Interface pinning of immiscible gravity-exchange flows in porous media

We study the gravity-exchange flow of two immiscible fluids in a porous medium and show that, in contrast with the miscible case, a portion of the initial interface remains pinned at all times. We elucidate, by means of micromodel experiments, the pore-level mechanism responsible for capillary pinning at the macroscale. We propose a sharp-interface gravity-current model that incorporates capillarity and quantitatively explains the experimental observations, including the $x\sim t^{1/2}$ spreading behavior at intermediate times and the fact that capillarity stops a finite-release current. Our theory and experiments suggest that capillary pinning is potentially an important, yet unexplored, trapping mechanism during CO${}_2$ sequestration in deep saline aquifers.

Figure 1

Lock-exchange flow in a porous medium with (a) miscible and (b) immiscible fluids. (a) The miscible fluids are water (blue) spreading over a denser, more viscous mixture of glycerol and water. A smooth macroscopic interface tilts around a stationary point with fixed height $h_s$. (b) The immiscible fluids are air (dark) spreading over the same glycerol-water mixture. Part of the initial interface remains pinned, which leads to sharp kinks or “hinges” in the macroscopic interface. We denote the height of the lower hinge by $h^{\prime }$. Both experiments were conducted in a transparent cell packed with 1 mm glass beads. The red curves correspond to the predictions of sharp-interface models.

Figure 2

Experimental setup of the lock-exchange experiments in quasi-two-dimensional flow cells packed with glass beads.

Figure 3

Bead size distribution for two of the nominal bead sizes used in our experiments: (a) bead size 300 to 425 $\mu$m, (b) bead size 425 to 600 $\mu$m. We take the arithmetic mean of the endpoints of each range as the characteristic grain size $d$.

Figure 4

Measurements of the advancing contact angle (left) and receding contact angle (right) on a glass surface for the different fluid pairs used in the experiments: (a) silicone oil, (b) propylene glycol, (c) glycerol-water mixture. The glycerol-water mixture and propylene glycol are both partially wetting to glass with respect to air, and exhibit contact angle hysteresis. Silicone oil is perfectly wetting to glass with respect to air, having advancing and receding contact angles of $0^{\circ }$.

Figure 5

Scaling of the pinned interface height in an immiscible lock-exchange flow. We measure the deviation of the immiscible lock exchange from the miscible lock exchange via the normalized difference between the lower hinge height $h^{\prime }$ and the height of the miscible tilting point, $h_s$ (inset), at the same value of $\mathcal{M}$. This quantity scales linearly with the strength of capillarity relative to gravity, as measured by the inverse of the Bond number, up to a point when the entire interface is pinned ($h^{\prime }=0$). Here we show experimental measurements (symbols) and a best-fit straight line (solid black line).

Figure 6

(a) Presence of a pinned interface in lock-exchange flow. Due to hysteresis effects, the capillary pressure at the drainage front ($P_c^{\mathrm{dr}}$) is larger than the capillary pressure at the imbibition front ($P_c^{\mathrm{imb}}$). Along the pinned vertical interface, the capillary pressure transitions from $P_c^{\mathrm{imb}}$ at the bottom to $P_c^{\mathrm{dr}}$ at the top. (b) Snapshot of the pinned interface of an immiscible lock exchange experiment (air/silicone oil) in a thin acrylic cell with a regular pattern of cylindrical posts simulating the pore-scale microstructure of a porous medium. The increase in capillary pressure (drop in wetting fluid pressure) from left to right is visible via the decrease in the radius of curvature along the interface. This increase in capillary pressure along the pinned interface is offset by the drop in hydrostatic pressure. (c) The solid red line shows a simple interpretation of the wetting phase pressure along the interface, as a function of elevation $z$. We assume the pressure in the air is atmospheric at all times. The wetting-phase interface pressure $P_w^{\mathrm{I}}$ can be calculated by subtracting the capillary pressure from the air pressure ($P_w^{\mathrm{I}}=P_{\mathrm{atm}}-P_c$). It is constant along the active drainage and imbibition fronts, and varies hydrostatically along the pinned interface. (d) Steady state interface configuration on the drainage side (solid red) and imbibition side (solid blue) of the pinned interface, along with the interface progression on the imbibition side (dashed blue), for a perfectly wetting imbibing fluid. Capillary pressure hysteresis is completely controlled by the pore geometry, due to the fundamental differences in the pore-level invasion events between drainage and imbibition, as a result, it is apparent even in the absence of contact angle hysteresis.

Figure 7

Interface configuration (solid blue) on the imbibition side of the exchange flow. We solve for the progression of the interface arcs as they advance through the pore by enforcing pressure continuity between posts 1,2 and 3,4 (i.e., $r_1=r_2$). The interfaces merge and advance to the next set of posts when the two interface arcs touch (i.e., ${\varphi }_1+{\varphi }_2=3/4\pi$). The diameter of the posts is $d$, and each post is one diameter away from its closest neighbors, the same as the design of the micromodel [Fig. 6b].

Figure 8

Dependence of the dimensionless quantity $d/r_{\mathrm{dr}}-d/r_{\mathrm{imb}}$ controlling the magnitude of the pinned interface in our micromodel system, as a function of advancing contact angle ${\theta }_a$, assuming ${\theta }_r=0^{\circ }$.

Figure 9

We develop a sharp interface model for the exchange flow between a buoyant, nonwetting fluid and a dense, wetting fluid, separated initially by a vertical interface. We assume Darcy flow within each fluid and hydrostatic pressure distribution everywhere. We then solve for the model by enforcing conservation of volume in a control volume (dashed blue). The resulting model is a partial differential equation for the height of the interface as a function of time.

Figure 10

The dimensionless nonlinear diffusion coefficient $g(h;\mathcal{M})/H\equiv (1-f(h;\mathcal{M}\left)\right)h/H$ in Eq. (9) as a function of the dimensionless height of the interface $h/H$ for several values of viscosity ratio $\mathcal{M}$.

Figure 11

Time evolution of the nose positions of the (a) buoyant nonwetting current, and (b) dense wetting current, measured relative to the position of the initial vertical interface. We scale nose position by the cell height $H$ and time by the characteristic time $T=H/\kappa$. We show the data for five experiments with different values of ${\mathrm{Bo}}^{-1}$ (black symbols: $\times =0.021$; $*=0.028$; $\square =0.029$; $▵=0.041$). For a particular experiment (${\mathrm{Bo}}^{-1}=0.052$; $\mathcal{M}=2580$), we compare the nose positions from the experiment (red circles) with those computed with our model (red solid line). We also show nose positions from the numerical solution to the miscible flow model ($[[h_c]]\to 0$; dashed line).

Figure 12

Finite release of a buoyant, nonwetting fluid (air) in a porous medium filled with a dense, wetting fluid (silicone oil). The dense, wetting front hits the left boundary, changing the spreading behavior of the buoyant current. Capillary hysteresis is responsible for the pinning of the initial interface and, ultimately, for stopping the buoyant plume at a finite distance, in stark contrast with a miscible plume which would continue to spread indefinitely.