On Seismic Waves in Linearly Elastic, Anisotropic and Nonuniform Continua: Tutorial (original) (raw)
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Chapter 3: Elastic waves in isotropic, homogeneous rocks
IFP Energies nouvelles eBooks, 2014
This book can be considered as a natural continuation of the book entitled 'Acoustics of Porous Media', co-authored by Thierry Bourbié, Olivier Coussy and Bernard Zinszner, and issued by our laboratory in 1986 for the French version, and in 1987 for the English version. However, here the clear guideline is experimentation. In contrast to previous books, all the techniques, from the most conventional (using piezoelectric transducers) to the most recent spaceage methods (as laser ultrasonics) are detailed. Furthermore the book is mainly based on experimental data allowing to select the most appropriate theories for describing elastic wave propagation in rocks. Emphasis on Nonlinear elasticity and Seismic anisotropy are also originality of the book. A part of the book also focuses on the history of the different sub-fields dealt with, having in mind that the knowledge of the history of a field contributes to understanding the field itself. For instance, in spite of the clear anteriority of their work the names of the Persian mathematician, physicist and optics engineer Ibn Sahl, and of the English astronomer and mathematician Thomas Harriot are unfairly not, or rarely, associated with the law of refraction, compared to the names of the Dutch astronomer and mathematician Willebrord Snell van Royen, known as Snellius, and of the French philosopher and writer René Descartes, as detailed in the first chapter.
Seismic coda due to non-linear elasticity
Geophysical Journal International, 2008
Non-linear elastic response of rocks has been widely observed in laboratory, but very few seismic studies are reported in the literature, even though it is the most natural environment where this feature could be observed. Analytic solutions to the non-linear wave propagation phenomena are not readily available, and there is a need to use approximated techniques. It is clear that when a seismic wave propagates through a homogeneous non-linear elastic media, it will be perturbed by the non-linearity. This perturbation can be treated as a source of scattering, spreading the energy of the primary wave in space and time, contributing to the seismic coda. This is in some sense similar to the effect of heterogeneities. The properties of the coda due to the non-linearity depend on the amount of non-linearity and the seismic moment. Using a perturbation approach we calculate the amplitude of the scattered waves, and show that it can describe reasonably well the main features of real seismic codas.
Waves and Rays in Elastic Continua
Waves and Rays in Elastic Continua, 2014
WAVES AND RAYS IN ELASTIC CONTINUA "Professor Slawinski has written an introductory textbook that rigorously develops the foundations for elastodynamics and wave propagation in anisotropic elastic media in a clear and well written style. The main text is supplemented with many worked examples and suggestions for further reading. This text would be a useful textbook for a senior undergraduate or introductory graduate level, applied mathematics course in waves in elastic media, and as an introduction to research in theoretical seismology." Chris Chapman, Cambridge University "This impressive treatise contains a lucid and comprehensive description of material symmetry in the context of seismic wave propagation. These insights form the foundation of a careful and detailed analysis of seismic waves in anisotropic media. 'Waves & Rays' is very well-suited as a text for a graduate course and as a reference monograph for experts in the field."
Evidence that non-linear elasticity contributes to the seismic coda
Geophysical Journal International, 2010
Different factors might affect the propagation of seismic waves producing scattering, including heterogeneities and non-linear elasticity. A key difference between these two factors is the dependence of the strength of the scattered waves on the strength of the incident wave, being linear for the former and non-linear for the latter. A detailed study of the TIPTEQ data, where about a hundred explosions were recorded on 180 three-component stations in the distance range of approximately 0-100 km, shows that this dependence is non-linear. Data were analysed in the following way: (i) the envelope of bandpass filtered data between 10 and 40 Hz was obtained for a large number of stations from different distance ranges and charge sizes of shots, (ii) for these distances we modelled the envelope considering the non-linear elasticity. The shapes of the theoretical and observed envelopes were in general very similar. A scale factor for each case was obtained considering the best fit of its complete envelope and (iii) since this scale factor depends mainly on the size of the explosion, we computed the ratio (R) of the scale factor (sf) for different sizes of explosions at the same distance. Finally, varying the distance between 0 and 50 km and (iv) we computed the power (p) of the dependence of the ratio (R) on the ratio of the charge sizes [R = (s f 1 s f 2) = (charge 1 charge 2) p ]. For the complete data set we obtain a value of p = 2.5 ± 0.9, which is clearly greater than 1. This shows that non-linear elasticity is an important factor in the contribution to seismic wave scattering in the frequency range of 10-40 Hz.
