Scaling laws and mechanisms of hydrodynamic dispersion in porous media

We present a theory that quantifies the interplay between intrapore and interpore flow variabilities and their impact on hydrodynamic dispersion. The theory reveals that porous media with varying levels of structural disorder exhibit notable differences in interpore flow variability, characterised by the flux-weighted probability density function (PDF), $\hat {\psi }_\tau (\tau ) \sim \tau ^{-\theta -2}$ , for advection times $\tau$ through conduits. These differences result in varying relative strengths of interpore and intrapore flow variabilities, leading to distinct scaling behaviours of the hydrodynamic dispersion coefficient $D_L$ , normalised by the molecular diffusion coefficient $D_m$ , with respect to the Péclet number $Pe$ . Specifically, when $\hat {\psi }_\tau (\tau )$ exhibits a broad distribution of $\tau$ with $\theta$ in the range of $(0, 1)$ , the dispersion undergoes a transition from power-law scaling, $D_L/D_m \sim Pe^{2-\theta }$ , to linear scaling, $D_L/D_m \sim Pe$ , and eventually to logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$ , as $Pe$ increases. Conversely, when $\tau$ is narrowly distributed or when $\theta$ exceeds 1, dispersion consistently follows a logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$ . The power-law and linear scaling occur when interpore variability predominates over intrapore variability, while logarithmic scaling arises under the opposite condition. These theoretical predictions are supported by experimental data and network simulations across a broad spectrum of porous media.