Drainage in two-dimensional porous media: From capillary fingering to viscous flow

This paper reports some experimental results on two-phase flows in model two-dimensional porous media. Standard microfluidic techniques are used to fabricate networks of straight microchannels and to control the throat size distribution. We analyze both the invasion mechanism of the medium by a nonwetting fluid and the drainage after the percolation for capillary numbers lying between ${10}^{-7}$ and ${10}^{-2}$. We propose a crude model allowing a description of the observed capillary fingering that captures its scaling properties. This model is supported by numerical simulations based on a pore-network model. Numerical simulations and experiments agree quantitatively.

Figure 1

Example of a mask used to grid $\left(\sigma /l=0.125\right)$. The real size of the porous domain (the square lattice) is 1.3 cm.

Figure 2

Example of data analysis for an experiment made in grid a with ${\text{Ca}}_i=4.3\times {10}^{-7}$. The gray scale represents the time at which the nodes have been invaded (the darker the sooner), and the small lines indicate the path followed by the invading fluid.

Figure 3

The injected fluid is fluorinated oil; the captured fluid is water. Experiments are performed on a hydrophilic PDMS grid (grid a). Top figure: evolution of the captured fluid saturation as a function of time multiplied by the applied capillary number ${\text{Ca}}_i$. From top to bottom ${\text{Ca}}_i=9\times {10}^{-7}$ (1), $1.85\times {10}^{-5}$, $2.4\times {10}^{-4}$ (2). Bottom figures: typical pictures during the invasion process at two different applied capillary numbers (labeled 1 and 2 in the top figure). For the sake of clarity, the contrast of the picture has been inversed. The darkest fluid is the injected fluid.

Figure 4

Variations of the measured mean local capillary number $⟨{\text{Ca}}_l⟩$ as a function of the imposed one ${\text{Ca}}_i$. The full lozenges (◆) correspond to experiments where the water-glycerin mixture is pushing dodecane in grid b, the empty lozenges (◇) correspond to water pushing dodecane in grid b, and the triangles (△) correspond to fluorinated oil pushing water in grid a. The full line corresponds to $⟨{\text{Ca}}_l⟩={\text{Ca}}_i$. The dotted lines correspond to $⟨{\text{Ca}}_l⟩=A\sqrt{{\text{Ca}}_i}$, where the numerical constant $A$ equals from the top to the bottom to $1.8\times {10}^{-2}$ and $5\times {10}^{-3}$.

Figure 5

Variation of the residual oil saturation at the percolation $S_{o_r}$ as a function of the applied capillary number ${\text{Ca}}_i$. Data correspond to the water-dodecane system in grid b.

Figure 6

Maximum size (in the pressure gradient direction) of remaining oil clusters as a function of ${\text{Ca}}_i$ for the water-dodecane system in grid b. The size is reported in numbers of junctions. The line corresponds to a slope of $-1/2$. Note that the data dealing with number of $\lambda$ greater than 20 correspond to huge clusters where finite-size effect could play an important role.

Figure 7

Scheme of the pressure field in the case of partial wetting. The colored phase is the invading fluid.

Figure 8

Modified local capillary number $⟨{\widetilde{\text{Ca}}}_l⟩=⟨{\text{Ca}}_l⟩/\Gamma ϵ\left|\text{cos}{\theta }_a\right|$ as a function of the imposed one ${\widetilde{\text{Ca}}}_i$. The numerical results (small circles) are plotted together with the experimental data reported in Fig. 4 (big symbols). The dashed line is $⟨{\widetilde{\text{Ca}}}_l⟩={\widetilde{\text{Ca}}}_i$, holding for the viscous regime; the solid line is $⟨{\widetilde{\text{Ca}}}_l⟩=0.27\sqrt{{\widetilde{\text{Ca}}}_i}$, holding for the capillary regime; and the dashed-dotted line is $⟨{\widetilde{\text{Ca}}}_l⟩=N_w{\widetilde{\text{Ca}}}_i$ holding for the finite-size regime at ultralow ${\text{Ca}}_i$. The simulation parameters are ${\text{Ca}}_i$ (varied between ${10}^{-8}$ and ${10}^{-2}$), $ϵ$ (varied between $5\times {10}^{-3}$ and 0.1), $\Gamma$ (varied between ${10}^{-2}$ and 0.1), $M$ (varied between 0.7 and 10), $\text{cos}{\theta }_a=-1$, and $N_w=50$.

Figure 9

Scheme of a trapped cluster. The colored fluid is the injected fluid. The arrows indicate the flow direction.