Wave propagation in a general anisotropic poroelastic medium: Biot’s theories and homogenisation theory

Anisotropic wave propagation is studied in a fluid-saturated porous medium, using two different approaches. One is the dynamic approach of Biot’s theories. The other approach known as homogenisation theory, is based on the averaging process to derive macroscopic equations from the microscopic equations of motion. The medium considered is a general anisotropic poroelastic (APE) solid with a viscous fluid saturating its pores of anisotropic permeability. The wave propagation phenomenon in a saturated porous medium is explained through two relations. One defines modified Christoffel equations for the propagation of plane harmonic waves in the medium. The other defines a matrix to relate the relative displacement of fluid particles to the displacement of solid particles. The modified Christoffel equations are solved further to get a quartic equation whose roots represent complex velocities of the four attenuating quasi-waves in the medium. These complex velocities define the phase velocities of propagation and quality factors for attenuation of all the quasi-waves propagating along a given phase direction in three-dimensional space. The derivations in the mathematical models from different theories are compared in order to work out the equivalence between them. The variations of phase velocities and attenuation factors with the direction of phase propagation are computed, for a realistic numerical model. Differences between the velocities and attenuations of quasi-waves from the two approaches are exhibited numerically.     

Effect of initial stress on reflection at the free surface of anisotropic elastic medium

The propagation of plane waves is considered in a general anisotropic elastic medium in the presence of initial stress. The Christoffel equations are solved into a polynomial of degree six. The roots of this polynomial represent the vertical slowness values for the six quasi-waves resulting from the presence of a discontinuity in the medium. Three of these six values are identified with the three quasi-waves traveling in the medium but away from its boundary. Reflection at the free plane surface is studied for partition of energy among the three reflected waves. For post-critical incidence, the reflected waves are inhomogeneous (evanescent) waves. Numerical examples are considered to exhibit the effects of initial stress on the phase direction, attenuation and reflection coefficients of the reflected waves. The phase velocities and energy shares of the reflected waves change significantly with initial stress as well as anisotropic symmetry. The presence of initial stress, however, has a negligible effect on the phase directions of reflected waves.     

Polarisations of quasi-waves in a general anisotropic porous solid saturated with viscous fluid

Wave propagation is studied in a general anisotropic poroelastic solid saturated with a viscous fluid flowing through its pores of anisotropic permeability. The extended version of Biot’s theory is used to derive a system of modified Christoffel equations for the propagation of plane harmonic waves in such media. The non-trivial solution of this system is ensured by a biquadratic equation whose roots represent the complex velocities of four attenuating quasi-waves in the medium. These complex velocities define phase velocity and attenuation of each quasi-wave propagating along a given phase direction in three-dimensional space. The solution itself defines the polarisations of the quasi-waves along with phase shift. The variations of polarisations of quasi-waves with their phase direction, are computed for a realistic numerical model.     

Existence of longitudinal waves in pre-stressed anisotropic elastic medium

In a pre-stressed anisotropic elastic medium, three types of quasi-waves propagate along an arbitrary direction. In general, none of the waves is truly longitudinal. The present study finds the specific directions in a pre-stressed anisotropic elastic medium along which longitudinal waves may propagate. This paper demonstrates how the propagation of longitudinal waves is affected by various pre-stresses present in the medium. The study establishes the explicit expressions defining the existence and propagation of longitudinal waves in pre-stressed anisotropic elastic medium. These expressions involve not only the direction and elastic stiffness of the medium, but also the pre-stresses present in the medium. Changes in conditions for the existence of longitudinal waves in orthotropic, monoclinic and triclinic anisotropies are discussed in detail. The most important part of the paper is a practical aspect suggested to calculate the specific directions for the existence of longitudinal waves in pre-stressed anisotropic elastic medium. In this approach, only those parameters are used that can be observed by the receiver in a geophysical experiment of wave propagation. The existence of longitudinal waves has been shown graphically using a numerical example for three types of anisotropic symmetries in elastic medium.     

Propagation of Love wave in fiber‐reinforced medium lying over an initially stressed orthotropic half‐space

In this paper, the propagation of Love waves in a fiber‐reinforced layered medium lying over an elastic orthotropic half‐space under initial stress has been investigated. We have obtained the velocity equations for Love wave in this media. It is observed that propagation of Love wave is influenced by reinforced parameters and initial stressed parameter. The velocity of Love wave has been computed for three different cases. Our computed equation of Love wave coincides with the standard equation of Love wave for the case of homogeneous layer and homogeneous half‐space (AEH Love, 1911). To study the effect of reinforced and initial stressed parameters, we have computed the numerical values for phase velocity and plotted in several figures. It is observed that the phase velocity decreases with the increases of reinforced parameters and initial stressed parameter. Using MATLAB software, GUI has been developed to generalize the effect of various parameters. Copyright © 2014 John Wiley & Sons, Ltd.     

