ELASTIC WAVE PROPAGATION IN MEDIA WITH PARALLEL FRACTURES AND ALIGNED CRACKS1

A model of parallel slip interfaces simulates the behaviour of a fracture system composed of large, closely spaced, aligned joints. The model admits any fracture system anisotropy: triclinic (the most general), monoclinic, orthorhombic or transversely isotropic, and this is specified by the form of the 3 × 3 fracture system compliance matrix. The fracture system may be embedded in an anisotropic elastic background with no restrictions on the type of anisotropy. To compute the long wavelength equivalent moduli of the fractured medium requires at most the inversion of two 3 × 3 matrices. When the fractures are assumed on average to have rotational symmetry (transversely isotropic fracture system behaviour) and the background is assumed isotropic, the resulting equivalent medium is transversely isotropic and the effect of the additional compliance of the fracture system may be specified by two parameters (in addition to the two isotropic parameters of the isotropic background). Dilute systems of flat aligned microcracks in an isotropic background yield an equivalent medium of the same form as that of the isotropic medium with large joints, i.e. there are two additional parameters due to the presence of the microcracks which play roles in the stress-strain relations of the equivalent medium identical to those played by the parameters due to the presence of large joints. Thus, knowledge of the total of four parameters describing the anisotropy of such a fractured medium tells nothing of the size or concentration of the aligned fractures but does contain information as to the overall excess compliance due to the fracture system and its orientation. As the aligned microcracks, which were assumed to be ellipsoidal, with very small aspect ratio are allowed to become non-fiat, i.e. have a growing aspect ratio, the moduli of the equivalent medium begin to diverge from the standard form of the moduli for flat cracks. The divergence is faster for higher crack densities but only becomes significant for microcracks of aspect ratios approaching 0.3.     

裂隙参数对衰减各向异性影响的数值模拟

Propagation through stress-aligned fluid-filled cracks and other inclusions have been claimed to be the cause of azimuthal anisotropy observed in the crust and upper mantle.This paper examines the behavior of seismic waves attenuation caused by the internal structure of rock mass,and in particular,the internal geometry of the distribution of fluid-filled openings Systematic research on the effect of crack parameters,such as crack density,crack aspect ratio(the ratio of crack thickness to crack diameter),pore fluid properties(particularly pore fluid velocity),VP/VS ratio of the matrix material and seismic wave frequency on attenuation anisotropy has been conducted based on Hudson’s crack theory.The result shows that the crack density,aspect ratio,material filler,seismic wave frequency,and P-wave and shear wave velocity in the background of rock mass,and especially frequency has great effect on attenuation curves.Numerical research can help us know the effect of crack parameters and is a good supplement for laboratory modeling.However,attenuation is less well understood because of the great sensitivity of attenuation to details of the internal geometry.Some small changes in the characteristics of pore fluid viscosity,pore fluids containing gas and liquid phases and pore fluids containing clay can each alter attenuation coefficients by orders of magnitude.Some parameters controlling attenuation are therefore necessary to make reasonable estimations,and anisotropic attenuation is worth studying further.     

