CMG Research: Robust Numerical Methods for Multi-Phase Darcy-Stokes Flow in Heterogeneous and Anisotropic Partially Molten Materials

部分熔化和熔体分离是地球和其他行星体内部的一个基本过程。在地球上，它导致了洋中扩张中心的洋壳和俯冲带之上的大陆壳的形成，因此是造成地球大尺度成分不均匀性的重要过程。部分熔融材料是多孔介质，其相平衡、传热和力学变形通过多相流动紧密耦合。模拟这些系统面临的两个挑战是：(1)蠕变地幔的粘性流动与熔体和其他挥发物的多孔流动之间的非线性耦合，即多相达西-斯托克斯流动；(2)所有地质物质的普遍非均质性和各向异性将决定部分熔融的程度和熔体分离的途径。这个跨学科项目的主要目标是发展强大的数值方法，以解决地球科学中的现实问题。在过去的30年里，地球物理学和冰川学已经发展了多相达西-斯托克斯流的数学公式，而非均质和各向异性多相系统的数值方法已经发展到地热和油气储层的模拟。该提案将在三个方面扩展现有能力：1)将数学描述扩展到部分熔融系统的三相流动动力学，并在孔隙空间中添加额外的流体相，从而允许对俯冲带挥发性诱发熔融进行自一致建模。这需要将部分熔融材料的两相达西-斯托克斯方程与多孔介质中多相流动的巴克利-莱弗里特理论相结合。2)开发和分析了描述部分熔融多孔介质的多相Darcy-Stokes系统的鲁棒混合有限元离散化方法。这些方法在非均质和各向异性达西流以及不可压缩斯托克斯流这两种极限情况下都取得了成功。混合方法允许对反应输运所必需的通量进行离散的保守计算。3)建立了部分熔融体系的反应输运模型和部分熔融材料的色谱分析。高温使化学反应保持在接近平衡的状态，但同时也使我们有必要把液体和固体当作溶液来处理。因此，部分熔融系统最好使用具有相平衡的多相流技术进行建模。

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.