Phase Velocity Dispersion and Attenuation of Seismic Waves due to Trapped Fluids in Residual Saturated Porous Media

Propagation of seismic waves in partially saturated porous media depends on various material properties, including saturation, porosity, elastic properties of the skeleton, viscous properties of the pore fluids, and, additionally, capillary pressure and effective permeability. If the wetting fluid is in a discontinuous state (i.e., residual saturated configuration), phase velocities and frequency-dependent attenuation additionally depend on microscopical (pore-scale) properties such as droplet and/or ganglia size. To model wave propagation in residual saturated porous media, we developed a three-phase model based on an enriched continuum mixture theory capturing the strong coupling between the micro- and the macroscale. The three-phase model considers a continuous and a discontinuous part. The continuous part exhibits similar behavior as the poroelastic model introduced by Biot. The discontinuous part describes the movement of blobs/clusters of the wetting fluid and is based on an oscillator rheology. In comparison with other three-phase models, the presented one accounts for the heterogeneity of the discontinuous fluid clusters by use of their dynamic properties, i.e., their statistically distributed inertia, eigenfrequency, and damping effects. This heterogeneous and discontinuous distribution of the wetting fluid in the form of single blobs or fluid clusters is represented by a model-embedded distribution function of the cluster sizes. We define a dimensionless parameter that determines if the overall motion of the residual fluid is dominated by oscillations (underdamped, resonance) or not (overdamped). Our results show that the residual fluid has a significant impact on the velocity dispersion and attenuation no matter if it oscillates or not. For long wavelengths our model coincides with the Biot–Gassmann equations. We show under which conditions and how the classical biphasic models can be used to approximate the dynamic behavior of residual saturated porous media.