A porous flow approach to model thermal non-equilibrium applicable to melt migration
A porous flow approach to model thermal non-equilibrium applicable to melt migration
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用于模拟适用于熔体迁移的热非平衡的多孔流方法
DOI：
10.1093/gji/ggx406
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发表时间：
2017
影响因子：
2.8
通讯作者：
M. Grebe
中科院分区：
文献类型：
--
作者：
Schmeling;G. Marquart;M. Grebe
我们开发了一种方法，用于流体和多孔介质的固相之间的热交换，其中流体和基质的温度不处于热平衡。该制剂考虑了在欧拉坐标系中静止或变形的多孔基质内的流体的移动。该方法可以应用于，例如，部分熔融的系统或在多孔岩石中的盐水输送。我们从现有的热交换理论开始，其中流体和固相的能量守恒方程被分离并通过热交换项耦合。这一项被扩展以考虑热交换的全部历史。它取决于流体相的微观几何形状。对于含有热的固体的情况下，充满流体的通道，我们推导出一个表达式的基础上的周期性半波的时间依赖傅立叶方法。在宏观尺度上，热交换的时间演化导致沿着固体的流动路径的卷积积分，这在静止矩阵的情况下大大简化。通过在能量方程中加入热交换项，导出了两相温度随时间的演化。我们探讨流体和固体之间的热不平衡的影响，考虑简单的情况下，流体和固体之间突然的温度差作为初始或边界条件，并通过改变流体的速度相对于休息的多孔固体。我们的结果与非运动流体和固体的解析解吻合得很好。固体和流体之间的温差取决于基于达西速度的Peclet数。对于Peclet数大于1的情况，一次扩散后的温差达到5%以上（是一个定标温度，例如初始温差）。因此，我们的研究结果意味着，热非平衡可以发挥重要作用，熔体迁移通过部分熔融系统，熔体集中到熔体通道附近的过渡到熔体上升的堤。我们的方法是基于解决卷积积分的热交换在整个流动历史，这是数值昂贵的。我们尝试用瞬时近似项代替热交换项。我们发现在短时间尺度上存在相当大的误差，但如果使用适当的近似项参数，则在长时间尺度上会有很好的一致性。我们推导出这些参数，可以实现在完全动态的两相流配方在地球上的熔体迁移。
We develop an approach for heat exchange between a fluid and a solid phase of a porous medium where the temperatures of the fluid and matrix are not in thermal equilibrium. The formulation considers moving of the fluid within a resting or deforming porous matrix in an Eulerian coordinate system. The approach can be applied, for example, to partially molten systems or to brine transport in porous rocks. We start from an existing theory for heat exchange where the energy conservation equations for the fluid and the solid phases are separated and coupled by a heat exchange term. This term is extended to account for the full history of heat exchange. It depends on the microscopic geometry of the fluid phase. For the case of solid containing hot, fluid-filled channels, we derive an expression based on a time-dependent Fourier approach for periodic half-waves. On the macroscopic scale, the temporal evolution of the heat exchange leads to a convolution integral along the flow path of the solid, which simplifies considerably in case of a resting matrix. The evolution of the temperature in both phases with time is derived by inserting the heat exchange term into the energy equations. We explore the effects of thermal non-equilibrium between fluid and solid by considering simple cases with sudden temperature differences between fluid and solid as initial or boundary conditions, and by varying the fluid velocity with respect to the resting porous solid. Our results agree well with an analytical solution for non-moving fluid and solid. The temperature difference between solid and fluid depends on the Peclet number based on the Darcy velocity. For Peclet numbers larger than 1, the temperature difference after one diffusion time reaches 5 per cent ofor more (is a scaling temperature, e.g. the initial temperature difference). Thus, our results imply that thermal non-equilibrium can play an important role for melt migration through partially molten systems where melt focuses into melt channels near the transition to melt ascent by dykes. Our method is based on solving the convolution integration for the heat exchange over the full flow history, which is numerically expensive. We tested to replace the heat exchange term by an instantaneous, approximate term. We found considerable errors on the short timescale, but a good agreement on the long timescale if appropriate parameters for the approximate terms are used. We derived these parameters which may be implemented in fully dynamical two-phase flow formulations of melt migration in the Earth.
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