Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

Existing and commonly used in industry nowadays, closed‐form approximations for a P‐wave reflection coefficient in transversely isotropic media are restricted to cases of a vertical and a horizontal transverse isotropy. However, field observations confirm the widespread presence of rock beds and fracture sets tilted with respect to a reflection boundary. These situations can be described by means of the transverse isotropy with an arbitrary orientation of the symmetry axis, known as tilted transversely isotropic media. In order to study the influence of the anisotropy parameters and the orientation of the symmetry axis on P‐wave reflection amplitudes, a linearised 3D P‐wave reflection coefficient at a planar weak‐contrast interface separating two weakly anisotropic tilted tranversely isotropic half‐spaces is derived. The approximation is a function of the incidence phase angle, the anisotropy parameters, and symmetry axes tilt and azimuth angles in both media above and below the interface. The expression takes the form of the well‐known amplitude‐versus‐offset “Shuey‐type” equation and confirms that the influence of the tilt and the azimuth of the symmetry axis on the P‐wave reflection coefficient even for a weakly anisotropic medium is strong and cannot be neglected. There are no assumptions made on the symmetry‐axis orientation angles in both half‐spaces above and below the interface. The proposed approximation can be used for inversion for the model parameters, including the orientation of the symmetry axes. Obtained amplitude‐versus‐offset attributes converge to well‐known approximations for vertical and horizontal transverse isotropic media derived by Rüger in corresponding limits. Comparison with numerical solution demonstrates good accuracy.     

Converted PS-wave Reflection Coefficients in Weakly Anisotropic Media

—?I derive converted P S-wave reflection coefficients at a planar weak-contrast interface separating two weakly anisotropic half-spaces using first-order perturbation theory. The general expressions are further specified for the interface separating any of the two following media: isotropic, transversely isotropic with a vertical symmetry axis (VTI), transversely isotropic with a horizontal symmetry axis (HTI) and orthorhombic. Relatively simple forms of small-angle reflection coefficients are also obtained. The coefficients are expressed as functions of Thomsen-type medium parameters and incidence and azimuthal phase angles. Derived expressions, as well as their application, are more complicated than the corresponding expressions for P P-wave reflection coefficients. General characteristics and pitfalls are discussed. Numerical tests reveal a good agreement between exact and approximate coefficients for most models presented.     

Introduction to seismic travel time methods in anisotropic media

Summary Section 1 (and 11) develops the concepts of the front velocity, the front gradient, the travel time in space and on seismometric profiles, the profile velocity and the profile gradient in connection with the propagation of the fronts of elastic waves in solid isotropic and anisotropic media. The sectional velocity and the sectional gradient are defined in terms of the motion of the curve of intersection of a front with a fixed surface. Section 2 (and 12) relates the coefficients of elasticity of the medium, the front types, and their respective rays. In section 12, the theory of fronts of arbitrary shape and of the corresponding rays for any anisotropic, homogeneous or inhomogeneous solid medium is summarized. In section 3 (and 13), the law of reflection and refraction of fronts on surfaces of discontinuity of arbitrary shape is presented. Sections 4 to 6 (and 14 to 16) treat some elementary applications of seismic travel time methods to homogeneous, uniaxially anisotropic media (=transverse isotropy) in greater detail. In section 4 (and 14), the travel time of a direct front generated by a point source is considered and it is shown how the coefficients of elasticity of the medium can be found based on travel time measurements. The seismic prospection of a plane reflector and of a reflecting boundary of arbitrary shape and position are discussed in section 5 (and 15). In section 6 (and 16), the seismic refraction method is used to locate a plane boundary between a homogeneous, uniaxially anisotropic and a homogeneous isotropic medium, where the boundary is perpendicular or at an arbitrary angle to the direction of anisotropy.     

横向各向同性地层井孔伪瑞利波、弯曲波、螺旋波的低频极限速度

在多极源声测井中, 低频弯曲波或螺旋波被广泛应用于测量地层的横波速度. 前人的研究已证明在各向同性地层中井孔伪瑞利波、弯曲波及螺旋波的低频极限速度都等于地层横波速度. 大量的数值计算结果似乎表明此结论在横向各向同性（TI）地层情况下也能成立, 但缺乏理论证明. 本文在井孔平行于TI弹性地层对称轴的模型下, 考察了井内声源激发的波在流-固边界上的反射和透射情况, 阐述了非泄漏模式导波产生的必要条件并讨论了其速度上限值. 我们发现在各向异性参数满足一定条件的TI地层中,导波速度的低频极限值小于沿井孔方向上的横波速度. 通过对井孔导波的频散分析以及对时域全波列的数值模拟, 论证了在这类地层中进行多极源声测井时,不可能根据弯曲波或螺旋波的到达时间准确获取地层的横波速度值.     

