垂向不均匀介质中波传播特点的研究

本文由波传播问题的变分原理导出了垂向不均匀介质中的弹性波波动方程及纵波和横波波动方程。在不同的假设条件下得到了纵、横波波动方程的几种简化形式,其中包括地震勘探中常用的变系数波动方程。利用有限单元法,求得了垂向不均匀介质中波动方程的数值解。通过数值结果对纵、横波之间的耦合程度以及介质的不均匀性对波的衰减作用进行了讨论。     

Acquisition and processing of pure and converted shear waves generated by compressional wave sources

The most common source of seismic energy onshore is a vertical impact on the earth's surface or an explosion at some depth in a borehole. These sources produce mainly compressional waves. Here it is shown how these sources can be used to generate shear waves either by conversion in the depth or in the immediate vicinity of the source itself. The use of theoretical seismograms can help to identify the individual onsets especially on the horizontal components of the earth's movement. Due to the more complex raypath, converted waves need a special data processing. The spectral behaviour can be improved by spectral balancing followed by a spiking deconvolution. As the CDP-concept is no longer applicable for converted PS-waves a special sorting for a common conversion point (CCP) is applied. The identification and analysis of the individual waves can be simplified by a detailed polarization analysis taking into account the full dynamic behaviour of the observed waves. Prestack depthmigration of converted PS waves allows to deduce additional information on the material properties of reflecting horizons. The depth-migration of individual shot gathers is achieved in the frequency-space domain. Thus the kinematic and dynamic aspects of these secondary waves is a valuable tool for a better understanding of the elastic behaviour of the subsurface.     

Axisymmetric soil–structure interaction by global–local finite elements

A version of the global–local finite element method is presented for studying dynamic steady-state soil–structure interaction wherein the soil medium extends to infinity. Herein, only axisymmetric behaviour is considered. In this approach, conventional finite elements are used to model the structure and some portion of the surrounding soil medium considered to be homogeneous and isotropic. A complete set of outgoing waves in the form of spherical harmonics for the entire space is used to represent the behaviour in the half-space beyond the finite element mesh and these are termed the global functions. Full traction and displacement continuity is enforced at the finite element mesh interface with the outer region. On the free surface of the half-space in the outer field, traction-free surface conditions are enforced by demanding that a sequence of integrals of the weighted-average tractions must vanish. Numerical examples are presented for the response of different shaped foundations, resting on the free surface or at various submerged levels, due to a normal seismic plane compressional wave. Plots of differential scattering cross-sections show the angular distribution of the energy (its directional nature) of the scattered field.     

Harmonic wave diffraction by two circular cavities in a poroelastic formation

This study investigates the dynamic interaction of time harmonic plane waves with a pair of parallel circular cylindrical cavities of infinite length buried in a boundless porous elastic fluid-saturated medium. The novel features of Biot dynamic theory of poroelasticity along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorems for cylindrical wave functions are employed to develop a closed-form solution in the form of infinite series. The analytical results are illustrated with numerical examples in which two empty cavities are insonified by a fast compressional or a shear wave at end-on incidence. The basic dynamic field quantities such as the hoop stress amplitude and the radial displacement of the elastic frame are evaluated and discussed for representative values of the parameters characterizing the system. The effects of the proximity of the two cavities, the incident wave frequency and type are examined. Particular attention has been focused on multiple scattering interactions in addition to the slow wave coupling effects which is known to be the primary distinction of the scattering phenomenon in poroelasticity from the classical elastic case. Limiting case involving two empty cylindrical cavities in an elastic solid is considered and excellent agreement with a well-known solution is established.     

Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under seismic excitation

An exact analysis for two-dimensional dynamic interaction of monochromatic progressive plane compressional and shear seismic waves with a permeable circular tunnel lining of circumferentially varying wall thickness buried in a boundless porous elastic fluid-saturated formation is presented. The novel features of Biot dynamic theory of poroelasticity in conjunction with the translational addition theorems for cylindrical wave functions, along with the appropriate wave field expansions and the pertinent boundary conditions are employed to develop a closed-form solution in form of infinite series. The analytical results are illustrated with numerical examples in which an air-filled and water-saturated permeable tunnel lining of variable wall thickness, embedded within water-saturated surrounding formations of distinct frame properties (soft, stiff, and stiff viscoelastic soil), is insonified by fast compressional or shear waves at selected angles of incidence. The effects of liner eccentricity, interface permeability, formation material type, incident wave frequency, and angle of incidence on the hoop stress amplitude are evaluated and discussed for representative values of the parameters characterizing the system. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.     

