  Magma-driven fractures are the main mechanism for magma emplacement in the
crust. A fundamental question is how the released fluid controls the
propagation dynamics and fracture geometry (depth and breadth) in three
dimensions. Analog experiments in gelatin have shown that fracture breadth
remains nearly stationary when the process in the fracture head (where breadth
is controlled) is dominated by solid toughness, whereas viscous fluid
dissipation is dominant in the fracture tail. We model propagation of the
resulting buoyant, finger-like fracture of stationary breadth with a slowly
varying opening along the crack length. The elastic response to fluid loading
in a horizontal cross-section is local and can be treated similarly to the
classical Perkins-Kern-Nordgren (PKN) model of hydraulic fracturing. The
propagation condition for a finger-like crack is based on balancing the global
energy release rate due to a unit crack extension with the rock fracture
toughness. It allows us to relate the net fluid pressure at the tip to the
fracture breadth and rock toughness. Unlike laterally propagating PKN fracture,
where breadth is known a priori, the final breadth of a finger-like vertically
ascending fracture is a result of processes in the fracture head. Because the
head is much more open than the tail, viscous pressure drop in the head can be
neglected leading to a 3D analog of Weertman hydrostatic pulse. This requires
relaxing the local elasticity assumption of the PKN model in the fracture head.
As a result, we resolve the breadth, and then match the viscosity-dominated
tail with the three dimensions, toughness-dominated head to obtain a complete
closed-form solution. We then analyze the buoyancy-driven fracture propagation
in conditions of either continuous injection or finite volume release for sets
of parameters representative of low viscosity magma diking.
