Dike geometry and scaling controlled by kinetics rather than host rock toughness

A common method of characterising dikes is to plot their measured maximum thickness (T) against their horizontal length (L). This method has been applied widely to fault systems to determine critical mechanical controls on intraplate fault evolution, in which the maximum displacement D_max  is related to L by Dmax=γLⁿ, where typically n=1. This power law D_max-L relationship (with scatter) is inferred to represent scaling under constant driving stress. For dikes and other opening mode fractures (e.g., joints, veins, and sills) T-L scaling is typically shown as n=0.5 (i.e. T=αL^0.5 ) albeit with significant scatter in aspect ratio at all data-rich length scales. In contrast to the frictional control for shear faults, this square root scaling is consistent with growth under conditions of constant rock properties, including material fracture toughness K_IC (i.e., the ability of a material containing a crack to resist fracture). Understanding scaling relationships therefore has significant implications for the mechanics of intrusions and other opening mode fractures.

                Thickness versus length (T-L) data for dikes (and veins, sills, etc., but here we focus on dikes) are universally interpreted using a linear elastic 2D pressurised crack model. The model assumes mechanical equilibrium, such that the stress intensity, K , at the tip of the dike is equal to the mode I fracture toughness of the country rock, K_IC . Measured thickness to length ratios are generally consistent with reasonable magma excess pressure estimates, in the range of 1–10 MPa, but the large areas over which that pressure operates in a constant pressure model results in extremely large stress intensity at the tip, which then requires excessively large fracture toughness to stabilise the crack: for most dike sets, K_IC=300-3000 MPa.m^0.5, which is about 100–1000 times that of measured K_IC values for rocks at upper crustal depths.

Here we propose that solidified intrusions variably preserve internal pressure gradients (required for magma flow), representing cracks controlled by kinetics; they are non-equilibrated structures and cannot be treated in continuum with toughness-controlled, uniform pressure (equilibrium) structures such as veins, or many types of scaled analogue model. Early stages of dike growth (inflation) result in increasing length and thickness, but magma pressure gradients within the dike may serve to drive late-stage lengthening at the expense of maximum thickness (relaxation). For cracks in 2D, we find that inflation is controlled by the magma injection rate, viscosity, and host rock stiffness. Pressure relaxation in the dike is controlled by magma viscosity and host rock stiffness, with the timescale of operation controlled by host rock thermal diffusivity (i.e., cooling toward eventual solidification). This combination of parameters imposes conditions that are unique to individual dikes and dike systems of variable volume, magma type, host rocks, and depth of emplacement, hence we suggest there is no unique scaling law for solidified intrusions. Host rock fracture toughness has no impact on kinetics-controlled dike growth in the upper crust, with the key controls being the host rock compliance relative to the magma flow, which will change during dike emplacement