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.     

粘弹性VTI介质地震波模拟特征分析

本文首先利用有限差分法分别对弹性和粘弹性VTI介质进行地震波传播数值模拟,并针对波场快照和波场记录特征,分析不同品质因子组合对波场能量衰减和频率吸收作用的影响.结果表明：对应于膨胀滞弹性形变的品质因子变化主要影响qP波的能量衰减;对应于剪切滞弹性形变的品质因子变化主要影响qSV波的能量衰减;对于qSH波,两个品质因子分别对应于垂直和水平方向的能量衰减;品质因子较小时,qSV波和qSH波的频率向低频方向移动,qP波频率变化不明显.     

Dispersion splitting of Rayleigh waves in layered azimuthally anisotropic media

Rayleigh wave dispersion can be induced in an anisotropic medium or a layered isotropic medium. For a layered azimuthally anisotropic structure, traditional wave equation of layered structure can be modified to describe the dispersion behavior of Rayleigh waves. Numerical stimulation results show that for layered azimuthal anisotropy both the dispersion velocities and anisotropic parameters depend principally on anisotropic S-wave velocities. The splitting S-wave velocities may produce dispersion splitting of Rayleigh waves. Such dispersion splitting appears noticeable at azimuthal angle 45°. This feature was confirmed by the measured results of a field test. The fundamental mode splits into two branches at azimuthal angle 45° to the symmetry axis for some frequencies, and along the same direction the difference of splitting-phase velocities of the fundamental model reaches the maximum. Dispersion splitting of Rayleigh waves was firstly displayed for anisotropy study in dispersion image by means of multichannel analysis of surface waves, the image of which provides a new window for studying the anisotropic property of media.     

Plane-layered models for the analysis of wave propagation in reservoir environments1

The long-wavelength propagation and attenuation characteristics of three geological structures that frequently occur in reservoir environments are investigated using a theoretical model that consists of a stack of fine and viscoelastic plane layers, with the layers being either solid or fluid. Backus theory properly describes fine layering and a set of fluid-filled microfractures, under the assumption that interfaces between different materials are bonded. The effects of saturation on wave attenuation are modelled by the relative values of the bulk and shear quality factors. The anisotropic quality factor in a fine-layered system shows a variety of behaviours depending on the saturation and velocities of the single constituents. The wave is less attenuated along the layering direction when the quality factors are proportional to velocity, and vice versa when inversely proportional to velocity. Fractured rocks have very anisotropic wavefronts and quality factors, in particular for the shear modes which are strongly dependent on the characteristics of the fluid filling the microfractures. When the size of the boundary layer is much smaller than the thickness of the fluid layer, the stack of solid-fluid layers becomes a layered porous media of the Biot type. This behaviour is caused by the slip-wall condition at the interface between the solid and the fluid. As in Biot theory, there are two compressional waves, but here the medium is anisotropic and the slow wave does not propagate perpendicular to the layers. Moreover, this wave shows pronounced cusps along the layering direction, like shear waves in a very anisotropic single-phase medium.     

Frequency-dependent PP and PS reflection coefficients in fractured media

When a porous layer is permeated by mesoscale fractures, wave-induced fluid flow between pores and fractures can cause significant attenuation and dispersion of velocities and anisotropy parameters in the seismic frequency band. This intrinsic dispersion due to fracturing can create frequency-dependent reflection coefficients in the layered medium. In this study, we derive the frequency-dependent PP and PS reflection coefficients versus incidence angle in the fractured medium. We consider a two-layer vertical transverse isotropy model constituted by an elastic shale layer and an anelastic sand layer. Using Chapman's theory, we introduce the intrinsic dispersion due to fracturing in the sand layer. Based on the series coefficients that control the behaviour of velocity and anisotropy parameters in the fractured medium at low frequencies, we extend the conventional amplitude-versus-offset equations into frequency domain and derive frequency-dependent amplitude-versus-offset equations at the elastic–anelastic surface. Increase in fracture length or fracture density can enlarge the frequency dependence of amplitude-versus-offset attributes of PP and PS waves. Also, the frequency dependence of magnitude and phase angle of PP and PS reflection coefficients increases as fracture length or fracture density increases. Amplitude-versus-offset type of PP and PS reflection varies with fracture parameters and frequency. What is more, fracture length shows little impact on the frequency-dependent critical phase angle, while the frequency dependence of the critical phase angle increases with fracture density.     

