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

Abstract

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 , the spatial average of the local front velocities over a window of lateral size . We have varied 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 (t) present very strong fluctuations which evolve systematically with the parameters μ, v, and . We have verified that the non-Gaussian fluctuations of the global velocity V (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 = / c (the ratio of the lateral size of the measuring window to the correlation length c ∼ 1/ √ μv). The large correlated excursions of V (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 . 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 / c , like the amplitude fluctuations of V (t), ξ T and the temporal correlations of V (t) evolve much more strongly with imposed flow rate v than with fluid viscosity μ.

Avalanches and extreme value statistics in interfacial crackling dynamics

“…Moreover, it is important to note that the values of those exponents characterizing the scaling behaviour of the global avalanches are different when considering local crack front bursts [12,48]. Our new experimental results clearly confirm that high local velocity clusters and global avalanches measured on crackling time series are different observables with different statistical properties, as already reported in numerical simulations [46,48], and in imbibition experiments [23,55,56]. Finally, we can also remark that those scaling exponents are also different from those reported for a three-dimensional crack front propagating in a bulk artificial rock [59].…”

“…In order to characterize this jerky front propagation, we have developed an original procedure [12], based on the measurement of the local waiting time fluctuations of the crack front during its propagation. Such method has been successfully applied to quantify the avalanche dynamics of various systems, such as fluid imbibition fronts [55,56] or biological cell migration [33]. Specifically, we first computed the local front velocities v(x, y) from measurements of the local waiting times wt(x, y) elapsed while the crack front was located at a given position (x, y), and defining v(x, y) = r/wt(x, y), where r is the pixel resolution.…”

Spatiotemporal Organization of Correlated Local Activity within Global Avalanches in Slowly Driven Interfaces

“…17 in Ref. [31]. To conclude, we have presented an analysis of how global velocity fluctuations build up from local activity clusters in dynamic growth processes.…”

“…2 shows the ratios maxðD i Þ=T and P i D i =T computed from our experimental data for a given clip level and different injection velocities. Durations are measured in units of ξ T ∼ l d =v, where l d ¼ 0.6 mm is the characteristic length of the disorder patches in the experimental cell [31].…”

Experimental study of stable imbibition displacements in a model open fracture. I. Local avalanche dynamics

“…In this paper (Part I) we focus our attention on the spatiotemporal behavior of the instantaneous velocities v(x,t) ≡ ∂h/∂t at a local scale, defined by our spatial resolution. A corresponding study of the dynamics through increasing length scales, up to the lateral system size L, is reported in the accompanying paper (Part II) [21].…”