Theoretical foundation for conductivity scaling

Scaling of conductivity with the support volume of experiments has been the subject of many recent experimental and theoretical studies. However, to date there have been few attempts to relate such scaling, or the lack thereof, to microscopic properties of porous media through theory. We demonstrate that when a pore network can be represented as a collection of hierarchical trees, scalability of the pore geometry leads to scalability of conductivity. We also derive geometrical and topological conditions under which the scaling exponent takes on specific values 1/2 and 3/4. The former is consistent with universal scaling observed by Neuman [1994], while the latter agrees with the allometric scaling laws derived by West et al. [1997].