Linking the viscous grain-shearing mechanism of wave propagation in marine sediments to fractional calculus

This work is an extension of Pandey and Holm [Abstract, JASA (2015)] and builds on Buckingham's viscous grain-shearing (VGS) model [JASA (2007)]. The VGS model is an extension to the grain-shearing mechanism initially proposed to explain wave propagation in marine sediments. We find that the material impulse response function obtained from the VGS model is characterized by power-law terms which are inherent to the framework of fractional calculus. The VGS model in the fractional framework yields two equations; a fractional-order wave equation for the compressional wave and a modified fractional-order diffusion-wave equation for the shear wave. The order of the fractional equations as well as the two characteristic time constants are identified as the physical parameters of the medium. The viscous dissipation incurred due to inter-granular sliding across the pore-fluid modifies the low frequency behavior of the dispersion plots significantly. The low frequency behavior (<1 Hz) for both compressional and shear wave show normal dispersion of decreasing phase velocity with increasing frequency. These features were not observed in the original VGS model. The overall goal is to establish a physically based fractional framework, which can effectively incorporate the role of different material parameters in wave propagation.