Geometrical interpretation of long-time tails of first-passage time distributions in porous media with stagnant parts

Using a combined experimental-numerical approach, we study the first-passage time distributions (FPTD) of small particles in two-dimensional porous materials. The distributions in low-porosity structures show persistent long-time tails, which are independent of the Péclet number and therefore cannot be explained by the advection-diffusion equation. Instead, our results suggest that these tails are caused by stagnant, i.e., quiescent areas where particles are trapped for some time. Comparison of measured FPTD with an analytical expression for the residence time of particles, which diffuse in confined regions and are able to escape through a small pore, yields good agreement with our data.

Figure 1

(a) Microscope image of a transparent porous structure made from PDMS, which is filled with dyed (blue) water. Afterwards, pure water is injected from the left with a flow velocity of about $6\mu \text{m}/\text{s}$ and displaces the dyed water. Snapshots are shown after (b) $5\text{min}$, (c) $10\text{min}$, and (d) $33\text{min}$. Even after more than 1 h, dyed water remains trapped in the stagnant regions of the structure.

Figure 2

(a) Grayscale (color) maps of experimentally obtained velocity fields in porous ROMC structures with porosities (a) ${\varphi }_{\mathrm{o}}=0.900$, (b) ${\varphi }_{\mathrm{o}}=0.582$, and (c) ${\varphi }_{\mathrm{o}}=0.232$. The plots show the local average velocity magnitude on a logarithmic scale in units of $\mu$m/s. With decreasing porosity, the velocity fields become more heterogeneous until only a few principal pathways for liquid flow remain, which are surrounded by large stagnant areas (dark).

Figure 3

(a) Experimental particle trajectories obtained during 30 min inside a porous ROMC structure with ${\varphi }_{\mathrm{o}}=0.232$. The flow is from left to right. The inset shows a magnification of a diffusing particle in a stagnant region. (b) Corresponding particle trajectories obtained by simulations with observation time of 20 h. Obviously, during this time the particles are able to explore much larger regions of the structure and in particular deeper parts of the stagnant areas.

Figure 4

Measured FPTD for different $\mathrm{Pe}$ for three different structures (solid lines). The insets show the same data but on a logarithmic time scale. (a) Corresponds to ${\varphi }_{\mathrm{o}}=0.900$, (b) to ${\varphi }_{\mathrm{o}}=0.582$, and (c) to ${\varphi }_{\mathrm{o}}=0.232$. The open symbols correspond to fits to the ADE. The highest $\mathrm{Pe}$ (black line) was 128 and each consecutive $\mathrm{Pe}$ decreases by a factor of two.

Figure 5

Flowing parts (dark blue) and stagnant parts (white) of (a) the structure with ${\varphi }_{\mathrm{o}}=0.582$ and (b) ${\varphi }_{\mathrm{o}}=0.232$. (c) Calculated FPTD due to stagnant parts for ${\varphi }_{\mathrm{o}}=0.582$ (gray) filled bars and ${\varphi }_{\mathrm{o}}=0.232$ (red) open bars. Solid (black) and dashed lines (red) show the measured FPTD for $\mathrm{Pe}=128$ for the structure with ${\varphi }_{\mathrm{o}}=0.582$ and ${\varphi }_{\mathrm{o}}=0.232$, respectively. (Inset c) Distribution of calculated mean residence times in stagnant parts.

Figure 6

(a) Distribution of number of visited stagnant parts $n$, (b) fraction $\epsilon$ of time spent in largest stagnant part for ${\varphi }_{\mathrm{o}}=0.232$ and $\mathrm{Pe}=128$. (Inset) Average number of visited stagnant zones $\overline{n}$ vs. $\mathrm{Pe}$ for ${\varphi }_{\mathrm{o}}=0.232$.