包含Biot机制和分数阶黏弹性机制的波传播模型

含流体孔隙介质中的波能量耗散通常由多种力学机制造成.传统Biot理论中的能量耗散仅仅考虑了固流两相相对运动引起的摩擦耗散,无法准确预测波在孔隙介质中低频段出现的高频散与强衰减现象.为了建立一个能准确预测地震波频段高频散与强衰减现象的动力学模型,我们在Biot理论的基础上引入黏弹性机制,并利用分数阶导数刻画黏弹性本构关系,最终获得了一种新的孔隙介质波传播模型.与传统的Biot模型相比,新模型考虑了含流体孔隙介质中固体骨架的内耗散,对波能量耗散的刻画更为精准.通过数值算例,我们研究了分数阶导数的阶数参数对快P波和S波频散和衰减的影响,并通过来自不同地区且具有不同物理性质的几组流体饱和岩芯实验数据,对比研究了新模型的有效性.结果表明,文章提出的新模型能更准确地预测快P波和S波在低频段出现的高频散和强衰减现象.     

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.

     

Seismic wave propagation in viscoelastic and saturated rock

Considering engineering practice, the viscoelastic two-phase model is adopted, seismic wave propagation in saturated rock is studied. Not only the effect of the viscosity of rock skeleton but also the effect of ground water on the propagation of the seismic wave can be considered by this model, the propagation characteristics of seismic wave in saturated rock can be understood comprehensively and the model is more reasonable than other model by which seismic wave propagation is studied. The effect of frequency, water content and viscosity constant on the wave velocity and attenuation are studied by numerical examples and some valuable conclusions are drawn.     

Seismic wave propagation in viscoelastic and saturated rock

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Biot流动和喷射流动耦合作用下波传播的FCT紧致差分模拟

本文从含流体多孔隙介质中同时包含Biot流动和喷射流动两种力学机制的BISQ (Biot-Squirt)方程出发, 利用FCT (Flux-Corrected Transport)紧致差分方法数值模拟了在Biot流和喷射流共同作用下的波在含流体多孔隙各向同性介质中的传播. 通过与仅受Biot流动作用下的波场结果对比, 我们研究了耗散系数b和慢P波速度以及能量衰减之间的关系. 同时, 本文也研究了波在双层双相孔隙介质中的传播规律. 数值结果表明, 由于喷射流动的影响, Biot流动和喷射流动共同作用下的快P波传播速度比仅受Biot流动影响下的快P波传播速度慢、慢P波衰减得更快, 表明具有局部特征的喷射流动对压缩波的衰减和频散具有重要影响;而对于剪切波, 两种模型得到的剪切波速几乎一致, 说明在各向同性情况下喷射流对剪切波的影响不明显. 这一结果与波衰减和频散的理论分析结果一致, 同时本文结果也表明FCT紧致差分方法能有效压制粗网格条件下模拟弹性波场所引起的数值频散以及源噪声, 适合于双相孔隙介质中的弹性波场模拟.     

适用于更广渗透率范围含泥质砂岩的广义粘弹性BISQ模型

To simultaneously take into account the Biot-flow mechanism, the squirt-flow mechanism, and the frame-viscoelasticity mechanism, a generalized viscoelastic BISQ （Biot/squirt） model is developed for wave propagation in clay-bearing sandstones based on Dvorkin＇s elastic BISQ model. The present model is extended to a wide range of permeabilities （k 〉 0.05 mD） by introducing a dimensionless correction factor for viscoelastic parameters, defined as a function of the permeability and the clay content. We describe the frame＇s stress-strain relationship of the clay-bearing sandstones by the differential constitutive equations of generalized viscoelasticity and then derive the viscoelastic-wave dynamic equations. With the assumption of a plane-wave solution, we finally yield the phase velocities and the attenuation coefficients by solving the dynamic wave equations in the frequency and wave number domain. The comparison of numerical results and experimental data shows that the generalized viscoelastic BISQ model is applicable for modeling the wave propagation in most of the sandstones mainly bearing kaolinite clay.     

