Finite element equations and numerical simulation of elastic wave propagation in two—phase anisotropic media

Based on Biot theory of two-phase anisotropic media and Hamilton theory about dynamic problem,finite element equations of elastic wave propagation in two-phase anisotopic media are derived in this paper.Numerical solution of finite element equations is given.Finally,properties of elastic wave propagation are observed and analyzed through FEM modeling.     

井眼条件下弹性波传播问题的三维有限差分数值模拟

推导了二阶改进Higdon吸收边界条件．利用空间上具有四阶精度、时间上具有二阶精度的中心差分方法。数值模拟了几种复杂井眼条件下的波场．给出了正交各向异性介质地层条件和椭圆井眼条件下波场模拟的结果．考查并分析了单极子源和偶极子源条件下的波场特性，所得结论与弹性波传播理论一致．通过计算表明，利用改进的Higdon吸收边界条件，声波测井三维模拟程序可用于复杂井眼各向异性地层条件的波场传播模拟．     

三维各向异性裂缝介质正演模拟的三种交错网格适应性比较及Lebedev方法的改进

研究了三维各向异性裂缝介质的正演模拟的三种不同有限差分法,即标准交错网格（SSG）、旋转交错网格（RSG）和Lebedev方法（LS）,详细对比分析这三种交错网格方法在模拟复杂各向异性介质的优势与局限.提出一种新的改进方法,简化了LS有限差分法在任意各向异性介质中的正演模拟.为了模拟三维大规模复杂各向异性介质的地震响应,提出一个优化的正演模拟计算流程：将模型参数分为模型介质参数和模型构造参数.该计算流程适用于三种有限差分法中的任何一种.使用LS方法实现任意三维各向异性裂缝介质地震响应的三维全波场模拟,通过使用三种不同的有限差分模拟方法进行二维和三维模型数值模拟试验,验证了所提出方法有效.

     

一种模拟非均匀介质中弹性波传播新的k-space方法

传统的伪谱（PS）方法,采用傅里叶变换（FT）计算空间导数具有很高的精度,每个波长仅需要两个采样点,而时间导数采用有限差分（FD）近似因而精度较低.当采用大时间步长时,由于时空精度不平衡,PS法存在不稳定性问题.原始的k-space方法可以有效地克服这些问题但是却无法适用于非均匀介质.为了提高原始k-space方法模拟非均匀介质波动方程的精度,我们提出了一种新的k-space算子族.它是用非均匀介质的变速度代替原k-space算子中的常数补偿速度构造得到,引入低秩近似可以高效求解.我们将构造的新的k-space算子应用于耦合的二阶位移波动方程,而不是交错网格一阶速度应力波动方程,使模拟弹性波的计算存储量减少.我们从数学上证明了基于二阶波动方程的k-space方法与基于一阶波动方程的k-space方法是等价的.数值模拟实验表明,与传统的PS、交错网格PS和原始的k-space方法相比,我们的新方法可以在时间和空间步长较大的均匀和非均匀介质中,为弹性波的传播提供更精确的数值解.在保持稳定性和精度的同时,采用较大的时空采样间隔,可以大大降低数值模拟的计算成本.

     

Pseudo‐spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media

Staggering grid is a very effective way to reduce the Nyquist errors and to suppress the non‐causal ringing artefacts in the pseudo‐spectral solution of first‐order elastic wave equations. However, the straightforward use of a staggered‐grid pseudo‐spectral method is problematic for simulating wave propagation when the anisotropy level is greater than orthorhombic or when the anisotropic symmetries are not aligned with the computational grids. Inspired by the idea of rotated staggered‐grid finite‐difference method, we propose a modified pseudo‐spectral method for wave propagation in arbitrary anisotropic media. Compared with an existing remedy of staggered‐grid pseudo‐spectral method based on stiffness matrix decomposition and a possible alternative using the Lebedev grids, the rotated staggered‐grid‐based pseudo‐spectral method possesses the best balance between the mitigation of artefacts and efficiency. A 2D example on a transversely isotropic model with tilted symmetry axis verifies its effectiveness to suppress the ringing artefacts. Two 3D examples of increasing anisotropy levels demonstrate that the rotated staggered‐grid‐based pseudo‐spectral method can successfully simulate complex wavefields in such anisotropic formations.     