Torsional wave propagation in non‐homogeneous layer between non‐homogeneous half‐spaces

The study of surface wave in a layered medium has their possible application in geophysical prospecting. In the present work, dispersion equation for torsional wave in an inhomogeneous isotropic layer between inhomogeneous isotropic half‐spaces has been derived. Two cases are discussed separately for torsional wave propagation in inhomogeneous layer between homogeneous and non‐homogeneous half‐spaces, respectively. Further, two possible modes for torsional wave propagation are obtained in case of inhomogeneous layer sandwiched between non‐homogeneous half‐spaces. Closed form solutions for displacement in the layer and half‐spaces are obtained in each case. The study reveals that the layer width, layer inhomogeneity, frequency of inhomogeneity, as well as inhomogeneity in the half‐space has significant effect on the propagation of torsional surface waves. Displacement and implicit dispersion equation for torsional wave velocities are expressed in terms of Heun functions and their derivatives. Effects of inhomogeneity on torsional wave velocity are also discussed graphically by plotting the dimensionless phase velocity against dimensionless and scaled wave number for different values of inhomogeneity parameter. Copyright © 2012 John Wiley & Sons, Ltd.     

致密岩石纵横波波速各向异性的比较研究

通过对板岩、千枚岩、糜棱岩和变质砂岩等4种岩石在平行和垂直层理、板理的3个正交方向上的纵、横波波速试验,提出了致密岩石纵波波速的各向异性与横波波速的各向异性普遍存在着一致性和差异性两种特征。一致性特征主要表现为致密岩石的横波各向异性随纵波各向异性的增强而增强;差异性特征主要表现为在平行和垂直层理、板理的两个方向上,纵波波速的各向异性指数大于横波,亦即纵波波速的各向异性比横波更为显著。从横观各向同性弹性介质的波速方程出发,通过具体的算例对这两种特征进行了较好的理论分析。     

各向异性弹性波动交错网格高阶差分法的误差研究

研究了各向异性介质中弹性波动方程交错网格高阶差分法求解的数值相速度色散和数值相速度误差。从而表明了交错网格差分法一般情况下要求每个最短波长离散网格点数约为６。同时表明了高阶差分（４，６）阶的数值各向异性要比低阶的（２，２）阶小得多。     

三维方位各向异性介质数值模拟及波场分析

采用交错网格技术将速度-应力方程中的速度对时间的导数转化为应力对空间的导数, 将弹性波动方程表示为与二阶双曲方程等价的一阶应力-速度公式, 以实现三维三分量地震波场模拟。对获得的波场快照、VSP记录图像和地面记录图像进行波场分析, 发现在波场图像中存在明显的拟P波、拟快横波和慢横波, 还出现了横波分裂、横波分裂盲区及波面三分叉等地震波在方位各向异性介质传播时产生的特殊现象。研究结果表明, 用交错网格三维模拟方法研究方位各向异性介质对方位各向异性介质中地震波传播反演能起到很强的辅助作用。     

单轴加载过程中砂岩声学特性研究

岩体的声学特性与应力状态和破坏程度密切相关,通过岩体声学特性的变化来分析岩体应力状态进而评价工程稳定性是一种行之有效的工程措施。针对砂岩开展了单轴压缩试验,并在加载过程中同步进行3个方向的声波测试,获得了砂岩加载过程中3个不同方向声波波速与应力的演化规律。试验结果表明：随着应力的增加,轴向波速逐渐增大,横向波速表现出先增后减的趋势。考虑到不同方向声波测试结果的差异性,采用含不同倾角裂隙的石膏试样进行声波试验。结果表明,当裂隙方向与声波传播方向一致时,波速最大,与声波传播方向垂直时,波速最小;此外,为分析岩样波速与应力状态的相关性,建立了波速与体应变的关系,结果表明,随着体应变的增加,平均波速逐渐增大,在体应变达到最大值附近时,平均波速达到最大值,在体应变下降阶段,波速开始下降;根据轴向波速与应力的变化规律,得到了应力与波速的指数函数拟合公式,据此可以通过现场测试获得的波速预测现场岩体的应力范围,进而评价工程岩体稳定性。     

