Foam-driven fracture

Significance

Hydraulic fracturing plays an important role in meeting today’s energy demands. However, the substantial use of fresh water in fracturing and wastewater returning to the surface pose risks to the environment. Alternative technology has been developed that reduces the water-related risks by injecting aqueous foam instead of water to fracture shale formations, but the mechanism is poorly understood. Here, we show, using laboratory experiments, that the injection of foam instead of water dramatically changes the fracture dynamics when the foam compressibility is important. We develop a scaling argument for the fracture dynamics that exhibits excellent agreement with the experimental results. Our findings extend to other systems involving compressible foams, including fire-fighting, energy storage using compressed foams, and enhanced oil recovery.

Abstract

In hydraulic fracturing, water is injected at high pressure to crack shale formations. More sustainable techniques use aqueous foams as injection fluids to reduce the water use and wastewater treatment of conventional hydrofractures. However, the physical mechanism of foam fracturing remains poorly understood, and this lack of understanding extends to other applications of compressible foams such as fire-fighting, energy storage, and enhanced oil recovery. Here we show that the injection of foam is much different from the injection of incompressible fluids and results in striking dynamics of fracture propagation that are tied to the compressibility of the foam. An understanding of bubble-scale dynamics is used to develop a model for macroscopic, compressible flow of the foam, from which a scaling law for the fracture length as a function of time is identified and exhibits excellent agreement with our experimental results.

The flow of compressible aqueous foam has a broad range of applications, such as fire-fighting (1–3), compressed air energy storage (4), materials processing (5), and enhanced oil recovery, where the injection of foam instead of water (6–12) suppresses viscous fingering at the fluid–fluid interface. In hydraulic fracturing (13, 14), water is injected at high pressure to crack shale formations, releasing trapped oil and natural gas. Alternative techniques using foams as injection fluids have been developed to reduce the water use of conventional hydrofracture (15–17). Here we show that when a foam is injected at high pressures to fracture an elastic medium, the foam compressibility produces a time-dependent flow that controls the dynamics of fracture propagation.

Although steady flow of foam in pipes (3, 10, 18–22) and 2D channels (6, 23) has been studied extensively, time-dependent foam flows resulting from the compressibility of bubbles are poorly understood. Here, we quantify these unsteady flows using one-dimensional model experiments, which we rationalize using mechanical principles. Using these results, we develop scaling relations for the propagation of foam-driven brittle fractures that are in quantitative agreement with our experiments.

A Qualitative Observation for Foam-driven Fractures

We fracture an elastic solid matrix by injecting into it an aqueous foam from a syringe, through a tube and needle (Fig. 1A). The elastic matrix is chosen to be gelatin since it models the brittle and elastic properties of rocks (24–29) and allows us to visualize the fracture dynamics (24, 26, 27, 30–32). We use Gillette® Foamy shaving foam in our experiments due to its well-known and robust properties (33–36). Initially, the syringe, tube, and needle are filled with foam that is at equilibrium with atmospheric pressure $p_{\infty }$. During the experiment, the volume of the syringe decreases with time at a rate $Q$. Initially, the injection causes the foam in the entire system to be compressed. Once the foam in the syringe reaches a certain volumetric strain ($\approx$ 30$\%$ for the experiment shown in Fig. 1 C and D), the foam fractures the elastic matrix in a lens shape (37) [also referred to as a penny shape (38)] and propagates along a plane (Fig. 1 C and D). We are interested in the growth of the crack radius $R\left(t\right)$ for a constant volumetric rate of injection $Q$. For an incompressible flow, the volume of the crack $V$ grows as $Qt$. However, we find that the dynamics of the crack growth are altered by the compressibility of the foam, as we discuss below.

Fig. 1.

(A) Schematic of the experimental setup. Foam is injected from a syringe (initial volume $V_0=25$ mL) through a tube (radius, 0.89 mm; length, 0.32 m; initial volume, 0.8 mL) and needle (radius, 1.08 mm; length, 0.11 m; initial volume, 0.4 mL) into an elastic gelatin matrix. (B) A microscopic view of the foam (Gillette^® Foamy), whose constituents include water and hydrocarbon gases. The bubble radii range from 6 to 47 $\mathrm{\mu }$m (polydisperse). The growth of a lens-shaped crack driven by foam injection is observed from both (C) top and (D) side. In this experiment, $t_0=40$ s, where $t_0$ denotes the time at which foam first enters the elastic matrix.

