CMG Collaborative Proposal: Multiphysics and multiscale modeling, computations, and experiments for Karst aquifers

喀斯特蓄水层是公共和私人用水的重要水源。喀斯特含水层，除了多孔的石灰岩基质外，通常还有大型的洞穴状管道，已知这些管道在很大程度上控制着含水层内的地下水流动和污染物运输。我们将开发一种新的管道/基质系统中水流和污染物输送的建模方法。管道内的流动用Navier-Stokes方程来描述，矩阵中的流动用Darcy定律来描述。用对流扩散方程描述了这两个区域的溶质输运。导管/基质界面处的条件是新颖的。我们将对新的Navier-Stokes/Darcy模型进行严格的数学分析，研究解的存在性和对数据的连续依赖等问题。这是此类研究的第一次。分析还将利用喀斯特含水层流动的多尺度特征，得出完全合理的模型简化，从而降低计算算法的成本。我们还将开发用于耦合Navier-Stokes/ Darcy模型的混合Galerkin和最小二乘有限元方法，以及用于污染物传输的耦合对流-扩散模型。实验将是拟议项目的一个关键方面。它们将用于获取管道/基质系统中流量和污染物迁移的信息，这些信息不仅对水资源管理者和决策者有用，而且对数学模型和计算算法的开发和验证也有用。当然，我们对喀斯特含水层定量知识的进步将极大地帮助它们的管理，包括划定公共供水的水源保护区域，设计监测程序来评估污染物羽流的停留和命运。更一般地说，我们将产生的方法也可以影响其他地下流体流动问题，包括石油工业中出现的一些问题。拟议的项目将为学生和博士后研究人员提供真正跨学科培训的工具。他们将获得对增水剂进行准确定量评估的能力，从而能够向工业管理人员和政府官员提供作出合理决策所需的资料。

Karst aquifiers represent a very significant source of water for public and private use. A Karst aquifer, in addition to a porous limestone matrix, typically has large cavernous conduits that are known to largely control groundwater flow and contaminant transport within the aquifer. We will develop a new modeling approach for water flow and contaminant transport in conduit/matrix systems. The flow in the conduits will be described by the Navier-Stokes equations and that in the matrix by Darcy's law. Convection-diffusion equations are used to describe the solute transport in the two regions. The conditions applied at the conduit/matrix interface are novel. We will subject the new Navier-Stokes/Darcy model to rigorous mathematical analyses, studying such issues as existence of solutions and continuous dependence on data. This represents the first such study of this kind. Analyses will also be used to take advantage of the multiscale character of flows in Karst aquifiers to derive fully justified model simplifications that will result in less costly computational algorithms. We will also develop mixed Galerkin and least-squares finite element methods for the coupled Navier-Stokes/ Darcy model along with a coupled convection-diffusion model for the transport of contaminants. Experiments will be a crucial aspect of the proposed project. They will be used to obtain information about flows and contaminant migration in conduit/matrix systems that can not only be useful to water resource managers and policy makers, but also for the development and validation of mathematical models and computational algorithms.Certainly, advances in our quantitative knowledge of Karst aquifers will greatly help in their administration, including the delineation of source-water protection areas for public water supplies and the design of monitoring programs to evaluate the residence and fate of contaminant plumes. More generally, the methodologies we will produce can also impact other underground fluid flow problems including some arising in the petroleum industry. The proposed project will provide a vehicle for the bona fide interdisciplinary training of students and postdoctoral researchers. They will gain the ability to make accurate quantitative assessments of aquifiers and thus can provide industrial administrators and governmental officials with the information they need to make sound policy decisions.