The fractal dimension (FD) of sedimentary rocks has become an important petrophysical parameter. The importance is partly because capillary presure (Pc) -and other transport coefficients-scale as power laws of fluid saturation. The scaling exponents are often related to the supporting media's FD. -Water Pc, in particular, is known to scale as a power law of water saturation, in imbibition at low saturation of the wetting phase, and the scaling exponent is related to the FD of the pore-rock surface. Here we show results of mercury intrusion tests, in drainage at high capillary presure, which clearly show power law dependence between intrusion volume and Pc. We relate the scaling exponent to the pore bulk FD. We also report results of two dimensional image analysis which indicate that, in fact, both pore bulk and pore-rock interfaces are fractal.

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The self averaging properties of the conductance g are explored in Random Resistor Networks (RRN) with a broad distribution of bond strengths P (g) ∼ g µ−1 . The RRN problem is cast in terms of simple combinations of random variables on hierarchical lattices. Distributions of equivalent conductances are estimated numerically on hierarchical lattices as a function of size L and the distribution tail strength parameter µ. For networks above the percolation threshold, convergence to a Gaussian basin is always the case, except in the limit µ → 0. A disorder length ξ D is identified, beyond which the system is effectively homogeneous. This length scale diverges as ξ D ∼ |µ| −ν , (ν is the regular percolation correlation length exponent) when the microscopic distribution of conductors is exponentially wide (µ → 0). This implies that exactly the same critical behavior can be induced by geometrical disorder and by strong bond disorder with the bond occupation probability p ↔ µ. We find that only lattices at the percolation threshold have renormalized probability distributions in a Levy-like basin. At the percolation threshold the disorder length diverges at a critical tail strength µ c as |µ − µ c | −z with z ∼ 3.2 ± 0.1, a new exponent. Critical path analysis is used in a generalized form to give the macroscopic conductance in the case of lattices above p c .

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