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PAGE 1 INVESTIGATION OF SALTWATER INTRUSION BASED ON THE HENRY PROBLEM AND A FIELD SCALE PROBLEM By CHANYUT KALAKAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014 PAGE 2 2014 Chanyut Kalakan PAGE 3 To my family PAGE 4 4 ACKNOWLEDGMENTS Born men are we all and one, Brown, black by the sun cultured. Knowledge can be won alike. Only the heart differs from man to man. The poem of His Majesty King Chulalongkorn, Rama V Translated by Mr. Seni Pramoj, the sixth Prime Minister of Thailand First of all, I would like to express my genuine gratitude to my supervisory committee chair, Dr. Louis H. Motz, for his mentorship while I have been at the University of Florida (UF). My gratitude is also extended to my c ommittee members namely Dr. Kirk Hatfield, Dr. Mark Newman, Dr. Robert J. Thieke and Dr. Elizabeth J. Screaton, whose assistance have been invaluable in the completion of my study. My admiration goes out to Anthony Murphy, Carolyn Carpenter and Nancy E. McIlrath Glanville who have helped me with my study. Another thank you go es to my friends in the Thai Student Association (TSA) at UF for their kind friendship. I wish to express my heartfelt appreciation to Dr. Charoen Chinwanitcharoen and D r. Mang Tia for their kindly help I would like to express my most sincere thanks to Dr. Suwatana Chittaladakorn and Dr. Bancha Kwanyuen for their invaluable advice and support for my study. I would like to extend my esteem to Dr. Arnon Wongkaew for his kind recommendations, Dr. Tieng Jivacate and Dr. Kitti Manokhoon for their helpful suggestions Dr. Thamnoon Rasmeemasmuang for his valuable comments, Dr. Wancheng Sittikijyothin for her cordial support, and the staffs of the Department of Civil PAGE 5 5 Engineerin g and Faculty of Engineering at Burapha University for their helpful asssitance. I would like to express my extreme thanks to my friends for their wonderful friendship especially Ms. P imara T hongsaeng (Dr. to be) for her welcoming help when I first arrive d in t he United States and Dr. Worasit Kanjanakijkasem for his kind support when I began my graduate study at UF. I also thank the staffs in the Office of t he Civil Service Commission in Thailand, and the Office of Educational Affairs at the Royal Thai Em bassy, in Washington D.C. for their assistance which served me during my time in the s cholarship p rogram Last but not least, the most cordial appreciation is expressed to my family, namely my father (Mr. Sompong Kalakan), my mother (Mrs. Cholao Kalakan) and my sisters and brother in law (Ms. Chatsuda Kalakan, Mrs. Parichat, and Mr. Jirakorn Cheunsuchon ), for their great encoura gement and support, and to Dr. Kotchaphan Chooluck for her patience and understanding. My Ph D study in UF was financially supported by the Royal Thai Government within the framework of the National Science and Technology Development Agency in the Science and Technology Scholarships Program (Grant: 50KA003). Additional funding was provided by the Burapha University, Chonburi, Thailand In actuality, all my student financial re sources have come from the to all of you PAGE 6 6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ .......... 13 LIST OF FIGURES ................................ ................................ ................................ ........ 20 LIST OF ABBREVIATIONS ................................ ................................ ........................... 42 ABSTRACT ................................ ................................ ................................ ................... 45 CHAPTER 1. INTRODUCTION ................................ ................................ ................................ .... 47 Back ground ................................ ................................ ................................ ............. 47 Saltwater Intrusion Problem ................................ ................................ .................... 49 Immiscible Fluids (Ghyben Herzberg Approximation) ................................ ...... 50 Miscible Fluids ................................ ................................ ................................ .. 52 ................................ ................................ .......................... 56 ................................ ................................ ............ 57 He ................................ ................................ ..................... 58 2. PREVIOUS INVESTIGATIONS ................................ ................................ .............. 60 Saltwater Intrusion Problem in the Physical and Numerical Model ......................... 60 Saltwater Intrusion Problem in the Ecology View ................................ .................... 64 Extent of Saltwater Intrusion ................................ ................................ ................... 65 Saltwater Recirculation ................................ ................................ ........................... 66 Henry Constant Dispersion Problem ................................ ................................ ....... 67 Henry Velocity dependent Dispersion Problem ................................ ...................... 69 Field scale Problem ................................ ................................ ................................ 72 3. STATEMENT OF PROBLEMS ................................ ................................ ............... 75 Henry Constant Dispersion Problem ................................ ................................ ....... 75 Statement of the Henry Constant Dispersion Problem ................................ ..... 75 Objectives of the Henry Constant Dispersion Problem ................................ ..... 75 Henry Velocity dependent Dispersion Problem ................................ ...................... 76 Statement of the Henry Velocity dependent Dispersion Problem ..................... 76 Objectives of the Henry Velocity dependent Dispersion Problem .................... 76 Field scale Problem ................................ ................................ ................................ 77 Statement of the Field scale Problem ................................ .............................. 77 Objectives of the Fiel d scale Problem ................................ .............................. 77 PAGE 7 7 Solutions of the Three Problems ................................ ................................ ............. 78 Statement of Solutions of the Three Problems ................................ ................. 78 Objectives of Solutions of the Three Problems ................................ ................. 78 4. BACKGROUND EQUATIONS ................................ ................................ ................ 79 Governing Equations for Flow and Transport Model ................................ ............... 79 Governing Equations for Flow Model ................................ ............................... 79 Governing Equations for Transport Model ................................ ........................ 80 ................................ ................................ ................................ ...... 83 Accuracy and Sta bility Criteria for Numerical Model ................................ ............... 84 Grid Pclet number ................................ ................................ .......................... 84 Courant number ................................ ................................ ............................... 85 5. DIMENSIONAL ANALYSIS ................................ ................................ ..................... 86 Dime nsional Analysis for Saltwater Intrusion Problem ................................ ............ 86 Dimensionless Ratios for the Exploration of Saltwater Intrusion Problem ........ 86 Flow Domain ................................ ................................ ................................ .... 88 Considered Variables ................................ ................................ ....................... 89 Determin ation of PI Terms for the Investigation and Quantitation of Saltwater Intrusion Problem ................................ ................................ ................................ 91 Dimensionless Extent of Saltwater Intrusion ) ................................ .............. 91 Dimensionless Recirculation of Seawater ( ) ................................ ................. 98 ................................ ................................ 98 Henry Constant Dispersion Problem ................................ ................................ ..... 100 Henry Velocity dependent Dispersion Problem ................................ .................... 101 Field scale Problem ................................ ................................ .............................. 102 6. DESIGN OF NUMERICAL EXPERIMENTS AND PARAMETERS ....................... 104 Numerical Modeling ................................ ................................ .............................. 104 Numerica l Solutions ................................ ................................ ....................... 105 Uncoupled solution ................................ ................................ .................. 105 Coupled solution ................................ ................................ ...................... 106 Boundary Conditions ................................ ................................ ...................... 108 Equivale nt freshwater heads at the downgradient boundary ................... 108 Concentration condition at the downgradient and upgradient boundaries 110 Finite difference Grid and Boundary Conditions ................................ ............. 110 Henry constant dispersion and Henry velocity dependent dispersion problems ................................ ................................ ............................... 110 Field scale problem ................................ ................................ .................. 112 Methods for Henry Constant Dispersion Problem ................................ ................. 114 Assumption of the Henry Constant Dispersion Problem ................................ 114 Investigation Procedures of the Henry Constant Dispersion Problem ............ 114 Aquifer Parameters of the Henry Constant Dispersion Problem ..................... 117 PAGE 8 8 Methods for Henry Velocity dependent Dispersion Problem ................................ 118 Assumption of the Henry Velocity dependent Dispersion Problem ................ 118 Investigation Procedures of the Henry Velocity dependent Dispersion Problem ................................ ................................ ................................ ....... 118 Aquifer Parameters of the Henry Velocity dependent Dispersion Problem .... 119 Methods of Field scale Problem ................................ ................................ ........... 122 Assumption of the Field scale Problem ................................ .......................... 122 Investigation Procedures of the Field scale Problem ................................ ..... 122 Aquifer Parameters of the Field scale Problem ................................ .............. 124 7. HENRY CONSTANT DISPERSION PROBLEM ................................ ................... 125 Benchmark Problem for the Henry Constant Dispersion Problem ........................ 125 Aquifer Parameters for Simpson and Clement (2004) ................................ .... 127 Methods and Schemes for Simpson and Clement (2004) .............................. 127 SEAWAT Version 4 Results for Simpson and Clement (2004) ....................... 128 Henry Constant Dispersion Problem ................................ ................................ ..... 129 Modified Method and Sche mes for the Henry constant dispersion problem ... 130 Input Parameters for the First Set of Investigations ................................ ........ 134 Results of the First Set of Investigations ................................ ........................ 136 Comparison of the 0.5 isochlor results between uncoupled and coupled solutions for the same value of the dimensionless ratio ..................... 136 Comparison of the 0.5 isochlor results between uncoupled and coupled solutions for the same value of the dimensionless ratio ..................... 138 Comparison of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) between uncoupled and coupled solutions for the first set of investigations ................................ ................................ ....... 140 Input Parameters for the Second Set of Investigations ................................ .. 150 Results of the Second Set of Investigations ................................ ................... 152 Comparison of the 0.5 isochlor results between unc oupled and coupled solutions for the same value of the dimensionless ratio ..................... 152 Comparison of the 0.5 isochlor results between u ncoupled and coupled solutions for the same value of the dimensionless ratio ..................... 154 Comparison of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) between uncoupled and coupled solutions for the second set of investigations ................................ ................................ .. 157 Comparison Results between the First and Second Sets of Investigations .... 167 Comparison patterns of the 0.5 isochlors between the first and second sets of investigations for the same value of the dimensionless ratio .. 16 7 Comparison patterns and values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) between the first and second sets of investigations ................................ ................................ 171 Summary of Comparison of Results between the First and Second Sets of Investigations of the Henry Constant Dispersion Problem .......................... 175 PAGE 9 9 Accuracy and Stability Criteria for the Henry Constant Dispersion Problem .. 176 Grid Pclet number ................................ ................................ .................. 176 Grid Pclet number of the first set of investigations ................................ 176 Grid Pclet number of the second set of investigations ........................... 179 Summary of Grid Pclet Number for the Henry Constant Dispersion Problem ................................ ................................ ................................ ....... 182 Discussion and Conclusions for the Henry Constant Dispersion Pro blem ............ 183 8. HENRY VELOCITY DEPENDENT DISPERSION PROBLEM .............................. 190 Benchmark Problems for the Henry Velocity dependent Dispersion Problem ...... 190 Notice of Sgol (1993) and Abarca et al. (2007) ................................ ............ 191 Aquifer Parameters for Sgol (1993) and Abarca et al. (2007) ....................... 198 problem for Sgol (1993) and Abarca et al. (2007) ............................... 198 Aquifer parameters for the coupled solutions of Abarca et al. (2007) reference cases ................................ ................................ .................... 198 Methods and Schemes for Sgol (1993) and Abarca et al. (2007) ................. 199 Sgol (1993) and Abarca et al. (2007) ................................ ........................ 200 (1964) Problem of Sgol (1993) and Abarca et al. (2007) Works ................ 203 Problem of Sgol (1993) and Abarca et al. (2007) Works, including Abarca et al. (2007) Reference Ca ses ................................ ........................ 204 and Abarca et al. (2007) works ................................ ............................. 204 The coupled solutions of Abarca et al. (2007) reference cases ............... 204 Henry Velocity dependent Dispersion Problem ................................ .................... 205 Methods and Scheme s for the Henry Velocity dependent Dispersion Problem ................................ ................................ ................................ ....... 205 Input Parameters for the Henry Velocity dependent Dispersion Problem ...... 206 Results of Henry Velocity dependent Dispersion Investigations ........................... 210 Comparison of the 0.5 Isochlor Results between Uncoupled and Coupled Solutions ................................ ................................ ................................ ..... 210 Results for the Henry Velocity dependent Dispersion Problem: Dispersivity Ratio ( ) = 0.01; and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ............... 211 Results for the Henry Velocity dependent Dispersion Problem: Dispersivity Ratio ( ) = 0.10; and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ............... 213 Results for the Henry Velocity dependent Dispersion Problem: Dispersivity Ratio ( ) = 1.00; and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ............... 215 Results for the Henry Velocity dependent Dispersion Problem: Hydraulic Conductivity Ratio ( ) = 0.01 where = 0.1, 1.0 and 10.0 ; and Dispersivity Ratio s ( ) = 0.01, 0.10 and 1.00 ................................ 217 PAGE 10 10 Results for the Henry Velocity dependent Dispersion Problem: Hydraulic Conductivity Ratio ( ) = 0.10 where = 0.1, 1.0 and 10.0; and Dispersivity Ratio s ( ) = 0.01, 0.10 and 1.00 ................................ 218 Results for the Henry Velocity dependent Dispersion Problem: Hydraulic Conductivity Ratio ( ) = 1.00 where = 0.1, 1.0 and 10.0; and Dispersivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ 219 Comparison of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) between Uncoupled and Coupled Solutions ........................ 220 Results of Extent of S altwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Dispersion Problem when Dispersivity Ratio ( ) = 0.01 and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ..... 221 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Dispersion Problem when Dispersivity Ratio ( ) = 0.10 and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ..... 228 Results of Extent of Saltwater Intrusion ( ) and Re circulation of Seawater ( ) for the Henry Velocity dependent Dispersion Problem when Dispersivity Ratio ( ) = 1.00 and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ..... 235 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Dispersion Problem when Hydraulic Conductivity Ratio ( ) = 0.01 and Dispersivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ..... 242 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawate r ( ) for the Henry Velocity dependent Dispersion Problem when Hydraulic Conductivity Ratio ( ) = 0.10 and Dispersivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ..... 247 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Dispersion Problem when Hydraulic Conductivity Ratio ( ) = 1.00 and Dispersivity Ratios ( ) = 0.01, 0.10 and 1.00 ................................ ................................ ..... 252 Discussion and Conclusions for the Henry Velocity dependent Dispersion Problem ................................ ................................ ................................ ............. 257 9. FIELD SCALE PROBLEM ................................ ................................ .................... 261 Benchmark Problem for the Field scale Problem ................................ .................. 261 Aquifer Parameters for Motz and Sedighi (2013) ................................ ........... 262 Methods and Schemes for Motz and Sedighi (2013) ................................ ...... 262 SEAWAT Version 4 Results for Motz and Sedighi (2013) .............................. 263 Field scale Problem ................................ ................................ .............................. 264 Methods and Schemes ................................ ................................ ................... 264 Input Parameters ................................ ................................ ............................ 265 PAGE 11 11 Results of the Field scale Problem ................................ ................................ ....... 268 Comparison of the 0.5 isochlor results of = 0.1, 1.0 and 10.0 with the dispersivity ratio ( ) = 0.01 where = 0.01, 0.10 and 1.00. ...................... 268 Comparison of the 0.5 isochlor results of the dispersivity ratio ( ) = 0.01 where = 0.1, 1.0 and 10.0, and = 0.01, 0.10 and 1.00. ....................... 270 Comparison of the Extent of Saltwater Intrusion ( ) and the Recirculation of Seawater ( ) betwee n Uncoupled and Coupled Solutions .................... 271 Comparison of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) when Decreasing the Value of Hydraulic Conductivity ............................... 279 The Verification of the Consistency of Results for the Saltwater Numerical Model in case of = 0.01 ................................ ................................ ........... 279 The regular and verifi ed sets for the field scale problem in case of = 0.01 ................................ ................................ ................................ ....... 279 Aquifer parameters and dimensionless ratios in case of = 0.01 ........... 279 Results of the regular and the verified sets in case of = 0.01 .............. 280 The Verification of the Consistency of Results for the Saltwater Numerical Model in case of = 0.10 ................................ ................................ ........... 289 The regular and verified sets from the field scale problem in case of = 0.10 ................................ ................................ ................................ .... 289 Aquifer parameters and dimensionless ratios in case of = 0.10 ........... 289 Results of the regular and the verified sets in case of = 0.10 .............. 290 The Verification of the Consistency of Results for the Saltwater Numerical Model in case of = 1.00 ................................ ................................ ........... 299 The regular and verified sets for the field scale problem in case of = 1.00 ................................ ................................ ................................ ....... 299 Aquifer parameters and dimensionless ratios in case of = 1.00 ........... 299 Results of the regular and the verified sets in case of = 1.00 .............. 300 Discussion and Conclusions for the Field scale Problem ................................ ..... 309 10. CONCLUSIONS OF INVESTIGATIONS ................................ ............................... 313 Summary of Contributions ................................ ................................ .................... 313 Opportunities of Future Work ................................ ................................ ................ 317 APPENDIX A. BENCHMARK AND HENRY CONSTANT DISPERSION PROBLEMS ................ 319 Benchmark Problem ................................ ................................ ............................. 319 Optimal Finite Difference Grid ................................ ................................ ............... 319 Finite difference Grid Optimization ................................ ................................ 320 Boundary Conditions and Solutions ................................ ............................... 323 Aquifer Parameters ................................ ................................ ........................ 324 Methods and Schemes ................................ ................................ ................... 324 PAGE 12 12 Results of Alternate Finite difference Grids ................................ .................... 325 Isochlors and velocity distribution ................................ ............................ 325 Instabilities and oscillations ................................ ................................ ...... 329 Transient 0.5 isochlors movement at the base of the domain .................. 333 Changing of the concentration of total dissolved solids (tds) at the edge of the domain ................................ ................................ ........................ 335 Compared alternative results to benchmark problems in terms of the 0.5 isochlors ................................ ................................ ............................... 338 Optimal Finite Difference Discretization ................................ ......................... 339 Aquifer Parameters and Dimensi onless ratios for the First Set of Investigations of the Henry Constant Dispersion Problem ................................ ........................ 340 Results of the First Set of Investigations of the Henry Constant Dispersion Problem ................................ ................................ ................................ ............. 345 Aquifer Parameters and Dimensionless ratios for the Second Set of Investigations of the Henry Constant Dispersion Problem ................................ 350 Results of the Second Set of Investigations of the Henry Constant Disp ersion Problem ................................ ................................ ................................ ............. 355 Comparison the Values of the Extent of Saltwater Intrusion ( ) and the Degree of saltwater recirculation ( ) between the First and Second Sets of Investigations for the Henry Constant Dispersion Problem ................................ 360 Accuracy and Stability Cri teria for the Henry Constant Dispersion Problem of the Henry Constant Dispersion Problem ................................ ............................ 370 B. BENCHMARK AND HENRY VELOCITY DEPE NDENT DISPERSION PROBLEMS ................................ ................................ ................................ .......... 380 Benchmark Problems ................................ ................................ ............................ 380 SEAWAT Version 4 Results of Benchmark Problems ................................ .......... 381 Aquifer Parameters and Dimensionless Variables of the Henry Velocity dependent Dispersion Problem ................................ ................................ ......... 386 Results of Extent of Saltwater Intrusion ( ) and Degree of saltwater recirculation ( ) of the Henry Velocity dependent Dispersion Problem ........... 395 C. BENCHMARK AND FIELD SCALE PROBLEMS ................................ ................. 404 Benchmark Problem ................................ ................................ ............................. 404 SEAWAT Version 4 Results of Benchmark Problem ................................ ............ 404 Aquifer Parameters and Dimensionless Variables of the Field scale Problem ..... 408 Results of Extent of Saltwater Intrusion ( ) and Degree of Recirculation of Seawater ( ) for the Field scale Problem ................................ ........................ 411 LIST OF REFERENCES ................................ ................................ ............................. 414 BIOGRAPH ICAL SKETCH ................................ ................................ .......................... 422 PAGE 13 13 LIST OF TABLES Table page 5 1 Variables affecting saltwater intrusion. ................................ ............................... 89 5 2 Dependent variable ( ), independent variables and their dimensions to determine the dimensionless extent of saltwater intrusion ( ). ......................... 92 6 1 Aquifer parameters used for the Henry constant dispersion problem. .............. 117 6 2 Aquifer parameters for the Henry velocity dependent dispersion problem. ...... 121 6 3 Aquifer parameters used in the field scale problem. ................................ ......... 124 7 1 Aquifer parameters used in the first set of investigations. ................................ 134 7 2 Summation of dimensionless ratios used for the first set of investigations. ...... 135 7 3 Aquifer parameters for the second set of investigations. ................................ .. 151 7 4 Summation of dimensionless ratios for the second set of investig ations. ......... 152 7 5 Summation of the grid Pclet number of the first set of investigations. ............ 177 7 6 Summation of the grid Pclet number of the second set of investigations. ....... 179 8 1 Aquifer parameters used for Abarca et al. (2007) reference cases with values of parameters from Povich et al. (2013). ................................ .......................... 199 8 2 Summation of dimensionless ratios used for the Henry velocity dependent dispersion problem. ................................ ................................ .......................... 209 9 1 Aquifer parameters used for the field scale problem to reproduce the work of Motz and Sedighi (2013). ................................ ................................ ................. 262 9 2 Aquifer parameters used in the field scale investigations. ................................ 266 9 3 Summation of dimensionless ratios used for the field scale problem. .............. 268 9 4 Aquifer parameters and dimensionless ratios for six selected regular cases to verify the consistency of results for = 0.01. ................................ .................. 281 9 5 Aquifer parameters and dimensionless ratios for six verified cases to verify the consistency of results when decreasing the value of hydraulic con ductivity by a factor of two for = 0.01. ................................ ..................... 281 PAGE 14 14 9 6 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing the value of hydraulic conductivity by a factor of two and the hydraulic conductivity ratio ( ) = 0.01 for uncoupled solutions. .......... 282 9 7 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing the value of hydraulic conductivity by a factor of two and the hydraulic conductivity ratio ( ) = 0.01 for coupled solutions. .............. 282 9 8 Aquifer parameters and dimensionless ratios for six selected regular cases to verify the consistency of results for = 0.10. ................................ .................. 291 9 9 Aquifer parameters and dimensionless ratios for six verified cases to verify the consistency of results when decreasing the value of hydraulic conductivity by a factor of two for = 0.10. ................................ ..................... 291 9 10 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing the value of hydraulic conductivity by a fact or of two and the hydraulic conductivity ratio ( ) = 0.10 for uncoupled solutions. .......... 292 9 11 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing the value of hydraulic conductivity by a factor of two and the hydraulic conductivity ratio ( ) = 0.10 for coupled solutions. .............. 292 9 12 Aquifer parameters and dimensionless ratios for six selected regular cases to verify the consistency of results for = 1.00. ................................ .................. 301 9 13 Aquifer parameters and dimensionless ratios for six verified cases to verify the consistency of results when decreasing the value of hydraulic conductivi ty by a factor of two for = 1.00. ................................ ..................... 301 9 14 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing the value of hydraulic conductivity by a factor of two and the hydraulic conductivity ratio ( ) = 1.00 for uncoupled solutions. .......... 302 9 15 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing the value of hydraulic conductivity by a factor o f two and the hydraulic conductivity ratio ( ) = 1.00 for coupled solutions. .............. 302 A 1 Alternative finite difference grids used for for the optimal finite difference grid. 322 A 2 Aquifer parameters and dimensionless ratios in case of = 0.1 for the first set of investigations. ................................ ................................ ......................... 340 A 3 Aquifer parameters and dimensionless ratios in case of = 0.2 for the first set of investigations. ................................ ................................ ......................... 341 A 4 Aquifer parameters and dimensionless ratios in case of = 0.3 for the first set of investigations. ................................ ................................ ......................... 342 PAGE 15 15 A 5 Aquifer parameters and dimensionless ratios in case of = 0.4 for the first set of investigation s. ................................ ................................ ......................... 343 A 6 Aquifer parameters and dimensionless ratios in case of = 0.5 for the first set of investigations. ................................ ................................ ......................... 344 A 7 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.1 of the first set of investigations. ................................ ................................ .. 345 A 8 Exte nt of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.2 of the first set of investigations. ................................ ................................ .. 346 A 9 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.3 of the first set of investigations. ................................ ................................ .. 347 A 10 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.4 of the first set of investigations. ................................ ................................ .. 348 A 11 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.5 of the first set of investigations. ................................ ................................ .. 349 A 12 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.1 for the second set of investigations. ................................ ............. 350 A 13 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.2 for the second set of investigations. ................................ ............. 351 A 14 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.3 for the second set of investigations. ................................ ............. 352 A 15 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.4 for the second set of investigati ons. ................................ ............. 353 A 16 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.5 for the second set of inves tigations. ................................ ............. 354 A 17 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.1 of the second set of investigations. ................................ ............................. 355 A 18 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.2 of the second set of investigations. ................................ ............................. 356 PAGE 16 16 A 19 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.3 of the second set of investigations. ................................ ............................. 357 A 20 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.4 of the second set of investigations. ................................ ............................. 358 A 21 Extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.5 of the second set of investigations. ................................ ............................. 359 A 22 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of investigations in case of = 0.1. ................................ .............. 360 A 23 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.1. ................................ .............. 361 A 24 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of investigations in case of = 0.2. ................................ .............. 362 A 25 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation for coupled solutions between the first and se cond sets of investigations in case of = 0.2. ................................ .............. 363 A 26 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of investigations in case of = 0.3. ................................ .............. 364 A 27 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.3. ................................ .............. 365 A 28 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first an d second sets of investigations in case of = 0.4. ................................ .............. 366 A 29 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.4. ................................ .............. 367 A 30 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of investigations in case of = 0.5. ................................ .............. 368 PAGE 17 17 A 31 Comparison the values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.5. ................................ .............. 369 A 32 Summation of the grid Pclet number of = 0.1 for the first set of investigations. ................................ ................................ ................................ ... 370 A 33 Summation of the grid Pclet number of = 1.0 for the first set of investig ations. ................................ ................................ ................................ ... 371 A 34 Summation of the grid Pclet number of = 10.0 for the first set of investigations. ................................ ................................ ................................ ... 372 A 35 Summation of the grid Pclet number of the first set of investigations. ............ 373 A 36 Summation of the grid Pclet number of uncoupled and coupled solutions for the first set of investigations in case of = 0.1. ................................ ................ 374 A 37 Summation of the grid Pclet number of = 0.1 for the second set of investigations. ................................ ................................ ................................ ... 375 A 38 Summation of the grid Pclet number of = 1.0 for the second set of investigations. ................................ ................................ ................................ ... 376 A 39 Summation of the grid Pclet number of = 10.0 for the second set of investigations. ................................ ................................ ................................ ... 377 A 40 Summation of the grid Pclet number of the second set of investigations. ....... 378 A 41 Summation of the grid Pclet number of uncoupled and coupled solutions for the second set of investigations in case of = 0.1. ................................ .......... 379 B 1 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.01 and = 0.01. ................... 386 B 2 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.01 and = 0.10. ................... 387 B 3 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.01 and = 1.00. ................... 388 B 4 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.10 and = 0.01. ................... 389 B 5 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.10 and = 0.10. ................... 390 PAGE 18 18 B 6 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.10 and = 1.00. ................... 391 B 7 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 1.00 and = 0.01. ................... 392 B 8 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 1.00 and = 0.10. ................... 393 B 9 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 1.00 and = 1.00. ................... 394 B 10 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.01 and = 0.01. ................................ ..... 395 B 11 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.01 and = 0.10. ................................ ..... 396 B 12 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.01 and = 1.00. ................................ ..... 397 B 13 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.10 and = 0.01. ................................ ..... 398 B 14 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.10 and = 0.10. ................................ ..... 399 B 15 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.10 and = 1.00. ................................ ..... 400 B 16 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 1.00 and = 0.01. ................................ ..... 401 B 17 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 1.00 and = 0.10. ................................ ..... 402 B 18 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 1.00 and = 1.00. ................................ ..... 403 PAGE 19 19 C 1 Reproduced results of SEAWAT for uncoupled and coupled solutions of Motz and Sedighi (2013). ................................ ................................ .......................... 407 C 2 Aquifer parameters and dimensionless variables for the field scale problem in case o f = 0.01 and = 0.01. ................................ ................................ ........ 408 C 3 Aquifer parameters and dimensionless variables for the field scale problem in case of = 0.01 and = 0.1. ................................ ................................ .......... 409 C 4 Aquifer parameters and dimensionless variables for the field scale problem in case of = 0.01 and = 1.0. ................................ ................................ .......... 410 C 5 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the field scale problem in case of = 0.01 and = 0.01. ................................ ................................ ........................ 411 C 6 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncou pled and coupled solutions of the field scale problem in case of = 0.01 and = 0.1. ................................ ................................ .......................... 412 C 7 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the field scale problem in case of = 0.01 and = 1.0. ................................ ................................ .......................... 413 PAGE 20 20 LIST OF FIGURES Figure page 1 1 Trends in population and freshwater withdrawals by source in the U.S.A., 1950 2000 (Kenny et al. 2009). ................................ ................................ .......... 47 1 2 The geographic distribution of total surface water and groundwater withdrawals in the U.S.A., 2005 (Kenny et al. 2009). ................................ .......... 48 1 3 Ghyben Herzberg approximation model (Custodio 1987). ................................ 51 1 4 Typical cross section of the saltwater intrusion problem (Custodio 1987). ......... 52 1 5 Isochlor in the Biscayne aquifer near Miami, Florida (Cooper 1964). ................. 54 1 6 Isochlor in the New Jersey Coastal Plain in New Jersey and on the Cont inental Shelf (Meisler 1989). ................................ ................................ ....... 54 1 7 Cooper 1970). ................................ ................................ ................................ ..... 56 1 8 Flow and salt concentration patterns of Henry idealized mathematical model A) Streamlines and B) Isochlors (Henry 1964). ................................ .................. 58 2 1 Results obtained from Simpson and Clement (2004) A) the standard and B) the modified Henry problems. ................................ ................................ ............. 68 2 2 Isochlors of A) Sgol (1993) and B) Abarca et al. (2007) for Henry problem. ..... 69 2 3 Isochlors of Abarca et al. (2007) for the solutions of the reference cases, namely A) diffusive case and B) dispersive case. ................................ ............... 71 2 4 Results of Motz and Sedighi (2013) A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for both uncoupled and coupled solutions. ................................ ................................ ................................ ............ 74 5 1 Investigated dimensionless ratios which related to extent of saltwater intrusion ( ) and deg ree of saltwater recirculation ( ) ................................ ... 87 5 2 Domain of the saltwater intrusion problem. ................................ ........................ 88 6 1 Generalized flow chart of the SEAWAT program for the uncoupled solution (modified from Guo and Langevin 2002). ................................ ......................... 106 6 2 Generalized flow chart of the SEAWAT program for the coupled solution (modified from Guo and Langevin 2002). ................................ ......................... 107 PAGE 21 21 6 3 Piezometer to calculate equivalent freshwater head (Guo and Langevin. 2002). ................................ ................................ ................................ ............... 109 6 4 Boundary conditions of the numerical modeling. ................................ .............. 109 6 5 Finite difference grid used for the Henry constant dispersion and the Henry velocity dependent dispersion problems. ................................ ......................... 111 6 6 Two dimensional finite difference grid and boundary conditions used for the Henry constant dispersion and the Henry velocity dependent dispersion problems. ................................ ................................ ................................ .......... 112 6 7 Finite difference grid used in field scale problem. ................................ ............ 113 6 8 Two dimensional finite difference grid and boundary conditions used for the field scale problem. ................................ ................................ .......................... 113 6 9 Schematic of investigation procedures of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 116 6 10 Schematic of investigation procedures of the Henry velocity dependent dispersion problem. ................................ ................................ .......................... 119 6 11 The relation of longtidutinal dispersivity and length scale with data classified by reliability (modified from Gelhar et al. 1992). ................................ ............... 120 6 11 Schematic of the investigation procedures of the field scale problem. ............. 123 7 1 Results obtained from Simpson and Clement (2004) A) the standard and B) the modified Henry problems. ................................ ................................ ........... 126 7 2 The work of Langevin and Guo (2006) A) the standard and B) the modified Henry problems. ................................ ................................ ............................... 126 7 3 Results of Motz and Sedighi (2013) A) the standard and B) the modified Henry problems. ................................ ................................ ............................... 126 7 4 Comparison between the 0.5 isochlors of SEAWAT results and the work of Simpson and Clement (2004) for A) the standard and B) the modified Henry problems b oth uncoupled and coupled solutions. ................................ ............. 129 7 5 Comparison of results when using TTSMULT = 1.2 and 1.5 for A) extent of saltwater in trusion ( ) and B) degree of saltwater recirculation ( ) of uncoupled solutions for the first set. ................................ ................................ 133 7 6 Comparison of results when using TTSMULT = 1.2 and 1.5 for A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) of unco upled solutions for the second set. ................................ ........................... 133 PAGE 22 22 7 7 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.1 for the first set of investigations. ............... 136 7 8 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.2 for the first set of investigations. ............... 137 7 9 The 0.5 isoc hlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.3 for the first set of investigations. ............... 137 7 10 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.4 for the first set of investigations. ............... 137 7 11 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.5 for t he first set of investigations. ............... 138 7 12 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1 in case of increasing from 0.1 to 0.5 for the first set of investigations. ..... 138 7 13 The 0.5 isochlors of A) uncoupled s olutions and B) coupled solutions for = 1.0 in case of increasing from 0.1 to 0.5 for the first set of investigations. ..... 139 7 14 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 10.0 in case of increasing from 0.1 to 0.5 for the first set of investigations. ... 139 7 15 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.1. 141 7 16 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.2. 141 7 17 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 i n case of = 0.3. 142 7 18 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.4. 142 7 19 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 i n case of = 0.5. 143 7 20 Extent of saltwater intrusion ( ) of the first set of investigations for A) uncoupled and B) coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5. ................................ ........... 143 PAGE 23 23 7 21 Degree of saltwater recirculation ( ) of the first set of investigations A) uncoupled and B) coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5. ................................ ........... 144 7 22 Extent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color f loods and velocity vectors for the first set of investigations. ................................ ........................... 146 7 23 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the first set of investigations. ................................ ........................... 147 7 24 Degree of saltwater recirculation ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the first set of investigations. ................................ ........................... 148 7 25 Degree of saltwater recirculation ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors f or the first set of investigations. ................................ ........................... 149 7 26 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.1 for the second set of investigations. ......... 153 7 27 The 0.5 isochlors of A) uncoupled solutio ns and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.2 for the second set of investigations. ......... 153 7 28 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.3 for the second set of investigations. ......... 153 7 29 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.4 for the second set of investigations. ......... 154 7 30 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.5 for the second set of investigations. ......... 154 7 31 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1 in case of increasing from 0.1 to 0.5 for the second set of investig ations. ................................ ................................ ................................ ... 155 7 32 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 1.0 in case of increasing from 0 .1 to 0.5 for the second set of investigations. ................................ ................................ ................................ ... 155 PAGE 24 24 7 33 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 10.0 in case of increasing from 0.1 to 0.5 for the second set of investigations. ................................ ................................ ................................ ... 155 7 34 Results of the second set of in vestigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.1. 157 7 35 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and cou pled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.2. 158 7 36 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.3. 158 7 37 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.4. 159 7 38 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.5. 159 7 39 Extent of saltwater intrusion ( ) of the second set of investigations for A) uncoupled and B) coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5. ................................ ........... 160 7 40 Degree of saltwater recirculation ( ) of the second set of investigations for A) uncoupled and B) coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5. ................................ ........... 160 7 41 Extent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration col or floods and velocity vectors for the second set of investigations. ................................ ..................... 163 7 42 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of investigations. ................................ ..................... 164 7 43 Degree of saltwater recirculation ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of investigations. ................................ ..................... 165 PAGE 25 25 7 44 Degree of saltwater recirculation ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of investigations. ................................ ..................... 166 7 45 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.1 between A) the first set and B) the second set of investigations. ................................ ................................ ............. 167 7 46 Comparison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.1 between A) the first set and B) the second set of investigations. ................................ ................................ ............. 167 7 47 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.2 between A) the first set and B) the second set of the Henry constant dispersion problem. ................................ ..... 168 7 48 Comparison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.2 between A) the first set and B) the second set of investigations. ................................ ................................ ............. 168 7 49 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.3 between A) the first set and B) the second set of investigations. ................................ ................................ ............. 168 7 50 Comparison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.3 between A) the first set and B) the second set of investigations. ................................ ................................ ............. 169 7 51 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.4 between A) the first set and B) the second set of investigations. ................................ ................................ ............. 169 7 52 Comparison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.4 between A) the first set and B) the second set of investigations. ................................ ................................ ............. 169 7 53 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.5 between A) the first set and B) th e second set of investigations. ................................ ................................ ............. 170 7 54 Comparison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.5 between A) the first set and B) the second set of investigations. ................................ ................................ ............. 170 PAGE 26 26 7 55 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.1. ................................ ................................ ... 171 7 56 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) f or uncoupled and coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.2. ................................ ................................ ... 172 7 57 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions between the first and second sets of investigations versus the dimens ionless ratio from 0.1 to 10.0 in case of = 0.3. ................................ ................................ ... 172 7 58 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.4. ................................ ................................ ... 173 7 59 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions between the first a nd second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.5. ................................ ................................ ... 173 7 60 Comparison of values of extent of saltwater intrusion ( ) for A) uncoupled and B) coupled solutions between the first and second sets of investigation s versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5. ................................ ................................ ................................ ......... 174 7 61 Comparison of values of d egree of saltwater recirculation ( ) for A) uncoupled and B) coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5. ................................ ................................ ............. 174 7 62 Isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the uncoupled solution in case of the dimensionless ratios = 0. 1 and = 0.1 for the first set of investigations. ................................ ................................ 177 7 63 Isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the coupled solution in case of the dimensionless ratios = 0.1 and = 0.1 for the first set of investigations. ................................ ................................ 178 7 64 Comparison of the grid Pclet numbers for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.1 for the first set of investigati ons. ................................ ................................ .................. 178 PAGE 27 27 7 65 Isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the uncoupled solution in case o f the dimensionless ratios = 0.1 and = 0.1 for the second set of investigations. ................................ ........................... 180 7 66 Isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the coupled solution in case of the dimensionless ratios = 0.1 and = 0.1 for the second set of investigations. ................................ ........................... 181 7 67 Comparison of the grid Pclet numbers for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.1 for the second set of investigations. ................................ ................................ ............. 181 7 68 Comparison of the grid Pclet number between the first and second sets of the Henry constant dispersion problem in case of = 0.1. ............................... 182 8 1 The works of A) Sgol (1993) and B) Abarca et al. (2007) illustrate isochlors ................................ ..................... 190 8 2 Abarca et al. (2007) isochlors for the solutions of the reference cases, namely A) diffusive case (constant dispersion case) and B) dispersive case (velocity dependent dispersion case) using the equivalent freshwater head and the varied concentration of seawater at the coastal boundary. .................. 191 8 3 con stant concentration of seawater at the coastal boundary (Simpson and Clement 2004). ................................ ................................ ................................ 192 8 4 Isochlors of Henry idealized mathemati cal model using the constant concentration of seawater at the coastal boundary (Henry 1964). .................... 