Equivalent time-average and effective medium for periodic layers

The propagation of acoustic waves through a periodic layered medium is analyzed by an eigenvalue decomposition of the propagator matrix. This reveals how the velocity and attenuation of the layered medium vary as function of the periodic structure, material parameters and frequency. There are two important parameters which control the wave propagation in the periodic medium: the reflection coefficient and the ratio between one‐way traveltimes of the two parts of the cyclic layered medium. For low frequencies (large values of wavelength to layer thickness), the layered structure behaves as an effective medium, then there is a transition zone, and for higher frequencies (small values of wavelength to layer thickness) the medium is described by the time‐average velocity. In this paper we mostly concentrate on the transition zone between an effective medium and time‐average medium regimes. The width of the transition zone increases with larger values of the reflection coefficient. The transition zone corresponds to a blocking regime for which the transmission response of the layered structure is close to zero. For even higher frequencies, the time‐average medium is replaced by a new transition zone, and then again a time‐average medium. This pattern is periodically repeated with higher frequencies. For small values of the reflection coefficient, the transition between effective medium and time‐average medium occurs around a value of wavelength to layer thickness equal to 4.     

Caustics in a periodically layered transversely isotropic medium with vertical symmetry axis

Wave propagation in a finely layered medium is a very important topic in seismic modelling and inversion. Here we analyse non‐vertical wave propagation in a periodically layered transversely isotropic (VTI) medium and show that the evanescent (attenuation) zones in the frequency‐horizontal slowness domain result in caustics in the group velocity domain. These caustics, which may appear for both the quasi‐compressional (qP) and quasi‐shear (qSV) wave surfaces are frequency dependent but display weak dependence at low frequencies. The caustics computed for a specific frequency differ from those observed at the low‐ and high‐frequency limits. We illustrate these caustics with a few numerical examples and snapshots computed for both qP‐ and qSV‐wave types.     

Low-frequency layer-induced dispersion in a weak-contrast vertically heterogeneous orthorhombic medium

In this paper, we consider wave propagation in a periodically layered medium with orthorhombic symmetry. The weak-contrast approximation is utilized to derive the low-frequency dispersion in effective properties for P, S1 and S2 waves. We show that the dispersion term for all effective properties is controlled by the second-order contrasts in elastic properties from the layers. We also compute the sensitivity matrices for second- and fourth-order coefficients from eigenvalues of frequency-dependent system matrix associated with kinematic parameters for individual wave modes.     

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Compensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.     

Slowness domain offset and traveltime approximations in layered vertical transversely isotropic media

Considering horizontally layered transversely isotropic media with vertical symmetry axis and all types of pure‐mode and converted waves we present a new wide‐angle series approximation for the kinematical characteristics of reflected waves: horizontal offset, intercept time, and total reflection traveltime as functions of horizontal slowness. The method is based on combining (gluing) both zero‐offset and (large) finite‐offset series coefficients. The horizontal slowness is bounded by the critical value, characterised by nearly horizontal propagation within the layer with the highest horizontal velocity. The suggested approximation uses five parameters to approximate the offset, six parameters to approximate the intercept time or the traveltime, and seven parameters to approximate any two or all three kinematical characteristics. Overall, the method is very accurate for pure‐mode compressional waves and shear waves polarised in the horizontal plane and for converted waves. The application of the method to pure‐mode shear waves polarised in the vertical plane is limited due to cusps and triplications. To demonstrate the high accuracy of the method, we consider a synthetic, multi‐layer model, and we plot the normalised errors with respect to numerical ray tracing.     

