黏弹性层状多孔介质中地震波传播方程的传递矩阵解法
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摘要
基于Biot理论,饱和多孔介质中波的传播是一个耦合问题。由于此模型的复杂性,难以给出高效的数值解法。为此本文提出了地震波在层状饱和多孔黏弹性介质中传播的一种实用求解方法,即在所研究的层状三维模型中,对时间域用拉氏变换,对空间域用傅氏变换,将原问题变换为具有六个独立变量的常微分方程组,每个变量只是深度坐标和一个水平坐标的函数,进而得到关于每一层中波的传播问题的传递矩阵,再用数值方法求出传递矩阵的特征值和特征向量。此算法的核心是首先求出在变换后的空间中的波场,然后用FFT方法进行傅氏和拉氏反变换得到在时间域的波场。文中给出了算法的详细推导和数值算例。
Based on Biot theory, the wave propagation in saturated porous media is coupling issue. Because of the complexity of the model, it's difficult to give high-efficient numeric solution. For that reason, the paper gave a practical solution that the seismic wave propagates in viscoelastic saturated layered and porous media, which is to use Laplace transform in time domain and Fourier transform in space domain for studied 3-D layered media and to transform the original issue into ordinary differential equation set with six independent variables, and each variable is function only having depth coordinate and one horizontal coordinate, and the transmission matrix of wave propagation issue in each layer can be further got, using numeric approach to compute the eigenvalues and the eigenvectors of transmission matrix. The kernel of the algorithm is first to get the wavefield in post-transform space, and then to use FFT to carry out inverse Fourier and Laplace transforms and to get wavefield in time domain. The paper gave the detailed deduce of algorithm and numeric cases.
Based on Biot theory, the wave propagation in saturated porous media is coupling issue. Because of the complexity of the model, it's difficult to give high-efficient numeric solution. For that reason, the paper gave a practical solution that the seismic wave propagates in viscoelastic saturated layered and porous media, which is to use Laplace transform in time domain and Fourier transform in space domain for studied 3-D layered media and to transform the original issue into ordinary differential equation set with six independent variables, and each variable is function only having depth coordinate and one horizontal coordinate, and the transmission matrix of wave propagation issue in each layer can be further got, using numeric approach to compute the eigenvalues and the eigenvectors of transmission matrix. The kernel of the algorithm is first to get the wavefield in post-transform space, and then to use FFT to carry out inverse Fourier and Laplace transforms and to get wavefield in time domain. The paper gave the detailed deduce of algorithm and numeric cases.
引文
[1]Zhang W and Li X.Semi-analytic axisymmetric pressure distribution in a consolidating porous medi-um.Transport in Porous Media,1997,29:179~189
[2]Zhang Wand Li X.Opti mumsource frequency in vi-bration production technology.Petroleum Science&Engineering,1998,21(3):239~248
[3]Biot M A.Theory of propagation of elastic waves in saturated porous solid,II,High-frequency range.J Acoust Soc Am,1956,28:179~191
[4]Biot M A.Mechanics of deformation and acoustic propagation in porous media.J Appl Phys,1962,33(4):1482~1498
[5]Dai N,Vafidis Aand Kanasewich E R.Wave propa-gation in heterogeneous,porous media:A velocity stress,finite-difference method.Geophysics,1995,60(2):327~340
[6]Gajo A,Saetta A and Vitaliani R.Silent boundary conditions for wave propagation in saturated porous media.Int J of Numer and anal methods in geo-mech,1996,20:253~273
[7]Zienkwicz O C,Chang C T and Bettess P.Drained consolidating and dynamic behavior assumptionsin soils.G啨otechnique,1980,30(4):385~395
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[9]Prevost J H.Wave propagation in fluid-saturated porous media:An efficient finite element procedure.Soil dynamics and earthquake engineering,1985,4(4):
[10]李晓江,张文飞.三维波动方程的一维化有限元法.石油地球物理勘探,1996,31(1):21~27
[2]Zhang Wand Li X.Opti mumsource frequency in vi-bration production technology.Petroleum Science&Engineering,1998,21(3):239~248
[3]Biot M A.Theory of propagation of elastic waves in saturated porous solid,II,High-frequency range.J Acoust Soc Am,1956,28:179~191
[4]Biot M A.Mechanics of deformation and acoustic propagation in porous media.J Appl Phys,1962,33(4):1482~1498
[5]Dai N,Vafidis Aand Kanasewich E R.Wave propa-gation in heterogeneous,porous media:A velocity stress,finite-difference method.Geophysics,1995,60(2):327~340
[6]Gajo A,Saetta A and Vitaliani R.Silent boundary conditions for wave propagation in saturated porous media.Int J of Numer and anal methods in geo-mech,1996,20:253~273
[7]Zienkwicz O C,Chang C T and Bettess P.Drained consolidating and dynamic behavior assumptionsin soils.G啨otechnique,1980,30(4):385~395
[8]Prevost J H.Nonlinear transient phenomena in satu-rated porous media.Comput methods in appl mech and eng,1982,20:3~18
[9]Prevost J H.Wave propagation in fluid-saturated porous media:An efficient finite element procedure.Soil dynamics and earthquake engineering,1985,4(4):
[10]李晓江,张文飞.三维波动方程的一维化有限元法.石油地球物理勘探,1996,31(1):21~27
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