Azimuth moveout correction for transversely isotropic media

The azimuth moveout (AMO) operator in homogeneous transversely isotropic media with a vertical symmetry axis (VTI), as in isotropic media, has an overall skewed saddle shape. However, the AMO operator in anisotropic media is complicated; it includes, among other things, triplications at low angles. Even in weaker anisotropies, with the anisotropy parameter η= 0.1 (10% anisotropy), the AMO operator is considerably different from the isotropic operator, although free of triplications. The structure of the operator in VTI media (positive η) is stretched (has a wider aperture) compared with operators in isotropic media, with the amount of stretch being dependent on the strength of anisotropy. If the medium is both vertically inhomogeneous, i.e. the vertical velocity is a function of depth (v(z)), and anisotropic, which is a common combination in practical problems, the shape of the operator again differs from that for isotropic media. However, the difference in the AMO operator between the homogeneous and the v(z) cases, even for anisotropic media, is small. Stated simply, anisotropy influences the shape and aperture of the AMO operator far more than vertical inhomogeneity does.     

Lithology-dependent seismic anisotropic amplitude variation with offset correction in transversely isotropic media

Analysis of amplitude variation with offset is an essential step for reservoir characterization. For an accurate reservoir characterization, the amplitude obtained with an isotropic assumption of the reservoir must be corrected for the anisotropic effects. The objective is seismic anisotropic amplitude correction in an effective medium, and, to this end, values and signs of anisotropic parameter differences (Δδ and Δε) across the reflection interfaces are needed. These parameters can be identified by seismic and well log data. A new technique for anisotropic amplitude correction was developed to modify amplitude changes in seismic data in transversely isotropic media with a vertical axis of symmetry. The results show that characteristics of pre-stack seismic data, that is, amplitude variation with offset gradient, can be potentially related to the sign of anisotropic parameter differences (Δδ and Δε) between two layers of the reflection boundary. The proposed methodology is designed to attain a proper fit between modelled and observed amplitude variation with offset responses, after anisotropic correction, for all possible lithofacies at the reservoir boundary. We first estimate anisotropic parameters, that is, δ and ε, away from the wells through Backus averaging of elastic properties resulted from the first pass of isotropic pre-stack seismic inversion, on input data with no amplitude correction. Next, we estimate the anisotropic parameter differences at reflection interfaces (values and signs of Δδ and Δε). We then generate seismic angle gather data after anisotropic amplitude correction using Rüger's equation for the P-P reflection coefficient. The second pass of isotropic pre-stack seismic inversion is then performed on the amplitude-corrected data, and elastic properties are estimated. Final outcome demonstrates how introduced methodology helps to reduce the uncertainty of elastic property prediction. Pre-stack seismic inversion on amplitude-corrected seismic data results in more accurate elastic property prediction than what can be obtained from non-corrected data. Moreover, a new anisotropy attribute (ν) is presented for improvement of lithology identification.     

THE INSENSITIVITY OF REFLECTED SH WAVES TO ANISOTROPY IN AN UNDERLYING LAYERED MEDIUM1

Propagation in the plane of mirror symmetry of a monoclinic medium, with displacement normal to the plane, is the most general circumstance in anisotropic media for which pure shear-wave propagation can occur at all angles. Because the pure shear mode is uncoupled from the other two modes, its slowness surface in the plane is an ellipse. When the mirror symmetry plane is vertical the pure shear waves in this plane are SH waves and the elliptical SH sheet of the slowness surface is, in general, tilted with respect to the vertical axis. Consider a half-space of such a monoclinic medium, called medium M, overlain by a half-space of isotropic medium I with plane SH waves incident on medium M propagating in the vertical symmetry plane of M. Contrary to the appearance of a lack of symmetry about the vertical axis due to the tilt of the SH-wave slowness ellipse, the reflection and transmission coefficients are symmetrical functions of the angle of incidence, and further, there exists an isotropic medium E with uniquely determined density and shear speed which gives exactly the same reflection and transmission coefficients underlying medium J as does monoclinic medium M. This means that the underlying monoclinic medium M can be replaced by isotropic medium E without changing the reflection and transmission coefficients for all values of the angle of incidence. Thus no set of SH seismic experiments performed in the isotropic medium in the symmetry plane of the underlying half-space can reveal anything about the monoclinic anisotropy of that underlying half-space. Moreover, even when the underlying monoclinic half-space is stratified, there exists a stratified isotropic half-space that gives the identical reflection coefficient as the stratified monoclinic half-space for all angles of incidence and all frequencies.     

