Combined paraxial-consistent boundary conditions finite element model for simulating wave propagation in elastic half-space media

In this paper, a finite element model of a soil island is coupled to both a consistent transmitting boundary and a paraxial boundary, which are then used to model the propagation of waves in semi-infinite elastic layered media. The formulation is carried out in the frequency domain while assuming plane strain conditions. It is known that a discrete model of this type, while providing excellent results for a wide range of physical parameters in the context of a half-space problem, may deteriorate rapidly at low frequencies of excitation. This is so because at low frequencies the various waves in the model eventually attain characteristic wavelengths which exceed the distance of the bottom boundary, which then causes that boundary to fail. Also, the paraxial boundaries themselves break down at very low frequencies. In this paper, this difficulty is overcome and the model׳s performance is improved upon dramatically by incorporating an artificial buffer layer sandwiched between the bottom of the soil medium and the underlying elastic half-space. Applications dealing with rigid foundations resting on homogenous or layered half-space media are shown to exhibit significant improvement. Following extensive simulations, clear guidelines are provided on the performance of the coupled model and an interpretation is given on the engineering significance of the findings. Finally, clear recommendations are provided for the practical use of the proposed modelling strategy.     

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.     

双相介质中地震波的频率-空间域数值模拟

本文利用优化的25点频率-空间域有限差分算法对基于BISQ模型双相各向同性介质中的地震波进行了数值模拟.通过与经典的Biot模型理论模拟结果进行对比,分析了Biot流动（宏观流体流动）和Squirt流动（微观流体流动）耦合作用对地震波在孔隙介质中传播特性的影响.数值模拟在地震频段进行,结果显示：在理想相界和黏滞相界情况下,Squirt流动机制都比Biot流动机制产生了更大的速度频散和能量衰减.其中,在Biot流动和Squirt流动耦合作用下的快P波的速度和振幅小于仅考虑Biot流动影响下快P波速度和振幅,而且慢P波的衰减也更加强烈.本文还研究了地震波在双层双相各向同性介质分界面处的反射和透射特征,双相介质中波的反射与透射现象类似于单相介质的情况.模拟结果表明,利用优化25点频率-空间域有限差分法模拟双相孔隙介质中的地震波场是可行的,这为开展双相孔隙介质全波形反演问题的研究提供了可能.     

含流体孔隙介质中面波的传播特性及应用

基于单相介质中地震波理论的高频面波法已广泛应用于求取浅地表S波的速度.然而水文地质条件表明, 普遍的浅地表地球介质富含孔隙.孔隙中充填的流体会显著地影响面波在浅地表的传播, 进而造成频散和衰减的变化.本文研究了地震勘探频段内针对含流体孔隙介质边界条件的面波的传播特性.孔隙流体在自由表面存在完全疏通、完全闭合以及部分疏通的情况.孔隙单一流体饱和时, 任何流体边界条件下存在R1模式波, 与弹性介质中的 Rayleigh 波类似, 相速度稍小于S波并在地震记录中显示强振幅.由于介质的内在衰减, R1在均匀半空间中也存在频散, 相速度和衰减在不同流体边界下存在差异.Biot 固流耦合系数(孔隙流体黏滞度与骨架渗透率之比)控制频散的特征频率, 高耦合系数会在地震勘探频带内明显消除这种差异.介质的迂曲度等其他物性参数对不同流体边界下的R1波的影响也有不同的敏感度.完全闭合和部分疏通流体边界下存在R2模式波, 相速度略低于慢P波.在多数条件下, 如慢P波在时频响应中难以观察到.但是在耦合系数较低时会显现, 一定条件下甚至会以非物理波形式接收R1波的辐射, 显示强振幅.浅表风化层低速带存在, 震源激发时的运动会显著影响面波的传播.对于接收点径向运动会造成面波的 Doppler 频移, 横向运动会造成面波的时频畸变.孔隙存在多相流体时, 中观尺度下不均匀斑块饱和能很好地解释体波在地震频带内的衰减.快P波受到斑块饱和显著影响, R1波与快P波有更明显关联, 与完全饱和模型中不同, 也更易于等效模型建立.频散特征频率受孔隙空间不同流体成分比例变化的控制, 为面波方法探测浅地表流体分布与迁移提供可能性.通常情况孔隙介质频散特征频率较高, 标准线性黏弹性固体可以在相对低频的地震勘探频带内等效表征孔隙介质中R1波的传播特征, 特别在时域, 可在面波成像反演建模中应用.     