Anisotropic attenuation of stratified viscoelastic media

An important cause of seismic anisotropic attenuation is the interbedding of thin viscoelastic layers. However, much less attention has been devoted to layer‐induced anisotropic attenuation. Here, we derive a group of unified weighted average forms for effective attenuation from a binary isotropic, transversely isotropic‐ and orthorhombic‐layered medium in the zero‐frequency limit by using the Backus averaging/upscaling method and analyse the influence of interval parameters on effective attenuation. Besides the corresponding interval attenuation and the real part of stiffness, the contrast in the real part of the complex stiffness is also a key factor influencing effective attenuation. A simple linear approximation can be obtained to calculate effective attenuation if the contrast in the real part of stiffness is very small. In a viscoelastic medium, attenuation anisotropy and velocity anisotropy may have different orientations of symmetry planes, and the symmetry class of the former is not lower than that of the latter. We define a group of more general attenuation‐anisotropy parameters to characterize not only the anisotropic attenuation with different symmetry classes from the anisotropic velocity but also the elastic case. Numerical tests reveal the influence of interval attenuation anisotropy, interval velocity anisotropy and the contrast in the real part of stiffness on effective attenuation anisotropy. Types of effective attenuation anisotropy for interval orthorhombic attenuation and interval transversely isotropic attenuation with a vertical symmetry (vertical transversely isotropic attenuation) are controlled only by the interval attenuation anisotropy. A type of effective attenuation anisotropy for interval TI attenuation with a horizontal symmetry (horizontal transversely isotropic attenuation) is controlled by the interval attenuation anisotropy and the contrast in the real part of stiffness. The type of effective attenuation anisotropy for interval isotropic attenuation is controlled by all three factors. The magnitude of effective attenuation anisotropy is positively correlated with the contrast in the real part of the stiffness. Effective attenuation even in isotropic layers with identical isotropic attenuation is anisotropic if the contrast in the real part of stiffness is non‐zero. In addition, if the contrast in the real part of stiffness is very small, a simple linear approximation also can be performed to calculate effective attenuation‐anisotropy parameters for interval anisotropic attenuation.     

TTI介质弹性波相速度与偏振特征分析

相速度和偏振方向是研究地震波传播规律和描述介质特性的重要参数,在理论研究和实际应用中有重要作用.本文假定倾斜横向各向同性(TTI)介质对称轴位于观测坐标系XOZ面内,在此观测坐标系下直接推导了TTI介质弹性波相速度和偏振方向的解析表达式,再进一步利用Thomsen弱各向异性理论,推导了弱各向异性近似条件下弹性波相速度以及qP波和qSV波偏振方向表达式.理论分析和数值试例表明,在相速度方面,随着各向异性介质参数改变,qP波和qSH波速度变化较为平缓,qSV波速度变化较为剧烈.弹性波相速度近似式误差均较小,能较好地近似精确相速度.在偏振方向方面,SH波偏振方向只是传播方向和对称轴倾角的函数,而与各向异性参数无关,SH波偏振方向既垂直于传播方向,又垂直于TTI介质对称轴方向.除特定方向外,qP波和qSV波的偏振方向与传播方向均成一定角度,并且随TTI介质对称轴倾角的改变而改变;在精确和近似情况下,qP波和qSV波的偏振方向始终垂直;在精度允许范围内,偏振方向的弱各向异性近似式与理论解析式吻合较好.     

粘弹各向异性介质中地震波场模拟与特征

通过引入记忆变量,可以避免粘弹性应力-应变关系中的褶积运算,使波场数值模拟易于实现.通过伪谱法对粘弹各向异性介质中的qP波、qS波数值模拟,结合理论分析,研究了粘弹各向异性介质中速度各向异性和衰减各向异性.衰减各向异性要比速度各向异性更为显著,并且qS波比qP波的衰减各向异性明显.粘弹各向异性介质中,粘弹性对波的影响主要在于波的衰减,各向异性主要影响波前面形状.     

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.     

