Dispersive behavior and acoustic scaling in granular rocks

Handling and making decisions based on data taken at different scales is a critical issue in the design of exploration and production tasks in the oil industry. Acoustic data is the classical example of the integration of dissimilar scales (i.e. seismic, well logs, lab data) where there is a scale dependent velocity. An understanding of the acoustic dispersion phenomenon in granular samples is needed. A detailed numerical work was conducted in order to establish the relationship between frequency and propagation speed for an acoustical pulse induced in simulated granular materials. The granular samples were generated with different grain size distributions while porosity and pressure were targeted and kept invariant using the grain radii expansion method. A sinusoidal burst with frequencies from 10Hz to 1MHz was applied and the corresponding acoustical speeds were estimated for each frequency. A coherent sigmoid dispersion relationship was obtained for each granular sample. The asymptotic boundaries for the dispersion function reflect the limiting cases for the wavelength/heterogeneity ratio in the granular pack. The lower speed asymptote was explained as the mean field value while upper speed asymptote can be understood based on a ray theory approximation scaled by a parameter we defined as the "acoustic tortuosity factor". This factor reflects the intricate acoustical path due to the texture of the stress network developed in the granular samples and can be used together with the sigmoid dispersive relationship to describe and clarify the scale discrepancy between different source acoustic data in granular materials.