The impact of different clay minerals on the anisotropy of clay matrix in shale

Although clay is composed of disconnected anisotropic clay platelets, many rock physics models treat the clay platelets in shale as interconnected. However, the clay matrix in shales can be modelled as anisotropic clay platelets embedded within a soft isotropic interplatelet region, allowing the influence of disconnected clay platelets on the elastic properties of the clay matrix to be analysed. In this model, properties of the interplatelet region are governed by its effective bulk and shear moduli, whereas the effective properties of the clay platelets are governed by their volume fraction, aspect ratio and elastic stiffness tensor. Together, these parameters implicitly account for variations in clay and fluid properties, as well as fluid saturation. Elastic stiffnesses of clay platelets are obtained from the literature, including both experimental measurements and first-principles calculations of the full anisotropic (monoclinic or triclinic) elastic stiffness tensors of layered silicates. These published elastic stiffness tensors are used to compile a database of equivalent transverse isotropic elastic stiffness tensors, and other physical properties, for eight common varieties of layered silicates. Clay matrix anisotropy is then investigated by examining the influence of these different elastic stiffnesses, and of varying model parameters, upon the effective transverse isotropic elastic stiffness tensor of the clay matrix. The relationship between the different clay minerals and their associated anisotropy parameters is studied, and their impact on the resulting anisotropy of the clay matrix is analysed.     

Reference transversely isotropic medium approximating a given generally anisotropic medium

For a given stiffness tensor (tensor of elastic moduli) of a generally anisotropic medium, we can estimate the extent to which the medium is transversely isotropic, and determine the direction of its reference symmetry axis. In this paper, we rotate the given stiffness tensor about this reference symmetry axis, and determine the reference transversely isotropic (uniaxial) stiffness tensor as the average of the rotated stiffness tensor over all angles of rotation. The obtained reference transversely isotropic (uniaxial) stiffness tensor represents an analytically differentiable approximation of the given generally anisotropic stiffness tensor. The proposed analytic method is compared with a previous numerical method in two numerical examples.     

Ambiguous Moment Tensors and Radiation Patterns in Anisotropic Media with Applications to the Modeling of Earthquake Mechanisms in W-Bohemia

Anisotropic material properties are usually neglected during inversions for source parameters of earthquakes. In general anisotropic media, however, moment tensors for pure-shear sources can exhibit significant non-double-couple components. Such effects may be erroneously interpreted as an indication for volumetric changes at the source. Here we investigate effects of anisotropy on seismic moment tensors and radiation patterns for pure-shear and tensile-type sources. Anisotropy can significantly influence the interpretation of the source mechanisms. For example, the orientation of the slip within the fault plane may affect the total seismic moment. Also, moment tensors due to pure-shear and tensile faulting can have similar characteristics depending on the orientation of the elastic tensor. Furthermore, the tensile nature of an earthquake can be obscured by near-source anisotropic properties. As an application, we consider effects of inhomogeneous anisotropic properties on the seismic moment tensor and the radiation patterns of a selected type of micro-earthquakes observed in W-Bohemia. The combined effects of near-source and along-path anisotropy cause characteristic amplitude distortions of the P, S1 and S2 waves. However, the modeling suggests that neither homogeneous nor inhomogeneous anisotropic properties alone can explain the observed large non-double-couple components.The results also indicate that a correct analysis of the source mechanism, in principle, is achievable by application of anisotropic moment tensor inversion.     

Inferring the Orientation-distribution Function from Observed Seismic Anisotropy: General Considerations and an Inversion of Surface-wave Dispersion Curves

—A general relation linking the elasticity tensor of an anisotropic medium with that of the constituting single crystals and the function describing the orientation distribution of the crystals is derived. By expanding the orientation distribution function (ODF) into tensor spherical harmonics and using canonical components of the elasticity tensors, it is shown that the elastic tensor of the medium is completely determined by a finite number of expansion coefficients, namely those with harmonic degree l≤ 4. The number of expansion coefficients actually needed to determine the elastic constants of the medium depends on the symmetry of the single crystals. For hexagonal symmetry of the single crystals it is shown that only 8 real numbers are required to fix the 13 elastic constants which are for example needed to determine the azimuthal dependence of surface wave velocities. Thus, inversions of observations of seismic anisotropy are feasible which do not make any a priori assumptions on the orientation of the crystals. As a byproduct of the derivation, a formula is given which allows the easy calculation of the elastic constants of a medium composed of hexagonal crystals obeying an arbitrary ODF. An application of the theoretical results to the inversion of surface wave dispersion curves for an anisotropic 1D-mantle model is presented. For the S-wave velocities the results are similar to those of previous inversions but the new approach also yields P-wave velocities consistent with the assumption of oriented olivine. Moreover it provides a hint of the orientation distribution of the crystals.     