ANISOTROPIC Q AND VELOCITY DISPERSION OF FINELY LAYERED MEDIA1

When a seismic signal propagates through a finely layered medium, there is anisotropy if the wavelengths are long enough compared to the layer thicknesses. It is well known that in this situation, the medium is equivalent to a transversely isotropic material. In addition to anisotropy, the layers may show intrinsic anelastic behaviour. Under these circumstances, the layered medium exhibits Q anisotropy and anisotropic velocity dispersion. The present work investigates the anelastic effect in the long-wavelength approximation. Backus's theory and the standard linear solid rheology are used as models to obtain the directional properties of anelasticity corresponding to the quasi-compressional mode qP, the quasi-shear mode qSV, and the pure shear mode SH, respectively. The medium is described by a complex and frequency-dependent stiffness matrix. The complex and phase velocities for homogeneous viscoelastic waves are calculated from the Christoffel equation, while the wave-fronts (energy velocities) and quality factor surfaces are obtained from energy considerations by invoking Poynting's theorem. We consider two-constituent stationary layered media, and study the wave characteristics for different material compositions and proportions. Analyses on sequences of sandstone-limestone and shale-limestone with different degrees of anisotropy indicate that the quality factors of the shear modes are more anisotropic than the corresponding phase velocities, cusps of the qSV mode are more pronounced for low frequencies and midrange proportions, and in general, attenuation is higher in the direction perpendicular to layering or close to it, provided that the material with lower velocity is the more dissipative. A numerical simulation experiment verifies the attenuation properties of finely layered media through comparison of elastic and anelastic snapshots.     

Elastic properties of saturated porous rocks with aligned fractures

Elastic properties of fluid saturated porous media with aligned fractures can be studied using the model of fractures as linear-slip interfaces in an isotropic porous background. Such a medium represents a particular case of a transversely isotropic (TI) porous medium, and as such can be analyzed with equations of anisotropic poroelasticity. This analysis allows the derivation of explicit analytical expressions for the low-frequency elastic constants and anisotropy parameters of the fractured porous medium saturated with a given fluid. The five elastic constants of the resultant TI medium are derived as a function of the properties of the dry (isotropic) background porous matrix, fracture properties (normal and shear excess compliances), and fluid bulk modulus. For the particular case of penny-shaped cracks, the expression for anisotropy parameter ε has the form similar to that of Thomsen [Geophys. Prospect. 43 (1995) 805]. However, contrary to the existing view, the compliance matrix of a fluid-saturated porous-fractured medium is not equivalent to the compliance matrix of any equivalent solid medium with a single set of parallel fractures. This unexpected result is caused by the wave-induced flow of fluids between pores and fractures.     

Bayesian Markov chain Monte Carlo inversion for anisotropy of PP- and PS-wave in weakly anisotropic and heterogeneous media

A single set of vertically aligned cracks embedded in a purely isotropic background may be considered as a long-wavelength effective transversely isotropy (HTI) medium with a horizontal symmetry axis. The crack-induced HTI anisotropy can be characterized by the weakly anisotropic parameters introduced by Thomsen. The seismic scattering theory can be utilized for the inversion for the anisotropic parameters in weakly anisotropic and heterogeneous HTI media. Based on the seismic scattering theory, we first derived the linearized PP- and PS-wave reflection coefficients in terms of P- and S-wave impedances, density as well as three anisotropic parameters in HTI media. Then, we proposed a novel Bayesian Markov chain Monte Carlo inversion method of PP- and PS-wave for six elastic and anisotropic parameters directly. Tests on synthetic azimuthal seismic data contaminated by random errors demonstrated that this method appears more accurate, anti-noise and stable owing to the usage of the constrained PS-wave compared with the standards inversion scheme taking only the PP-wave into account.     

