It has been proven by J.C.Criscione that constitutive relations(mixed approach) based on a set of five strain invariants (Beta-1, Beta-2, Beta-3, Beta-4, Beta-5) are useful and stable for experimentally determining response terms for transversely isotropic material. On the other hand, Rivlin’s classical model is an unsuitable choice for determining response terms due to the co-alignment of the five invariants (I1, I2, I3, I4, I5). Despite this, however, a mixed (Lagrangian and Eulerian) approach causes unnecessary computational time and requires intricate calculation in the constitutive relation. Through changing the way to approach the derivation of a constitutive relation, we have verified that using an Eulerian approach causes shorter computational time and simpler calculation than using a mixed approach does. We applied this approach to a boundary value problem under specific deformation, i.e. finite extension and torsion to a fiber reinforced circular cylinder. The results under this deformation show that the computational time by Eulerian is less than half of the time by mixed. The main reason for the difference is that we have to determine two unit vectors on the cross fiber direction from the right Cauchy Green deformation tensor at every radius of the cylinder when we use a mixed approach. On the contrary, we directly use the left Cauchy Green deformation tensor in the constitutive relation by the Eulerian approach without defining the two cross fiber vectors. Moreover, the computational time by the Eulerian approach is not influenced by the degree of deformation even in the case of computational time by the Eulerian approach, possibly becoming the same as the computational time by the mixed approach. This is from the theoretical thought that the mixed approach is almost the same as the Eulerian approach under small deformation. This new constitutive relation by Eulerian approach will have more advantages with regard to saving computational time as the deformation gets more complicated. Therefore, since the Eulerain approach effectively shortens computational time, this may enhance the computational tools required to approach the problems with greater degrees of anisotropy and viscoelasticity.

Made available in DSpace on 2014-10-09T12:24:40Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:05:12Z (GMT). No. of bitstreams: 1 00387.pdf: 1743145 bytes, checksum: 809e9f2fe10221e21059eec072c0659d (MD5) / Dissertacao (Mestrado) / IEA/D / Escola Politecnica, Universidade de Sao Paulo - POLI/USP

Made available in DSpace on 2014-10-09T12:24:40Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:05:12Z (GMT). No. of bitstreams: 1 00387.pdf: 1743145 bytes, checksum: 809e9f2fe10221e21059eec072c0659d (MD5) / Dissertacao (Mestrado) / IEA/D / Escola Politecnica, Universidade de Sao Paulo - POLI/USP

The objective of this thesis is to provide a detailed analysis of a new invariant for Grothendieck topoi in the special case of the topos of continuous G-sets and continuous G-equivariant maps. We use a well-known site to present the isotropy group in elementary terms, as systems of right cosets of open subgroups of G. We establish properties of the the isotropy group for an arbitrary topological group and use the developed theory to compute the isotropy group for the Schanuel topos.

>Magister Scientiae - MSc / The Cosmological Principle forms part of one of the most fundamental hypotheses of modern Cosmology. So it is very important to assess whether it holds true using observational data, or whether it consists of a mathematical simplification. We probe the statistical isotropy of the Universe using the existing radio continuum data, by means of a local variance estimator. In order to investigate this, we analyse the number count variance of the radio catalog by looking at patches of approximately 10, 15, 20 & 25 degrees in radii, and thus comparing it to mock catalogs which reproduce the matter density power spectrum, as well as the same sky coverage of the real data. We establish criteria for accepting patches that have more than 90%, 70% & 50% of their pixels not masked. We make use of the NRAO VLA Sky Survey (NVSS), whose operational frequency is 1.4 GHz. We perform statistical tests for detecting possible departures from statistical isotropy using galaxy number counts with flux limits of 20 < SNVSS < 1000 mJy. We also compare the real data to the mock catalogs of the radio data in order to assess the statistical significance of our results. We use the local variance estimator for testing the statistical isotropy of our data sample. We find that the statistical properties of our sample are in reasonable agreement with the standard cosmological model. The mean of the distribution for the data falls well within the 95% confidence interval of the average of the simulated mocks. For all the radii and acceptance criteria for the patches, we found no significant deviations beyond those allowed by the standard model. As expected there were no large discrepancies between our mocks and the data. The results are consistent with statistical isotropy.

Hydraulic fracturing of transverse isotropic reservoirs is of major interest for reservoir stimulation and in-situ stress estimation. Rock fabric anisotropy not only causes in-situ stress anisotropy, but also affects fracture initiation from the wellbore. In this study a semi-analytical method is used to investigate these effects with particular reference to shale stimulation. Using simplifying assumptions, equations are derived for stress distribution around the wellbore's walls. The model is then used to study the fracture initiation pressure variations with anisotropy. A sensitivity analysis is carried out on the impact of Young's modulus and Poisson's ration, on the fracture initiation pressure. The results are useful in designing hydraulic fractures and also can be used to develop information about in-situ rock properties using failure pressure values observed in the field. Finally, mechanical and permeability anisotropy are measured using Pulse Permeameter and triaxial tests on Pierre shale.

