Modeling Effective Interface Laws for Transport Phenomena Between an Unconfined Fluid and a Porous Medium Using Homogenization
Modeling Effective Interface Laws for Transport Phenomena Between an Unconfined Fluid and a Porous Medium Using Homogenization
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DOI：
10.1007/s11242-009-9354-9
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发表时间：
2009-07-01
影响因子：
2.7
通讯作者：
Mikelic, Andro
中科院分区：
文献类型：
--
作者：
Jaeger, Willi;Mikelic, Andro
在“多孔介质中的传输”杂志的这一章中，主题为“可渗透区域上方的流动和传输”，我们介绍了使用均匀化方法对可渗透区域上的流动和传输进行建模。我们的目标是开发一种启发式方法，可以被工程界用于处理这类问题，并且具有坚实的数学背景。在作者的相应文章中给出了所给结果的严格的数学证明。我们的计划如下：我们从“介绍”部分开始，在那里我们给出一个概述，并与使用其他方法获得的界面条件进行比较。在第二节中，我们利用典型孔径参数epsilon的双尺度展开，通过均匀化给出了Darcy定律的一个非常简短的推导。给出了各种辅助函数的定义和渗透率等典型的有效性质。在第三节中，我们通过一个简单的一维例子介绍了我们的有效界面定律的方法。利用两步策略启发式地获得了近似解。对于一维问题，我们显式地计算了它的近似值和有效界面律，并证明它在O阶(epsilon(2))是有效的。然后，在第四节中，我们用齐次化的方法给出了Beivers-Joseph-Saffman界面条件和压力跳跃条件的推导。我们构造了相应的边界层，并给出了一种启发式计算，引出了界面定律，并以严格的数学结果为基础。此外，我们还证明了该定律对于界面位置选择的微小变化的不变性。最后是一个简短的结束语部分。
In this chapter of the special issue of the journal "Transport in Porous Media," on the topic "Flow and transport above permeable domains," we present modeling of flow and transport above permeable domains using the homogenization method. Our goal is to develop a heuristic approach which can be used by the engineering community for treating this type of problems and which has a solid mathematical background. The rigorous mathematical justification of the presented results is given in the corresponding articles of the authors. The plan is as follows: We start with the section "Introduction" where we give an overview and comparison with interface conditions obtained using other approaches. In Sect. 2, we give a very short derivation of the Darcy law by homogenization, using the two-scale expansion in the typical pore size parameter epsilon. It gives us the definition of various auxiliary functions and typical effective properties as permeability. In Sect. 3, we introduce our approach to the effective interface laws on a simple 1D example. The approximation is obtained heuristically using the two steps strategy. For the 1D problem we calculate the approximation and the effective interface law explicitly and show that it is valid at order O(epsilon (2)). Next, in Sect. 4 we give a derivation of the Beavers-Joseph-Saffman interface condition and of the pressure jump condition, using homogenization. We construct the corresponding boundary layer and present a heuristic calculation, leading to the interface law and being based on the rigorous mathematical result. In addition, we show the invariance of the law with respect to the small variations in the choice of the interface position. Finally, there is a short concluding section.