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

Abstract

This article focuses on the mathematical problem of reconstructing the dynamic permeability and dynamic tortuosity of poroelastic composites from permeability data at different frequencies, utilizing the analytic structure of the Stieltjes function representation of derived by Avellaneda and Tortquato (1991 Phys. Fluids A 3 2529), which is valid for all pore space geometry. The integral representation formula (IRF) for dynamic tortuosity is derived and its analytic structure exploited for reconstructing the function from a finite data set. All information of pore-space microstructure is contained in the measure of the IRF. The theory of multipoint Padé approximates for Stieltjes functions guarantees the existence of relaxation kernels that can approximate the dynamic permeability function and the dynamic tortuosity function with high accuracy. In this paper, a numerical algorithm is proposed for computing the relaxation time and the corresponding strength for each element in the relaxation kernels. In the frequency domain, this approximation can be regarded as approximating the Stieltjes function by rational functions with simple poles and positive residues. The main difference between this approach and the curve fitting approach is that the relaxation times and the strengths are computed from the partial fraction decomposition of the multipoint Padé approximates, which is the main subject of the proposed approximation scheme. With the idea from dehomogenization, we also established the exact relations between the moments of the positive measures in the IRFs of permeability and tortuosity with two important parameters in the theory of poroelasticity: the infinite-frequency tortuosity for the general case and the weighted volume-to-surface ratio Λ for the JKD model, which is regarded as a special case of the general model. From these relations, we suggest a new way for evaluating these two microstructure-dependent parameters from a finite data set of permeability at different frequencies, without assuming any specific forms of the functions except the fact that they satisfies the IRFs. Numerical results for JKD permeability and tortuosity are presented.