192 8 5 = 0.1 using the constant concentration of seawater at the coastal boundary revised by Sgol (1993). ................................ ................................ ................................ .... 193 8 6 The steady state concentration distribution from the finite element numerical solution using a constant scalar value of hydrodynamic dispersion ( ) and the varied concentration of seawater at the coastal boundar y by Sgol personal communicated with Clifford I. Voss in 1992 Sgol (1993). ................. 193 8 7 The work of Voss and Souza (1987) presented 0.5 isochlors with numerical solutions using a constant scalar value of hydrodynamic dispersion ( ) = 18.8571x10 6 m 2 /s and the varied concentration of seawater at the coastal boundary. The result from = 6.6x10 6 m 2 /s numerical isochlors (dotted) is also shown (Sgol 1993; Voss and Souza 1987). ................................ ................................ ............................. 194 PAGE 28 28 8 8 The results of the standard Henry problem from Simpson and Clement (2004) present the comparison of coupled numerical and semi analytical results for = 0.263, = 0.1 and = 2.0 and the constant concentration of seawater at the coastal boundary. The uncoupled 50% numerical isochlors (dashed) is also shown. ................................ ................................ .................... 195 8 9 The work of Abarca et al. (2007) illustrates isochlors for the solution of ................................ ................................ ................... 195 8 10 Abarc a et al. (2007) isochlors for the solutions of the reference cases, namely A) diffusive case (constant dispersion case) and B) dispersive case (velocity dependent dispersion case) using the equivalent freshwater head and the varied concentration of seawate r at the coastal boundary. .................. 196 8 11 The numerical model setup to reproduce Abarca et al. (2007) Henry problem and reference cases using the bulk fluid mass and momentum balance equations and the solute mass balance equation were coupled through density and viscosity equations of state in the domain (Povich et al. 2013). .... 197 8 12 The work of Povich et al. (2013) presented isochlors for coupled solutions of Abarca et al. (2007) reference cases, namely A) diffusive case and B) dispersive case. ................................ ................................ ................................ 197 8 13 problem of Sgol (1993) and Abarca et al. (2007) works using the intern ally calculate equivalent freshwater heads and the varied concentration at the downgradient condition. ................................ ................................ .................... 201 8 14 SEAWAT Version 4 isochlors for the coupled solutions of Henry problem using the manually input equivalent freshwater heads and the varied concentration at the downgradient condition A) Sgol (1993) and B) Abarca et al. (2007). ................................ ................................ ................................ ..... 204 8 15 SEAWAT Version 4 isochlors for the coupled solutions of Abarca et al. (2007) reference cases, using the manually input equivalent freshwater heads and the varied conce ntration at the downgradient conditions A) diffusive case (constant dispersion case) and B) dispersive case (velocity dependent dispersion case). ................................ ................................ ............. 205 8 16 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 0.01. ............ 211 8 17 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 0.10. ............ 211 8 18 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 1.00. ............ 211 PAGE 29 29 8 19 Comparison of the pattern of the 0.5 isochlors of A) uncoup led B) coupled solutions for = 0.01 in case of = 0.1 and = 0.01, 0.10 and 1.00. .......... 212 8 20 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 1.0 and = 0.01, 0.10 and 1.00. .......... 212 8 21 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 10.0 and = 0.01, 0.10 and 1.00. ........ 212 8 22 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.10 and = 0.01. ............ 213 8 23 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in ca se of = 0.10 and = 0.10. ............ 213 8 24 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.10 and = 1.00. ............ 213 8 25 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 0.1 and = 0.01, 0.10 and 1.00. .......... 214 8 26 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled soluti ons for = 0.10 in case of = 1.0 and = 0.01, 0.10 and 1.00. .......... 214 8 27 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 10.0 and = 0.01, 0.10 and 1.00. ........ 214 8 28 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 1.00 and = 0.01. ............ 215 8 29 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 1.00 and = 0.10. ............ 215 8 30 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 1.00 and = 1.00. ............ 215 8 31 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 0.1 and = 0.01, 0.10 and 1.00. .......... 216 8 32 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 1.0 and = 0.01, 0.10 and 1.00. .......... 216 8 33 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 10.0 and = 0.01, 0.10 and 1.0 0. ........ 216 8 34 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 0.1 and = 0.01, 0.10 and 1.00. .......... 217 PAGE 30 30 8 35 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled soluti ons for = 0.01 in case of = 1.0 and = 0.01, 0.10 and 1.00. .......... 217 8 36 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 10.0 and = 0.01, 0.10 and 1.00. ........ 217 8 37 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 0.1, and = 0.01, 0.10 and 1.00. ......... 218 8 38 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 1.0, and = 0.01, 0.10 and 1.00. ......... 218 8 39 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 10.0, and = 0.01, 0.10 an d 1.00. ....... 218 8 40 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 0.1, and = 0.01, 0.10 and 1.0 0. ......... 219 8 41 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 1.0, and = 0.01, 0.10 and 1.00. ......... 219 8 42 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled soluti ons for = 1.00 in case of = 10.0, and = 0.01, 0.10 and 1.00. ....... 219 8 43 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01. ............................. 221 8 44 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.10. ............................. 221 8 45 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 1.00. ............................. 222 8 46 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. .......................... 222 8 47 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. .......................... 223 PAGE 31 31 8 48 Extent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 224 8 49 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 225 8 50 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 an d 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 226 8 51 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 227 8 52 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01. ............................. 228 8 53 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.10. ............................. 228 8 54 Results of A) extent of saltwater intrusion ( ) and B) recir culation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 1.00. ............................. 229 8 55 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1. 00. .......................... 229 8 56 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00. .......................... 230 PAGE 32 32 8 57 Extent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 231 8 58 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentr ation color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 232 8 59 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for t he second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 2 33 8 60 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant di spersion problem. ................................ ................................ ................................ ........... 234 8 61 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncouple d and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01. ............................. 235 8 62 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless r atio from 0.1 to 10.0 in case of = 1.00 and = 0.10. ............................. 235 8 63 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 1.00. ............................. 236 8 64 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00. .......................... 236 8 65 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for co upled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00. .......................... 237 PAGE 33 33 8 66 Extent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.0 1, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 238 8 67 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 239 8 68 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 240 8 69 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentr ation color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 241 8 70 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. .......................... 242 8 71 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. .......................... 242 8 72 Extent of saltwater intrusion ( ) for uncouple d solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 243 8 73 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentr ation color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 244 PAGE 34 34 8 74 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for t he second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 245 8 75 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant di spersion problem. ................................ ................................ ................................ ........... 246 8 76 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncouple d solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00. .......................... 247 8 77 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00. .......................... 247 8 78 Extent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 248 8 79 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 249 8 80 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 250 8 81 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 251 8 82 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0 .10 and 1.00. .......................... 252 PAGE 35 35 8 83 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00. .......................... 252 8 84 Extent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 253 8 85 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 254 8 86 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 255 8 87 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration co lor floods and velocity vectors for the second set of the Henry constant dispersion problem. ................................ ................................ ................................ ........... 256 9 1 The work of Motz and Sedighi (2013) A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for both uncoupled and coupled solutions. ... 261 9 2 Results of SEAWAT Version 4 with TVD scheme for the field scale problem A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for both uncoupled and coupled simulations comparing to the work of M otz and Sedighi (2013). ................................ ................................ ................................ 264 9 3 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 0.01. ................................ ................................ ................................ ................. 269 9 4 Comparison of the pattern of the 0.5 isochlors of A) uncoupl ed and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 0.1. 269 9 5 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 1.0 . 269 PAGE 36 36 9 6 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 0.1 in case of = 0.01, and = 0.01, 0.10 and 1.00. ................................ ................................ ................................ ................. 270 9 7 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled so lutions for = 1.0 in case of = 0.01, and = 0.01, 0.10 and 1.00. ................................ ................................ ................................ ................. 270 9 8 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00. ................................ ................................ ................................ ................. 271 9 9 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01. ............................. 272 9 10 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and co upled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.1. ............................... 272 9 11 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 1.0. ............................... 273 9 12 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00. ......................... 273 9 13 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00. ......................... 274 9 14 Extent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors. ................................ ................................ ......................... 275 9 15 Extent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors. ................................ ................................ ......................... 276 9 16 Degree of saltwater recirculation ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors. ................................ ................................ ......................... 277 PAGE 37 37 9 17 Degree of saltwater recirculation ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors. ................................ ................................ ......................... 278 9 18 Results of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01. .......................... 283 9 19 Results of the regular set A) unc oupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01. .......................... 284 9 20 Results of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01. .......................... 285 9 21 Results of the verified set A) un coupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01. .......................... 286 9 22 Results of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01. .......................... 287 9 23 Results of the verified set A) un coupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01. .......................... 288 9 24 Results of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1. ............................ 293 9 25 Results of the regular set A) uncou pled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1. ............................ 294 9 26 Results of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1. ............................ 295 9 27 Results of the verified set A) un coupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.10. .......................... 296 PAGE 38 38 9 28 Results of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.10. .......................... 297 9 29 Results of the verified set A) un coupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.10. .......................... 298 9 30 Results of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00. .......................... 303 9 31 Results of the regular set A) unc oupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00. .......................... 304 9 32 Results of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00. .......................... 305 9 33 Results of the verified set A) un coupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00. .......................... 306 9 34 Results of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00. .......................... 307 9 35 Results of the verified set A) un coupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00. .......................... 308 A 1 Results obtained by Simpson and Clement (2004) results A) the standard and B) the modified Henry problems. ................................ ............................... 321 A 2 Henry problems. ................................ ................................ ............................... 321 A 3 Results from Motz and Sedighi (2013) work A) the standard and B) the modified Henry problems. ................................ ................................ ................. 321 A 5 ................................ .... 323 A 6 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative I 0.1 results for standard Henry problem A) uncoupled and B) coupled solutions. ................................ ... 326 PAGE 39 39 A 7 Isochlors from 0.1 to 0.9 with concentration color floods and velocity dis tribution in the domain of SEAWAT alternative I 0.1 results for modified Henry problems A) uncoupled and B) coupled solutions. ................................ 326 A 8 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative II 0.05 results for standard Henry problem A) uncoupled and B) coupled solutions. ................................ ... 327 A 9 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative II 0.05 results for modified Hen ry problems A) uncoupled and B) coupled solutions. ................................ 327 A 10 Isochlors from 0.1 to 0.9 with concentration color floods and velocit y distribution in the domain of SEAWAT alternative III 0.025 results for standard Henry problem A) uncoupled and B) coupled solutions. .................... 329 A 11 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative III 0.025 results for modified Henry problem A) uncoupled and B) coupled solutions. .................... 329 A 12 The velocity contour lines (unit: m/day) in the domain from SEAWAT alternative I 0.1 results for standard Henry problem A) uncoupled and B) coupled sol utions. ................................ ................................ ............................. 330 A 13 The velocity contour lines (unit: m/day) in the domain from SEAWAT alternative I 0.1 results for modified Henry problem A) uncoupled and B) coupled solutions. ................................ ................................ ............................. 330 A 14 The velocity contour lines (unit: m/day) in the domain from SEAWAT alternative II 0.05 results for standard Henry problem A) uncoupled and B) coupled solutions. ................................ ................................ ............................. 331 A 15 The velocity contour line s (unit: m/day) in the domain from SEAWAT alternative II 0.05 results for modified Henry problem A) uncoupled and B) coupled solutions. ................................ ................................ ............................. 331 A 16 The velocity contour lines (unit: m/day) in the domain from SEAWAT alternative III 0.025 results for standard Henry problem A) uncoupled and B) coupled solutions. ................................ ................................ ............................. 332 A 17 The velocity contour lines (unit: m/day) in the domain from SEAWAT alternative III 0.025 results for modified Henry problem A) uncoupled and B) coupled solutions. ................................ ................................ ............................. 332 A 18 Transient position of the intersection of the 0.5 isochlors with the base of the domain of SEAWAT alternative I 0.1 results for A) the standard and B ) the modified Henry problems both uncoupled and coupled solutions. .................... 333 PAGE 40 40 A 19 Transient position of the intersection of the 0.5 iso chlors with the base of the domain of SEAWAT alternative II 0.05 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. .................... 334 A 20 Transient position of the intersection of the 0.5 isochlors with the base of the domain of SEAWAT alternative III 0.025 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. .............. 335 A 21 Changing of the TDS at the edge of the domain for SEAWAT alternative I 0.1 results for A) the stand ard and B) the modified Henry problems both uncoupled and coupled solutions. ................................ ................................ .... 336 A 22 Changing of the TDS at the edge of the doma in for SEAWAT alternative II 0.05 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. ................................ ................................ .... 336 A 23 Changing of the TDS at the edge of the domain for SEAWAT alternative III 0.025 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. ................................ ................................ .... 33 7 A 24 Comparison the 0.5 isochlors of SEAWAT alternative I 0.1 results and Simpson and Clement (2004) solutions for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. .................... 338 A 25 Comparison the 0.5 isochlors of SEAWAT alternative II 0.05 results and Simpson and Clement (2004) solutions for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. .................... 339 A 26 Comparison the 0.5 isochlors of SEAWAT alternative III 0.025 results and Simpson and Clement (2004) solutions for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. .................... 339 B 1 The works of A) Sgol (1993) and B) Abarca et al. (2007) present the coupled solutions of Henry problem. ................................ ................................ 381 B 2 The coupled solutions for the reference cases A) diffusive case (constant dispersion case) and B) dispersive case (velocity dependent dispersion case), as Abarca et al. ( ................................ ..... 381 B 3 The SEAWAT Version 4 results for the coupled solutions of Henry problem using the manually input equivalent freshwater heads and the varied concentration at the downgradient conditions A) Sgol (1993) and B) Abarca et al. (2007). ................................ ................................ ................................ ..... 382 PAGE 41 41 B 4 The SEAWAT Version 4 results for the coupled solutions of Henry problem using the manually input equivalent freshwater heads and the varied concentration at the downgradient conditions in shape of isochlors for A) Sgol (1993) and B) Ab arca et al. (2007) with concentration color floods and velocity vectors. ................................ ................................ ................................ 383 B 5 The SEAWAT Version 4 results for the coupled sol utions of Abarca et al. (2007) reference cases, using the manually input equivalent freshwater heads and the varied concentration at the downgradient conditions A) diffusive case (constant dispersion case) and B) dispersive case (velocity dependent dispers ion case). ................................ ................................ ............. 384 B 6 The SEAWAT Version 4 results for the coupled solutions of Abarca et al. (2007) reference cases, using the manually input equivalent freshwater heads and the varied concentration at the downgradient conditions in shape of isochlors for A) diffusive case (constant dispersion case) and B) dispersive case (velocity dependent dispersion case) with concentration col or floods and velocity vectors. ................................ ................................ ......................... 385 C 1 The work of Motz and Sedighi (2013) A) extent of saltwater intrusion ( ) and B) recircu lation of seawater ( ) for both uncoupled and coupled solutions. ... 404 C 2 Reproduced results of SEAWAT A) extent of saltwater intrusion and B) recirculation of seawater for both uncoupled and coupled simulations comparing to the work of Motz and Sedighi (2013). ................................ ......... 405 C 3 Reproduced results of SEAWAT simulations A) uncoupled and B) coupled solutions with conc entration color floods and velocity vectors when = 0.2 of Motz and Sedighi (2013). ................................ ................................ ............. 405 C 4 Reproduced results of SEAWAT A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 of Motz and Sedighi (2013). ................................ ................................ ................................ 406 C 5 Reproduced results of SEAWAT A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 of Motz and Sedighi (2013). ................................ ................................ ................................ 406 PAGE 42 42 LIST OF ABBREVIATIONS Dimensionless ratio Dimensionless ratio for an isotropic and homogeneous medium Dimensionless ratio Dimensionless ratio for an anisotropic and homogeneous medium Dimensionless ratio T ransverse vertical dimension the medium V ertical dimension of the medium Dispersivity ratio = H ydraulic conductivity ratio = H ydrodynamic dispersion H ydrodynamic dispersion coefficient Molecular diffusion coefficient E ffective molecular diffusion coefficient C oefficient of permeability for an isotropic and homogeneous medium Hydraulic conductivity for an isotropic and homogeneous medium L ongitudinal hydraulic conductivity i.e., parallel to groundwater flow ( direction) T ransverse horizontal hydraulic conductivity i.e., t ransverse horizontal to groundwater flow direction ( direction) T ransverse v ertical hydraulic conductivity i.e., t ransverse v ertical to groundwater flow direction ( direction) B uoyancy flux D ensity driven b uoyancy flux PAGE 43 43 B uoyancy flux D ensity driven buoyancy flux D ensity driven vertical buoyancy flux L ongitudinal dimension of the medium Landward extend of the = 0.5 concentration contour along the base of the aquifer E xtent of s altwater i ntrusion = Freshwater inflow per unit width at the upgradient boundary Freshwater inflow per unit width Freshwater advective flux F reshwater inflow per unit width and depth Specific discharge Freshwater mass inflow at the upgradient boundary Saltwater inflow per unit width at the downgradient boundary Saltwater inflow per unit width S altwater mass inflow at the coastal boundary D egree of saltwater recirculation = L ongitudinal dispersivity of the medium i.e., parallel to groundwater flow ( direction) T ransverse horizontal dispersivity of the medium i.e., transverse horizontal to groundwater flow direction ( direction) T ransverse v ertical dispersivity of the medium i.e., t ransverse v ertical to groundwater flow direction ( direction) Longitudinal discretization Transverse horizontal discretization Transverse vertical discretization Absolute or dynamic viscosity of freshwater Absolute or dynamic viscosity of saltwater PAGE 44 44 Density of freshwater = 1,000 kg/m 3 Density of saltwater = 1,025 kg/m 3 Henry (1964) d ensity difference ratio Density contrast parameter = = 0.025 Porosity of medium Aspect ratio PAGE 45 45 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INVESTIGATION OF SALTWATER INTRUSION BASED O N THE HENRY PROBLEM AND A FIELD SCALE PROBLEM By Chanyut Kalakan May 2014 Chair: Louis H. Motz Major: Civil Engineering Three problems regarding saltwater intrusion, namely the Henry constant dispersion and velocity dependent dispersion problems and a larger, field scale problem, have been investigated to determine quantitatively how saltwater intrusion and the recirculatio n of seawater at a coastal boundary are related to the f reshwater advective flux ( ) and the density driven buoyancy flux ( ) Based on dimensional analysis, it was determined that saltwater intrusion and the recirculation of seawater are depende nt functions of the independent ratio of freshwater advective flux ( ) relative to the density driven vertical buoyancy flux ( ) defined as the dimensionless ratio (or for an isotropic aquifer) and the ratio of horizontal ( ) and verti cal ( ) dimensions of the cross section. For the Henry constant dispersion problem, in which the aquifer is isotropic, saltwater intrusion and recirculation are related to an additional independent dimensionless parameter that is the ratio of the constant dispersion coefficient treated as a scalar quantity, the porosity and the freshwater advective flux, defined as the dimensionless ratio For the Henry velocity dependent dispersion problem, the PAGE 46 46 dimensionless ratio is zero, and saltwater int rusion and recirculation are related to an additional independent dimensionless parameter that is the ratio of the vertical and horizontal dispersivities, or the dispersivity ratio For an anisotropic aquifer, saltwater intrusion and recirc ulation are also dependent on the ratio of vertical and horizontal hydraulic conductivities, or the hydraulic conductivity ratio For the field scale problem, saltwater intrusion and recirculation are dependent on the same independent ratio s as the Henry velocit y dependent dispersion problem. In the two dimensional cross section for all three problems, freshwater inflow occurs at an upgradient boundary, and recirculated seawater outflow occurs at a downgradient coastal boundary. The upgradi ent boundary is a specified flux boundary with zero freshwater concentration, and the downgradient boundary is a specified head boundary with a specified concentration equal to seawater. Equivalent freshwater heads are specified at