On possible fluid detection in equivalent transversely isotropic media

We consider a transversely isotropic medium that is long-wave equivalent to a stack of thin, parallel, isotropic layers and is obtained using the Backus average. In such media, we analyse the relations among anisotropy parameters; Thomsen parameters, ε and δ, and a new parameter ϕ. We discuss the last parameter and show its essential properties; it equals 0 in the case of isotropy of equivalent medium and/or constant Lamé coefficient λ in layers. The second property occurs to make ϕ sensitive to variations of λ in thin-bedded sequences. According to Gassmann, in isotropic media the variation of fluid content affects only the Lamé coefficient λ, not μ; thus, the sensitivity to changes of λ is an essential property in the context of possible detection of fluids. We show algebraically and numerically that ϕ is more sensitive to these variations than ε or δ. Nevertheless, each of these parameters is dependent on the changes of μ; to understand this influence, we exhibit comprehensive tables that illustrate the behaviour of anisotropy parameters with respect to specific variations of λ and μ. The changes of μ in layers can be presented by the Thomsen parameter γ that depends on them solely. Hence, knowing the values of elasticity coefficients of equivalent transversely isotropic medium, we may compute ϕ and γ, and based on the aforementioned tables, we predict the expected variation of λ; in this way, we propose a new method of possible fluid detection. Also, we show that the prior approach of possible detection of fluids, proposed by Berryman et al., may be unreliable in specific cases. To establish our results, we use the Monte Carlo method; for the range and chosen variations of Lamé coefficients λ and μ – relevant to sandstones – we generate these coefficients in thin layers and, after the averaging process, we obtain an equivalent transversely isotropic medium. We repeat that process numerous times to get many equivalent transversely isotropic media, and – for each of them  – we compute their anisotropy parameters. We illustrate ϕ, ε and δ in the form of cross-plots that are relevant to the chosen variations of λ and μ. Additionally, we present a table with the computed ranges of anisotropy parameters that correspond to different variations of Lamé coefficients.     

SYSTEMATIC CLASSIFICATION OF LAYER-INDUCED TRANSVERSE ISOTROPY*

Waves propagating through a sequence of layers that are thin compared with the wavelength show effects of anisotropy: velocity and displacement direction depend on the angle between the plane of layering and the wave normal, and shear waves split up into two distinct types of different velocity. The layered medium can thus be replaced by a transversely isotrophic medium the parameters of which depend on the parameters of the individual constituent layers. A survey of the anisotropy effects possible in such a medium is generally done by varying the layer parameters in order to obtain different replacement media. This approach guarantees that the replacement medium is realistic, but it does not guarantee adequate sampling of the set of replacement media. To this end one has to begin by selecting the replacement media and then check whether the chosen media possess stable (and eventually realistic) representations by layer sequences. In general, there is an infinite number of layer representations for any transversely isotropic medium that can at all be represented. However, if one restricts the solutions to those requiring the minimal number of layers and the minimum number of different layer parameters, the set of solutions has only one free parameter (i.e., it is a one-dimensional manifold), and an important subset even has a unique solution. A simple algorithm exists for the determination of these “simplest representations”. Aside from sampling the set of representable transversely isotropic media for survey purposes, the method can be applied to the problem of determining the cause of observed anisotropy effects (or lateral changes in such effects). If this method can be applied to real data, it would for instance allow to determine changes in relative thickness or lithology on a scale smaller than the limit of resolution of the seismic method.     

各向异性介质中的AVO

分析了横向各向同性和方位各向异性介质的本构关系,由此讨论弹性波在两种各向异性介质中的传播特点,提出可表征这两种介质各向异性程度的广义参数．以此为基础讨论了两种各向异性介质中存在水平界面时的反射系数近似式,将Dely等人推导的横向各向同性介质中的反射系数公式推广到方位各向异性介质的主轴方向上．根据算例讨论修正的Banik和Thomsen的近似式,着重分析两种各向异性介质中的AVO关系及其对实际勘探的影响和指导意义．     

Elastic properties of double-porosity rocks using the differential effective medium model

