Function In Anisotropic Media Research Articles

Microseismic Full Waveform Modeling in Anisotropic Media with Moment Tensor Implementation

Seismic anisotropy which is common in shale and fractured rocks will cause travel-time and amplitude discrepancy in different propagation directions. For microseismic monitoring which is often implemented in shale or fractured rocks, seismic anisotropy needs to be carefully accounted for in source location and mechanism determination. We have developed an efficient finite-difference full waveform modeling tool with an arbitrary moment tensor source. The modeling tool is suitable for simulating wave propagation in anisotropic media for microseismic monitoring. As both dislocation and non-double-couple source are often observed in microseismic monitoring, an arbitrary moment tensor source is implemented in our forward modeling tool. The increments of shear stress are equally distributed on the staggered grid to implement an accurate and symmetric moment tensor source. Our modeling tool provides an efficient way to obtain the Green’s function in anisotropic media, which is the key of anisotropic moment tensor inversion and source mechanism characterization in microseismic monitoring. In our research, wavefields in anisotropic media have been carefully simulated and analyzed in both surface array and downhole array. The variation characteristics of travel-time and amplitude of direct P- and S-wave in vertical transverse isotropic media and horizontal transverse isotropic media are distinct, thus providing a feasible way to distinguish and identify the anisotropic type of the subsurface. Analyzing the travel-times and amplitudes of the microseismic data is a feasible way to estimate the orientation and density of the induced cracks in hydraulic fracturing. Our anisotropic modeling tool can be used to generate and analyze microseismic full wavefield with full moment tensor source in anisotropic media, which can help promote the anisotropic interpretation and inversion of field data.

Double-plane-wave reverse time migration in the frequency domain

We have developed an efficient, accurate, and flexible plane-wave migration algorithm in the frequency domain by using a compressed and coupled-plane-wave data set, known as the double-plane-wave (DPW) data set. The DPW data set obtained by slant stacking of seismic shot profiles over source and receiver/offset represents seismic data in a fully decomposed plane-wave domain, which is called the DPW domain. A new DPW migration algorithm is derived under the Born approximation in the frequency domain, and it is referred to as the frequency-domain DPW reverse time migration (RTM). Frequency plane-wave Green’s functions need to be constructed and used during the migration. Time dips in shot profiles help to estimate the range of plane-wave decomposition. Therefore, the number of frequency plane-wave Green’s functions required for migration is limited. Furthermore, frequency plane-wave Green’s functions can be used for imaging each set of plane waves — either source or receiver/offset plane waves. As a result, the computational burden of computing Green’s function is substantially reduced; this results in increasing the migration efficiency. A selected range of plane-wave components can be migrated independently to image specific targets. Ray-parameter common-image gathers can be generated after migration without extra effort. The algorithm was tested on several synthetic data sets to show its feasibility and usefulness. The frequency-domain DPW RTM can also include anisotropy by constructing plane-wave Green’s function in anisotropic media.

Interaction of short single-component vector solitons

We study interaction of different-polarization single-component vector solitons of the envelope function in anisotropic media within the framework of the system of two coupled third-order nonlinear Schrodinger equations which allow for the third-order linear dispersion, nonlinear dispersion, nonlinear cross-phase modulation, and cross-nonlinear dispersion. The regimes of mutual reflection, passage, and asymptotic approach of the solitons are obtained. It is shown that the character of interaction of such solitons is determined by the initial relationship of their amplitudes and phases. The stationary mutual locations of interacting solitons and their coupled, the so-called breather states are discussed. The roles of the cubic nonlinearity, cubic cross-nonlinearity, and cross-nonlinear dispersion during interaction of solitons are studied.

