Multiple scattering of seismic waves in fractured media: Velocity and effective attenuation of the coherent components ofP waves andS waves

We show that the multiple scattering by small fractures of seismic waves with wavelengths long compared to the fracture size and fracture spacing is indistinguishable from multiple-scattering effects produced by regular porosity, except for an orientation factor due to fracture alignment. The fractures reduce theP-wave andS-wave velocities and produce an effective attenuation of the coherent component of the seismic waves. The attenuation corresponds to 1000/Q of about unity for a Gaussian spectrum of fractures, and it varies with frequencyf asf ³. For a Kolmogorov spectrum of fractures of spectral index the attenuation is an order of magnitude or so larger and varies with frequency asf ^3-v The precise degree of attenuation depends upon the matrix properties, the fracture porosity, the degree of fracture anisotropy, the type of fluid filling the fractures, and the incidence angle of the wave.For fracture porosities less than about 15% theP-wave andS-wave velocities are decreased by the order of 5–10% with a lesser dependence on the type of fluid filling the fractures (gas, oil, or brine) and with a dependence on both the degree of anisotropy and the incident angle made by the wave. The tendency of fractures to occur perpendicularly to bedding suggests that the best way to measure seismically fractured rock behavior in situ is by using the travel-time delay and reflection amplitude. As both the offset and the azimuth of receivers vary from a shot, the travel-time delay and reflection amplitude should both show an elliptical pattern of behavior—the travel-time delay in response to the varying seismic speed, and the reflection amplitude in response to angular variations in the multiple scattering. Observations of attenuation at several frequencies should permit (a) determination of the spectrum of fractures (Gaussian versus Kolmogorovian) and (b) determination of the contribution of viscous damping to the effective attenuation.     

Multiple scattering ofSH waves in 2-D media with many cavities

The full waveform synthetic seismogram of multiple scatteredSH waves by many cylindrical cavities in two-dimensional homogeneous elastic media is computed. We used the so-called single-layer potential integral representation of the scattered field and a discretization scheme with line source distribution for each cavity. The total field is the sum of the incident wave plus the field radiated from all sources, each multiplied by an unknown complex constant representing its strength. These constants are determined by imposing the appropriate boundary conditions in the least-squares sense. Here we solve scattering problems involving one, two, four, twelve and fifty cavities regularly distributed in a half-space. The seismograms computed along the free-surface show regions where the incident wave is strongly attenuated, as well as the arrivals of all multiple scattered phases. The accuray of the method is estimated from the degree of agreement of our solution for one cavity with the corresponding analytical solution, and also from the magnitude of the residual tractions along the boundaries of two cavities separated at various distances. Finally we apply the method to compute the case of fifty cylindrical cavities, each of radiusa, randomly distributed in a region 80a wide by 30a deep in a half-space. The value of scattering loss is obtained from the amplitude decay of the primary wave with distance for wavelengths in the range from 1.7a to 13.3a, using the synthetic seismogram calculated for the same distribution of 50 cavities as above, but in full-space.     

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.     

Elementary seismograms of seismic body waves in dissipative media

Summary Approximate expressions for elementary seismograms of seismic body waves propagating in media with small causal absorption are derived. Special attention is devoted to modulated signals with a smooth envelope, for which especially simple formulae were obtained. The derived expressions give a good picture of all important effects of causal absorption, viz., the frequency dependent exponential decrease of amplitudes, the velocity dispersion related to absorption, and the decrease of the prevailing frequency.     

Effect of seismic waves on the hydro-mechanical properties of fractured rock masses

The transmission of seismic waves in a particular region may influence the hydraulic properties of a rock mass, including permeability, which is one of the most important. To determine the effect of a seismic wave on the hydraulic behavior of a fractured rock mass, systematic numerical modeling was conducted. A number of discrete fracture network(DFN) models with a size of 20 m × 20 m were used as geometrical bases, and a discrete element method(DEM) was employed as a numerical simulation tool. Three different boundary conditions without(Type Ⅰ) and with static(Type Ⅱ) and dynamic(Type Ⅲ) loading were performed on the models, and then their permeability was calculated. The results showed that permeability in Type Ⅲ models was respectively 62.7% and 44.2% higher than in Type I and Type Ⅱ models. This study indicates that seismic waves can affect deep earth, and, according to the results, seismic waves increase the permeability and change the flow rate patterns in a fractured rock mass.     

