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## Electrical conductivity and

The DC electrical properties of the model porous media are determined using a finite-difference method of solving Laplace's equation in the pore space. This procedure is equivalent to setting up a random conductor network and solving the Kirchhoff's law network equations [20]. A node is placed in the center of each pixel, and unit conductance bonds are placed between adjacent pore space pixels. All other nearest-neighbor pairs are assigned zero conductance. Bond assignment is carried out according to periodic boundary conditions. A unit voltage difference is applied in the x-direction, and is maintained throughout the conjugate-gradient relaxation process [21] by requiring that nodes separated by the system size L in the x-direction differ in voltage by one.

Once the voltage at every node was obtained, the total current was computed to determine the bulk conductivity σ, equal in this case to Γ because σo was normalized to one. The electric fields were computed from the voltages, and used in Eq. (5) to compute Λ directly. We have checked the accuracy of this method against the exact result for a single insulating circle in an L x L conducting sheet. The relative conductivity for a single isolated circle [15], with diameter much less than L, is Λ = 2φ − 1 , which can be used to show, by using a technique described in Ref. 3 involving adding a a very thin conducting layer to the insulating circle, that Λ = L 2 / (πd) , where d is the diameter of the circle and φ is the porosity. In this limit, h and Λ become equal. Using Eq. (5) with our electric field results, we found that the computed values of Λ were off by the multiplicative factor π/4. This is due to the fact that the exact perimeter of a digital circle is 4D, instead of πD (see Fig. 1), no matter how large the diameter D. The surface integral part of the calculation is then too large. Correcting our Λ results by 4/π gave the exact results within 2%. All subsequent computations were corrected by the same factor.

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