Drying and percolation in correlated porous media

We study how the dynamics of a drying front propagating through a porous medium are affected by small-scale correlations in material properties. For this, we first present drying experiments in microfluidic micromodels of porous media. Here, the fluid pressures develop more intermittent dynamics as local correlations are added to the structure of the pore spaces. We also consider this problem numerically, using a model of invasion percolation with trapping, and find that there is a crossover in invasion behavior associated with the length scale of the disorder in the system. The critical exponents that describe large enough events are similar to the classic invasion percolation problem, while the addition of a finite correlation length significantly affects the exponent values of avalanches and bursts, up to some characteristic size. We find that even a weak local structure can interfere with the universality of invasion percolation phenomena. This has implications for a variety of multiphase flow problems, such as drying, drainage, and fluid invasion.

Figure 1

Invasion into correlated porous media. (a) A flat pore space is designed and fabricated in a microfluidic chip by soft lithography techniques. The drying cell consists of an array of pillars, which surround pores. The pores are initially filled through channels along one edge of the cell and then dry through the opposite edge, which is open to the atmosphere. (b) Correlations between pillar sizes, and hence pore sizes, are randomly introduced with different correlation lengths $\zeta$. (c) The drying pattern (shown here at breakthrough, when the invading phase connects to the back of the chip) and sequence of invaded pores reflect the underlying structure of pillar sizes.

Figure 2

Local correlations in the sizes of pores and throats dramatically affect how the fluid pressure fluctuates as a porous medium is dried, in both experiments and a complimentary invasion percolation model. When there are no correlations $\left(\zeta =0\right)$ the pressure sequence is similar to white noise, centered around the average invasion pressure of the porous medium $\left(P=1\right)$. In contrast, when the correlation length $\zeta$ is 15 times the typical pore size, the pressure sequence shows strong intermittency.

Figure 3

Model and statistics of the pressure-saturation fluctuations. (a) The invasion percolation model tracks an interfacial line between wet (blue) and unsaturated (white) regions. At every time step $i$ the interfacial throat with the lowest absolute invasion pressure $P_i$ is broken, and air invades the pore behind it. Two example invasions are shown, highlighted in yellow and green. The difference $\mathrm{\Delta }P$ measures the absolute value of the change in the fluid pressure between the successive invasion of two pores. (b) Bursts are defined by record-breaking values of the pressure (highlighted in red). Their size is measured by the number of subsequent events required until the previous maximum pressure is exceeded. Forward and backward avalanches are defined, in contrast, for every invasion step (as in Refs. [2, 30]). Their sizes, $S_f$ and $S_b$, give the number of invasion events required before that moment's invasion pressure is next exceeded, or was last seen, respectively.

Figure 4

The power spectra of the pressure signals from (a) experiments and (b) simulations respond to local correlations. For low frequencies, in particular representing fluctuations over tens or more invasion events, the uncorrelated porous media show relatively flat power spectra, characteristic of white noise. The introduction of correlations into the pore geometry adds significantly to the low-frequency power, which corresponds to the jumps of the pressure shown in Fig. 2. For comparison, the dashed lines show a $1/q$ response, which is classically associated with intermittency [15].

Figure 5

Distributions of the pressure changes between subsequent pore invasions are shown for (a) drying experiments and (b) simulations. As the correlation length $\zeta$ increases, there is a tendency to have fewer extreme events, as more invasion events will occur within correlated regions of a similar pore size. This is clearer in the simulation data, as uncertainties in exactly determining the experimental pressure signal introduce high-frequency random noise.

Figure 6

A burst covers the invaded area between any two sequential record-breaking values of the saturation pressure. For pressure-controlled invasion, as in Refs. [4, 5], this would be a rapid “burst” of activity between stable pressure values, while for our volume-controlled invasion it can be measured from the invasion pressure series, as demonstrated in Fig. 3. In the absence of any correlations in pore size, the bursts follow a power-law distribution with an exponent that is similar to values reported for other examples of invasion percolation [4, 35, 36]. When correlations are introduced the distribution shifts to a steeper power law, representing more frequent, but smaller, burst events, such as those confined to a single correlated patch of pores.

Figure 7

Backward avalanches measure the number of events, going backwards in time, until the pressure of any particular invasion was last exceeded (see Fig. 3). The simulated distributions (a) collapse onto a master curve (b) when scaled by the area of a correlated patch of pores, ${\zeta }^2$. The crossover between these two scaling regimes, marked by a dashed line, occurs at around five times the correlation length, $\zeta$. Essentially, while small bursts are strongly affected by local correlations, events large enough to average over many such correlated patches show a mean-field distribution.

Figure 8

Backward avalanches were measured in the experimental pressure signals. (a) There is no clear effect of the correlation length on the avalanche statistics, despite its leading to intermittency in the original signal (Fig. 2) and power spectra (Fig. 4). We find, however, that the measured avalanche size distribution is strongly affected by noise or uncertainties in the invasion pressure of each throat. (b) Adding 3% random noise to each measurement, comparable to our experimental manufacturing tolerance [10], is enough to interfere with seeing the crossover behavior of our simulations. As a guide to the eye, curves show the fit from Fig. 7 for the noise-free $\zeta =0$ simulations.

Figure 9

Forward avalanches measure the number of events, going forward in time, until any particular invasion pressure is next exceeded (see Fig. 3). (a) In simulations the distribution for an uncorrelated porous medium follows a power-law distribution with exponent ${\theta }_f=1.99\pm 0.05$. This is consistent with a “superuniversal” distribution, where ${\theta }_f=2$, seen in a large class of invasion models [16, 30]. The local correlations modify this distribution, lowering the exponent to ${\theta }_f=1.66\pm 0.04$. (b) The forward avalanches in the experiment show little variation with $\zeta$ and are consistent with ${\theta }_f=2$. (c) As with backward avalanches, the introduction of as little as 3% noise into the pressure signal interferes with the clear observation of the crossover behavior.