Absorption of strain waves in porous media at seismic frequencies

An understanding of strain wave propagation in fluid containing porous rocks is important in reservoir geophysics and in the monitoring in underground water in the vicinity of nuclear and toxic waste sites, earthquake prediction, etc. Both experimental and theoretical research are far from providing a complete explanation of dissipation mechanisms, especially the observation of an unexpectedly strong dependence of attenuationQ ^–1 on the chemistry of the solid and liquid phase involved. Traditional theories of proelasticity do not take these effects into account. In this paper the bulk of existing experimental data and theoretical models is reviewed briefly in order to elecidate the effect of environmental factors on the attenuation of seismic waves. Low fluid concentrations are emphasized. Thermodynamical analysis shows that changes in surface energy caused by weak mechanical disturbances can explain observed values of attenuation in real rocks. Experimental dissipation isotherms are interpreted in terms of monolayered surface adsorption of liquid films as described by Langmuir's equation.In order to describe surface dissipation in consolidated rocks, a surface tension term is added to the pore pressure term in the O'Connell-Budiansky proelastic equation for effective moduli of porous and fractured rocks. Theoretical calculations by this modified model, using reasonable values for elastic parameters, surface energy, crack density and their geometry, lead to results which qualitatively agree with experimental data obtained at low fluid contents.     

基于地震正交各向异性的裂缝介质岩石力学建模研究

岩石力学模型是描述地层原地应力状态的基础,而常规各向同性模型难以刻画地层本征横观各向同性(VTI)和天然高角度裂缝的耦合作用,建立更为准确的正交各向异性模型变得尤为重要.本文利用VTI介质中发育单组垂直缝的Schoenberg裂缝等效模型,简化了一般正交各向异性介质的表征参数,推导了由背景VTI介质弹性参数和裂缝参数的各向异性杨氏模量、泊松比和水平地应力的精确方程.借鉴Thomsen弱各向异性近似思路,舍去裂缝弱度参数高阶扰动项,推导了正交各向异性介质岩石力学近似方程,由裂缝弱度表示的近似方程更为直观地反映了裂缝对背景介质力学性质的影响,杨氏模量和泊松比值随裂缝法向弱度的增大而显著减小.选取页岩实验数据进行数值实验,结果表明本文方程与实际物理规律相吻合,随裂缝弱度的增加,形成相同应变所需的水平地应力值降低,且近似方程与模拟结果相对误差小于5%,有助于提高岩石力学参数估算和地应力预测精度.     

Bayesian seismic inversion for estimating fluid content and fracture parameters in a gas-saturated fractured porous reservoir

Understanding the effects of in situ fluid content and fracture parameters on seismic characteristics is important for the subsurface exploration and production of fractured porous rocks. The ratio of normal-to-shear fracture compliance is typically utilized as a fluid indicator to evaluate anisotropy and identify fluids filling the fractures, but it represents an underdetermined problem because this fluid indicator varies as a function of both fracture geometry and fluid content. On the bases of anisotropic Gassmann's equation and linear-slip model, we suggest an anisotropic poroelasticity model for fractured porous reservoirs. By combining a perturbed stiffness matrix and asymptotic ray theory, we then construct a direct relationship between the PP-wave reflection coefficients and characteristic parameters of fluids(P-and S-wave moduli) and fractures(fracture quasi-weaknesses), thereby decoupling the effects of fluid and fracture properties on seismic reflection characterization.By incorporating fracture quasi-weakness parameters, we propose a novel parameterization method for elastic impedance variation with offset and azimuth(EIVOA). By incorporating wide-azimuth observable seismic reflection data with regularization constraints, we utilize Bayesian seismic inversion to estimate the fluid content and fracture parameters of fractured porous rocks. Tests on synthetic and real data demonstrate that fluid and fracture properties can be reasonably estimated directly from azimuthal seismic data and the proposed approach provides a reliable method for fluid identification and fracture characterization in a gas-saturated fractured porous reservoir.     

