Critical Bursts in Filtration

Particle detachment bursts during the flow of suspensions through porous media are a phenomenon that can severely affect the efficiency of deep bed filters. Despite the relevance in several industrial fields, little is known about the statistical properties and the temporal organization of these events. We present experiments of suspensions of deionized water carrying quartz particles pushed with a peristaltic pump through a filter of glass beads measuring simultaneously the pressure drop, flux, and suspension solid fraction. We find that the burst size distribution scales consistently with a power law, suggesting that we are in the presence of a novel experimental realization of a self-organized critical system. Temporal correlations are present in the time series, like in other phenomena such as earthquakes or neuronal activity bursts, and also an analog to Omori’s law can be shown. The understanding of burst statistics could provide novel insights in different fields, e.g., in the filter and petroleum industries.

Figure 1

Experimental setup. The suspension is pumped from a beaker through two pressure oscillation dampeners and through the granular filter.

Figure 2

Filter phase diagram. At low solid fractions, the flux is continuous with no indications for clogging (green area). With an increasing solid fraction, jumps are observed (red area). For even higher values of $\mathrm{\Phi }$, the filter reaches complete clogging (yellow area). Every point in the diagram indicates an experiment, and error bars represent variations of $Q$ and $\mathrm{\Phi }$ ($\pm 1.96\sigma$, where $\sigma$ is the data standard deviation). The black point with white error bars indicates the parameters of the experiments whose statistical analysis is shown in the following. The insets show the typical pressure loss vs time evolution for experiments of the nonclogging (upper inset) and clogging regime (lower inset).

Figure 3

Time evolution of pressure loss $P$ through the filter during three experiments. Successive enlargements into a temporal interval are shown in the balloons.

Figure 4

Time evolution of the flow rate $Q$ and the suspension solid fraction $\mathrm{\Phi }$ during the three experiments. The inset shows the temporal evolution of $Q$ and $\mathrm{\Phi }$ (top) and $P$ (bottom) during a single jump of experiment 3 indicated by the arrows (units are the same as in the main figure, and $P$ is in kPa).

Figure 5

Size distribution of pressure loss (blue), flow rate (red), and solid fraction (black) jumps. Circles are data from experiment 1, triangles from experiment 2, and squares from experiment 3. The jump size $S$ is normalized by the size of the largest jump. The error of the exponents $\alpha$ is evaluated as the maximum difference between the fitted value for data from all experiments and from single experiments.

Figure 6

Pressure loss jump rates for the three experiments. Dashed lines are the average Poissonian rates, and dash-dotted lines are the 95% confidence intervals.

Figure 7

(a) Jump series for experiment 1 during a phase in which $\mu$ is higher than average. Main jumps are events above the red line. The green line indicates the minimum jump size. (b) Quiet time distributions are fitted by a Gamma distribution. (c) The occurrence rate of after jumps can be fitted with the modified Omori’s law.