Separation of diffracted waves in transversely isotropic media

Imaging diffracted waves can provide useful information about complex subsurface geology and fracture networks. Separation of diffractions from typically more intensive reflected events can be done based on specularity, which measures deviation from Snell’s law. Here, we analyze two formulations of specularity and their applicability to diffraction processing in the presence of anisotropy. We show that the most common definition of specularity, originally introduced for pure modes in isotropic media, remains valid for both pure and converted waves in arbitrarily anisotropic models. The other formulation operates directly with the difference between the slowness projections onto the reflector for the incident and reflected waves. Testing on a VTI (transversely isotropic with a vertical symmetry axis) diffraction ramp model demonstrates that both formulations produce satisfactory results for anisotropic media with appropriate tapering of the specularity gathers. Then separation and imaging of diffractions is performed for the structurally complex VTI Marmousi model. We also analyze the sensitivity of diffractions in the specularity gathers to errors in the symmetry-direction velocity and anellipticity parameter η.     

Reflection and transmission of plane SH-waves in an initially stressed inhomogeneous anisotropic magnetoelastic medium

This study investigates the reflection and transmission of plane SH-waves in two semi-infinite anisotropic magnetoelastic media. The lower half-space is considered as initially stressed and inhomogeneous. The density of lower half-space is taken exponentially varying with depth. The solutions for half-spaces are obtained analytically. The expressions for reflection and transmission coefficient are obtained in the closed form subject to continuity conditions at the interfaces of anisotropic magnetoelastic half-spaces and the Snell’s law. It is found that these coefficients depend on the initial stress, inhomogeneity parameter, the magnetoelastic coupling parameter, and the angle at which wave crosses the magnetic field of the half-spaces. Numerical computations are performed for these coefficients for a specific model of two different anisotropic magnetoelastic half-spaces. The numerical results are illustrated by the graph of reflection and transmission coefficient versus the angle of incidence. In general, as the initial stress increases the reflection and transmission coefficient increases, the affect is more prominent for more than 10 GPa. Inhomogeneity in the density of the material also increases the reflection and transmission coefficient. The anisotropic magnetoelastic parameter and the angle at which the wave crosses the magnetic field for both the half-spaces have a quite significant effect on the reflection and transmission coefficient.     

水平单轴应力与横波反射系数

地球物理勘探中,地下应力的估测是非常重要的.本文利用应力诱导的弱各向异性的各向异性参数,借助已有的具有水平对称轴横向各向同性介质顾及高阶小量的横波反射系数近似公式,针对下介质为各向同性受单轴应力作用情况随应力的增加求取应力精度降低的问题,从横波反射系数近似公式的推导出发,联合SH波与SV波反射系数给出新的改进的单轴应力与横波反射系数的直观表达式;计算并对比了受单轴应力作用的各向同性介质的横波近似和严格的反射系数,并利用理论得到的横波反射系数对应力进行了估值.结果表明,利用改进后横波反射系数与应力的直观表达式,当应力增加时,应力估值精度有了明显的改善,这为利用横波反射数据直观估测应力提供理论依据.     

各向异性弹性波场的有限元数值模拟

研究了各向异性弹性波有限元正演系统的精度和效率问题，提出了一种透射加衰减的组合人工边界方案（吸收边办条件），它对各向异性波具有较好的吸收效果，并且有较好的稳定性能，均匀TI介质中的模拟获得了非常清晰的波场快照，其波场特征与理论分析能够准确吻合，各向异性介质模型的地表地震记录表明，各向异性波炮集记录在波的类型、同相轴形态、能量分布和相位等方面与各向同性波都有很大差别。     

THE APPLICATION OF PML BOUNDARY CODITIONS IN THE SIMULATION OF SEISMIC WAVE BY THE HIGH-ORDER STAGGERED-GRID FINITE DIFFERENCES AT THE TRANSVERSELY ISOTROPIC MEDIA

