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.     

LAYERED PERMEABLE SYSTEMS1

Permeability is a second rank tensor relating flow rate to pressure gradient in a porous medium. If the permeability is a constant times the identity tensor the permeable medium is isotropic; otherwise it is anisotropic. A formalism is presented for the simple calculation of the permeability tensor of a heterogeneous layered system composed of interleaved thin layers of several permeable constituent porous media in the static limit. Corresponding to any cumulative thickness H of a constituent is an element consisting of scalar H and a matrix which is H times a hybrid matrix function of permeability. The calculation of the properties of a medium equivalent to the combination of permeable constituents may then be accomplished by simple addition of the corresponding scalar/matrix elements. Subtraction of an element removes a permeable constituent, providing the means to decompose a permeable medium into many possible sets of permeable constituents, all of which have the same flow properties. A set of layers of a constituent medium in the heterogeneous layered system with permeability of the order of 1/h as h→ 0, where h is that constituent's concentration, acts as a set of infinitely thin channels and is a model for a set of parallel cracks or fractures. Conversely, a set of layers of a given constituent with permeability of the order of h as h→ 0 acts as a set of parallel flow barriers and models a set of parallel, relatively impermeable, interfaces, such as shale stringers or some faults. Both sets of channels and sets of barriers are defined explicitly by scalar/matrix elements for which the scalar and three of the four sub-matrices vanish. Further, non-parallel sets of channels or barriers can be ‘added’ and 'subtracted’ from a background homogeneous anisotropic medium commutatively and associatively, but not non-parallel sets of channels and barriers reflecting the physical reality that fractures that penetrate barriers will give a different flow behaviour from barriers that block channels. This analysis of layered media, and the representations of the phenomena that can occur as the thickness of a constituent is allowed to approach zero, are applicable directly to layered heat conductors, layered electrostatic conductors and layered dielectrics.     

Reflected waves in finely layered tilted orthorhombic media

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

一种弱各向异性介质地震波群速度的近似表示新方法

地震各向异性介质的群速度是关于相角的复杂函数,将其表示成射线角形式较为困难,这给地震各向异性分析以及走时正演模拟等带来诸多不便;另一方面,观测资料表明实际地球介质的地震各向异性通常较弱,这为用射线角近似表示地震波群速度提供了可能.本文基于以射线角近似表示相角的思想,提出了一种弱各向异性条件下,群速度射线角近似表示的新方法.计算表明,在弱地震各向异性条件下,新方法在很宽的射线角范围内,对三种地震波的群速度都能很好地近似,在准SV波计算精度方面显著优于目前通常使用的近似方法.     

Dispersion splitting of Rayleigh waves in layered azimuthally anisotropic media

Rayleigh wave dispersion can be induced in an anisotropic medium or a layered isotropic medium. For a layered azimuthally anisotropic structure, traditional wave equation of layered structure can be modified to describe the dispersion behavior of Rayleigh waves. Numerical stimulation results show that for layered azimuthal anisotropy both the dispersion velocities and anisotropic parameters depend principally on anisotropic S-wave velocities. The splitting S-wave velocities may produce dispersion splitting of Rayleigh waves. Such dispersion splitting appears noticeable at azimuthal angle 45°. This feature was confirmed by the measured results of a field test. The fundamental mode splits into two branches at azimuthal angle 45° to the symmetry axis for some frequencies, and along the same direction the difference of splitting-phase velocities of the fundamental model reaches the maximum. Dispersion splitting of Rayleigh waves was firstly displayed for anisotropy study in dispersion image by means of multichannel analysis of surface waves, the image of which provides a new window for studying the anisotropic property of media.     

Generalized Dix equation and analytic treatment of normal-moveout velocity for anisotropic media*

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common-midpoint (CMP) spreads is usually well described by the normal-moveout (NMO) velocity defined in the zero-offset limit. In their recent work, Grechka and Tsvankin showed that the azimuthal variation of NMO velocity around a fixed CMP location generally has an elliptical form (i.e. plotting the NMO velocity in each azimuthal direction produces an ellipse) and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for the NMO velocity in anisotropic media of arbitrary symmetry. For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The contribution of anisotropy to NMO velocity is contained in the slowness components of the zero-offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) — quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix-type average of the matrices responsible for the ‘interval’ NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrarily anisotropic media. For models with a throughgoing vertical symmetry plane (i.e. if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi-axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions. Modelling of normal moveout in general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero-offset) ray. Remarkably, the expressions for geometrical spreading along the zero-offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi-azimuth, multi-offset ray tracing and, therefore, can be used efficiently in traveltime inversion and in devising fast dip-moveout (DMO) processing algorithms for anisotropic media. This technique becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces. The high accuracy of our NMO expressions is illustrated by comparison with ray-traced reflection traveltimes in piecewise-homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P-wave moveout can be used to find the depth-dependent fracture orientation and to evaluate the magnitude of azimuthal anisotropy.     

