Length Scales, Multi-fractals and Non-fickian Diffusion

In single phase flow of a passive tracer through heterogeneous porous media, a mixing layer develops between the tagged and untagged regions of the fluid. The mixing region expands as the time evolves. We have developed a multilength scale theory for the growth of the mixing region induced by a general random velocity field (multi-fractal field). The theory relates the statistics of the mixing layer to the statistics of the random field and derives an effective equation which governs the statistical properties of the mixing layer. The theory provides an analytic prediction for the growth of the size of the mixing layer. The scaling behavior of the mixing layer in a general random velocity field is determined over all length scales. The analysis of the multi-length scale theory shows that the growth rate (i.e. the scaling exponent) of the mixing layer at length scale l depends on the statistical properties of the random field on all length scales smaller than l. In general, the scaling exponent of the mixing layer is non-Fickian on all finite length scales. The asymptotic diffusion is non-Fickian when the correlation function of the random field decays slowly at large length scales, and Fickian when the correlation function of the random field decays rapidly at large length scales. Let γ∞ be the asymptotic scaling exponent of the mixing layer, and β∞ be the asymptotic scaling exponent of the correlation function of the velocity (or permeability) field. Then γ∞ = max{ 2 1 hh , 1+ 2 β∞ hhh}. Furthermore the theory explains why the effective macroscopic diffusion is a non-decreasing function of the length scale.