A new method for generating fully isotropic laminates
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Using matricant method elastic moduli of occasionally heterogeneous isotropic and anisotropic elastic media were received. Anisotropic behaviour and conditions for change in anisotropy of media associated with averaging of one-dimensional periodic structures was determined.
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Abstract Properties of a new parameter, ηκ, that is recently introduced by Kawakatsu et al. for transverse isotropy are examined. It is illustrated that the parameter nicely characterizes the incidence angle dependence of bodywave phase velocities for vertical transverse isotropy models that share the same P- and S-wave anisotropy. When existing models of upper-mantle radial anisotropy are compared in terms of this new parameter, PREM shows a distinct property. Within the anisotropic layer of PREM (a depth range of 24.4–220 km), ηκ < 1 in the upper half and ηκ > 1 in the lower half. If ηκ > 1, anisotropy cannot be attributed to a layering of homogeneous isotropic layers, and thus requires the presence of intrinsic anisotropy.
Layering
Seismic anisotropy
Transverse isotropy
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In inhomogeneous isotropic regions, S-waves can be modeled using the ray method for isotropic media. In inhomogeneous strongly anisotropic regions, the independently propagating qS1- and qS2-waves can similarly be modeled using the ray method for anisotropic media. The latter method does not work properly in inhomogenous weakly anisotropic regions, however, where the split qS-waves couple. The zeroth‐order approximation of the quasi‐isotropic (QI) approach was designed for just such inhomogeneous weakly anisotropic media, for which neither the ray method for isotropic nor anisotropic media applies. We test the ranges of validity of these three methods using two simple synthetic models. Our results show that the QI approach more than spans the gap between the ray methods: it can be used in isotropic regions (where it reduces to the ray method for isotropic media), in regions of weak anisotropy (where the ray method for anisotropic media does not work properly), and even in regions of moderately strong anisotropy (in which the qS-waves decouple and thus could be modeled using the ray method for anisotropic media). A modeling program that switches between these three methods as necessary should be valid for arbitrary‐strength anisotropy.
Shear waves
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Linearization
USable
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Summary To solve an anisotropic flow problem, the problem is transformed into an equivalent isotropic problem. It is expected that an anisotropic solution can be readily obtained from the equivalent isotropic solution. However, sometimes that is not the case. Using the drawdown equation of a fully penetrated vertical well test and the drawdown and buildup equations for a probe test, this paper discusses when it is both possible and impossible to directly obtain an anisotropic solution from an isotropic solution. Because of the fact that transformation causes the changes in flow geometry and the shape of the wellbore or probe, the equivalent isotropic solution must be modified to obtain the anisotropic solution. Finally, a case of how horizontal well productivity was incorrectly derived in a published paper --caused by the misuse of the concept of anisotropy--is presented. The full manuscript of SPE 114504, Discussion of Permeability Anisotropy Effect, is available as a supplement to this article.
Drawdown (hydrology)
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Susceptibility measurements have been done on a random mixture of two anisotropic antiferromagnets with different orientations of easy axes, FeCl 2 ·2H 2 O and CoCl 2 ·2H 2 O. An isotropic state in which the anisotropy in the susceptibility becomes zero is found in the intermediate concentration region. A preliminary specific heat measurement suggests that the isotropic state lies within an ordered phase of the random mixture. The present experiment is discussed in connection with the existing theory.