Plane-layered models for the analysis of wave propagation in reservoir environments1

The long-wavelength propagation and attenuation characteristics of three geological structures that frequently occur in reservoir environments are investigated using a theoretical model that consists of a stack of fine and viscoelastic plane layers, with the layers being either solid or fluid. Backus theory properly describes fine layering and a set of fluid-filled microfractures, under the assumption that interfaces between different materials are bonded. The effects of saturation on wave attenuation are modelled by the relative values of the bulk and shear quality factors. The anisotropic quality factor in a fine-layered system shows a variety of behaviours depending on the saturation and velocities of the single constituents. The wave is less attenuated along the layering direction when the quality factors are proportional to velocity, and vice versa when inversely proportional to velocity. Fractured rocks have very anisotropic wavefronts and quality factors, in particular for the shear modes which are strongly dependent on the characteristics of the fluid filling the microfractures. When the size of the boundary layer is much smaller than the thickness of the fluid layer, the stack of solid-fluid layers becomes a layered porous media of the Biot type. This behaviour is caused by the slip-wall condition at the interface between the solid and the fluid. As in Biot theory, there are two compressional waves, but here the medium is anisotropic and the slow wave does not propagate perpendicular to the layers. Moreover, this wave shows pronounced cusps along the layering direction, like shear waves in a very anisotropic single-phase medium.     

Determination of the Cole-Cole parameters in polarizable media from logging data

The paper presents the solution to an axially symmetric problem of the electromagnetic field excited by an electric dipole and a current in a half-infinite cable in a cylindrically layered polarizable medium. The polarization of the medium is described by the Cole-Cole formula, including such parameters as the conductance, polarizability, response time, and attenuation coefficient. The spatiotemporal structures of the field in polarizable and nonpolarizable homogeneous media are compared. The apparent Cole-Cole parameters are defined and determined for a model problem of a cylindrically layered medium.     

Influence of frequency-dependent anisotropy on seismic amplitude-versus-offset signatures for fractured poroelastic rocks

Frequency-dependent amplitude variation with offset offers an effective method for hydrocarbon detections and analysis of fluid flow during production of oil and natural gas within a fractured reservoir. An appropriate representation for the frequency dependency of seismic amplitude variation with offset signatures should incorporate influences of dispersive and attenuating properties of a reservoir and the layered structure for either isotropic or anisotropic dispersion analysis. In this study, we use an equivalent medium permeated with aligned fractures that simulates frequency-dependent anisotropy, which is sensitive to the filled fluid of fractures. The model, where pores and fractures are filled with two different fluids, considers velocity dispersion and attenuation due to mesoscopic wave-induced fluid flow. We have introduced an improved scheme seamlessly linking rock physics modelling and calculations for frequency-dependent reflection coefficients based on the propagator matrix technique. The modelling scheme is performed in the frequency-slowness domain and can properly incorporate effects of both bedded structure of the reservoir and velocity dispersion quantified with frequency-dependent stiffness. Therefore, for a dispersive and attenuated layered model, seismic signatures represent a combined contribution of impedance contrast, layer thickness, anisotropic dispersion of the fractured media and tuning and interference of thin layers, which has been avoided by current conventional methods. Frequency-dependent amplitude variation with offset responses was studied via considering the influences of fracture fills, layer thicknesses and fracture weaknesses for three classes amplitude variation with offset reservoirs. Modelling results show the applicability of the introduced procedure for interpretations of frequency-dependent seismic anomalies associated with both layered structure and velocity dispersion of an equivalent anisotropic medium. The implications indicate that anisotropic velocity dispersion should be incorporated accurately to obtain enhanced amplitude variation with offset interpretations. The presented frequency-dependent amplitude variation with offset modelling procedure offers a useful tool for fracture fluid detections in an anisotropic dispersive reservoir with layered structures.     

Nonlinear ray perturbation theory with its applications to ray tracing and inversion in anisotropic media

A comprehensive approach, based on the general nonlinear ray perturbation theory (Druzhinin, 1991), is proposed for both a fast and accurate uniform asymptotic solution of forward and inverse kinematic problems in anisotropic media. It has been developed to modify the standard ray linearization procedures when they become inconsistent, by providing a predictable truncation error of ray perturbation series. The theoretical background consists in a set of recurrent expressions for the perturbations of all orders for calculating approximately the body wave phase and group velocities, polarization, travel times, ray trajectories, paraxial rays and also the slowness vectors or reflected/transmitted waves in terms of elastic tensor perturbations. We assume that any elastic medium can be used as an unperturbed medium. A total 2-D numerical testing of these expressions has been established within the transverse isotropy to verify the accuracy and convergence of perturbation series when the elastic constants are perturbed. Seismological applications to determine crack-induced anisotropy parameters on VSP travel times for the different wave types in homogeneous and horizontally layered, transversally isotropic and orthorhombic structures are also presented. A number of numerical tests shows that this method is in general stable with respect to the choice of the reference model and the errors in the input data. A proof of uniqueness is provided by an interactive analysis of the sensitivity functions, which are also used for choosing optimum source/receiver locations. Finally, software has been developed for a desktop computer and applied to interpreting specific real VSP observations as well as explaining the results of physical modelling for a 3-D crack model with the estimation of crack parameters.     

