The physical realizability issue is an important one, but not one we have dwelt much on in this paper. Because of the complex nature of the dispersion equations and the existence of three compressional modes for this theory, it will require some substantial amount of effort to clarify the proper analytic structure for the theory so that unphysical results are not generated. The main issues here are whether or not it is always appropriate to use the results presented in Appendix B. It is known that these types of results are valid for single-porosity analysis, but when used together with some of the approximations we have made here -- such as (for example) the neglect of cross-terms in the quasistatic permeability equations -- may introduce some unphysical behavior at high frequencies for some combinations of parameters. We leave this analysis to future work.

Another important practical issue involves the careful comparison of these results with the extensive literature on wave propagation and attenuation in fractured earth materials. These comparisons will also be left to the future.

We conclude that the double-porosity dual-permeability analysis that has been presented here has the capability to explain both wave propagation and attenuation in earth materials when the attenuation is due to out-of-phase motion of pore fluids in storage and fracture porosity. However, there remains quite a lot of work to do yet both on the theory and on its applications to real data before we can consider the story to be complete.

4/20/1999