An approach to determining the effective elastic moduli of rocks with double porosity is presented. The double‐porosity medium is considered to be a heterogeneous material composed of a homogeneous matrix with primary pores and inclusions that represent secondary pores. Fluid flows in the primary‐pore system and between primary and secondary pores are neglected because of the low permeability of the primary porosity. The prediction of the effective elastic moduli consists of two steps. Firstly, we calculate the effective elastic properties of the matrix with the primary small‐scale pores (matrix homogenization). The porous matrix is then treated as a homogeneous isotropic host in which the large‐scale secondary pores are embedded. To calculate the effective elastic moduli at each step, we use the differential effective medium (DEM) approach. The constituents of this composite medium – primary pores and secondary pores – are approximated by ellipsoidal or spheroidal inclusions with corresponding aspect ratios. We have applied this technique in order to compute the effective elastic properties for a model with randomly orientated inclusions (an isotropic medium) and aligned inclusions (a transversely isotropic medium). Using the special tensor basis, the solution of the one‐particle problem with transversely isotropic host was obtained in explicit form. The direct application of the DEM method for fluid‐saturated pores does not account for fluid displacement in pore systems, and corresponds to a model with isolated pores or the high‐frequency range of acoustic waves. For the interconnected secondary pores, we have calculated the elastic moduli for the dry inclusions and then applied Gassmann's tensor relationships. The simulation of the effective elastic characteristic demonstrated that the fluid flow between the connected secondary pores has a significant influence only in porous rocks containing cracks (flattened ellipsoids). For pore shapes that are close to spherical, the relative difference between the elastic velocities determined by the DEM method and by the DEM method with Gassmann's corrections does not exceed 2%. Examples of the calculation of elastic moduli for water‐saturated dolomite with both isolated and interconnected secondary pores are presented. The simulations were verified by comparison with published experimental data.     

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.     

Quantifying Damage, Saturation and Anisotropy in Cracked Rocks by Inverting Elastic Wave Velocities

Crack damage results in a decrease of elastic wave velocities and in the development of anisotropy. Using non-interactive crack effective medium theory as a fundamental tool, we calculate dry and wet elastic properties of cracked rocks in terms of a crack density tensor, average crack aspect ratio and mean crack fabric orientation from the solid grains and fluid elastic properties. Using this same tool, we show that both the anisotropy and shear-wave splitting of elastic waves can be derived. Two simple crack distributions are considered for which the predicted anisotropy depends strongly on the saturation, reaching up to 60% in the dry case. Comparison with experimental data on two granites, a basalt and a marble, shows that the range of validity of the non-interactive effective medium theory model extends to a total crack density of approximately 0.5, considering symmetries up to orthorhombic. In the isotropic case, Kachanov's (1994) non-interactive effective medium model was used in order to invert elastic wave velocities and infer both crack density and aspect ratio evolutions. Inversions are stable and give coherent results in terms of crack density and aperture evolution. Crack density variations can be interpreted in terms of crack growth and/or changes of the crack surface contact areas as cracks are being closed or opened respectively. More importantly, the recovered evolution of aspect ratio shows an exponentially decreasing aspect ratio (and therefore aperture) with pressure, which has broader geophysical implications, in particular on fluid flow. The recovered evolution of aspect ratio is also consistent with current mechanical theories of crack closure. In the anisotropic cases—both transverse isotropic and orthorhombic symmetries were considered—anisotropy and saturation patterns were well reproduced by the modelling, and mean crack fabric orientations we recovered are consistent with in situ geophysical imaging. Our results point out that: (1) It is possible to predict damage, anisotropy and saturation in terms of a crack density tensor and mean crack aspect ratio and orientation; (2) using well constrained wave velocity data, it is possible to extrapolate the contemporaneous evolution of crack density, anisotropy and saturation using wave velocity inversion as a tool; 3) using such an inversion tool opens the door in linking elastic properties, variations to permeability.     

Experimental verification of effective anisotropic crack theories in variable crack aspect ratio medium