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.     

Numerical modelling of wave propagation in anisotropic soil using a displacement unit-impulse-response-based formulation of the scaled boundary finite element method

An efficient method for modelling the propagation of elastic waves in unbounded domains is developed. It is applicable to soil–structure interaction problems involving scalar and vector waves, unbounded domains of arbitrary geometry and anisotropic soil. The scaled boundary finite element method is employed to derive a novel equation for the displacement unit-impulse response matrix on the soil–structure interface. The proposed method is based on a piecewise linear approximation of the first derivative of the displacement unit-impulse response matrix and on the introduction of an extrapolation parameter in order to improve the numerical stability. In combination, these two ideas allow for the choice of significantly larger time steps compared to conventional methods, and thus lead to increased efficiency. As the displacement unit-impulse response approaches zero, the convolution integral representing the force–displacement relationship can be truncated. After the truncation the computational effort only increases linearly with time. Thus, a considerable reduction of computational effort is achieved in a time domain analysis. Numerical examples demonstrate the accuracy and high efficiency of the new method for two-dimensional soil–structure interaction problems.     

基于分数阶时间导数的黏弹性衰减VTI介质中平面波传播

目前在地震勘探频带范围内通常假设品质因子Q与频率无关,且呈衰减各向同性.事实上,相比较速度各向异性,介质的衰减各向异性同样不可忽视.本文将衰减各向异性和速度各向异性二者与常Q模型相结合,建立了黏弹性衰减VTI介质模型,并基于分数阶时间导数理论,给出了对应的本构关系和波动方程.利用均匀平面波分析和Poynting定理,推导出准压缩波qP、准剪切波qSV和纯剪切波SH的复速度、相速度、能量速度以及品质因子的解析表达式.对模型的正确性进行了数值验证,并分析了qP,qSV和SH波在介质中的传播特性.数值试验结果表明：本模型能够实现理想的恒定Q行为,表现了品质因子和速度的各向异性特征,显示出黏弹性增强将导致能量速度和相速度的频散曲线变化剧烈;速度和衰减各向异性参数与传播角度之间的耦合效应对qP,qSV和SH波的速度和能量影响明显;qP,qSV和SH波的频散曲线和波前面随着衰减各向异性强度的改变发生显著变化,其中耦合在一起的qP和qSV波变化趋势相同,而SH波与它们呈现相反的变化规律.本研究为从常Q模型角度分析地震波在衰减各向异性黏弹性介质中的传播特征奠定了理论基础.

     

Train-induced wave propagation in layered soils using finite/infinite element simulation

In this paper, the transmissibility of soils for vibrations induced by trains moving at different speeds is studied. The 2.5 D finite/infinite element approach adopted herein allows us to consider the load-moving effect of the train in the direction normal to the two-dimensional profile of the soils considered, and, therefore, to obtain three-dimensional responses for the soils using only plane elements. The moving train is simulated by a sequence of moving wheel loads that may vibrate with certain frequency. Two train speeds are considered, one is smaller and the other is greater than the Rayleigh wave speed of the layered soils, to represent the effects of speed in the sub-critical and super-critical ranges. In order to evaluate the effect of each parameter on the ground response induced by moving trains, parametric studies are conducted for the following parameters: the shear wave speed, damping ratio and stratum depth of the supporting soils, and the moving speed and vibration frequency of the traveling trains. Conclusions concerning the mechanism of wave propagation in layered soils are drawn from the parametric studies, which should prove useful to practicing engineers.     

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.

