Partial melting and melt segregation is a fundamental process in the interior of the Earth and other planetary bodies. In the Earth it leads to the formation of the oceanic crust at mid-ocean spreading centers and the continental crust above subduction zones, and is therefore an important process creating the large scale compositional heterogeneity of the Earth. Partially molten materials are porous media in which phase equilibrium, heat transfer, and mechanical deformation are tightly coupled by multi-phase flow. Two challenges in modeling these systems are: (1) The non-linear coupling between the viscous flow of the creeping mantle and the porous flow of the melt and other volatiles, i.e., multi-phase Darcy-Stokes flow; and (2) The pervasive heterogeneity and anisotropy of all geological materials that will determine the extent of partial melting and the pathways of melt segregation. The main goal of this interdisciplinary project is to develop robust numerical methods that allow the solution of realistic problems in the geosciences.
In the last 30 years, mathematical formulations for multi-phase Darcy-Stokes flow have been developed in geophysics and glaciology, while robust numerical methods for heterogeneous and anisotropic multi-phase systems have been developed for the simulation of geothermal and hydrocarbon reservoirs. This proposal will extend current capabilities in three areas: 1) Extension of the mathematical description to the dynamics of three-phase flow in partially molten systems with an additional fluid phase in the pore space, to allow self-consistent modeling of volatile induced melting in subduction zones. This requires the coupling of both the two-phase Darcy-Stokes equations for the partially molten material with the Buckley-Leverett theory used in multi-phase flow in porous media. 2) The development and analysis of a robust mixed finite element discretization for the multi-phase Darcy-Stokes system that describes partially molten porous media. These methods have been successful, in both limiting cases: heterogeneous and anisotropic Darcy flow as well as incompressible Stokes flow. Mixed methods allow discretely conservative computations of the fluxes that are essential for reactive transport. 3) The development of a reactive transport model for partially molten systems and the analysis of chromatography in partially molten materials. High temperatures allow chemical reactions to remain close to equilibrium, but also make it necessary to treat the liquids as well as the solid as solutions. Partially molten systems are therefore best modeled using techniques for multi-phase flow with phase equilibrium.