Computing the Response of Multi‐Component Induction Logging in Layered Anisotropic Formation by the Recursive Matrix Method with Magnetic‐Current‐Source Dyadic Green's Function

AbstractA recursive matrix method is adopted to compute the magnetic‐current‐source dyadic Green's function in anisotropic media. The response of multi‐component induction logging in layered and deviated anisotropic formation is simulated by using the obtained dyadic Green's function. The influence of coil spacing, bed thickness, deviation angle and shoulder bed on the response is analyzed. The computation result shows that the changing pattern of coplanar apparent conductivity is more complex than coaxial apparent conductivity and can not reflect the formation's real conductivity. The coaxial apparent conductivity decreases with the increasing deviation angle in anisotropic formation, while the coplanar apparent conductivity increases with growing deviation angle. The tool's vertical resolution and the influence of shoulder bed's anisotropy on the tool's response in the target formation bed are determined by coil spacing. The influence of the shoulder bed's anisotropy on the response in high‐conductivity target formation is bigger than that in high‐resistivity target formation.

Theory and numerical simulation of fluid-pressure diffusion in anisotropic porous media

The physics of fluid diffusion in anisotropic media was studied, based on Biot’s theory of poroelasticity and using wave propagation concepts. Diffusion and elastic strain can be uncoupled fully, being a good approximation in many situations. We have used a correction to the stiffness of the rock under conditions of transverse isotropy and uniaxial strain to model borehole conditions. The concepts of phase, group, and energy velocities were analyzed to describe the location of the diffusion front, and the attenuation and quality factors were obtained to quantify the amplitude decay. We have found that the location of the front is described correctly by the energy velocity. The Green’s function in anisotropic media can be obtained by applying a change of coordinates to the isotropic solution. We have simulated the diffusion in inhomogeneous media using a time-domain spectral explicit scheme and the staggered Fourier pseudospectral method to compute the spatial derivatives. The method is based on a spectral Chebychev expansion of the evolution operator of the system. The scheme allows the solution of linear periodic parabolic equations, having accuracy within the machine precision, in time and in space. The results match the analytic solution obtained from the Green’s function. The performance of the algorithm is confirmed in the case of a pressure field generated by a fluid-injection source in a hydrocarbon reservoir where the properties vary fractally.

Local discretization error bounds using interval boundary element method

AbstractIn this paper, a method to account for the point‐wise discretization error in the solution for boundary element method is developed. Interval methods are used to enclose the boundary integral equation and a sharp parametric solver for the interval linear system of equations is presented. The developed method does not assume any special properties besides the Laplace equation being a linear elliptic partial differential equation whose Green's function for an isotropic media is known. Numerical results are presented showing the guarantee of the bounds on the solution as well as the convergence of the discretization error. Copyright © 2008 John Wiley & Sons, Ltd.

Two-dimensional time-harmonic elastodynamic Green's functions for anisotropic media

An exact matrix formulation for 2D time-harmonic elastodynamic Green's functions for anisotropic media is presented. In this formulation, a Fourier transform with respect to spatial coordinates is employed. The displacement and stress in the Fourier transform domain is obtained by using the method of modal expansion. A technique is suggested to evaluate the 2D inverse integral in the polar coordinate system to obtain the solution for the displacement and stress. In this technique, the inverse integration is carried out in a very efficient way along the wave number axis over a semi-infinite region by changing integral paths. Therefore a numerical integration is needed only for the polar angle over a finite region. Numerical examples are presented to demonstrate the present method, and wave fields for the displacement are investigated for isotropic and anisotropic media.

A 3-D boundary element formulation of anisotropic elasticity with gravity

This paper presents a new boundary element formulation for three-dimensional (3-D) linear elastic bodies with anisotropic properties (transverse isotropy with any orientation) subject to gravity. Green's functions for such anisotropic media are derived in exact closed-form. Particular stresses and displacements related to the gravity body force are also presented in exact closed-form for generalized anisotropic media. The boundary integral equation with only weak singularities is formulated based on the principle of superposition and is such that no numerical integration associated either with the Green's functions for anisotropic media or with body forces is necessary. This new 3-D boundary element formulation has been programmed, and several numerical examples are presented and checked with existing closed-form solutions. It is found that even with a very coarse discretization, excellent results can be obtained with the proposed method.

Electromagnetic scattering from anisotropically coated impedance cylinder

The electromagnetic scattering from an anisotropically coated impedance cylinder is formulated by using the wave functions for anisotropic media and the boundary-value method. This composite structure is assumed to be infinite in length, and it is illuminated by a plane wave or a cylindrical wave from a line source. The problem is two dimensional and the solutions to both types of polarisation (TE and TM) are presented. The validity of this solution is verified by comparing the numerical results with those in literature. Numerical results for the effects of various geometrical and electrical parameters on the bistatic radar cross-section are presented.

