Shear and Compressional Waves

Shear waves were not studied explicitly by Berryman and Wang (2000), but Eq. (5) of that paper can be used for that purpose simply by applying the curl operator to all three of the equations in the set. When this is done, the result is that the first equation describes the actual shear mode, while the other two equations provide constraints on the relative motion of the pore fluid in each type of porosity versus the displacement of the solid frame. In particular, the shear components of the differences in fluid and solid displacements can be uniquely related by complex factors (that are known explicitly) to the displacement of the solid alone. Furthermore, as in the case for single-porosity poroelasticity, all of the interesting behavior of the shear mode -- at least for isotropic media -- comes from the inertial terms. The form of the resulting dispersion relation at low frequencies is identical to (20) with the replacement

$$\begin{eqnarray} \kappa \to \kappa_1 + \kappa_2 \simeq \kappa_1, \end{eqnarray}$$ (27)

since we assume here that $\kappa_1 \gt\gt \kappa_2$. A similar result follows for the compressional wave. Thus, as for single-porosity, the attenuation of the shear and compressional waves is dominated by the largest permeability present in the system. However, this leads to no contradiction in the double-porosity formulation. Thus, the problem inherent in up-scaling with single-porosity poroelasticity is resolved in an intellectually satisfying way in the double-porosity approach.