Physical modelling of cracked/fractured media using downscaled laboratory experiments has been used with great success as a useful alternative for understanding the effect of anisotropy in the hydrocarbon reservoir characterization and in the crustal and mantle seismology. The main goal of this work was to experimentally verify the predictions of effective elastic parameters in anisotropic cracked media by Hudson and Eshelby–Cheng's effective medium models. For this purpose, we carried out ultrasonic measurements on synthetic anisotropic samples with low crack densities and different aspect ratios. Twelve samples were prepared with two different crack densities, 5% and 8%. Three samples for each crack density presented cracks with only one crack aspect ratio, whereas other three samples for each crack density presented cracks with three different aspect ratios in their composition. It results in samples with aspect ratio values varying from 0.13 to 0.26. All the cracked samples were simulated by penny‐shaped rubber inclusions in a homogeneous isotropic matrix made with epoxy resin. Moreover, an isotropic sample for reference was constructed with epoxy resin only. Regarding velocity predictions performed by the theoretical models, Eshelby–Cheng shows a better fit when compared with the experimental results for samples with single and mix crack aspect ratio (for both crack densities). From velocity values, our comparisons were also performed in terms of the ε, γ, and δ parameters (Thomsen parameters). The results show that Eshelby–Cheng effective medium model fits better with the measurements of ε and γ parameters for crack samples with only one type of crack aspect ratio.     

Research on propagation properties of elastic waves in two-phase anisotropic media

IntroductionItiswellknownthatanisotropylieswidelyintheundergroundmedia.Anisotropicmediawhicharemetintheseismicengineeringandseismicexplorationofenergyaremainlycausedbytheperiodicthinlayers(PTL)andextensivedilatancyanisotropy(EDA).Insuchmedia,anisotropyleadstomorecomplicatepropagationofseismicwave,thesignificantfeatureinanisotropicmediaisvelocityanisotropy.Infact,undergroundstrataareverycomplicated,whichareusuallycomposedofsolidframeandfluid(suchasoil,gasesorwater)inpores.Inordertostudyseism…     

MODELLING CHANNEL WAVES WITH SYNTHETIC SEISMOGRAMS IN AN ANISOTROPIC IN-SEAM SEISMIC SURVEY1

In-seam seismic surveys with channel waves have been widely used in the United Kingdom and elsewhere to map coal-seams and to detect anomalous features such as dirt bands, seam thinning and thickening, and particularly in-seam faulting. Although the presence of cleat-induced anisotropy has been recognized in the past, almost all previous analyses have assumed homogeneous isotropic or transversely isotropic coal-seams. Channel waves, however, exhibit properties which cannot be fully explained without introducing anisotropy into the coal-seam. In particular, Love-type channel waves are observed for recording geometries where, in a homogeneous isotropic or transversely isotropic structure, the source would not be expected to excite transverse motion. Similarly, modes of channel-wave propagation display the coupled three-component motion of generalized modes in anisotropic substrates, which would not be expected for Rayleigh and Love wave motion in isotropy or in transversely isotropic media with azimuthal isotropy. We model the observed in-seam seismic channel waves with synthetic seismograms to gain an understanding of the effects of cleat-induced anisotropy on the behaviour of channel waves. The results show a reasonable good match with the observations in traveltime, relative amplitudes, dispersion characteristics and particle motions. We demonstrate that anisotropy in the surrounding country rocks contributes significantly to the coupling of channel wave particle motion, although its effect is not as strong as the anisotropy in the coal-seam. We conclude that the effects of cleat- and stress-induced anisotropy are observed and can be modelled with synthetic seismograms, and that anisotropy must be taken into account for the detailed interpretation of channel waves.     

利用伪谱法模拟横向各向同性介质中的波

介质的弹性常数为三维四阶张量的分量,共有８１个,由于应力张量和应变张量的对称性及能量密度是应变的二次函数,一般各向异常性介质的独立弹性常数可减为２１个,如果介质具有较高的对称性,独立弹性常数的数目会更少。 对于地壳和上地幔,具有５个独立弹性常数的横向各向同性介质是一个非常好的近似,本研究中横向各向同性介质的对称轴方向可以是任意的（即对称轴可以不平等于铅直方向）,在此情况下,需要进行坐标变换,如果已知介质在某一坐标系（其坐标轴平行或垂直于介质的对称轴）中的弹性常数,我们能够容易地利用变换公式得到变换后新坐标系中的弹性常数。 本文提出了一种方案,利用伪谱法既能模拟横向各向同性介质中的平面波,也能模拟点源激发的波场。在勘探地球物理和地震学中,模拟横向各向同性介拮中传播的平面波及区域源产生的波是最重要的研究课题之一。然而在一般各向异性介质中,很难或不可能确定弹性波的相速度和偏振方向,但在横向各向同性介质中,则可以通过坐标变换来实现,这里我们所提出的方法可以用于横向各向同性介质中弹性波的模拟。     

