On the logarithmic nature of axial dispersion in Darcy flow through heterogeneous porous media

The present study provides novel insights in how spatial velocity variations in a heterogeneous porous medium cause the dispersion of a passive tracer. The study consists of two parts. The first part describes a series of numerical computations of the axial dispersion in the flow through heterogeneous porous media, idealized as Darcy flow through two-dimensional and three-dimensional patchwork geometries of zones with randomized permeability fields. Data on the axial dispersion were obtained using the mean age theory, which transforms the transient advection-diffusion equation into the steady-state mean age field equation, thus reducing the required computational effort by multiple orders of magnitude. This allowed to consider a sufficiently large number of randomizations to obtain a statistically representative ensemble average, as well as to consider sufficiently large systems to reduce the influence of boundary conditions. In the second part, it is shown that the relation between the axial dispersion coefficient and the velocity can be represented as a series, summing up the effect of velocity differences on all length scales, assuming the velocity differences are analogous to white noise. The sum can be closely fitted by a logarithmic law containing only two parameters with a well-defined physical meaning. A similar logarithmic dependency was also obtained by Saffman, Koch, and Brady. However, the logarithmic dependency obtained in the present work emerges from the heterogeneity of the porous medium, whereas the logarithmic dependency in the aforementioned works emerged from the no-slip boundary conditions at solid surfaces.