流体饱和多孔隙介质弹性波方程边界元解法研究
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摘要
基于流体饱和多孔隙各向同性介质模型,本文首先推导了流体饱和多孔隙介质中弹性波传播的频率域系统动力方程及边界积分方程,然后给出了流体饱和多孔隙介质弹性波方程的基本解,最后,利用本文给出的边界元方法对流体饱和多孔隙各向同性介质中的弹性波传播进行了数值模拟.结果表明:不论是从固相位移,还是液相位移的地震合成记录都能看到明显的慢速P波,本文提出的流体饱和多孔隙介质弹性波边界元法是有效可行的.
Based on the fluid-saturated porous model this paper analyses and presents the dynamic equation of elastic wave propagation and boundary integral equation formulation of fluid saturated porous media in the frequency domain. The analogy between dynamic poroelasticity and thermoelasticity in the frequency domain is used to obtain the fundamental solution. At last, the numerical simulation of the elastic wave propagation in two-phase isotropic media is carried out by using the boundary element method. The results show that a slow quasi P-wave can be seen in both solid and fluid wave-field synthetic seismograms. It indicates that the boundary element method presented in this paper is effective and feasible.
Based on the fluid-saturated porous model this paper analyses and presents the dynamic equation of elastic wave propagation and boundary integral equation formulation of fluid saturated porous media in the frequency domain. The analogy between dynamic poroelasticity and thermoelasticity in the frequency domain is used to obtain the fundamental solution. At last, the numerical simulation of the elastic wave propagation in two-phase isotropic media is carried out by using the boundary element method. The results show that a slow quasi P-wave can be seen in both solid and fluid wave-field synthetic seismograms. It indicates that the boundary element method presented in this paper is effective and feasible.
引文
[1] Biot M A . Theory of propagation of elastic waves in a fluid-saturated porous solid. Part I: Low frequency range. J.Acoust. Soc. Am. , 1956 ,28:168-178
[2] Biot M A. Theory of propagation of elastic waves in a fluid-saturated porous solid. Part Ⅱ: Higher frequency range. J.Acoust. Soc.Am. ,1956,28 : 179-191
[3] Simon B R, Wu J S-S, Zienkiewicz O C. Evaluation of u-w and urn finite element of saturated porous media using one dimensional models. Int. J. Numer. Anal. Methods Geomech. , 1986 ,10:461-482
[4] Simon B R, Wu J S-S, Zienkiewicz O C. Evaluation of higher order, mixed and hermitean finite element procedures for dynamic analysis of saturated porous media using one dimensional models. Int. J. Numer. Anal. Methods Geomech. ,1986,10:483-499
[5] Atalla N, Panneton R, Debergue P. A mixed Displacement-pressure formulation for Biot ' s poroelastic equation. J.Acoust. Soc. Am. ,1996,104:1444-1452
[6] Panneton R, Atalla N. An efficient finite element scheme for solving the three dimensional poroelasticity problem in acoustics. J. Acoust. Soc Am. ,1997,101:328-329
[7] 王尚旭.双相介质中弹性波问题有限元数值解和AVO问题[博士论文].北京:石油大学,1990 Wang S X. Numerical solutions of the finite element method for the elastic wave problem and AVO problems in two-phase media [Ph. D. thesis]. Beijing: Petroleum University, 1990
[8] 徐文骏,郭建.流体饱和多孔介质中弹性波的数值模拟.地球物理学报,1994,37(增刊Ⅰ):424-433 Xu W J, Guo J. Numerical simulation of elastic waves in fluid-satuated porous media. Chinese J. Geophys. ( Acta Geophysica Sinica) ,1994 ,37(Suppl,Ⅰ ):424-433