DIFFRACTION BY A WEDGE IN AN ACOUSTIC CONSTANT DENSITY MEDIUM1

A new method is presented for solving the 2D problem of diffraction of a plane wave by a wedge of arbitrary angle in a purely acoustic, constant-density medium with different constant compressional wave speeds inside and outside the wedge. The diffraction problem is formulated as integral equations, and a wavenumber–frequency representation of the scattered field is obtained. With the aid of the Cagniard–de Hoop method, exact analytical expressions in the space–time domain are obtained for the different wave constituents, i.e. geometric optical scattered waves and edge diffracted waves including head waves. These expressions can be computed to any degree of accuracy within reasonable computation times on a computer, and the semi-analytical method of solution presented thus constitutes a means of constructing reference solutions for wedge configurations. Such highly accurate reference solutions are of importance for verification of results that include diffraction phenomena modelled by general numerical approximate methods, e.g. finite differences, finite elements and spectral methods. Examples of such applications of the method of solution are given.     

声波测井中的纵波和横波

充流体井中声传播理论是声波测井的理论基础。Blot（1952）绘出了充流体井中简正模式和斯通利波的频散曲线,White等（1968）首先计算了合成全波列波形,Peterson（1974）和余寿绵（1984）导出了简正模式和斯通利波的振幅表达式。余寿绵还预言纵波和横波是一种共振现象,为沿轴向传播存在特征频率的不衰减波。但是,现行理论认为纵     

Diffraction of a compressional wave by a rigid barrier in a liquid layer — Part II

Summary The problem of diffraction of compressional waves striking obliquely at a rigid barrier fixed in the surface of a liquid layer lying over a solid halfspace has been studied. Exact values for the amplitudes of the waves reflected in the solid halfspace have been obtained. The method of solution is based on the Wiener-Hopf technique.     

REFLECTION OF ELASTIC WAVES FROM PERIODICALLY STRATIFIED MEDIA WITH INTERFACIAL SLIP*

A periodically stratified elastic medium can be replaced by an equivalent homogeneous transverse isotropic medium in the long wavelength limit. The case of a homogeneous medium with equally spaced parallel interfaces along which there is imperfect bonding is a special instance of such a medium. Slowness surfaces are derived for all plane wave modes through the equivalent medium and reflection coefficients for a half-space of such a medium are found. The slowness surface for the SH mode is an ellipsoid. The exact solution for the reflection of SH-waves from a half-space with parallel slip interfaces is found following the matrix method of K. Gilbert applied to elastic waves. Explicit results are derived and in the long wavelength limit, shown to approach the results for waves in the equivalent homogeneous medium. Under certain conditions, a half-space of a medium with parallel slip interfaces has a reflection coefficient independent of the angle of incidence and thus acts like an acoustic reducing mirror. The method for the reflection of P- and SV-waves is fully outlined, and reflection coefficients are shown for a particular example. The solution requires finding the eigenvalues of a 4 × 4 transfer matrix, each eigenvalue being associated with a particular wave. At higher frequencies, unexpected eigenvalues are found corresponding to refracted waves for which shear and compressional parameters are completely coupled. The two eigenvalues corresponding to the transmitted wavefield give amplitude decay perpendicular to the stratification along with up- and downgoing phase propagation in some other direction. Much of this work was performed while the author was at the Department of Geophysics and Planetary Sciences, Tel-Aviv University, Ramat-Aviv, Israel. The author is grateful for illuminating discussions with K. Helbig and K. Gilbert.     