First-order perturbation approximation for rock elastic moduli in transversely isotropic media

Seismic anisotropy is a relatively common seismic wave phenomenon in laminated sedimentary rocks such as shale and it can be used to investigate mechanical properties of such rocks and other geological materials. Young’s modulus and Poisson’s ratio are the most common mechanical properties determined in various rock engineering practices. Approximate and explicit equations are proposed for determining Young’s modulus and Poisson’s ratio in anisotropic rocks, in which the symmetry plane and symmetry axis of the anisotropy are derived from the constitutive equation of transversely isotropic rock. These equations are based on the media decomposition principle and seismic wave perturbation theory and their accuracy is tested on two sets of laboratory data. A strong correlation is found for Young’s modulus in two principal directions and for Poisson’s ratio along the symmetry plane. Further, there is an underprediction of Poisson’s ratio along the symmetry axis, although the overall behavior follows the trend of the measured data. Tests on a real dataset show that it is necessary to account for anisotropy when characterizing rock mechanical properties of shale. The approximate equations can effectively estimate anisotropic Young’s modulus and Poisson’s ratio, both of which are critical rock mechanical data input for hydraulic fracturing engineering.     

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.     

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.     

点力源在横向各向同性介质中激发的弹性波

点力源在均匀横向各向同性介质中激发的弹性波可以用透射-反射矩阵法进行求解。假定点力源位于一假想的各向同性层中,并利用各向同性介质中等效于势函数的过源面间断,从而易于求出介质中各个点上的弹性波场;再令假想的各向同性层的厚度趋近于0,则可求出原始问题的解,即横向各向同性介质中的格林函数。最后得出的结果和假想的各向同性层的性质无关,表明这一方法用于求解点力源在各向异性介质中激发的弹性波场是可行的。     

Aligned vertical fractures, HTI reservoir symmetry and Thomsen seismic anisotropy parameters for polar media

Certain crack-influence parameters of Sayers and Kachanov are shown to be directly related to Thomsen's weak-anisotropy seismic parameters for fractured reservoirs when the crack/fracture density is small enough. These results are then applied to the problem of seismic wave propagation in polar reservoirs, i.e., those anisotropic reservoirs having two axes that are equivalent but distinct from the third axis), especially for horizontal transversely isotropic seismic wave symmetry due to the presence of aligned vertical fractures and resulting in azimuthal seismic wave symmetry at the Earth's surface. The approach presented suggests one method of inverting for fracture density from wave speed data. A significant fraction of the technical effort in the paper is devoted to showing how to predict the angular location of the true peak (or trough) of the quasi-SV-wave for polar media and especially how this peak is related to another angle that is very easy to compute. The axis of symmetry is always treated here as the x ₃-axis for either vertical transversely isotropic symmetry (due, for example, to horizontal cracks), or horizontal transversely isotropic symmetry (due to aligned vertical cracks). Then, the meaning of the stiffnesses is derived from the fracture analysis in the same way for vertical transversely isotropic and horizontal transversely isotropic media, but for horizontal transverse isotropy the wave speeds relative to the Earth's surface are shifted by  90^o  in the plane perpendicular to the aligned vertical fractures. Skempton's poroelastic coefficient B is used as a general means of quantifying the effects of fluids inside the fractures. Explicit Biot-Gassmann-consistent formulas for Thomsen's parameters are also obtained for either drained or undrained fractures resulting in either vertical transversely isotropic or horizontal transversely isotropic symmetry of the reservoir.