Flow geometry controls viscous fingering

In the early 20th century, petroleum engineers noticed that water pumped into porous, oily rock does not displace the oil uniformly but instead penetrates it through a process now known as viscous fingering—the tentaclelike flow of one fluid into another. When Philip Saffman and G. I. Taylor analyzed the problem in 1958, they realized that instabilities at the fluids’ interface naturally emerge as long, propagating fingers. Small perturbations at an otherwise flat interface create local pressure gradients that force the less viscous fluid into regions just ahead of each perturbation. As the fluid speed locally increases, so do the pressure gradients, and the positive feedback drives the fingers’ growth.

Saffman and Taylor also realized that they could capture the essential physics of the flow in porous rock using a two-dimensional channel known as a Hele–Shaw cell, in which the different fluids are confined between two parallel glass plates separated by a small gap. Although an imperfect analogue for porous media, the cell offers a simple framework for visualizing the rich set of patterns that can appear (see figure 1 and the article by Thomas Halsey in Physics Today, November 2000, page 36 ). Further, the cell’s planar geometry and the viscous nature of the flow within it allow researchers to use a simplified set of equations derived from the more formidable 3D Navier–Stokes equations. And ever since 1958, the Hele–Shaw cell has taken on a life of its own in studies of viscous fingering in narrow passages, ¹ a phenomenon that arises, it turns out, in many real-world systems beyond the oil well.

Convenient as the framework is, though, the parallel-plate configuration may be too limiting. Whether the passages consist of pores in rock, airways in the lungs, or microfluidic channels, they’re rarely uniform. While working on the breakup of droplets through obstructed microfluidic channels last year, Princeton University’s Howard Stone wondered how one might design a channel’s shape in order to control—in some

cases suppress, in others trigger—the emergence of finger-forming instabilities.

After all, viscous fingers are a blessing or a curse depending on the application. They drastically reduce the efficiency of oil recovery if they pervade a well. But the fingers also improve mixing in fluidic devices whose channels are small enough, or their flows slow enough, that turbulence is nonexistent (see Physics Today, January 2012, page 72 ).

In a study combining theory with experiments, Stone, Talal Al-Housseiny, and Peichun Amy Tsai (now at the University of Twente) found that the simplest adjustment to the Hele–Shaw cell—introducing a tiny gradient in channel depth—could lead to fundamentally different displacement behavior. ² To see how, consider the case of gently converging plates as illustrated in figure 2. As the gap narrows, the tip of an incipient viscous finger experiences an increased surface-tension force—or more precisely, greater capillary pressure—that opposes the finger’s advance. The effect is to reduce the pressure gradient and thus damp out the initial perturbation.

Stone’s team wasn’t the first to introduce a taper into the Hele–Shaw cell. Researchers at the University of Pittsburgh in Pennsylvania had done so 20 years ago, and that work recently prompted Eduardo Dias and Jose Miranda from the Federal University of Pernambuco in Brazil to theoretically explain the taper’s effect on the fingers’ width and branching behavior with flow speed. ³ In both studies, the focus was on the nonlinear evolution of the interfacial morphologies.

Stone and company, in contrast, focused entirely on the task of calculating the conditions required to trigger or prevent the initial onset of the instability. To do so, they recast the Young–Laplace equation, which describes the pressure drop across the interface, to include changing boundary conditions seen by the advancing finger. The equation thus became the sum of two opposing terms: One accounts for the lateral curvature (the perturbation) that drives the fingers’ growth, and the other accounts for the curvature due to depth, which can inhibit it.

For two immiscible fluids, the researchers derived a stability condition set by just four parameters: the angle of taper, the fluids’ wetting angle, the viscosity ratio, and the capillary number. As the ratio of viscous forces to surface-tension forces, the capillary number varies with the flow speed, which turns out to be a convenient experimental knob to turn. In an air–oil experiment, the team established a critical speed—about 0.05 cm/s for a slight taper of 10⁻³ radians, or a decrease of 1.5 mm in the cell’s depth over its 50-cm length—for the transition between stable and unstable displacements. The result is in good agreement with prediction.

As for applications, Stone and Al-Housseiny envision that low-viscosity solvents can better flush out contaminants in a sensor whose channels are subtly tapered and whose flow rate is kept below a critical speed. The results may also bear on oil recovery, but that connection is more tenuous. The efficiency may increase, they argue, if injection and production wells were strategically positioned to sweep oil in the direction of decreasing permeability—through regions whose pore sizes become progressively smaller.