忽略TTI介质对称轴倾角的可行性

假设横向各向同性(TI)介质的对称轴是垂直的(VTI)或者水平的(HTI)能给实际资料处理带来便利,然而实际TI介质的对称轴往往是倾斜的(TTI),忽略对称轴倾角可能给各向异性参数提取和成像带来偏差,因此需要研究是否能、以及什么条件下能忽略TTI介质对称轴倾角.本文通过理论研究和数值分析研究了与TTI介质弹性性质最接近的VTI介质(OAVTI)的弹性常数和各向异性参数与原TTI介质的弹性常数和各向异性参数之间的联系与差别.结果表明:OAVTI介质各向异性参数与原TTI介质各向异性参数之间的差别可统一表示成F(α₀/β₀,ε,δ,γ)ξ²的形式,其中F(α₀/β₀,ε,δ,γ)是无量纲各向异性参数(ε, δ, γ)的线性函数,ξ是对称轴倾角;ξ的大小对各参数的误差起主导作用,一般不建议忽略20°~25°以上的对称轴倾角;当ξ较小时,即使是对强各向异性的TTI介质作VTI近似,引起的P波各向异性参数误差也很小,因此在纵波资料处理中忽略TTI介质对称轴倾角通常是可行的;即使在小ξ条件下,倾斜对称轴对SV波也有显著影响,因此在转换波资料处理中,不建议忽略TTI介质的对称轴倾角.本文的研究为分析忽略TTI介质对称轴倾角的可行性提供了理论依据和简便的判据.     

Determination of the reference symmetry axis of a generally anisotropic medium which is approximately transversely isotropic

For a given stiffness tensor (tensor of elastic moduli) of a generally anisotropic medium, we estimate to what extent the medium is transversely isotropic (uniaxial) and determine the direction of its reference symmetry axis expressed in terms of the unit reference symmetry vector. If the medium is exactly transversely isotropic (exactly uniaxial), we obtain the direction of its symmetry axis. We can also calculate the first–order and second–order spatial derivatives of the reference symmetry vector which may be useful in tracing the reference rays for the coupling ray theory. The proposed method is tested using various transversely isotropic (uniaxial) and approximately transversely isotropic (approximately uniaxial) media.     

Spatially non‐local model of inelastic deformations: applications for rock failure problem

A spatially non‐local model for inelastic deformation of solids is proposed and studied. The non‐locality of deformation is taken into account by the additional parameter of state beyond the classical parameters such as stress and strain tensors. This additional parameter is the curvature tensor expressed in terms of the metric strain tensor, and it is called the failure parameter. In the case of small deformation, it is equivalent to the Saint‐Venant incompatibility tensor. Thermodynamic properties of the model are studied, and governing differential equations for spatially non‐local model are formulated, which are composed by the elasticity equations and parabolic equation for the failure parameter. The model can be applied to the study of the rock failure problem, and as an example, the one‐dimensional problem for the deformation of half‐plane loaded by the normal surface stress is studied. Stationary and non‐stationary formulations of the problem are considered, and qualitative agreement with available experimental data is observed.     

On the retrieval of moment tensors from borehole data

The complete moment tensors of seismic sources in homogeneous or vertically inhomogeneous isotropic structures cannot be retrieved using receivers deployed in one vertical borehole. The complete moment tensors can be retrieved from amplitudes of P‐waves, provided that receivers are deployed in at least three boreholes. Using amplitudes of P‐ and S‐waves, two boreholes are, in principle, sufficient. Similar rules also apply to transversely isotropic media with a vertical axis of symmetry. In the case of limited observations, the inversion can be stabilized by imposing the zero‐trace constraint on the moment tensors. However, this constraint is valid only if applied to observations of shear faulting on planar faults in isotropic media, which produces double‐couple mechanisms. For shear faulting on non‐planar faults, for tensile faulting, and for shear faulting in anisotropic media, the zero‐trace constraint is no longer valid and can distort the retrieved moment tensor and bias the fault‐plane solution. Numerical modelling simulating the inversion of the double‐couple mechanism from real data reveals that the errors in the double‐couple and non‐double‐couple percentages of the moment tensors rapidly decrease with increase in the number of boreholes used. For noisy P‐ and S‐wave amplitudes with noise of 15% of the top amplitude at each channel and for a velocity model biased by 10%, the errors in the double‐couple percentage attain 25, 13 and 6% when inverting for the double‐couple mechanism from one, two and three boreholes.     

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.     

