Anisotropic scaling of Berea sandstone log air permeability statistics

Minipermeameter Berea sandstone data and their increments are found to be consistent with sub-Gaussian random fields subordinated to truncated fractional Gaussian noise. Their statistics and scaling behavior are shown to depend on direction. We analyze directional dependence and scaling of air log permeability statistics characterizing minipermeameter measurements by Tidwell and Wilson on the faces of a 30 by 30 by 30 cm³ size unsaturated block of Berea sandstone. We find distinct differences between the statistics and scaling behaviors of data measured on faces parallel and normal to bedding, and of incremental measurements in three orthogonal directions along these faces. Whereas the distribution of data and their increments parallel to bedding are heavy tailed, those of increments normal to bedding are Gaussian. Order q sample structure functions of increments parallel to bedding scale as powers ξ(q) of directional lag, sd, over limited ranges of sd. Using moment and extended self-similarity methods of analysis we find ξ(q) to be generally nonlinear in q, tending to be concave in q on faces normal to bedding and convex on faces parallel to bedding. Whereas the literature attributes nonlinear scaling of ξ(q) with q to multifractals or fractional Laplace motions, we find the data to be consistent with sub-Gaussian random fields subordinated to truncated (monofractal, self-affine) fractional Gaussian noise. The increments exhibit negative statistical dependence (antipersistence) that varies with direction parallel to bedding and is more pronounced on faces parallel than normal to bedding.