Frequency-scale effect in elastic periodic muti-layered media



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In general, a reservoir is formed by laminated layers of shale, sand, and other types of lithologies. Each layer has different elastic properties that are described by its own stiffness tensor, density, and thickness. A periodic medium is defined as the stack of homogeneous layers with different elastic properties that it repeats after some length (d), where d is called the period of the medium. If a wave-field travels through this periodic medium composed of isotropic layers, the wave shows two mutually exclusive behaviors. When the wavelength is smaller than d, the wave behaves as if in an isotropic medium but when wavelength is bigger than d, the wave behaves as if in an equivalent transversely anisotropic medium (Postma,1955; Rytov,1956; Rich,2006). The goals of this thesis are three-fold. The first one is to describe the physical behavior of wave velocity as a function of thickness d and the impedance contrast between constituents when P-, Sv-, and Sh-wave travels in a periodic medium. The second goal is to quantify, in term of thickness of the period of the medium, the frequency and wavelength values at which the medium behaves as an effective anisotropic one. The third goal is to compute the seismic response of this elastic periodic medium.

The description of how the wavefield propagates through the medium is given by the solution of the wave equation for elastic media. This solution allows the definition of Brillouin zones whose width is equal to pi/d and shows that the periodic medium exhibits a range of stop-bands frequencies where the wave does not propagate. These stop-bands are located at the boundary of the Brillouin zone for the all of the types of waves (P, Sv, and Sh). In the case of P- and Sv-waves, there are also stop-bands inside the Brillouin zone that depend on the frequency and angle of incidence of the wavefield. As a result, seismic response and dispersion relationships show that the medium can be considered anisotropic when wavelength>10d and this anisotropic behavior is also a function of the wavefield frequency, for instance if the medium has a period of d=30m, the medium is anisotropic for frequency less than 10hz.



Wavelength, Wavelength of traveling wave, Frequency, Frequency of the wave