Nonlinear spectral analysis of flow through multifractal porous media

In previous work, we have considered flow in a saturated D-dimensional aquifer when the hydraulic conductivity K is an isotropic lognormal multifractal field. We found that the resulting hydraulic gradient H and specific flow are also multifractal fields and obtained their scaling properties in analytical form. Here we use these scaling properties to derive the spectral density tensors of H and and the macrodispersivities. The analysis is nonlinear and accounts for the local random rotation of the H and fields as the resolution to which the ln(K) field is developed increases. The spectra show that H and are anisotropic at large scales, but become increasingly isotropic at smaller scales. The rate at which isotropy is approached depends on the space dimension D and the multifractal characteristics of K. Conventional first- and second-order methods do not capture this important feature and further give incorrect amplitudes and power decays of the spectral density tensors. At short distances, the macrodispersivities from nonlinear theory are significantly larger than those from linear perturbation analysis.