Quasistatic fluid-fluid displacement in porous media: Invasion-percolation through a wetting transition

We study the influence of wettability on the morphology of fluid-fluid displacement through analog porous media in the limit of vanishing flow rates. We introduce an invasion-percolation model that considers cooperative pore filling and corner-flow mechanisms and captures interface motion at the pore scale for all quasistatic flow regimes between strong drainage and strong imbibition. We validate the method against recent experimental observations of wetting transition in microfluidic cells patterned with circular posts and we use it to explore the sensitivity of fluid invasion to wettability heterogeneity, post spacing, and post height. Our model therefore extends the Cieplak-Robbins description of quasistatic fluid invasion by reproducing the wetting transition in strong imbibition, a feature that requires incorporating three-dimensional effects.

Figure 1

(a) Invasion front configuration between two posts. (b) Burst event: An unstable interface (red line) advances into the pore. (c) Touch event: The interface touches the nearest post. (d) Overlap event: Two fronts (green lines) coalesce on the post surface and fill the pore. (e) Corner-flow event: The corner meniscus touches and coats the neighboring post. (f) Capillary bridge event: The corner menisci coalesce mid-post before reaching the next post. (g) Invading front configuration with post IDs: Red, blue, and green interfaces correspond to burst, touch, overlap critical interfaces, respectively.

Figure 2

(a) Schematic diagram of meniscus coalescence away from the post surface. (b) Drainage overlap at $\theta >{90}^{\circ }$: Menisci coalesce inside the pore space, leaving trapped oil on the wall of the invaded post.

Figure 3

Generated pore geometry along with post diameter and pore throat size histograms. Posts were placed on irregular triangular lattice generated with matlab's $\mathsf{pdemesh}$ and post radii were assigned to 45% of the smallest connected edge.

Figure 4

Immiscible fluid invasion simulation results: The algorithm presented in Sec. 2 covers the full range of pore wettabilities, from strong drainage ($\theta ={160}^{\circ }$) to strong imbibition ($\theta ={10}^{\circ }$). Dark blue regions represent fully invaded pores; light blue regions represent partially invaded pores with coated post corners. We include video files of the invasion process at different wettabilities in the Supplemental Material [38].

Figure 5

Cooperative pore-filling and fractal dimension plots, showing invasion patterns for different realizations at each fixed global contact angle. As the system moves from strong drainage ($\theta ={160}^{\circ }$) to weak imbibition ($\theta ={45}^{\circ }$), the percentage of cooperative pore-filling events gradually increases. The transition from weak to strong imbibition is marked with a sharp drop in both fraction of cooperative pore-filling events and fractal dimension.

Figure 6

(a) Schematic diagrams of the drainage overlap and overlap event of Cieplak and Robbins [23, 24]. (b) Percentage of cooperative pore-filling events for posts spaced out by a factor of $\lambda$ from the original post geometry. The colored circles represent simulations with drainage overlap considered; solid lines represent simulations with the original overlap definition of Cieplak and Robbins [23, 24]. (c) Schematic diagram of post spacing and the fluid-fluid interface.

Figure 7

Figure showing how post heights alter the shapes of the invasion fronts: The onset of the corner-flow-dominated regime is heavily influenced by the out-of-plane curvature correction of the burst, touch, and overlap instabilities. The magnitude of the correction is controlled by the height of the posts. Invasion fronts on the right are plotted for different $h$ at $\theta ={10}^{\circ }$.

Figure 8

Shape of the corner meniscus around the post with radius $r_1$. Here $AM$ and $AN$ are the principal radii of curvature at point $A$, where $AM$ is in plane and $AN$ is perpendicular to the plane.

Figure 9

Interface shape of a corner meniscus outside a post. Equation (A7) is solved with $\theta ={40}^{\circ }$, $r_1=500\mu \text{m}$, and $r_n=1200\mu \text{m}$. Here $\mathrm{\Delta }{p}_f$ represents the Laplace pressure obtained from the force balance equation (7). The condition $\dot{x}\left(r_n\right)=tan{\theta }_1$ is exactly satisfied when $\mathrm{\Delta }p=\mathrm{\Delta }{p}_f$.

Figure 10

Laplace pressure for a growing corner meniscus. Initially, the invading liquid is confined to the corners. As the horizontal extent of the liquid grows, the top and bottom corners meet at the mid-height of the post and the shape changes into a capillary bridge. This transition point corresponds to the negative jump in Laplace pressure. (a) Evolution of the corner menisci for contact angles between $4^{\circ }$ and ${44}^{\circ }$ and $r_1=500\mu \text{m}$. (b) Evolution of the corner menisci for post radii between 300 and $800\mu \text{m}$ and a contact angle of ${40}^{\circ }$.

Figure 11

Fractal dimensions calculated with the box-counting method: (a) color image ($1200\times 1200$ pixels) produced by the invasion algorithm for $\theta ={40}^{\circ }$, (b)–(d) black and white versions of the invasion pattern placed on a grid of size $\epsilon =[1,n_{\text{pix}}]$, and (e) fractal dimension measured as a slope of number $N$ of filled boxes against $\epsilon$ on a log-log plot, with the slope calculated from points with $\epsilon =[1,\frac{n_{\text{pix}}}{8}]$.

Figure 12

Ratio of the invading pattern finger width to mean pore size, estimated in analogy to the work of Cieplak and Robbins [23, 24].