A multiscale model of partial melts: 2. Numerical results
A multiscale model of partial melts: 2. Numerical results
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部分熔化的多尺度模型：2.数值结果
DOI：
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发表时间：
2009
期刊：
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通讯作者：
M. I. Weinstein
中科院分区：
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作者：
Gideon Simpson;M. Spiegelman;M. I. Weinstein
在论文中，利用双尺度均匀化理论推导了部分熔融介质的方程。这种方法从粒度描述开始，然后通过多个尺度扩展将其粗化为宏观模型。均质化的一个优点是，有效的材料特性，如渗透性、两相介质的剪切和体粘度，是由细胞问题表征的，即在具有代表性的微结构细胞上提出的边界值问题。可以对这些问题的解进行平均，得到与给定微观结构一致的宏观参数。这对于估计“压实长度”尤其重要，压实长度取决于渗透率和体粘度的乘积，是粘性可变形两相流的固有长度尺度。本文用数值方法解决了几种几何图形的单元方程组问题。我们从简单的相交管开始，因为这是一个具有众所周知的渗透率结果的单参数问题族。利用这些数据，通过曲线拟合估计了孔隙度与所有有效参数之间的关系。对于交叉管模型，渗透率尺度如预期的那样为ϕn， n ~ 2，体积粘度尺度为φ - m， m ~ 1，这是推测的，但从未直接显示在可变形多孔介质中。第二组细胞问题增加了管相交处的球形内含物。对于这些几何形状，渗透率是由孔喉控制的，而不是像预期的那样由总孔隙度控制。然而，体积粘度仍然与孔隙率成反比，我们推测这个量对特定的微观结构不敏感。所开发的计算机制可以应用于更一般的几何形状，例如纹理平衡孔隙形状。然而，我们怀疑我们简化模型的定性行为在这些更现实的结构中仍然存在。特别是，我们的混合数值分析模型预测，对于微观尺度上的纯机械耦合，所有均质化模型的压实长度将随着孔隙率趋于零而消失。这对数值模拟有影响，它表明这些模型可能无法抵抗完全压实。
[1] In the companion paper, equations for partially molten media were derived using two-scale homogenization theory. This approach begins with a grain-scale description and then coarsens it through multiple scale expansions into a macroscopic model. One advantage of homogenization is that effective material properties, such as permeability and the shear and bulk viscosity of the two-phase medium, are characterized by cell problems, boundary value problems posed on a representative microstructural cell. The solutions of these problems can be averaged to obtain macroscopic parameters that are consistent with a given microstructure. This is particularly important for estimating the “compaction length” which depends on the product of permeability and bulk viscosity and is the intrinsic length scale for viscously deformable two-phase flow. In this paper, we numerically solve ensembles of cell problems for several geometries. We begin with simple intersecting tubes, as this is a one parameter family of problems with well-known results for permeability. Using the data, we estimate relationships between the porosity and all of the effective parameters by curve fitting. For the model of intersecting tubes, permeability scales as ϕn, n ∼ 2, as expected, and the bulk viscosity scales as ϕ−m, m ∼ 1, which has been speculated but never shown directly for deformable porous media. The second set of cell problems adds spherical inclusions where the tubes intersect. For these geometries, the permeability is controlled the pore throats and not by the total porosity, as expected. However, the bulk viscosity remains inversely proportional to the porosity, and we conjecture that this quantity is insensitive to the specific microstructure. The computational machinery developed can be applied to more general geometries, such as texturally equilibrated pore shapes. However, we suspect that the qualitative behavior of our simplified models persists in these more realistic structures. In particular, our hybrid numerical-analytical model predicts that for purely mechanical coupling at the microscale, all homogenized models will have a compaction length that vanishes as porosity goes to zero. This has implications for numerical simulations, and it suggests that these models might not resist complete compaction.