The complex and frequency-dependent stiffness components of a transversely isotropic viscoelastic (TIV) medium are determined by performing compressibility and shear oscillatory experiments, which allow us to obtain the phase velocity and quality factor of homogeneous viscoelastic waves as a function of the propagation angle. The numerical tests are defined by a collection of boundary-value problems (BVP) formulated in the space-frequency domain, which are solved using a Galerkin finite-element procedure. In particular, we apply the method to determine the set of complex and frequencydependent coefficients defining a TIV medium equivalent to a finely layered medium in the long-wavelength approximation. The results are compared to the analytical phase velocities and quality factors predicted by the Backus/Carcione theory. A sequence of shale and limestone thin layers is considered as an example.
Many geological systems can be modeled as effective TIV media. Fine layering is a typical example which refers to the case when the dominant wavelength of the traveling waves are much larger than the average thicknesses of the single layers. When this occurs, the medium is effectively anisotropic with a TI symmetry. In the case when the single layers are transversely isotropic and elastic, with the symmetry axis perpendicular to the layering plane, (Backus, 1962) obtained the average elastic constants. Later Backus theory was extended for anisotropic single constituents (Schoenberg, M. and Muir, F., 1989). Backus averaging was verified numerically in Carcione, J. M. et al. (1991), and generalized to the anelastic case in Carcione, J. M. (1992), in what constitutes the Backus/Carcione (BC) theory to describe anisotropic attenuation. As an example of fine layering, we mention the sedimentary record of the oceans, seas and lakes produced by a series of depositional events occurring on seasonal, annual or decade timescales (Kempt (1996)). When preserved in reservoir formations, laminated sediments record these short-time scale processes, in the form of 3-4 cm thick thin layers composed by a shale-limestone or a sandstone-limestone stratification. We present a new procedure to determine the complex coefficients defining a TIV medium. The methodology consists of applying time-harmonic oscillatory tests to a numerical rock sample at a finite number of frequencies. Each test is defined as a BVP stated in the space frequency domain, using the viscoelastic equation of motion with appropriate boundary conditions, and is solved employing a finite-element method (FEM). These tests can be regarded as an upscaling method to obtain the effect of the fine layering on the macroscale. The method is illustrated in Santos, J. E. et al. (2009) for poroelastic isotropic media and it is generalized here for anisotropic media. The procedure is applied to compute the phase velocities and quality factors of a sequence of shale and limestone layers.
Let u(x,w) denote the time Fourier transform of the displacement vector, where w is the angular frequency, and let si j(u) and ei j(u) denote respectively the time Fourier transforms of the stress and strain tensors of the viscoelastic material.
