOS N,等.双相介质中波的播[M].许云,译.北京:石油工业出版社,1986.(Amos Nur.Wavepropagation in the two-Phasemedia[M].Xu Yun,Translated.Beijing:Petroleum Industry Press,1986.(in Chinese))
[5]GHASBOUSSI J,WILLSON E L.Variational for-mulation of dynamics of fluid-saturated porous and e-lastic solid[J].J Eng Mech Div,ASCE,1972,98:947-963.
[6]郭建.双相介质中P波波场的有限差分模拟[J].石油地球物理勘探,1992,27(2):182-199.(GUOJian.Finite difference modeling of-Pwavefield intwo-phase medium[J].Oil Geophysical Prospec-ting,1992,27(2):182-199.(in Chinese))
[7]徐文俊,郭建.流体饱和多孔隙介质中弹性波的数值模拟[J].地球物理学报,1994,37:424-433.(XUWen-jun,GUO Jian.Numerical simulation of elasticwaves in fluid-saturated porous media[J].Chinese JGeophys,1994,37:424-433.(in Chinese))
[8]YAZDCHI M,KHALILI N,VALLIPPAN S.Non-linear seismic behavior of concrete gravity dams usingcoupled finite element-boundary element method[J].International Journal for Numerical Methods inEngineering,1999,44:101-130.
[9]KHALILI N,YAZDCHI M,VALLIAPPAN S.Wave propagation analysis of two-phase saturatedporous media using coupled finite-infinite elementmethod[J].Soil Dynamics and Earthquake Engi-neering,1999,18:533-553.
[10]马坚伟,朱亚平.多尺度有限差分方法求解波动方程[J].计算力学学报,2002,19(4):379-383.(MAJian-wei,ZHU Ya-ping.A new method of multireso-lution finite difference for wave equation[J].ChineseJournal of Computational Mechanics,2002,19(4):379-383.(in Chinese))
[11]DAUBECHIES I.Ten lecture on wavelet[J].Phila-dephia SIAM,1992,26:793-834.
[12]ROBIRT&CRAMER.A multiresolution approachto fast summation and regularization of singular oper-ators[D].Department of Applied Mathematics.Uni-versity of Colorado,1996.
[2]BIOT M A.Theory of propagation of elastic waves ina fluid-saturated porous solid:higher frequence range[J].J Acoust Soc Amer1956b,28:179-191.
[3]PLONA T J.Observation of a second bulk compres-sional wave in porous media at frequences[J].ApplPhys,1980,36:259-261.
[4]AMOS N,等.双相介质中波的播[M].许云,译.北京:石油工业出版社,1986.(Amos Nur.Wavepropagation in the two-Phasemedia[M].Xu Yun,Translated.Beijing:Petroleum Industry Press,1986.(in Chinese))
[5]GHASBOUSSI J,WILLSON E L.Variational for-mulation of dynamics of fluid-saturated porous and e-lastic solid[J].J Eng Mech Div,ASCE,1972,98:947-963.
[6]郭建.双相介质中P波波场的有限差分模拟[J].石油地球物理勘探,1992,27(2):182-199.(GUOJian.Finite difference modeling of-Pwavefield intwo-phase medium[J].Oil Geophysical Prospec-ting,1992,27(2):182-199.(in Chinese))
[7]徐文俊,郭建.流体饱和多孔隙介质中弹性波的数值模拟[J].地球物理学报,1994,37:424-433.(XUWen-jun,GUO Jian.Numerical simulation of elasticwaves in fluid-saturated porous media[J].Chinese JGeophys,1994,37:424-433.(in Chinese))
[8]YAZDCHI M,KHALILI N,VALLIPPAN S.Non-linear seismic behavior of concrete gravity dams usingcoupled finite element-boundary element method[J].International Journal for Numerical Methods inEngineering,1999,44:101-130.
[9]KHALILI N,YAZDCHI M,VALLIAPPAN S.Wave propagation analysis of two-phase saturatedporous media using coupled finite-infinite elementmethod[J].Soil Dynamics and Earthquake Engi-neering,1999,18:533-553.
[10]马坚伟,朱亚平.多尺度有限差分方法求解波动方程[J].计算力学学报,2002,19(4):379-383.(MAJian-wei,ZHU Ya-ping.A new method of multireso-lution finite difference for wave equation[J].ChineseJournal of Computational Mechanics,2002,19(4):379-383.(in Chinese))
[11]DAUBECHIES I.Ten lecture on wavelet[J].Phila-dephia SIAM,1992,26:793-834.
[12]ROBIRT&CRAMER.A multiresolution approachto fast summation and regularization of singular oper-ators[D].Department of Applied Mathematics.Uni-versity of Colorado,1996.
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