Home About us Contact
|
Home About us Contact | ||
|
|
||
Isotropic Media (isotropic + media)
Selected AbstractsEnergy flux in viscoelastic anisotropic mediaGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2006Vlastislav, ervený SUMMARY We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P, SV and SH waves. [source] Surface waves in a general anisotropic poroelastic solid half-spaceGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2004M. D. Sharma SUMMARY A method is introduced for studying surface waves in a general anisotropic poroelastic medium. The method is analogous to the one used for isotropic media and derives a complex secular equation to represent the propagation of surface waves at the stress-free plane surface of a non-dissipative porous medium. The point of importance is that the derived equation is, analytically, separable into real and imaginary parts and hence can be solved by iterative numerical methods. A root of this secular equation represents the existence of surface waves and calculates the apparent phase velocity along a given direction on the surface. Numerical work is carried out for the model of a crustal rock. The propagation of surface waves is studied numerically for the top three anisotropies (i.e. triclinic, monoclinic, orthorhombic). [source] Vertically fractured transversely isotropic media: dimensionality and deconstructionGEOPHYSICAL PROSPECTING, Issue 2 2009Michael A. Schoenberg ABSTRACT A vertically fractured transversely isotropic (VFTI) elastic medium is one in which any number of sets of vertical aligned fractures (each set has its normal lying in the horizontal x1, x2 -plane) pervade the medium and the sets of aligned fractures are the only features of the medium disturbing the axi-symmetry about the x3 -axis implying that in the absence of fractures, the background medium is transversely isotropic (TI). Under the assumptions of long wavelength equivalent medium theory, the compliance matrix of a fractured medium is the sum of the background medium's compliance matrix and a fracture compliance matrix. For sets of parallel rotationally symmetric fractures (on average), the fracture compliance matrix is dependent on 3 parameters , its normal and tangential compliance and its strike direction. When one fracture set is present, the medium is orthorhombic and the analysis is straightforward. When two (non-orthogonal) or more sets are present, the overall medium is in general elastically monoclinic; its compliance tensor components are subject to two equalities yielding an 11 parameter monoclinic medium. Constructing a monoclinic VFTI medium with n embedded vertical fracture sets, requires 5 TI parameters plus 3×n fracture set parameters. A deconstruction of such an 11 parameter monoclinic medium involves using its compliance tensor to find a background transversely isotropic medium and several sets of vertical fractures which, in the long wavelength limit, will behave exactly as the original 11 parameter monoclinic medium. A minimal deconstruction, would be to determine, from the 11 independent components, the transversely isotropic background (5 parameters) and two fracture sets (specified by 2 × 3 = 6 parameters). Two of the background TI medium's compliance matrix components are known immediately by inspection, leaving nine monoclinic components to be used in the minimal deconstruction of the VFTI medium. The use of the properties of a TI medium, which are linear relations on its compliance components, allows the deconstruction to be reduced to solving a pair of non-linear equations on the orientations of two fracture sets. A single root yielding a physically meaningful minimum deconstruction yields a unique minimal representation of the monoclinic medium as a VFTI medium. When no such root exists, deconstruction requires an additional fracture set and uniqueness is lost. The boundary between those monoclinic media that have a unique minimal representation and those that do not is yet to be determined. [source] Migration velocity analysis for tilted transversely isotropic mediaGEOPHYSICAL PROSPECTING, Issue 1 2009Laxmidhar Behera ABSTRACT Tilted transversely isotropic formations cause serious imaging distortions in active tectonic areas (e.g., fold-and-thrust belts) and in subsalt exploration. Here, we introduce a methodology for P-wave prestack depth imaging in tilted transversely isotropic media that properly accounts for the tilt of the symmetry axis as well as for spatial velocity variations. For purposes of migration velocity analysis, the model is divided into blocks with constant values of the anisotropy parameters , and , and linearly varying symmetry-direction velocity VP0 controlled by the vertical (kz) and lateral (kx) gradients. Since determination of tilt from P-wave data is generally unstable, the symmetry axis is kept orthogonal to the reflectors in all trial velocity models. It is also assumed that the velocity VP0 is either known at the top of each block or remains continuous in the vertical direction. The velocity analysis algorithm estimates the velocity gradients kz and kx and the anisotropy parameters , and , in the layer-stripping mode using a generalized version of the method introduced by Sarkar and Tsvankin for factorized transverse isotropy with a vertical symmetry axis. Synthetic tests for several models typical in exploration (a syncline, uptilted shale layers near a salt dome and a bending shale layer) confirm that if the symmetry-axis direction is fixed and VP0 is known, the parameters kz, kx, , and , can be resolved from reflection data. It should be emphasized that estimation of , in tilted transversely