In the realm of the numerical simulation, finite difference method and finite element method are more intuitive and effective than other simulation methods. In the process of simulating seismic wave propagation, the finite differences method is widely used because of its high computational efficiency and the advantage of the algorithm is more efficient. With the demand of precision, more and more researchers have proposed more effective methods of finite differences, such as the high-order staggered-grid finite differences method, which can restore the actual process of wave propagation on the premise of ensuring accuracy and improving the efficiency of operation. In the past numerical simulation of seismic wave field, different models of isotropic medium are mostly used, but it is difficult to reflect the true layer situation. With the research demand of natural seismology and seismic exploration, the research on anisotropic media is more and more extensive. Transversely isotropic(TI)media can well simulate the seismic wave propagation in the formation medium, such as gas-bearing sandstone, mudstone, shale et al., the character of TI media is reflected by introducing the Thomsen parameters to reflect its weak anisotropy of vertical direction by using Thomson parameter. Therefore, studying the process of seismic wave propagation in TI media can restore the true information of the formation to the greatest extent, and provide a more reliable simulation basis for the numerical simulation of seismic wave propagation. In the geodynamic simulation and the numerical simulation of the seismic wave field, under the limited influence of the calculation area, if no boundary conditions are added, a strong artificial boundary reflection will be generated, which greatly reduces the validity of the simulation. In order to minimize the influence of model boundaries on the reflection of seismic waves, it is often necessary to introduce absorbing boundary conditions. At present, there are three types of absorption boundary conditions: one-way wave absorption boundary, attenuation absorption boundary, and perfectly matched layer(PML)absorption boundary. In terms of numerical simulation of seismic waves, the boundary absorption effect of PML is stronger than the first two, which is currently the most commonly used method, and it also represents the cutting-edge development direction of absorption boundary technology. The perfectly matched layer absorbing boundary is effectively applied to eliminating the reflective waves from model boundaries, but for transversely isotropic medium, the effect of the absorbing is not very well. For this reason, the elastic dynamic wave equations in transversely isotropic media are derived, and we describe a second-order accurate time, tenth-order accurate space, formulation of the Madariaga-Virieux staggered-grid finite difference methods with the perfectly matched layer(PML)are given. In addition, we have established vertical transversely isotropic(VTI)media and arbitrary inclined tilted transversely isotropic(TTI)media models, using a uniform half-space velocity model and a two-layer velocity model, respectively. By combining the actual geoscience background, we set the corresponding parameters and simulation conditions in order to make our model more research-oriented. When setting model parameters, different PML thickness, incident angle, source frequency and velocity layer models were transformed to verify the inhibition of boundary reflection effect by PML absorption boundary layer. The implementations of this simulation show that the formula is correct and for the transversely isotropic(TI)media of any angular symmetry axis, when the thickness of the PML layer reaches a certain value, the seismic wave reflection effect generated by the artificial boundary can be well suppressed, and the absorption effect of PML is not subject to changes in incident angle and wave frequency. Therefore, the results of our study indicate that our research method can be used to simulate the propagation process of seismic waves in the transversely isotropic(TI)media without being affected by the reflected waves at the model boundary to restore the actual formation information and more valuable geological research.     

TI介质局部角度域射线追踪与叠前深度偏移成像

研究与实践表明,对于长偏移距、宽方位地震数据,忽略各向异性会明显降低成像质量,影响储层预测与描述的精度.针对典型的横向各向同性(TI)介质,本文面向深度域构造成像与偏移速度分析的需要,研究基于射线理论的局部角度域叠前深度偏移成像方法.它除了像传统Kirchhoff叠前深度偏移那样输出成像剖面和炮检距域的共成像点道集,还遵循地震波在成像点处的局部方向特征、基于扩展的脉冲响应叠加原理获得入射角度域和照明角度域的成像结果.为了方便快捷地实现TI介质射线走时与局部角度信息的计算,文中讨论和对比了两种改进的射线追踪方法:一种采用从经典各向异性介质射线方程演变而来的由相速度表征的简便形式;另一种采用由对称轴垂直的TI(即VTI)介质声学近似qP波波动方程推导出来的射线方程.文中通过坐标旋转将其扩展到了对称轴倾斜的TI(即TTI)介质.国际上通用的理论模型合成数据偏移试验表明,本文方法既适用于复杂构造成像,又可为TI介质深度域偏移速度分析与模型建立提供高效的偏移引擎.     

黏弹各向异性介质中波的反射与透射问题分析

黏弹各向异性介质中传播不均匀波,其反射、透射模式不仅与介质分界面两侧速度对比有关,还与品质因子Q的对比有关. 用伪谱技术模拟黏弹各向异性介质分界面上波的反射、透射,并与弹性各向异性介质、黏弹各向同性介质和弹性各向同性介质的模拟结果做比较. 计算平面波的反射、透射系数,分析介质的黏弹性和各向异性对反射、透射系数的影响. 数值模拟了一个三层介质模型中的波场,分析两个分界面上产生的反射波的特征. 黏弹各向异性介质中,qS波比qP波衰减程度大.     