任意各向异性介质相(群)速度的计算

反映弹性波在各向异性介质中传播特性的两个基础的物理量是相速度和群速度.本文在总结前人工作的基础上,提出任意各向异性介质相(群)速度的计算方案:首先推导各自计算公式,其次考虑剪切波奇点的特殊性,再次令其遵循相应约束条件,最后,采用三个计算实例检验该方案的正确性和有效性.通过对计算结果的分析以及各向异性理论预测可以加深对各向异性特有性质(如剪切波奇点、群速度多值性)的理解,有助于增强我们对任意各向异性理论的基本认识.     

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.     

THE POTENTIAL OF SHEAR-WAVE OBSERVATIONS*

Shear waves can today be generated and observed, though not with the flexibility and the technical standard of compressional waves, and they can be identified in seismograms by various means. Their potential lies not so much in their lower velocity (corresponding—for the same frequency—to shorter wavelength and higher resolution) but in the fact that they probe the earth with stresses and strains that differ from those of compressional waves. Full utilization of the information potential of shear waves, therefore, requires combined use of P-and S-waves. Complications in the combined use of different wave types should be regarded as opportunities to obtain additional information. A typical example is the observation that the depth of one and the same interface estimated on the bases of P- and SH-reflections, respectively, can differ significantly. This discrepancy may be due to the anisotropy of a finely layered medium. Under favorable circumstances some of the parameters describing this anisotropy can be deduced from the different depth estimates and the curvature of the squared-offset/squared-time representation of the different reflections. Since in anisotropic media vertically polarized shear waves are significantly different from horizontally polarized ones, the combined observation of all three waves opens up additional possibilities.     

各向异性TI介质qP反射波走时层析成像

地震走时层析成像是反演地层各向异性参数分布的有效方法,但是关于地震各向异性介质走时层析成像的研究并不多,其技术远远没有达到成熟的阶段.在野外数据采集时,地表反射波观测方式相对井间和垂直地震剖面观测方式的成本更低,利用qP反射波走时反演各向异性参数具有更加广泛的实用价值.本文实现的TI介质地震走时层析成像方法结合了TI介质反射波射线追踪算法、走时扰动方程和非线性共轭梯度算法,它可以对任意强度的TI介质模型进行反演,文中尝试利用qP反射波走时重建TI介质模型的参数图像.利用qP反射波对层状介质模型和块状异常体模型进行走时反演,由于qP波相速度对弹性模量参数和Thomsen参数的偏微分不同,所以可以分别反演弹性模量参数和Thomsen参数.数值模拟结果表明:利用qP反射波可以反演出TI介质模型的弹性模量参数与Thomsen参数,不同模型的走时迭代反演达到了较好的收敛效果,与各向同性介质走时反演结果相比较,各向异性介质走时反演结果具有较好的识别能力.     

Physical modeling observations of wave velocity and apparent attenuation in isotropic and anisotropic two-phase media

We study wave propagation through isotropic and anisotropic scatterer distributions in order to observe azimuthal variations in velocity and apparent attenuation. Using thin aluminum plates as physical models, we obtained seismograms for compressional and shear wave propagation through heterogeneous media. Three random distributions of scatterers are studied: circular scatterers in isotropic distributions (modeling circular scatterers), elongated scatterers in isotropic distributions (modeling randomly oriented elliptical scatterers), and elongated scatterers in anisotropic distributions (modeling aligned elliptical scatterers). All scatterers had approximately the same cross-sectional area and were filled with epoxy in order to reduce the impedance contrast. In addition to seismograms recorded for no scatterers, seismograms were recorded for several scatterer volume fractions. Azimuths were measured relative to the long axis of the aligned elongated scatterers. Velocities were calculated using travel times and phase shifts at low frequencies. The velocities measured from the data were compared to simple low-frequency average-velocity theories based on thin lamellae or on distributions of penny-shaped cracks. The apparent attenuation for different scatterer distributions was computed using spectral ratios.Comparisons of the results for circular and randomly oriented elongated scatterers were made to determine the effects of scatterer shape. As expected, the circular and randomly oriented elongated scatterers showed no systematic azimuthal variation in velocity. The velocity anomalies were systematically larger for the randomly oriented elongated scatterers than for the circular scatterers. Both methods of theoretical estimation for the isotropic velocities produced velocities significantly larger than those measured. The spectral ratios showed more apparent attenuation for the randomly oriented elongated scatterers than for the circular scatterers.Comparisons of the results for the randomly oriented and aligned elongated scatterers were made to determine the effects of anisotropy in the scatterer distribution. Compressional waves for the aligned elongated scatterers with wave propagation parallel to the scatterers had larger velocities than for the aligned elongated scatterers with wave propagation perpendicular to the scatterers for all velocity calculations. Shear wave velocities were complicated by an anomalous phase change in the shear wave seismograms for azimuths less than 40° and were not as conclusive. The general trend of the theoretical velocities is similar to the velocities calculated from the data. There are, however, what appear to be significant differences. The spectral ratios showed more apparent attenuation for the randomly oriented elongated scatterers than for the aligned elongated scatterers with wave propagation parallel to the scatterers, and less attenuation than for the aligned elongated scatterers with wave propagation perpendicular to the scatterers.     

