Modeling Dike Propagation in Both Vertical Length and Horizontal Breadth

We present analog experiments on dike propagation, followed by a numerical model of horizontal and vertical growth, which is partially analytical and partially based on empirical observations. Experimental results show that the growth rates are similar until buoyancy becomes significant and, afterward, vertical growth dominates. The numerical model is defined for different conditions in a homogeneous medium: (a) constant flux, fracture‐limited propagation; (b) constant flux, viscous‐limited propagation; and (c) variable flux dependent on the driving pressure and dike dimensions. These conditions distinguish between cases when the influx depends on the deeper source of magma (e.g., a conduit, independent of the dike geometry) and when it depends on the dike, so the influx can change as it grows. In all cases, the ratio of vertical to horizontal propagation is proportional to the ratio of buoyancy pressure to source pressure, in which buoyancy drives vertical propagation. We test the numerical model on dikes observed at Piton de la Fournaise, in which the dimensions were estimated using geodetic and seismic data. The results show that the final dimensions can be reproduced using magma‐crust density differences of 50–300 kg/m3, viscosities of 30–300 Pa·s, influxes of 50–750 m3/s and shear moduli of ∼10 GPa. The modeled magma and host rock parameters agree with previous studies of the volcano, while the flux is higher than what is typically observed during eruption. This implies a variable injection condition, in which the flux peaks during propagation and diminishes by the onset of eruption. Plain Language Summary Magma‐filled cracks, or “dikes,” grow through the Earth's crust and can feed volcanic eruptions. Building reliable models of how they grow is important for forecasting future eruptions. Mathematical models of dike growth commonly consider growth in length only, which allows for simplifications that make it efficient to run simulations. More‐complex 3D simulations tend to require powerful computers and can take a long time to complete. We present a new model that simplifies the dike to an ellipsoidal shape, like a flattened sphere. This model allows dikes to grow in a way that matches experiments and shows that the growth depends on physical forces like buoyancy. When dikes have a low buoyancy, they can grow both vertically and horizontally, but as they become more buoyant, they grow primarily vertically. We test our model against past eruptions at P