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 $\mu$ and nearly identical oil-air surface tension, and forced them to slowly invade a model open fracture at very different flow rates $\mathrm{v}$. 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 $\mu$ and $\mathrm{v}$ on the statistical properties of the velocity $V_{\ell }$, the spatial average of the local front velocities over a window of lateral size $\ell$. We have varied $\ell$ 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 $V_{\ell }\left(t\right)$ present very strong fluctuations which evolve systematically with the parameters $\mu$, $\mathrm{v}$, and $\ell$. We have verified that the non-Gaussian fluctuations of the global velocity $V_{\ell }\left(t\right)$ 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 $N_{\text{eff}}=\ell /{\ell }_c$ (the ratio of the lateral size of the measuring window $\ell$ to the correlation length ${\ell }_c\sim 1/\sqrt{\mu \mathrm{v}}$). The large correlated excursions of $V_{\ell }\left(t\right)$ 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 $\mu$, $\mathrm{v}$, and $\ell$. Nevertheless, the exponential upper cutoffs of the distributions evolve systematically with those parameters. We have found, moreover, that maximum sizes ${\xi }_S$ and maximum durations ${\xi }_T$ of global avalanches are not controlled by the same mechanism. While ${\xi }_S$ are also determined by $\ell /{\ell }_c$, like the amplitude fluctuations of $V_{\ell }\left(t\right)$, ${\xi }_T$ and the temporal correlations of $V_{\ell }\left(t\right)$ evolve much more strongly with imposed flow rate $\mathrm{v}$ than with fluid viscosity $\mu$.

Figure 1

Spatiotemporal activity maps of the same three experiments shown in Fig. 2 of Part I [20]. $\mathrm{v}=0.13$ mm/s in all cases but $\mu =10,50,350$ cP from top to bottom. The graycode for $v(x,t)$ is the same in all cases. The black vertical line on the bottom right corner of each panel gives the nominal correlation length ${\ell }_c$ for each experimental condition (${\ell }_c=15.3,6.7,2.6$ mm from top to bottom).

Figure 2

Examples of spatially averaged (global) velocities of the front as functions of time. Six limiting cases are shown to identify the effect of the window of observation $\ell$ (top), the dynamic viscosity $\mu$ (middle), and the imposed mean velocity $\mathrm{v}$ (bottom) on the actual form of the signal. Each panel (top, middle, bottom) corresponds to setting two of the parameters $(\mathrm{v},\mu ,\ell )$ constant and changing the third one (smaller value on the left than on the right plots). The insets show $V_{\ell }\left(t\right)$ corresponding to the same mean advancement of the front (30 s at $\mathrm{v}=0.13$ mm/s). Dashed lines indicate the time-averaged velocity $\langle V_{\ell }\left(t\right)\rangle$ in each experiment. System size $L=136$ mm.

Figure 3

Evolution of the distributions of the rescaled global velocity with window of observation $\ell$ (left), dynamic viscosity $\mu$ (middle), or imposed mean velocity $\mathrm{v}$ (right). In the left panel, data for $\ell =L/n$ with $n=1$, 8, 11, 16, 27, 50 are plotted. Data in the middle panel correspond to $\mu =10$, 50, 100, 169, 350 cP. Data in the right panel correspond to $\mathrm{v}=0.036$, 0.053, 0.13, 0.23, 0.35, 0.55 mm/s. The solid lines correspond to a normal distribution.

Figure 4

Skewness of the velocity distributions as a function of $\ell$. Different symbols correspond to different viscosities. The curves obtained for different $\mu$ and $\mathrm{v}$ collapse when plotted as a function of $\ell /{\ell }_c$ (inset). The solid line is a guide to the eye.

Figure 5

Distributions of the normalized fluctuations of the global velocity, $Y=\left[(V_{\ell }\left(t\right)-\langle V_{\ell }\left(t\right)\rangle \right]/{\sigma }_{V_{\ell }}$, for various values of $\ell /{\ell }_c$. Top panels show $P\left(Y\right)$ for all the experiments with different $(\mu ,\mathrm{v})$ compatible with $\ell /{\ell }_c=10$, 2, 1, and 0.5. Distributions for each experiment separately and for data sets containing data from all experiments are shown. Solid lines correspond to generalized Gumbel distributions with the value of $Sk$ of the experimental distributions. Dashed-dotted lines correspond to a normal distribution.

Figure 6

Autocorrelation functions of $V_{\ell }\left(t\right)$ as a function of the time lag $\mathrm{\Delta }t$. Top: Evolution with $\ell$ for $\ell /{\ell }_c=10,$ 5, 2, 1, 0.75, 0.5 and two different values of $\mu$. Middle: Evolution with $\mu =10,$ 50, 100, 169 cP for $\ell /{\ell }_c=1$. A zoom of the minima is also shown. Bottom: Evolution with $\mathrm{v}$ from 0.036 to 0.55 mm/s for two values of $\ell /{\ell }_c$.