     

Modelling of viscoacoustic wave propagation in transversely isotropic media using decoupled fractional Laplacians

A new wave equation is derived for modelling viscoacoustic wave propagation in transversely isotropic media under acoustic transverse isotropy approximation. The formulas expressed by fractional Laplacian operators can well model the constant-Q (i.e. frequency-independent quality factor) attenuation, anisotropic attenuation, decoupled amplitude loss and velocity dispersion behaviours. The proposed viscoacoustic anisotropic equation can keep consistent velocity and attenuation anisotropy effects with that of qP-wave in the constant-Q viscoelastic anisotropic theory. For numerical simulations, the staggered-grid pseudo-spectral method is implemented to solve the velocity–stress formulation of wave equation in the time domain. The constant fractional-order Laplacian approximation method is used to cope with spatial variable-order fractional Laplacians for efficient modelling in heterogeneous velocity and Q media. Simulation results for a homogeneous model show the decoupling of velocity dispersion and amplitude loss effects of the constant-Q equation, and illustrate the influence of anisotropic attenuation on seismic wavefields. The modelling example of a layered model illustrates the accuracy of the constant fractional-order Laplacian approximation method. Finally, the Hess vertical transversely isotropic model is used to validate the applicability of the formulation and algorithm for heterogeneous media.     

Mesh grading approach for wave propagation in high velocity-contrast media

A mesh grading approach based on investigated lump method has been presented for simulating wave propagation in high velocity-contrast media. Unstructured fine grids are used to discretize the low wave-velocity medium in order to ensure the accuracy of numerical computation, and unstructured coarse grids are used for the high wave-velocity medium in order to substantially reduce the computational cost. On the interface, one coarse grid can match the fine grids of arbitrary odd number. The key feature of the proposed method is the constructions of investigated lumps on the interfaces of media. The transition zone, which is commonly used in the discontinuous grid scheme based on the staggered-grid finite-difference method, will not be used any more. Moreover, the computational instability that the discontinuous grid schemes frequently encountered does not arise in the proposed method. The comparisons with the analytical solutions and the application in studying the effects of sedimentary basin demonstrated that the mesh grading approach is a valid, accurate, convenient and flexible algorithm in simulating wave propagations in high velocity-contrast media with irregular interfaces.     

油水两相渗流多孔介质弹性波传播动力学模型研究

弹性波在储层渗流场中的传播与衰减规律是研究波场强化采油动力学机理的重要基础.基于等效流体理论和饱和静态流体弹性波传播Biot理论,建立油水两非混相流体渗流条件下储层多孔介质中弹性波传播的动力学模型,通过算例求解与分析,发现含油水两相渗流储层多孔介质中同时存在着3种纵波P1、P2、P3和1种横波S;受频率和含水饱和度的影响,各波波速和品质因子呈现出不同变化规律,4种体波波速与频率、饱和度正相关,P1、P2波品质因子与饱和度正相关,P3和S波品质因子与饱和度负相关;最后,通过与传统静态弹性波模型结果对比,进一步分析了宏观渗流场对弹性波传播特征的影响规律,为揭示低频人工地震波辅助强化采油技术的动力学机理和工艺参数优化提供了重要理论依据.

     

流体饱和多孔隙介质弹性波方程边界元解法研究

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.     

Transient analysis of 1D inhomogeneous media by dynamic inhomogeneous finite element method

The dynamic inhomogeneous finite element method is studied for use in the transient analysis of one dimensional inhomogeneous media. The general formula of the inhomogeneous consistent mass matrix is established based on the shape function. In order to research the advantages of this method, it is compared with the general finite element method. A linear bar element is chosen for the discretization tests of material parameters with two fictitious distributions. And, a numerical example is solved to observe the differences in the results between these two methods. Some characteristics of the dynamic inhomogeneous finite element method that demonstrate its advantages are obtained through comparison with the general finite element method. It is found that the method can be used to solve elastic wave motion problems with a large element scale and a large number of iteration steps.     

