Aperture correlation of a fractal fracture

A rough-walled fracture is modeled by fractal geometry. In the fractal fracture model, the rock surfaces are characterized by a fractal dimension D between 2 and 3, with lower D for smoother surfaces and higher D for rougher surfaces. The mismatch due to shear displacement between two mirror-image fractal surfaces determines the fracture aperture distribution. An analytic equation is derived for the variogram γƒ(r) describing the spatial correlation of the aperture of a fractal fracture. The aperture of a smooth fracture with low D is highly correlated over distances much larger than the shear displacement. The aperture of a rough fracture with high D becomes uncorrelated within a range shorter than the shear displacement. Near the origin, r = 0, of the variogram, the variogram is proportional to r6–2D. For the special case of Brownian fractal with D = 2.5, the variogram is proportional to r, which is the same r dependence exhibited by the spherical model widely used in geostatistical analyses.

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