Zero (linguistics)
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Media containing aligned rotationally symmetrical inclusions show transverse isotropy with respect to elastic wave propagation. The characteristics of this type of anisotropy have been investigated in the first part of this thesis (chapters 2, 3, and 4) while its implications on Vertical Seismic Profiling have been investigated in the second part of this thesis (chapters 5, 6, and 7). Transverse isotropy due to aligned inclusions has been studied for inclusions ranging from flat cracks (very small aspect ratio a.) up to spheres (a. = 1) using Nishizawa's model (chapter 2). The resultant anisotropy as described by this model (which is based on a static approach) is identical to the anisotropy described by Hudson's crack model (based on the scattering of elastic waves) for inclusions with aspect ratios up to 0.3. This result, which is surprising because Hudson's model has been derived for small aspect ratios (a. < 1), implies (assuming the validity of Nishizawa's model) that Hudson's model which is often used to interpret anisotropy observations can be applied to much larger aspect ratios than the aspect ratios for which it has been derived. Another characteristic of the anisotropy as described by Nishizawa's model is that almost spherical inclusions (a. =:: 1) result in elliptical anisotropy, which is a type of anisotropy that can never be due to sequences of thin isotropic layers. Sequences of isotropic layers and systems of large aligned fractures are just like aligned inclusions possible causes of transverse isotropy. Although these fractures and sequences of isotropic layers have a geometry that is different from the geometry of aligned inclusions they may result in the same anisotropy as aligned inclusions (chapters 3 and 4). The model describing the anisotropy due to large aligned fractures turns out to be identical to Hudson's crack model, whereas the model describing the anisotropy due to fine layering is identical to Hudson's model for ranges of aspect ratios that strongly depend on the fluid inside the inclusions. For the situations that these models are similar observed anisotropy can only be interpreted in terms of crack distributions if additional information shows the existence of cracks. However, for the situations where the similarity does not hold it is possible to distinguish between the causes of transverse isotropy (chapter 4). It should be realized, however, that other causes of transverse isotropy exist. Therefore, the 'separation' method described in this thesis should only be considered as a first step towards distinguishing between the causes of transverse isotropy. Because the 'representability' of cracked media by finely layered media strongly depends on the fluid inside the cracks, this 'representability' might not only be an interesting way to distinguish between the causes of transverse isotropy, but might also be an useful tool to investigate the nature of the fluid. Considering an uniformly cracked medium monitoring the nature of the fluid as a function of time or space could be very important for earthquake prediction or gas exploration, respectively. Studying anisotropy observations is a powerful way to obtain information about the internal structures of the rocks (such as aligned inclusions, thin layering) which have dimensions much smaller than the seismic wavelength used. Studying crack-induced anisotropy offers the possibility to monitor the stress-field that aligns the cracks. There is evidence that a changing stress strongly affects the aspect ratio of the cracks. The results of the first part of this thesis on the effect of a changing aspect ratio on crack-induced anisotropy have been used to develop methods to interpret anisotropy observations in multi-offset shear-wave VSPs in terms of a changing aspect ratio. Changes in the aspect ratio can be monitored in such VSPs (chapter 6) by studying the changes in the direction of wave propagation at which there is no shear-wave splitting. This technique which has been applied to synthetic shear-wave VSPs could become important if repeated VSPs are carried out to analyze temporal changes in anisotropy in terms of a changing stress-field. Although shear-wave splitting is often used as a key identifier of anisotropy one should be aware that shear-wave splitting can also be caused by transmission effects at interfaces in isotropic media (chapter 5). This effect should be taken into account first before shear-wave splitting is interpreted in terms of anisotropy. Anisotropy may give valuable information about the internal structures of rocks, but it may also lead to erroneous interpretations, when it is not properly taken into account. In chapter 7 this has been shown for an isotropic traveltime inversion scheme which, when applied to multi-offset VSP traveltime data in layered transversely isotropic media, may introduce errors in the depths of the interfaces separating the layers. Therefore, anisotropy should be included in inversion schemes. In a first attempt to develop inversion schemes that do take anisotropy into account a transversely isotropic traveltime inversion scheme has been developed and successfully applied to synthetic multi-offset VSP-data. The method developed is a robust method and further research is necessary to develop more elegant methods. Despite the robustness of the method the results of the transversely isotropic traveltime inversion scheme show, when compared with the results of the isotropic inversion scheme, that both an isotropic and a transversely isotropic model can explain the same traveltime data set (consisting of the arrival times of P- and first arriving S-waves). To attack this problem of non-uniqueness additional information (such as polarization, shear-wave splitting) should be incorporated in the inversion.
Aspect ratio (aeronautics)
Transverse isotropy
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Variogram
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Using matricant method elastic moduli of occasionally heterogeneous isotropic and anisotropic elastic media were received. Anisotropic behaviour and conditions for change in anisotropy of media associated with averaging of one-dimensional periodic structures was determined.
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