Reflected waves in finely layered tilted orthorhombic media

Upscaling in seismics is a homogenization of finely layered media in the zero-frequency limit. An upscaling technique for arbitrary anisotropic layers has been developed by Schoenberg and Muir. Applying this technique to a stack of layers of orthorhombic (ORT) symmetry whose vertical symmetry planes are aligned, results in an effective homogeneous layer with orthorhombic symmetry. If the symmetry planes in a horizontal orthorhombic layer are rotated with respect to vertical, the medium is referred to as tilted orthorhombic (TOR) medium, and the stack composed of TOR layers in zero-frequency limit will produce an effective medium of a lower symmetry than orthorhombic. We consider a P-wave that propagates through a stack of thin TOR layers, then it is reflected (preserving the mode) at some interface below the stack, and then propagates back through the same stack. We propose to use a special modified medium for the upscaling in case of this sequential down- and up-propagation: each TOR layer in the stack is replaced by two identical TOR layers whose tilt angles have the opposite algebraic sign. In this modified medium, one-way propagation of a seismic wave (any wave mode) is equivalent to propagation of a pure-mode reflection in the original medium. We apply this idea to study the contribution from an individual layer from the stack and show how the approach can be applied to a stack of TOR layers. To demonstrate the applicability of the model, we use well log data for the upscaling. The model we propose for the upscaling can be used in well-seismic ties to correct the effective parameters obtained from well log data for the presence of tilt, if latter is confirmed by additional measurements (for example, borehole imaging).     

层状介质对破裂过程的影响

震源动力学中破裂产生的地震动在层状介质中的传播模拟,是地震学以及地震工程学研究的前沿课题之一。本文通过建立精确的三维模型,选取具备灵活网格、高精度高效率计算性能的谱元法,利用有效抑制伪震荡的时间域离散方法——加权速度Newmark方法以及多次透射人工边界条件,进行了SCEC/USGS基准项目中TPV5模型的地震破裂过程模拟,得到基于层状介质模型和均匀介质模型（后者采用相同破裂模型）的埋深2km的震源参数结果。将二者进行对比,并具体分析破裂面位错、地震矩、破裂传播时间、上升时间和地表位移,发现层状介质对破裂过程的传播影响较为明显:① 层状介质的存在整体增加了破裂面上的位错,在层状介质模型下计算得到的地震矩约是均匀介质模型结果的1.3倍,因此认为层状介质增强了地震破裂过程中的能量释放;② 层状介质的存在使得破裂传播至地表的速度减慢,并缩短了地表各点的上升时间,增强了地表的地震动响应;③ 层状介质对于地表位移有着明显的增加作用,同时协同破裂面上的初始应力异常区域对位移峰值中心的改变有显著影响。④ 介质分异面附近地震动强烈。对结果进行整理后发现,在具有地下层状介质的地区要充分考虑层状介质产生的场地效应,否则可能会低估该地区的地震危险性。     

COMPUTATION OF ZERO-OFFSET VERTICAL SEISMIC PROFILES INCLUDING GEOMETRICAL SPREADING AND ABSORPTION*

Synthetic vertical seismic profiles (VSP) provide a useful tool in the interpretation of VSP data, allowing the interpreter to analyze the propagation of seismic waves in the different layers. A zero-offset VSP modeling program can also be used as part of an inversion program for estimating the parameters in a layered model of the subsurface. Proposed methods for computing synthetic VSP are mostly based on plane waves in a horizontally layered elastic or anelastic medium. In order to compare these synthetic VSP with real data a common method is to scale the data with the spherical spreading factor of the primary reflections. This will in most cases lead to artificial enhancement of multiple reflections. We apply the ray series method to the equations of motion for a linear viscoelastic medium after having done a Fourier transformation with respect to the time variable. This results in a complex eikonal equation which, in general, appears to be difficult to solve. For vertically traveling waves in a horizontally layered viscoelastic medium the solution is easily found to be the integral along the ray of the inverse of the complex propagation velocity. The spherical spreading due to a point source is also complex, and it is equal to the integral along the ray of the complex propagation velocity. Synthetic data examples illustrate the differences between spherical, cylindrical, and plane waves in elastic and viscoelastic layered media.     

Analytical solution for the electric potential in arbitrary anisotropic layered media applying the set of Hankel transforms of integer order

The analytical solution and algorithm for simulating the electric potential in an arbitrarily anisotropic multilayered medium produced by a point DC source is here proposed. The solution is presented as a combination of Hankel transforms of integer order and Fourier transforms based on the analytical recurrent equations obtained for the potential spectrum. For the conversion of the potential spectrum into the space domain, we have applied the algorithm of the Fast Fourier Transform for logarithmically spaced points. A comparison of the modelling results with the power‐series solution for two‐layered anisotropic structures demonstrated the high accuracy and computing‐time efficiency of the method proposed. The results of the apparent‐resistivity calculation for both traditional pole‐pole and tensor arrays above three‐layered sequence with an azimuthally anisotropic second layer are presented. The numerical simulations show that both arrays have the same sensitivity to the anisotropy parameters. This sensitivity depends significantly on the resistivity ratio between anisotropic and adjacent layers and increases for the models with a conductive second layer.     