任意各向异性介质中三维可控源音频大地电磁正演模拟

在一些地层层理发育的地区,地下介质存在显著的电各向异性,此时基于各向同性模型解释含各向异性效应的可控源音频大地电磁（CSAMT）测深观测数据会导致错误的结果.本文通过引入3×3的对称正定张量表征电导率各向异性,采用非结构四面体网格和矢量有限元方法离散电场满足的矢量Helmholtz方程,并将电磁场源等效为系列电偶极子,实现任意各向异性介质中CSAMT高效数值模拟.本文首先通过层状各向异性模型检验三维有限元算法的精度和有效性,进一步建立三维地电模型研究异常体各向异性和围岩各向异性对CSAMT响应的影响,最后使用视电阻率极性图来识别各向异性电导率主轴方向.数值模拟结果表明,各向异性电导率对CSAMT视电阻率幅值及分布规律都有很大影响,视电阻率极性图能够很好地识别各向异性主轴方向.     

Retrieval of source parameters of an event of the 2000 West Bohemia earthquake swarm assuming an anisotropic crust

We propose an inversion scheme for retrieval of characteristics of seismic point sources, which in contrast to common practice, takes into account anisotropy. If anisotropy is neglected during inversion, the moment tensors retrieved from seismic waves generated by sources situated in anisotropic media may be biased. Instead of the moment tensor, the geometry of the source is retrieved directly in our inversion; if necessary, the moment tensor can be then determined from the source geometry aposteriori. The source geometry is defined by the orientation of the slip vector and the fault normal as well as the strength of the event given by the size of the slip and the area of the fault. This approach allows direct interpretation of the source geometry in terms of shear and tensile faulting. It also makes possible to identify volumetric source changes that occur during rupturing. We apply the described algorithm to one event of the 2000 West Bohemia earthquake swarm episode. For inversion we use information of the direct P waves. The structure is approximated by three different models determined from travel-time observations. The models are inhomogeneous isotropic, inhomogeneous anisotropic, and homogeneous anisotropic. For these models we obtain seismic moments M_T = 3.2 − 3.8 × 10¹⁴ Nm and left-lateral near-vertical oblique normal faulting on a N-S trending rupture surface. The orientation of the rupture surface is consistent with fault-plane solutions of earlier studies and with the spatial distribution of other events during this swarm. The studied event seems to be accompanied by a small amount of crack opening. The amount of crack opening is slightly reduced when the inhomogeneous anisotropic model is assumed, but it persists. These results and additional independent observations seem to indicate that tensile faulting occurs as a result of high fluid pressure.     

各向异性ATI介质剪切位错源地震矩张量

考虑震源区为各向异性ATI介质情形下,给出了剪切位错源地震矩张量解析表达式并模拟了震源区各向异性对双力偶分量(DC)、补偿线性偶极子分量(CLVD)和各向同性分量(ISO)的影响,结果表明,即使剪切位错源仍能产生非双力偶各向同性分量,其导致沿着ATI介质对称轴方向体积变化;当断层面位于ATI介质对称平面或者震源区为各向...     

Moment Tensor Solutions and Triggering Environment for Earthquakes in Koyna-Warna Water Reservoirs Region,India

We consider nine earthquakes in the Koyna-Warna reservoir region on the western side of the Peninsular India. The deviatoric moment tensors of these earthquakes have been evaluated by minimizing the least-squares misfit between observed and synthetic seismograms. We use broadband seismograms of observatories at KARD and PUNE which are at distances of nearly 50 and 150 km, respectively, from the epicenters. Both surface wave inversion and the difference between the arrival times of SH and SV show the presence of an anisotropic crust. However, we have obtained an equivalent isotropic structure by improving the published crustal structures of this area through inversion of surface wave group velocity data. The deviatoric moment tensors of the earthquakes are decomposed into two components: double-couple and compensated linear vector dipoles (CLVD). The double-couple components of all the nine earthquakes show normal faulting with minor strike slip; the T axis is consistently subhorizontal with an average azimuth of 260.6° and the P axis is nearly vertical. The fault planes of six events give average strike direction and dip, respectively as 194.0° and 51.8° and are associated with the main fault of the area. The other three events lie in the southern part of this area and have strike direction between SSE and SE which is parallel to the tectonic features in this part. The CLVD component is generally within 20 percent of the total moment tensor. Recent studies show that anisotropy can produce source mechanism with CLVD up to 30 percent and can also cause high pore fluid pressure leading to fault instability more rapidly compared to conventional mechanism in an isotropic medium. It appears that the anisotropic crust, noted in the present work, is generating the CLVD component and also gives the proper environment to trigger earthquakes by reservoirs through pore fluid pressure.     