The pressure drop along the tube and needle can be estimated from a balance of stresses in the foam. The pressure gradient along the foam flow balances the shear stress gradient across the tube; the inertia of the foam is negligible for the injection rates studied here. Therefore, the shear stress at the tube wall is ${\tau }_w=-\frac{a}{2}\frac{\partial p}{\partial z}$, where $z$ is in the flow direction and $a$ is the tube radius. For smooth walls and ${\tau }_w$ smaller than the yield stress, the foam moves as a plug with velocity $u$, lubricated by a thin film of liquid with viscosity $\mu$ near the wall (39). Denkov et al. (39) showed that ${\tau }_w\approx \kappa {\left(\mu \gamma u\right)}^{1/2}/\left(2\overline{R}\right)$, where $\gamma$ is the interfacial tension and $\overline{R}$ is the mean radius of the bubbles (40). Here, $\kappa$ is a dimensionless resistance that depends on the liquid volume fraction $ϵ$ of the foam and is related to the fractional area of the tube wall wetted by liquid films (39, 41, 42) (see SI Appendix). Combining these approximations, the average velocity of foam in the tube of length $\ell$ obeys

$$u=u_D{\left(\frac{\partial \left(p/p_{\infty }\right)}{\partial \left(z/\ell \right)}\right)}^2,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{u}_D=\frac{{\left(\overline{R}a{p}_{\infty }\right)}^2}{{\gamma \mu \left(\kappa \ell \right)}^2}$$ [1]

is the typical foam velocity when the pressure drop along the tube is on the order of atmospheric pressure $p_{\infty }$—that is, $\partial \left(p/p_{\infty }\right)/\partial \left(z/\ell \right)=\mathcal{O}\left(1\right)$. Relative motion between liquid and bubbles [i.e., drainage (43–45)] is negligible for the pressure gradients used in our experiments.

To estimate the pressure drop along the tube, we measured the liquid properties, $\gamma =29\pm 2$ mN/m and $\mu =3\pm 1$ mPa$\cdot$s, after separating the liquid phase using a centrifuge (see SI Appendix). We measured the mean bubble radius $\overline{R}=12\pm 1\mathrm{\mu }$m using optical microscopy (e.g., Fig. 1B). We weighed the foam at $p_{\infty }$ to determine its liquid volume fraction ${ϵ}_{\infty }\approx 0.1$, and note that $\kappa \left(ϵ={ϵ}_{\infty }\right)\approx 4$ (see SI Appendix). For a typical measured foam velocity in the tube $u\approx 1.5$ cm/s, we estimate using Eq. 1 that the typical pressure drop along the injection tube is $\mathrm{\Delta }{p}_t\approx {\left(\gamma \mu u\right)}^{1/2}\kappa \ell /\left(\overline{R}a\right)\approx 1.8\times 10^5$ Pa. We measured the foam pressure during the fracturing experiment and also found the typical foam pressure drop along the tube to be $\mathcal{O}\left(10^5\right)$ Pa (see SI Appendix). Since $\mathrm{\Delta }{p}_t=\mathcal{O}\left(p_{\infty }\right)$, compressibility effects are important. The foam does not coarsen significantly for the range of pressures in our experiments (see SI Appendix).

When a foam-driven fracture of a typical radius $R\approx 20$ mm and thickness $2W\approx 8$ mm is generated in the gelatin matrix (Young’s modulus $E\approx 66$ kPa, Poisson’s ratio $\sigma \approx 0.5$), the typical elastic stress due to the elastic deformation around the fracture is $\mathrm{\Delta }{p}_e\approx EW/\left(2\left(1-{\sigma }^2\right)R\right)\approx 8.8\times 10^3$ Pa (37, 46, 47). Since $\mathrm{\Delta }{p}_t\gg \mathrm{\Delta }{p}_e$, the stresses related to fracture formation are negligible compared with the pressure drop in the foam along the tube. Therefore, to better understand the dynamics, the foam flow can be modeled with a one-dimensional experiment where a tube of length $\ell$ and radius $a$ is connected to a syringe. The outlet of the tube, rather than connecting to an elastic matrix, is directly exposed to atmospheric pressure (Fig. 2A). Note that $\ell$ represents the combined length of the tube and needle in the fracture experiments (Fig. 1A).

