Eruption versus intrusion? Arrest of propagation of constant volume, buoyant, liquid-filled cracks in an elastic, brittle host

When a volume of magma is released from a source at depth, one key question is whether or not this will culminate in an eruption or in the emplacement of a shallow intrusion. We address some of the physics behind this question by describing and interpreting laboratory experiments on the propagation of cracks filled with fixed volumes of buoyant liquid in a brittle, elastic host. Experiments were isothermal, and the liquid was incompressible. The cracks propagated vertically because of liquid buoyancy but were then found to come to a halt at a configuration of static mechanical equilibrium, a result that is inconsistent with the prediction of the theory of linear elastic fracture mechanics in two dimensions. We interpret this result as due to a three-dimensional effect. At the curved crack front, horizontal cracking is necessary in order for vertical propagation to take place. As the crack elongates and thins, the former becomes progressively harder and, in the end, impossible to fracture. We present a scaling law for the final length and breadth of cracks as a function of a governing dimensionless parameter, constructed from the liquid volume, the buoyancy, and host fracture toughness. An important implication of this result is that a minimum volume of magma is required for a volcanic eruption to occur for a given depth of magma reservoir.