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Reactive Contaminant Transport Modeling Using Analytic Element Flow Solutions

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Reactive Contaminant Transport Modeling Using Analytic Element Flow Solutions REACTIVE CONTAMINANT TRANSPORT MODELING USING ANALYTIC ELEMENT FLOW SOLUTIONS by James R. Craig September 21, 2004 A dissertation submitted to the Faculty of the Graduate School of The State University of New York at Buffalo in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Civil, Structural, and Environmental Engineering Copyright by James R. Craig 2004 ii Acknowledgements I could not have performed any of this work without the companionship and support of my fianc´e Dawn. She helped me deal with many nights of dissertation-induced insomnia and sacrificed many Sundays, evenings, and even a few Saturdays to the cause, doing everything she could to ensure that when I got home I could forget all about the stress, even when she had plenty of her own. Alan has repeatedly lent me his insight, remarkable editing skills, and about half his library (which I will eventually return). He gave me free reign over my work, allowing me to direct my curiosity wherever it roamed (and it roamed into quite a few time-consuming side projects and dead ends). More importantly, he has been the quintessential mentor: teaching me not just about contaminant transport and numerical methods, but about the procedures and politics of academic life, for which I am grateful. Plus he taught me a few chords. Igor has been my sounding board for ideas and the ultimate StarbucksTM brainstorming com- panion. He has been my window into the analytic element method, being both a patient instructor (when I didn’t understand his answer) and an enjoyable intellectual sparring partner (when I didn’t like his answer). Whether I was listening or debating, I was always learning, and more often than not, learning that he was right. He has also taught me more about nickel metal hydride batteries than I could ever wish to know. Thanks to my officemates Shawn, Mary, James, and Karl for not being upset when I zoned out in the middle of a conversation and generally just being good people to be around. I enjoyed learning with you. Thanks to Raghu Suribhatla who derived the expression for the elliptical element discharge derivative in appendix C, within hours of me asking for it. Thanks to StarbucksTM for keeping me awake. I would also like to thank my patient committee members, Matt Becker, John Van Benschoten, and Doug Flewelling. Your willingness to abide exhaustively long proposals and act as a reality check is duly appreciated. Finally, I would like to acknowledge the NCGIA and the National Science Foundation, who graciously funded my graduate education through IGERT award DGE-987066. iii Contents Acknowledgements iii Front Matter iv TableofContents.................................... .. iv ListofFigures ...................................... ix ListofTables....................................... xi ListofSymbols...................................... xii Abstract xix 1 Introduction 1 1.1 Motivation ...................................... ... 1 1.2 OverviewofResearch .............................. ..... 5 1.3 OutlineofContents ............................... ..... 7 2 Background 9 2.1 LiteratureReview ................................ ..... 9 2.1.1 GroundwaterFlowModeling . .... 9 2.1.2 Contaminant Transport Modeling . ...... 13 2.1.3 Discretization of Transport Parameters . .......... 24 2.2 MathematicalBackground. ....... 29 2.2.1 TheAnalyticElementMethod . 29 2.2.2 Vertically-AveragedTransport . ....... 31 2.2.3 TheFiniteDifferenceMethod. ..... 33 2.2.4 TheFiniteElementMethod. 34 iv CONTENTS v 2.2.5 Discretization Constraints: Peclet and Courant Numbers ........... 35 3 Methods 39 3.1 Object-Orientation ............................... ...... 40 3.1.1 Motivation .................................... 40 3.1.2 Flow Modeling Library: Bluebird ....................... 40 3.1.3 Transport Modeling Library: Cardinal .................... 48 3.2 MassBalanceAccounting . ...... 52 3.3 Using AEM for Finite Difference Transport Simulation . .............. 54 3.3.1 Finite Difference Approximation of the Vertically-AveragedADE . 55 3.3.2 FluxDiscretization. ..... 59 3.3.3 Pseudo-3DFluxDiscretization . ...... 71 3.3.4 Backwards Method of Characteristics . ........ 73 3.4 The Effective Parameter Formulation . .......... 74 3.4.1 Overview ...................................... 74 3.4.2 TheDischargeDerivative . ..... 76 3.4.3 Dispersion Coefficients and Their Spatial Derivatives .............. 77 3.4.4 EffectiveVelocities . ..... 79 3.4.5 Effectivevelocitiesin3D. ...... 81 3.4.6 Computational Considerations . ....... 83 3.5 ImplementationofEPVAMethods . ....... 84 3.5.1 EPVAApplicability ............................. 84 3.5.2 EPVA Limitations: Finite Difference Methods . ......... 