the downstream boundary to account for density differences between freshwater and saltwater at the downstream boundary. The three problems were solved using the numerical groundwater flow and transport code SEAWAT for both uncoupled (constant density) and coupled (variable dens ity) solutions. PAGE 47 47 CHAPTER 1 INTRODUCTION Background Freshwater is one of the essentials of human life and land ecosystems. There are two main sources of freshwater for supplying them which have enough both quantity and quality, namely surface and groundwater sources. Figures 1 1 and 1 2 present trends in population and freshwater withdrawals by source in the U.S. A. and the g eographic distribution of total surface water and groundwater withdrawals in the U.S. A. (Kenny et al. 2009) respectively. Some places perhaps have a plentiful supply of one and/or the other, but some For many rural areas around the world, groundw ater might be the main source of water supply. Figure 1 1 Trends in population and freshwater withdrawals by source in the U.S.A., 1950 2000 (Kenny et al. 2009) PAGE 48 48 Figure 1 2 T he geographic distribution of total surface water and groundwater withdrawa ls in the U.S.A., 2005 (Kenny et al. 2009) PAGE 49 49 In the case of groundwater sources, normally, if there are potential contamination sources in an aquifer, the contaminants will usually travel very slowly from the source to a downgradient boundary and may degra de the groundwater quality for a very long time are actually many sources of groundwater contamination such as human waste, industrial toxic waste, agricultural pesticide s and sea salt (Barbash and Resek 1996; Hallberg 1989; Patrick et al. 1987) The operating and maintenance costs of remediation ar e always expensive This work focuses on the contamination of groundwater by sea salt i.e., saltwater intrusion problems S altwater intrusion is one of the most common contaminations found in groundwater aquifers along coastlines all over the world (Barlow and Reichard 2010) Furthermore, these areas usually have high consumption of groundwater (Bricker 2009) water supplies, a large number of abandoned groundwater sources and new water resources investments which can be serious prob lems Therefore, learning how to manage maint ain and save freshwater sources is the best way to reduce the contamination problems. Saltwater Intrusion Problem reporting the saltwater intrusion problem was in 1855 in Lon don and Liverpool, England by Braithwaite (1855) He presented the problem due to well pumping and mentioned that this problem originated from lowering t he groundwater level below sea level (Kashef 1972) However, Braithwaite because it was found that although the groundwater level is high er than the sea level s eawater can still PAGE 50 50 and contaminate into the coastal aquifer Currently, that is found to result from the slightly different densities of fresh and sea water In 1888, the high impact paper regarding the saltwater intrusion was presented by Drabbe and Badon Ghyben (1888) which seems to be the beginning for understanding the saltwater intrusion problem. In 1901, the second high impact paper was presented by Herzberg (1901) Both of them independently attempted to explain th e relationship of the interface between freshwater and saltwater in a coastal aquifer and to determine the shape and position of the interface (Bear 1979; Carlston 1963; Reilly and Goodman 1985) Now, their principles are united in what is known as the Ghyben Herzberg approximation. Immiscible Fluids (Ghyben Herzber g Approximation) The Ghyben Herzberg approximation assumes that freshwater and saltwater are immiscible The position of the interface ( ) in the coastal aquifer was determined by neglecting hydrodynamic dispersion and applying the U tube equilibrium of freshwater and saltwater as shown in Figure 1 3. The Ghyben Herzberg approximation relates the elevation of the freshwater table written in Equa tion 1 1 (Bear 1979; Reilly and Goodman 1985) : (1 1) where: is the saltwater density ; is the freshwater density ; and is the depth below mean sea level to a point on the freshwater and saltwater interface vertically below where is the measured height PAGE 51 51 Figure 1 3 Ghyben Herzberg approximation model (Custodio 1987) The Ghyben Herzberg app roximation should only be applied under the following conditions (Bear 1979; Essink 2001; Reilly and Goodman 1985; Verruijt 1968) : 1. The system is at static equilibrium and there is a hydrostatic pressure distribution in the freshwater region; 2. The sea water is stationary, or dy namic equilibrium can be applied to the system, 3. 4. (Hydrodynamic dispersion is assumed to be negligible); 5. The freshwat 6. In the freshwater regime, equipotential lines are strictly vertical, and flow lines are firmly horizontal; PAGE 52 52 7. The groundwater discharge is proportional to the thickness of the aquifer, i.e., the Dupuit assumption can be applied to the system; and 8. The Ghyben Herzberg approximation results in a sharp interface as shown in Figure 1 3. Although it is always stated that the Ghyben Herzberg approximation is somewhat incorrect, the use of the equation still gives a rather good approximation in the real situation (Essink 2001; Verruijt 1968) Especially in the case of small discharge of freshwater, the Ghyben Herzberg approximation yields adequate results (Bear and Dagan 1964; Verruijt 1968) Miscible Fluids When salt water intrusion is considered more realistically, freshwater and saltwater are treated as The typical cross section of saltwater intrusion is similar to a wedge as shown in Figure 1 4 (Custodio 1987) Figure 1 4 Typical c ross section of the saltwater intrusion problem (Custodio 1987) PAGE 53 53 T he density of water is the key variable of the saltwater intrusion s ince the problem results from the slightly different densi ty of freshwater and saltwater. The temperature and pressure, but the density of sea water salinity as well The temperature and the pressure are considered as constant values i n order to simplify the problem. Therefore, the freshwater typically has a density of 1,000 kg/m 3 at 4C The density of sea water is subject to the chloride ion ( ) concentration, which is the predominant negative ion in seawater. The chloride ion ( ) co ncentration is represented in the term of total dissolved solids (TDS) The TDS of seawater is generally equal to 35 kg/m 3 ( or 35,000 ppm) and an average density of saltwater is approximately 1,025 kg/m 3 Although the difference in density of sea water an d freshwater is quite small, it still can have the movement of groundwater and saltwater in coastal aquifers (Bear 1979; Essink 2001; Reilly and Goodman 1985) D ue to the density differ ence between freshwater and saltwater t he lighter freshwater above to the sea coast while the heavier underneath landward which creates the bala nce of inflow and outflow and the zone of contact between freshwater and saltwater, as shown in Figure s 1 4 through 1 6. The zone of contact or mixing zone, always stays between the interface of freshwater and saltwater due to hydrodynamic dispersion (Bear 1979) At the mixing zone, some of freshwater flows o ut into the sea mixes with the seawater and moves seaward causing the (Lee and Cheng 1974; Reilly and Goodman 1985) PAGE 54 54 Figure 1 5 Isochlor in the Biscayne aquifer near Miami, Florida (Cooper 1964) Figure 1 6 Isochlor in the New Jersey Coastal Plain in New Jersey and on the Continental Shelf (Meisler 1989) PAGE 55 55 The recirculated seawater can include components due to wave set up, tidally driven oscillations and convection caused by density or thermal differences between the saltwater and freshwater. The sum of the recirculated seawater and fresh groundwater discharge is called submarine groundwater discharge or SGD (Motz and Sedighi 2013; Taniguchi et al. 2002; Younger 1996) This recirculation of seawater and mixing with freshwater is characterized by a zone of contact between the freshwater and the saltwater, which genera lly takes the form of a finite width transition zone caused by hydrodynamic dispersion across which the density of the mixed water varies from those of freshwater and saltwater (Bear 1979; Motz and Sedighi 2013) The horizontal thic kness of the mixing zone can be narrow or broad in width. For example, in the Biscayne aquifer in Florida, the horizontal width of the mixing zone is approximately 1,500 feet as shown in Figure 1 5 (Cooper 1964) However, in the New Jersey Coastal Plain in New Jersey the horizontal width of mixing is several miles in width as shown in Figure 1 6 (Meisler 1989) The vertical thickness of the mixing zone also varies among aquifers but generally is much smaller than the horizontal width since it is limited by the total thickness of the aquifer (Barlow 2003; Reilly and Goodman 1985) This results from the characteristics and structure of the aquifer. PAGE 56 56 Henry (1964) problem likely is the beginning of quantitative analysis of the saltwater intrusion problem which is one of the benchmarks of steady state density dependent groundwater flow and transport models for a coastal aquifer. This work 1960 as suggested by Hilton H. Cooper Jr. of the U.S. Geological Survey. Henry (1964) solved the problem in term s of a steady state solution by developing a Fourier Galerkin method f or an idealized mathematical model The effects of dispersion and density determined by using the advection diffusion equation (mi following a ssumption s and is shown schematically in Figure 1 7 (Henry 1964; Pinder and Cooper 1970) Figure 1 7 Domain and b (1964) problem (Pinder and Cooper 1970) PAGE 57 57 Dom (19 64) problem i s shown in Figure 1 7 and assumptions to solve the problem are described as shown below : 1. Two depth and 2 meter in length); 2. The confined aquifer was treated as a homogeneous, isotropic aquifer; 3. Impermeable boundaries were located at the top and bottom of the aquifer; 4. A constant freshwater flux entered the aquifer over the vertical dimension at and a constant saltwater head was maintained at ; and 5. The hydrodynamic dispersion ( ) was assumed to be a constant scalar throughout the flowfield, which d oes not destroy the essential features of the problem. Henry (1964) derived analytical expressions for the stream function and the salt concentration in the form of a Fourier series (Henry 1964; Simpson and Clement 2004; Sgol 1993) He adapted a mathematical solution that was developed by Poots (1958) and originally used for modeling heat transfer processes in order to solve nonlinear boundary value equations (Goswami and Clement 2007; Reilly and Goodman 1985) namely the aspect ratio and the dimensionless ratios and to solve the problem The Fourier series and nu merical techniques were applied to determine the solution in terms of ow and salt concentration patterns in the domain which use s the aspect ratio = 2.0 and the dimensionless ratios = 0.263 and = 0.1. Henry mentioned the assumptions on page C75 in U S Geological Survey Water Supply Paper 1613 C ffects of dispersion on salt encroachment in coastal aquifers, in Sea Water in Coastal Aquifers The hydrodynamic dispersion ( ) is also known as the hydrodynamic dispersion coefficient. PAGE 58 58 The salt concentration patterns in the idealized mathematical model are shown in Figure 1 8 (Henry 1964) Figure 1 8 Flow and salt concentration patterns of Henry idealized mathematical model A) Streamlines and B) Isochlors (Henry 1964) From the past 50 years until now, t here have been many investigations using (1964) problem as a reference for both physical an d numerical models. A lot of the discussion of various developments serves to illustrate the significance of PAGE 59 59 improvements in computer hardware in solving the Henry problem. Furthermore, although many physical and numerical experiments have been performed to compare and with each other These can be explained by natural differen ces between the fundamentals of each approach which is one of the significant reasons Also cu rrently, (1964) solution has be en found to contain some errors which might come from sources such as follows : 1. It has been surmised that Henry (1964) started with the wrong guess, i.e. isochlors were too far inland and did not reach a full equilibrium solution (Sgol 1993) ; 2 When Henry (1964) solved the system of equations it was solved for the Fourier coefficients that were greatly sensitive to the c omponents of the system matrices and intolerant of any mistake (Sgol 1993) ; 3 Due to the lack of an adequate computing hardware to perform simulations in that era, Henry (1964) solved this complex problem with the use of a slide rule, a desk calculator and, finally, an IBM 650 Magnetic Drum Calculato r which might have cause d (Frind 1982; Henry 1964; Sgol 1993) ; and 4 Henry (1964) expanded the salt concentration and the stream function in d ouble Fourier series by using 78 terms which i s not an adequate number of terms to obtain the correct solution (Voss and Souza 1987; Zidane et al. 2013) Although there are some errors in his solution, Henry (1964) created definitely not only a challeng ing problem but also a valuable solution (Sgol 1993) PAGE 60 60 CHAPTER 2 PREVIOUS INVESTIGATIONS S altwater I n trusion P roblem in the Physical and Numerical Model (1964) problem has been used not only as a frame work to investigate saltwater intrusion but also as a benchmark of steady state density dependent groundwater flow and transport models for a coastal aquifer This problem is one of the most popular as a test case of saltwater intrusion. However, a lot of discussion of various developments serves to illustrate the significance of improvements in solving (1964) problem. From 50 years ago until now, t here are a number of work s attempting to revise and reproduce this problem but their results have been compared to show some agreement. The comparisons normally have been flawed because the fundamentals of two solutions were naturally different. This work does not provide the entire history of the saltwater intrus ion and Henry (1964) problem s, but a brief discussion of them is includ ed in this work The history of them is provided in many works, such as Cooper (1964) Pinder and Cooper (1970) Lee and Cheng (1974) Reilly and Goodman (1985) and Sgol et al. (1993) A brief discussion of the saltwater intrusion and Henry (1964) problems starts with Croucher and O'Sullivan (1995) They presented the numerical solution of the Henry problem that combined the highly accurate Arakawa diffe rence approximation for the Jacobian term in the salt advection diffusion equation with the Buneman algorithm for efficient solution of the stream function equation. They also used very fine finite difference grids to maintain computational costs at a rea sonable level. Benson et al. (1998) showed the differe nce between the solution of Henry's (1964) problem obtained by particle tracking (Lagrangian) algorithms, a finit e difference Lagrangian and an PAGE 61 61 Eulerian finite element variable density flow and transport code. They mentioned that the velocity field in many flow and transport problems is highly variable and that the error associated with velocity approximation strong ly affects the solution. The Lagrangian transport algorithms gain by using coarse grids is lost when modeling transport within highly variable velocity fields. Simpson and Clement (2003; 2004) developed a method and applied it to solve (1964) problem in the same density coupled mode as Henry (1964) and proposed a density uncoupled mode, also known as the standard Henry problem for both uncoupled and coupled solutions. They concluded that the standard Henry problem has limited usefulne ss in benchmarking density dependent flow models. They recommended that decreasing the freshwater recharge by a factor of two can increase the relative importance of the density dependent effects of the problem also known as the modified Henry problem f or both uncoupled and coupled solutions. These made their work become one of the main benchmarks of saltwater intrusion problems in terms of the constant dispersion problem Held et al. (2005) proposed the upscaling of density dependent flow for the two (1964) problem. Homogenization theory was applied to determine effective flow and transport parameters for saltwater intrusion in statistically isotropic and anisotropic heterogeneous permeability fields Results presented indicated that the appropriate dispersion coefficients for the problem corresponded to the local dispersion coefficients, rather than macro dispersion coefficients. Dentz et al. (2006) described H problem in terms of two dimensionless groups, a coupling parameter ( ), which is the ratio of the buoyancy and viscous forces PAGE 62 62 and the Pclet number ( ), which is the ratio of advective ( purely convective ) and dispersive transport. Actually, t he c oupling parameter ( ) dimensionless ratio and the Pclet number ( ) dimensionless ratio Dentz et al. (2006) conducted a systematic analysis of the Henry problem for a full range of the Pclet number from small to large values, including coupled and uncoupled solutions (the latter are called pseudo coupled solutions in their study). They determin ed that the perturbation approach was applicable to a wide range of v ariable Goswami and Clement (2007) conducted laboratory scale experiments and numerical modeling to develop steady state and transient salt wedge data sets. Their work represented benchmark problems for testing d ensity coupled groundwater flow models. The finite difference model SEAWAT (Langevin et al. 2003) was used to simulate steady state and transient experiments. They also used the finite element model for saturated unsaturated transport (SUTRA) (Voss and Souza 1987) and a modified version of the MO DFLOW model with the sharp interface package Seawater Intrusion (SWI) (Bakker and Schaars 2003) to simulate the experiment results. The investig ation of Goswami and Clement (2007) included comparing the results of coupled and uncoupled steady state and transient SEAWAT (Langevin et al. 2003) numerical simulations to investigate density coupling effects, similar to Simpson and Clement (2004) Goswami and Clement (2007) also proposed reducing the freshwater inflow as a more robust alternative to the traditional Henry problem. Abarca et al. (2007) is one of the well known revised (1964) works in terms of the anisotropic domain They reproduced the solu tion of (1964) PAGE 63 63 problem and intr oduced the two reference cases in shape of the revised problem by accounting for anisotrop y of hydraulic conductivity and dispersivity of the domain, i.e., diffusive and dispersive cases They performed a number of cases to explore the role of ea ch parameter but some of them would be unusual for field conditions e.g. the ratio of dispersivity ( ) was greater than 1.0 and /or the ratio of hydraulic conductivity ( ) also was greater than 1.0 in some cases. A lattice Boltzmann method (LBM) with a two relaxation time collision operator (LTRT) was developed by Servan Camas and Tsai (2010) to solve saltwater intrusion problems The LTRT was verified with the coupled solutions of both standard and modified Henry problem from the work of Simpson and Clement (2004) and the semi (1964) problem from Sgol (1993) work. The numerical results show ed good agreement with both of them. Recently, Po vich et al. (2013) developed a numerical model using finite element methods to solve variable density flow and solute transport The developed f inite element cod e could solve the variable density flow and solute transport problem and also reproduced the reference cases of Abarca et al. (2007) namely diffusive and dispersive in order to verify their code. The obtained results showed a good agreement with Abarca et al. (2007) Additionally, Povich et al (2013) noted that Abarca et al. (2007) utilized a hydrodynamic dispersion tensor and the bounda ry conditions which are more physical than (1964) problem Actually, to reflect on the assumption of each case of Abarca et al. (2007), the diffusive case should be depe ndent Sgol (1993) and Simpson and Clement (2004). PAGE 64 64 S altwater I ntrusion P roblem in the Ecology View The saltwater intrusion problem has been also presented in the view of ecology. Gssling (2001) studied the consequences of problems of tourism at the east coast of Zanzibar, Tanzania where a large number of tourists have visited. The freshwater has been poor and resources are not enough. The freshwater originates from seasonal rains and is stored in less efficient aquifers. Tourism developments have been expected to put additional pressure on the freshwater resources. The consequences of overexploitation can include the lowering of the groundwater table, land subsidence, deteriorating groundwater quality and saltwater intrusion. T hat has had a significant effect on the local populations and the tourist industry. Burak et al. (2004) found that the impact of urbanization and tourism on the coastal environment at two coastal settlements in Turkey, namely o n th e Aegean sea and Mersin located on the Mediterranean sea, included t he loss of agricultural land and over pumping of groundwater. This significantly accelerated the saltwater intrusion problem, particularly in the late summer and autumn, and this was geological analyses. Zektser et al. (2005) studied the e nvironmental impacts of groundwater overdraft in the southwestern United States which is one of the large st urban and agro industrial economic centers This regi on is also one of the driest in North America, with highly variable seasonal and inter annual precipitation regimes and frequent droughts The combination of a large demand for usable water and semi arid climate has led to groundwater overdraft in this re gion Groundwater overdraft normally develops when long term groundwater extraction exceeds aquifer recharge, producing declining trends in aquifer storage and hydraulic head This region has faced composite problems which PAGE 65 65 were g roundwater overdraft, dec line d surface water levels and streamflow, reduction or elimination of vegetation, land subsidence and seawater intrusion Mcleod et al. (2010) reviewed coastal impact models to determine sea level rise vulnerability and provides guidance to help managers and policy makers determine the appropriateness of various models at local, regiona l and global scales. There are a variety of models, each with strengths and weaknesses, which are suited for different management objectives. Ex tent of Saltwater Intrusion The investigation and quantitation of saltwater intrusion can be represented by two dimensionless ratios, namely the extent of saltwater intrusion ( ) and the recirculation of seawater ( ) A lot of saltwater intrusion problem s have been studied in both models and field scale problem s, but they might be quite complicated to present the extent of saltwater intrusion and the degree of saltwater recirculation Especially in the field scale problems, there are a lot of factors for each site, such as the h ydrogeologic system and freshwater and saltwater inflow outflow. It is very challenging to investigate all sufficient good quantity and quality of field data. Therefore, the saltwater intrusion problems in the field scale problem have been frequently presented by estimated isochlors on maps or specific cross sections at eac h site. Most of works in the last decades about s altwater intrusion in the field scale problem were analyzed by using data from observation wells since the computing tools at that time had not sufficient power. Cooper (1964) proposed cross sections of i so chlors in the Biscayne aquifer near Miami, Florida, which is investigated from observation wells Meisler (1989) also pre sented cross sections of isochlors in the aquifer in New Jersey and on the PAGE 66 66 Continental Shelf that were investigated from wells. Stumm (2001) and Stumm et al. (2002) presented isochlor maps from investigation field data at Great Neck, Long Island, New York and Manhasset Neck, Nassau County, New York respectively, which are analyzed data from observation wells. The analyzed saltwater intrusion problems currently use high performance tools and programming codes Obtained results can be developed and calculated in two or three dimensional analysis, such as Zheng (2006) Langevin et al. (2003) and Langevin et al. (2008) The r esults can estimate isochlors on specific cross sections at each site and certainly the accurate of results depend on assumptions and input parameters that can be verified by observation wells. Saltwater Recirculation The recirculation of seawater has been investigated in terms of submarine groundwater discharge (SGD) Zektzer et al. (1973) investigated the problem of direct groundwater discharge to the seas, such as the role of submarine groundwater discharge in the world water balance, the effect of groundwater on forming the water and salt balances of the seas, the interrelationship between sea and groundwaters in coastal areas, the e ffect of groundwater discharge on forming mineral deposits at sea and ocean floors and points of fresh groundwater discharge on the sea floor for water supply purposes. Johannes (1980) reviewed the early background of submarine groundwater discharge and mentioned that, o n a glo b al scale the impact of submarine groundwater discharge on marine and estuarine communities is less than that of surface r unoff. However, It is obvious that SGD is widespread and, in some areas, of greater ecological s i gnificance than surface runoff SGD influences productivity, biomass, species PAGE 67 67 composition and zonation The c oastal groundwater quantity and quality might be changed, which alters salinities, dissolved nutrient concentrations and levels of dissolved pollutants Li et al. (1999) pr esented results from a theoretical model of flow and chemical transport processes in subterranean estuaries The model show ed that groundwater circulation and oscillating flow which are cause d by wave setup and tide, may constitute up to 96% of submarine groundwater discharge compared with 4% due to the net groundwater discharge Their results were consistent with the experimental findings of Moore (1996) and have important implications for coastal resour ces management. T aniguchi et al. (2002) mentioned that SGD might be both volumetrically and chemically important to coastal water and chemical budgets Additionally SGD has a significant influence o n the environmental condition of many nearshore marine environments and provides a strong motivation for improved assessments SGD should be given more attention with regard to water and dissolved material budgets at the local regional and global scales. Henry Constant Dispersion Problem (1964) problem have been for the constant dispersion problem (1964) work. One of them can be used as a benchmark, namely Simpson and Clement (2004) Simpson and Clement (2004) re investigated Henry's (1964) semi analytical solution and performed numerical solutions for both uncoupled and coupled solutions also known as the standard (or original) Henry problem They concluded that the Henry problem has limited usefulness in benchmarking density dependent flow models since the pattern of flow wa s dictated by the boundary conditions (Simpson and Clement PAGE 68 68 2003) They propos ed that decreasing the freshwater recharge by a factor of two (i.e., decreasing the dimensionle ss ratio and increasing the dimensionless ratio ) in order to increase the relative importance of density dependent effects could improve the worthiness of Henry's problem Thus, they formulated and performed both semi analytical and numerical solutions to address th is issue for both the uncoupled and coupled solutions also known as the modif ied Henry p roblem as shown in Figure 2 1 A B Figure 2 1 R esults obtained from Simpson and Clement (2004) A) the standard and B) the modified Henry problems. Simpson and Clement (200 4) us ing both semi analytical and numerical methods preformed not only the standard Henry problem but also the modif ied Henry problem for a homogeneous and isotropic aquifer in terms of two dimensionless ratio s, i.e. and which are the ratio of the f reshwater advective flux ( ) to the density driven buoyancy flux ( ) and the ratio of the porosity and the coefficient of dispersion ( ) to the f reshwater advective flux ( ), respectively; and a third dimensio nless parameter that represents the ratio of the horizontal ( ) and vertical ( ) dimensions of the problem, also called the aspect ratio ( ). The coastal boundary was assigned to be the constant concentration of seawater (1964) variables and assumption The s imulations were achieved on identical grids with a constant dispersion coefficient PAGE 69 69 Henry Velocity dependent Dispersion Problem (1964) problem in terms of the constant dispersion problem, some of them changed the view by considerin g the domain as an anisotropic and homogeneous domain Among these works, there are two works which are considered as benchmarks namely Sgol (1993) and Abarca et al. (2007) Sgol (1993) reviewed the background from basic to advanced solutions of the groundwater problem She