Fig. 2.

(A) Foam flow in a tube of length $\ell$ and radius $a$. The tube inlet connects to a syringe filled with foam (volume $V_0$), and the tube outlet is exposed to atmospheric pressure. The syringe pump reduces the syringe volume with a constant injection rate $Q$. Initially no foam is observed to exit the outlet of the tube, and the foam in the entire system is compressed. At $t_0$, foam exits the tube outlet. (B) The volume of foam collected at the outlet of the tube $V$ is measured as a function of time for different $Q$, $\ell$, $a$, $V_0$, $\mu$, and ${ϵ}_{\infty }$. The experimental parameters are shown in Table 1. Two flow regimes are observed. When $\phi \ll 1$ (Expt. J), where $\phi$ is defined in Eq. 2, $V$ approaches the steady-state incompressible results, $V=Q\left(t-t_0\right)$, as shown by the dashed line. When $\phi =\mathcal{O}\left(1\right)$, foam compressibility affects the flow and a nonlinear dependence of $V$ on $t$ is observed (experiments A–I). (C) The dimensionless volume $\mathcal{V}$ versus dimensionless time $\tau$ for the fast-injection experiments (experiments A–I) collapses onto a universal curve. For simplicity, we fit a power-law function to the dimensionless curve $\mathcal{V}\left(\tau \right)$, as shown by the solid line (Eq. 3).

A One-Dimensional Model Experiment

In the one-dimensional model experiment (Fig. 2A), the entire tube and the syringe are filled with foam at atmospheric pressure, and the syringe is compressed at a volumetric rate $Q$ (see Table 1). As with the fracture experiment, no foam flow is observed at the tube outlet until time $t_0$. To characterize the flow, we measure the volume of foam collected at the outlet of the tube as a function of time $V\left(t\right)$ (Fig. 2B). The experiment ends when the foam in the syringe is completely injected into the tube. Experimental parameters are summarized in Table 1.

Table 1.

Experimental parameters for the one-dimensional model experiments (experiments A–K) shown in Fig. 2

Experiment	$Q$, mL/min	$\ell$, m	$a$, mm	$V_0$, mL	$\mu$, mPa$\cdot$ s	${ϵ}_{\infty }$	Fluid
A	10	0.43	1.19	25	3	0.1	Foam
B	10	0.43	0.89	25	3	0.1	Foam
C	20	0.43	0.89	25	3	0.1	Foam
D	30	0.43	0.89	25	3	0.1	Foam
E	10	0.43	0.89	15	3	0.1	Foam
F	10	0.43	0.89	37	3	0.1	Foam
G	10	1.20	0.89	25	3	0.1	Foam
H	10	0.43	0.89	25	9	0.2	Foam
I	10	0.43	0.89	25	3	0.2	Foam
J	0.5	0.01	0.89	25	3	0.1	Foam
K	0.5	0.01	0.89	25	1	—	Water

We varied the injection flowrate Q, the tube length $\ell$ and radius a, the initial volume $V_0$ of the foam-filled syringe, the viscosity $\mu$ of the liquid in the foam, and the liquid volume fraction ${ϵ}_{\infty }$ of the foam.

For incompressible flows, mass conservation necessitates that $V\left(t\right)=Q\left(t-t_0\right)$ (see experiment K and the dashed line in Fig. 2B). However, in the foam experiments (experiments A–J), we observed $V\left(t\right)$ to be significantly different from that of incompressible flows. Since $Q$ is constant, the nonlinear dependence of $V$ on time (Fig. 2B) indicates that the foam is compressed throughout the experiment. The experimental results vary with $Q$, $\ell$, $a$, and the initial volume of the syringe $V_0$. We also change the foam properties, in particular $\mu$ and ${ϵ}_{\infty }$, by mixing glycerol and water with the foam. The effects of both $\mu$ and ${ϵ}_{\infty }$ on the experimental results are negligible, as shown by experiments B, H, and I in Fig. 2B.

