Connectivity enhancement due to film flow in porous media

We study the effects of connectivity enhancement due to film flow phenomena on the drainage of an artificial porous medium and the relative influence of gravity for such effects. The medium is initially fully saturated with a liquid (wetting phase), which is displaced by air (nonwetting phase). Our setup allows us to directly visualize the dynamics of the flow and, in particular, to pinpoint which pore invasion events are due to film flow phenomena. We have observed the formation of an active zone behind the liquid-air interface, inside which film flow drainage events are more likely to occur. Understanding the basic mechanisms of film flow in artificial porous media is of relevance for the analysis of real three-dimensional porous systems, in which gravity cannot be neglected and film flow is bound to be present.

Figure 1

Diagram of the experimental setup showing the cell with the porous medium made of a single layer of glass beads. The camera, light box, and cell are all connected to a common frame that can be inclined by an angle $\theta$, as shown. The liquid outlet on the left is connected to a syringe pump that withdraws the liquid. The water inlet for the cushion (on the right) is connected to a bottle which is kept at a high level ($\approx 3\mathrm{m}$). This is to keep the cushion pressurized. There is no connection between the water in this cushion and the fluids in the porous medium.

Figure 2

Sequence of images showing the drainage of a supposedly trapped liquid cluster. The average flow direction is from top to bottom and the system is inclined by ${20}^{\circ }$, thus yielding a vertical component of the acceleration of gravity $g_y$ as shown. The time difference between frames I and III is $\approx 10\mathrm{h}$. The zoomed inset on the left (corresponding to the red rectangle in frame I) shows the path of interconnected capillary bridges through which film flow takes place, with some of the bridges marked by green arrows.

Figure 3

Macro image of an experiment showing some capillary bridges (three of them are marked by arrows). The diameter of the glass beads is $\approx 1\mathrm{mm}$.

Figure 4

Diagram showing the algorithm employed to track the capillary bridges. For each pore throat we compute the color intensity plot along a perpendicular bisector segment (solid blue lines), a line passing through the center of the pore throat along a direction perpendicular to the direction connecting the center of the beads (dashed blue lines). There are four possible outcomes for this plot. If the pore throat lies in the bulk of one of the phases, the color intensity is approximately constant, giving a higher value for the air-filled pores (lighter color) and a lower value for the liquid-filled pores (darker color), as shown in the two diagrams on the top. If the pore throat belongs to a cluster boundary, it is partially liquid and partially air filled, and the plot shows a transition from lower to higher values or vice versa (bottom left). When a capillary bridge is present, the color intensity goes from high to low to high, indicating the thin liquid bridge in between air-filled pore bodies (bottom right). It is this kind of pattern that we track for all pore throats in the model in order to pinpoint the position of the capillary bridges.

Figure 5

Identification of film flow events. (a) A typical snapshot of the experiment with an inclination of ${20}^{\circ }$, after thresholding to isolate the invading air phase (white). (b) Detail from (a) (red square) showing a differential image between two snapshots separated by $\sim 2\mathrm{h}$. The area in cyan denotes pores that have been invaded in this interval. The majority of the invasion happens along the front (red line), but some events occur behind the front [film flow events, isolated in (c)]. In (d) we show the invasion map at breakthrough. The green area shows pores that were invaded via the usual mechanisms while the yellow area denotes those pores invaded via film flow effects. In this experiment, film flow effects were responsible for an increase in saturation of $8.3\%$. A full spatiotemporal map of the invasion is shown in (e), where the color code denotes the invasion time (in hours) of a given pore.

Figure 6

Capillary bridges tracking. The red lines connect the centers of beads through which a continuous capillary bridge connection exists. The bridges positions are marked by an asterisk on the lines. The liquid in the cluster C1 can eventually be withdrawn to the defending cluster C0 through the capillary bridges pathway b1, marked in yellow. Similarly, the liquid in cluster C2 can flow into the cluster C1 through the capillary bridges in b2, also marked in yellow. From there, it could also flow into the defending cluster C0 through b1. This interconnection mechanism, through bridges and intermediate clusters, effectively enhances the overall connectivity of the liquid phase.

Figure 7

Histogram showing the total frequency of pore throats (brown) and capillary bridges (blue) in an experiment with ${20}^{\circ }$ inclination. In the inset it is shown an estimate of the conditional probability that a pore throat of a given size will host a capillary bridge (ratio of the histograms in the main figure). Notice that if a pore throat is wider than a given threshold ($\approx 0.5\mathrm{mm}$ in this case) the formation of a capillary bridge in that throat is very unlikely.

Figure 8

Drainage of a model with an inclination of ${20}^{\circ }$. The average flow direction and gravity component are from top to bottom, along the $y$ axis. In red we mark the pores that were invaded due to film flow phenomena between the instant of the picture and another image taken $\approx 2\text{h}$ earlier. The time difference between the first and second frames is $\approx 24\text{h}$. A film flow active zone (shown schematically as the blue bands in the figure) follows the liquid-air interface. The vertical positions $y_{az}$ of the active zone and $y_f$ of the invading front are shown schematically.

Figure 9

Temporal development of the film flow active zone and the main liquid-air interface (the invading front) for an experiment with a ${20}^{\circ }$ inclination. In the main plot we show the average vertical position of the invading front ($y_f$, green circles) and the film flow active zone ($y_{az}$, black crosses). After a transient time (brown vertical dashed line), the film flow active zone follows the invading front, moving at the same speed. In the inset we show the development of the widths of the invading front ($w_f$, green circles) and of the film flow active zone ($w_{az}$, black crosses). While the invading front maintains a roughly constant width throughout the experiment, the width of the active zone develops for a much longer time.

Figure 10

Temporal evolution of the (a) active zone and (b) invading front width, as measured from its standard deviation with respect to the mean position. Measurements for $0^{\circ }$, ${10}^{\circ }$, ${20}^{\circ }$, and ${30}^{\circ }$. The higher the gravitational component (higher angles), the more compact the active zone and invading front are (smaller width). In the absence of the stabilizing effect from gravity, breakthrough happens at an early stage in the experiment with $0^{\circ }$, which is reflected in the sharp increase in the front width.

Figure 11

Probability distribution of film flow events computed in the reference frame of the moving front. The measured data is shown for three inclination angles (see legend). Notice that apart from a few tail events, most of the film flow drainage occurs inside an active zone in the vicinity of the front. The position of the front is marked by the vertical dashed red line at the origin and the film flow active zone follows its motion.

Figure 12

Histogram showing the spatial distributions of capillary bridges. In (a) we show the essentially constant distribution inside subwindows in the inlet-outlet direction, as a function of the distance $y$ to the inlet. In the inset we show the orientation of the $y$ axis with respect to the model and a subwindow (made approximately three times taller than the ones used in the analysis, to aid visualization). In (b) we show the distribution in the perpendicular direction. Again, apart from fluctuations and boundary effects, the distribution is essentially constant.

Figure 13

Air saturation in width-spanning subwindows for an experiment with ${30}^{\circ }$ inclination. The subwindows are located at fixed positions with respect to the inlet for different times (shown in the legend). In the inset we can see that the increment in saturation due to late film flow effects can be very significant, amounting to almost $10\%$ of the total saturation in the subwindow.