Free surface models of partially molten rock with visco-elasto–plastic rheology: numerical solutions using a staggered-grid finite-difference discretisation

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. Here we present our progress in consistent numerical modelling of poro-viscoelastic–plastic modelling of deformation with a free surface.

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 (Spiegelman et al. 2016). The elastic deformation of rocks can also affect the propagation of internal failure. Furthermore, deformation and plastic failure drives topographic change, which imposes a significant static stress field. 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. 

We present a new finite difference staggered grid framework for solving partial differential equations (FD-PDE) for single-/two-phase flow magma dynamics (Pusok et al., 2020). Staggered grid finite-difference methods are mimetic, conservative, inf-sup stable and with small stencil — thus they are well suited to address these problems. The FD-PDE framework uses PETSc (Balay et al., 2020) and aims to separate the user input from the discretization of governing equations. The core goals for the FD-PDE framework is to allow for extensible development and implement a framework for rigorous code validation. Here, we present simplified model problems using the FD-PDE framework for two-phase flow visco-elasto-plastic models designed to characterise the solution quality and assess both the discretisation and solver robustness. We also present results obtained using the phase-field method (Sun and Beckermann, 2007) for representing the free surface. Verification of the phase-field approach will be shown via simplified problems previously examined in the geodynamics community (Crameri et al, 2012).

Balay et al. (2020), PETSc Users Manual, ANL-95/11 - Revision 3.13.

Pusok et al. (2020) https://doi.org/10.5194/egusphere-egu2020-18690 

Spiegelman et al. (2016) https://doi.org/10.1002/2015GC006228

Sun and Beckermann (2007) https://doi.org/10.1016/j.jcp.2006.05.025

Crameri et al. (2012) https://doi.org/10.1111/j.1365-246X.2012.05388.x