Numerical upscaling in 2‐D heterogeneous poroelastic rocks: Anisotropic attenuation and dispersion of seismic waves

The presence of stiffness contrasts at scales larger than the typical pore sizes but smaller than the predominant seismic wavelengths, can produce seismic attenuation and velocity dispersion in fluid-saturated porous rocks. This energy dissipation mechanism is caused by wave-induced fluid-pressure diffusion among the different components of the probed geological formations. In many cases, heterogeneities have elongated shapes and preferential orientations, which implies that the overall response of the medium is anisotropic. In this work, we propose a numerical upscaling procedure that permits to quantify seismic attenuation and phase velocity considering fluid-pressure diffusion effects as well as generic anisotropy at the sample's scale. The methodology is based on a set of three relaxation tests performed on a 2D synthetic rock sample representative of the medium of interest. It provides a complex-valued frequency-dependent equivalent stiffness matrix through a least-squares procedure. We also derive an approach for computing various poroelastic fields associated with the considered sample in response to the propagation of a seismic wave with arbitrary incidence angle. Using this approach, we provide an energy-based estimation of seismic attenuation. A comprehensive numerical analysis indicates that the methodology is suitable for handling complex media and different levels of overall anisotropy. Comparisons with the energy-based estimations demonstrate that the dynamic-equivalent viscoelastic medium assumption made by the numerical upscaling procedure is reasonable even in presence of high levels of overall anisotropy. This work also highlights the usefulness of poroelastic fields for the physical interpretation of seismic wave phenomena in strongly heterogeneous and complex media.