Comparison of numerical calculations of two Biot coefficients with analytical solutions

Abstract

The Biot theory for the acoustics of porous media contains drag and virtual mass coefficients that depend on the physical properties of the fluid and solid constituents, the frequency, and the microstructure of the porous medium. Biot derived an equation for the drag coefficient as a function of frequency by assuming cylindrical pores. In this paper, the finite element method is used to obtain values of the drag and virtual mass coefficients for face-centered cubic granular materials with three different porosities and compare our numerical results to Biot’s analytical solutions. By making appropriate choices of three parameters in Biot’s analytical solution for cylindrical pores—the pore size parameter, the Kozeny parameter, and the tortuosity—the analytical solution matches these numerical results very well. This suggests that with appropriate choices of these parameters the analytical approach can predict the dependence of the drag and virtual mass coefficients on frequency for an arbitrary pore geometry. These results support a relation suggested by Hovem and Ingram for approximating the pore-size parameter for spherical grains, and they also agree with a relation suggested by Berryman for the dependence of the tortuosity on the porosity.

The accuracy of some models to predict the acoustical properties of granular media

“…b = 1/2 for spherical grain packing and 0<b<1 for other shapes. Alternative values for b were introduced for spherical or uniform cylindrical shapes by [25] and [26], respectively. The above equation suggests that the tortuosity for this type of material is basically independent of the effective pore size, but depends on the 90 way particles are arranged in the stack.…”

Study on the Biot-Stoll model for porous marine sediments

“…where r ¼ 0:5 for isolated spherical particles and lies between 0 and 1 for other ellipsoidal shapes [23,29]. According to the Berryman equation, the structure factor depends on the porosity and the shape of the grains.…”

Determination of the Parameters of Cancellous Bone Using High Frequency Acoustic Measurements Ii: Inverse Problems

“…Figure 2 shows that if pore size is indeed a linear function of porosity as indicated by Fig. 1, then permeability, as predicted by (13), will deviate significantly from log-linearity. We took the uncertainty in this parameter to be ±10 0.75 so that the Kozeny-Carmen values would be encompassed as well as the regression values.…”

“…A value s = 0.227 was suggested by the finite element analysis of Yavari and Bedford. 13 We will calculate α assuming 0.2 ≤ s ≤ 0.3.…”