isotropic media requires using nonhyperbolic moveout for long offsets reaching at least twice the reflector depth. We also demonstrate that application of processing algorithms designed for a vertical symmetry axis to data from tilted transversely isotropic media may lead to significant misfocusing of reflectors and errors in parameter estimation, even when the tilt is moderate (30°). The ability of our velocity analysis algorithm to separate the anisotropy parameters from the velocity gradients can be also used in lithology discrimination and geologic interpretation of seismic data in complex areas. [source] A laboratory study of seismic velocity and attenuation anisotropy in near-surface sedimentary rocksGEOPHYSICAL PROSPECTING, Issue 5 2007Angus I. Best ABSTRACT The laboratory ultrasonic pulse-echo method was used to collect accurate P- and S-wave velocity (±0.3%) and attenuation (±10%) data at differential pressures of 5,50 MPa on water-saturated core samples of sandstone, limestone and siltstone that were cut parallel and perpendicular to the vertical borehole axis. The results, when expressed in terms of the P- and S-wave velocity and attenuation anisotropy parameters for weakly transversely isotropic media (,, ,, ,Q, ,Q) show complex variations with pressure and lithology. In general, attenuation anisotropy is stronger and more sensitive to pressure changes than velocity anisotropy, regardless of lithology. Anisotropy is greatest (over 20% for velocity, over 70% for attenuation) in rocks with visible clay/organic matter laminations in hand specimens. Pressure sensitivities are attributed to the opening of microcracks with decreasing pressure. Changes in magnitude of velocity and attenuation anisotropy with effective pressure show similar trends, although they can show different signs (positive or negative values of ,, ,Q, ,, ,Q). We conclude that attenuation anisotropy in particular could prove useful to seismic monitoring of reservoir pressure changes if frequency-dependent effects can be quantified and modelled. [source] Paraxial ray methods for anisotropic inhomogeneous mediaGEOPHYSICAL PROSPECTING, Issue 1 2007Tijmen Jan Moser ABSTRACT A new formalism of surface-to-surface paraxial matrices allows a very general and flexible formulation of the paraxial ray theory, equally valid in anisotropic and isotropic inhomogeneous layered media. The formalism is based on conventional dynamic ray tracing in Cartesian coordinates along a reference ray. At any user-selected pair of points of the reference ray, a pair of surfaces may be defined. These surfaces may be arbitrarily curved and oriented, and may represent structural interfaces, data recording surfaces, or merely formal surfaces. A newly obtained factorization of the interface propagator matrix allows to transform the conventional 6 × 6 propagator matrix in Cartesian coordinates into a 6 × 6 surface-to-surface paraxial matrix. This matrix defines the transformation of paraxial ray quantities from one surface to another. The redundant non-eikonal and ray-tangent solutions of the dynamic ray-tracing system in Cartesian coordinates can be easily eliminated from the 6 × 6 surface-to-surface paraxial matrix, and it can be reduced to 4 × 4 form. Both the 6 × 6 and 4 × 4 surface-to-surface paraxial matrices satisfy useful properties, particularly the symplecticity. In their 4 × 4 reduced form, they can be used to solve important boundary-value problems of a four-parametric system of paraxial rays, connecting the two surfaces, similarly as the well-known surface-to-surface matrices in isotropic media in ray-centred coordinates. Applications of such boundary-value problems include the two-point eikonal, relative geometrical spreading, Fresnel zones, the design of migration operators, and more. [source] Traveltime computation with the linearized eikonal equation for anisotropic mediaGEOPHYSICAL PROSPECTING, Issue 4 2002Tariq Alkhalifah A linearized eikonal equation is developed for transversely isotropic (TI) media with a vertical symmetry axis (VTI). It is linear with respect to perturbations in the horizontal velocity or the anisotropy parameter ,. An iterative linearization of the eikonal equation is used as the basis for an algorithm of finite-difference traveltime computations. A practical implementation of this iterative technique is to start with a background model that consists of an elliptically anisotropic, inhomogeneous medium, since traveltimes for this type of medium can be calculated efficiently using eikonal solvers, such as the fast marching method. This constrains the perturbation to changes in the anisotropy parameter , (the parameter most responsible for imaging improvements in anisotropic media). The iterative implementation includes repetitive calculation of , from traveltimes, which is then used to evaluate the perturbation needed for the next round of traveltime calculations using the linearized eikonal equation. Unlike isotropic media, interpolation is needed to estimate , in areas where the traveltime field is independent of ,, such as areas where the wave propagates vertically. Typically, two to three iterations can give sufficient accuracy in traveltimes for imaging applications. The cost of each iteration is slightly less than the cost of a typical eikonal solver. However, this method will ultimately provide traveltime solutions for VTI media. The main limitation of the method is that some smoothness of the medium is required for the iterative implementation to work, especially since we evaluate derivatives of the traveltime field as part of the iterative approach. If a single perturbation is sufficient