Ascent rates of 3D fractures driven by a finite batch of buoyant fluid

Abstract

Propagation of fluid-filled fractures by fluid buoyancy is important in a variety of settings, from magmatic dykes and veins to water-filled crevasses in glaciers. Industrial hydro-fracturing utilises fluid-driven fractures to increase the permeability of rock formations; the effect of buoyancy on fracture pathways in this context is typically neglected. Analytical approximations for the buoyant ascent rate facilitate quantitative estimates of buoyant effects. Such analysis exists for two-dimensional fractures, but real fractures are 3D. Here we present novel analysis to predict the buoyant ascent speed of 3D fractures containing a fixed-volume batch of fluid. We provide two estimates of the ascent rate: an upper limit applicable at early time, and an asymptotic estimate (proportional to t^(-2/3)) describing how the speed decays at late time. We infer and verify these predictions by comparison with numerical experiments across a range of scales and analogue experiments on liquid oil in solid gelatin. We find the ascent speed is a function of the fluid volume, density, viscosity and the elastic parameters of the host medium. Our approximate solutions can predict the ascent rate of fluid-driven fractures across a broad parameter space, including cases of water injection in shale and magmatic dykes. Our results demonstrate that both dykes and industrial hydro-fractures can ascend by buoyancy over a kilometre within a day.