Seismic attenuation and velocity dispersion due to squirt flow in cracks with rough walls

We numerically study the effects that roughness in the walls of cracks has on the P-wave modulus dispersion and attenuation due to squirt flow. We emulate the deformation caused by a seismic P-wave by applying an oscillatory relaxation test on numerical rock models having two perpendicular fluid-filled cracks interconnected and embedded in a cubic elastic background. The deformation caused by the P-wave induces a fluid pressure gradient and then, during the consequent fluid pressure diffusion process, the friction between fluid particles dissipate seismic wave energy. In this work, we consider P-wave deformation normal to one of the cracks. We first consider binary aperture distribution for the cracks to analyse where the energy dissipation process takes place. Then, more complex geometries for the roughness of the walls are also considered. In both cases, the cracks have finite length and square-shape and no contact areas between the walls of the cracks were allowed to occur. We show that the arithmetic mean of the apertures controls the P-wave modulus magnitudes at the low- and high-frequency limits. Additionally, two attenuation peaks and modulus dispersion regimes may occur associated with squirt flow. In general, at low-frequencies, the energy dissipation tends to happen inside the minimum aperture of the cracks, and consequently, the minimum aperture determines the frequency at which the low-frequency attenuation peak occurs. For the considered models, we observed that when the percentage of minimum aperture in the cracks is lower than 10$\%$, a second attenuation peak at high frequencies become dominant. The characteristic frequency of this attenuation process is controlled by an effective hydraulic aperture. Finally, we simulate an increase in confining pressure by reducing the crack apertures by a constant value, allowing for contact areas occurrence. In this scenario, the stiffness of the cracks can not longer be explained with the arithmetic mean of the aperture, as the stiffening effect of the distribution of the contact areas plays a much stronger role. In general, from the analysis of the local energy dissipation, different apertures seem to control the energy dissipation process at each frequency, which means that a frequency-dependent hydraulic aperture might be needed to describe the squirt flow process in cracks with rough walls.