Computation of ray amplitudes of seismic body waves in vertically inhomogeneous media

Summary Standard methods of interpolating the velocity-depth distribution v=v(z) do not guarantee the continuity of the first and second derivatives of velocity and generate false interfaces of a higher order. These false interfaces cause anomalies in the amplitude-distance curves. It is suggested to apply the smoothed spline approximation to the depth-velocity distribution z=z(v). In this case, the ray integrals can be evaluated in a closed form. The amplitude-distance curves become quite smooth and stable. All necessary formulae and numerical examples are presented.     

Evidence for scattering of seismic PKP waves near the mantle-core boundary

The hypothesis of scattering of PKP waves near the mantle-core boundary provides a comprehensive interpretation of the observed precursors to PKIKP, certain features of which have not been adequately accounted for by any alternative interpretation. These features include the variation with distance of the times and slownesses of precursor onsets, and the variations in amplitude, azimuth and slowness observed in precursor wavetrains.The observed times and slownesses of the earliest precursor onsets are in close agreement with the theoretical least time curve for singly scattered waves.Amplitudes and slownesses of scattered waves have been calculated for earth models which are spherically symmetrical except for random variations in density and elastic parameters in a layer 200 km thick at the base of the mantle. The calculations show that observed precursor amplitudes and slownesses can be accounted for by random variations of about one percent having a correlation distance of about 30 km in this layer. In particular, it is shown that scattering structures up to 900 km above the mantle-core boundary inferred by Doornbos and Vlaar (1973) are not required by their data. There is a suggestion that the main scattering may actually occur inside a layer much less than 200 km thick at the base of the mantle.     

横向各向同性(VTI)介质中非线性地震波场模拟

数值计算方法是考察非线性弹性波在介质中（如岩石）传播特征的重要手段.非线性弹性波的数值模拟存在陡峭间断面（点）、数值振荡以及误差的指数级增长等现象而破坏数值解的稳定性、收敛性,能否消减上述现象的不利影响成为制约数值方法有效与否的重要因素.文中同时引入了FCT算子和幅值限制器,采用中心差分格式对具有垂直对称轴的横向各向同性（VTI）介质中的二维非线性弹性波进行数值模拟,从而克服了上述困难;介绍了适用于非线性弹性波的吸收边界条件,给出了差分方程的稳定性条件.在验证了方法的有效性后成功地获取了二维VTI介质中非线性弹性波的三分量地震正演记录,表明非线性波在传播过程中会发生波形畸变等现象.     

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.     

双排弹性桩隔振屏障对平面SH波的多重散射

提出了一种新的求解任意排列、任意半径的弹性桩对平面SH波的多重散射的理论方法,以解决以往假设单重散射的计算方法中不考虑桩列作为整体屏障从而忽略桩间相互干涉关系的不足,并且可用于分析多排弹性桩对平面SH波的散射性状。在数值计算分析中讨论了散射重数,排间距,桩土剪切模量比,桩数等对双排弹性桩屏障隔离效果的影响,从而对实际工程中利用排桩进行振动污染的治理和屏障隔振设计提出了重要的理论依据。     

Multiple scattering and energy transfer of seismic waves—Separation of scattering effect from intrinsic attenuation II. Application of the theory to Hindu Kush region

