Symmetric wetting heterogeneity suppresses fluid displacement hysteresis in granular piles

We investigate experimentally the impact of heterogeneity on the capillary pressure hysteresis in fluid invasion of model porous media. We focus on symmetric heterogeneity, where the contact angles the fluid interface makes with the oil-wet (${\theta }_1$) and the water-wet (${\theta }_2$) beads add up to $\pi$. While enhanced heterogeneity is usually known to increase hysteresis phenomena, we find that hysteresis is greatly reduced when heterogeneities in wettability are introduced. On the contrary, geometric heterogeneity (like bidisperse particle size) does not lead to such an effect. We provide a qualitative explanation of this surprising result, resting on rather general geometric arguments.

Figure 1

(a) Sketch of the experimental apparatus. The sample cell is sealed at the top and the bottom with an oleophilic (om) and a hydrophilic (hm) membrane, respectively. This prevents the fluid interface from leaving the cell. The hydrostatic pressure difference between water and oil is controlled by the height of the water reservoir using the translation stage. The mass of the displaced oil (and thereby the water saturation) is measured by a high-precision balance. (b) Paradigmatic saturation curve. Open triangles show the first run and closed triangles all consecutive runs.

Figure 2

Matric suction measured with samples of mixed-wet beads of equal size. The scaled quantity $p_{s}R/\gamma$ is plotted on the ordinate; the abscissa is ${\varphi }_{\mathrm{oi}l}={\varphi }_2=({\mathrm{\Phi }}_2-{\mathrm{\Phi }}_1)/2\mathrm{\Phi }+\frac{1}{2}$. Closed circles show $R=116\pm 9\mu \mathrm{m}$ and open circles $R=195\pm 16\mu \mathrm{m}$. The inset shows a log-log plot of $p_s$ vs the bead size, measured with monodisperse samples of water-wet beads. The straight line has a slope of unity.

Figure 3

(a) Hysteresis measured with bidisperse piles of oil-wet (open symbols) and water-wet (closed symbols) beads, as a function of the relative fraction $\varphi$ of large beads. Within scattering, the data sets are almost in parallel, the decreasing trend stemming from the change in average length scale [small beads and high pressures (left-hand side) to large beads and low pressures (right-hand side)]. (b) Hysteresis as a function of the capillary pressure scale $\gamma /\langle R\rangle$. Open symbols show monodisperse beads of variable size and closed symbols bidisperse mixtures of beads: circles, $R_1=116\mu \mathrm{m}$ and $R_2=195\mu \mathrm{m}$; triangles, $R_1=58\mu \mathrm{m}$ and $R_2=231\mu \mathrm{m}$. The dotted line with a slope of unity represents what would be expected if the bead size were uniformly varying between the largest (left) and the smallest (right) size.

Figure 4

Hysteresis measured with bidisperse piles of beads, as a function of the relative fraction $\varphi$. (a) Mixtures of beads of different size ($R_1=116\pm 9\mu \mathrm{m}$ and $R_2=195\pm 16\mu \mathrm{m}$). Closed symbols show beads of equal wettability and open symbols beads of different wettability: squares, large water-wet beads and small oil-wet beads; diamonds, large oil-wet beads and small water-set beads. (b) Mixtures of beads of different wettability. Squares and diamonds indicate data as above; upward pointing triangles, large beads only; downward pointing triangles, small beads only. For an explanation of the dashed curve, see the text.

Figure 5

(a) Interface (solid) between two fluids (1 and 2) near a contact between two beads. Shown on the left is contact between two beads wettable by fluid 1. The latter is likely to form a capillary bridge between both beads, with fluid 2 as the surrounding phase. Shown on the right is contact between two beads of different wettability. A capillary bridge is not likely to form. For the symmetric case ${\theta }_i=\pi /2\pm \eta$ (as for our system), there is not even an axisymmetric solution. Instead, the interface goes flat through the contact point, making an angle $\eta$ with the line connecting the centers of the beads (dash-dotted line). (b) Thresholds for site percolation (closed symbols) and bond percolation (open symbols) for a range of ordered (squares) and random (circles) lattices (see [26] and references therein).