Fractal Structure of Hydrodynamic Dispersion in Porous Media

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 $D\simeq 1.42\pm 0.05$. The dispersion front may on the average be described by the hydrodynamic dispersion with a longitudinal dispersion coefficient $D_{\parallel }=U{d}_{\parallel }$, where $U$ is the average flow velocity and $d_{\parallel }$ is a characteristic length of the order of a pore diameter. This result is valid for dispersion at high Péclet numbers $\mathrm{Pe}=\frac{U{d}_{\parallel }}{D_m}$, where $D_m$ is the molecular diffusion coefficient of the dye.