Scattering of shear waves by an elliptical cavity in a radially inhomogeneous isotropic medium

Complex function and general conformal mapping methods are used to investigate the scattering of elastic shear waves by an elliptical cylindrical cavity in a radially inhomogeneous medium. The conformal mappings are introduced to solve scattering by an arbitrary cavity for the Helmholtz equation with variable coefficient through the transformed standard Helmholtz equation with a circular cavity. The medium density depends on the distance from the origin with a power-law variation and the shear elastic modulus is constant. The complex-value displacements and stresses of the inhomogeneous medium are explicitly obtained and the distributions of the dynamic stress for the case of an elliptical cavity are discussed. The accuracy of the present approach is verified by comparing the present solution results with the available published data. Numerical results demonstrate that the wave number, inhomogeneous parameters and different values of aspect ratio have significant influence on the dynamic stress concentration factors around the elliptical cavity.     

两种不同势函数下半空间饱和多孔介质中Rayleigh波求解比较

分别对"考虑两种压缩波和幅值比例系数"和"考虑一种压缩波(P1或P2波)但不考虑幅值比例系数"两种不同势函数下的半空间饱和多孔介质中Rayleigh波求解进行详细推导,理论分析表明"考虑两种压缩波和幅值比例系数"下Rayleigh波求解推导更为严密,与饱和多孔介质中存在两种压缩波的事实相一致。在研究半空间饱和多孔介质中Rayleigh波时应采用"考虑两种压缩波和幅值比例系数"的势函数。     

Infinite elements for elastodynamics

An axisymmetric infinite element and a three-dimensional infinite element are developed to solve three-dimensional elastic wave propagation problems in unbounded media. The elements are capable of transmitting Rayleigh, shear and compressional waves in the frequency domain. A scheme to integrate numerically the characteristic matrices of the elements is formulated based upon Gauss—Laguerre quadrature. Finally, the axisymmetric infinite element is used to find the compliance functions of a rigid circular plate subjected to harmonic loading on a semi-infinite medium. By using infinite elements, the size of the near field may be kept small. Consequently, the system is characterized by relatively few degrees of freedom, thus providing the analyst with an inexpensive solution.     

SECONDARY SHEAR WAVES FROM SOURCE BOREHOLES1

Previous studies of radiation from point sources in fluid-filled boreholes have most often been based on far-field, stationary phase analysis. In these papers, the explicit contribution of the borehole itself acting as a waveguide has not been properly considered, with a few exceptions. In general, these studies accurately describe S-wave radiation in high-velocity rocks such as granites and limestones and P-wave radiation in most rocks, and experiments have confirmed this. However, tube waves directly influence the external wavefield and in fact create a shear-wave ‘wake’ outside the borehole due to constructive interference of tube-wave emission if a velocity condition is met. This constructive interference or wake is generated when the tube-wave velocity is greater than the shear-wave velocity. When this happens, a tube-wave complex pole invalidates the mathematical assumptions for stationary phase analysis and the stationary phase predictions do not agree with experimentally derived radiation patterns. Shales at shallow depths and other soft sediments characteristically have tube-wave velocities greater than shear-wave velocities. Because the tube-wave is of relatively high amplitude compared to body waves generated directly by the source, these secondary shear waves can be the highest amplitude arrivals on receiver arrays. The shape and properties of these secondary shear waves are calculated and shown to have identical properties to Mach waves of aerodynamics and seismology. For instance, these waves are geometrically conical and the aperture of the cone and the moveout velocity can be calculated. This paper also demonstrates the important effect that casing has on the Mach waves and provides predictions about when these waves are likely to be observed. Finally, evidence of Mach waves in data sets is examined and it is shown how these waves have been confused with receiver borehole tube waves. It is possible, though rare, that the tube-wave velocity of the borehole is greater than the compressional-wave velocity of the surrounding medium. In this case secondary compressional or compressional Mach waves would be generated although this problem is not addressed here.     

