Fractional viscoelastic modelling of wave propagation in fluid-saturated marine sediments

Abstract

It is shown that Buckingham's grain-shearing (GS) model [JASA (2000)] as well as its improved viscous-GS (VGS) model [JASA (2007)] can be expressed using the mathematical framework of fractional calculus. The fractional version of the standard fluid model is adopted to independently arrive at the wave equations and dispersion relations derived by Buckingham. The fractional-order wave equations obtained for the compressional waves and shear waves are relatively easier to analyse due to their closed-form representation in the fractional framework. It is also shown that the fractional calculus approach may help in bridging the disparate fields of non-Newtonian rheology and sediment acoustics, which may have actually developed independently of each other. Further, the experimental data relating wave dispersion and attenuation in marine sediments is found to match with the predictions from the fractional framework. The overall goal is to show that fractional calculus is not just a mathematical framework that can only be applied to curve-fit the observational data for complex media. In fact, it has an inherent connection to real physical processes that needs to be explored more.