### Title page for ETD etd-07042009-224338

Type of Document Dissertation
Author Hua, Fei
URN etd-07042009-224338
Title Modeling, Analysis and Simulation of the Stokes-Darcy System with Beavers-Joseph Interface Condition
Degree Doctor of Philosophy
Department Mathematics, Department of
Max Gunzburger Committee Chair
Xiaoming Wang Committee Co-Chair
Brian Ewald Committee Member
Rudy Horne Committee Member
Xiaolong Hu Outside Committee Member
Keywords
• Solute Transport
• Well-Posedness
• Stokes and Darcy System
• Beavers-Joseph Interface Boundary Condition
• Groundwater
• Fluid and Porous Media Flow
• Pipe Flow
• Contamination
• Karst Aquifer
• Error Bound
• Initial Boundary Value Problem
• Mass Exchange Coefficient
• Finite Element Approximation
Date of Defense 2009-06-12
Availability unrestricted
Abstract
In this dissertation, the coupling phenomenon of porous media flow and free flow is extensively studied, with an application to studying the solute transport in the groundwater system of karst aquifers.

This dissertation consists of three major parts. The first part focuses on the modeling and well-posedness of the mathematical equations. The Stokes equations are used for the free flow part and the Darcy's law is used for the porous media flow part. This is called the Stokes-Darcy system. To couple the two spatially non-overlapping yet neighboring regions, Beavers-Joseph interface condition is used and studied. We show that the transient Stoke-Darcy system with Beavers-Joseph condition is well-posed when a proper scaling parameter is introduced that essentially brings the two physical processes to the same time scale. The steady state problem with Beavers-Joseph condition is also studied and the well-possedness is obtained under some assumptions of the Beavers-Joseph parameter α.

We then, in the second part, put the Stokes-Darcy system under finite element analysis. The analysis is conducted under the framework of a Gårding type inequality established in the first part. First, the spatially semi-discretized problem is studied. Then, a Stokes type projection is devised to aid the convergence rate analysis for the fully discretized system. We obtain a rigorous error bound on the L2 norm with suitable regularity assumptions. The rate is then verified by numerical tests using the Taylor-Hood and quadratic elements pair for the free flow and porous media flow respectively.

In the third part, the research is further carried on to compare the Stokes-Darcy system with conventional models that is used to study the karst aquifer, an important and susceptible groundwater system. In particular, the coupled continuum pipe flow (CCPF) model, the most conventionally used one, is picked as a the counter part to the Stokes-Darcy model. By using the laboratory experiment results as reference, we compare the two models in the aspects of flow rate, head distribution and ability to predict contaminant transport. We reject the conventional wisdom in choosing the exchange coefficient αex in the CCPF model and propose a new region where the coefficient should fall in. In the new parameter region, we compare the Stokes-Darcy model and CCFP model with different boundary and geometry setting to motivate the necessity of switching away from the crude and less physically reasonable CCPF model.

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