Generalized anisotropy parameters and approximations of attenuations and velocities in viscoelastic media of arbitrary anisotropy type – theoretical and experimental aspects*

Seismic anisotropy in geological media is now widely accepted. Parametrizations and explicit approximations for the velocities in such media, considered as purely elastic and moderately anisotropic, are now standards and have even been extended to arbitrary types of anisotropy. In the case of attenuating media, some authors have also recently published different parametrizations and velocity and attenuation approximations in viscoelastic anisotropic media of particular symmetry type (e.g., transversely isotropic or orthorhombic). This paper extends such work to media of arbitrary anisotropy type, that is to say to triclinic media. In the case of homogeneous waves and using the so‐called ‘correspondence principle’, it is shown that the viscoelastic equations (for the phase velocities, phase slownesses, moduli, wavenumbers, etc.) are formally identical to the corresponding purely elastic equations available in the literature provided that all the corresponding quantities are complex (except the unit vector in the propagation direction that remains real). In contrast to previous work, the new parametrization uses complex anisotropy parameters and constitutes a simple extension to viscoelastic media of previous work dealing with non‐attenuating elastic media of arbitrary anisotropy type. We make the link between these new complex anisotropy parameters and measurable parameters, as well as with previously published anisotropy parameters, demonstrating the usefulness of the new parametrization. We compute the explicit complete directional dependence of the exact and of the approximate (first and higher‐order perturbation) complex phase velocities of the three body waves (qP, qS1 and qS2). The exact equations are successfully compared with the ultrasonic phase velocities and phase attenuations of the three body waves measured in a strongly attenuating water‐saturated sample of Vosges sandstone exhibiting moderate velocity anisotropy but very strong attenuation anisotropy. The approximate formulas are checked on experimental data. Compared to the exact solutions, the errors observed on the first‐order approximate velocities are small (<1%) for qP‐waves and moderate (<10%) for qS‐waves. The corresponding errors on the quality factor Q are moderate (<6%) for qP‐waves but critically large (up to 160%) for the qS‐waves. The use of higher‐order approximations substantially improves the accuracy, for instance typical maximum relative errors do not exceed 0.06% on all the velocities and 0.6% on all the quality factors Q, for third‐order approximations. All the results obtained on other rock samples confirm the results obtained on this rock. The simplicity of the derivations and the generality of the results are striking and particularly convenient for practical applications.     

黏弹性EDA介质中地震波的衰减特性研究

本文首先由Christoffel方程推导出黏弹性EDA介质中均匀、 非均匀P波、 SV波和SH波的相速度表达式, 然后参照极端各向异性介质的相关计算方法, 推导出EDA介质中均匀、 非均匀地震波相衰减系数和群衰减系数的表达式, 并通过数值计算分析了相速度、 相衰减系数、 群衰减系数与裂隙方位的关系. 结果表明: 均匀介质中SH波的相速度和相衰减系数均可指示裂隙的走向; 非均匀介质中SH波相衰减系数随非均匀角的增大而增大, 且其对称轴与介质对称轴的夹角也相应增加; 由于地震波振幅的衰减随岩石物理性质的变化比地震波速度的变化更为灵敏, 而且携带了更多的岩石物理性质信息, 因此可用来探明裂隙走向、 密度及含水特性, 进而应用于预测、 预防地下工程地质灾害事故.      

