Multifractal anisotropic scaling of the hydraulic conductivity

We analyzed the scaling properties of the hydraulic conductivity K at three sites in Northern America: MADE, Borden, and Cape Cod. We found that K at all sites exhibits multifractality (fractal and multiscaling) in both the vertical and horizontal directions, though the multiscaling was within a range smaller than that of the maximum distance between measurements. In the vertical direction, the K data for MADE, Borden, and Cape Cod were multiscaling from 0.15 to 1.35 m, 0.05 to 0.5 m, and 0.15 to 0.9 m, respectively. They were multiscaling in the horizontal direction from 9 to 45 m, 1 to 10 m, and 1 to 17 m, respectively. The multiscaling was also anisotropic. Evidence of scaling was poorest for the horizontal direction of the MADE site, and it spanned half an order of magnitude. Such results compel one to treat multifractality in the horizontal direction of MADE as supported by heuristic arguments, rather than by pure statistical evaluation. We fitted a multifractal model to the data and estimated its parameters. We found the underlying statistics of all data to be non-Gaussian, and the model capable of reproducing the probability distribution of K data, especially the negative skewness of log K. We also generated two-dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model.