Influence of coupling through porosity changes on the propagation of acoustic waves in linear poroelastic materials

Abstract

In earlier works it has been shown that linearization of thermodynamical models of poroelastic materials yields a contribution to stresses in the form βΔ_n, Δ_n:=n-n_E, where n is the current porosity, n_E denotes its value in the thermodynamical equilibrium and β is a material parameter. It has been also claimed on the basis of rough estimates that this parameter gives only negligible contributions to the properties of acoustic waves. The purpose of this work is twofold. We investigate the influence of β on the propagation of acoustic waves in more details. We use the full linear model of saturated poroelastic materials in which an additional coupling to porosity changes appears. This results from the presence of a term with the porosity gradient in the momentum sources. Such a model becomes identical with Biot's model without any added mass contribution in the limit if infinite relaxation time of porosity τ→∞ and for the parameters β=0, and N=0, where the latter is describing the above mentioned influence of porosity gradient on the momentum sources. The analysis of influence of the parameter N, i.e. the influence of the porosity gradient on properties of acoustic waves, is the second purpose of this work. We also indicate a correction of the permeability contribution to the momentum source. The permeability coefficient π is assumed in Biot's model to be dependent on the frequency. This is inconsistent with other temporal contributions to field equations. In order to obtain such a dependence in the Fourier space, one has to assume a viscous effect to enter the momentum source and we do so in the general equations. However, we do not use this general relation in the analysis of monochromatic waves.