Unconventional oil and gas, which is of major interest in petroleum industry, often occur in reservoirs with transversely isotropic rock properties such as shales. Overlooking transverse isotropy may result in deviation in stress distribution around wellbore and inaccurate estimation of fracture initiation pressure which may jeopardize safe drilling and efficient fracturing treatment. In this work, to help understand the behavior of transversely isotropic reservoirs during drilling and fracturing, the principle of generalized plane-strain finite element formulation of anisotropic poroelastic problems is explained and a finite element model is developed from a plane-strain isotropic poroelastic model. Two numerical examples are simulated and the finite element results are compared with a closed form solution and another FE program. The validity of the developed finite element model is demonstrated. Using the validated finite element model, sensitivity analysis is carried out to evaluate the effects of transverse isotropy ratios, well azimuth, and rock bedding dip on pore pressure and stress distribution around a horizontal well. The results show that their effect cannot be neglected. The short term pore pressure distribution is sensitive to Young’ modulus ratio, while the long term pore pressure distribution is only sensitive to permeability ratio. The total stress distribution generally is not sensitive to transverse isotropy ratios. The effective stress and fracture initiation are very sensitive to Young’ modulus ratio. As the well rotates from minimum horizontal in-situ stress to maximum horizontal in-situ stress, the pore pressure and stress distributions tend to be more unevenly distributed around the wellbore, making the wellbore easier to fracture. The pore pressure and stress distributions tend to "rotate" in correspondence with the rock bedding plane. The fracture initiation potential and position will alter when rock bedding orientation varies.

Cognition of the hydraulic properties of soils is important in civil and environmental projects such as for water supply, geotechnical investigations and evaluation of pollutant spreading. This pilot project aims to develop and test a new method for characterization of the homogeneity and isotropy of different soils by using geophysical measurements. The method is based on geoelectrical (resistivity) measurements using a radial array. Using a radial array with electrode lines in various orientations crossing a midpoint at which water was infiltrated, it was possible to build a 3D-block model showing the water penetration into a soil body. Based on the infiltration pattern it was possible to evaluate the hydraulic heterogeneity and anisotropy of the material. Several common Swedish types of soils were tested. Measurement and evaluation techniques including software were developed. The project shows that geoelectrical measurements using a radial array and geoelectrical instruments available on the market can be used for rapid evaluation of the hydraulic heterogeneity and anisotropy of soils and rock.

Since Schwarzschild found the first solution of the Einstein’s equations, more than 800 solutions were found. Solutions of Einstein’s equations are classified according to their Lie algebras of isometries and their isotropy subalgebras. Solutions were taken from the USU electronic library of solutions of Einstein’s field equations and the classification used Maple code developed at USU. This classification adds to the data contained in the library of solutions and provides additional tools for addressing the equivalence problem for solutions to the Einstein field equations. In this thesis, homogeneous spacetimes, hypersurface-homogeneous spacetimes, Robinson-Trautman solutions, and some famous black hole solutions have been classified.

The estimation of anisotropic parameters in the shallow subsurface becomes increasingly important for 4C seismic data processing in order to obtain accurate images in both time and depth domain. I focus on two approaches to evaluate anisotropy in seismic data: using P-wave data and PS-converted (C-wave) data. To gain better insight into the accuracy and sensitivity of anisotropic parameters to for instance layering and compaction gradients, I undertake numerical modelling studies and verify the results with full-wave modelling as well as findings from the real data from a 4C data set from the Alba field. The focus of this thesis is on vertical transverse isotropy (VTI) which widely occurs in marine sediments and cannot be neglected in seismic processing. P-wave data alone cannot constrain the vertical velocity and the depth scale of the earth model for a VTI medium. Therefore, the joint inversion of non-hyperbolic P- and converted wave (C-wave) or S-wave data from long offsets has been suggested. I carried out a detailed analysis of the resolution and accuracy of non-hyperbolic moveout inversion for P-, S- and C-waves for a single VTI layer in two parts. First, I introduce the concept of the inherited error delta inh as a measure of the possible resolution of the moveout approximations for the different wave types. The range of this error stays constant regardless of the magnitude of the anisotropic parameter for each wave type. Second, I analyse the accuracy of non-hyperbolic moveout inversion. I find that for anisotropy parameter eta the error of estimation from C-wave data is in most cases about half that from P-wave data. Inversion of non-hyperbolic S-wave moveout data does not resolve the anisotropy parameter due to the presence of cusps in the data. The study is then extended to a multilayered medium considering only P- and C-waves. The results confirm the findings from the single layer case. Furthermore, I investigate phase effects on parameter estimation for P- and C-waves. It is suggested that eta estimated from C-wave data gives a better description of the anisotropy found in a medium than the eta values picked from P-wave data. To verify the above findings near-surface effects are studied on the 4C data from the Alba field and accompanied by a full-waveform modelling study. I find that the picked eta values from P-wave data are distinctly larger than the eta values from C-wave data and also larger than the eta values from VSP data. The full-wave modelling study shows that picked eta values from P-wave data may account for influence of structure such as velocity gradients in the near-surface and are influenced by high velocity ratios and phase reversals. Finally, I have carried out an integrated analysis of the Alba 4C data to demonstrate how seismic anisotropy can be estimated from 4C seismic data and how such information can be used to improve subsurface imaging. The results are presented in two parts. The first part deals with non-hyperbolic moveout analysis for estimating anisotropic parameters to gain improved stacked sections. The second part describes migration model building and final imaging. The models are evaluated by comparison with VSP data results and with a synthetic modelling study for three events of the overburden. The evaluation confirms that the anisotropy parameter obtained from C-wave moveout corresponds better with the VSP data than the values directly estimated from P-wave data.