Transients in porous media: asymptotic time-domain Green functions and limits of current frequency-domain models

TL;DR

This paper compares time-domain Green functions of porous media using different models, highlighting the properties of the response and the accuracy of asymptotic and exact formulas, with implications for understanding material behavior.

Contribution

It introduces an asymptotic high-frequency expansion and compares it with exact formulas, clarifying the limits of current frequency-domain models for porous media.

Findings

Green function maximum depends on a single dimensionless parameter

PL model provides the best full frequency description but has discrepancies at short times

Asymptotic expansion shows the Green function decreases as 1/ξ^2 then exponentially

Abstract

Time domain responses of porous media have been studied by some authors, but generally the possible descriptions have been given in the frequency domain. The aim of this paper, limited to materials with rigid skeleton considered as equivalent fluids, is to compare the descriptions by Johnson-Allard ($JA$%) as well as by Pride-Lafarge ($PL$) with i) some analytical, approximate formulas, based upon asymptotic high frequency expansion ; ii) the exact formula by Zwikker and Kosten for the case of cylindrical pores. The paper starts with a short summary of the statement of the different general full frequency models ($JA$ and $PL).$ The Green function in the time domain is shown to exhibit interesting properties of materials. In particular the maximum response depends on one dimensionless parameter only, which is denoted $\xi $ and is the ratio of the travelled distance to the product of the \textquotedblleft frozen\textquotedblright\ sound speed and a characteristic viscous relaxation time. The distance $\xi $ is related to a time domain Stokes number. The numerical computation of the Green function is done by FFT, with some precautions, because of the importance of the higher frequencies on the response shape. The $PL$ description is shown to be the best full frequency general model, but some discrepancies with the exact model appear at short times or short distances. When the distance $\xi $ increases from zero, the asymptotic expansion shows that the maximum of the Green function decreases first as $1/\xi ^{2}$, then exponentially.