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A Discontinuous Galerkin Finite Element Method Solution of One-Dimensional Richards’ Equation

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A Discontinuous Galerkin Finite Element Method Solution of One-Dimensional Richards’ Equation A Discontinuous Galerkin Finite Element Method Solution of One-Dimensional Richards’ Equation THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Yilong Xiao Graduate Program in Civil Engineering The Ohio State University 2016 Master's Examination Committee: Dr. Gajan Sivandran, Advisor Dr. Ethan J. Kubatko Dr. Yu-Ping Chin © Copyright by Yilong Xiao 2016 Abstract Unsaturated flows in porous media, attributing to hydraulic conductivity, capillary pressure and gravity, are governed by the Richards’ equation. Due to high non-linearity in the relations among it variables, the Richards’ equation does not have a general closed- form solution. Past numerical Richards’ equation solvers were predominantly predicated on conforming finite element methods or finite different methods, of which the requirement on continuity in solutions occasionally led to convergence failure when abrupt change occurs in boundary conditions or in hydraulic properties of the porous media or both. Moreover, past infiltration schemes influenced by different user-defined vertical discretization could produce significantly different runoff partitioning results even with a same set of parameters. In light of such challenges, hereby presented is a one- dimensional numerical Richards’ equation solver based on the discontinuous Galerkin finite element method (DG-FEM) – a class of finite element method that operates on piecewise continuous trial spaces and allows discontinuity in solutions. Such characteristics make DG-FEM solutions fit for high-intensity moisture inputs and natural heterogeneities. This thesis outlines the steps of DG-FEM formulation on a one- dimensional Richards’ equation. Soil moisture readings under laboratory settings were ii obtained to validate accuracy, while a long term precipitation dataset was used to contrast the capability of this DG-FEM model to that of a continuous Galerkin FEM model in handling extreme rainfall events and partitioning runoff. iii Acknowledgments Words cannot express my gratitude enough to Gaj, my advisor, for broadening my horizon in the realm of hydrology, dragging me through dungeons and dens. Learning DG from Ethan has been a joyful journey of stumbling and bruising. The way you put up with such a numerical modeling noob in me is beyond reckoning. It is to my greatest delight that Yo accepted my sudden request to be on the committee. Your perspectives as a scientist have always been refreshing and educating. iv Vita 2014................................................................B.S. Environmental Engineering, The Ohio State University 2015 – Present ................................................Graduate Teaching Assistant, Department of Engineering Education, The Ohio State University Fields of Study Major Field: Civil Engineering v Table of Contents Abstract ............................................................................................................................... ii Acknowledgments.............................................................................................................. iv Vita ...................................................................................................................................... v Table of Contents ............................................................................................................... vi List of Tables ..................................................................................................................... ix List of Figures ..................................................................................................................... x Chapter 1: Introduction ...................................................................................................... 1 1.1 Motivation and Objective ..................................................................................... 2 Chapter 2: Literature Review .............................................................................................. 5 2.1 The Green-Ampt Equation ................................................................................... 5 2.2 The Richards’ Equation ........................................................................................ 9 2.2.1 Solution of the Richards’ Equation – Analytical ........................................ 12 2.2.2 Solution of the Richards’ Equation – Numerical ........................................ 14 2.3 Discontinuous Galerkin Finite Element Method ................................................ 15 Chapter 3: Discontinuous Galerkin Formulation ............................................................. 17 vi 3.1 Spatial Discretization ......................................................................................... 17 3.1.1 Weak Formulation ...................................................................................... 17 3.1.2 Numerical Flux ........................................................................................... 19 3.1.3 Basis Polynomials ....................................................................................... 21 3.1.4 Master Element Transformation ................................................................. 26 3.1.5 Numerical Integration ................................................................................. 28 3.1.6 Global Equations ......................................................................................... 31 3.1.7 Hydraulic Conductivity and Diffusivity ..................................................... 33 3.1.8 Numerical Representation of Diffusivity .................................................... 35 3.1.9 Equation Summary ...................................................................................... 39 3.2 Temporal Discretization ..................................................................................... 40 3.2.1 Initial Condition .......................................................................................... 41 3.3 Boundary Conditions and Sink/Source Terms ................................................... 43 Chapter 4: Materials and Methods ................................................................................... 45 4.1 Data Collection ................................................................................................... 45 4.2 Model Comparison ............................................................................................. 48 Chapter 5: Results and Discussion ................................................................................... 50 5.1 Parameter Fitting ................................................................................................ 50 5.2 Rainfall Simulation ............................................................................................ 56 vii 5.2.1 Times Series ................................................................................................ 56 5.2.2 Point-to-Point .............................................................................................. 66 5.2.3 Mass Conservation and Sensitivity Study................................................... 70 5.3 Model Comparison: DG vs. CG ......................................................................... 77 Chapter 6: Conclusion...................................................................................................... 79 References ......................................................................................................................... 81 Appendix A: Setup ........................................................................................................... 89 Appendix B: Procedures for Determining Soil Parameters .............................................. 91 Appendix C: Pseudo-Code ................................................................................................ 95 viii List of Tables Table 1: Numerical Fluxes ................................................................................................ 21 Table 2: Default Settings of Simulation I ......................................................................... 49 Table 3: Default Settings of Simulation II ........................................................................ 49 Table 4: Soil Parameters for Play Sand ............................................................................ 51 Table 5: Rainfall/Hiatus Regime ...................................................................................... 56 Table 6: Coefficients of Determination ............................................................................ 68 Table 7: Comparison of R2 and Error ............................................................................... 72 Table 8: Summary of Simulation II .................................................................................. 78 Table 9: Location of Sensors in Sand Column ................................................................. 90 ix List of Figures Figure 1: Scenario of the Green-Ampt Equation ................................................................ 6 Figure 2: Concept of Numerical Fluxes ............................................................................ 20 Figure 3: Concept of the Master Element ......................................................................... 26 Figure 4: Vertical Discretization and Moisture Content ................................................... 42 Figure 5: Soil Depth vs. Moisture
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