Long-wavelength propagation of waves in jointed rocks - study using resonant column experiments and model material

The wave propagation in jointed rock mass depends on many factors, in particular the wave frequency, existing stress conditions and the induced strain levels. When the wavelength of the propagating wave is much longer than the spacing of the joints, it is referred to as long-wavelength condition. This is a dominant condition in seismological studies due to the presence of closely spaced joints in rocks. A comprehensive study on long-wavelength propagation of shear and compression waves in very weak rocks has been carried out with the help of a resonant column apparatus using a model material (plaster of Paris) at various strain levels. The working principle and suitability of the apparatus for testing stiff samples are discussed. The velocity reductions of shear and compression waves across joints are obtained. The influence of frictional and filled joints in attenuating stress waves under various strain levels is analysed. Wave velocities are found to be reduced with increasing strain levels and decreasing joint spacing. The joints with gouge material are more efficient at damping the waves. It is recognised that wave velocity reductions and damping across joints are functions of confining stress and strain levels induced in very weak rocks.     

各向异性介质相速度、群速度与克利斯托费尔方程

本文详细回顾了克利斯托费尔方程(Christoffel)的推导.在此基础上,提炼出时空域和频率域克利斯托费尔方程,前者可作正演计算介质的相速度,后者可计算介质的群速度.并计算了横观各向同性介质的相速度和群速度.最后对无损耗各向异性介质的相速度和群速度的关系进行了简单讨论.     

Directional dependence of P- and S-wave propagation and polarization in foliated rocks from the Kola superdeep well: Evidence from laboratory measurements and calculations based on TOF neutron diffraction

We have measured P- and S-wave velocities on two amphibolite and two gneiss samples from the Kola superdeep borehole as a function of pressure (up to 600 MPa) and temperature (up to 600 °C). The velocity measurements include compressional (Vp) and shear wave velocities (Vs1, Vs2) propagating in three orthogonal directions which were in general not parallel to inherent rock symmetry axes or planes. The measurements are accompanied by 3D-velocities calculations based on lattice preferred orientation (LPO) obtained by TOF (Time Of Flight) neutron diffraction analysis which allows the investigation of bulk volumes up to several cubic centimetres due to the high penetration depth of neutrons. The LPO-based numerical velocity calculations give important information on the different contribution of the various rock-forming minerals to bulk elastic anisotropy and on the relations of seismic anisotropy, shear wave splitting, and shear wave polarization to the structural reference frame (foliation and lineation). Comparison with measured velocities obtained for the three propagation directions that were not in accordance with the structural frame of the rocks (foliation and lineation) demonstrate that for shear waves propagating through anisotropic rocks the vibration directions are as important as the propagation directions. The study demonstrates that proper measurement of shear wave splitting by means of two orthogonal polarized sending and receiving shear wave transducers is only possible when their propagation and polarization directions are parallel and normal to foliation and lineation, respectively.     

Rayleigh waves in liquid layer resting over an initially stressed orthotropic half-space under self-weight

The study of surface waves (Rayleigh wave) finds their virtuous applications in a numerous geological and geophysical fields including water, oil, gas, and other subsurface geological probing and exploration. The present paper efforts to investigate the influence of initial stress, Earth magnetism, and gravity on propagation of Rayleigh waves. Considered model is consist of a liquid layer lying over a magnetoelastic orthotropic half-space under self-weight and initial stress. Method of separation of variable is used to solve the equation of motion. Solutions of governing equations are obtained in terms of displacement. Frequency relation for Rayleigh wave has been obtained and matched with classical Rayleigh wave equation. In addition to classical case, some existing results have been deduced as particular case of the present study. Obtained results have been shown through numerical illustrations. It is found that the considered parameters (initial stress, Earth magnetism, and gravity) have prominent effect on phase velocity of Rayleigh wave. Graphical representations have been made to exhibit the velocity profile of Rayleigh waves for different cases with the help of MATHEMATICA. The present study may be useful for seismologists and engineers who are concern with applications of wave propagation in magnetoelastic orthotropic medium.     

