Discussion

These observations show that there is a significant problem with up-scaling Biot's theory, i.e., that the resulting system of equations is no longer of the same form as Biot's theory. This is certainly no failing of Biot's theory, but rather a failing of any attempted application of Biot's theory directly to the up-scaled macro-system. Biot's theory predicts correctly tha compressional and shear wave attenuation both depend on the integral of the permeability $\kappa$ along the path of each wave. But the permeability itself along the same path averages as the inverse of the permeability (harmonic mean). Thus, the overall permeability depends most strongly on the smallest permeabilities present in the system, while the wave attenuation depends most strongly on the largest permeabilities in the system (Berryman, 1988). When we try to up-scale under these circumstances, we have an inherent problem due to the fact that Biot's theory contains only one permeability; yet, for heterogeneous systems, there are two very distinct measures of permeability (the mean and the harmonic mean) that play significant roles.

  

Figure 2: The elements of a double porosity model are: porous rock matrix intersected by fractures. Three types of macroscopic pressure are pertinent in such a model: external confining pressure p_c, internal pressure of the matrix pore fluid p_f⁽¹⁾, and internal pressure of the fracture pore fluid p_f⁽²⁾.