Perfectly matched layer boundary conditions for frequency-domain acoustic wave simulation in the mesh-free discretization

Mesh-free discretization, flexibly distributing nodes without computationally expensive meshing process, is able to deal with staircase problem, oversampling and undersampling problems and saves plenty of nodes through distributing nodes suitably with respect to irregular boundaries and model parameters. However, the time-domain mesh-free discretization usually exhibits poorer stability than that in regular grid discretization. In order to reach unconditional stability and easy implementation in parallel computing, we develop the frequency-domain finite-difference method in a mesh-free discretization, incorporated with two perfectly matched layer boundary conditions. Furthermore, to maintain the flexibility of mesh-free discretization, the nodes are still irregularly distributed in the absorbing zone, which complicates the situation of artificial boundary reflections. In this paper, we implement frequency-domain acoustic wave modelling in a mesh-free system. First, we present the perfectly matched layer boundary condition to suppress spurious reflections. Moreover, we develop the complex frequency shifted–perfectly matched layer boundary condition to improve the attenuation of grazing waves. In addition, we employ the radial-basis-function-generated finite difference method in the mesh-free discretization to calculate spatial derivatives. The numerical experiment on a rectangle homogeneous model shows the effectiveness of the perfectly matched layer boundary condition and the complex frequency shifted–perfectly matched layer boundary condition, and the latter one is better than the former one when absorbing large angle incident waves. The experiment on the Marmousi model suggests that the complex frequency shifted–perfectly matched layer boundary condition works well for complicated models.     

A viscous-spring transmitting boundary for cylindrical wave propagation in saturated poroelastic media

Based on the u–U formulation of Biot equation and the assumption of zero permeability coefficient, a viscous-spring transmitting boundary which is frequency independent is derived to simulate the cylindrical elastic wave propagation in unbounded saturated porous media. By this viscous-spring boundary the effective stress and pore fluid pressure on the truncated boundary of the numerical model are replaced by a set of spring, dashpot and mass elements, and its simplified form is also given. A u–U formulation FEA program is compiled and the proposed transmitting boundaries are incorporated therein. Numerical examples show that the proposed viscous-spring boundary and its simplified form can provide accurate results for cylindrical elastic wave propagation problems with low or intermediate values of permeability or frequency content. For general two dimensional wave propagation problems, spuriously reflected waves can be greatly suppressed and acceptable accuracy can still be achieved by placing the simplified boundary at relatively large distance from the wave source.     

An explicit method for numerical simulation of wave equations: 3D wave motion

In this paper, an explicit method is generalized from 1D and 2D models to a 3D model for numerical simulation of wave motion, and the corresponding recursion formulas are developed for 3D irregular grids. For uniform cubic grids, the approach used to establish stable formulas with 2M-order accuracy is discussed in detail, with M being a positive integer, and is illustrated by establishing second order (M=1) recursion formulas. The theoretical results presented in this paper are demonstrated through numerical testing.     

二维各向异性介质中地震波场的高阶同位网格有限差分模拟

本文将DRP/opt MacCormack有限差分格式用于模拟二维各向异性介质中的地震波传播.DRP/opt MacCormack是一种同位网格下的差分格式,避免了传统的交错网格在计算各向异性问题时由于变量插值而导致的误差.而且相对于低阶同位网格差分格式,它具有低色散、低耗散的优点.此格式将中心差分算子分成前向和后向两个空间单边差分,然后在4-6步Runge-Kutta时间积分中使用单边差分组合.在具有垂直对称轴的横向各向同性（VTI）模型下,通过对比DRP/opt MacCormack有限差分和谱元方法的模拟结果,验证了前者具有很高的精度和稳定性.由于实际地质条件下TI介质的对称轴通常是倾斜的(TTI),本文在二维三分量框架下模拟TTI介质中的地震波场.结果显示横波分裂和切平面/反平面运动耦合的特征.数值实验表明DRP/opt MacCormack是一种有效的研究各向异性介质中地震波传播规律的差分格式.     

一种优化的频率域三维声波有限差分模拟方法

为提高频率域有限差分（FD,finite-difference）正演模拟技术的计算精度和效率,基于旋转坐标系统的优化差分格式被广泛应用,但是只应用于正方形网格的情况.基于平均导数法（ADM）的优化差分格式,应用于正方形和长方形网格模拟.这些频率域有限差分算子,各自具有不同的差分格式和对应的优化系数求解表达式.本文基于三维声波方程发展了一种新的优化方法,只要给定FD模板形式,可直接构造频散方程,求取FD模板上各节点的优化系数.此方法的优点在于频率域FD算子的优化系数对应各个节点,可扩展优化其他格式.运用此优化方法,计算得到了不同空间采样间距比情况下27点和7点格式的优化系数.数值实验表明,优化27点格式与ADM 27点格式具有相同的精度,优化7点格式比经典的7点格式具有更小的数值频散.

