Patrick Kurzeja

About the transition frequency in Biot’s theory

Biots 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.

Variational formulation of oscillating fluid clusters and oscillator-like classification. I. Theory

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.

Variational formulation of oscillating fluid clusters and oscillator-like classification. II. Numerical study of pinned liquid clusters

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 depe...

The Critical Frequency in Biphasic Media: Beyond Biots Approach

Biots theory for wave propagation in biphasic media is still of great influence on current research and its applications in engineering and geosciences. In particular, the characteristic frequency introduced by Biot is used to distinguish a viscosity-dominated low frequency range from an inertia-dominated high frequency range. An understanding of the transition between these ranges of contrasting dominance of mechanisms is of vital importance for the interpretation of experiments and the basis of new theories. Biot derived the characteristic frequency on the microscale and questions remain regarding its correct transformation to the macroscale. Specifically, three aspects are neglected due to simplifying assumptions on the microscale: inertia of the solid, elasticity of the solid, and a frequency-dependent momentum interaction. Corrections accounting for these aspects are particularly significant for systems with a weak solid skeleton and a rather incompressible fluid. Neglection of these corrections appears however often justified for typical systems of rocks or soils. Experiments were conducted in which waves propagate through elastic tubes of different materials (steel, silicone) filled with various fluids (air, water, Na-Polywolframat). These experiments are supposed to represent the micro-scale mechanisms in single pores. The mismatch between experimental records and theoretical predictions suggests that some of the assumptions made in current theoretical treatments of the problem require further consideration before a reliable upscaling can be undertaken.

Oscillatory fluid flow in deformable tubes: Implications for pore-scale hydromechanics from comparing experimental observations with theoretical predictions

Oscillatory flow of four fluids (air, water, two aqueous sodium-tungstate solutions) was excited at frequencies up to 250 Hz in tubes of two materials (steel, silicone) covering a wide range in length, diameter, and thickness. The hydrodynamical response was characterized by phase shift and amplitude ratio between pressures in an upstream (pressure excitation) and a downstream reservoir connected by the tubes. The resulting standing flow waves reflect viscosity-controlled diffusive behavior and inertia-controlled wave behavior for oscillation frequencies relatively low and high compared to Biots critical frequency, respectively. Rigid-tube theories correspond well with the experimental results for steel tubes filled with air or water. The wave modes observed for silicone tubes filled with the rather incompressible liquids or air, however, require accounting for the solids shear and bulk modulus to correctly predict speed of pressure propagation and deformation mode. The shear mode may be responsible for significant macroscopic attenuation in porous materials with effective frame-shear moduli lower than the bulk modulus of the pore fluid. Despite notable effects of the ratio of densities and of acoustic and shear velocity of fluid and solid, Biots frequency remains an approximate indicator of the transition from the viscosity to the inertia controlled regime.

Wave dispersion in highly deformable, fluid-filled structures: Numerical and experimental study of the role of solid deformation and inertia

The application and scientific interpretation of wave measurements in fluid-filled structures strongly depend on the frequency regime of interest. This includes, for example, absorption bands, inverse calculation of elastic moduli, and non-destructive crack localization. The wave properties significantly differ between the low-frequency regime (where viscous forces couple fluid and solid) and the high-frequency regime (where inertia forces allow for multiple decoupled wave modes with individual speeds and attenuation). Thus, knowledge of the separative transition frequency is crucial for a reliable prediction, but respective approximations like Biot’s characteristic frequency still originate from stiff structures. Soft materials such as biological tissues or synthetic materials are neglected regarding their high deformability. Hence, this presentation demonstrates the change of wave properties from low to high frequencies in soft, fluid-filled structures and highlights the influence of solid deformability...

Wave propagation in residual saturated porous rocks - A multiscale approach

Phase velocities and dispersion of acoustic waves in residual saturated porous rocks are affected by material properties of the solid skeleton (porosity, skeleton and grain stiffness) and properties of the inherent pore fluids (viscosities, bulk moduli) as well as the morphology of saturation. A multiscale continuum model describing acoustic waves in porous rocks saturated with a continuous gas phase and a discontinuous liquid (water or oil) phase is analysed. Dispersion relations for the P-waves are shown and compared with Biot’s poroelastic approach. It is observed, that negative partial densities occur close to the resonance frequency of the liquid phase. The analogy with acoustic metamaterials is discussed.