investigated a lot of significant groundwater pr oblems, such as ground water flow, contaminant transport and saltwater intrusion problem. She stated and reviewed (1964) problem, namely finite elements (Huyako rn et al. 1987) consistent velocities (Voss and Souza 1987) Eulerian Lagrangian formulation (Gambolati et al. 1992) and stream function formulation (Fogg and Senger 1985; Senger and Fogg 1990) Moreover, she revised Henr (1964) analytical solution and poin (1964) solution She (1964) problem using the varied concentration of seawater at the coastal boundary as shown in Figure 2 2 (A). A B Figure 2 2 Isochlors of A) Sgol (1993) and B) Abarca et al. (2007) f or Henry problem. PAGE 70 70 Abarca et al. (2007) was se lected as another selected problem to verify the numerical model H owever, there were th re e main differences from the original Henry (1964) problem: first, they imposed the mass flux boundary condition at the seaside instead of a constant concentration boundary condition ; second, the salt di spersion is velocity dependent and anisotropic with no molecular diffusion; and third, the hydraulic conductivity is anisotropic The aquifer was initially filled with freshwater with a corresponding hydrostatic freshwater head distribution (Servan Camas and Tsai 2010) Abarca et al. (2007) used the saturated unsaturated transport (SUTRA) finite element code (Voss and Provost 2002) to (1964) problem which was compared to the 50% isochlors from Sgol (199 3 ) work, as shown in Figure 2 2 (B) They also introduced two solutions of reference cases, which they called diffusive and dispers ive cases (1964) problem Their modifications were the adjustments of hydraulic conductivities and dispersivities values of the medium to be representative of an anisotropic medium T hey performed solutions of two reference cases, as shown in Figure 2 3 (A) and (B). Add itionally, their 152 case of anisotropic medium (60 diffusive and 92 dispersive) were completed S ome of these cases would b e unusual for field conditions, e.g., the ratio of dispersivity ( = ) was greater than 1.0 and/or the ratio of hydraul ic conductivity ( = ) also was greater than 1.0 in some cases ( Abarca et al. 2007 ) Actually, to reflect on the assumption of each case of Abarca et al. (2007), the diffusive case should be rsive case of Abarca et al. (2007) should be called the Sgol (1993) and Simpson and Clement (2004). PAGE 71 71 A B Figure 2 3 I sochlor s of Abarca et al. (2007) for the solutions of the reference cases namely A) diffusive case and B ) dispersive case. However, these simulations were performed to explore the role of each parameter Since the small values of and required a large domain to get rid of the boundary effect of the domain, the aspect ratio ( ) was adjusted to be equal to 2, 4, 8 and 10 depending on the extent of saltwater intrusion ( ) needed They proposed reducing the dimensionless ratio ( Pclet number) which affected the results since it made them sensitive to density variations within the do main. That was suitable for testing seawater intrusion codes where stable density profiles extend throughout most of the domain The reduced diffusion problem affected seawater intrusion consistent with widely accepted concepts and concentration profiles that are similar to those observed in the field. (2004) recommendation to decrease the f reshwater inflow per unit width ( ), thereby reducing the dimensionless ratio in the Henry problem. Actually, for the two reference cases of Abarca et al. (2007) it seems reasonable means of a constant scalar value of hydrodynamic dispersion ( ) (Henry 1964; Simpson and Clement 2004) dependent PAGE 72 72 (Simpson and Clement 2004; Sgol 1993) because the hydrodynamic dispersion effect was presented using t he summation of the product of dispersivity ( ) and velocity ( ) and the constant of molecular diffusion coeffic ient ( ) in each axis That makes the dispersion effect vary with them in each axis Field s cale Problem The field scale saltwater intrusion problems have been investigated from many places around the word, such as southeastern Florida U.S.A. (Sgol and Pinder 1976) the Nile Delta, Egypt (Kashef 1983) the Lower Mary River, Northern Territory, Australia (Mulrennan and Woodroffe 1998) the eastern shore of Virginia, U.S. A. (Nowroozi et al. 1999) the Mersin (Demirel 2004) and the Yangtze River Estuary, China (An et al. 2009) The problems have been regularly investigated with specific assumptio ns and methods which might be suitable for a particular site depended on the characteristic data and problems and occasionally relate s to the assumptions of (1964) proble m. However, this works attempted to apply assumptions of (1964) proble m to saltwater intrusion problems since (1964) assumptions are flexible and practical in order to develop a nd apply to any general coastal aquifer so as to get results. Among many works about the field scale saltwater intrusion problem, Motz and Sedighi (2009 a ; 2009 b 2013) investigated saltwater intrusion and recirculation of seawater at a coastal boundary us ing parameters based on data from the upper part of the Floridan aquifer system at Hilton Head Island in South Carolina, U.S.A. (Bush 1988) Motz and Sedighi ( 2009 a ; 2009 b 2013 (1964) assumption s. Additionally, t hei r work could be applied to a coastal aquifer in order to PAGE 73 73 solve the problem of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) b ased on a couple basic aquifer parameters, such as freshwater inflow, hydraulic conduct ivity and porosity Their investigation cover ed a wide range of possible freshwater inflow which can demonstrate the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) in the realistic view. Th erefore the work of Motz and Sedighi (2013) can serve as a good benchmark problem for the field scale problem Motz and Sedighi (2013) presented results for two condition s, i.e., uncoupled and coupled conditions, for the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) as shown in Figure 2 4. For both condition s, the results demonstrate that the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) are decreased significantly as the dimensionless ratio of the freshwater advective flux relative to the density driven buoyancy flux ( ) is increased T he significant differences can occur between uncoupled and coupled simulations especially at smaller v alues of the dimensionless ratio A lthough the results of extent of saltwater intrusion ( ) at the smaller values of dimensionless ratio are somewhat affected by the flow domain size, both the extent of saltwater intrusion ( ) and the degr ee of saltwater recirculation ( ) are still related to the dimensionless ratio which is the ratio of the freshwater advective flux ( ) to the density driven vertical buoyancy flux ( ) for both the uncoupled and coupled solutions, as sho wn in Figure 2 4 (A). PAGE 74 74 A B Figur e 2 4 Results of Motz and Sedighi (2013) A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for both uncoupled and coupled solutions. PAGE 75 75 CHAPTER 3 STATEMENT OF PROBLEMS Three problems regarding saltwater intrusion, namely the Henry constant dispersion and velocity dependent dispersion problems and a larger, field scale problem, have been investigated to determine quantitatively how saltwate r intrusion and the recirculation of seawater at a coastal boundary are related to the freshwater advective flux ( ), the density driven vertical buoyancy flux ( ), the dispersion coefficient ( ), the molecular diffusion coefficient ( ) the aspect ratio = the dimensionless ratio = the dimensionless ratio = the dispersivity ratio and the hydraulic conductivity ratio Henry Constant Dispersion Problem Statement of the Henry Constant Dispersion Problem Henry ( 1964) assumed that the dispersion coefficient ( ) w as a constant scalar throughout the flowfield which does not destroy the essential features of the problem. That is synonymous with the assumption of the Henry constant dispersion problem Actually, although Henry (1964) assum ption doe s not correspond with current flow and transport theory it still has been useful for the basic knowledge of the saltwater intrusion problem Objectives of the Henry Constant Dispersion Problem The purposes of the Henry constant dispersion problem were to investigate saltwater intrusion and determine quantitatively how saltwater intrusion and the I n the case of a homogeneous and isotropic domain ( the Henry constant dispersion problem ): T he density driven buoyancy flux ( ) ; and T he dimensionless ratio PAGE 76 76 recirculation of saltwater are related to the freshwater inflow per unit width ( ), the density driven buoyancy flux ( ) and a constant dispersio n coefficient ( ) over a broader range than previously considered for the dimensionless ratio and the dimensionless ratio Henry Velocity dependent Dispersion Problem Statement of the Henry Velocity dependent Dispersion Problem Conversely to the Henry constant dispersion problem, the Henry velocity dependent dispersion problem was investigated by using velocity dependent dispersion coefficient ( ) throughout the flowfield. These assumptions are consistent with the curr ent theory of flow and transport in a domain, which makes the Henry velocity dependent dispersion problem more realistic than the Henry constant dispersion problem Objectives of the Henry Velocity dependent Dispersion Problem The purposes of the Henry vel ocity dependent dispersion problem were to investigate saltwater intrusion and determine quantitatively how saltwater intrusion and the recirculation of saltwater are related to the freshwater advective flux ( ) and the density driven vertical buoyancy flux ( ) over a broader range than previously considered for the dimensionless ratio = the dispersivity ratio = and the hydraulic conductivity ratio = PAGE 77 77 Field s cale Problem Statement of the Field s cale Problem Both the Henry constant dispersion and the Henry velocity dependent dispersion problems had some limit of assumptions, which is inappropriate so as to apply them to the field scale problem, namely: 1. The assumptions of the Henry constant dispersi on problem do not correspond with the current theory of flow and transport in a domain, and values of the (1964) solution; 2. Values of the parameters are used in simulating the results of Henry constant dispersion (1964) problem. Actually, some of values of the parameters a re not suitable in order to apply in the realistic problem. Especially, t he flow domain is an i sotropic and homogenous aquifer ; 3. For the assumptions of the Henry velocity dependent dispersion problem although they are consistent with the current theory of flow and transport, values (1964) problem; and 4 Although the dimensionless ratio in the Henry velocity dependent dispersion problem is extended into the large spans of value s some of results are controlled by the flow domain size, i.e., the aspect ratio ( ). The flow domain size, which i s 1.0 m in depth and 2.0 m in length i s not large enough to demon s trate the full extent of the saltwater intrusion problem. Thus, the field scale problem was created to investigate the saltw ater intrusion problem which is based on the works of Bush (1988) and Motz and Sedighi ( 2009b; 2013 ) Objectives of the Field s cale Problem The purposes of field scale problem were to investigate saltwater intrusion and determine quantitatively how saltwater intrusion and the recirculation of saltwater are PAGE 78 78 associated with the freshwater advective flux ( ) and the density driven vertical buoyancy flux ( ) over a broader range than previously considered for the dimensionless ratio = and the hydraulic conductivity ratio = The aspect ratio = = 100 of the field scale problem is more realistic than the aspect ratio (1964) problem. T he dispersivity ratio = is held equal to 0.01, which is based on (1988) work. Solutions of the Three Problem s Statement of Solutions of the Three Problem s For all three problems, s imulations were solved numerically in two solutions, i.e. a density uncoupled solution (a less computationally intensive method) and a density coupled solution (a more computationally intensive method). Certainly, the density coupled solution, in which t he fluid density is a function of t he concentration of total dissolved solids (TDS) is more accurate than the density uncoupled solution, in which the density of the fluid is considered constant. However, comparing between the accuracy of the results and computational costs should be cons idered. Objectives of Solutions of the Three Problems The purposes of the solutions of the three problems were to present the results of these problems in two solutions, i.e., the uncoupled and coupled solutions and to compare how significantly different the results are between the uncoupled and coupled solutions. PAGE 79 79 CHAPTER 4 BACKGROUND EQUATIONS Governing Equations for Flow and Transport Model The details of the governing equations for flow and transport used in this study are described by SEAWAT (Guo and Langevin. 2002) SEAWAT 2000 (Langevin et al. 2003) MODFLOW 2000 (Harbaugh et al. 2000) MODFLOW 2005 (Harbaugh 2005) MT3DMS (Zheng 2006; Z heng and Wang 1999) and SEAWAT Version 4 (Langevin et al. 2008) However, a brief accou nt of the governing equations i s restated here Governing Equations for Flow Model T hree dimensi onal flow through an anisotropic heterogeneous confined porous medium with constant density fluid can be described by the partial differential equation as shown in Equation 4 1 (McDonald and Harbaugh 1988) : = ( 4 1 ) where : and = values of hydraulic conductivity along the and coordinate axes, which are assumed to be parallel to the major axes of hydraulic conductivity ; = potentiometric head ] ; = a volumetric flux per unit volume representing sources and/or sinks of water, with 0.0 for flow out of the ground water system and 0.0 for flow into the system ; = specific storage of the aquifer, ; and = time In general, th e specific storage of the aquifer the hydraulic conductivity and sources and/or sinks of water can be function s of space and time. PAGE 80 80 Equation 4 1 describes ground water flow under nonequilibrium conditions in a heterogeneous and anisotropic medium, provided the principal axes of hydraulic conductivity are aligned with the coordinate directions. Governing Equations for Transport Model The partial differential equation describing the fate and transport of contaminants of species in 3 D, transient fl ow systems can be written as Equation 4 2 (Zheng and Wang 1999) : = ( 4 2 ) where : = porosity of the domain, ; = dissolved concentration of species ; = time, ; = distance along the respective Cartesian coordinate axis, ; = dispersion coefficient tensor, ; = seepage or linear pore water velocity, ; it is related to the specific discharge or Darcy flux through the relationship, ; = volumetric flow rate per unit volume of aquifer representing fluid sources (positive) and sinks (negative), ; = concentration of the source or sink flux for species ; and = chemical reaction term, The left hand side of Equation 4 2 can be expanded into two terms namely = = ( 4 3 ) PAGE 81 81 where : = is the rate of change in transient groundwater storage, The chemical reaction term in the Equation 4 2 can be used to include the effect of general biochemical and geochemical reactions on contaminant fate and transport Considering only two basic types of chemical reactions, i.e., aqueous solid surface reaction (sorption) and first order rate reaction, the chemical reaction term can be expressed as follows: = ( 4 4 ) where : = bulk density of the domain, ; = concentration of species sorbed on the medium, ; = first order reaction rate for the dissolved phase ; and = first order reaction rate for the sorbed (solid) phase Substituting Equatio ns 4 3 and 4 4 into Equation 4 2 and dropping the species index for simplicity of presentation, Equation 4 2 can be arranged and written as Equation 4 5 : = ( 4 5 ) Equation 4 5 is essentially a mass balance statement, i.e., the change in the mass storage (both dissolved and sorbed phases) at any given time is equal to the difference in the mass inflow and outflow due to dis persion, advection, sink/sour ce and chemical reactions. PAGE 82 82 The dispersion is caused by mechanical dispersion, a result of deviations of actual velocity on a microscale from the average groundwater velocity and by molecular diffusion driven by concentration gradients Molecular diffusion is generally secondary and negligible, compared with the effects of mechanical dispersion, and only becomes important when groundwater velocity is very small. The sum of mechanical dispersion and molecular diffusion is termed hydrodynamic dispersion ( ) or simply dispersion (Zheng and Wang 1999) The hydrodynamic dispersion ( ) or dispersion for anisotropic porous medium is defined in terms of a tensor as shown below (Zheng and Bennett 2002): = (4 6) = (4 7) = (4 8) = (4 9) = (4 10) = (4 11) where: = principal components of the dispersion tensor, ; = cross terms of the dispersion tensor, ; The hydrodynamic dispersion ( ) is also known as the coefficient of hydrodynamic dispersion ( Gillham and Cherry 1982) PAGE 83 83 = longitudinal dispersivity (some references use ), ; = transverse horizontal dispersivity (some references use ), ; = transverse vertical dispersivity (some references use ), ; = molecular diffusion coefficient or effective molecular diffusion coefficie nt, ; = components of the velocity vector along the x, y and z axes ; and = magnitude of the velocity vector, When the velocity vector is aligned with one of the coordinate axes, all the cross t erms become zero (Zheng and Wang 1999) There are two equations namely the governing equations for flow and transport models, which connect s with each other in calculation transport equations via the seepage or linear pore water velocity ( ) (Brown 2002; Darcy 1856; Ingebritsen and Sanford 1999) : = ( 4 12 ) where: = seepage or linear pore water velocity, ; = values of hydraulic conductivity along the coordinate axes ; = potentiometric head ] ; and = hydraulic head gradient in the direction. The seepage or linear pore water velocity ( ) is also called advective regional velocity (Leap and Kaplan 1988), advective velocity (Kaplan and Leap 1985), average interstitial velocity (Konikow and Reilly 2006), average linear groundwater velocity (Hall et al. 1991), average linear vel ocity (Fetter 2001, Konikow and Reilly 2006, Ward and Trimble 2003), average pore velocity (Delleur 2006), and (average) seepage velocity (Lambe and Whitman 1969, Watson and Burnrett 1993, Konikow and Reilly 2006, and Zheng and Bennett 2002) PAGE 84 84 Accuracy and Stability Criteria for Numerical Model When numerical methods are used to approximate the differential flow and transport equations, then generally there are two problems, namely accuracy (or numerical dispersion) and stability (or oscillation) (Peaceman 1977; Pinder and Gray 1977) Both of them relate to the finite difference grid. There are two general guidelines that need to be followed in order to ob tain sufficiently accurate solutions, namely limits placed on the grid Pclet number ( ) and the Courant number ( ) (Essink Oude 2003; Zheng and Bennett 2002) Grid Pclet number The grid Pclet number ( ) is used to preserve the sharpness of the concentration front. The general form of the Pclet number is defined as the ratio of the coefficients of the advective and dispersive terms, as shown in Equation 4 13: = ( 4 13 ) where : = seepage or linear pore water velocity, ; = the characteristic length defined as the grid spacing in any direction, ; and = dispersion in the direction Upon substituting the d ispersion term with the Pclet number becomes the grid Pclet number ( ) as shown in Equation 4 1 4: = ( 4 1 4) Voss and Provost (2002) recommend that g rid Pclet number ( ) in the direction should be less than 4.0, and the g rid Pclet number in the direction should be less than 10 .0. However, Daus et al. (1985) and Zheng and Bennett (2002) PAGE 85 85 recommend that the g rid Pclet number should not be greater than 2.0 (Essink Oude 2003) Courant number The Courant number ( ) can be interpreted as the number of cells (or the fraction of a cell) that a solute particle will be allowed to move in one transport step in any direction (Zheng 2006; Zheng and Bennett 2002) The Courant number is defined as show n in Equation 4 1 5 : = ( 4 1 5 ) where: = seepage or linear pore water velocity, ; = time step ; and = characteristic length defined as the grid spacing in any direction, Generally, to ob tain appropriately accurate solutions the Courant number should be less than 1.0 if the problem i s advection dominated (Delleur 2006; Zheng and Wang 1999) PAGE 86 86 CHAPTER 5 DIMENSIONAL ANALYSIS Dimensional analysis is a method that can be used to simplify complicated problems by reducing the number of system parameters that need to be considered (Bolster et al. 2011) A study of any complex problem, such as fluid flow in pipe, open channel or porous media, has a number of variables that have an effect on the flow phenomena. It can be too challenging to study and understand every variable After monitoring the problem, getting experience and doing experiments, each of the problems naturally have some variables that dominate the results. Some of variables can be grouped to make them represent physical meaning and values, but some others c an be neglected. Dimensional analysis using the Buckingham PI theorem can deduce from a study of the dimensions of the variables in any complex problems certain limitations on the form of any possible relationship between variables et al. 2010; Sonin 2001) Dimensional A nalysis for Sal twater Intrusion Problem To investigate the saltwater intrusion problem, the investigation and quantitation of saltwater intrusion were determined based on the physical parameters of fluids and the domain by using d imensional a nalysis Dimensionless Ratios for the Exploration of Saltwater Intrusion Problem The simulations considered the steady flow of incompressible fluids, namely freshwater and saltwater, through a confined domain, i.e., a coastal aquifer. Dimensionless ratios, namely the extent of saltwa ter intrusion ( ) and the recirculation PAGE 87 87 of seawater ( ) were chosen to illustrate the investigat ion and quantitati on of saltwater intrusion as shown in Figure 5 1. Figure 5 1 Investigated dimensionless ratios which related to extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) The extent of saltwater intrusion ( ) is the dimensionless length representing how far the saltwater intrudes into the coastal aquifer. It is the ratio of the toe of sa ltwater intrusion interface ( ) to the depth of the aquifer ( ). The toe of the saltwater intrusion interface ( ) can be determined by measuring the landward extend of the 0.5 concentration contour along the base of the aquifer, wher e is the concentration of mixed fresh and salt water, i.e., brackish water, in the coastal aquifer and is the concentration of seawater (35 kg/m 3 or 35,000 ppm). The degree of saltwater recirculation ( ) is the dimensionless ratio of mass flow ra te representing how much circulated saltwater is entrained within the overlying freshwater in the coastal aquifer and returned to the sea at the downgradient boundary (Barlow 2003; Bush 1988; Motz and Sedighi 2013; Simpson and Clement 2004) It is measured as the ratio of the saltwater mass in flow ( ) at the downgradi ent boundary to the freshwater mass in flow ( ) at the upgradient boundary. PAGE 88 88 The saltwater mass in flow ( ) and the freshwater mass in flow ( ) are the products of inflow rates per unit width ( ) and the respective densities ( ), whe re the densities of saltwater and freshwater are 1,025 and 1,000 kg/m 3 respectively. Flow Domain T he domain of the saltwater intrusion problem was considered a two dimensional cross section in the direction. T he width of the domain perpendicular to the direction of flow in the cross section was arbitrarily set to 1.0 m, as shown in Figure 5 2. Figure 5 2 Domain of the saltwater intrusion problem The domain wa s simulated in which freshwater inflow occurred at the upgradient boundary, and saltwater inflow and recirculate d seawater outflow occurred at the downgradient coastal boundary The upgradient boundary wa s a specified flux boundary with a zero freshwater concentration, and the downgradient boundary wa s a specified head boundary with a specified saltwater concentration The top and bottom of the domain were both no flow boundaries PAGE 89 89 Consider ed Variables = 0.5 concentration contour at base of the domain is the location representing the extent of s altwater intrusion, called the toe of the saltwater intrusion interface ( ) illustrating the extent of saltwater intrusion ( ). The saltwater mass in flow ( ) and the freshwater mass in flow ( ) represent the recirculation of seawa ter ( ), as shown in Figure 5 2. The location of the toe of saltwater intrusion interface ( ) illustrating the extent of saltwater intrusion ( ), the saltwater mass in flow ( ), and the freshwater mass in flow ( ) naturally are r elated to the characteristics of the freshwater, saltwater and medium, as shown in Table 5 1. Table 5 1 Variables affecting saltwater intrusion. Parameter Description Dimensions Porosity of medium Density contrast parameter = Density of freshwater Density of saltwater Horizontal dimension of the medium Vertical dimension of the medium Horizontal dispersivity Vertical dispersivity Freshwater inflow per unit width at upgradient boundary Saltwater inflow per unit width at downgradient boundary Horizontal hydraulic conductivity Vertical hydraulic conductivity Molecular diffusion coefficient or effective molecular diffusion coefficient Absolute or dynamic viscosity of freshwater Absolute or dynamic viscosity of saltwater Henry (1964 ) called this parameter the density difference ratio. PAGE 90 90 The porosity of the medium ( ) is defined as the ratio of void volume ( ) to total volume ( ). Moreover, the p orosity of the medium i s often expressed as the percentage of rock or soil void of material which can be expressed by multiply ing the ratio by 100 (Fetter 2001) It was assumed in this work that the porosity of the medium ( ) was a constant value through the domain The density contrast parameter ( ) is defined as: ( 5 1 ) where : is the density of saltwater ; and is the density of freshwater. The density contrast parameter ( ) depends on both the saltwater density ( ) and the freshwater density ( ), as shown in Equation 5 1 The effects of pressure and temperature in this work were neglected since both freshwater and saltwater were considered as incompressible fluids The density of saltwater ( ) and the density of freshwater ( ) were constant values Therefore, for the density of saltwater ( ) = 1,025 kg/m 3 and the density of freshwater ( ) = 1,000 kg/m 3 the density contrast parameter ( ) is a constant va lue, 0.025, based on Equation 5 1. The hydraulic conductivity ( ) in any direction relates to the properties of the medium and fluids, which is expressed as: ( 5 2 ) where: is the intrinsic permeability or the permeability which is defined depending solely upon properties of the medium (Bear 1979; Fetter 2001) Childs (1969) noted that the physical properties of the porous matrix might be changed by chemical reactions from fluids. However, this work neglected any chemical reactions PAGE 91 91 occurring in the domain, and thus the intrinsic permeability is a constant value throughout th e flowfield The specific weight is the specific weight of fluid which is equal to the density of the fluid ( ) multiplied by the acceleration of gravity ( ), and is the absolute or dynamic viscosity of fluid As mentioned above in neglect ing the effects of pressure and temperature, the a bsolute or dynamic viscosity ( ) of both saltwater ( ) and freshwater ( ) were considered to be constant values. Determination of PI Terms for the I nvestigation an d Q uantitation of Saltwater Intrusion Problem Dimensionless Extent of Saltwater Intrusion ( ) The dimensionless extent of saltwater intrusion ( ) is defined as the ratio of the toe of the saltwater intrusion interface ( ) to the depth of the a quifer ( ) Therefore, the saltwater intrusion interface ( ) is the variable of interest. The saltwater intrusion interface ( ) was treated as a dependent variable to determine the d imensionless extent of saltwater intrusion ( ) The i n dependent variable s that may have an effect on the dependent variable, i.e., the toe of the saltwater intrusion interface ( ) are shown in Table 5 1. The dimensional analysis was carried out as follows: 1. The relationship between the dependent var iable ( ) and a complete set of the in dependent variable s is: ( 5 3 ) The dimensionless parameters are not included in the dimensional analysis, namely the po rosity of the medium ( ) and the density contrast parameter ( ). PAGE 92 92 Therefore, the relationship between the dependent variable ( ) and the set of in dependent variable s is: ( 5 4 ) with 14 dimensional variables 2. These primary dimensions were selected: M ass, L ength and T ime ( ) system. 