Experimental study of stable imbibition displacements in a model open fracture. I. Local 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 different constant flow rates $\mathrm{v}$. In this first part of the study we have focused on the local dynamics at a scale below the size of the quenched disorder. Changing $\mu$ and $\mathrm{v}$ independently, we have found that the dynamics is not simply controlled by the capillary number $\text{Ca}\sim \mu \mathrm{v}$. Specifically, we have found that the wide statistical distributions of local front velocities, and their large spatial correlations along the front, are indeed controlled by the capillary number $\text{Ca}$. However, local velocities exhibit also very large temporal correlations, and these correlations depend more strongly on the mean imposed velocity $\mathrm{v}$ than on the viscosity $\mu$ of the invading fluid. Correlations between local velocities lead to a burstlike dynamics. Avalanches, defined as clusters of large local velocities, follow power-law distributions—both in size and duration—with exponential cutoffs that diverge as $\text{Ca}\to 0$, the pinning-depinning transition of stable imbibition displacements. Large data sets have led to reliable statistics, from which we have derived accurate values of critical exponents of the relevant power-law distributions. We have investigated also the dependence of their cutoffs on $\mu$ and $\mathrm{v}$ and related them to the autocorrelations of local velocities in space and time.

Figure 1

Sketch of the model open fracture. The liquid is driven at constant flow rate from the inlet(I) displacing the previously resident air. The model open fracture consists of two horizontal, parallel glass plates (G) and a disorder fiber-glass plate (FG). Squared copper patches (Cu) of lateral sizes $0.4\times 0.4{\mathrm{mm}}^2$ are randomly distributed occupying 35% of the surface of the FG plate. The region of interest (ROI) defines the lateral system size $L=136$ mm.

Figure 2

Velocity maps of three experiments performed at the same $v=0.13$ mm/s but different $\mu =10,$ 50, 350 cP (top, middle, bottom). The color code (gray scale) bar for $v(x,y)$ is the same in all cases. The bottom-right white line in each panel gives the nominal correlation length ${\ell }_c=15.3,$ 6.7, 2.6 mm for each experimental condition.

Figure 3

Distributions of local velocities rescaled by their mean value $P(v/\langle v\rangle )$. Main plot: Distributions corresponding to experiments performed with the same oil but different $\mathrm{v}=0.036,$ 0.053, 0.13, 0.23, 0.35, 0.55 mm/s. Inset: Distributions corresponding to experiments with different oils of viscosities $\mu =10,$ 50, 100, 169, 350 cP. Dashed and dotted lines are guides to the eye.

Figure 4

Distributions $P(v/\langle v\rangle )$ for displacements at various $\text{Ca}=6.3\times {10}^{-5}, 2.5\times {10}^{-4}, 3.3\times {10}^{-4}$, and $2.2\times {10}^{-3}$. Different symbols correspond to different dynamic viscosities.

Figure 5

Distributions of normalized local velocities—at the scale of spatial resolution—in a semilog plot (main) and a log-log plot for the positive values (inset). Results are shown for all experiments, but different $\mathrm{v}$ are not differentiated.

Figure 6

Autocorrelation of local velocities along the front. Left: Experiments at different imposed mean velocities $\mathrm{v}=0.036,$ 0.053, 0.13, 0.23, 0.35, 0.55 mm/s. Right: Experiments using different oils of dynamic viscosities $\mu =10,$ 50, 100, 169, 350 cP.

Figure 7

Lateral length scales of the maximum anticorrelation ($\mathrm{\Delta }{\ell }^{*}$, solid symbols) and decorrelation ($\mathrm{\Delta }{\ell }^{\min }$, open symbols), extracted from Fig. 6, vs the nominal correlation length ${\ell }_c$. Different velocities are plotted for each $\mu$, with ${\ell }_c\sim 1/\sqrt{\mathrm{v}}$. Dashed lines are guides to the eye. The dotted line has slope 1.

Figure 8

Statistical width of local velocities as a function of the window of observation $\ell$. Top: Experiments using different oils of viscosities $\mu =10,$ 50, 100, 169, 350 cP. Bottom: Experiments performed at different imposed mean velocities $\mathrm{v}=0.036,$ 0.053, 0.13, 0.23, 0.35, 0.55 mm/s.

Figure 9

Autocorrelation of local velocities in the propagation direction $y$ for experiments with different oil viscosities. The spatial correlation of the disorder is also shown (dashed line). Inset: $\mathrm{\Delta }{y}^{*}$ (open symbols) and $\mathrm{\Delta }{y}^{\min }$ (full symbols) vs Ca. The dotted line corresponds to $\mathrm{\Delta }{y}_d$ (the length scale of decorrelation of the disorder). The dashed line is a guide to the eye proportional to ${\text{Ca}}^{-0.4}$.

Figure 10

Maximum anticorrelation time $\mathrm{\Delta }{t}^{*}$ as a function of the capillary number Ca. Dashed and dotted lines are guides to the eye proportional to ${\mathrm{v}}^{-1.4}$ and ${\mu }^{-0.4}$, respectively. Open symbols correspond to experiments at $\mathrm{v}=0.13$ mm/s.