     

Transient mechanical wave propagation in semi-infinite porous media using a finite element approach

In this study, we propose a numerical investigation in the time domain of the mechanical wave propagation of an impulsional load on semi-infinite soil. The ground is modelled as a porous saturated viscoelastic medium involving complete Biot theory. All the couplings and a hysteretic Rayleigh damping are taken into consideration. An accurate and efficient finite element method using a matrix-free technique and an expert multigrid system are applied. Our results present the displacements of the fluid and solid particles over the surface and in depth. The arrival times of body and surface waves are studied. Particularly, the compressional wave of the second kind is highlighted. The influence of the different couplings and more specifically, the influence of the permeability on the response of the soil are analyzed.     

基于BISQ模型的三维双相各向异性介质数值模拟

Biot-flow and squirt-flow are the two most important fluid flow mechanisms in porous media containing fluids. Based on the BISQ （Biot-Squirt） model where the two mechanisms are treated simultaneously, the elastic wave-field simulation in the porous medium is limited to two-dimensions and two-components （2D2C） or two-dimensions and three-components （2D3C）. There is no previous report on wave simulation in three- dimensions and three-components. Only through three dimensional numerical simulations can we have an overall understanding of wave field coupling relations and the spatial distribution characteristics between the solid and fluid phases in the dual-phase anisotropic medium. In this paper, based on the BISQ equation, we present elastic wave propagation in a three dimensional dual-phase anisotropic medium simulated by the staggered-grid high-order finite-difference method. We analyze the resulting wave fields and show that the results are an improvement.     

A staggered-grid high-order finite-difference modeling for elastic wave field in arbitrary tilt anisotropic media

Introduction The real Earth usually presents anisotropy. Therefore, it is of theoretical and practical sig- nificance for many fields as oil and gas, seismic exploration and production, earthquake prediction, detection of deep structure and so on to study on seismic wave theory, numerical simulation method and its applications in the anisotropic media (Crampin, 1981, 1984; Crampin et al, 1986; Hudson et al, 1996; Liu et al, 1997; Thomsen, 1986, 1995; TENG et al, 1992; HE and ZHANG, 1996)…     

Extended reflectivity method for modelling the propagation of diffusive–viscous wave in dip‐layered media

The reflectivity method plays an important role in seismic modelling. It has been used to model different types of waves propagating in elastic and anelastic media. The diffusive–viscous wave equation was proposed to investigate the relationship between frequency dependence of reflections and fluid saturation. It is also used to describe the attenuation property of seismic wave in a fluid‐saturated medium. The attenuation of diffusive–viscous wave is mainly characterised by the effective attenuation parameters in the equation. Thus, it is essential to obtain those parameters and further characterise the features of the diffusive–viscous wave. In this work, we use inversion method to obtain the effective attenuation parameters through quality factor to investigate the characteristics of diffusive–viscous wave by comparing with those of the viscoacoustic wave. Then, the reflection/transmission coefficients in a dip plane‐layered medium are studied through coordinate transform and plane‐wave theory. Consequently, the reflectivity method is extended to compute seismograms of diffusive–viscous wave in a dip plane multi‐layered medium. Finally, we present two models to simulate the propagation of diffusive–viscous wave in a dip plane multi‐layered medium by comparing the results with those in a viscoacoustic medium. The numerical results demonstrate the validity of our extension of reflectivity method to the diffusive–viscous medium. The numerical examples in both time domain and time–frequency domain show that the reflections from a dip plane interface have significant phase shift and amplitude change compared with the results of horizontal plane interface due to the differences in reflection/transmission coefficients. Moreover, the modelling results show strong attenuation and phase shift in the diffusive–viscous wave compared to those of the viscoacoustic wave.     

频散介质中地质雷达波传播的数值模拟

地质雷达所探测的地球介质常常具有频散性．为了研究地质雷达在频散介质中的探测能力,提出了频散介质中时间域有限差分法计算麦克斯韦方程的方法,给出了满足Debye关系的频散介质中的电位移和磁场的迭代算法,以及由电位移计算电场的算法．只有在电场计算时才用到介质的物性参数．提出一种新的吸收边界条件的算法,通过增加假想的介电常数和磁导率,实现了吸收层中波的无反射衰减,克服了以往Berenger完全匹配层计算时对场进行分裂带来的麻烦,从而提高了计算效率．计算实例表明,频散介质中电磁波的衰减更快,测量信号变得很弱．     