表面波有效相速度近似分析方法

分层介质中瑞利面波有多个模态,表面瞬态响应是多个模态响应的叠加。在近场,面波模态响应传播速度随传播距离而变化;在远场,其趋于模态相速度。由分层介质表面两不同点响应互谱分析(SASW)得到的有效相速度并不对应于面波基阶模态相速度,它与波场中高阶模态能量分配比例有关。有效相速度随传播距离而变化,近场体波对有效相速度影响较大。对分层介质在简谐荷载下表面质点位移响应进行了互谱分析,得到了有效相速度理论值,通过理论值与测试值匹配分析可估算分层参数,该分析方法考虑了近场及高阶面波模态对有效相速度的影响。     

Aligned fractures in carbonate rocks: laboratory and in situ measurements of seismic anisotropy

By vertical seismic profiling and shear wave analysis we show that a packet of carbonate reservoir rocks, found at nearly 3000 m depth in the North German Basin, is seismically anisotropic. For vertical paths of wave propagation the estimated velocity difference of the split shear waves is 10%. No shear wave birefrigence is observed within the hangingwall which, therefore, has to be regarded as isotropic or transversely isotropic. Additional laboratory investigations of the petrography of drilled carbonate samples and of their seismic velocities show that the anisotropy is most probably caused by subvertical fractures with preferred azimuthal orientation. The strike direction of the aligned fractures determined by analysis of split shear waves is approximately N55°E. This value agrees with recently published directions of maximum horizontal tectonic stress in pre-Zechstein sediments in the eastern part of the North German Basin, but it is in contrast to the world stress map. Received: 20 April 1999 / Accepted: 25 August 1999     

Wave propagation in thermoelastic saturated porous medium

Biot’s theory for wave propagation in saturated porous solid is modified to study the propagation of thermoelastic waves in poroelastic medium. Propagation of plane harmonic waves is considered in isotropic poroelastic medium. Relations are derived among the wave-induced temperature in the medium and the displacements of fluid and solid particles. Christoffel equations obtained are modified with the thermal as well as thermoelastic coupling parameters. These equations explain the existence and propagation of four waves in the medium. Three of the waves are attenuating longitudinal waves and one is a non-attenuating transverse wave. Thermal properties of the medium have no effect on the transverse wave. The velocities and attenuation of the longitudinal waves are computed for a numerical model of liquid-saturated sandstone. Their variations with thermal as well as poroelastic parameters are exhibited through numerical examples.     

Evaluation of equivalent medium methods for stress wave propagation in jointed rock mass

This paper evaluates the existing equivalent medium methods for jointed rock mass and further develops the equivalent viscoelastic medium method proposed by the authors. The advantages and limitations of different equivalences to the discontinuous rock mass are discussed. Theoretical derivation of stress wave propagation through the equivalent viscoelastic medium is carried out by adopting the Fourier transformation method, and the parameters of the equivalent viscoelastic medium method are determined analytically. The frequency dependence and the wave attenuation phenomenon can be properly described when the imaginary terms of the complex moduli of the rock mass are included. The results show that the equivalent viscoelastic medium method is able to predict the effective velocity and the stress wave transmission coefficient in a rock mass more accurately than the conventional effective elastic moduli methods. An example of the stress wave propagation through rock mass with parallel joints shows that the equivalent viscoelastic medium method is promising and worthy to be further explored for application in practical rock engineering problems. Copyright © 2011 John Wiley & Sons, Ltd.     

裂隙型单斜介质中弹性系数的计算及波的传播特性研究

根据Hudson等关于裂隙介质弹性系数计算的扰动理论及Bond变换矩阵原理，给出了各向同性介质中含多组垂直裂隙时等效弹性系数的计算方法。计算了含两组斜交的垂直裂隙形成的单斜各向异性介质中的等效弹性系数，并根据Christoffel方程推导、得出水平面内平面波传播的相速度和群速度随方位变化的特性。     

Propagation of Rayleigh waves in unsaturated porothermoelastic media

Based on the frame of elastic theory for unsaturated porous medium, considering the influence of thermal effect, the propagation characteristics of Rayleigh wave in unsaturated porous media are studied. Firstly, the thermoelastic wave equations for three-phase porous media are established, in which the mass balance equations, generalized Darcy law, momentum balance equations, and generalized non-Fourier heat conduction law are taken into account. Secondly, through theoretical derivation, the dispersion equation of Rayleigh wave for unsaturated porothermoelastic media is given by introducing the potential functions. Finally, the variations of the phase velocity of Rayleigh wave are analyzed with numerical examples. The results show that the thermal conductivity has little effect on the phase velocity of Rayleigh wave. The phase velocity of Rayleigh wave increase with increasing of the thermal expansion coefficient and media temperature.