  Hydraulic fractures propagating at depth are subjected to buoyant forces
caused by the density contrast between fluid and solid. This paper is concerned
with the analysis of the transition from an initially radial towards an
elongated buoyant growth -- a critical topic for understanding the extent of
vertical hydraulic fractures in the upper Earth crust. Using fully coupled
numerical simulations and scaling arguments, we show that a single
dimensionless number governs buoyant hydraulic fracture growth: the
dimensionless viscosity of a radial hydraulic fracture at the time when
buoyancy becomes of order one. It quantifies if the transition to buoyancy
occurs when the growth of the radial hydraulic fracture is either still in the
regime dominated by viscous flow dissipation or is already in the regime where
fracture energy dissipation dominates. A family of fracture shapes emerge at
late time from finger-like (toughness regime) to inverted elongated cudgel-like
(viscous regime). 3D toughness dominated buoyant fractures exhibit a
finger-like shape with a constant volume toughness dominated head and a viscous
tail having a constant uniform horizontal breadth: there is no further
horizontal growth past the onset of buoyancy. However, if the transition to
buoyancy occurs while in the viscosity dominated regime, both vertical and
horizontal growths continue to match scaling arguments. As soon as the fracture
toughness is not strictly zero, horizontal growth stops when the dimensionless
horizontal toughness becomes of order one. The horizontal breadth follows the
predicted scaling.
