Tortuosity and the Averaging of Microvelocity Fields in Poroelasticity

Abstract

The relationship between the macro-and microvelocity fields in a poroelastic representative volume element (RVE) has not being fully investigated. This relationship is considered to be a function of the tortuosity: a quantitative measure of the effect of the deviation of the pore fluid streamlines from straight (not tortuous) paths in fluid-saturated porous media. There are different expressions for tortuosity based on the deviation from straight pores, harmonic wave excitation, or from a kinetic energy loss analysis. The objective of the work presented is to determine the best expression for tortuosity of a multiply interconnected open pore architecture in an anisotropic porous media. The procedures for averaging the pore microvelocity over the RVE of poroelastic media by Coussy and by Biot were reviewed as part of this study, and the significant connection between these two procedures was established. Success was achieved in identifying the Coussy kinetic energy loss in the pore fluid approach as the most attractive expression for the tortuosity of porous media based on pore fluid viscosity, porosity, and the pore architecture. The fabric tensor, a 3D measure of the architecture of pore structure, was introduced in the expression of the tortuosity tensor for anisotropic porous media. Practical considerations for the measurement of the key parameters in the models of Coussy and Biot are discussed. In this study, we used cancellous bone as an example of interconnected pores and as a motivator for this study, but the results achieved are much more general and have a far broader application than just to cancellous bone.

The In Situ Mechanics of Trabecular Bone Marrow: The Potential for Mechanobiological Response

“…The results support the use of poroelasticity [54] or mixture [20] theory to identify pressure gradients in bone marrow, as the very low velocity gradients are consistent with the underlying Darcy flow assumption [20,54], However, the permeability of tra becular bone is anisotropic [57], and the relationship between the shear stress and the permeability is complicated by the tortuosity that also gives rise to fluid resistance [58], As such, while poroe lasticity theory would likely provide reasonable results for the macroscopic level, interpretation of the shear stress field within the marrow may require microscale models as used here. Fig.…”

Difficulties arising from different definitions of tortuosity for wave propagation in saturated poroelastic media models

“…For the time being we prefer the Coussy [14] approach to both these questions, answers we view as conservative. Coussy's [14] approach to the definition of tortuosity was described in the previous section where its connection to the Biot [8] consideration of averaging local velocity fields is developed by Souzanchi et al [36]. We can find no suggestions from Coussy on how permeability and tortuosity are related, which leaves the issue open and a question for research.…”

“…Coussy [14] does not consider a relationship between tortuosity and permeability and, in his analysis, treats them as independent. The connection between the tortuosity approach of Coussy [14] to the consideration of Biot [8] of averaging local velocity fields is developed by Souzanchi et al [36].…”

On reconstruction of dynamic permeability and tortuosity from data at distinct frequencies

“…For a rigid porous medium filled with Newtonian pore fluid with density ρ f and dynamic viscosity η, a key effective property is the fluid permeability tensor K, which is described by the so-called Darcy's law, [6,43,7,62] …”