3. V ariable s and d imension al considerations are shown in Table 5 2. Table 5 2 D ependent variable ( ), in dependent variable s and their dimensions to determine the d imensionless extent of saltwater intrusion ( ) V ariable Dimension 3 primary dimensions 4. R epeating parameters were selected, namely and Thus, PAGE 93 93 5. Dimensionless groups were formed: = 14 3 = 11 dimensionless groups will result. (5 5) 6. D imensionless equations were set up: = = Therefore, : = then : = (5 6) : = = = = Therefore, : = then : = (5 7) : = = PAGE 94 94 = = Therefore, : = then : = ( 5 8 ) : = = = = Therefore, : = then : = ( 5 9 ) : = = = = PAGE 95 95 Therefore, : = then : = ( 5 10 ) : = = = = Therefore, : = then : = ( 5 11 ) : = = = = Therefore, : = then : = ( 5 12 ) : = = PAGE 96 96 = = Therefore, : = then : = ( 5 13 ) : = = = = Therefore, : = then : = ( 5 14 ) : = = = = PAGE 97 97 Therefore, : = then : = ( 5 15 ) : = = = = Therefore, : = then : = ( 5 16 ) : = = The results of dimensional analysis for the d imensionless extent of saltwater intrusion ( ) can be expressed in the form as shown in Equations 5 17 to 5 19: = (5 17) = (5 18) = (5 19) The other preferred dimensionless groups can be determined by the combination of the product of selected terms. PAGE 98 98 Dimensionless Re circulation of Seawater ( ) The d imensionless recirculation of seawater ( ) i s defined as the ratio of the saltwater mass in flow ( ) to the freshwater mass in flow ( ). Therefore, the d imensionless recirculation of seawater ( ) can be determined by the combination of the product of the and terms: = = = = (5 20) D imensionless Variables (1964) problem considered three main dimensionless variables, na mely the aspect ratio ( ) and the dimensionless ratios and to determine solutions. The first dimensionless variable, the aspect ratio ( ), is a ratio of the overall horizontal ( ) to vertical ( ) length scales of the domain. The term represents the aspect ratio ( ) as follows: Therefore, the aspect ratio ( ) is: (5 21) PAGE 99 99 The second dimensionless variable, the dimensionless ratio is the ratio of the freshwater advective flux ( ) to the density driven buoyancy flux ( ) (Motz and Sedighi 2013; Simpson and Clement 2004) It can be determined by using the density contrast parameter ( ) and or (1964) problem is a homogeneous and isotropic aquifer, = = the or terms can be written as follows: or or B y a combination of the product of the term and the dimensionless density contrast parameter: ( 5 22 ) The third dimensionless variable, the dimensionless ratio is the ratio of the porosity and the coefficient of dispersion ( ) to the freshwater advective flux ( ). It can be obtained by combining the term and the porosity of the medium ( ) as follows: (5 23) PAGE 100 1 00 Therefore, the d imensionless function of the three independent dimensionless ratio s for (1964) problem to investigate saltwater intrusion and to determine quantitatively the e xtent of s altwater Intrusion ( ) is: (5 24) where : is a function of three independent dimensionless ratios namely the aspect ratio ; the dimensionless ratio ; and the dimensionless ratio Henry Constant Dispersion Problem The Henr y constant dispersion problem i (1964) problem This problem used the aspect ratio ( ) and the dimensionless ratios and to r epresent values of the parameters to determine solutions. D imensionless function s of the three independent dimensionless ratio s for the Henry c onstant d ispersion p roblem to investigate saltwater intrusion and to determine quantitatively the e xtent of s altwater Intrusion ( ) and the d egree of saltwater recirculation ( ) are written as shown in Equations 5 25 and 5 26. (5 25) (5 26) where : and are function s of three independent dimensionless ratios namely the aspect ratio ; the dimensionless ratio ; and the dimensionless ratio PAGE 101 101 Henry Velocity dependent Dispersion Problem The Henry velocity dependent dispersion problem consider s five dimensionless parameters, namely the aspect ratio ( ), the dimensionless ratios and the dispersivity ratio ( ) and the hydraulic conductivity ratio ( ). By considering these additional parameters ( and ) this problem seem s to be more realistic than (1964) constant dispersion problem. D imensionless function s of the f ive independent dimensionless ratio s to investigate saltwater intrusion and to determine quantitatively the e xtent of s altwater Intrusion ( ) and the d egree of saltwater recirculation ( ) are written as shown in Equations 5 27 and 5 28. (5 27) (5 28) where : and are function s independent dimensionless ratios namely the aspect ratio ; the dimensionless ratio ; the dimensionless ratio ; the dispersivity ratio ; and the hydraulic conductivity ratio In this investigation, the effect of mole cular diffusion coefficient ( ) was so small when compar ed to the summation of the coefficient of dispersion ( ) that it could be considered zero That makes t he dimensionless ratio equal to zero. Therefore, the dimensionless functi ons for the Henry velocity dependent dispersion problem to investigate saltwater intrusion and to determine quantitatively the PAGE 102 102 e xtent of s altwater Intrusion ( ) and the d egree of saltwater recirculation ( ) can be rewritten as shown in Equation s 5 29 and 5 30. (5 29) (5 30) Field s cale Problem The field scale problem considers five dimensionless parameters, namely the aspect ratio ( ), the dimensionless ratios and the dispersivity ratio ( ) and the hydraulic conductivity ratio ( ), which is synonymous with the Henry velocity dependent dispersion problem. D imensionless function s of the f ive independent dimensionless ratio s to investigate saltwater intrusion and to determine quantitatively the e xtent of s altwater Intrusion ( ) and the d egree of saltwater recirculation ( ) are written as shown in Equations 5 31 and 5 32. (5 31) (5 32) where : and are function s independent dimensionless ratios namely the aspect ratio ; the dimensionless ratio ; the dimensionless ratio ; the dispersivity ratio ; and the hydraulic conductivity ratio PAGE 103 103 In this investigation, the effect of molecular diffusion coefficient ( ) was very small when compar ed to the summation of th e coefficient of dispersion ( ) and thus it was assigned to be zero That makes t he dimensionless ratio equal to zero. Therefore, the dimensionless functions for the field scale problem to investigate saltwater intrusion and to deter mine quantitatively the e xtent of s altwater Intrusion ( ) and the d egree of saltwater recirculation ( ) can be rewritten as shown in Equation s 5 33 and 5 3 4 (5 33 ) (5 3 4 ) PAGE 104 104 CHAPTER 6 DESIGN OF NUMERICAL EXPERIMENTS AND PARAMETERS Numerical experiments were performed to investigate saltwater intrusion and the components of the submarine groundwater discharge consisting of the fresh groundwater d ischarge and recirculation of seawater due to the density difference between saltwater and freshwater. The investigation wa s undertaken to determine quantitatively how saltwater intrusion and the recirculation of seawater are related to the freshwater adv ective flux ( ) and density driven buoyancy flux ( ) over a broad range of values for the dimensionless ratio = for an isotropic aquifer and the freshwater advective flux ( ) and density driven vertical buoyancy flux ( ) over a broad range of values for the dimensionless ratio = for an anisotropic aquifer Also, the numerical experiments were conducted to determine the differences and similarities between the less computationally intense uncoupled solu tions versus the more computationall y intensive coupled solutions. Numerical Modeling Solution s of the problems were obtained by running SEAWAT Version 4 with the variable density flow (VDF) package to steady state conditio ns. SEAWAT wa s developed by comb ining MODFLOW 2000 and MT3DMS into a single program to simulate three dimensional variable density groundwater flow coupled with multi species solute and heat transport (Lange vin et al. 2008) Since SEAWAT is written in FORTRAN, all input and output files are processed by using specific position of rows and columns in files Therefore, the positions of each variable in the files are very important and represent specific meaning If any parameter value is positioned incorrectly, the simulation fail s, and it is too difficult to find the incorrect value and PAGE 105 105 position. In order to simplify the input and outpu t files process, the solutions wer e performed using a commercial software, i.e., G roundwater V istas Version 5 (GV5) which provid es a user friendly W indows based graphical interface (Rumbaugh and Rumbaugh 2012) GV5 was used to create input files, run SEAWAT and analyze simulation results. Numerical Solutions S altwater intrusion consists of a dense front be ing transported into a less dense region due to the forced convection generated from boundary forcing and due to the free convection generated from internal density variations (Gebhart et al. 1988; Goswami and Clement 2007; Schincariol and Schwartz 1990; Simpson and Clement 2003) Therefore, the three problems were solved numerically for two types of solutions namely a density uncoupled solution (a less co mputationally intensive method) and a density coupled solution ( a more computationally intensive method). For t he density uncoupled solution the fluid density is constant, and thus, the flow and transport equations are uncoupled in a constant density flowfield, which accounts for the forced convection process. F or the density coupled condition the fluid density is a function of the concentrat ion of total dissolved solids (TDS) and thus, the flow and transport equations are coupled in a variable density flowfield which is depended on both the forced and free convection process. Uncoupled s olution The uncoupled solution can be conceptually vi ewed as a method to isolate the transport due to internal free convection caused by density dependent flow processes from the transport due to forced convection caused by external force, i.e., the saltwater PAGE 106 106 hydrostatic heads at the downgradient boundary (Goswami and Clement 2007; Simpson and Clement 2003) Thus, f or the uncoupled solution, the density of the fluid i s considered constant, i.e. no concentration change within a timestep in the domain, to represent constant density flow and transport in a coastal aquifer T he flow and transport equations are uncoupled in the calculation which is a less computationally intensive method. The flowchar t of the SEAWAT program for the uncoupled solution i s shown in Figure 6 1 Figure 6 1 Generalized flow chart of the SEAWAT program for the uncoupled solution (modified from Guo and Langevin 2002) Coupled s olution The transport of saltwater of the coupl ed solution is similar to that of the uncoupled solution. A dense front will be transported into a less dense region due to It is represented by e quivalent freshwater heads PAGE 107 107 the forced convection generated from the saltwater hydrostatic heads at the downgradient boundary and due to the free convection g enerated from internal density variations (Goswami and Clement 2007; Simpson and Clement 2003) The fluid density is a function of the concentration of tota l dissolved solids (TDS) Thus, it result s that the flow and transport equations are coupled in the flow field which is a more computationally intensive method and requires significantly more computation cost s than the unc oupled solution The schematic diagram of the SEAWAT program for the coupled solution i s shown in Figure 6 2 Figure 6 2 Generalized flow chart of the SEAWAT program for the coupled solution (modified from Guo and Langevin 2002) It is represented by e quivalent freshwater heads PAGE 108 108 Boundary Conditions To solve the three problems domains of each problem were assigned a constant flux at the upgradient inland boundary with the concentration of TDS equal to zero and equivalent freshwater heads at the downgradient boundary with a constant concentration of TDS equal to 35 kg/m 3 Equival ent freshwater h eads at the downgradient b oundary Equivalent freshwater heads were specified at the downgradient boundary, which represents saltwater hydrostatic heads, for both the uncoupled and coupled solutions to account for the density differences bet ween freshwater and saltwater at the downstream boundary. For the uncoupled solutions, equivalent freshwater heads were specified over the depth of the coastal boundary with 0 m, 1,025 kg/m 3 and 1,000 kg/m 3 based on Equation 6 1 (Guo and Langevin 2002) : 6 1 where: = equivalent freshwater head, ; and = elevation of given point above or below a datum (sea level), i.e., = 0 at sea level as shown in Figure 6 3. For the coupled solutions, equivalent freshwater heads were calculated internally in SEAWAT Version 4 PAGE 109 109 Figure 6 3 Piezometer to calculat e equivalent freshwater head (Guo and Langevin. 2002) The boundary c onditions for the Henry constant dispersion and the Henry velocity dependent dispersion problems are shown in Figure 6 4. Figure 6 4 Boundary con ditions of the numerical modeling PAGE 110 110 Concentration condition at the downgradient and upgradient boundaries For the Henry constant dispersion and Henry velocity dependent dispersion problems the upgradient inland boundary on the left hand side of the domain was specified as a constant flux boundary with the concentration of TDS equal to zero, representing the freshwater inflow The downgradient coastal boundary on the right hand side of the domain was simulated by specifying a concentration boundary condition with a concentration of TDS equal to 35 kg/m 3 representing the concentration of TDS in seawater, as shown in Figures 6 5 and 6 6. The field scale problem used the same c oncentration of TDS as the Henry constant dispersion and the Henry velocity dependent dispersion problems but the upgradient inland boundary with freshwater inflow was on the right hand side of the domain. The downgradient coastal boundary with seawater inflow was on t he left hand side of the domain, as shown in Figures 6 7 and 6 8. The flow rates at the coastal boundary were calculated internally by SEAWAT, which was shown in the o utput file s, i.e., the LST files. Finite difference Grid and B oundary C onditions Henry co nstant dispersion and H enry velocity dependent dispersion problems The Henry constant dispersion and the Henry velocity dependent dispersion problems used a slightly modified version (1964) domain to solve the numerical modeling From the finite difference grid optimization as sho wn in Appendix A when comparing SEAWAT results to the work of Simpson and Clement (2004) setting and equal to 0.05 m was determined to be the optimal finite difference grid ( Optimal Finite Difference Grid Appendix A) PAGE 111 111 Therefore, the domain discretization was a three dimensional model which represent s a coastal aquifer discretized into one row, 41 columns and 20 layers The columns w ere equally spaced with 0.05 m for a total length of 2.05 m The 41 st column was used to precisely locate the saltwater hydrostatic heads at a distance of 2.0 m since the Block Center ed Flow (BCF) package was applied in the SEAWAT calculations All layers were specified as confined aquifer layers and were equally spaced with 0.05 m for a total depth of 1.0 m The width of the row perpendicular to the direction of flow in the cross section was arbitrarily set to 1.0 m as shown in Fig ure 6 5 To simplify the numeri cal model, the finite difference grid and b oundary conditions can be represented in a two dimensional cross section, as shown in Figure 6 6. Figure 6 5 Finite difference grid used for the Henry constant dispersion and th e Henry velocity dependent dispersion problems. PAGE 112 112 Figure 6 6 Two dimensional f inite difference grid and b oundary c onditions used for the Henry constant dispersion and the Henry velocity dependent dispersion problems. Field scale p roblem A slightly modifi ed version (2013) domain was used for the domain of the field scale problem to solve the numerical modeling A three dimensional domain was discr etized into one row, 81 columns and 40 layers The columns were equally spaced with = 250 m for a total length of 20,250 m The 81 st column was used to precisely locate the saltwater hydrostatic heads at a distance of 20,000 m since the Block Center ed Flow (BCF) package was applied in the SEAWAT calculations. All layers were specified as confined aquifer layers and were equally spaced with = 5 m for a total depth of 200 m The width of the row perpendicular to the direction of flow in the cross section was arbitrarily set to = 250 m, as shown in Figure 6 7 PAGE 113 113 Figure 6 7 Finite difference grid used in field scale problem. The simplified numerical model can be re presented in a two dimensional cross section as shown in Figure 6 8. Figure 6 8 Two dimensional f inite difference grid and b oundary c onditions used for the field scale problem. PAGE 114 114 Methods f or Henry Constant Dispersion Problem Assumption of the Henry C onstant D ispersion P roblem (1964) problem i.e., the dispersion coefficient or the hydrodynamic dispersion ( ) was treated as a constant scalar throughout the flowfield (Henry 1964) Thus, the molecular diffusion coefficient ( ) in SEAWAT was used to represent all of the effects of the dispersion coefficient ( ), i.e. neglecting all of the effects of velocity dependent dispersivity ( ). The aquifer parameters in this problem were synonymous with those of Guo and Langevin (2002) and Simpson and Clement (2004) Investigation Procedures of the Henry Constant Dispersion Problem First of all, the model w as verified by comparing the results to the benchmark problem, i.e., Simpson and Clement (2004) Then, as shown in Chapter 5 25 and 5 26, the investigation results of the Henry constant dispersion problem were pres ented via two dimensionless dependent parameters, i.e., the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) The two dimensionless dependent parameters i.e., and were functions of three independent dimensionless ratios, namely the aspect ratio the dimensionless ratio and the dimensionless ratio The aspect ratio was assigned a value equal to 2.0 However, the dimensionless ratios and were varied. The freshwater advective flux ( ) was selected as a repeating variable for both dimensionless ratios. The d ensity contrast parameter ( ) and the t ransverse vertical dimension or the vertical dimension of the cross section ( ) were considered constant PAGE 115 115 values equal to 0.025 and 1.0 respectively. Therefore, t he investigation w as determined to solve the Henry constant dispersion problem in two sets of investigation since there were two independent aquifer p arameters in the dimensionless ratios and which could be varied, namely the hydraulic conductivity ( ) and the molecular diffusion coefficient ( ) In the first set of investigations as shown in Figure 6 9 the value of the isotropic hydraulic conductivity ( ) was considered as a constant value, which was equal to 864 m/day The aspect ratio ( ) was assigned a value equal to 2.0 The dimensionless ratios and were considered independent var iables The freshwater advective flux ( ) was selected as a repeating variable The dimensionless ratio was varied from 0.1 to 10.0 for each of five investigations in which the values of the dimensionless ratio were varied from 0.1 to 0.5 Thi s was accomplished by varying the freshwater inflow per unit width ( ) from 2.16 to 216 m 2 /day to achieve a range of dimensionless ratio = 0.1 to 10.0 and varying the value of molecular diffusion coefficient ( ) from 0.617142857 to 308.57142860 m 2 /day to achieve a range of dimensionless ratio = 0.1 to 0.5 As shown in Figure 6 9, t he second set of investigations was similar to the first set of investigations, but the value of molecular diffusion coefficient ( ) was considered as a constant v alue, which was equal to 1.62924998 m 2 /day The aspect ratio ( ) was also assigned a value equal to 2.0 The dimensionless ratios and were considered independent variables The freshwater advective flux ( ) was selected as a repeating variable The dimensionless ratio was varied from 0.1 to 0.5 for each of nineteen investigations in which the values of the dimensionless ratio were PAGE 116 116 varied from 0.1 to 10.0 This was accomplished by varying the freshwater inflow per unit width ( ) from 5.7024 to 1.14048 m 2 /day to achieve a range of dimensionless ratio 0.1 to 0.5 and varying the value of Isotropic hydraulic conductivity ( ) from 2,282.1010 to 4.5620 m/day to achieve a range of dimensionless ratio from 0.1 to 10.0 A schematic of the investigation procedures is shown in Figure 6 9 Figure 6 9 Schematic of investigation procedures of the Henry constant dispersion problem PAGE 117 117 Aquifer Parameters of the Henry Constant Dispersion Problem Until now, m an y reproduce d work s have published either a different method or have used different aquifer parameter values for comparing to Henry (1964) problem One of the most interested parameters is the molecular diffusion coefficient ( ) Many reproduced works adjusted the value of molecular diffusio n coefficient ( ) so as to make their (1964) work They considered the value of molecular diffusion coefficient ( ) to be two cases, namely 1.62925 m 2 /d and 0.57024 m 2 /d (Guo and Langevin 2002; Voss and Souza 1987) Some reproduced works adjusted the b oundary conditions of the domain to obtain results, which w ere similar to (1964) re sults (Frind 1982; Gambolati et al. 1992; Huyakorn et al. 1987) Some values of the aquifer parameters used i n the Henry constant dispersion problem were synonymous with Guo and Langevin (2002) and Simpson and Clement (2004) as shown in Table 6 1. Table 6 1 Aquifer parameters used for the Henry constant dispersion problem Parameter Value Unit Density contrast parameter ( ) 0.025 Porosity ( ) 0.35 Total length of the domain 2.05 m Horizontal dimension ( ) 2.0 m Vertical dimension ( ) 1.0 m Dispersivity ( ) 0 m For the first set: I sotropic hydraulic conductivity ( ) 864 m/day M olecular diffusion coefficient ( ) 0.617 to 308.571 m 2 /day For the second set: I sotropic hydraulic conductivity ( ) 2,282.101 to 4.562 m/day M olecular diffusion coefficient ( ) 1.62924998 m 2 /day PAGE 118 118 Methods for Henry Velocity dependent Dispersion Problem Assumption of the Henry Velocity dependent Dispersion Problem The a ssumption of Henry velocity dependent dispersion problem is different from (1964) original problem in that the H enry velocity dependent dispersion problem assumes that the flow and transport in an aquifer are associat ed with velocity dependent hydrodynamic dispersion ( ), in which all the effects o f molecular diffusion coefficient ( ) are included in the hydrodynamic dispersion term This corresponds with the current flow and transport theory. Investigation Procedures of the Henry Velocity dependent Dispersion Problem First of all, the model w as verified by comparing the results to the bench mark problems, i.e., Sgol (1993) and Abarca et al. (2007) Then, as shown in Chapter 5 2 9 and 5 30 the investigation results of the Henry velocity dependent dispersion problem were presented via two dimensionless parameters, i.e., the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) The two dimensionless dependent varia bles i.e., and were functions of four independent dimensio nless ratios, namely the aspect ratio the dimensionless ratio the dispersivity ratio = and the hydraulic conductivity ratio = The aspect ratio was assigned a value = 2.0. The investigation was classified into three main sets by the dispersivity ratio namely 0.01, 0.10 and 1.00 Each set of the dispersivity ratios ( ) included three sets of the hydraulic conductivity ratio = namely 0.01, 0.10 and 1.00. A schematic of the investigation procedures is shown in Figure 6 10. PAGE 119 119 Figure 6 10 Schematic of investigation procedures of the Henry velocity dependent dispersion problem Aquifer Parameters of the Henry V elocity dependent D ispersion P roblem Many modified H enry problem works attempted to perform solutions of the problem in other ways such as modifying equations by using a density dependent transport (Frind 1982) editing the b oundary conditions of the domain (Gambolati et al. 1992; Huyakorn et al. 1987) changing the medium properties by using a heterogeneous domain (Held et a l. 2005) applying to another approach by using an i nverse modeling (Sanz and Voss 2006) or adjusting the aquifer parameters by using an a nisotropic dispersive (Abarca et al. 2007) etc. PAGE 120 120 Henry velocity dependent dispersion problem adjusted the domain of Henr y (1964) problem in order to be an anisotropic, homogeneous, confined aquifer The solution for this problem corresponds with the current flow and transport theory However, (1964) problem consisted of three parameters namely the aspect ratio = the dimensionless ratios = and = and he did not give any details to determine the other aquifer parameters especially the value of dispersivity ( ). Therefore, the work of Gelhar et al. (1992) was applied to determine the dispersivity ( ) in the longitudinal direction for the the Henry velocity dependent dispersion problem Figure 6 11 The relation of longtidutinal dispersivity and length scale with data classified by reliability (modified from Gelhar et al. 1992). PAGE 121 121 Gelhar et al. (1992) concluded about the t ransverse horizontal dispersivity ( ) and the t ransverse vertical dispersivity ( ) that: 1. The t ransverse horizontal dispersivity values ( ) are an order of the magnitude s maller than the longitudinal ( horizontal ) dispersivity values ( ) which was the blue line in Figure 6 11, i.e., 0.1 ; and 2. The t ransverse vertical dispersivity values ( ) are two orders of the magnitude smaller than both the l ongitudinal d ispersivity values ( ) and the t ransverse horizontal dispersivity values ( ), i.e., 0.01 Thus, the value of l ongitudinal dispersivity ( ) was assigned to correspond with the work of Gelhar et al. (1992) which was equal to one tenth of the overall length scale in the longitudinal ( horizontal ) flow direction The values of l ongitudinal dispersivity ( ) of the Henry velocity dependent dispersion problem was equal to 0.2 m The values of aquifer parameters for this problem are shown in Table 6 2. Table 6 2 Aquifer parameters for the Henry velocity dependent dispersion problem Parameter Value Unit Density contrast parameter ( ) 0.025 Porosity ( ) 0.35 Total length of the domain 2.05 m Horizontal dimension of the domain ( ) 2.0 m Vertical dimension of the domain ( ) 1.0 m Longitudinal dispersivity ( ) 0.2 m Transverse horizontal dispersivity ( ) 0.2 m Transverse vertical dispersivity ( ) 0.002, 0.02 and 0.2 m Longitudinal hydraulic conductivity ( ) 864 m/day Transverse horizontal hydraulic conductivity ( ) 864 m/day Transverse vertical hydraulic conductivity ( ) 8.64, 86.4 and 864 m/day M olecular diffusion coefficient ( ) 0.0 m 2 /day Note: Although the transverse horizontal dispersivity ( ) was assigned equal to the longitudinal dispersivity ( ) and the transverse horizontal hydraulic conductivity value ( ) were assigned equal to the longitudinal hydraulic conductivity ( ), the investigations were still accurate since the investigations were considered in a two dimensional cross section model. PAGE 122 122 As shown in Table 6 2, some of the aquifer parameters were similar to the work of Guo and Langevin (2002) but the dispersivity values were adjusted by using the work of Gelhar et al. (1992) T he transverse vertical dispersivity values ( ) were in the range of Gelhar et al. (1992) i.e., 0.002 m. That made the dispersivity ratio s = equal to 0.01. Moreover, the dispersivity ratio s = were expanded equal to 0.10 and 1.00, which cover ed a wide range of possible dispersivity values. Methods of Field s cale Problem Assumption of the Field s cale Problem The assumptions of the field scale problem w ere similar to the Henry velocity dependent dispersion problem in terms of velocity dependent dispersion. The field scale problem assumes that the flow a nd transport in an aquifer are associat ed with velocity dependent hydrodynamic dispersion ( ), in which all the effects of molecular diffusion coefficient ( ) are included in the hydrodynamic dispersion term This corresponds with the current flow and transport theory T he values of the aspect ratio = and other aquifer parameter s used in this problem were applied from Motz and Sedighi ( 2009b; (1988) investigation o f the upper part of the Floridan aquifer system at Hilton Head Island, South Carolina, U.S.A. Investigation Procedures of the Field s cale Problem First of all, the model was verified by comparing the results to the benchmark problem i.e., Motz and Sedighi (2013) 33 and 5 34 the investigation results of the Henry velocity dependent dispersion problem were presented via two dimensionless parameters, i.e., the extent o f saltwater intrusion ( ) and the degree of saltwater recirculation ( ) PAGE 123 123 The two dimensionless dependent varia bles i.e., and were functions of four independent dimensionless ratios, namely the aspect