Figure 7

Evolution of the maximum anticorrelation time $\mathrm{\Delta }{t}^{*}$ with the capillary number Ca. The solid line $\mathrm{\Delta }{t}^{*}\sim {\mathrm{v}}^{-1.4}$ follows the results for various velocities but one viscosity $\mu =50$ cP (solid circles). The dashed line $\mathrm{\Delta }{t}^{*}\sim {\mu }^{-0.4}$ passes through data for various viscosities but one velocity $\mathrm{v}=0.13$ mm/s (open symbols). Inset: Evolution with $\ell /{\ell }_c$ for one set $(\mu ,\mathrm{v})$.

Figure 8

The main panel shows an example of the global velocity signal for $\ell =L$, clipped by its average value. The inset is a single avalanche of size $S$ and duration $T$.

Figure 9

Effect of the clip level $c$ on $P\left(S\right)$ (main plot) and $P\left(T\right)$ (inset) for experiments with $\mathrm{v}=0.13$ mm/s, $\mu =50$ cP, and $\ell =2{\ell }_c$. Here $c=-0.5$, $-0.25$, 0, 0.25, 0.5. Solid lines are guides to the eye corresponding to a power law with exponent $\alpha =0.96$ for sizes and $\tau =1.15$ for durations.

Figure 10

Distributions of sizes (left) and durations (right) of the global avalanches for velocities analyzed at $\ell \ge {\ell }_c$. The top, middle, and bottom panels show the evolution of the distributions with $\ell$, $\mu$, and $\mathrm{v}$ respectively when the other two parameters are not changed. The top panels show the results for $\ell =136,$ 34, 17, 12.4, 8.5, and 5 mm and the bottom panels for $\mathrm{v}=0.036,$ 0.053, 0.13, 0.23, 0.35, 0.55 mm/s. Dashed lines are guides to the eye corresponding to power laws with exponents $\alpha =0.96$ for sizes and $\tau =1.15$ for durations.

Figure 11

Distributions of sizes (left) and durations (right) of the global avalanches for velocities computed at $\ell <{\ell }_c$. The top, middle, and bottom panels show the evolution of the distributions with $\ell$, $\mu$, and $\mathrm{v}$ respectively when the other two parameters are not changed. The top panels show results for $\ell =6.5,$ 5, 4.3, 3.4, and 2.7 mm, and the bottom panels for $v=0.036,$ 0.053, 0.13, 0.23, 0.35, 0.55 mm/s. The inset shows the results for the lowest and highest velocity in the main plot. Dashed lines are guides to the eye corresponding to power laws with exponents $\alpha =0.96$ for sizes and $\tau =1.15$ for durations and dotted lines a power law with exponent 0.8.

Figure 12

Distributions of avalanche sizes for $\ell /{\ell }_c>1$. Experiments compatible with $\ell /{\ell }_c=2$ (solid symbols) and $\ell /{\ell }_c=10$ (open symbols) are shown. $\mu$ ranges from 10 to 350 cP and $\mathrm{v}$ from 0.036 to 0.55 mm/s. The dashed line gives a power law exponent $\alpha =0.96$.

Figure 13

Distributions of avalanche duration for two experimental conditions: $\mu =100$ cP and $\mathrm{v}=0.051$, 0.13 mm/s, for three different values of $\ell /{\ell }_c$. The line gives a power-law exponent $\tau =1.15$.

Figure 14

Data collapse of the probability distributions of sizes (top) and durations (bottom) of the global avalanches for experiments analyzed at $\ell /{\ell }_c>1$. Different symbols correspond to different viscosities. The best fits to a power law with an exponential cutoff are achieved with $\alpha =0.96$ and $\tau =1.15$ (solid lines). The insets show the error of the fits as a function of $\alpha$ or $\tau$.

Figure 15

Top: Distributions of avalanche sizes rescaled by $S^{\alpha }$ for $\ell /{\ell }_c=$ 2, 5, 10, 20. Solid lines are exponential fits of the data for each $\ell /{\ell }_c$. Bottom: Distributions of avalanche durations rescaled by $T^{\tau }$ for $\ell /{\ell }_c=2$ and $\mathrm{v}=0.052$, 0.13, 0.22, $0.35$ mm/s. In both cases different symbols correspond to experiments with different viscosities—we do not distinguish different $\mathrm{v}$ for the same $\mu$. Solid lines are exponential fits of the data for each $\mathrm{v}$. Insets: Evolution of the cutoff values obtained from the previous fits with $\ell /{\ell }_c$ (top) and $\mathrm{v}$ (bottom). Solid lines are guides to the eye.

Figure 16

Dependence of the mean value and the cutoff of the distributions of sizes (left) and durations (right) on $\ell /{\ell }_c$ and $\mathrm{v}$, respectively. The solid line in the left panel is a power-law fit to $\langle S\rangle$ vs $\ell /{\ell }_c$ down to $\ell ={\ell }_c$. In the right panel, the experimental values are compared to ${\tau }_c={\ell }_d/\mathrm{v}$ with ${\ell }_d=0.6$ mm.

Figure 17

Probability density plots of the joint distribution of rescaled variables $S^{*}$ and $T^{*}$ for $\ell /{\ell }_c=1$ (left) and 10 (right). Vertical dashed lines show the scaling regime chosen to extract the power-law exponent $\gamma$. The solid straight lines are the best fits $S^{*}\sim {\left(T^{*}\right)}^{\gamma }$ (statistical error of the fit quoted in the figure). The probability density of points increases from the sides to the center.