Permeability of self-affine aperture fields.

We introduce a model that allows for the prediction of the permeability of self-affine rough channels (one-dimensional fracture) and two-dimensional fractures over a wide range of apertures. In the lubrication approximation, the permeability shows three different scaling regimes. For fractures with a large mean aperture or an aperture small enough to the permeability being close to disappearing, the permeability scales as the cube of the aperture when the zero level of the aperture is set to coincide with the disappearance of the permeability. Between these two regimes, there is a third regime where the scaling is due to the self-affine roughness. For rough channels, the exponent is found to be 3-1/H, where H is the Hurst exponent. For two-dimensional fractures, it is necessary to introduce an equivalent aperture b(c) to make the scaling regime apparent. b(c) is defined as the hydraulic aperture of the most restrictive barrier crossing the fracture normal to the flow direction. This regime is characterized by an exponent higher than that for the one-dimensional case: it is 2.25 for H=0.8 and 2.16 for H=0.3.