ratio the dimensionless rat io the dispersivity ratio = and the hydraulic conductivity ratio = In the field scale p roblem t he aspect ratio was assigned equal to 100, and t he dispersivity ratio = was held equal to 0.01 T he investigation was categorized into three main sets based on the hydraulic conductivity ratio = namely 0.01, 0.10 and 1.00, which cover ed a wide range of possible hydraulic conductivity values in the field scale A schematic of t he investigation pro cedures is shown in Figure 6 11 Figure 6 11 Schematic of the investigation procedures of the field scale p roblem PAGE 124 124 Aquifer Parameters of the Field s cale Problem This problem adjusted the a quifer p arameters from the works of Bush (1988) and Motz and Sedighi (2009b ; 2013) The flow field was modified to be an anisotropic, homogeneous, confined aquifer by varying the values of dispersivity and hydraulic conductivity in the longitudinal ( ), the transverse horizontal ( ) and the transverse vertical dispersivity ( ) direction. The case of the hydraulic conductivity ratio = = 0.01 and the dispersivity ratio = = 0.01 was similar to Bush (1988) and Motz and Sedighi (2009a; 2009b; 2010 ; 2013) The values of the aquifer parameters used in the field scale problem are shown in Table 6 3. Table 6 3 Aquifer parameters used in the field scale p roblem Parameter Value Unit Density contrast parameter ( ) 0.025 Porosity ( ) 0.30 Total length of the domain 20,250 m Longitudinal dimension of the domain ( ) 20,000 m Longitudinal discretization ( ) 250 m Transverse horizontal dimension of the domain ( ) 250 m Transverse horizontal discretization ( ) 250 m Transverse vertical dimension of the domain ( ) 200 m Transverse vertical discretization ( ) 5 m Longitudinal dispersivity ( ) 125 m Transverse horizontal dispersivity ( ) 125 m Transverse vertical dispersivity ( ) 1.25 m Longitudinal hydraulic conductivity ( ) 50 m/day Transverse horizontal hydraulic conductivity ( ) 50 m/day Transverse vertical hydraulic conductivity ( ) 0.5, 5.0 and 50 m/day Freshwater inflow per unit width ( ) 0.25 to 2,500 m 2 /day Note: Although the transverse horizontal dispersivity ( ) was assigned equal to the longitudinal dispersivity ( ), and the transverse horizontal hydraulic conductivity value ( ) were assigned equal to the longitudinal hydraulic conductivity ( ), the investigations were still accurate since the inve stigations were considered in a two dimensional cross section model PAGE 125 125 CHAPTER 7 HENRY CONSTANT DISPERSION P ROBLEM Benchmark Problem for the Henry Constant Dispersion Problem In this study, the use of SEAWAT and Groundwater Vistas 5 was verified by comparing results to the previous work obtained by Simpson and Clement (2003; 2004) The work of Simpson and Clement (2004) compared results for the original (or standard) Henry problem obtained by re calculating Henry (1964) semi analytical solution for the dimensionless ratios 0.263, 0.1 and the aspect ratio 2.0 with results obtained numerically using a Galerkin finite element numerical solution The y also obtained a Galerkin finite element numerical solution for t he uncoupled (constant density) case, as shown In Figure 7 1 ( A) They proposed a modified Henry problem with the dimensionless ratios 0.1315, 0.2 and the aspect ratio 2.0 and compared the results with results obtained by re (1964) semi analytical solution They also obtained a Galerkin finite element numerical solution for the uncoupled (constant density) case, as shown In Figure 7 1 ( B). Langevin and Guo (2006) compared results obtained using SEAWAT 2000 (Langevin et al. 2003) (1964) semi analytical solution as re calculated by Simpson and Clement (2004) for both the standard and modified Henry problems The results of Langevin and Guo (2006) were in quite good agr eement with the work of Simpson and Clement (2004) as shown in Figure 7 2 Motz and Sedighi (2013) compared results obtained us ing SEAWAT Version 4 (Langevin et al. 2008) with the numercal solutions obtained by Simpson and Clement (2004) for the standard and modified Henry problems for both the uncoupled and PAGE 126 126 coupled solutions The results of Motz and Sedighi (2013) also were in quite favorable agreement with the work of Simpson and Cleme nt (2004) as shown in Figure 7 3. A B Figure 7 1 R esults obtained from Simpson and Clement (2004) A) the standard and B) the modified Henry problems. A B Figure 7 2 T he work of Langevin and Guo (2006) A) the standard and B) the modified Henry problems. A B Figure 7 3 R esults of Motz and Sedighi (2013) A) the standard and B) the modified Henry problems. PAGE 127 127 Aquifer Parameters for Simpson and Clement (2004) The domain was assigned to be an isotropic, homogeneous, confined aquifer with the hydraulic conductivit y ( ) = 864 m/day and the porosity ( ) = 0.35 The dispersivities ( and ) were set equal to zero The molecular diffusion coefficient ( ) in SEAWAT was us ed to represent all the effects of the dispersion coefficient ( ) and set equal to 1.6 2925 m 2 /day (Guo and Langevin. 2002) which represent s the constant dispersion coefficient original (1964) problem The freshwater inflow per unit width ( (1964) s tandard and modified Henry problems were equal to 5.702 and 2.851 m 2 /day, respectively (Simpson and Clement 2004) M ethods and Schemes for Simpson and Clement (2004) Solutions were obtained by running SEAWAT Version 4 to steady state results, which is similarly to Langevin et al. (2008) The f low and transport components in SEAWAT were solved for both the standard and modified Henry problems for both the uncoupled and coupled solutions with identical solution procedures. Th e flow model was solved using the Pre Conditioned Conjugate Gradient (PCG2) package. The advective term of the solute transport equation was solved using the implicit finite difference scheme with the central in space weighting. The remaining terms (disp ersion, source/sink mixing and reaction) were solved using the iterative implicit Generalized Conjugate Gradient (GCG) solver with the Modified Incomplete Cholesky (MIC) pre conditioner. The convergence criterion in terms of relative concentration (CCLOSE ) was set equal to 1x10 6 The transport time step size within each time step of the flow solution (DT0) was assigned a value of 1x10 4 days, and a PAGE 128 128 timestep multiplier (TTSMULT) of 1.2 was used to increase subsequent timesteps The Variable Density Flow (VDF) package options were applied with the firs t transport time step (FIRSTDT) = 1x10 4 days. Each solution w as run to steady state in the transient mode for the 2 day simulation period which required 46 timesteps The o utput file s, i.e., the LST files, w ere checked to ensure that steady state was reached. The uncoupled flow and transport option in SEAWAT was us ed to represent constant density flow and trans port for the uncoupled solution (the less computationally intensive method). T he coupled fl ow and transport option was us ed to represent density dependent flow and transport for the coupled solution (the more computationally intensive method) SEAWAT Version 4 Results for Simpson and Clement (2004) The SEAWAT results of the standard and the modified Henry problems ( both the uncoupled and coupled solutions ) were presented by using the 0.5 isochlors. These results were compared to the b enchmark p roblem i.e., the work of Simpson and Clement (2004) as shown in Figure 7 4. As illustrated by the 0.5 isochlors, t he comparisons indicate very good ag reement with Simpson and Clement (2004) and demonstrate that SEAWAT Version 4 c an solve numerically the standard and the modified Henry problem s for both the uncoupled and coupled solutions. The comparisons also indicate that the m ethods and the s chemes are suitable to simulate the solutions of Henry constant dispersion problem A value of TTSMULT between 1.0 and 2.0 is generally adequate (Zheng and Wang 1999); The g rateful appreciation is expressed to Drs. Simpson and Clement, who provided copies of the output files, standard and modified Henry problem PAGE 129 129 A B Figure 7 4 Comparison between the 0.5 isochlors of SEAWAT results and the work of Simpson and Clement (2004) for A) the standard and B) the modified Henry problems both uncoupled and coup led solutions Henry Constant Dispersion Problem A 25 and 5 26, the investigation results of the Henry constant dispersion problem were presented via two dimensionless dependent parameters, i.e., the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) The two dimensionless dependent parameters i.e., and were functions of three independent dimensionless ratios, namely the aspect ratio the dimensionless ratio and the dimensionless ratio The aspect ratio was assigned a value equal to 2.0. However, the dimensionless ratios and were varied. The freshwater advective fl ux ( ) was selected as a repeating variable for both dimensionless ratios. The d ensity contrast parameter ( ) and the t ransverse vertical dimension or the vertical dimension of the cross section ( ) were considered constant values equal to 0.025 and 1.0, respectively. Therefore, t he investigation w as determined to solve the Henry constant dispersion problem in two sets of investigation since there were two independent aquifer parameters in the dimensionless ratios PAGE 130 130 and which could be varied, namely the hydraulic conductivity ( ) and the molecular diffusion coefficient ( ). For the first set of investigation, the value of the isotropic hydraulic conductivity ( ) was considered as a constant value, which was equal to 864 m/day. The aspect ratio ( ) was assigned a value equal to 2.0. The dimensionless ratios and were considered independent variables. The dimensionless ratio was varied from 0.1 to 10.0 for each of five investigations i n which the values of the dimensionless ratio were varied from 0.1 to 0.5. For t he second set of investigations, the value of molecular diffusion coefficient ( ) was considered as a constant value, which was equal to 1.62924998 m 2 /day The aspect r atio ( ) was also assigned a value equal to 2.0 The dimensionless ratios and were considered independent variables The dimensionless ratio was varied from 0.1 to 0.5 for each of nineteen investigations in which the values of the dimensionless ratio were varied from 0.1 to 10.0 Modified Method and Schemes for the Henry constant dispersion problem Methods and Sc hemes for Simpson and Clement (2004) the m ethod and the s chemes were verif ied by comparing the results to the benchmark problem, i .e., Simpson and Clement (2004) and then apply them to solve the Henry constant dispersion problem for both the first and second sets of investigations Afte r investigating the first and second sets the results indicate that t he transport stepsize multiplier within a flow model timestep (TTSMULT) = 1.2 did not make all PAGE 131 131 numerical cases reach steady state conditio ns at the end of the 2 day simulation period e specially the uncoupled solutions Accordingly, a satisfying value of TTSMULT was determined for all cases of the solutions The first set was selected to find the satisfying TTSMULT in five approaches: 1. When TTSMULT was equal to 1.2, SEAWAT could not calculate the steady state results at the end of the 2 day simulation period for some uncoupled solution cases at a large value of the dimensionless ratio and a large value of the dimensionless ratio such as 9.0 and 0.3; 2. If TTSMULT was assigned a value of 1.5, SEAWAT could not calculate the steady state results at the end of the 2 day simulation period for some coupled solution cases at a small value of the dimensionless ratio and a small value of the dimensionless ratio such as 0.4 and 0.1; 3. Upon s etting DT0 = 0 to let SEAWAT determine the timestep of models, SEAWAT could not calculate the steady state results at the end of the 2 day simulation period for some coupled solution cases at a small value of the dimensionles s ratio and a large value of the dimensionless ratio such as 0.1 and 0.3; 4 Upon d ecreasing the transport time step size within each time step of the flow solution (DT0) from 1x10 4 to 1x10 8 days, SEAWAT could not calculate the steady state results at the end of the 2 day simulation period for some uncoupled solution cases at a large value of the dimensionless ratio and a small value of the dimensionless ratio such as 10.0 and 0.2 ; and 5 T he third order total variation diminishing (TVD) scheme with Courant number ( ) = 0.01 SEAWAT could not calculate the steady state results at the end of the 2 day simulation period for some coupled solution cases at a large value of T he first set of the Henry constant dispersion investigations had 19 cases for the uncoupled solutions and another 19 c ases for the coupled solutions with the 5 set numerical model of the dimensionless ratio The second set of the Henry constant dispersion i nvestigations had the sa me amount of numerical models as the first set. Thus, the total of number of numerical mod els was equal to 380 cases. PAGE 132 132 the dimensionless ratio and a small value of t he dimensionless ratio such as 5.0 and 0.1 Therefore, based on these results, it was necessary to separate the transport stepsize multiplier within a flow model timestep (TTSMULT) into two groups, i.e., a TTSMULT value for the uncoupled solut ions and a TTSMULT value for the coupled solutions. The value of TTSMULT for the uncoupled solutions was equal to 1.5 and for the coupled solutions, it was equal to 1.2. There might be a question regarding the accuracy of the results between TTSMULT = 1.2 and 1.5 for the uncoupled solutions After investigating, it was determined that SEAWAT could perform the complete sets of uncoupled solutions for the first and second sets by using TTSMULT = 1.2 and 1.5 as shown in Figure s 7 5 and 7 6. At steady state the value of TTSMULT did not have any effect on the results The comparison presented indicates there was no significant difference in either patterns or values for the first and second sets of the coupled solutions wh en using TTSMULT = 1.2 and 1.5 as shown in Figure s 7 5 and 7 6 Thus, TTSMULT for the uncoupled solution s of both sets was assigned equal to 1.5 which required 23 timesteps and for the coupled solutions of both sets, it was equal to 1.2 which required 46 timesteps The o utput file s, i.e ., the LST files, w ere checked to ensur e that steady state was reached at the end of the 2 day simulation period PAGE 133 133 A B Figure 7 5 C omparison of results when using TTSMULT = 1.2 and 1.5 for A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) of uncoupled solutions for the first set. A B Figure 7 6 C omparison of results when using TTSMULT = 1.2 and 1.5 for A) extent of saltwater intrusion ( ) and B) degree of salt water recirculation ( ) of uncoupled solutions for the second set. PAGE 134 134 Input Parameters for the First Set of Investigations In t he first s et of inves tigations t he d imensionless r atio and the d imensionless r atio were considered i ndependent v ariables T he d imensionless r atio was varied from 0.1 to 10.0 for each of five investigations in which values of the d imensionless r atio were varied from 0.1 to 0.5 This was accomplished by varying the freshwate r inflow per unit width ( ) from 2.16 to 216 m 2 /day to achieve a range of a = 0.1 to 10.0. Values of the molecular diffusion coefficient ( ) were selected so that the value of the d imensionless r atio (which includes as the repeating variable ) in each of the five investigations could be held constant as was varied to change t he d imensionless r atio The value of hydraulic conductivity ( ) was considered as a constant value equal to 864 m/day. For both the uncoupled and coupled solutions, input parameters were divided into two groups, namely a quifer parameters and dimensionless ratios, which are shown in Table 7 1 and 7 2, respectively. Table 7 1 Aquifer parameters used in the first set of investigations Parameter Value Density contrast parameter ( ) 0.025 Porosity ( ) 0.35 Horizontal dimension of the domain ( ) 2.0 m Vertical dimension of the domain ( ) 1.0 m Dispersivity ( ) 0 m I sotropic hydraulic conductivity ( ) 864 m/day A quifer parameters as shown in Table 7 1 were specified to represent an isotropic, homogeneous, confined aquifer PAGE 135 135 The dimensionless ratios were assigned to determine quantitatively how saltwater intrusion and the recirculation of seawater are related to the three independent ra tios: The aspect ratio was equal to 2.0; The dimensionless ratio was varied from 0.1 to 10.0 by varying the freshwater inflow per unit width and depth ( ) from 2.16 to 216 .0 m/day ; Using the freshwater inflow per unit width and depth ( ) f rom calculating the dimensionless ratio the dimensionless ratio was varied from 0.1 to 0.5 by varying the molecular diffusion coefficient ( ); The molecular diffusion coefficient ( ) represented all of the effect s of the dispersion coeffic ient ( ) ; and All of the effects of the d ispersivity were neglected ( = 0). A summation of the dimensionless ratios for the first set used in the Henry constant dispersion problem is shown in Table 7 2 and the details are shown in Appendix A. Table 7 2 Summation of dimensionless ratios used for the first set of investigations. Molecular diffusion coefficient ( ) m 2 /day m 2 /day 2.0 0.1 to 10.0 2.16 to 216.0 0.617142857 to 61.71428571 0.1 2.0 0.1 to 10.0 2.16 to 216.0 1.234285714 to 123.4285714 0.2 2.0 0.1 to 10.0 2.16 to 216.0 1.851428571 to 185.1428571 0.3 2.0 0.1 to 10.0 2.16 to 216.0 2.468571429 to 246.8571429 0.4 2.0 0.1 to 10.0 2.16 to 216.0 3.085714286 to 308.5714286 0.5 PAGE 136 136 Result s of the First Set of Investigations Comparison of the 0.5 isochlor results between uncoupled and coupled solutions for the same value of the dimensionless ratio In a ll cases of the comparison of the 0.5 isochlor results between the uncoupled and coupled solutions for the same value of the dimensionless ratio the toe of saltwater intrusion interface ( ) was determined by measuring the landward extent of saltwater intrusion along the base of the aquifer i.e., the 0.5 isochlors, for the d imensionless ratio s 0.1, 1.0 and 10.0, as shown in Figure s 7 7 through 7 11, respectively. The results indicate that the 0.5 isochlors decrease significantly as the dimensionless ratio increase from 0.1 to 10.0 at the same value of the dimensionless ratio from 0.1 to 0.5 for both the uncoupled and coupled solutions. The results seem as if the 0.5 isochlors for the coupled solutions asymptotically approach the corresponding values for the uncoupled solutions. A B Figure 7 7 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.1 for the first set of investigations. PAGE 137 137 A B Figure 7 8 The 0.5 isochlors of A) uncoupled solutions and B) coup led solutions for = 0.1, 1.0 and 10.0 in case of = 0.2 for the first set of investigations A B Figure 7 9 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.3 for the first set of investigations. A B Figure 7 10 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.4 for the first set of investigations PAGE 138 138 A B Figure 7 11 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.5 for the first set of investigations Comparison of the 0.5 isochlor results between uncoupled and coupled solutions for the same value of the dimensionless ratio In a ll cases of the comparison of the 0.5 isochlor results between the uncoupled and coupled solutions for the same value of the dimensionless ratio the toe of saltwater i ntrusion interface ( ) was determined by measuring the landward extent of saltwater intrusion along the base of the aquifer i.e., the 0.5 isochlors, for the dimensionless ratio s 0.1, 1.0 and 10.0, as shown in Figure s 7 12 through 7 14, respecti vely. A B Figure 7 12 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1 in case of increasing from 0. 1 to 0.5 for the first set of investigations. PAGE 139 139 A B Figure 7 13 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 1.0 in case of increasing from 0. 1 to 0.5 for the first set of investigations. A B Figure 7 14 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 10.0 in case of increasing from 0. 1 to 0.5 for the first set of investigations. The results indicate that as shown in Figure 7 12 (A) the location of the of the 0.5 isochlor s do not change by any significant amount of the dimensionless ratio f or the uncoupled solution of the dimensionless ratio = 0.1 when the dimensionless ratio is increased from 0.1 to 0.5 but the location of the 0.5 isochlors is slightly different at the upper right hand corner of the domain The coupled solution of the dimensionless ratio = 0.1 when the dimensionless ratio is increased from 0.1 to 0.5 is shown in Figures 7 12 (B). T he of the 0.5 isochlor s indicate a high difference in values which is much more significant than the results of the uncoupled solution. The of the 0.5 isochlor s for the dimensionless PAGE 140 140 ratios = 0.1 and = 0.1 i s the greatest results and the results for the dimensionless ratios = 0.1 and = 0.5 are the least results. A t the upper right hand corner of the domain the 0.5 isochlors reverse location patterns The 0.5 isochlor s for the dimensionless ratios = 0.1 and = 0. 5 become the greatest results and the 0.5 isochlor s for the dimensionless ratios = 0.1 and = 0. 1 become the least results. The stationary point of the 0.5 isochlor s is approximately a half distance of v ertical dimension of the domain (0.5 m) and the distance from the downgradient coastal boundary to the stationary point is roughly 0.5 m. For the dimensionless ratio = 1.0 and = 10.0, both the uncoupled and coupled results, as shown in Figures 7 13 and 7 14 indicate that the of the 0.5 isochlors increase significantly when the dimensionless ratio i s increased from 0.1 to 0.5 and the patterns of results are stable. Both the uncoupled and coupled solutions, the of the 0.5 isochlors for the dimensionless ratios = 0.1 and = 0. 5 was the greatest results and the results for the dimensionless ratios = 0.1 and = 0. 1 were the least results Comparison of e xtent of s altwater i ntrusion ( ) and degree of saltwater recirculation ( ) between uncoupled and coupled solutions for the first set of investigations In a ll cases of the Henry constant dispersion problem, the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) versus the dimensionless ratio from 0.1 to 10.0 were used to determine quantitatively the degree of saltwater intrusion between the uncoupled and coupled solutions for the first set of investigations as shown in Figures 7 15 through 7 21 and Appendix A PAGE 141 141 A B Figure 7 15 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solut ions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 1 A B Figure 7 16 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.2. PAGE 142 142 A B Figure 7 17 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of sal twater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.3. A B Figure 7 18 Results of the first set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.4. PAGE 143 143 A B Figure 7 19 Results of the first set of investigations A) exten t of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.5. A B Figure 7 20 E xtent of saltwater intrusion ( ) of the first set of investigations for A) uncoupled and B) coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 PAGE 144 144 A B Figure 7 21 D egree of saltwater reci rculation ( ) of the first set of investigations A) uncoupled and B) coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0. 5 The results are shown in Figures 7 15 through 7 21 and Appendix A, which indicate that: 1. For both the uncoupled and coupled solutions for the first set of investigations, the extent of saltwater intrusion ( ) significantly decrease s and the degree of saltwater recirculation ( ) also significantly decrease s when the dimensionless ratio is increased from 0.1 to 10.0 at a value of the dimensionless ratio as shown in Figures 7 15 through 7 19; 2. When comparing the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) as the dimensionless ratio value is increased from 0.1 to 0.5 the extent of saltwater intrusion ( ) is also significantly decrease d for the uncoupled and coupled solutions as shown in Figures 7 20; 3. However, the results of the degree of saltwater re circulation ( ) indicate that t here i s no significant difference in the values of results for the uncoupled solution but t here i s slight ly difference in the values of results for the coupled as shown in Figures 7 21 (A) and (B), respectively; PAGE 145 145 4 For b oth the uncoupled and coupled solutions from the dimensionless ratio = 0.1 and approaching = 0.3, the decreasing rates of the extent of the saltwater intrusion ( ) were significant. When the dimensionless ratio > 0.3, the decreasing rates of t he extent of saltwater intrusion ( ) were less significant than that from the dimensionless ratio = 0.1 and approaching = 0.3 for both the uncoupled and coupled solutions. Also, the saltwater intrusion ( ) values were approximately the same wh en the dimensionless ratio 1.0 for both the uncoupled and coupled sol utions, as shown in Figure 7 20; and 5. For both the uncoupled and coupled solutions when the dimensionless ratio 1 .0, the saltwater recirculation ( ) was equal to 0.0 since the large amount of freshwater advective flux ( ) dominated the flow in the domain, as shown in Figure 7 21 : a) For the uncoupled solution, increasing the dimensionless ratio value did not have any effect on the degree of saltwater recirculation ( ) i.e., the results for the saltwater recirculation ( ) were the same value of the dimensionless ratio The degree of saltwater recirculation ( ) was negligible for the dimensionless ratio 1.0 as shown in Figure 7 21 (A) ; and b) For the co upled solution, increasing the dimensionless ratio value had a small effect on the degree of saltwater recirculation ( ) T he degree of saltwater recirculation ( ) was nearly negligible for the dimensionless ratio 1.0 as shown in Figure 7 21 (B). Figures 7 22 through 7 25 also presented the investigation results of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) of the Henry constant dispersion problem The upgradient inland boundary on the left hand side of the domain was specified as the freshwater inflow (represented by the blue color) The downgradient coastal boundary on the right hand side of the domain was assigned as seawater (represented by the red color). PAGE 146 146 Figure 7 22 E xtent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the first set of investig ations PAGE 147 147 Figure 7 23 E xtent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the first set of investigations PAGE 148 148 Figure 7 24 D egree of saltwater recirculation ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentrat ion color floods and velocity vectors for the first set of investigations PAGE 149 149 Figure 7 25 D egree of saltwater recirculation ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochl ors from 0.1 to 0.9 with concentration color floods and velocity vectors for the first set of investigations PAGE 150 150 Input Parameters for the Second Set of Investigations For th e second set of investigations the dimensionless ratio and the dimensionless ratio were considered independent variables which was similar to the first set of investigations However, t he value of molecular diffusion coefficient ( ) was a constant but the hydraulic conductivity ( ) was varied to change the value of dimensionless ratio T he dimensionless ratio was varied from 0.1 to 0.5 for each of nineteen investigations in which values of the dimensionless ratio were varied from 0.1 to 10.0 This was accomplished by varyin g the freshwater inflow per unit width ( ) from 5.7024 to 1.14048 m 2 /day to achieve a range of the dimensionless ratio = 0.1 to 0.5 Values of the freshwater inflow per unit width ( ) were selected so that the value of the dimensionless ratio (which includes as the repeating variable) in each of the five investigations could be held constant as was varied to change the value of dimensionless ratio The value of molecular diffusion coefficient ( ) was a constant value equal to 1.62924998 m 2 /day. For both the uncoupled and coupled solutions, input parameters were divided into two groups, namely a quifer parameters and dimensionless ratios, which are shown in Table 7 3 and 7 4, respectively. A quifer parameters as shown in Table 7 3 were specified to represent an isotropic, homogeneous, confined aquifer PAGE 151 151 Table 7 3 Aquifer parameters for the second set of investigations. Parameter Value Density contrast parameter ( ) 0.025 Porosity ( ) 0.35 Horizontal dimension of the domain ( ) 2 m Vertical dimension of the domain ( ) 1 m Dispersivity ( ) 0 m M olecular diffusion coefficient ( ) 1.62924998 m 2 /day The dimensionless ratios were assigned to quantitatively determine how the extent of saltwater intrusion and the recirculation of seawater are related to the three independent ratios : The aspect ratio was equal to 2.0 ; The dimensionless ratio was varied from 0.1 to 0.5 by decreasing freshwater inflow per unit width and depth ( ) from 5.7024 to 1.14048 m/day; Using the freshwater inflow per unit width and depth ( ) f rom calculating the dimensionless ratio the dimensionless ratio was varied from 0.1 to 10.0 as the hydraulic conductivity ( ) was decreased from 2,282.101 to 4.562 m/day ; The m olecular diffusion coefficient ( ) represented all of the effect s of the dispersion coefficient ( ) ; and All of the effects of the d ispersivity were neglected ( = 0). A summation of the dimensionless ratios for the second set used in the Henry constant dispersion problem is shown in Table 7 4, and the details are shown in Appendix A. PAGE 152 152 Table 7 4 Summation of dimensionless ratios for the second set of investigations. Hydraulic Conductivity ( ) m 2 /day m/day 2.0 0.1 5.70240 0.1 to 10.0 2,282.1010 to 22.8100 2.0 0.2 2.85120 0.1 to 10.0 1,141.0500 to 11.4050 2.0 0.3 1.90080 0.1 to 10.0 760.7000 to 7.6030 2.0 0.4 1.42560 0.1 to 10.0 570.5250 to 5.7020 2.0 0.5 1.14048 0.1 to 10.0 456.4200 to 4.5620 Results of the Second Set of Investigations Comparison of the 0 .5 isochlor results between uncoupled and coupled solutions for the same value of the dimensionless ratio In all cases of the comparison of the 0.5 isochlor results between the uncoupled and coupled solutions for the same value of the dimensionless ratio the toe of saltwater intrusion interface ( ) was determined by measuring the landward extent of saltwater intrusion along the base of the aquifer, i.e., the 0.5 isochlors, for the dimensionless ratios 0.1, 1.0 and 10.0, as shown in Figures 7 26 through 7 30, respectively. The results indicate the same distributions of the 0.5 isochlors as the fir st set of investigations T he 0.5 isochlors decrease significantly as the dimensionless ratio increase from 0.1 to 10.0 at the same value of the dimensionless ratio from 0.1 to 0.5 for both the uncoupled and coupled solutions. The results seem as i f the 0.5 isochlors for the coupled solutions asymptotically approach the corresponding values for the uncoupled solutions. All 0.5 isochlor results of the second set of investigations look similar to that of the first set of investigations PAGE 153 153 A B Figure 7 26 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.1 for the second set of investigations A B Figure 7 27 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0. 