Analysis of the numerical dispersion of waves in saturated poroelastic media

This work presents the analysis of the numerical dispersion properties of a finite element method used to approximate the solution of the equations of motion for a fluid saturated porous elastic solid (a Biot medium) in the two-dimensional case and the low frequency regime. The finite element method employed comprises a nonconforming rectangular element for the approximation of each component of the displacement vector in the solid phase, and the Raviart–Thomas–Nedelec mixed finite element space of zero order for the fluid phase.The study is carried out by constructing and analyzing analytic and numerical dimensionless dispersion relations, and by evaluating derived quantities such as dimensionless phase and group velocities and dimensionless attenuation for the three type of waves predicted by Biot’s equations of motion as a measure of the numerical distortion. It is observed that the finite element procedure introduces both numerical dispersion and anisotropy in all three waves in a similar fashion; however, only the slow wave displayed a significant frequency dependent behavior. It was also observed that the loss of accuracy is more important for the dimensionless attenuation than for the dimensionless group or phase velocities. The analysis presented yields lower bounds for the number of points per wavelength needed to reach a desired accuracy in the dimensionless phase and group velocities and attenuation coefficients in Biot media.