基于Kelvin-Voigt黏弹性骨架的含非饱和流体孔隙介质BISQ模型(英文)

本文综合考虑了在波传播过程中孔隙介质的三种重要力学机制——"Biot流动机制一squirt流动机制-固体骨架黏弹性机制",借鉴等效介质思想,将含水饱和度引入波动力学控制方程,并考虑了不同波频率下孔隙流体分布模式对其等效体积模量的影响,给出了能处理含粘滞性非饱和流体孔隙介质中波传播问题的黏弹性Biot/squirt(BISQ)模型。推导了时间-空间域的波动力学方程组,由一组平面谐波解假设,给出频率-波数域黏弹性BISQ模型的相速度和衰减系数表达式。基于数值算例分析了含水饱和度、渗透率与频率对纵波速度和衰减的影响,并结合致密砂岩和碳酸盐岩的实测数据,对非饱和情况下的储层纵波速度进行了外推,碳酸盐岩储层中纵波速度对含气饱和度的敏感性明显低于砂岩储层。     

声波在两种多孔介质界面上的反射和透射

本文导出了声波在两种多孔介质界面上反射、透射的一般计算公式.作为例子,数值计算了P₁波入射于界面时,P₁、P₂和S波的反射、透射系数与声波频率、入射角等量之间的关系.结果表明,各种模式波的反射、透射系数与入射角、多孔介质性质有关,在Biot特征频率附近与频率有关,并用界面两侧的法向能流相等验证了结果的正确性.若把多孔介质当作均匀固体处理,将会得到显著不同的结果.     

Propagation of seismic waves in patchy-saturated porous media: double-porosity representation

Propagation of harmonic plane waves is studied in a patchy-saturated porous medium. Patchy distribution of the two immiscible fluids is considered in a porous frame with uniform skeletal properties. A composition of two types of patches, connected through continuous paths, constitutes a double-porosity medium. Different compressibilities of pore-fluids in two porous phases facilitate the wave-induced fluid-flow in this composite material. Constitutive relations are considered with frequency-dependent complex elastic coefficients, which define the dissipative behaviour of porous aggregate due to the flow of viscous fluid in connected patches. Relevant equations of motion are solved to explain the propagation of three compressional waves and one shear wave in patchy-saturated porous solids. A numerical example is solved to illustrate dispersion in phase velocity and quality factor of attenuated waves in patchy-saturated porous materials. Role of fluid–solid inertial coupling in Darcy's law is emphasized to keep a check on the dispersion of wave velocities in the porous composite. Effects of patchy saturation on phase velocities and quality factors of attenuation are analysed using the double-porosity formulation as well as the reduced single-porosity equivalents.     

Diffraction of a compressional wave by a rigid barrier in a liquid layer — Part I

Summary The problem of diffraction of compressional waves incident obliquely on a barrier of finite height fixed at a boundary between a liquid layer and a solid halfspace has been studied. The problem has been solved by employing the Wiener-Hopf technique. The displacement potential in the liquid layer has been obtained in terms of Fourier integrals, whose evalutation along an appropriate contour yields the transmitted, reflected and diffracted waves. Approximate values of the amplitudes of the waves reflected in the solid medium have also been found.     

Using refracted shear waves for velocity estimation

The most difficult part of multicomponent processing is the estimation of the shear-wave velocity map for migration. We used refracted shear waves and a simple iterative method called wavefield continuation (WFC) to evaluate the shallow shear-wave velocity profile on a real data example. The WFC was developed in 1981 by Clayton and McMechan to determine compressional-wave velocity profiles from refracted compressional waves. The application to refracted shear waves is straightforward. The real data example shows that shear structure can be easily determined independently of the compressional structure.     

Dynamic response of 3-D rigid surface foundations by time domain boundary element method

The dynamic response of three-dimensional rigid surface foundations of arbitrary shape is numerically obtained. The foundations are placed on a linear elastic, isotropic and homogeneous half-space representing the soil medium and are subjected to either external dynamic forces or seismic waves of various kinds and directions, with a general transient time variation. The problem is formulated in the time domain by the boundary element method and the response is obtained by a time step-by-step integration. Two examples dealing with three-dimensional rectangular foundations are presented in detail, together with comparisons with other methods, in order to document the accuracy of the method. The main advantages of the proposed method are that, unlike frequency domain techniques, it provides directly the transient response and forms the basis for extension to the case of non-linear behaviour.