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

本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的介观描述开始，并考虑广泛的非均质性。我们计划研究：(1)大规模的异构问题，使得具有同质性质的子域可以以子域之间的接口条件为代价引入；(2)对于尺度不可分割的问题，可以利用适当的多尺度离散化技术，得到富含中尺度信息的宏观解；(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高效求解方法的基础。在情况(1)具有明确的尺度分离的情况下，我们希望适应和应用均质化技术，以证明和分析有效的模型和有效的近似方案。我们希望发展数值方法，利用问题的多尺度特征，并希望在实际实施中进行严格的分析。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；这可以导致一相和两相流方程之间的耦合条件，有效的狄利克雷条件或流出条件。此外，在(2)更为复杂的非均质性情况下，当尺度无法分离但仍然过于昂贵或无法解决完整的中尺度问题时，我们希望进一步发展和分析多尺度离散化技术，如异构多尺度方法HMM和多尺度有限体积方法MSFV。如(3)所示，当目标是完整的中尺度解决方案时，我们计划开发适当的多尺度预调节器。考虑到对于具有不同异质性水平的问题，离散化/预处理组件和分析工具将会有显著的协同效应和重用，我们建议在一个共同的研究项目中考虑它们。我们还计划从连续多尺度问题的研究、离散化方法和求解离散化问题的鲁棒方法之间的紧密联系中获益。特别是，我们还旨在数值评估某些多尺度方案的适用性范围，并提出一个广义框架，以便能够在共同设置中处理广泛的异质性。

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.