Wave Propagation In Elastic And Viscoelastic Media With The Study Of Green’s Function And Fractional Viscoelasticity



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Viscoelasticity is a great example of how properties of real-life objects in Fourier space are manifested through complex numbers consisting of real and imaginary components. In Fourier space, the stiffness/compliance constants are real for elastic materials but complex for viscoelastic materials. In Geosciences, it is common occurring that the host rock and the fluid inside the fractures both have viscoelastic characteristics. Thus, a scientific study taking the physics of the system into account is necessary to derive the constitutive equations responsible for wave propagation in such media. I have dealt with such problems in detail in this thesis. I start the thesis by discussing elastic wave propagation and a detailed derivation of the far-field Green’s function in elastic transversely anisotropic media. Validation of the Green’s function is done by considering the stiffness tensor of Barnett shale cores. I use the generalized singular approximation method of effective medium theory to model the effective stiffness of the core samples from microstructural properties. The behaviour of Green’s function, which reflects the behaviour of the media, is studied in the static, low, and high-frequency domains and under different physical parameters. It is observed that the variations of crack induced porosity produces different trends in Green’s function for vertically transverse isotropic and horizontally transverse isotropic media. I start with wave propagation in viscoelastic media in the next part and move on to derive the anisotropic far-field Green’s function for a viscoelastic media. I have derived the frequency-dependent Green’s function by using the spectral theorem for matrices, thus simplifying the process of computing Green’s function and obtaining analytical solutions. Based on the degeneracy of the eigenvalues of the Green-Christoffel tensor, the inverse of the Green-Christoffel tensor is expressed in the form of partial fractions. Finally, I touch on the topic of fractional viscoelasticity. Fractional calculus helps explain the behaviour of a wide range of viscoelastic materials and thus helps bridge the gap between theory and observed behaviour of media. I end the thesis by discussing this topic and derivation of equations of traditional viscoelastic models by involving fractional calculus.



Green's function, Viscoelasticity, Wave Propagation, Attenuation, Elasticity


Portions of this document appear in: Ghosh, A., and Morshed, S. (2021) A Green’s function approach to the study of effective anisotropic properties of the Barnett Shale. Geophysical Prospecting, 69, 968-983; and in: Ghosh, A., Albeshr, I., and Morshed, S. (2020) Green’s Function of the Wave Equation for a Fractured Dissipative HTI Medium Taking the Viscoelasticity of the System into Account. Journal of Physics Research and Applications, 4, 1000115.