Analytical Solutions for Steady-State Gas Flow in Layered Soils with Field Applications

Soil vapor extraction (SVE) is widely used to remove volatile organic compounds from the vadose zone. Design of SVE systems rely largely upon vacuum responses and limited vapor concentration data measured during short-term soil gas extraction tests performed in single extraction wells. Interpretation of such vacuum data is often simply a rule of thumb as most field sites have layering complexity negating applicability of existing analytical models. This paper provides the derivation of an analytical model for steady, axisymmetric gas flow in heterogeneous (layered) soils from a single well. A general, variable flow boundary condition along the well screen represents actual conditions more closely than a uniform flow or uniform well pressure condition. Each soil layer is assumed homogeneous with anisotropic gas permeability. The solution is derived using the generalized integral transform technique and includes expressions for vacuum, velocities, and streamlines. The model is applied to the interpretation of multiple well tests at a field site and uses linear superposition to extend the flow model to multi-well extraction. The demonstration site included an array of vacuum monitoring data collected during nine individual well flow tests. A method of normalizing the vacuum data is illustrated that allowed the full data set to be employed in a single calibration effort. The test site also included a surface cap with an apparent vertical permeability two to three orders of magnitude smaller than the sands of the vadose zone. This large permeability contrast posed no difficulties in evaluating the solution.     

Ground water whirls

Numerical experiments with steady-state ground water flow models show that spiraling flow lines occur in layered aquifers that have different anisotropic horizontal hydraulic conductivities in adjacent layers. Bundles of such flow lines turning in the same direction can be referred to as ground water whirls. An anisotropic layered block in a field of uniform horizontal flow results in one or more whirls with their axes in the uniform flow direction. The number of whirls depends on the number of interfaces between layers with different anisotropic properties. For flow to a well in an aquifer consisting of two anisotropic layers, with perpendicular major principal directions, whirls are found to occur in quadrants that are bounded by the principal directions of the hydraulic conductivity. The combined effect of flow to a well and a layered anisotropy implies that a single well in a system with a single anisotropic layer within an otherwise isotropic aquifer causes eight whirls. All adjacent whirls rotate in opposite directions.     

3D description and inversion of reflection moveout of PS‐waves in anisotropic media

Common‐midpoint moveout of converted waves is generally asymmetric with respect to zero offset and cannot be described by the traveltime series t²(x²) conventionally used for pure modes. Here, we present concise parametric expressions for both common‐midpoint (CMP) and common‐conversion‐point (CCP) gathers of PS‐waves for arbitrary anisotropic, horizontally layered media above a plane dipping reflector. This analytic representation can be used to model 3D (multi‐azimuth) CMP gathers without time‐consuming two‐point ray tracing and to compute attributes of PS moveout such as the slope of the traveltime surface at zero offset and the coordinates of the moveout minimum. In addition to providing an efficient tool for forward modelling, our formalism helps to carry out joint inversion of P and PS data for transverse isotropy with a vertical symmetry axis (VTI media). If the medium above the reflector is laterally homogeneous, P‐wave reflection moveout cannot constrain the depth scale of the model needed for depth migration. Extending our previous results for a single VTI layer, we show that the interval vertical velocities of the P‐ and S‐waves (V_P0 and V_S0) and the Thomsen parameters ε and δ can be found from surface data alone by combining P‐wave moveout with the traveltimes of the converted PS(PSV)‐wave. If the data are acquired only on the dip line (i.e. in 2D), stable parameter estimation requires including the moveout of P‐ and PS‐waves from both a horizontal and a dipping interface. At the first stage of the velocity‐analysis procedure, we build an initial anisotropic model by applying a layer‐stripping algorithm to CMP moveout of P‐ and PS‐waves. To overcome the distorting influence of conversion‐point dispersal on CMP gathers, the interval VTI parameters are refined by collecting the PS data into CCP gathers and repeating the inversion. For 3D surveys with a sufficiently wide range of source–receiver azimuths, it is possible to estimate all four relevant parameters (V_P0, V_S0, ε and δ) using reflections from a single mildly dipping interface. In this case, the P‐wave NMO ellipse determined by 3D (azimuthal) velocity analysis is combined with azimuthally dependent traveltimes of the PS‐wave. On the whole, the joint inversion of P and PS data yields a VTI model suitable for depth migration of P‐waves, as well as processing (e.g. transformation to zero offset) of converted waves.     

