Roughness and intermittent dynamics of imbibition fronts due to capillary and permeability disorder

Abstract

We follow the propagation of an air–liquid interface during forced-flow imbibition of a viscous wetting liquid by a random medium, using a high resolution fast camera. Our model disordered medium mimics an open fracture by a Hele-Shaw cell with a two-valued gap spacing randomly distributed in the fracture (or Hele-Shaw) plane. By systematically varying the imposed flow rate we achieve average imbibition front velocities in the range 0.057 < v < 0.648 mm/s, allowing the investigation of two different regimes. Up to v ≃ 0.2 mm/s the front roughening is dominated by capillary pressure fluctuations (capillary regime). Above this value it is influenced also by fluctuations of the permeability of the medium (mixed capillary and permeability regime). We analyze the scaling properties of the front morphology in these two regimes, and find that the local roughness exponent changes from α loc ≃ 0.8 in the capillary regime to 0.5 in the mixed capillary–permeability regime. In contrast, the statistics of the spatially-averaged (global) velocity of the front V l(t) does not seem to be influenced by the randomness in the permeability. The fluctuations of V l(t) follow a non-Gaussian distribution, whose skewness γ ~ 1 / v in both regimes. Similarly, sizes and durations of the avalanches of the global velocity present the same statistical distributions in both regimes. In the mixed capillary–permeability regime, the power-law exponents that characterize these distributions take the values α = 0.97 ± 0.05 for the sizes, τ = 1.16 ± 0.10 for the durations, and x = 1.3 ± 0.1 for the joint distribution of sizes vs. durations. These values are in excellent agreement with the ones measured previously in the capillary regime (Planet et al., 2009). The dynamics therefore appear to be equivalent in both regimes, a result that we relate to the fact that the correlation length of the local velocities along the front is observed to follow the same dependence l c ~ 1 / v in both regimes.