Transient solution for viscoacoustic wave propagation in a double porosity medium, and its limitations

Abstract

It is well known that Biot theory (Biot, 1956;1962) of porous-media acoustics ignores all wave-induced flow at mesoscopic scales, i.e. scales greater than the grain size but less that the wavelength. Biot?s theory can not explain high level attenuation observed in natural porous media such as fluid-filled sands or sandstone over the seismic frequency range ( 10- 200 Hz). This attenuation is successfully described by the mesoscopic heterogeneity models (e.g., Gurevich and Lopatnikov, 1995; Gelinksy and Shapiro, 1997; Johnson, 2001). By applying the volume averaging theory to the local Biot poroelastic law, Pride an Berryman (2003a,b) developed the double-porosity, dual permeability (DPDP) model It gives a theoretical framework, including the field equations governing the linear acoustics of composites with two isotropic porous constituents (phase 1 and phase 2), to model acoustic wave propagation through heterogeneous porous structures. Under the assumption that phase 2 is entirely embedded in phase 1, the double-porosity theory is reduced to the effective Biot theory with the complex frequency-dependent elastic moduli in which the internal mesoscopic flow is incorporated. This theory provides good agreement with measurements of attenuation over the seismic and ultrasonic frequency bands (Pride et al., 2004). In this paper, the analytical transient solution and dispersion characteristics for the double-porosity model are obtained over the whole frequency range for a homogeneous medium. A homogeneous poro-viscoacoustic model is constructed and analytically solved to approximate the double porosity model. The comparison between the results of the two models shows the likely validity and limitations of numerical solutions using a poro-viscoacoustic model to represent a double porosity medium in the heterogeneous case. Our calculations show that the dissipation by local mesoscopic flow of the double porosity model is very hard to fit over the entire frequency range by a single Zener element. We choose the relaxation function which just approximates the dispersion behaviour of the double porosity model around the source centre frequency. It is shown that if the frequency is much lower than the peak attenuation frequency of the double porosity model, then wave propagation can be well described by the poro-viscoacoustic model with a single Zener element. For most water-filled sandstones having a double porosity structure, this holds true across the seismic frequency range.