A SANISAND model with anisotropic elasticity

Numerous experimental studies indicate that as a result of shear stress, the elastic behavior of granular media becomes both non-linear and anisotropic. This paper presents a simple constitutive model for sands with respect to anisotropic elasticity. To this aim, using the concept of second order fabric tensor, a simplified elasticity theory is presented which is capable of considering the effect of induced anisotropy on the elastic response. SANISAND is the name used for a family of simple anisotropic sand models developed in the framework of critical state soil mechanics and bounding surface plasticity. An existing SANISAND model is modified in order to include the proposed anisotropic elasticity. The modified model simulations are compared with those obtained from the other members of this family. It is shown that considering anisotropic elasticity can take part in explanation of drastic loss of mean principal stress when sand is subjected to reverse loading in dilative branch of behavior and as a result, improve the liquefaction simulations.     

Classes of Anisotropic Media: A Tutorial

The purpose of the present article is to give a precise definition and analysis from first principals of anisotropy, as the term applies to elastic media, taking care to avoid unnecessary assumptions. Two fundamental concepts, material invariance and symmetry group of a material, are defined purely in terms of the stress-strain relation. The implications of material symmetry, or in other words, of anisotropy, for the structure of the stiffness tensor are then investigated. Using the reduced notation of Voigt, these results are presented as the well-known simplifications in the form taken by the six-by-six stiffness matrix that represents the material's stiffness tensor. A new, simple proof is given for the remarkable fact that an elastic medium cannot have rotational symmetry by an angle of less than 90° without being transversely isotropic. In addition, the mutual relation that the notions of elastic symmetry and crystal symmetry have with respect to the so-called orthogonal group is sketched. Despite the historical association between anisotropic elastic materials and the study of crystals, the given presentation shows that conceptually the notion of anisotropy in elastic media is entirely independent of that of crystal symmetry.     

Exact elastic impedance tensor for isotropic media

The existing expressions of elastic impedance,as the generalized form of acoustic impedance,represent the resistance of subsurface media to seismic waves of non-normal incidence,and thus include information on the shear-wave velocity.In this sense,conventional elastic impedance is an attribute of the seismic reflection and not an intrinsic physical property of the subsurface media.The derivation of these expressions shares the approximations made for reflectivity,such as weak impedance contrast andisotropic or weakly anisotropic media,which limits the accuracy of reflectivity reconstruction and seismic inversion.In this paper,we derive exact elastic impedance tensors of seismic P-and S-waves for isotropic media based on the stress-velocity law.Each componentof the impedance tensor represents a unique mechanical property of the medium.Approximations of P-wave elastic impedance tensor components are discussed for seismic inversion and interpretation.Application to synthetic data and real data shows the accuracy and robust interpretation capability of the derived elastic impedance in lithology characterizations.     

Release-Rebound Processes: Vector Motions

The tensor relations describing the shear deviatoric strains and rotation strains may be presented as vector relations in a special coordinate system, e.g., in the diagonal or off-diagonal one. However, these fields can be also presented in the 4D invariant forms by means of invariant Dirac tensors. We present 4D relativistic relations for the invariant shear deviatoric strain and rotation strain vectors closely related to a fracture process in solids and to the molecular strains (shear and rotational) in fluids. These shear and rotation strains may interact with the radial derivatives of pressure along the propagation directions.     

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.     

Determination of shale stiffness tensor from standard logs

A technique allowing inversion of the shale stiffness tensor from standard logging data: sonic velocities, density, porosity and clay content is developed. The inversion is based on the effective medium theory. The testing of the technique on laboratory measurements of the elastic wave velocities in shale samples shows that the inversion makes it possible to predict the elastic wave velocities V_P, V_S1 and V_S2 in any direction within an error of a few per cent. The technique has been applied for the stiffness tensor inversion along a well penetrating a shale formation of the Mississippian age altered by thin layers of limestone. It is demonstrated that the symmetry of a stiffness tensor inverted at the sonic frequency (2 kHz) is slightly orthorhombic and taking into account the experimental errors, can be related to the vertical transverse isotropy symmetry. For the productive interval of the shale formation, the Thomsen parameters ?, γ, and δ average, respectively, 0.32, 0.25 and 0.21, which indicate anelliptic behaviour of the velocities in this shale. The coefficients of anisotropy of this shale interval are around 24% and 20% for the compressional and shear waves, respectively. The values of the inverted velocities in the bedding plane for this interval are in good agreement with the laboratory measurements. The technique also allows inversion of the water saturation of the formation (Sw) and the inverted values are in agreement with the Sw values available for this formation. A Backus‐like upscaling of the inverted stiffness tensors is carried out for the lower and upper bounds of the frequency band used in the crosswell tomography (100 Hz and 500 Hz). These results can serve as an initial velocity model for the microearthquake location during hydrofracking of the shale formation.