Geophysical Journal International, 1987
The differential system for elastic waves in stratified anisotropic media is, as in the isotropic case, most simply solved by constructing a propagator from the eigenvalues and eigenvectors. In general, the eigensolutions must be found numerically, which may lead to an enormous amount of computation if the ultimate goal is to construct synthetic seismograms. However, if the stiffness tensor for each layer displays a horizontal plane of symmetry, simple analytical solutions exist and the expense of numerical solution can be avoided. We consider monoclinic, orthorhombic, hexagonal (including transversely isotropic), and istropic symmetries and give analytical expressions for the eigensolutions for each of these forms when the alignment is such that a horizontal symmetry plane exists. LThe analytical solutions are concise enough to make the construction of complete synthetic seismograms (a task which involves the construction and manipulation of 3-D functions) a realistic endeavour. While not applicable to arbitrary anisotropy, these results have widespread application as most of the anisotropies hypothesized in seismology require a horizontal plane of elastic symmetry.
The effect of microstructure and nonlinear stress on anisotropic seismic velocities
GEOPHYSICS, 2008
Recent work in hydrocarbon reservoir monitoring has focused on developing coupled geomechanical/fluid-flow simulations to allow production-related geomechanical effects, such as compaction and subsidence, to be included in reservoir models. To predict realistic time-lapse seismic signatures, generation of appropriate elastic models from geomechanical output is required. These elastic models should include not only the fluid saturation effects of intrinsic, shapeinduced, and stress-induced anisotropy, but also should incorporate nonlinear stress-dependent elasticity. To model nonlinear elasticity, we use a microstructural effective-medium approach in which elasticity is considered as a function of mineral stiffness and additional compliance is caused by the presence of low-aspect ratio displacement discontinuities. By jointly inverting observed ultrasonic P-and S-wave velocities to determine the distribution of such discontinuities, we assessed the appropriateness of modeling them as simple, planar, penny-shaped features. By using this approximation, we developed a simple analytical approach to predict how seismic velocities will vary with stress. We tested our approach by analyzing the elasticity of various sandstone samples; from a United Kingdom continental shelf ͑UKCS͒ reservoir, some of which display significant anisotropy, as well as two data sets taken from the literature.
A Study of the Elastic Behaviour Presented by Different Types of Sedimentary Rocks
Revista Brasileira de Geofísica, 2015
ABSTRACT. The purpose of this paper is to study the elastic behavior of the different kinds of sedimentary rocks from outcrops of the Middle-West region of the United States of America, among those, Berea sandstone from Bedford formation (Ohio), Indiana limestone from Salem formation (Indiana) and Silurian dolomite from Thornton formation (Illinois). To do so, it has been made in the Laboratório de Física de Rochas from Cenpes (Centro de Pesquisas e Desenvolvimento da Petrobras ), measurements of porosity, density and elastic wave propagation velocity presented by each type of studied rock. The wave propagation velocities were estimated by measuring the transit time ultrasonic pulses transmitted through the samples. From the results obtained, it was possible to compare the measured velocities with predictions from theoretical models, as well as to observe correlations between the petrophysical properties of rocks and its seismic behavior. Understanding these correlations helps to im...
SN Applied Sciences
Different geomechanical properties such as elastic modulus/deformation modulus, Poisson's ratio (υ), uniaxial compressive strength, shear strength properties, tensile strength and point load index are widely used for rock mass characterisation in geological and geotechnical engineering. However, there are no such direct methods by which these properties can be obtained in the laboratory or in situ, without following time-consuming and a laborious procedure. Thus, ultrasonic technique, an indirect method, was found reliable to determine these properties in rocks. Many researchers have studied the correlation between the compressional wave velocity (Vp) and the geomechanical properties of sedimentary rocks such as sandstones and carbonates. However, these correlations have not considered the change in rock mineralogy, porosity and saturation conditions when deriving relationships. Hence, in this study, the authors attempt to understand the variability of Vp with the help of a fragmented analysis of major mineral constituents, porosity variation and saturation conditions for rock mass classification based on Vp. A review of the existing studies on the relationship between the rock properties and Vp has been used to perform this analysis. The resulting template can be a basis for interpreting more realistic lithology-based geomechanical behaviours and thus highlights the importance of an integrated study involving geological, petrophysical and engineering data. The results derived from fragmented analysis indicate that with an increase in quartz content and a subsequent decrease in feldspar content in sandstones, Young's modulus (E) and Vp increase. With an increase in porosity from 2 to 40%, there is a decrease in E and Vp values for all types of saturation scenarios (water, gas, brine and oil saturated) for sandstones and carbonates.