Acoustic wave propagation in saturated porous media: reformulation of the Biot/Squirt flow theory

A model of wave propagation in fluid-saturated porous media is developed where the principal fluid/solid interaction mode affecting the propagation of the acoustic wave results from the conjunction of the Biot and the Squirt flow mechanism. The difference between the original Biot/Squirt (BISQ) flow theory and the new theory, which we call the reformulated BISQ, is that the average fluid pressure term appearing in the dynamic equation for a two component solid/fluid continuum is independent of squirt flow length. P-velocity and attenuation relate to measurable rock physical parameters: the Biot's poroelastic constants, porosity, permeability, pore fluid compressibility and viscosity. Modelling shows that velocity and attenuation dispersion obtained using the reformulated BISQ theory are of the same order of magnitude as those obtained using the original BISQ theory. Investigation on permeability effect on velocity and attenuation dispersion indicate that the transition zone in velocity and attenuation peak, occurring both at the relaxation frequency, shifts toward high frequency when permeability decreases. This behaviour agrees with Biot's theory prediction.     

Complex rays applied to wave propagation in a viscoelastic medium

In order to trace a ray between known source and receiver locations in a perfectly elastic medium, the take-off angle must be determined, or equialently, the ray parameter. In a viscoelastic medium, the initial value of a second angle, the attenuation angle (the angle between the normal to the plane wavefront and the direction of maximum attenuation), must also be determined. There seems to be no agreement in the literature as to how this should be done. In computing anelastic synthetic seismograms, some authors have simply chosen arbitrary numerical values for the initial attenuation angle, resulting in different raypaths for different choices. There exists, however, a procedure in which the arbitrariness is not present, i.e., in which the raypath is uniquely determined. It consists of computing the value of the anelastic ray parameter for which the phase function is stationary (Fermat's principle). This unique value of the ray parameter gives unique values for the take-off and attenuation angles. The coordinates of points on these stationary raypaths are complex numbers. Such rays are known as complex rays. They have been used to study electromagnetic wave propagation in lossy media. However, ray-synthetic seismograms can be computed by this procedure without concern for the details of complex raypath coordinates. To clarify the nature of complex rays, we study two examples involving a ray passing through a vertically inhomogeneous medium. In the first example, the medium consists of a sequence of discrete homogeneous layers. We find that the coordinates of points on the ray are generally complex (other than the source and receiver points which are usually assumed to lie in real space), except for a ray which is symmetric about an axis down its center, in which case the center point of the ray lies in real space. In the second example, the velocity varies continuously and linearly with depth. We show that, in geneneral, the turning point of the ray lies in complex space (unlike the symmetric ray in the discrete layer case), except if the ratio of the velocity gradient to the complex frequency-dependent velocity at the surface is a real number. We also present a numerical example which demonstrates that the differences between parameters, such as arrival time and raypath angles, for the stationary ray and for rays computed by the above-mentioned arbitrary approaches can be substantial.     

A two-dimensional horizontal wave propagation and mud mass transport model

The present study offers a two-dimensional horizontal wave propagation and morphodynamic model for muddy coasts. The model can be applied on a general three-dimensional bathymetry of a soft muddy coast to calculate wave damping, fluid mud mass transport and resulting bathymetry change under wave actions. The wave propagation model is based on time-dependent mild slope equations including the wave energy dissipation due to the wave-mud interaction of bottom mud layers as well as the combined effects of the wave refraction, diffraction and breaking. The constitutive equations of the visco-elastic–plastic model are adopted for the rheological behavior of fluid mud. The mass transport velocity within the fluid mud layer is calculated combining the Stokes’ drift, the mean Eulerian velocity and the gravity-driven mud flow. The results of the numerical model are compared against a series of conducted wave basin experiments, wave flume experiments and field observations. Comparisons between the computed results with both the field and laboratory data reveal the capability of the proposed model to predict the wave transformation and mud mass transport.     