Below, we use physical arguments to identify the important dimensionless groups and rationalize our experimental observations. Assuming that $ϵ$ is uniform throughout the syringe, liquid mass conservation requires that $\mathrm{d}\left[ϵ\left(V_0-Qt\right)\right]/\mathrm{d}t+ϵ\pi a^{2}u=0$ at the tube inlet. The first term represents the rate of change in the mass of foam in the syringe due to the injection. The second term is the mass flow rate of foam vented into the tube. The first term gives a characteristic time of injection $t^{*}=V_0/Q$ over which relative changes of $ϵ$ are of $\mathcal{O}\left(1\right)$. Then, liquid mass conservation inside the tube, $\partial ϵ/\partial t+\partial \left(ϵu\right)/\partial z=0$, establishes a characteristic foam speed $u^{*}=\ell /t^{*}=\ell Q/V_0$. The ratio between the characteristic foam speed $u^{*}$ in the tube and the speed $u_D$, at which compressibility effects dominate (Eq. 1), defines a dimensionless injection rate

$$\phi \equiv \frac{u^{*}}{u_D}=\frac{Q\gamma \mu {\kappa }^2{\ell }^3}{V_0{\left(\overline{R}a{p}_{\infty }\right)}^2}.$$ [2]

The dimensionless parameter $\phi$ is also a measure of the magnitude of the pressure drop $\mathrm{\Delta }{p}_t$ along the tube relative to atmospheric pressure $p_{\infty }$. Combining Eqs. 1 and 2, where the foam velocity in the tube is $u=\mathcal{O}\left(u^{*}\right)$, the dimensionless pressure drop along the tube is $\mathrm{\Delta }{p}_t/p_{\infty }=\mathcal{O}\left(\sqrt{\phi }\right)$. Thus, a larger injection rate $\phi$ results in greater compressibility effects in the foam flow.

Fast-Injection Regime.

The value of $\phi$ for each experiment is calculated and shown in Fig. 2C. For experiments A–I, $\phi =\mathcal{O}\left(1\right)$ so that $\mathrm{\Delta }{p}_t/p_{\infty }=\mathcal{O}\left(1\right)$. Therefore, the injection is fast enough so that compressibility effects are important and the experiments are in the fast-injection regime. The foam volume at the outlet $V\left(t\right)=\pi a^2\int u\mathrm{d}t$ therefore has a characteristic scale $\pi a^2{u}^{*}{t}^{*}=\pi a^2\ell$, suggesting a dimensionless volume $\mathcal{V}=V/\left(\pi a^2\ell \right)$. A dimensionless time $\tau$ can be defined as time rescaled with the characteristic injection time $t^{*}$—that is, $\tau \equiv \left(t-t_0\right)/t^{*}$. After nondimensionalizing our data in Fig. 2B using the dimensionless groups ($\mathcal{V}$ and $\tau$) obtained above, we find that the dimensionless data of volume $\mathcal{V}$ and time $\tau$ collapse onto a universal curve over a range of injection rate $Q$, initial volume $V_0$, tube length $\ell$ and radius $a$, liquid viscosity $\mu$, and liquid volume fraction ${ϵ}_{\infty }$, as shown in Fig. 2C.

The dimensionless universal curve is well-fit by a power law at the late times, as shown by the solid line in Fig. 2C:

$$\mathcal{V}=\beta {\tau }^{\alpha }\mathrm{f}\mathrm{o}\mathrm{r}\phi =\mathcal{O}\left(1\right),$$ [3]

where $\beta =39\pm 6$ and $\alpha =2.5\pm 0.3$ are dimensionless fitting parameters averaged over all fast-injection experiments. Near the end of some experiments ($\tau \approx 1$), the data deviate from the power law. Note that Eq. 3 shows no dependence of foam flow dynamics on $\mu$ and ${ϵ}_{\infty }$, which agrees with the observations (experiments B and H). The experimental system contains 10 parameters ($Q,\ell ,a,V_0,\mu ,\gamma ,{ϵ}_{\infty },\overline{R},V,t$) yet can, for the parameter range of our experiments, be adequately described with a simple power law involving two dimensionless parameters ($\mathcal{V}\text{and}\tau$).

Slow-Injection Regime.