Determining maximum shear stress in confined substrate from elastic wave reflection coefficient

The relationship between the maximum shear stress in a substrate solid and the elastic wave reflection coefficient from the interface between the substrate solid and an overlying solid half-space is investigated. Both substrate and overlying solid media are assumed to be initially isotropic and stress-free. Then as the substrate is subjected to horizontal confined stresses it becomes anisotropic. It is shown that longitudinal and shear wave reflection coefficients are related to the degree of stress induced anisotropy in the substrate medium. From this relation the confined stress level and the maximum shear stress generated on the vertical planes of the substrate are estimated. Authors in their previous investigation computed plane wave reflection coefficient in a biaxially compressed solid substrate immersed in a fluid. This paper reports for the first time how the maximum shear stress in a biaxially compressed substrate medium can be measured from the plane wave reflection coefficients when the overlying medium is also a solid half-space.     

Anomalous Behaviour of Seismic Waves at Interfaces between Anisotropic Media

An account of possible anomalous effects in reflection and refraction of elastic waves at an interface between anisotropic media is presented. These effects are due to anisotropy and they cannot occur at an interface between isotropic media. The shape of the slowness surface (its local deviations from spherical symmetry) is the decisive factor for appearance of these effects. A numerical example of such anomalous behaviour of elastic waves at a free boundary of the crystal of spinel is presented.     

Anomalous behaviour of seismic waves at an interface between anisotropic media

Summary An account of possible anomalous effects in reflection and refraction of elastic waves at an interface between anisotropic media is presented. These effects are due to anisotropy and they cannot occur at an interface between isotropic media. The shape of the slowness surface (its local deviations from spherical symmetry) is the decisive factor for appearance of these effects. A numerical example of such anomalous behaviour of elastic waves at a free boundary of the crystal of spinel is presented.     

由VSP反射波旅行时重建各向异性层速度结构

本文提出-种利用有偏VSP资料反射波旅行时信息重建椭圆各向异性介质中水平向与垂直向速度的方法。其中,地下介质假定为层状椭圆各向异性介质。反射波旅行时间采用射线追踪理论及几何关系计算得到,反演中的线性方程组采用奇异值分解(SVD)技术进行求解。 方法检测时,我们对各向同性介质及椭圆各向异胜介质情况下有限差分法正演模拟的深井有偏移距VSP地震资料分别进行各向同性和各向异性方法反演成像。结果表明,本文所述方法较之各向同性介质模型反演方法对介质类型有很好的适用性,同时也说明了本方法的司行性。最后,我们分别介绍了对实际有偏VSP资料反演得到的地下介质的速度结构图像。     

含两项不混合/单项黏性流体孔隙介质分界面上反透射系数特征研究

弹性孔隙介质分界面上的反透射系数特征,在岩性划分、流体识别、储层边界判识等方面有重要的应用.本文研究上层为含两项不混合黏性流体孔隙介质、下层为含单项黏性流体孔隙介质分界面上的反透射理论.首先根据两种孔隙介质分界面上的能量守恒得到边界条件,再将波函数、位移、应力与应变关系代入边界条件,推导出完全连通孔隙情况下,第一类纵波入射到孔隙介质分界面上的反透射系数方程.通过建立砂岩孔隙介质模型,分别分析不同孔隙流体类型、不同含油饱和度及不同入射角情况下,各类波的反透射系数特征.研究表明,第二、三类纵波反透射系数数值比第一类纵波小多个数量级,且两者对入射角的变化不敏感,但对孔隙流体性质、含油饱和度的变化较敏感,而横波反透射系数特征恰好与此相反;第一类纵波反透射系数特征比较复杂,入射角、孔隙流体的性质及含油饱和度的变化都对其产生影响.不同孔隙流体弹性物性的差异、孔隙介质中含油饱和度的变化及不同入射角引起垂向和切向应力分量的变化都会影响各类波的反透射系数特征,分析这些特征可以为研究储层含油气性提供理论基础.     

基于GSLS模型TI介质衰减拟声波方程

随着计算机硬件技术的发展以及高分辨率勘探需求的增加,我们希望能够更准确地模拟地下介质,得到更丰富的地层信息.然而,传统的声学假设并不能描述实际地层所存在各向异性和黏滞性,使得成像分辨率较低.为了实现深部储层的高精度成像,本文同时考虑了介质的各向异性和黏滞性,从TI介质弹性波的基本理论出发,结合各向异性GSLS理论,并通过声学近似方法导出基于GSLS模型的各向异性衰减拟声波方程.数值模拟表明该方程既能准确地描述各向异性介质下的准P波运动学规律,又能体现地层的吸收衰减效应;模型逆时偏移结果表明,在实现成像过程中考虑各向异性和黏滞性的影响,能对高陡构造清晰成像,且剖面振幅相对均衡,分辨率较高.