Wave equations for porous media described by the Biot model.

Single-mode equivalent space-time representations of the acoustic wave propagating in a Biot poroelastic medium have previously been found only for asymptotic cases: In the low frequency regime, where the viscous skin depth is greater than the characteristic pore size, the time domain equivalent is represented with integer order temporal and spatial loss terms, whereas in the high frequency regime, it is represented with fractional order temporal and spatial loss terms. In the current work, a time domain representation in terms of a partial differential equation is proposed for all three wave solutions of the Biot model across all frequencies, and it is derived from the material response function of the Biot poroelastic medium with suitable approximations for the compressional modes and the dynamic permeability. The dynamic permeability in the time domain is represented by a fractional pseudo-differential operator.

A multiple relaxation interpretation of the extended Biot model.

The biphasic extended Biot poroviscoelastic model takes into account the squirt flow in grain-grain contacts and introduces the bulk and shear relaxation modes associated with it. This model has been criticized for its empirical approach, but here the constitutive equations and the time domain wave equations of the model are derived. This also makes it possible to find single phase viscoelastic equivalents for all three wave solutions of the extended Biot model.