Figure 11

Definition of local avalanches. Bottom: Velocity map after clipping. Local velocities $v<v_c$ are all plotted in blue (dark gray). The first and last interfaces in contact with the avalanche highlighted are shown. Top left: Close-up of the selected avalanche. Top middle: Definition of area, $A$, and lateral sizes, $L_x$ and $L_y$, of the avalanche. Top right: Close-up of the same avalanche on the activity map $v(x,t)$. The avalanche duration is $D$. Experimental conditions: $\mathrm{v}=0.13$ mm/s, $\mu =50$ cP.

Figure 12

Comparison of local avalanches with the disorder patches in the plate underneath. Bottom panel: Velocity map in $(x,y)$ space. Top left panel: Close-up of a single avalanche. Local avalanches are depicted in white. The color code (gray scale) for the disorder ranges from orange (dark gray) (no disorder patch in the pixel) to black (disorder occupying the whole pixel). Top right panel: Areas of disorder swept by local avalanches vs the areas of the corresponding avalanches in double-logarithmic scale and in lin-lin scale (inset).

Figure 13

Distributions of sizes of local avalanches. Left: Evolution of the pdfs with the imposed mean velocity $\mathrm{v}=0.036$, 0.053, 0.11, 0.16, 0.22, 0.35, 0.55 mm/s. Open symbols correspond to ill-behaved distributions. Right: Evolution of the pdfs with oil viscosity $\mu$. In both cases ${\alpha }_A=1.09\pm 0.08$.

Figure 14

Distributions of durations of local avalanches. Left: Evolution of the pdfs with the imposed mean velocity $\mathrm{v}=0.036$, 0.053, 0.11, 0.16, 0.22, 0.35, 0.55 mm/s. Right: Evolution of the pdfs with oil viscosity $\mu$. In both cases ${\alpha }_D=1.03\pm 0.10$.

Figure 15

Distributions of lateral extents $L_x$ (left) and $L_y$ (right) of local avalanches. The range of velocities shown in the top panels is $\mathrm{v}=0.036$, 0.053, 0.13, 0.23, 0.35, 0.55 mm/s. The bottom panels show the evolution of the pdfs with viscosity. The exponents obtained are ${\alpha }_{L_x}=0.83\pm 0.10$ and ${\alpha }_{L_y}=1.08\pm 0.15$.

Figure 16

Left: Joint distributions of $P(L_x,L_y)$. Results for all experiments are shown, but different velocities are not distinguished. Full symbols represent data used to obtain $H$. Vertical dotted lines show the lower and upper cutoffs of the range of $L_x$ considered. Right: Rescaled joint distributions of the data within the scaling range. The power-law exponent is obtained from a linear fit of the log-log data. The color code (gray scale) corresponds to the density of events and increases from dark blue (dark gray, outer region) to red (dark gray, inner region).

Figure 17

Top panels: Evolution of the cutoffs of $P\left(L_x\right)$ (left) and $P\left(L_y\right)$ (right) with capillary number in double-logarithmic scale. Dashed straight lines are guides to the eye with the exponents shown in the figure. Bottom left: Cutoffs ${\xi }_{L_x}$ and maximum lateral anticorrelation lengths $\mathrm{\Delta }{\ell }^{*}$ vs the nominal correlation length ${\ell }_c$. Bottom right: Cutoffs ${\xi }_{L_y}$ and decorrelation lengths in the direction of propagation $\mathrm{\Delta }{y}^{\min }$ vs ${\ell }_c^H$, where $H=0.82$ is the anisotropy exponent. Dashed straight lines are guides to the eye. The dotted line has slope 1. Different symbols correspond to different viscosities.

Figure 18

Evolution of the cutoffs of $P\left(A\right)$ (left) and $P\left(D\right)$ (right) with the capillary number in log-log scale. Dashed and dotted straight lines are guides to the eye with the exponents shown in the figure. Different symbols (either open or solid) correspond to different viscosities. The values for experiments performed at $\mathrm{v}=0.13$ mm/s have been distinguished by using open symbols.

Figure 19

Cutoffs of $P\left(D\right)$ (solid symbols) and maximum anticorrelation time $\mathrm{\Delta }{t}^{*}$ (open symbols) vs ${\ell }_c^H/\mathrm{v}$, an estimation of the maximum duration of local avalanches. The dashed line is a guide to the eye. Different symbols correspond to different viscosities. For the same viscosity, slower experiments have larger ${\ell }_c^H/\mathrm{v}$.

Figure 20

The top and middle panels show the joint probability distributions of $A$ and $L_x$ with $D$, rescaled by the cutoffs of their individual pdfs. The bottom panel shows the joint probability distribution of $A$ with $L_x$, rescaled by the cutoffs of their individual pdfs. The color code (gray scale) ranges from dark blue (dark gray, outer region) to red (dark gray, inner region). The different solid lines are power law fits.