About the transition frequency in Biot’s theory

Abstract

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.

Ultrasonic properties in marrow-filled and water-filled bovine femoral trabecular bones invitro

“…Table 3 shows the input parameters of the Biot model for marrow-filled and water-filled trabecular bones. 16,17 As seen from Fig. 1, reasonable agreements can be found between the measurements and the predictions of the Biot model.…”

Frequency‐dependent properties of multiphase materials: origins, needs, and effective use in geophysics and modern synthetic materials.

“…At high frequencies, inertia terms dominate at the central axis and result in a decoupled zone with a flat profile and different interaction stress at the solid wall [5]. Referring to two main contributors, this effect is described by the Womersley number or, alternatively, by Biot's characteristic frequency [1,4,10]. It relates viscous forces to inertia forces that increase with frequency, cf.…”

“…Information on the flow profile vanishes by transformation to a larger scale for multiphase systems. Typical approaches such as Mixture Theory or homogenization techniques do not consider such smaller-scale flow phenomena [4,8]. Instead, the relationship between averaged flow and the viscous interaction between solid and fluid phases need to be extended by a frequency-dependent correction term.…”

“…These extensions can be expressed by a new or modified characteristic frequency, cf. [4] for detailed analysis and formulas. Moreover, fluid-fluid interfaces play an extra role in the case of partially and residually saturated porous media [9].…”

“…For example, changing flow profiles can lead to modified attenuation regimes and multiple oscillations modes yield distinct stimulation regimes of enclosed fluid clusters [6,9]. Moreover, wrong assumptions can lead to errors of several hundred percent when estimating characteristic frequencies, for instance, in the case of osteoporotic bone or high-porosity foams [4]. However, frequency dependence may be negligible if the dynamic processes of interest do not cover a wide frequency range.…”