Seismic analysis of structures with a fractional derivative model of viscoelastic dampers

Viscoelastic dampers are now among some of the preferred energy dissipation devices used for passive seismic response control. To evaluate the performance of structures installed with viscoelastic dampers, different analytical models have been used to characterize their dynamic force deformation characteristics. The fractional derivative models have received favorable attention as they can capture the frequency dependence of the material stiffness and damping properties observed in the tests very well. However, accurate analytical procedures are needed to calculate the response of structures with such damper models. This paper presents a modal analysis approach, similar to that used for the analysis of linear systems, for solving the equations of motion with fractional derivative terms for arbitrary forcing functions such as those caused by earthquake induced ground motions. The uncoupled modal equations still have fractional derivatives, but can be solved by numerical or analytical procedures. Both numerical and analytical procedures are formulated. These procedures are then used to calculate the dynamic response of a multi-degree of freedom shear beam structure excited by ground motions. Numerical results demonstrating the response reducing effect of viscoelastic dampers are also presented.     

Seismic wave propagation in a viscoelastic porous solid saturated by viscous liquid

A general solution is deduced of the differential equations describing the propagation of elastic waves in a dissipative liquid-filled viscoelastic porous solid. The velocities of three existing waves have been expressed in convenient form using the moduli of the solid phase and by introducing the frequency-dependent equivalent mass densities. The solution is then used to examine some of the phenomena which arise when each of the three-body waves, in turn, are incident on a traction-free plane boundary. Analytic expressions for the reflection coefficients are obtained. Numerical calculations have been made, for a particular model, in case of incidentP _I wave. Effect of viscoelasticity and viscosity on the reflection coefficients has also been exhibited.     

用传输函数构建的大气重力波传播理论模式

本文根据考虑大气热传导和黏滞的重力波复色散关系,采用传输函数的概念,基于重力波的线性理论,构建了用于研究对流层内重力波激发源与电离层响应之间的传输函数数值模式.在相空间中讨论了传输函数振幅的分布特性,并以地面单位脉冲源为例,分析了从地面到300 km高空的响应,得到了物理量的时空分布特征.结果表明：（1）对内重力波的传播而言,大气相当于一个滤波器,只有波动周期在15～30 min,水平波长在200～450 km之间的重力波扰动才最容易到达300 km电离层高度;（2）电离层的响应主要在与地面的激发源之间相隔较远的水平距离上发生;（3）黏滞和热传导系数在低层对上传重力波的影响较小,随着高度的增加它们对重力波的影响越来越大;（4）在低层计算的波动频率与Row理论的计算结果比较一致,然而到了高层却相差较大.     

A thin-layer interface model for wave propagation through filled rock joints

The present study essentially employs a thin-layer interface model for filled rock joints to analyze wave propagation across the jointed rock masses. The thin-layer interface model treats the rough-surfaced joint and the filling material as a continuum medium with a finite thickness. The filling medium is sandwiched between the adjacent rock materials. By back analysis, the relation between the normal stress and the closure of the filled joint are derived, where the effect of joint deformation process on the wave propagation through the joint is analyzed. Analytical solutions and laboratory tests are compared to evaluate the validity of the thin-layer interface model for filled rock joints with linear and nonlinear mechanical properties. The advantages and the disadvantages of the present approach are also discussed.     

弹性波在含双裂纹岩体中的传播分析

岩石和岩体是具有复杂细微观结构的非均匀介质,弹性波在岩体中传播时,与岩体细微观缺陷相互作用表现出弹性波的频散效应.为研究岩体内部细观结构对弹性波频散效应的作用,本文采用双裂纹模型:在模型内部,考虑裂纹间的相互作用对弹性波的影响,以分析弹性波在双裂纹体系间的多次散射作用;在双裂纹体系间,采用线性叠加分析法,以考虑岩体缺陷影响的局部化.对波动方程应用Green函数基本解,利用边界积分方法,将双裂纹体系作为内边界处理,得到相应的频散方程,由此对比分析了双裂纹体系在上述两种分析方法下的区别,进一步探讨了双裂纹体系参数、孔隙流体压力和卸荷对岩体频散特性的影响.     

