Correct description of S-wave propagation across material discontinuities in poroelasticity

Abstract

The shear motion in Newtonian fluids, i.e., fluid vorticity, represents an intrinsic loss mechanism. The vorticity is governed by a diffusion equation, and its description involves the trace-free part of the fluid viscous stress tensor. This part is, however, missing in the widely accepted Biot theory of poroelasticity. As a result, the fluid vorticity is not captured, and only one shear wave is predicted in the Biot theory. The missing fluid vorticity has implications for the propagation of shear waves across discontinuities, and we show that it results in unphysical predictions. This becomes most apparent in the simple problem of shear wave propagation across the welded contact of an elastic solid with a porous medium, as presented in this work. The no-slip condition between the elastic solid and the constituent parts of the porous medium, the solid-frame, and the pore-fluid, must hold at the boundary. This requirement translates into a vanishing relative motion of the fluid with respect to the solid-frame, i.e., filtration field, at the contact. Nevertheless, our analysis shows that for the Biot theory, in the low-frequency regime, a non-zero, although insignificantly small, filtration field exists at the contact. More importantly, the filtration field is noticeable when the transition to the high-frequency regime occurs. This constitutes a disagreement with the requirement of a no-slip boundary condition. This unphysical prediction can be circumvented by including the fluid viscous stress tensor into the poroelastic constitutive relations, as is suggested in the de la Cruz and Spanos poroelasticity theory. A second shear wave is then predicted, which manifests the fluid vorticity at the macro-scale. This process is distinct from the fast shear wave, the other predicted shear wave, akin to the Biot shear wave. We find that the generation of this process at the contact induces a filtration field equal and opposite to that associated with the fast shear wave. Therefore, the no-slip condition is fully satisfied, and the shear wave reflection/transmission across a material discontinuity becomes physically meaningful.