S‐wave kinematics in acoustic transversely isotropic media with a vertical symmetry axis

Acoustic transversely isotropic models are widely used in seismic exploration for P‐wave processing and analysis. In isotropic acoustic media only P‐wave can propagate, while in an acoustic transversely isotropic medium both P and S waves propagate. In this paper, we focus on kinematic properties of S‐wave in acoustic transversely isotropic media. We define new parameters better suited for S‐wave kinematics analysis. We also establish the travel time and relative geometrical spreading equations and analyse their properties. To illustrate the behaviour of the S‐wave in multi‐layered acoustic transversely isotropic media, we define the Dix‐type equations that are different from the ones widely used for the P‐wave propagation.     

On shear‐wave triplications in a multilayered transeversely isotropic medium with vertical symmetry axis

The presence of triplications (caustics) can be a serious problem in seismic data processing and analysis. The traveltime curve becomes multi‐valued and the geometrical spreading correction factor tends to zero due to energy focusing. We analyse the conditions for the qSV‐wave triplications in a homogeneous transversely isotropic medium with vertical symmetry axis. The proposed technique can easily be extended to the case of horizontally layered vertical symmetry axis medium. We show that the triplications of the qSV‐wave in a multilayered medium imply certain algebra. We illustrate this algebra on a two‐layer vertical symmetry axis model.     

The influence of sea water velocity variation on seismic traveltimes, ray paths, and amplitude

The main factors affecting seismic exploration is the propagation velocity of seismic waves in the medium. In the past, during marine seismic data processing, the propagation velocity of sea water was generally taken as a constant 1500 m/s. However, for deep water exploration, the sound velocity varies with the season, time, location, water depth, ocean currents, and etc.. It also results in a layered velocity distribution, so there is a difference of seismic traveltime, ray paths, and amplitude, which affect the migration imaging results if sea water propagation velocity is still taken as constant for the propagation wavefield. In this paper, we will start from an empirical equation of seismic wave velocity in seawater with changes of temperature, salinity, and depth, consider the variation of their values, build a seawater velocity model, and quantitatively analyze the impact of seawater velocity variation on seismic traveltime, ray paths, and amplitude in the seawater velocity model.     

Extended reflectivity method for modelling the propagation of diffusive–viscous wave in dip‐layered media

The reflectivity method plays an important role in seismic modelling. It has been used to model different types of waves propagating in elastic and anelastic media. The diffusive–viscous wave equation was proposed to investigate the relationship between frequency dependence of reflections and fluid saturation. It is also used to describe the attenuation property of seismic wave in a fluid‐saturated medium. The attenuation of diffusive–viscous wave is mainly characterised by the effective attenuation parameters in the equation. Thus, it is essential to obtain those parameters and further characterise the features of the diffusive–viscous wave. In this work, we use inversion method to obtain the effective attenuation parameters through quality factor to investigate the characteristics of diffusive–viscous wave by comparing with those of the viscoacoustic wave. Then, the reflection/transmission coefficients in a dip plane‐layered medium are studied through coordinate transform and plane‐wave theory. Consequently, the reflectivity method is extended to compute seismograms of diffusive–viscous wave in a dip plane multi‐layered medium. Finally, we present two models to simulate the propagation of diffusive–viscous wave in a dip plane multi‐layered medium by comparing the results with those in a viscoacoustic medium. The numerical results demonstrate the validity of our extension of reflectivity method to the diffusive–viscous medium. The numerical examples in both time domain and time–frequency domain show that the reflections from a dip plane interface have significant phase shift and amplitude change compared with the results of horizontal plane interface due to the differences in reflection/transmission coefficients. Moreover, the modelling results show strong attenuation and phase shift in the diffusive–viscous wave compared to those of the viscoacoustic wave.     