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.     

基于临界孔隙度模型的地震波传播

基于岩石物里学中临界孔隙度模型,建立一种简洁的均匀弹性流体饱和孔隙介质模型,进行地震波传播研究.首先定义了构建目标模型的基本力学模型:介绍了全孔隙度区间内基本力学模型和目标孔隙介质的含义,其中基本力学模型除了完全弹性固体模型S和完全弹性流体模型F还包括临界孔隙模型C.然后通过等效力学模型推出了目标力学模型介质本构关系的组分表达形式.文中分别通过直接求取弹性参数的表达形式和运用应力应变关系两种方法得到介质模型的本构关系,进而得到该模型波动方程的组分表达形式.最后对这种介质模型进行了地震波传播的数值模拟,结合模拟结果分析孔隙对地震波传播的影响.     

近地表强衰减介质中的地震波传播模型

建立符合油气储层近地表复杂介质强衰减性质的数学-物理模型是油气勘探和开发的重要课题.本文针对D'Alembert黏弹性介质模型存在的量纲不统一和不能充分刻画孔隙介质结构的不足,通过引入孔隙度和渗透率等参数修改耗散项,改进了原D'Alembert模型,获得了能精细刻画具有强吸收衰减特征的近地表复杂介质模型,即改进的近地表黏弹性模型.基于这种改进的模型,推导了波频散和衰减的表达式,并研究了孔隙度、流体黏度等物理参数对波频散和衰减的影响,获得了相应的规律性认识.为了验证新模型预测近地表介质中波衰减的有效性,本文将新模型应用于胜利油田YX工区的近地表实测数据,同时与弹性Biot模型和BISQ模型以及黏弹性BISQ模型进行了比较.结果表明,与其它三个模型比较,改进的近地表黏弹性模型能够很好地刻画近地表介质的强衰减性,而且新模型所涉及到的物理参数明显少于其它几种模型,有利于新模型在油气储层近地表复杂介质地震勘探的实际应用.

     

砾岩储层地震波传播方程:三重孔隙结构模型

针对砾岩储层的砂、砾、泥三重孔隙结构特征,本文分析砾岩孔隙区域、砂岩孔隙区域以及泥岩孔隙区域相互之间的孔隙流体流动机制,将静态的砾岩骨架本构方程与动态的孔隙流体运动方程联立,提出了复杂砾岩储层的弹性波传播理论方程.采用实测砾岩储层参数,在算例中与双重孔隙介质理论进行对比分析,验证了本文理论方程的合理性;基于三重孔隙介质模型,分析不同储层环境下纵波的传播特征,结果显示：随流体黏滞系数增大,在衰减-频率轴坐标系中,砾与砂、砂与泥孔隙区域间局域流导致的两个衰减峰向低频端移动,而Biot全局流导致的衰减峰向高频端移动;嵌入体尺寸及背景相介质渗透率的变化,主要影响纵波速度频散曲线沿频率轴左、右平移,不影响波速低频、高频极限幅值;嵌入体含量及孔隙度的变化改变了岩石干骨架的弹性、密度参数,不仅影响速度频散曲线沿频率轴平移,而且影响其上、下限幅值;砾包砂包泥三重孔隙介质模型所预测的衰减曲线中,低频段“第一个衰减峰”主要由砾岩孔隙区域与砂岩孔隙区域之间的局域流导致,中间频段“第二个衰减峰”主要由砂岩孔隙区域与泥岩孔隙区域之间的局域流导致,超声频段“第三个衰减峰”由Biot全局流导致.对慢纵波传播特征的分析显示,砂岩骨架（局部孔隙度较大）内部的宏观孔隙流体流动造成的耗散明显强于砾岩与泥岩骨架.

     

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.