Extension of Biot’s theory of wave propagation to frozen porous media

An extension of Biot’s theory is proposed for frozen porous media where the solid substrate, ice particles, and unfrozen water can coexist. Elastic, kinetic, and dissipation energy densities are written using the results of continuum mechanics, then the equations of propagation are deduced with the help of Lagrange’s equations and Hamilton’s least-action principle. The ice parameters are introduced in the model in addition to those used in Biot’s theory. It appears that only the percolation theory is able to describe the transition of the ice matrix between the continuous state and the discontinuous state during a freezing or a thawing process. The resolution of the equations of propagation lead to the existence of three longitudinal and two transverse modes. Their velocities and attenuations are calculated as functions of the physical parameters of the medium. Independently, a thermodynamical argument is developed which allows the mechanical properties to be related to temperature. Experimental results are briefly presented to confirm the theoretical predictions.