Properties of wave functions in homogeneous anisotropic media.

The general solutions of the first, second, third, and fourth kinds to the wave equation in homogeneous anisotropic media are expressed by integrals over a finite range. The convergence of the series solution of wave functions in homogeneous anisotropic media [Phys. Rev. E 47, 664 (1993)] is discussed. The use of the wave functions in anisotropic media is demonstrated. The theory is expounded via an illustrative example of a two-dimensional scalar case. The analytical solution of plane-wave scattering by a conducting circular cylinder coated with anisotropic materials is formulated in terms of the series of wave functions for anisotropic media. Numerical results show that the solution in terms of wave functions of various kinds in anisotropic media gives essentially the same radar cross sections as obtained by Beker, Umashankar, and Taflove [Electromagnetics 10, 387 (1990)] using a different approach. Numerical results in the resonance region are presented for reference purposes. The analysis of this paper can be easily generalized to vector and tensor wave functions in homogeneous anisotropic media.

An alternative solution of the scattering from an anisotropic cylindrical dielectric shell

A method based on the approximate wave functions for anisotropic media and the mode-matching approach is developed to solve the problem of the electromagnetic scattering from an anisotropic cylindrical dielectric shell. The cylindrical shell is assumed to be infinite in length, and it is illuminated by a plane wave or a cylindrical wave from a line source. The problem is two-dimensional and the solutions to both types of polarization (TE and TM) are presented. The validity of this solution is verified by comparing the numerical results with those in literatures and the previous calculations based on the exact wave functions for anisotropic media. Numerical results show the higher computational efficiency of the present method for bounded anisotropic media.

Axial slot antenna on an anisotropic dielectric-coated circular cylinder

This paper presents an analytical solution of an axial slot antenna on a conducting circular cylinder coated with an anisotropic material in terms of the series of wave functions for anisotropic media. This solution is formally similar to the solution of an axial slot antenna on a conducting elliptic cylinder with an isotropic dielectric coating. In the anisotropic dielectric coating, and in the exterior free space region, the fields are expanded by the wave functions in anisotropic media and the well known wave functions in isotropic media, respectively. Numerical results for the far-field radiation patterns and the aperture conductance against coating thickness for different anisotropic media are presented for the transverse electric (TE) and transverse magnetic (TM) polarisations. Numerical results for the gyrotropic type media are used to check our calculations, and those for general anisotropic media are presented to provide references for future computations by using other methods, respectively.

Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media.

An electromagnetic wave theory of bounded homogeneous anisotropic media is developed by using the method of angular spectrum expansion. The series and integral representations of the circular cylindrical wave functions and the spherical wave functions of the first, second, third, and fourth kind for homogeneous anisotropic media are obtained. Each coefficient of the Fourier series of a circular cylindrical wave function is a one-dimensional finite range of integration and every coefficient of the spherical-harmonic-function series is a two-dimensional finite range of integration. The addition theorem of wave functions for anisotropic media can be derived from that of wave functions for isotropic media. Weyl's method of deriving the scalar Green's function in isotropic media is generalized to the study of the dyadic Green's function in anisotropic media. The cold homogeneous magnetoplasma is considered as an illustrative example. For a cold homogeneous magnetoplasma, simplified series representations of wave functions and dyadic Green's functions are given. The distributional singular behavior of the dyadic Green's functions in the source region is investigated and taken into account by solving the static problem and the boundary integral equation is derived.

On the application domain of the Green’s function approximation for mild anisotropic media [Comments to the article ‘‘On Green’s functions for elastic waves in anisotropic media,’’ J. Acoust. Soc. Am. 8 3, 118–121 (1988)

A Green’s function for anisotropic media was developed for materials having no cusps in the energy velocity profile and exhibiting a mild anisotropy, that is with a relatively small deviation of the quasilongitudinal wave velocity from its averaged value. A scaling transform is introduced that allows severe anisotropy to be considered.