[9] 张新明,刘克安,刘家琦.流体饱和多孔隙介质二维弹性波方程正演模拟的小波有限元法.地球物理学报,2005,48(5) ,1156-1166 Zhang X M, Liu K A, Liu J Q. A wavelet finite element method for the 2-D wave equation in fluid-saturated porous media. Chinese J. Geophys. (in Chinese), 2005,48(5) : 1156-1166
[10] 牟永光,裴正林.三维复杂介质地震数值模拟.北京:石油工业出版社,2004 Mou Y G, Pei Z L. 3-D Seismic Numerical Simulation Complex Media. Beijing:Petroleum Industry Press, 2004
[11] Dai N, Vafidis A. Wave propagation in heterogeneous,porous media: A velocity-stress, finite-difference method.Geophysics, 1995,60(2) :327-340
[12] 王秀明,张海澜.利用高阶交错网格有限差分法模拟地震波在非均匀孔隙介质中的传播.地球物理学报,2003,46(6) :842-849 Wang X M, Zhang H L. Modeling of seismic wave propagation in Hetero geneous poroelastic media using a high-order staggered finite-difference method. Chinese J.Geophys. (in Chinese) , 2003, 46(6) :842-849
[13] 裴正林.三维双相各向异性介质弹性波方程交错网格高阶有限差分法模拟.中国石油大学学报,2006,30(2) :16-20 Pei Z L. A staggered-grid high-order difference method for modeling seismic wave equation in 3-D dual-phase anisotropic media. Journal of China University of Petroleum (in Chinese), 2006 ,30(2) : 16-20
[14] 刘洋,李承楚.双相各向异性介质中弹性波传播伪谱法数值模拟研究.地震学报,2000,22(2) :132-138
[15] Komatisch D, Vilotte J P. The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological strucstures. Bull. Seism. Soc. Am. , 1 998 ,88 : 368-392
[16] Burridge R, Vargas C A. The fundamental solution in dynamic poroelasticity. Geophys. J. Royal Astron. Soc. ,1979． 58(1) :61-90
[17] Manolis G V, Beskos D E. Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity. Acta Mech. ,1989,76:89-104
[18] Bonnet G. Basic singular solutions for a poroelastic medium in the dynamic range. J. Acost. Aoc. Am. , 1987,82; 1758-1762
[19] Cheng H D. A direct boundary element method for plane strain poroelasticity. Int. J. Numer. Anal. Meth.Geomech. ,1988,12:551-572
[20] 符力耘,牟永光.弹性波边界元法正演模拟.地球物理学报,1994,37(4) :521-529 Fu L Y, Mou Y G. Boundary element method for elastic wave forward modeling. Chinese J. Geophys. (in Chinese),1994,37(4) ;521-529
[21] Predeleanu M. Reciprocal theorem in the consolidation theory of porous media. Appl. Math. Modelling,1968,8:378-382
[22] Cheng A H D, Predeleanu M. Transient boundary element formulation for poroelasticity. Appl. Math. Modelling. ,1987,11:285-290
[23] Kupradze V D, Gegelia T G. Three-dimensional problems of Mathematical Theory of Elasticity and Thermoelasticity.North-Holland, Amsterdam, 1979
[24] Nowacki W. Dynamics Problems of Thermoelasticity.Noordhoff, Leyden, the Netherlands, 1975
[2] Biot M A. Theory of propagation of elastic waves in a fluid-saturated porous solid. Part Ⅱ: Higher frequency range. J.Acoust. Soc.Am. ,1956,28 : 179-191
[3] Simon B R, Wu J S-S, Zienkiewicz O C. Evaluation of u-w and urn finite element of saturated porous media using one dimensional models. Int. J. Numer. Anal. Methods Geomech. , 1986 ,10:461-482
[4] Simon B R, Wu J S-S, Zienkiewicz O C. Evaluation of higher order, mixed and hermitean finite element procedures for dynamic analysis of saturated porous media using one dimensional models. Int. J. Numer. Anal. Methods Geomech. ,1986,10:483-499
[5] Atalla N, Panneton R, Debergue P. A mixed Displacement-pressure formulation for Biot ' s poroelastic equation. J.Acoust. Soc. Am. ,1996,104:1444-1452
