Experimental study of stable imbibition displacements in a model open fracture. II. Scale-dependent avalanche dynamics.

We report the results of an experimental investigation of the spatiotemporal dynamics of stable imbibition fronts in a disordered medium, in the regime of capillary disorder, for a wide range of experimental conditions. We have used silicone oils of various viscosities μ and nearly identical oil-air surface tension, and forced them to slowly invade a model open fracture at very different flow rates v. In this second part of the study we have carried out a scale-dependent statistical analysis of the front dynamics. We have specifically analyzed the influence of μ and v on the statistical properties of the velocity V_{ℓ}, the spatial average of the local front velocities over a window of lateral size ℓ. We have varied ℓ from the local scale defined by our spatial resolution up to the lateral system size L. Even though the imposed flow rate is constant, the signals V_{ℓ}(t) present very strong fluctuations which evolve systematically with the parameters μ, v, and ℓ. We have verified that the non-Gaussian fluctuations of the global velocity V_{ℓ}(t) are very well described by a generalized Gumbel statistics. The asymmetric shape and the exponential tail of those distributions are controlled by the number of effective degrees of freedom of the imbibition fronts, given by N_{eff}=ℓ/ℓ_{c} (the ratio of the lateral size of the measuring window ℓ to the correlation length ℓ_{c}∼1/sqrt[μv]). The large correlated excursions of V_{ℓ}(t) correspond to global avalanches, which reflect extra displacements of the imbibition fronts. We show that global avalanches are power-law distributed, both in sizes and durations, with robustly defined exponents-independent of μ, v, and ℓ. Nevertheless, the exponential upper cutoffs of the distributions evolve systematically with those parameters. We have found, moreover, that maximum sizes ξ_{S} and maximum durations ξ_{T} of global avalanches are not controlled by the same mechanism. While ξ_{S} are also determined by ℓ/ℓ_{c}, like the amplitude fluctuations of V_{ℓ}(t), ξ_{T} and the temporal correlations of V_{ℓ}(t) evolve much more strongly with imposed flow rate v than with fluid viscosity μ.