用传播矩阵法研究层状交错地层中的多分量感应测井

为了实现交错沉积等复杂环境中的电磁场数值模拟,本文在常规横向同性模型的基础上引入了电导率主轴坐标系相对地层坐标系的层理方位角和倾角,建立了交错地层模型.并利用传播矩阵法建立了一维层状交错地层模型中的多分量感应测井仪器响应的正演模拟算法.首先将频率-波数域中的电磁场分解为上行和下行模式波,给出了任意朝向的磁偶极子在无限大地层中模式波的解析解.进一步通过引入地层界面上的透射、局部反射以及广义反射系数矩阵,推导了一维层状地层中的模式波表达式.在此基础上,利用二维Gauss-Legendre积分实现了Fourier逆变换,得到了可用于多分量感应测井模拟的频率-空间域磁场并矢格林函数.最后,通过多个数值模拟结果考察了井眼倾角、层理方位角和倾角变化对多分量感应测井响应的影响.

     

Interaction of multiple courses of wave‐induced fluid flow in layered porous media

Different theoretical and laboratory studies on the propagation of elastic waves in layered hydrocarbon reservoir have shown characteristic velocity dispersion and attenuation of seismic waves. The wave‐induced fluid flow between mesoscopic‐scale heterogeneities (larger than the pore size but smaller than the predominant wavelengths) is the most important cause of attenuation for frequencies below 1 kHz. Most studies on mesoscopic wave‐induced fluid flow in the seismic frequency band are based on the representative elementary volume, which does not consider interaction of fluid flow due to the symmetrical structure of representative elementary volume. However, in strongly heterogeneous media with unsymmetrical structures, different courses of wave‐induced fluid flow may lead to the interaction of the fluid flux in the seismic band; this has not yet been explored. This paper analyses the interaction of different courses of wave‐induced fluid flow in layered porous media. We apply a one‐dimensional finite‐element numerical creep test based on Biot's theory of consolidation to obtain the fluid flux in the frequency domain. The characteristic frequency of the fluid flux and the strain rate tensor are introduced to characterise the interaction of different courses of fluid flux. We also compare the behaviours of characteristic frequencies and the strain rate tensor on two scales: the local scale and the global scale. It is shown that, at the local scale, the interaction between different courses of fluid flux is a dynamic process, and the weak fluid flux and corresponding characteristic frequencies contain detailed information about the interaction of the fluid flux. At the global scale, the averaged strain rate tensor can facilitate the identification of the interaction degree of the fluid flux for the porous medium with a random distribution of mesoscopic heterogeneities, and the characteristic frequency of the fluid flux is potentially related to that of the peak attenuation. The results are helpful for the prediction of the distribution of oil–gas patches based on the statistical properties of phase velocities and attenuation in layered porous media with random disorder.     

用传播矩阵法研究层状交错地层中的多分量感应测井

为了实现交错沉积等复杂环境中的电磁场数值模拟,本文在常规横向同性模型的基础上引入了电导率主轴坐标系相对地层坐标系的层理方位角和倾角,建立了交错地层模型.并利用传播矩阵法建立了一维层状交错地层模型中的多分量感应测井仪器响应的正演模拟算法.首先将频率-波数域中的电磁场分解为上行和下行模式波,给出了任意朝向的磁偶极子在无限大地层中模式波的解析解.进一步通过引入地层界面上的透射、局部反射以及广义反射系数矩阵,推导了一维层状地层中的模式波表达式.在此基础上,利用二维Gauss-Legendre积分实现了Fourier逆变换,得到了可用于多分量感应测井模拟的频率-空间域磁场并矢格林函数.最后,通过多个数值模拟结果考察了井眼倾角、层理方位角和倾角变化对多分量感应测井响应的影响.     

Reflection from a deep interface in a strongly heterogeneous layered medium1

We consider the problem of acoustic pulse propagation through a layered medium with a reflector at one end. The fluctuations in the medium properties are assumed to be strong, i.e. of finite amplitude, rapid in comparison to the typical wavelength and to have statistical structure. The depth of the reflector is assumed to be large in comparison to the wavelength. In this regime, simple formulae for the reflected pulse and its arrival time at the surface are obtained. The amplitude of the pulse is broadened and attenuated as a result of multiple scattering: the fine-layered structure of the medium can be characterized by a single constant which appears in the formula for the limiting waveform and which measures the size of the fluctuations in the medium. Within the theory, the commonly observed discrepancy between the integrated sonic traveltime and the seismic traveltime can be studied and understood. The theory is a natural extension of the long-wavelength effective medium theory of Backus. The analysis is rigorous and based on the invariant embedding technique.     