84 3.5.3 EPVA Limitations: Finite Element Methods . ........ 85 3.5.4 Effective Parameter Random Walk Technique . ........ 86 3.6 Implementation of a Graded Finite Element Method . ............ 89 3.6.1 FiniteElementFormulation . ...... 89 3.6.2 FiniteElementDiscretization . ........ 92 3.6.3 FiniteElementFluxDiscretization . ......... 94 3.6.4 Finite Element Material Integral Evaluation . ............ 96 3.6.5 MeshGeneration ................................ 102 CONTENTS vi 3.7 Discontinuous Concentration Conditions . .............105 3.7.1 MeshSplitting ................................. 107 3.7.2 LeakyWallFluxConditions. 110 3.7.3 Transport Beneath a Pseudo-Partially Penetrating River............111 3.8 Modular and Adaptive Reactive Transport Modeling . .............114 3.8.1 Object-OrientedReactionLibrary . .......114 3.8.2 Adaptive Enabling of Reactions . .......114 4 Numerical Testing 116 4.1 Comparison with Analytic and Numerical Solutions . .............117 4.1.1 Benchmark Test 1: Cleary and Ungs Solution . ........117 4.1.2 Benchmark Test 2: Hunt Solution . 120 4.1.3 Benchmark Test 3: MT3DMS Comparison . 122 4.2 Including Continuous Gradation in Transport Parameters ...............133 4.2.1 Verification of Analytic Effective Velocity Expressions .............133 4.2.2 EPVARandomWalkSimulation . 136 4.2.3 Finite Element Integral Evaluation . .........142 4.2.4 Velocity Interpolation Effects in Characteristic Methods ............146 4.3 DiscontinuousConditions . ........149 4.3.1 Transport Across a Low Permeability Barrier . ..........149 4.3.2 Transport Around an Impermeable Wall . .......153 4.4 Field Scale Reactive Transport Through a PRB . ...........155 5 Discussion 162 5.1 LinkingAEMandTransport . 162 5.2 Reducing the Size of the Transport Problem . ...........165 5.3 Using Continuous Parameters . ........168 5.3.1 FiniteDifferenceMethods . 168 5.3.2 FiniteElementMethods . 170 5.3.3 Lagrangian/Eulerian-Lagrangian Methods . ..........171 5.3.4 ImplicationsforOtherMethods. .......171 5.4 Multi-scaleTransportModeling . .........173 CONTENTS vii 5.5 Computational Considerations . .........173 5.6 SoftwareDevelopments. .......175 5.6.1 Products ...................................... 175 5.6.2 ObjectOrientation. 176 5.6.3 UserInterface................................. 176 5.7 Conclusions ..................................... 178 5.8 FutureWork ...................................... 179 6 Summary 183 A Parameter Derivatives 185 A.1 Saturated Thickness Spatial Derivatives . .............185 A.2 VelocitySpatialDerivatives . ..........186 A.3 Velocity Magnitude Spatial Derivatives . .............186 B Pseudo-3D Parameter Derivatives 188 B.1 VerticalVelocityComponent . ........188 B.2 VerticalVelocitySpatialDerivatives . .............189 B.3 Velocity Magnitude Spatial Derivatives . .............190 C Element Discharge Derivatives 191 C.1 AWell........................................... 191 C.2 LaurentSeries................................... 191 C.3 TaylorSeries .................................... 192 C.4 AHigh-orderDoublet ............................... 192 C.5 EllipticalElement ............................... 193 Appendices 185 D Triangular Finite Elements 195 D.1 LocalCoordinateSystem . 195 D.2 UpstreamWeighting ............................... 197 D.3 StreamlineUpwindPetrov-Galerkin . .........198 CONTENTS viii D.4 NumericalIntegration . .......198 E Mass Balance Calculations 200 E.1 FiniteDifferenceMassBalance . ........200 E.1.1 TotalSystemMass............................... 200 E.1.2 Advection through system boundaries . ........200 E.1.3 SourceTerms................................... 201 E.1.4 SinkTerms ..................................... 201 E.1.5 DirichletSource/SinkFluxes . .......201 E.2 FiniteElementMassBalance . .......202 E.2.1 TotalSystemMass............................... 202 E.2.2 BoundaryLoss .................................. 202 E.2.3 SourceTerms................................... 202 E.2.4 SinksandSources ............................... 202 E.2.5 DirichletSource/SinkFluxes . .......203 E.3 Characteristic Methods Mass Balance . ..........204 F Adaptive Particle Tracking 205 G Cation Exchange Formulation 207 H Contents of Digital Appendix 209 References 209 List of Figures 3.1 Multiple levels of inheritance subclasses for the master analytic element abstraction CAnalyticElem ........................................ 42 3.2 Bluebird libraryclassorganization . 43 3.3 The abstraction of an analytic element model . ............ 44 3.4 Bluebird library class awareness. Analytic elements are only aware of the most ab- stract form of the layer that they are in, through which they may request information such as the potential from other elements or the hydraulic conductivity. 45 3.5 Flow chart of generic Bluebird iterative solution algorithm . 46 3.6 Flow chart of discharge
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