Lebedev scheme for the numerical simulation of wave propagation in 3D anisotropic elasticity‡

This paper presents a Lebedev finite difference scheme on staggered grids for the numerical simulation of wave propagation in an arbitrary 3D anisotropic elastic media. The main concept of the scheme is the definition of all the components of each tensor (vector) appearing in the elastic wave equation at the corresponding grid points, i.e., all of the stresses are stored in one set of nodes while all of the velocity components are stored in another. Meanwhile, the derivatives with respect to the spatial directions are approximated to the second order on two‐point stencils. The second‐order scheme is presented for the sake of simplicity and it is easy to expand to a higher order. Another approach, widely‐known as the rotated staggered grid scheme, is based on the same concept; therefore, this paper contains a detailed comparative analysis of the two schemes. It is shown that the dispersion condition of the Lebedev scheme is less restrictive than that of the rotated staggered grid scheme, while the stability criteria lead to approximately equal time stepping for the two approaches. The main advantage of the proposed scheme is its reduced computational memory requirements. Due to a less restrictive dispersion condition and the way the media parameters are stored, the Lebedev scheme requires only one‐third to two‐thirds of the computer memory required by the rotated staggered grid scheme. At the same time, the number of floating point operations performed by the Lebedev scheme is higher than that for the rotated staggered grid scheme.     

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

本文将小波有限元法引入到流体饱和多孔隙介质二维波动方程的正演模拟中,以二维Daubechies小波的尺度函数代替多项式函数作为插值函数,构造二维张量积小波单元.引入一类特征函数解决了Daubechies小波没有显式解析表达式所带来的基函数积分值计算问题,并推导出计算分数节点上Daubechies小波函数值的递推公式,从而构造出由小波系数空间到波场位移空间的快速小波变换.数值模拟结果表明该方法是有效的.     

Modeling of dam break wave propagation in a partially ice-covered channel

During the last four decades, several numerical formulations and specialized software have been developed in response to studies about dam break (DB) wave propagation and its hydraulic and environmental impacts on downstream hydraulic structures and valleys. These methods cannot, however, be used to predict wave propagation within partially covered channels or reservoirs located upstream of hydraulic structures. In fact, such problems require the modelling of the complex transition from a free surface flow into a pressurized one. Because rivers or channels partially covered with ice sheets are typical examples commonly met in winter in northern climates, it is vitally important to assess ice-cover effects on the DB wave propagation and develop appropriate tools to predict resulting hydrodynamic loads on hydraulic structures downstream. This paper proposes an original numerical formulation to model wave propagation and hydrodynamic pressure in partially covered channels. The proposed formulation uses one-dimensional St. Venant equations to simulate open-water flow and water hammer equations to simulate pressure flow within the partially covered channel. To illustrate the use of the hydrodynamic pressures obtained, a case study is presented where a channel cover and a dam located downstream are modelled using finite elements to investigate their dynamic structural response.     

Multi-physics modeling of flow and transport in porous media using a downscaling approach

Multi-phase flow and transport processes generally occur on different spatial and temporal scales. Very often also, within a physical system, they vary in space meaning that different kinds of processes might take place in different parts of the system. In order to account for the variety of processes and to take their scale-dependence into account, the development of multi-scale multi-physics techniques can be envisaged.     

基于非均匀化多尺度方法的自组织介质波前愈合效应波场模拟

随机介质表征的地球介质自组织性,体现了地球内部复杂介质的统计性特征,对理解地球内部构造和动力学演化有重要的意义.波前愈合效应是自组织介质散射效应的体现,会导致高频近似射线理论的计算走时和真实波场到时有一定的差异.为了研究射线理论在自组织介质中的适应性范围,本文选取高斯型和指数型自相关函数来描述自组织介质,采用非均匀化多尺度方法进行大尺度地球模型的波场模拟.利用互相关方法求取背景速度场与附加自组织介质速度场之间的波场走时差,并与由射线理论得到的走时差进行比较.结果表明,非均匀化多尺度方法在节省计算时间的同时,又可保持计算精度.介质相关长度越小、波长越长且传播距离越远时,波前愈合效应越强.当相关长度a、波长λ以及传播距离L之间满足a/(λL)1/2≤0.5时,波前愈合效应显著,且随着比值减小两者差异增大,波前愈合效应在增加,在该范围内射线理论计算走时的误差较大.