Propaga on of seismic waves in par ally saturated porous media depends on various material proper es, including satura on, porosity, elas c proper es of the skeleton, viscous proper es of the pore fl uids, and, addi onally, capillary pressure and eff ec ve permeability. If the we ng fl uid is in a discon nuous state (i.e., residual saturated confi gura on), phase veloci es and frequency-dependent a enua on addi onally depend on microscopical (pore-scale) proper es such as droplet and/or ganglia size. To model wave propaga on in residual saturated porous media, we developed a three-phase model based on an enriched con nuum mixture theory capturing the strong coupling between the micro-and the macroscale. The three-phase model considers a con nuous and a discon nuous part. The con nuous part exhibits similar behavior as the poroelas c model introduced by Biot. The discon nuous part describes the movement of blobs/clusters of the we ng fl uid and is based on an oscillator rheology. In comparison with other three-phase models, the presented one accounts for the heterogeneity of the discon nuous fl uid clusters by use of their dynamic proper es, i.e., their sta s cally distributed iner a, eigenfrequency, and damping eff ects. This heterogeneous and discon nuous distribu on of the we ng fl uid in the form of single blobs or fl uid clusters is represented by a model-embedded distribu on func on of the cluster sizes. We defi ne a dimensionless parameter that determines if the overall mo on of the residual fl uid is dominated by oscilla ons (underdamped, resonance) or not (overdamped). Our results show that the residual fl uid has a signifi cant impact on the velocity dispersion and a enua on no ma er if it oscillates or not. For long wavelengths our model coincides with the Biot-Gassmann equa ons. We show under which condi ons and how the classical biphasic models can be used to approximate the dynamic behavior of residual saturated porous media.Abbrevia ons: PDF, probability density func on; REV, representa ve elementary volume.To understand and characterize the dynamical behavior of partially saturated porous media, such as soils, rocks, or organic materials, the partial properties of the solid skeleton, of the inherent pore fl uids and, in addition, the main physical coupling eff ects between these interacting constituents have to be taken into account. Since the seminal work of Biot (1956a,b) and Frenkel (1944) in the middle of the last century, many theoretical and numerical studies about wave propagation phenomena in fully saturated porous media have been published (e.g., Bourbié et al., 1987;Stoll, 1989;Carcione, 2007, and references therein). To describe the macroscopic behavior of seismic waves propagating through partially saturated media (i.e., porous media saturated with a wetting and a nonwetting pore fl uid), the biphasic Biot-type approach was extended to take into account quasistatic (Santos et al., 1990;Tuncay and Corapcioglu, 1996;Wei and Muraleetharan, 2002;Carcione et al., 2004;Lo and Sposito, 2005;...

Biot’s theory of wave propagation in porous media includes a characteristic frequency which is used to distinguish the low-frequency from the high-frequency range. Its determination is based on an investigation of fluid flow through different pore geometries on a smaller scale and a subsequent upscaling process. This idea is limited due to the assumptions made on the smaller scale. It can be enhanced for a general two-phase system by three properties: Inertia of the solid, elasticity of the solid, and frequency dependent corrections of the momentum exchange. They become important for highly porous media with liquids.

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The present work develops the theoretical framework to describe oscillations of fluid clusters. The basic physical phenomena are presented and justified assumptions lead to the final set of equations for different types of oscillations (pinned/sliding). The special combination of a liquid cluster surrounded by a rigid solid matrix and a gas is investigated in more detail. Furthermore, a classification of oscillating fluid clusters is presented using a one-dimensional oscillator model. This classification includes three dynamic properties: mass, eigenfrequency, and damping whereas conceptual implementation and limitations for use in multiphase theories are clearly indicated. The frequency dependent flow profile leads to frequency dependence of the dynamic parameters. This is discussed and represented by dimensionless numbers.

A numerical study of pinned, oscillating water clusters is presented. Two main models represent a liquid bridge between the walls of two particles and a water column enclosed in a slender pore channel, respectively. Variations include material properties (density, viscosity, surface tension, contact angle) and geometric properties (volume, slenderness, winding, interfacial areas). They are initially based on water clusters in 1 mm pore-space, which are weakly damped at eigenfrequencies around a few hundred Hz. Stiffness and damping are characterized by eigenfrequency and damping coefficient of an equivalent 1-dim. harmonic-oscillator model. Finally, frequency dependence of the dynamical properties is demonstrated. The comprehensive quantitative analysis extends and explains relationships between geometric and material properties and the response to harmonic stimulation. Furthermore, interpolation functions of characteristic dynamic properties are provided for use in multiphase theories. The frequency dependence of cluster stiffness and damping was proven and of limited influence on the stimulation of two typical, weakly damped liquid clusters.