Roughness of wetting fluid invasion fronts in porous media

Abstract

We report an imbibition experiment that involves water displacing air in a Hele-Shaw cell filled with glass beads. We varied the flow rate such that the capillary number (${\mathit{C}}_{\mathit{a}}$) spans the range ${10}^{\mathrm{-}5}$${\mathit{C}}_{\mathit{a}}$${10}^{\mathrm{-}2}$. The invasion front exhibits self-affine roughness and the total width ${\mathit{W}}_{\mathrm{tot}}$ increases with decreasing ${\mathit{C}}_{\mathit{a}}$. A heuristic argument suggests that ${\mathit{W}}_{\mathrm{tot}}$∝${\mathit{C}}_{\mathit{a}}^{\mathrm{-}2\mathrm{\alpha }/(2\mathrm{}\mathrm{\alpha }+1)}$, where α is the roughness scaling exponent. We also find that ${\mathit{W}}_{\mathrm{tot}}$ exhibits large fluctuations with time and the effective value of α varies over a wide range (0.65–0.91). These results suggests that the behavior of interfaces in random media is dominated by the existence of many metastable states in much the same way as the domain walls in the random-field Ising model.