for the traveltime calculation, which may be the case for weak anisotropy, no smoothness of the medium is necessary. Numerical tests demonstrate the robustness and efficiency of this approach. [source] Out-of-plane geometrical spreading in anisotropic mediaGEOPHYSICAL PROSPECTING, Issue 4 2002Norman Ettrich Two-dimensional seismic processing is successful in media with little structural and velocity variation in the direction perpendicular to the plane defined by the acquisition direction and the vertical axis. If the subsurface is anisotropic, an additional limitation is that this plane is a plane of symmetry. Kinematic ray propagation can be considered as a two-dimensional process in this type of medium. However, two-dimensional processing in a true-amplitude sense requires out-of-plane amplitude corrections in addition to compensation for in-plane amplitude variation. We provide formulae for the out-of-plane geometrical spreading for P- and S-waves in transversely isotropic and orthorhombic media. These are extensions of well-known isotropic formulae. For isotropic and transversely isotropic media, the ray propagation is independent of the azimuthal angle. The azimuthal direction is defined with respect to a possibly tilted axis of symmetry. The out-of-plane spreading correction can then be calculated by integrating quantities which describe in-plane kinematics along in-plane rays. If, in addition, the medium varies only along the vertical direction and has a vertical axis of symmetry, no ray tracing need be carried out. All quantities affecting the out-of-plane geometrical spreading can be derived from traveltime information available at the observation surface. Orthorhombic media possess no rotational symmetry and the out-of-plane geometrical spreading includes parameters which, even in principle, are not invertible from in-plane experiments. The exact and approximate formulae derived for P- and S-waves are nevertheless useful for modelling purposes. [source] Localized electrical current propagation in anisotropically perturbed atmospheresINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2010T. I. ZohdiArticle first published online: 29 MAR 2010 Abstract The trajectory of free atmospheric electrical currents, such as lightning and sparks, is strongly influenced by microscale events that occur at the current front. In particular, highly conductive pathways can occur at the free surface front due to dielectric breakdown. The specific directions of the local pathways are minutely perturbed, due to the gaseous, disordered, nature of the media at the small scale. This results in highly conductive, anisotropically perturbed, continuum-level properties at the electrical current front. In this work, a model is developed to investigate the role of the resulting anisotropically perturbed conductivity at the propagation front on the overall trajectory of free atmospheric electrical currents. The approach is to relate the electrical current velocity to the local anisotropic conductivity at the propagation front and the surrounding electric field. The conductive anisotropy is decomposed into an isotropic ,base state' and an anisotropic perturbation. The current trajectory is shown to be governed by a set of non-linear differential equations, for which a numerical solution scheme is developed. The difference between paths taken through anisotropically perturbed and isotropic media is analytically bounded and quantified numerically as a function of the magnitude of the anisotropic perturbation. The analysis and numerical experiments indicate that, in a statistical sense, the difference in the paths taken in anisotropically perturbed and isotropic media depends quasilinearly on the perturbation magnitude. Copyright © 2010 John Wiley & Sons, Ltd. [source] Local discretization error bounds using interval boundary element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009B. F. Zalewski Abstract In 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. [source] Synthesis and characterization of new alternating, amphiphilic, comblike copolymers of poly(ethylene oxide) macromonomer and N -phenylmaleimideJOURNAL OF POLYMER SCIENCE (IN TWO SECTIONS), Issue 3 2005Luminita Cianga Abstract A surface-active p -vinyl benzyloxy-,-hydroxy-poly(ethylene oxide) macromonomer containing 22 pendant structural units of ethylene oxide (St,PEO22) was synthesized with an initiation method. Because of its solubility in a large variety of solvents, the free-radical copolymerization with electron-acceptor N -phenylmaleimide (NPMI) was performed at 60 °C in benzene and tetrahydrofuran (THF) as isotropic media and in a water,THF mixture or water as a heterogeneous medium. Oil-soluble 2,2,-azobisisobutyronitrile and water-soluble 4,4,-azobis(4-cyanovaleric acid) were used as the initiators at fixed concentrations. Two different St,PEO22/NPMI comonomer ratios (1/1 and 3/7) at a fixed total comonomer concentration in the polymerization system were used. The structures, compositions, and microstructure peculiarities of the obtained alternating, amphiphilic, comblike copolymers were determined by NMR analysis. For the copolymers synthesized in hydrophilic media, differential scanning calorimetry showed, near the endothermic peak attributed to the melting of the poly(ethylene oxide) side chains, the presence of a second peak due to the partially ordered phase that could exist between the crystalline state and the isotropic melt. Also, the thermal stability of the obtained copolymers was studied with thermogravimetric analysis. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 479,492, 2005 [source] | ||||||||||