Continuum models of dykes: a comparison to discrete fracture models

Dykes are fluid-filled fractures that transport melt within the lithosphere. Their presence impacts lithospheric stresses and hence the forces of plate tectonics. Accurate representation of dykes in geodynamic models is crucial for modelling the dynamics of rifting. Li et al. (2023) approximated dyking as plastic tensile failure in a two-phase continuum with a poro-viscoelastic—viscoplastic (poro-VEVP) rheology. Li et al. only partially validated this model approach, and hence the extent to which it approximates a natural fracture remains unclear. In this study, we extend the comparison between our continuum formulation and the widely accepted theory of Linear Elastic Fracture Mechanics (LEFM) to the case of a buoyancy-driven dyke (Roper and Lister, 2007). We achieve this through detailed consideration of the dynamics and energetics.

 

Comparing the dynamics of the continuum and LEFM models is challenging due to their differing assumptions and the limitations of finite-difference numerical solutions. The continuum model treats the liquid phase as porous flow, while the LEFM model assumes a lubricated channel flow inside of an open fracture. A drawback of numerical computations is that the grid size can exceed the dyke width. In fact, one shortcoming of the previous computational framework is that the dyke width can grow to several grid cells, which further complicates the comparison. To overcome these challenges, we adjust both models to have the same geometry and mechanics: (1) we incorporate in the continuum model an anisotropic permeability treatment based on plastic strain components; (2) we introduce an intermediate 'poro-fracture' model that is a modified LEFM model for porous channel flow.

 

Our results show that the poro-VEVP model converges to the LEFM poro-fracture model at a toughness value that is large relative to natural rock and at a propagation speed that is slow relative to the standard LEFM formulation. Firstly, we attribute the slow propagation speed in the continuum model to the high Darcy’s drag force, which can be improved by augmenting the permeability. Secondly, we show that plasticity in the continuum model relates quantitatively to the toughness in the fracture model. It is evidenced by the good agreement between the two models in stress field prediction and also the dyke porosity profiles when a specific toughness value is applied to the fracture model. Intriguingly, this toughness is derived from equating the total energy dissipation rate in the continuum model to the energy required to open a fracture, thereby establishing an energy-based connection between the two models.

References

Li, Y., Pusok, A., Davis, T., May, D., and Katz, R., (2023). Continuum approximation of dyking with a theory for poro-viscoelastic–viscoplastic deformation, Geophysical Journal International.

Roper, S.M. and Lister, J.R., (2007). Buoyancy-driven crack propagation:the limit of large fracture toughness. Journal of Fluid Mechanics.