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

In this paper, two approaches - the host phase fields and the total fields, were respectively applied to formulate the effective Biot governing equations from the original double-porosity dual-permeability (DPDP) model. The host-phase formulation given previously in the literature is made under the assumption that the macroscopic fluid flux of the included phase is zero, so that this term can be ignored in the conservation of the momentum equation. The total-field formulation developed here has no such limiting assumption and gives rise to new and general governing equations that cover the host-field approach as a special case. By computing the phase velocity and attenuation dispersion curves of sample rocks, we show that the two sets of governing equations are consistent at a very low frequency but for larger volume fractions of phase 2, there is a significantly increasing discrepancy in the slow P-wave as the frequency increases. The slow P-wave, whilst difficult to observe, does exist and must be considered when computing the frequency-dependent reflection coefficients at an interface with a porous medium.