[6] Panneton R, Atalla N. An efficient finite element scheme for solving the three dimensional poroelasticity problem in acoustics. J. Acoust. Soc Am. ,1997,101:328-329
[7] 王尚旭.双相介质中弹性波问题有限元数值解和AVO问题[博士论文].北京:石油大学,1990 Wang S X. Numerical solutions of the finite element method for the elastic wave problem and AVO problems in two-phase media [Ph. D. thesis]. Beijing: Petroleum University, 1990
[8] 徐文骏,郭建.流体饱和多孔介质中弹性波的数值模拟.地球物理学报,1994,37(增刊Ⅰ):424-433 Xu W J, Guo J. Numerical simulation of elastic waves in fluid-satuated porous media. Chinese J. Geophys. ( Acta Geophysica Sinica) ,1994 ,37(Suppl,Ⅰ ):424-433
[9] 张新明,刘克安,刘家琦.流体饱和多孔隙介质二维弹性波方程正演模拟的小波有限元法.地球物理学报,2005,48(5) ,1156-1166 Zhang X M, Liu K A, Liu J Q. A wavelet finite element method for the 2-D wave equation in fluid-saturated porous media. Chinese J. Geophys. (in Chinese), 2005,48(5) : 1156-1166
[10] 牟永光,裴正林.三维复杂介质地震数值模拟.北京:石油工业出版社,2004 Mou Y G, Pei Z L. 3-D Seismic Numerical Simulation Complex Media. Beijing:Petroleum Industry Press, 2004
[11] Dai N, Vafidis A. Wave propagation in heterogeneous,porous media: A velocity-stress, finite-difference method.Geophysics, 1995,60(2) :327-340
[12] 王秀明,张海澜.利用高阶交错网格有限差分法模拟地震波在非均匀孔隙介质中的传播.地球物理学报,2003,46(6) :842-849 Wang X M, Zhang H L. Modeling of seismic wave propagation in Hetero geneous poroelastic media using a high-order staggered finite-difference method. Chinese J.Geophys. (in Chinese) , 2003, 46(6) :842-849
[13] 裴正林.三维双相各向异性介质弹性波方程交错网格高阶有限差分法模拟.中国石油大学学报,2006,30(2) :16-20 Pei Z L. A staggered-grid high-order difference method for modeling seismic wave equation in 3-D dual-phase anisotropic media. Journal of China University of Petroleum (in Chinese), 2006 ,30(2) : 16-20
[14] 刘洋,李承楚.双相各向异性介质中弹性波传播伪谱法数值模拟研究.地震学报,2000,22(2) :132-138
[15] Komatisch D, Vilotte J P. The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological strucstures. Bull. Seism. Soc. Am. , 1 998 ,88 : 368-392
[16] Burridge R, Vargas C A. The fundamental solution in dynamic poroelasticity. Geophys. J. Royal Astron. Soc. ,1979． 58(1) :61-90
[17] Manolis G V, Beskos D E. Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity. Acta Mech. ,1989,76:89-104
[18] Bonnet G. Basic singular solutions for a poroelastic medium in the dynamic range. J. Acost. Aoc. Am. , 1987,82; 1758-1762
[19] Cheng H D. A direct boundary element method for plane strain poroelasticity. Int. J. Numer. Anal. Meth.Geomech. ,1988,12:551-572
[20] 符力耘,牟永光.弹性波边界元法正演模拟.地球物理学报,1994,37(4) :521-529 Fu L Y, Mou Y G. Boundary element method for elastic wave forward modeling. Chinese J. Geophys. (in Chinese),1994,37(4) ;521-529
[21] Predeleanu M. Reciprocal theorem in the consolidation theory of porous media. Appl. Math. Modelling,1968,8:378-382
[22] Cheng A H D, Predeleanu M. Transient boundary element formulation for poroelasticity. Appl. Math. Modelling. ,1987,11:285-290
[23] Kupradze V D, Gegelia T G. Three-dimensional problems of Mathematical Theory of Elasticity and Thermoelasticity.North-Holland, Amsterdam, 1979
[24] Nowacki W. Dynamics Problems of Thermoelasticity.Noordhoff, Leyden, the Netherlands, 1975
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