Simulation of partially molten rocks with visco-elasto-viscoplastic rheology and a hyperbolic yield surface for plasticity
Authors: Li Yuan , Pusok Adina , May Dave , Katz Richard 
Published: March 2022

&lt;p&gt;It is broadly accepted that magmatism plays a key dynamic role in continental and oceanic rifting. However, these dynamics remain poorly studied, largely due to the difficulty of consistently modelling liquid/solid interaction across the lithosphere. The RIFT-O-MAT project seeks to quantify the role of magma in rifting by using models that build upon the two-phase flow theory of magma/rock interaction. A key challenge is to extend the theory to account for the non-linear rheological behaviour of the host rocks, and investigate processes such as diking, faulting and their interaction (Keller et al., 2013). Here we present our progress in consistent numerical modelling of poro-viscoelastic-viscoplastic (VEVP) flow. We show that a VEVP model with a new, hyperbolic yield surface can help to robustly simulate both shear and tensile modes of plastic failure in a two-phase system.&amp;#160;&lt;/p&gt;&lt;p&gt;Failure of rocks (plasticity) is an essential ingredient in geodynamics models because Earth materials cannot sustain unbounded stresses. However, plasticity represents a non-trivial problem even for single-phase flow formulations with shear failure only. In two-phase systems, tensile failure of rocks can also occur due to an overpressured liquid phase. Robustly solving a discretised model that includes this physics presents severe challenges, and many questions remain as to effective solvers for these strongly nonlinear systems.&lt;/p&gt;&lt;p&gt;An appropriate rheological model is required to meet this challenge. The most straightforward choice is a Maxwell visco-elasto-plastic model, but this leads to grid-scale localisation and hence mesh-dependence. To obtain mesh-independent shear localisation, we employ the visco-elasto-viscoplastic model by introducing a viscous dashpot in parallel to the plasticity element. Whilst this formulation has shown promise in regularising shear failures in a single-phase flow model (de Borst and Duretz, 2020), its incorporation within two-phase systems has not been examined. We will show that the linear Griffith criteria for the tensile failure can lead to convergence issues whereas a new, hyperbolic yield surface is proposed to resolve these numerical issues. This yield surface provides a smooth transition between the two modes of failure.&lt;/p&gt;&lt;p&gt;The underlying PDEs are discretised using a conservative, finite-difference, staggered-grid framework implemented with PETSc (FD-PDE) that supports single-/two-phase flow magma dynamics. Here, we present simplified model problems using the FD-PDE framework for poro-viscoelastic-viscoplastic models designed to characterise the solution quality and assess both the discretisation and solver robustness. It has been observed that employing the hyperbolic yield surface improved the robustness in simulating plastic failures in both modes.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;Keller, T., May, D. A., &amp; Kaus, B. J. P., (2013). Numerical modelling of magma dynamics coupled to tectonic deformation of lithosphere and crust,&amp;#160;Geophysical Journal International, v195, 1406-1442, https://doi.org/10.1093/gji/ggt306.&lt;/p&gt;&lt;p&gt;de Borst, R., Duretz, T., (2020). On viscoplastic regularisation of strain-softening rocks and soils. International Journal for Numerical and Analytical Methods in Geomechanics, v44, 890-903. https://doi.org/10.1002/nag.3046.&lt;/p&gt;