Velocity and attenuation dispersion relations for the effective Biot double porosity model: Total field formulation

Two approaches - the host phase fields and the total fields - have been applied to formulate the effective Biot governing equations from the double-porosity, dual permeability model. The host phase formulation, given previously by Pride and Berryman (2003) is made under the assumption that the volume fraction of phase 2 (the inclusions) is very small. The total-field formulation is new and has no such limitation and gives rise to more general equations which cover the host phase field approach as a special case. We have computed the phase velocity and attenuation dispersion curves for the fast and slow Biot compressional waves using both approaches for a sample rock of water-saturated sandstone having sand as the inclusions. The curves are consistent across a wide frequency range for the fast P wave but deviate at higher frequencies and for large volume fractions of the inclusions .For the slow P wave there is a significantly increasing discrepancy in wavespeed as frequency increases, even for low inclusion concentrations.