Collaborative Research: Data-enabled Modeling, Numerical Method, and Data Assimilation for Coupling Dual Porosity Flow with Free Flow

双重介质渗流和自由渗流的耦合在许多重要的应用中都出现了。然而，现有的Stokes-Darcy模型由于只考虑单一孔隙介质，不能准确地描述这类耦合问题。因此，在实验室实验数据的支持下，PIS发展了一种新的耦合多物理多尺度模型和相应的数值方法来精确描述这种耦合。此外，实验室和现场资料都提供了通过数据同化来提高模式预报精度的可能性。这个项目为学生提供了许多宝贵的培训机会，包括数据建模、数值方法和程序包的开发、数据同化、数学分析和工程应用。他们可以获得坚实的计算数学和数据科学基础，宝贵的研究经验，以及与工程师的广泛合作经验。从这次协作工作开始，研究人员计划将提出的模型、方法和包传播给更多的工程师和科学家，以解决他们的实际问题，并在专业会议和座谈会上介绍工作，并在会议中组织相关工作的专题会议。此外，该项目是计算和应用数学项目以及密苏里州计算和应用数学科学研究所在密苏里州S和密苏里州的扩展的一部分。这种面向系的扩展将使整个以工程为基础的大学受益，并帮助密苏里州加强其在计算数学方面相对较不活跃的研究。在怀俄明大学，数学系和统计系将在2017年合并，新的重点是数据科学、数学和统计。该项目将立即促进数据科学计划，并为招收新学生、科学家和教师提供理由。为新模型提出适当的接口条件以使两个流以物理有效的方式耦合是具有挑战性的。此外，两个组分模型的耦合导致了双重介质渗流和自由渗流中涉及不同尺度的复杂系统，这就需要准确、高效的数值方法。由于数据同化方法具有数据量大、迭代的特点，利用已有的数据来改进模式预报，将使预报的复杂性和计算量进一步大幅增加。当动态系统中的非线性、时变性、真实界面/边界条件和数据信息相互作用时，整个系统变得更加复杂，计算规模也变得更大。因此，对于复杂的多物理多尺度模型来说，将双重介质流动与自由流动相耦合仍然是一个巨大的挑战。本项目在实验室实验数据的支持下，提出了双孔隙率-N-S模式，发展了具有最优收敛速度的解耦非迭代多物理区分解方法，研究了具有新定义的代价函数的变分同化方法以改进界面模式预报，对模型和数值方法进行了数学分析，并将它们应用于一个或两个应用。这项研究将所有这些组件动态地结合到一个研究和开发的混合系统中，该系统将充分利用新的数学建模/方法/分析与验证/数据同化/应用方面的实际工程进展之间的内在联系，从而为涉及具有高导热管道的裂隙多孔介质中复杂流动的许多应用程序的可靠建模奠定基础。

The coupling of dual porosity flow and free flow arises in many important applications. However, the existing Stokes-Darcy types of models cannot accurately describe this type of coupled problem since they only consider single porosity media. Therefore, with the support of lab experiment data, the PIs develop a new coupled multi-physics multi-scale model and the corresponding numerical methods for accurately describing this coupling. Furthermore, both the lab and field datum provide the possibility to improve the accuracy of the model prediction through data assimilation. This project provides students many valuable training opportunities in data-enabled modeling, development of numerical methods and code packages, data assimilation, mathematical analysis, and engineering applications. They can gain solid foundation in computational math and data science, valuable research experience, and extensive collaboration experience with engineers. Starting from this collaboration work, the investigators plan to disseminate the proposed model, methods, and packages to more engineers and scientists for solving their realistic problems, present the work in professional conferences and colloquia, and organize special sessions in conferences for related works. Moreover, this project is part of the expansion of the computational and applied mathematics program and Missouri Institute for Computational and Applied Mathematical Sciences at Missouri S&T. This department-oriented expansion will benefit the entire engineering-based university and help state of Missouri enhance its relatively less active research in computational mathematics. At the University of Wyoming, the mathematics and statistics departments are merging in 2017 with a new emphasis on data sciences, mathematics, and statistics. This project will provide an immediate boost to the data science initiative and provide a justification for recruiting new students, scientists, and faculty.It is challenging to propose appropriate interface conditions for the new model in order to couple the two flows in a physically valid way. Moreover, coupling two constituent models leads to a complex system involving different scales in the dual porosity flow and the free flow, which demands accurate and efficient numerical methods. The use of existing data to improve the model prediction will even further increase the complexity and computational cost by a significant amount due to the big amount of data and iterative feature of the data assimilation methods. When the nonlinearity, time-dependence, realistic interface/boundary conditions, and data information interact with each other in a dynamic system, the whole system becomes much more complicated and much larger in computational scale. Therefore, significant challenges still remain for the intricate multi-physics multi-scale model to couple the dual porosity flow with the free flow. This project proposes a dual-porosity-Navier-Stokes model with the support of lab experiment data, develops the decoupled non-iterative multi-physics domain decomposition method with optimal convergence rates, study the variational data assimilation method with a newly defined cost function for improving the interface model prediction, carries out the mathematical analysis for the model and the numerical methods, and applies them to one or two applications. This research dynamically combines all of these components into a hybrid system of research and development that will take full advantage of the inherent relationship between the novel mathematical modeling/methods/analysis and the practical engineering advances in validation/data assimilation/applications, hence will lay the groundwork for reliable modeling of many applications involving complex flow in fractured porous media with highly-conductive conduits.