Critical Fluid Injection Volumes for Uncontrolled Fracture Ascent

Hydrofracturing is a routine industrial technique whose safety depends on fractures remaining confined within the target rock volume. Both observations and theoretical models show that, if the fluid volume is larger than a critical value, pockets of fluid can propagate large distances in the Earth's crust in a self-sustained, uncontrolled manner. Existing models for such critical volumes are unsatisfactory; most are two-dimensional and depend on poorly constrained parameters (typically the fracture length). Here we derive both analytically and numerically in three-dimensional scale-independent critical volumes as a function of only rock and fluid properties. We apply our model to gas, water, and magma injections in laboratory, industrial, and natural settings, showing that our critical volumes are consistent with observations and can be used as conservative estimates. We discuss competing mechanisms promoting fracture arrest, whose quantitative study could help to assess more comprehensively the safety of hydrofracturing operations.