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Multi-scale analysis of two-phase flow in porous media with complex heterogeneities

复杂非均质性多孔介质中两相流的多尺度分析

基本信息

批准号：
142009460
负责人：
Professor Dr. Oleg Iliev
金额：
--
依托单位：
Fraunhofer-Institut für Techno- und Wirtschaftsmathematik (ITWM)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2009
资助国家：
德国
起止时间：
2008-12-31 至 2011-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/142009460?language=en
关键词：
Multi scale analysis two phase

项目摘要

The goal of this project is the multi-scale analysis of porous media equations. We will start from a mesoscopic description of two phase flow in porous media and consider a wide range of heterogeneities. We plan to study: (1) problems with heterogeneities at a large scale such that subdomains with homogeneous properties can be introduced at the cost of interface conditions between the subdomains; (2) problems where the scales can not be separated but a proper multi-scale discretization technique can be exploited in order to obtain a macro-scale solution enriched by meso-scale information; (3) problems where the meso-scale solution is the target and multi-scale preconditioning techniques can serve as a basis for robust and efficient solution methods. In case (1) with a clear scale separation, we want to adapt and apply homogenization techniques in order to justify and analyze effective models and efficient approximation schemes. We want to develop numerical methods that exploit the multi-scale character of the problem and we wish to accompany the practical implementation with a rigorous analysis. As an extension of the basic model, we will consider interface conditions between different porous materials; this can lead to either coupling conditions between one- and two-phase flow equations, to effective Dirichlet conditions or to outflow conditions. Furthermore, in case (2) of more complex heterogeneities, when the scales can not be separated but it is still too expensive or impossible to solve a full meso-scale problem, we want to further develop and analyze multi-scale discretization techniques, such as the heterogeneous multiscale method, HMM, and the multiscale finite volume method, MSFV. When, as in (3), the full meso-scale solution is the target, we plan to develop proper multi-scale preconditioners. Having in mind that there will be a significant synergy effect and reuse of the discretization/ preconditioning components and of the analytical tools for problems with different levels of heterogeneity, we suggest to consider them in one common research project. We plan also to benefit from a tight connection between the studies of the continuous multiscale problems, the approaches for their discretization, and robust methods for solving the discretized problems. In particular, we also aim at evaluating numerically the range of applicability of certain multi-scale schemes and to come up with a generalized framework in order to be able to treat a wide range of heterogeneities within a common setting.

该项目的目标是多孔介质方程的多尺度分析。我们将从多孔介质中两相流动的介观描述开始，并考虑广泛的非均匀性。我们计划研究：（1）在大尺度上具有非均匀性的问题，可以引入具有均匀性质的子域，但需要牺牲子域之间的界面条件：（2）尺度不能分离，但可以利用适当的多尺度离散技术来获得由细观尺度信息丰富的宏观尺度解;（3）以中尺度解为目标的问题，多尺度预处理技术可以作为鲁棒和有效求解方法的基础。在情况（1）中，尺度分离明显，我们希望调整和应用均匀化技术，以证明和分析有效的模型和有效的近似方案。我们希望开发利用问题的多尺度特性的数值方法，我们希望伴随着严格的分析的实际实施。作为基本模型的扩展，我们将考虑不同多孔材料之间的界面条件;这可以导致单相和两相流方程之间的耦合条件，有效的Dirichlet条件或流出条件。此外，在情况（2）更复杂的非均匀性，当尺度不能分离，但它仍然是太昂贵或不可能解决一个完整的介观尺度的问题，我们想进一步发展和分析多尺度离散技术，如非均匀多尺度方法，HMM，和多尺度有限体积方法，MSFV。如（3）中所述，当目标是完整的中尺度解时，我们计划开发适当的多尺度预处理器。考虑到将有一个显着的协同效应和重用的离散化/预处理组件和分析工具的问题，不同层次的异质性，我们建议考虑在一个共同的研究项目。我们还计划受益于连续多尺度问题的研究之间的紧密联系，其离散化的方法，以及解决离散化问题的鲁棒方法。特别是，我们的目标也是在数值上评估某些多尺度计划的适用范围，并提出一个广义的框架，以便能够在一个共同的设置范围广泛的异质性治疗。