Constitutive model and wave equations for linear, viscoelastic, anisotropic media

Rocks are far from being isotropic and elastic. Such simplifications in modeling the seismic response of real geological structures may lead to misinterpretations, or even worse, to overlooking useful information. It is useless to develop highly accurate modeling algorithms or to naively use amplitude information in inversion processes if the stress-strain relations are based on simplified rheologies. Thus, an accurate description of wave propagation requires a rheology that accounts for the anisotropic and anelastic behavior of rocks. This work presents a new constitutive relation and the corresponding time-domain wave equation to model wave propagation in inhomogeneous anisotropic and dissipative media. The rheological equation includes the generalized Hooke’s law and Boltzmann’s superposition principle to account for anelasticity. The attenuation properties in different directions, associated with the principal axes of the medium, are controlled by four relaxation functions of viscoelastic type. A dissipation model that is consistent with rock properties is the general standard linear solid. This is based on a spectrum of relaxation mechanisms and is suitable for wavefield calculations in the time domain. One relaxation function describes the anelastic properties of the quasi-dilatational mode and the other three model the anelastic properties of the shear modes. The convolutional relations are avoided by introducing memory variables, six for each dissipation mechanism in the 3-D case, two for the generalized SH-wave equation, and three for the qP qSVwave equation. Two-dimensional wave equations apply to monoclinic and higher symmetries. A plane analysis derives expressions for the phase velocity, slowness, attenuation factor, quality factor and energy velocity (wavefront) for homogeneous viscoelastic waves. The analysis shows that the directional properties of the attenuation strongly depend on the values of the elasticities. In addition, the displacement formulation of the 3-D wave equation is solved in the time domain by a spectral technique based on the Fourier method. The examples show simulations in a transversely-isotropic clayshale and phenolic (orthorhombic symmetry). Modeling waves in a realistic medium involves several INTRODUCTION aspects. In the first place, the wave equation should be able to properly simulate the body waves and the complete set of waves produced by different types of interface, like for instance, Rayleigh waves at the free surface, Stoneley waves at the solid-solid interface, head waves, and guided waves. These interfaces could have irregular shapes, i.e., the input model should allow arbitrary variations of the physical properties. But these conditions alone are not enough to describe a realistic geological structure, for petrophysical and lithological properties play an important role, particularly in the targets of exploration geophysics, i.e., reservoir