In order to separate the scattering effect from intrinsic attenuation, we need a multiple scattering model for seismic wave propagation in random heterogeneous media. In paper I (Wu, 1985), radiative transfer theory is applied to seismic wave propagation and the energy density distribution (or the average intensity) in space for a point source is formulated in the frequency domain. It is possible to separate the scattering effect and the absorption based on the measured energy density distribution curves. In this paper, the data from digital recordings in the Hindu Kush region are used as an example of application of the theory. We also discuss two approximate solutions of coda envelope in the time domain: the single scattering approximation and the diffusion approximation and discuss the relation with the frequency domain solution. We point out that in only two cases can the apparent attenuation be expressed as an exponential decay form. One is thedark medium case, i.e., whenB ₀0.5, whereB ₀ = _s /( _s + _a ) is the seismic albedo, _s is the scattering coefficient, _a is the absorption coefficient. In this case the absorption is dominant, the apparent attenuationb can be approximated by the coherent wave attenuationb = _s + _a . The other case is thediffuse scattering regime, i.e., whenB ₀0.5 (bright medium) andRL _s ,t _s , whereR andt are the propagation distance and lapse time,L _s and _s are the scattering lengths (mean free path) and scattering time (mean free time), respectively. However, in this case the envelope decays with a rate close to the intrinsic attenuation, while the intensity decreases with distance with a coefficientb d ₀( _s + _a ) d _{s s} , whered ₀ andd _s are the diffusion multipliers (0<d ₀,d _s <1).For the Hindu Kush region, by comparing the theory with data from two digital stations of 53 events distributed up to depths of 350 km, we find that the scattering is not the dominant factor for the measured apparent attenuation ofS waves in the frequency range 2–20 Hz. From the observation on high frequency (f>20 Hz) seismograms, we suggest the existence of a stron-scattering surface layer with fine scale heterogeneities in the crust, at least for this region.     

Uncertainty analysis: one-dimensional radioactive nuclide transport in a single fractured media

Uncertainty analysis of radioactive nuclide transport for one-dimensional single fracture has been studied. First order differential analysis is applied to introduce analytical form of output expectation and variance for contaminant transport equation, by regarding uncertainty of dispersion coefficient and retardation factor. Breakthrough curve of dimensionless concentration is demonstrated by taking I-129 as radioactive nuclide in fracture transport. It is possible to pick up critical ranges in spatial and temporal domain from the output variance. From the viewpoint of preliminary performance assessment for nuclear waste disposal the parameter importance in such system can be substantially measured in the site characterization in future.     

The scattering of seismic waves through a crust and upper mantle with random lateral and vertical inhomogeneities

The propagation of seismic waves through Earth models with slightly random lateral and vertical inhomogeneities superimposed on one composed of layers with vertical velocity gradients was investigated. The maximum deviation of velocity from a mean value at a given depth and a correlation distance derived from a two-dimensional smoothing filter were two parameters used to vary the amplitude and size of the velocity anomalies. The resulting models show short discontinuous reflectors scattered about at various depths throughout the model, and are thus in agreement with many deep seismic reflection experiments. On the other hand numerical experiments using ray-tracing techniques showed that the effect of the lateral and vertical velocity anomalies is to scatter the energy, and break up the continuous travel-time lines from vertical gradient models into travel-time segments with different slopes similar to those observed in many long range seismic refraction experiments, and to those resulting from layering effects in the media. Many of the numerical experiments which modelled the random crust produced a Pg segment and a P* segment with an apparent Conrad discontinuity at a depth of 10–20 km, this apparent depth being related to the correlation distance.When a seismic wave propagates through a heterogeneous Earth the amount of its energy which is converted into scattered energy will be a function of the inhomogeneous characteristics of the medium through which it has passed. If a ray passes through a homogeneous Earth the energy arriving at an array station should be relatively coherent whereas if the ray encounters lateral and vertical inhomogeneities its energy will be incoherent and much more complex. A series of coherency measurements done on array recordings of earthquakes at various distances showed that large lateral and vertical variations in complexity exist for different ray paths through the Earth with the region below the 650 km discontinuity in the mantle tending to be much simpler than the region just below the lithosphere.