Modeling Dike Propagation in Both Vertical Length and Horizontal Breadth

Abstract

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.

Magma Flow Patterns in Dikes: Observations From Analogue Experiments

“…This can be a large far field stress (e.g., gelatin is incompressible, so a large dike generates a compressive stress field throughout the medium) or when the tip is significantly close to an interface. As we show in a previous publication (Pansino et al., 2022), which uses data from the C series experiments presented here, the far field stresses are always at least an order of magnitude lower than the dikes' internal stresses. Near‐field stresses ahead of a dike tip are assumed to be negligible following results from Rivalta and Dahm (2006), who show that constant‐volume dikes accelerate as they approach a free surface, when the distance of the tip is similar to the dike's half‐length.…”

“…Analogue experiments in gelatin tend to have dominant elastic and fracture pressures, and negligible viscous pressure drop (Menand & Tait, 2002; Pansino et al., 2022). In basaltic dikes, it is debatable whether the fracture pressure or viscous pressure drop limits propagation (Rivalta et al., 2015).…”

“…We argue that the Π P parameter effectively quantifies the ratio of buoyancy pressure to total driving pressure (source pressure, Δ P , plus buoyancy). This is because the total driving pressure in gelatin experiments will always be in balance with both the elastic and fracture pressures in the dike head (Menand & Tait, 2002; Pansino et al., 2022), so that P b + Δ P ∼ P e ∼ P f . By extension, as P b increases and comes into balance with P f , Δ P decreases accordingly.…”

Mechanical Modeling of Pre‐Eruptive Magma Propagation Scenarios at Calderas

“…Our model does not consider the viscous flow of magma within dikes and, as such, does not model dike velocity. The two approaches may be integrated by combining the pathways predicted by our model with existing models of dike velocity (Davis et al, 2023;Pinel et al, 2017) or growth, such as Zia and Lecampion (2020), introducing a numerical model of propagation of planar 3D hydraulic fractures, or Möri and Lecampion (2022); Pansino et al (2022). We also remark that different magma compositions may involve large differences in magma viscosity and density, and neglecting the viscous flow may undermine the predictive power of our dike models in case of high-viscosity magmas.…”

“…The simplest two‐dimensional (2D) trajectory models are streamlines perpendicular to the least‐compressive stress axis (Anderson, 1937; Pollard, 1987), while the most sophisticated approaches model dikes as cracks steered in the direction of maximum strain energy release rate (Dahm, 2000a; Maccaferri et al., 2010, 2011). Dike trajectory models have recently evolved from two‐dimensional (Anderson, 1937; Dahm, 2000a; O. H. Muller & Pollard, 1977; Pollard, 1987) to partially (Heimisson et al., 2015; Pansino et al., 2022; Sigmundsson et al., 2015) or fully three‐dimensional (3D) by Davis et al. (2020, 2021).…”