几种弱向异性介质中qSV波速度和偏振异常

研究了在具有生趣对称轴的横向各向同性介质（ＴＩＶ介质）中ｑＳＶ波群速度及偏振矢量,给出了相应的精确和近似公式,进一步讨论了用较简单的近似公式来代替复杂精确公式的可靠性;最后展示了地球内部几种常见各向异性岩石矿物中地震ｓＳＶ波速度各向异性因子、偏振信息及其与传播方向之间的偏差。     

各向异性介质中扭转波分裂的实验观测

实验室可以产生两种振动模式的横波,一种是剪切振动,另一种是扭转振动,在各向同性介质中两模式的横波速度是相同的,但它们的振动特性不一样,前者表现出很强的偏振特性,后者为无偏振特性.实验测试表明无偏振特性的扭转波在各向异性介质中传播时也会出现两种速度不同的扭转波,速度值与剪切横波的快慢横波速度值一致.用扭转波换能器接收时,这快慢扭转波的波形振幅不受各向异性方位影响.通过两块均匀的各向异性样品,用实验观测揭示了各向异性介质中扭转波的一些传播特征.     

Influence of frequency-dependent anisotropy on seismic amplitude-versus-offset signatures for fractured poroelastic rocks

Frequency-dependent amplitude variation with offset offers an effective method for hydrocarbon detections and analysis of fluid flow during production of oil and natural gas within a fractured reservoir. An appropriate representation for the frequency dependency of seismic amplitude variation with offset signatures should incorporate influences of dispersive and attenuating properties of a reservoir and the layered structure for either isotropic or anisotropic dispersion analysis. In this study, we use an equivalent medium permeated with aligned fractures that simulates frequency-dependent anisotropy, which is sensitive to the filled fluid of fractures. The model, where pores and fractures are filled with two different fluids, considers velocity dispersion and attenuation due to mesoscopic wave-induced fluid flow. We have introduced an improved scheme seamlessly linking rock physics modelling and calculations for frequency-dependent reflection coefficients based on the propagator matrix technique. The modelling scheme is performed in the frequency-slowness domain and can properly incorporate effects of both bedded structure of the reservoir and velocity dispersion quantified with frequency-dependent stiffness. Therefore, for a dispersive and attenuated layered model, seismic signatures represent a combined contribution of impedance contrast, layer thickness, anisotropic dispersion of the fractured media and tuning and interference of thin layers, which has been avoided by current conventional methods. Frequency-dependent amplitude variation with offset responses was studied via considering the influences of fracture fills, layer thicknesses and fracture weaknesses for three classes amplitude variation with offset reservoirs. Modelling results show the applicability of the introduced procedure for interpretations of frequency-dependent seismic anomalies associated with both layered structure and velocity dispersion of an equivalent anisotropic medium. The implications indicate that anisotropic velocity dispersion should be incorporated accurately to obtain enhanced amplitude variation with offset interpretations. The presented frequency-dependent amplitude variation with offset modelling procedure offers a useful tool for fracture fluid detections in an anisotropic dispersive reservoir with layered structures.     

Anisotropic velocity analysis1

Results from walkaway VSP and shale laboratory experiments show that shale anisotropy can be significantly anelliptic. Heterogeneity and anellipticity both lead to non-hyperbolic moveout curves and the resulting ambiguity in velocity analysis is investigated for the case of a factorizable anisotropic medium with a linear dependence of velocity on depth. More information can be obtained if there are several reflectors. The method of Dellinger et al. for anisotropic velocity analysis in layered transversely isotropic media is examined and is shown to be restricted to media having relatively small anellipticity. A new scheme, based on an expansion of the inverse-squared group velocity in spherical harmonics, is presented. This scheme can be used for larger anellipticity, and is applicable for horizontal layers having monoclinic symmetry with the symmetry plane parallel to the layers. The method is applied to invert the results of anisotropic ray tracing on a model Sand/shale sequence. For transversely isotropic media with small anisotropy, the scheme reduces to the method of Byun et al. and Byun and Corrigan. The expansion in spherical harmonics allows the P-phase slowness surface of each layer to be determined in analytic form from the layer parameters obtained by inversion without the need to assume that the anisotropy is weak.