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

双重介质渗流与自由渗流的耦合在许多重要的应用中出现。然而，现有的Stokes-Darcy类型的模型不能准确地描述这种类型的耦合问题，因为他们只考虑单一的孔隙介质。因此，在实验室实验数据的支持下，研究人员开发了一个新的多物理场多尺度耦合模型和相应的数值方法来精确描述这种耦合。此外，无论是实验室和现场数据提供了可能性，以提高模式的预测精度，通过数据同化。该项目为学生提供了许多宝贵的培训机会，包括数据建模，数值方法和代码包的开发，数据同化，数学分析和工程应用。他们可以在计算数学和数据科学方面获得坚实的基础，宝贵的研究经验以及与工程师的广泛合作经验。从这项合作工作开始，研究人员计划向更多的工程师和科学家传播所提出的模型、方法和软件包，以解决他们的现实问题，在专业会议和学术讨论会上介绍这项工作，并在相关工作的会议上组织特别会议。此外，该项目是计算和应用数学计划和密苏里州计算和应用数学科学研究所在密苏里州ST扩展的一部分。这种以部门为导向的扩张将有利于整个工程为基础的大学，并帮助密苏里州加强其在计算数学相对不活跃的研究。在怀俄明州大学，数学系和统计系将于2017年合并，新的重点是数据科学，数学和统计学。该项目将为数据科学计划提供直接的推动，并为招募新的学生，科学家和教师提供理由。为新模型提出适当的接口条件，以便以物理有效的方式耦合两个流是具有挑战性的。此外，耦合两个组分模型导致一个复杂的系统，涉及不同的尺度在双重孔隙流和自由流，这需要准确和有效的数值方法。由于数据同化方法的数据量大、迭代性强等特点，利用现有数据改进模式预报将进一步增加计算复杂度和计算成本。当动态系统中的非线性、时间依赖性、实际界面/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，复杂的多物理场多尺度模型如何耦合双重介质流和自由流仍然是一个重大的挑战。本项目在实验室试验数据的支持下提出了一个双孔隙度Navier-Stokes模型，发展了具有最佳收敛速度的解耦非迭代多物理场区域分解方法，研究了一种新定义的改进界面模型预报的代价函数的变分同化方法，对模型和数值方法进行了数学分析，并将其应用于一个或两个应用中。本研究动态地将所有这些组件结合成一个混合系统的研究和开发，将充分利用新的数学建模/方法/分析和验证/数据同化/应用程序的实际工程进展之间的固有关系，因此将奠定可靠的建模的许多应用程序，涉及复杂的流动在裂缝多孔介质与高导电管道。

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.