Effect of fracture fill on frequency‐dependent anisotropy of fractured porous rocks

In fractured reservoirs, seismic wave velocity and amplitude depend on frequency and incidence angle. Frequency dependence is believed to be principally caused by the wave‐induced flow of pore fluid at the mesoscopic scale. In recent years, two particular phenomena, i.e., patchy saturation and flow between fractures and pores, have been identified as significant mechanisms of wave‐induced flow. However, these two phenomena are studied separately. Recently, a unified model has been proposed for a porous rock with a set of aligned fractures, with pores and fractures filled with two different fluids. Existing models treat waves propagating perpendicular to the fractures. In this paper, we extend the model to all propagation angles by assuming that the flow direction is perpendicular to the layering plane and is independent of the loading direction. We first consider the limiting cases through poroelastic Backus averaging, and then we obtain the five complex and frequency‐dependent stiffness values of the equivalent transversely isotropic medium as a function of the frequency. The numerical results show that, when the bulk modulus of the fracture‐filling fluid is relatively large, the dispersion and attenuation of P‐waves are mainly caused by fractures, and the values decrease as angles increase, almost vanishing when the incidence angle is 90° (propagation parallel to the fracture plane). While the bulk modulus of fluid in fractures is much smaller than that of matrix pores, the attenuation due to the “partial saturation” mechanism makes the fluid flow from pores into fractures, which is almost independent of the incidence angle.     

Effect of anelastic patchy saturated sand layers on the reflection and transmission responses of a periodically layered medium

Based on analytic relations, we compute the reflection and transmission responses of a periodically layered medium with a stack of elastic shales and partially saturated sands. The sand layers are considered anelastic (using patchy saturation theory) or elastic (with effective velocity). Using the patchy saturation theory, we introduce a velocity dispersion due to mesoscale attenuation in the sand layer. This intrinsic anelasticity is creating frequency dependence, which is added to the one coming from the layering (macroscale). We choose several configurations of the periodically layered medium to enhance more or less the effect of anelasticity. The worst case to see the effect of intrinsic anelasticity is obtained with low dispersion in the sand layer, strong contrast between shales and sands, and a low value of the net‐to‐gross ratio (sand proportion divided by the sand + shale proportion), whereas the best case is constituted by high dispersion, weak contrast, and high net‐to‐gross ratio. We then compare the results to show which dispersion effect is dominating in reflection and transmission responses. In frequency domain, the influence of the intrinsic anelasticity is not negligible compared with the layering effect. Even if the main resonance patterns are the same, the resonance peaks for anelastic cases are shifted towards high frequencies and have a slightly lower amplitude than for elastic cases. These observations are more emphasized when we combine all effects and when the net‐to‐gross ratio increases, whereas the differences between anelastic and elastic results are less affected by the level of intrinsic dispersion and by the contrast between the layers. In the time domain, the amplitude of the responses is significantly lower when we consider intrinsic anelastic layers. Even if the phase response has the same features for elastic and anelastic cases, the anelastic model responses are clearly more attenuated than the elastic ones. We conclude that the frequency dependence due to the layering is not always dominating the responses. The frequency dependence coming from intrinsic visco‐elastic phenomena affects the amplitude of the responses in the frequency and time domains. Considering intrinsic attenuation and velocity dispersion of some layers should be analyzed while looking at seismic and log data in thin layered reservoirs.     

A semi-analytical approach for analyzing ground vibrations caused by trains moving over elevated bridges

A semi-analytic approach is presented for the three-dimensional analysis of ground vibrations induced by trains moving over elevated bridges. The train is modeled as two sets of moving loads, with one for the front wheels and the other for the rear ones, the elevated bridge as a series of elastically supported beams, and the ground as a viscoelastic half space. Three key elements are considered in the solution: (1) the analytic solution for the vibration of an elastically supported beam under a series of moving loads, (2) the impedance of the foundation–soil system, and (3) Green's function for an elastic half space under a harmonic point load. Such an approach allows us to consider the structural dynamics of the elevated bridge, the foundation–soil interaction, and the wave propagation characteristics in the half space. From the numerical examples studied, the proposed approach was demonstrated to be accurate and efficient. The framework of analysis described herein can be generalized to solve problems with complex foundations and layered soils.