Fractal Structure of Hydrodynamic Dispersion in Porous Media

Abstract

Concentration contours in the displacement of a clear fluid by a colored fluid of the same viscosity and density in a two-dimensional porous medium are shown to be self-affine fractal curves with a fractal dimension $D\ensuremath{\simeq}1.42\ifmmode\pm\else\textpm\fi{}0.05$. The dispersion front may on the average be described by the hydrodynamic dispersion with a longitudinal dispersion coefficient ${D}_{\ensuremath{\parallel}}=U{d}_{\ensuremath{\parallel}}$, where $U$ is the average flow velocity and ${d}_{\ensuremath{\parallel}}$ is a characteristic length of the order of a pore diameter. This result is valid for dispersion at high P\'eclet numbers $\mathrm{Pe}=\frac{U{d}_{\ensuremath{\parallel}}}{{D}_{m}}$, where ${D}_{m}$ is the molecular diffusion coefficient of the dye.