On the other hand, when $\phi \ll 1$, $\mathrm{\Delta }{p}_t/p_{\infty }=\mathcal{O}\left(\sqrt{\phi }\right)\ll 1$. Then, compressibility effects are small and the experiments are in the slow-injection regime. The flow approaches a steady state ($\partial ϵ/\partial t=0$) with $u=Q/\left(\pi a^2\right)$ and $V=Q\left(t-t_0\right)$ (see Expt. J where $\ell$ and $Q$ are sufficiently small so that $\phi \approx 4\times 10^{-7}$). The injection of water (Expt. K, $t_0=0$) in comparison with foam (Expt. J, $t_0=80$ s) at the same $Q,a,\ell ,V_0$ is shown in Fig. 2b. The crossover time at which foam flow approaches the incompressible result $V=Q\left(t-t_0\right)$ (dashed line in Fig. 2b) can be estimated by matching the initial unsteady velocity with the incompressible steady-state asymptote. At short times, the mass flux vented into the tube is small, so that the pressure increase $\delta p_s$ and volume decrease $-\delta V_s$ in the syringe for isothermal compression follows $\delta p_s/p_{\infty }\approx -\delta V_s/V_0\approx Qt/V_0=\tau$. The initial unsteady foam flow velocity (Eq. 1) is $u\approx u_D{\left(\delta p/p_{\infty }\right)}^2\approx u_D{\tau }^2$ and volume is $V\approx \frac{\pi }{3}{a}^2{u}_D{t}^{*}{\tau }^3$. The time for foam to reach the steady-state result $Q\left(t-t_0\right)=Q\tau t^{*}$ is therefore ${\tau }_c=\sqrt{3\phi V_0/\left(\pi a^2\ell \right)}$. For Expt. J in Fig. 2 the crossover time is $t_c-t_0=\sqrt{\phi V_0^3/\left(\pi a^2\ell Q^2\right)}\approx$ 100 s, which gives an order-of-magnitude estimate of the time when $V\left(t\right)$ approaches the incompressible limit. Note that although the fast-injection experimental results (Fig. 2c) are independent of $\mu$, the crossover time $t_c$ for the slow-injection experiment depends on $\phi$ and thus is affected by the foam properties.

A Quantitative Description for Foam-Driven Fractures

We can now apply our findings for $\mathcal{V}\left(\tau \right)$ from the one-dimensional experiments to foam-driven fractures in elastic solids (Fig. 1). We conduct fracturing experiments with $Q=5-30$ mL/min, $E=28-125$ kPa, and $\phi =\mathcal{O}\left(1\right)$ and, in each case, measure the radius of the lens-shaped crack $R\left(t\right)$ (Fig. 1D) as a function of time, as plotted in Fig. 3B. We observe a linear increase of crack radius $R\left(t\right)$ with time, which differs from the crack growth driven by incompressible flows (see SI Appendix).

Fig. 3.

(A) A snapshot of fracture driven by foam and water injection taken at $t-t_0=100$ s, where $t_0$ is the time when the fracture starts to grow. Although the experimental parameters are the same for both foam and water ($Q=5$ mL/min and $E=66$ kPa), the fracture size is visibly different. (B) The radius $R$ of a foam-driven crack measured in time for the fast-injection regime [$\mathrm{\Delta }{p}_t/p_{\infty }=\mathcal{O}\left(\sqrt{\phi }\right)=\mathcal{O}\left(1\right)$]. Different curves correspond to experiments with different $E$ and $Q$. The fracture radius grows linearly with time at the late times, which is different from the results of the incompressible fluid-driven cracks (see SI Appendix). (C) The collapse of data rescaled by Eq. 5 shows a good agreement between the experiments and the scaling law of fracture growth driven by compressible foam flow (the solid line). The dimensionless prefactor $A=0.8\pm 0.1$ is obtained by fitting Eq. 5 to each experimental curve at late times.

For a lens-shaped fracture of radius $R$ inside an elastic medium with energy per unit area required to open new crack surfaces $2{\gamma }_s$ (48, 49), the elastic stress around the fracture is $\mathrm{\Delta }{p}_e\approx EW/\left(2\left(1-{\sigma }^2\right)R\right)$ (37, 46, 47), and the pressure required to break the atomic/molecular bonds and extend the fracture is $\mathrm{\Delta }{p}_f\approx \sqrt{\pi {\gamma }_{s}E/\left(2R\right)}$ (47–51). Assuming the stored elastic energy in the solid matrix is instantaneously dissipated by creating new crack surfaces (31, 38, 48, 51), according to $\mathrm{\Delta }{p}_e\approx \mathrm{\Delta }{p}_f$, we find

$$\frac{E}{2\left(1-{\sigma }^2\right)}\frac{W}{R}\approx \sqrt{\frac{\pi {\gamma }_{s}E}{2R}}.$$ [4]

