A theory for foam-driven fracturing

We provide a theoretical model for foam-driven fracturing of an elastic solid. Based on recent experimental observations, we take into consideration the influence of foam rheology and compressibility. We arrive at a nonlinear non-local system that couples the dynamics of one-dimensional compressible plug flow of foam and elastic deformation of the infinite solid. The model leads to time-dependent solutions for the evolution of the fracture’s profile shape and the pressure distribution in the film of foam. The solutions are affected by a dimensionless control parameter $\varPi _c$ , which measures the influence of foam compressibility. An increase in $\varPi _c$ leads to more significant expansion of the gas phase in the fracture, and the spreading of the film of foam is enhanced. Asymptotic efforts were also made in the weakly compressible regime. The profile now admits a linear asymptote $h \propto \chi$ as $\chi \equiv x_f - x \to 0^+$ , compared with the standard viscous asymptote $h \propto \chi ^{2/3}$ for hydraulic fracturing using incompressible Newtonian fluids. The pressure singularity now follows $p \propto \mbox{ln}\, \chi$ , which is weaker than the standard $p \propto \chi ^{-1/3}$ singularity in the viscous regime. It is also observed that the profile shape and pressure distribution can be rescaled appropriately to collapse onto universal curves at intermediate times, with the length and pressure scales following the $x \propto t^{5/9}$ , $h \propto t^{4/9}$ and $p \propto t^{-1/9}$ scaling laws. Using foam as the fracturing fluid may save water in practical applications such as enhanced gas recovery from shale rocks.