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。     

Estimation of geological dip and curvature from time‐migrated zero‐offset reflections in heterogeneous anisotropic media

Starting from a given time‐migrated zero‐offset data volume and time‐migration velocity, recent literature has shown that it is possible to simultaneously trace image rays in depth and reconstruct the depth‐velocity model along them. This, in turn, allows image‐ray migration, namely to map time‐migrated reflections into depth by tracing the image ray until half of the reflection time is consumed. As known since the 1980s, image‐ray migration can be made more complete if, besides reflection time, also estimates of its first and second derivatives with respect to the time‐migration datum coordinates are available. Such information provides, in addition to the location and dip of the reflectors in depth, also an estimation of their curvature. The expressions explicitly relate geological dip and curvature to first and second derivatives of reflection time with respect to time‐migration datum coordinates. Such quantitative relationships can provide useful constraints for improved construction of reflectors at depth in the presence of uncertainty. Furthermore, the results of image‐ray migration can be used to verify and improve time‐migration algorithms and can therefore be considered complementary to those of normal‐ray migration. So far, image‐ray migration algorithms have been restricted to layered models with isotropic smooth velocities within the layers. Using the methodology of surface‐to‐surface paraxial matrices, we obtain a natural extension to smooth or layered anisotropic media.     

层状方位各向异性介质的视电阻率计算

从电性各向异性的欧姆定律出发，推导了直流电法层状方位各向异性介质中的电位分布、边界条件及视电阻率计算公式．以四极对称装置系统为例，对具有相同各向异性系数的4层模型采用核函数递推法作了理论数值模拟，得到了不同方向的电阻率测深曲线及其等值线形态．结果表明理论公式是正确的，测深曲线既反映了分层介质的电阻率差异，又反映了各层中电阻率的各向异性特征．      

COMPLETE INVERSION OF ZERO-OFFSET SEISMIC DATA*

The one-dimensional seismic inverse problem consists of recovering the acoustic impedance (or reflectivity function) as a function of traveltime from the reflection response of a horizontally layered medium excited by a plane-wave impulsive source. Most seismic sources behave like point sources, and the data must be corrected for geometrical spreading before the inversion procedure is applied. This correction is usually not exact because the geometrical spreading is different for primary and multiple reflections. An improved algorithm is proposed which takes the geometrical spreading from a point source into account. The zero-offset reflection response from a stack of homogeneous layers of variable thickness is used to compute the thickness, velocity and density of each layer. This is possible because the geometrical spreading contains additional information about the velocities.     

Apparent resistivity of azimuthal anisotropy layered media

Introduction The method of apparent resistivity with direct current (DC) measurement, adopted from geophysical exploration, is one of the most important precursory approaches in earthquake prediction, in which the geo-electric constitutive model is illustrated by the scalar theory Archies law and correspondingly a set of calculation methods for apparent resistivity of layered media has been established (QIAN, et al, 1985; YAO, 1989). But this is not consistent with the fact that in earthqu…     

Phase velocities of Rayleigh waves in southeastern Europe and Eastern Mediterranean Sea

Summary Measurements of phase velocities along several paths in southeastern Europe and along one path in the Eastern Mediterranean have been made in the period range 15 to 60 sec for the fundamental mode of Rayleigh waves. The data are fitted by layered models of the crust and upper mantle. The crust in the Balkan peninsula is thicker than the normal continental crust by about 10 km while the crust in the northern Aegean Sea has the normal continental thickness. The existence of a low shear velocity layer of thickness equal to 140 km with its top in a depth of 90 km interpretes the data well for the longer periods. The data for the eastern Mediterranean show that this region has an oceanic character. The mean crustal thickness in this area is of the order of 20 km.     

Love wave dispersion: A comparison of results for a semi-infinite medium with inhomogeneous layers and for its approximation by homogeneous layers

Love wave dispersion in various semi-infinite media consisting of inhomogeneous layers is discussed. The phase and group velocities are computed when shear wave velocity and density in each inhomogeneous layer are varying exponentially with depth. At the beginning one or two inhomogeneous layers over a homogeneous semi-infinite medium are considered. The dispersion results for these structures are compared with those for their approximations with homogeneous layers. Comparisons show that differences of phase and group velocities for the original models from those for their approximated models (i) increase with the increase of wave number and (ii) are larger for group velocity than for phase velocity. The difference is approximately proportional to the rate of change of parameters in the layers. Finally, dispersion curves are obtained for model IP3MC, which consists of many inhomogeneous and homogeneous layers over a homogeneous semi-infinite medium. The results are compared with the observed group velocity data across the Indian Peninsula.