An alternate algorithm of numerical simulation in anisotropic media

A possible problem of anisotropic medium at the internal interface for finite-difference method is investigated in this paper. Firstly, we show a finite-difference scheme for the 2-dimensional wave equations with 15 elastic parameters. Secondly, we mainly analyse the stress-continuity conditions for the finite-difference scheme at discontinuities in the 2-D transversely isotropic medium and suggest an alternate simulation technique. Finally, the seismic wavefield of theoretical model in transversely isotropic medium is simulated. The numerical computation shows that both computational results and theoretical analyse for the model are the same and that the algorithm is feasible and satisfactory. This study is supported by the National Natural Science Foundation of China.     

弱形式时域完美匹配层——滞弹性近场波动数值模拟

本文旨在构建适用于滞弹性近场时域波动有限元模拟的高精度人工边界条件：完美匹配层（Perfectly Matched Layer：PML）,其中阻尼介质时域本构基于广义标准线性体建立.与以往研究不同,本文采用复坐标延拓技术变换弱形式波动方程构建了可直接用有限元离散的弱形式时域PML,规避以往独立对无限域内波动方程及界面条件进行延拓可导致的PML场方程和界面条件匹配不合理引发数值失稳、计算精度低下等问题.其次,针对PML中多极点有理分式与频域函数乘积的傅里叶反变换难以计算的问题,利用PML精度对复坐标延拓函数中延拓参数微调不敏感这一特点,明确给出了参数微调准则以规避多重极点,进而利用有理分式分解给出了一种普适、简便的计算方法,极大地简化了PML计算.基于该方法可实现任意高阶PML.最后,将本文构建滞弹性PML与高阶勒让德谱元（高精度集中质量有限元）结合得到滞弹性近场波动谱元离散方案.基于算例验证了滞弹性PML的计算效率、精度及新离散方案的长持时稳定特性.新离散方案可应用于计入实际介质阻尼的地震波动正、反问题数值模拟,提高波形模拟的精度以及地下波速结构反演的精度和可靠性.

     

黏弹TTI介质中旋转交错网格高阶有限差分数值模拟

以Carcione黏弹各向异性理论为基础,给出了适用于黏弹性具有任意倾斜对称轴横向各向同性介质(黏弹TTI介质)的二维三分量一阶速度-应力方程,采用旋转交错网格任意偶数阶精度有限差分格式求解该方程,并推导出了二维黏弹TTI介质完全匹配层(PML)吸收边界条件公式和相应的旋转交错网格任意偶数阶精度有限差分格式,实现了该类介质的地震波场数值模拟.数值模拟结果表明:该方法模拟精度高,边界吸收效果好,可以得到高精度的波场快照和合成记录;并且波场快照和合成记录能较好地反映地下介质的各向异性特征和黏弹性特征.     

Application of the nearly perfectly matched layer for seismic wave propagation in 2D homogeneous isotropic media

Numerical modelling plays an important role in helping us understand the characteristics of seismic wave propagation. The presence of spurious reflections from the boundaries of the truncated computational domain is a prominent problem in finite difference computations. The nearly perfectly matched layer has been proven to be a very effective boundary condition to absorb outgoing waves in both electromagnetic and acoustic media. In this paper, the nearly perfectly matched layer technique is applied to elastic isotropic media to further test the method's absorbing ability. The staggered‐grid finite‐difference method (fourth‐order accuracy in space and second‐order accuracy in time) is used in the numerical simulation of seismic wave propagation in 2D Cartesian coordinates. In the numerical tests, numerical comparisons between the nearly perfectly matched layer and the convolutional perfectly matched layer, which is considered the best absorbing layer boundary condition, is also provided. Three numerical experiments demonstrate that the nearly perfectly matched layer has a similar performance to the convolutional perfectly matched layer and can be a valuable alternative to other absorbing layer boundary conditions.     

基于第二代小波自适应网格的二维声波方程波传模拟

本文利用第二代小波多尺度分解和快速变换的特点,构造自适应计算网格.对初始计算网格上的数值解进行第二代小波变换,得到数值解对应的小波系数空间.小波系数的大小表示相邻网格上数值变化率,小波系数大的区域网格点上的数值解变化梯度大.当小波系数大于等于预设的阈值时,在小波系数对应的网格点周围插入新的计算网格点,通过阈值可以实现网格的细化,得到多尺度下层层嵌套的细化自适应网格;由有限差分法得到相应网格点的空间导数.比较数值算例得到的波场快照和计算时间,验证了该方法的有效性.