Foam-driven fracture

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.

Keywords: fluid-driven cracks; fluid–structure interactions; foam fracturing; foams; hydraulic fracturing.

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.

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).

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.