For all of our experiments, the viscous stresses along the tube $\mathrm{\Delta }{p}_t$ are large compared with $\mathrm{\Delta }{p}_e$ and $\mathrm{\Delta }{p}_f$. However, the viscous stresses $\mathrm{\Delta }{p}_v$ due to flow within the fracture are negligible since $\mathrm{\Delta }{p}_v\ll \mathrm{\Delta }{p}_f$, as shown in SI Appendix. In addition, the foam-driven fractures are designed to grow horizontally so that the buoyancy due to the density difference between foam and gelatin does not affect the fracture dynamics.

We recall that the volumetric flux of foam in the tube is determined by the pressure drop along the tube $\mathrm{\Delta }{p}_t$ rather than the stresses related to fracture formation $\mathrm{\Delta }{p}_e$ since $\mathrm{\Delta }{p}_e\ll \mathrm{\Delta }{p}_t$. Thus, the volume of foam $V$ vented into the fracture at the outlet of the needle obeys the same power law as the volume collected at the tube outlet in the one-dimensional model experiments (Fig. 2 and Eq. 3). Since $\phi =\mathcal{O}\left(1\right)$ in the fracture experiments, we use the volumetric flux in the fast-injection regime—that is, $\mathcal{V}=\beta {\tau }^{\alpha }$. Combining Eqs. 3 and 4, the experimental value $\alpha =2.5$ (Fig. 2C), and the crack volume $V=4\pi /3W{R}^2$ for an elliptical fracture, we obtain

$$R=A{v}_f\left(t-t_0\right),v_f\equiv {\left(\frac{9{\beta }^2{a}^4{\ell }^{2}E}{32\pi {\left(1-{\sigma }^2\right)}^2{\gamma }_s}\right)}^{1/5}\frac{Q}{V_0},$$ [5]

where $A$ is a dimensionless numerical prefactor that depends only on the shape of the fracture and $t_0$ is the time when foam first enters the gelatin matrix. The collapse of the rescaled experimental data (symbols) at the late times in Fig. 3C shows excellent agreement with the prediction for fracture dynamics given by Eq. 5 (Fig. 3C, solid line). Note that the speed of crack propagation $v_f$ for foam injection is constant, in contrast with incompressible fluid-driven fractures where the fracturing velocity decreases with time.

Although foam consists of water and gas, the dynamics of foam-driven fractures for $\phi =\mathcal{O}\left(1\right)$, where the compressibility of foam is important, differs significantly from those of fractures driven by the injection of either water or gas. Water is effectively incompressible, so $V=Qt$. Gas is compressible but has a small resistance to flow in the tube, and so the pressure drop is not large enough to probe compressibility effects. Foam is as compressible as gas but has a large viscous resistance to flow along the tube. This produces a large pressure drop in the tube, $\mathrm{\Delta }{p}_t\ge p_{\infty }$, causing compression. We checked the foam-fracture experiments in the slow-injection regime ($\phi \ll 1$) where compressibility effects are small. The dynamics of fracture growth driven by slow injection of foam obeys the same scaling law as the classical results of crack growth driven by incompressible flows, $R\left(t\right)\approx {\left(9Q^{2}E/\left(32{\pi }^3{\left(1-{\sigma }^2\right)}^2{\gamma }_s\right)\right)}^{1/5}{t}^{2/5}$, as shown in SI Appendix.

Conclusion

In conclusion, we study the flow of compressible foam through a tube and its impact on fractures in elastic solids driven by foam injection. We found two flow regimes depending on whether or not the injection is fast enough to cause significant compression of the foam. In the fast-injection regime, a time-dependent flow was observed as a result of compressed bubbles in the foam. In the slow-injection regime, the flow approaches the incompressible results within the experimental timescale. Finally, we demonstrated that a scaling argument based on our empirical result of the mass balance of foam flow and the stress balance for fracture propagation exhibits excellent agreement with our experiments of foam-driven fractures. Our results could potentially inform other systems involving injection of compressible two-phase flows in channels with narrow geometries.