2 for the second set of investigations. A B Figure 7 28 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0. 3 for the second set of investigations. PAGE 154 154 A B Figure 7 29 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0. 4 for the second set of investigations. A B Figure 7 30 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0. 5 for the second set of investigations. Comparison of the 0.5 isochlor results b etween uncoupled and coupled solutions for the same value of the dimensionless ratio In a ll cases of the comparison of the 0.5 isochlor results between the uncoupled and coupled solutions for the same value of the dimensionless ratio the toe of saltwater intrusion interface ( ) was determined by measuring the landward extent of saltwater intrusion along the base of the aquifer i.e., the 0.5 isochlors, for the dimensionless ratio s 0.1, 1.0 and 10.0, as shown in Figure s 7 31 through 7 3 3, respectively. PAGE 155 155 A B Figure 7 31 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 0.1 in case of increasing from 0. 1 to 0.5 for the second set of investigations. A B Figure 7 32 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 1.0 in case of increasing from 0. 1 to 0.5 for the second set of investigations. A B Figure 7 33 The 0.5 isochlors of A) uncoupled solutions and B) coupled solutions for = 10.0 in case of increasing from 0. 1 to 0.5 for the second set of investigations. PAGE 156 156 T he results present the same patterns of t he 0.5 isochlors as the results of the first set of the i nvestigations The results indicate that as shown in Figure 7 31 (A) the location of the of t he 0.5 isochlor s do not change by any significant amount of the dimensionless ratio f or the uncoupled solution of the dimensionless ratio = 0.1 when the dimensionless ratio is increased from 0.1 to 0.5 but the location of the 0.5 isochlors is sli ghtly different at the upper right hand corner of the domain The coupled solution of the dimensionless ratio = 0.1 when the dimensionless ratio is increased from 0. 1 to 0.5 is shown in Figures 7 31 (B). T he of the 0.5 isochlor s indicate a high difference in values which is much more significant than the results of the uncoupled solution. The of the 0.5 isochlor s for the dimensionless ratios = 0.1 and = 0.1 i s the greatest results and the results for the dimensionless ratios = 0.1 and = 0.5 are the least results. A t the upper right hand corner of the domain, the 0.5 isochlors reverse location patterns The 0.5 isochlors for the dimensionless ratios = 0.1 and = 0. 5 become the greatest results and the 0.5 isochlor s for the dimensionless ratios = 0.1 and = 0. 1 become the least results. The stationary point of the 0.5 isochlor s is approximately a half distance of v ertical dimension of the domain (0.5 m) and the distance from the downgradient coastal boundary t o the stationary point is roughly 0.5 m. For the dimensionless ratio = 1.0 and = 10.0, both the uncoupled and coupled results, as shown in Figures 7 32 and 7 33, indicate that the of the 0.5 isochlors increase significantly when the dimensionless ratio i s increased from 0.1 to 0.5 and the patterns of results are stable. Both the uncoupled and coupled solutions, the of the 0.5 isochlors for the dimensionless ratios = 0.1 and = 0. 5 was the PAGE 157 157 greatest results and th e results for the dimensionless ratios = 0.1 and = 0. 1 were the least results Comparison of e xtent of s altwater i ntrusion ( ) and degree of saltwater recirculation ( ) between uncoupled and coupled solutions for the second set of investigations In a ll cases of the Henry constant dispersion problem, the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) versus the dimensionless ratio from 0.1 to 10.0 were used to determine quantitatively the deg ree of saltwater intrusion between the uncoupled and coupled solutions for the second set of investigations as shown in Figures 7 34 through 7 40 and Appendix A A B Figure 7 34 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 1 PAGE 158 158 A B Figure 7 35 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.2. A B Figure 7 36 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.3. PAGE 159 159 A B Figure 7 37 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.4. A B Figure 7 38 Results of the second set of investigations A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.5. PAGE 160 160 A B Figure 7 39 E xtent of saltwater intrusion ( ) of the second set of investigations for A) uncoupled and B) coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increas ing from 0.1 to 0.5 A B Figure 7 40 D egree of saltwater recirculation ( ) of the second set of investigations for A) uncoupled and B) coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increas ing from 0.1 to 0. 5 PAGE 161 161 The results as shown in Figures 7 34 through 7 40 and Appendix A, present the same distributions of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) as the first set of the i nvestigations which indicate that: 1. For both the uncoupled and coupled solutions for the second set of investigations, the extent of saltwater intrusion ( ) significantly decrease and the degree of saltwater recirculation ( ) also significantly decrease when the dimensionless ratio is increased from 0.1 to 10.0 at a value of dimensionless ratio as shown in Figures 7 34 through 7 38 ; 2. When comparing the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) as the dimensionless ratio value is increased from 0.1 to 0.5 the extent of saltwater intrusion ( ) is also significantly decrease d for the uncoupled and coupled solutions as shown in Figures 7 39 ; 3. However, the results of the degree of saltwater recirculation ( ) indicate that t here i s no significant difference in the values of results for the uncoupled solution but t here i s slight ly difference in the values of results for the coupled as shown in Figures 7 40 (A) and ( B), respectively; 4 For both the uncoupled and coupled solutions from the dimensionless ratio = 0.1 and approaching = 0.3, the decreasing rates of the extent of the saltwater intrusion ( ) were significant. When the dimensionless ratio > 0.3, the decreasing rates of the extent of saltwater intrusion ( ) were less significant than that from the dimensionless ratio = 0.1 and approaching = 0.3 for both the uncoupled and coupled solutions. Also, the saltwater intrusion ( ) values were approximately the same when the dimensionless ratio 1.0 for both the uncoupled and coupled sol utions, as shown in Figure 7 39; and 5. For both the uncoupled and coupled solutions when the dimensionless ratio 1 .0, the saltwater recirculation ( ) was equal to 0.0 since the large amount of freshwater advective flux ( ) dominated the flow in the domain, as shown in Figure 7 40: PAGE 162 162 a) For the uncoupled solution, increasing the dimensionless ratio value did not have any effect on the degree o f saltwater recirculation ( ) i.e., the results for the saltwater recirculation ( ) were the same value of the dimensionless ratio The degree of saltwater recirculation ( ) was negligible for the dimensionless ratio 1.0 as shown in Figur e 7 40 (A) ; and b) For the coupled solution, increasing the dimensionless ratio value had a small effect on the degree of saltwater recirculation ( ) T he degree of saltwater recirculation ( ) was nearly negligible for the dimensionless ratio 1.0 as shown in Figure 7 40 (B). Figures 7 41 through 7 44 also presented the investigation results of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) of the Henry constant dispersion problem The upgradient inland boundary on the left hand side of the domain was specified as the freshwater inflow (represented by the blue color) The downgradient coastal boundary on the right hand side of the domain was assigned as seawater (represented by the red color). PAGE 163 163 Figure 7 41 E xtent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity v ectors for the second set of investigations PAGE 164 164 Figure 7 42 E xtent of saltwater intrusion ( ) for c oupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentrat ion color floods and velocity vectors for the second set of investigations PAGE 165 165 Figure 7 43 D egree of saltwater recirculation ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of investigations PAGE 166 166 Figure 7 44 D egree of saltwater recirculation ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of investigations PAGE 167 167 Comparison Results between the First and Second Sets of Investigati ons Comparison patterns of the 0.5 isochlor s between the first and second sets of investigations for the same value of the dimensionless ratio Comparison pattern s of the 0.5 isochlors between the first and second sets of investigations for the same value of the dimensionless ratio were obtained by considering the dimensionless ratio and the dimensionless ratio as independent variables as shown in Figures 7 45 through 7 54 and Appendix A A B Figure 7 45 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.1 between A) the first set and B) the second set of investigations. A B Figure 7 46 Comparison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.1 between A) the first set and B) the second set of investigations. PAGE 168 168 A B Figure 7 4 7 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.2 between A) the first set and B) the second set of the Henry constant dispersion problem A B Figure 7 48 Com parison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.2 between A) the first set and B) the second set of investigations. A B Figure 7 49 Comparison of the pattern of th e 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.3 between A) the first set and B) the second set of investigations. PAGE 169 169 A B Figure 7 50 Comparison of the p attern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.3 between A) the first set and B) the second set of investigations. A B Figure 7 51 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.4 between A) the first set and B) the second set of investigations. A B Figure 7 52 Comparison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case o f = 0.4 between A) the first set and B) the second set of investigations. PAGE 170 170 A B Figure 7 53 Comparison of the pattern of the 0.5 isochlors of the uncoupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.5 between A) the first set and B) the second set of investigations. A B Figure 7 54 Comparison of the pattern of the 0.5 isochlors of the coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.5 between A) the first set and B) the second set of investigations. The c omparison as shown in Figure 7 45 to 7 54 and Appendix A, presented indicates a very good agreement for the location of the 0.5 isochlor distributions between the first and second sets of inve stigations for both the uncoupled and coupled solutions of the Henry constant dispersion problem PAGE 171 171 Comparison patterns and values of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) between the first and second sets of investigations A c omparison patterns of patterns and values of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) for the uncoupled and coupled solutions of the Henry constant dispersion problem between the first and second sets of investigations is shown in Figure 7 55 through 7 61 and Appendix A A B Figure 7 55 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 1. PAGE 172 172 A B Figure 7 56 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.2 A B Figure 7 57 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in c ase of = 0.3 PAGE 173 173 A B Figure 7 58 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoupled and coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 4. A B Figure 7 59 Comparison of values of A) extent of saltwater intrusion ( ) and B) degree of saltwater recirculation ( ) for uncoup led and coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 5. PAGE 174 174 A B Figure 7 60 Comparison of values of extent of saltwater intrusion ( ) for A) uncoupled and B) coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 A B Figure 7 61 Comparison of values of d egree of saltwater recirculation ( ) for A) uncoupled and B) coupled solutions between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 in case of increasing from 0.1 to 0.5 PAGE 175 175 Comparing patterns and value s of both the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) for the uncoupled and coupled solutions of the Henry constant dispersion problem between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 and for the dimensionless ratio from 0.1 to 0.5 indicate s no si gnificant differences in either the patterns or values as shown in Figure s 7 55 through 7 61 and Appendix A Summary of Comparison of Results between the First and Second Sets of Investigations of the Henry Constant Dispersion Problem The comparisons of the 0.5 isochlors pattern, patterns and values of extent of saltwater intrusion ( ) and patterns and values of degree of saltwater recirculation ( ) for the uncoupled and coupled solutions of the Henry constant dispersion problem between the first and second sets of investigations versus the dimensionless ratio from 0.1 to 10.0 and for the dimensionless ratio fro m 0.1 to 0.5 indicate that there were no significant differences in either the patterns or values Therefore, it can be concluded that dimensional analysis c an be applied to determine the solutions of the Henry constant dispersion problem and also (1964) problem by considering three dimensionless ratios as independent variables namely the aspect ratio ( ), the dimensionless ratio and the dimensionless ratio The extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) c an be treated as dependent dimensionless variables PAGE 176 176 Accuracy and Stability Criteria for the Henry Constant Dispersion Problem G rid Pclet n umber In this investigation of the H enry constant dispersion problem t he constant dispersion coefficient ( ) was represented by the constant value of the molecular diffusion coefficient ( ) in SEAWAT neglecting the effects of dispersivity ( ) and velocity dependent dispersion. Accordingly, the grid Pclet number ( ) was determined from instead of and since the total velocity along the sea boundary of the Henry problem is only composed of a horizontal flow component ( direction) because of the hydrostatic boundary condition as Simpson and Clement (2004) Grid Pclet number of the first set of investigations For both the uncoupled and coupled investigations, the grid Pclet numbers of the first set of investigations were d etermined at steady state which was the last timestep of the 2 day simulation period, i.e., the 23 rd timestep for the uncoupled solutions and the 46 th timestep for the coupled solutions, as shown in Table 7 5 and Appendix A. As shown in Table 7 5, althoug h both the uncoupled and coupled solutions i n case of = 0.1 and = 0.1 indicate d oscill a tory results ( the grid Pclet numbers > 4.0 ) they reached steady state conditions, as shown in Figures 7 62 through 7 63 and results in the LST files The remain ing solutions were s table because the grid Pclet numbers were less than 4.0 (Daus et al. 1985; Voss and Provost 2002; Voss and Souza 1987; Zheng and Bennett 2002) The comparison of the grid Pclet nu mber s showed that as the dimensionless ratios and in the numerical model were increased, the grid Pclet numbers were decreased and the more stable were the results of the model, as shown in Figure 7 64 and Appendix A. PAGE 177 177 Table 7 5 Summation of the grid Pclet number of the first set of investigations. = 0. 1 = 1.0 = 10.0 Case Max. Grid Pclet number Max. Grid Pclet number Max. Grid Pclet number ( ) ( ) ( ) Uncoupled solutions 0.1 7.8798 1.2380 0.5738 0.2 3.9399 0.6190 0.2869 0.3 2.6266 0.4127 0.1913 0.4 1.9699 0.3095 0.1434 0.5 1.5760 0.2476 0.1148 Coupled solutions 0.1 5.1477 1.1928 0.5586 0.2 2.4184 0.5158 0.2707 0.3 1.4766 0.3083 0.1771 0.4 1.0139 0.2132 0.1311 0.5 0.7339 0.1605 0.1039 Figure 7 62 Isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the uncoupled solution in case of the dimensionless ratios = 0.1 and = 0.1 for the first set of investigations. PAGE 178 178 Figure 7 63 Isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the coupled solution in case of the dimensionless ratios = 0.1 and = 0.1 for the first set of investigations. Figure 7 64 Comparis on of the grid Pclet number s for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.1 for the first set of investigations. PAGE 179 179 Grid Pclet number of the second set of investigations Similar to the first set of i nvestigations for both the uncoupled and coupled investigations, the grid Pclet numbers of the second set of inves tigations were determined at steady state which was the last timestep of the 2 day simulation period, i.e., the 23 rd timestep for t he uncoupled solutions and the 46 th timestep for the coupled solutions, as shown in Table 7 6 and Appendix A, respectively. Table 7 6 Summation of the grid Pcl et number of the second set of i nvestigations = 0.1 = 1.0 = 10.0 Case Max. Grid Pclet number Max. Grid Pclet number Max. Grid Pclet number ( ) ( ) ( ) Uncoupled solutions 0.1 7.8835 1.2383 0.5738 0.2 3.9420 0.6193 0.2870 0.3 2.6278 0.4127 0.1912 0.4 1.9709 0.3096 0.1435 0.5 1.5767 0.2476 0.1147 Coupled solutions 0.1 5.1469 1.1932 0.5586 0.2 2.4196 0.5162 0.2708 0.3 1.4811 0.3084 0.1770 0.4 1.0126 0.2131 0.1313 0.5 0.7442 0.1601 0.1040 A s ummation of the grid Pclet number of the second set of inves tigations was similar to the first set of inves tigations There were some differences when comparing each other, but t hey d id not appear to be significantly different. PAGE 180 180 I n case of = 0.1 and = 0.1 both the uncoupled and coupled solution s also presented oscillatory results but they reached steady state conditions after the 2 day simulation period as shown in Figure s 7 65 and 7 66. However, the remaining solutions were stable because the grid Pclet numbers were less than 4.0 which is similar to the grid Pclet number results of the first set. The comparison of the grid Pclet number s showed that the stability of the model results increased as the dimensionless ratios and assigned t o the numerical model were increased as shown in Figure 7 67 and Appendix A Figure 7 65 Isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the uncoupled solution in case of the dimensionless ratios = 0.1 and = 0. 1 for the second set of investigations. PAGE 181 181 Figure 7 66 Isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the coupled solution in case of the dimensionless ratios = 0.1 and = 0.1 for the second set of investigations. Figure 7 67 C omparison of the grid Pclet number s for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.1 for the second set of investigations. PAGE 182 182 Summary of Grid Pclet N umber for the H enry Constant Dispersion Problem The c omparison of the grid Pclet number s results between the first and second sets of the Henry constant dispersion problem for the uncoupled and coupled solutions, for example in case of the dimensionless ratio = 0.1, as shown in Figure 7 68, indicates that: 1. For example, i n case of = 0.1 and = 0.1 for both sets of investigation and both the uncoupled and coupled solution s, as shown in Figure 7 68, oscillatory results were present since the grid Pclet n umber s were greater than 4.0. However, the solutions reached steady state conditions at the end of the 2 day simulation periods, as shown in Figure 7 62 to 7 63 and 7 65 to 7 66; 2. As the dimensionless ratios and assigned to the numerical model wer e increased, the stability results of the models increased; and 3. There w as no significant difference between the first and second sets of the Henry constant dispersion problem for the uncoupled and coupled solutions as shown in Figure 7 68. Figure 7 68 C omparison of the grid Pclet number between the first and second sets of the Henry constant dispersion problem in case of = 0.1 PAGE 183 183 Discussion and Conclusions for the Henry Constant Dispersion Problem The results of the numerical experiments indicate that t he two dimensionless dependent varia bles that is the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) are functions of three independent dimensionless ratios : the aspect ratio the dimensionless ratio and the dimensionless ratio T he aspect ratio was assigned a value equal to 2.0 and both of the other independent dimensionless ratios, namely and were varied T he investigation involved solving the Henry constant dispersion problem in two sets of investigation s that were dependent on the value of hydraulic conductivity ( ) for the dimensionless ratio in the first set of investigation s, and the value of molecular diffusion coe fficient ( ) for the d imensionless r atio in the second set of investigation s. This involved t he comparison of the 0.5 isochlors between the first and second set s of investigations, which were obtained by considering the dimensionless ratio and the dimensionless ratio as independent variables demonstrates very good agreement between both sets of investigations for both the uncoupled and coupled solutions as shown in Figures 7 45 through 7 54 The results demonstrate that the toe o f the saltwater intrusion interface ( ) that is, the 0.5 isochlor, decreased significantly for both the first and second set s of investigations as the dimensionless ratio was increased from 0.1 to 10.0 and also as the dimensionless ratio was increased from 0.1 to 0.5 as shown in Figure s 7 45 through 7 54. When considering the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) for the uncoupled and coupled solutions between the first PAGE 184 184 and second set s of investigations versus the dimensionless ratio from 0.1 to 10.0 and the dimensionless ratio from 0.1 to 0.5 the results demonstrate that, a s shown in Figure s 7 55 through 7 61, t he extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) were significantly decreased by the dimensionless ratio increased The results of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) also indicate that there are significant differences between the uncoupled and coupled solutions, especially at smaller values of dimensionless ratio In the case of the same value of dimensionless ratio and at smaller values of dimensionless ratio such as 0.1, where the density driven buoyancy flux ( ) dominates the freshwater advective flux ( ), the extent of saltwater intrusion ( ) obtained from the uncoupled solutions was less than t he results from the coupled solutions. Conversely, the degree of saltwater recirculation ( ) derived from the uncoupled solutions was greater than the results from the coupled solutions In case of the same value of dimensionless ratio and at larger values of dimensionless ratio such as 10.0, where the freshwater advective flux ( ) dominates the density driven buoyancy flux ( ), the extent of saltwater intrusion ( ) simulated for the uncoupled solution becomes equal to the results for the coupled solutions and t he degree of saltwater recirculation ( ) for the uncoupled s olution asymptotically approaches the corresponding values for the coupled solutions The results indicate how important differences between the uncoupled and coupled solutions were observed over the investigated range of the dimensionless ratio values and that t he uncoupled constant density flow and transport code may be an PAGE 185 18 5 alternative to determine the results at larger values of dimensionless ratio instead of a coupled variable density flow and transport code. In addition, the uncoupled constant de nsity flow and transport code, which is a less computationally intense method, may be an alternative to determine the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) instead of a coupled variable density flow and trans port code which is a more computationally intense method as shown in Figure s 7 55 to 7 61. At smaller values of dimensionless ratio such as 0.1 there are significant difference s in the results for both the extent of saltwater intrusion ( ) an d the degree of saltwater recirculation ( ) between the uncoupled and coupled solutions, while a t larger values of dimensionless ratio such as 1.0, the uncoupled solution yields approximately the same results for both the extent of saltwater intr usion ( ) and the degree of saltwater recirculation ( ) as the coupled solution. When considering the extent of saltwater intrusion ( ) results ( Figure 7 60), as the dimensionless ratio which includes the effect of the molecular diffusion ( ) relative to the ratio of freshwater advective flux ( ), was increased from 0.1 to 0.5 for both sets of the investigations, the following occurred. First, for both the uncoupled and coupled solutions (Figure 7 60) the re sults f or the dimensionless ratio = 0.1 and approaching = 0.3 indicate that (1) t he decreasing rates of the extent of saltwater intrusion ( ) were more significant than the rates for the dimensionless ratio > 0.3 ; (2) the saltwater intrusion ( ) values were approximately the same when the dimensionless ratio was 1.0 ; and (3) t he difference in value of the saltwater intrusion ( ) results in the uncoupled solution were PAGE 186 186 less significant than the results in the coupled solution for each v alue of the dimensionless ratio Second, for t he uncoupled solution [Figure 7 60 (A)], the results for the dimensionless ratio = 0.1 and approaching = 0.3 indicate that the extent of saltwater intrusion ( ) was not significantly different for each result of the dimensionless ratio a lthough the dimensionless ratio value was increased from 0.1 to 0.5. However, the results for the dimensionless ratio 0.3 and approaching = 10.0 indicate that (1) t he difference in the results of t he extent of saltwater intrusion ( ) w ere more significant than the results for the dimensionless ratio = 0.1 and approaching = 0.3; (2) t he extent of saltwater intrusion ( ) was significantly different in values for each result of the dimensio nless ratio which was more expressively different in values than the results for the dimensionless ratio = 0.1 and approaching = 0.3; (3) each of the extent of saltwater intrusion ( ) results was approximately the same for the dimensionless ra tio 1.0; and (4) t he results of the extent of saltwater intrusion ( ) for the value of the dimensionless ratio = 0.5 were the greatest results and t he results for the value of the dimensionless ratio = 0. 1 w ere the least results Third, for t he coupled solution [Figure 7 60 (B)], the results for the dimensionless ratio = 0.1 and approaching = 0.3 indicate that (1) as the dimensionless ratio value was increased, the extent of saltwater intrusion ( ) was significantly decrease d ; (2) as the dimensionless ratio value was increased, the extent of saltwater intrusion ( ) was also meaningfully decrease d ; and (3) t he results of the extent of saltwater intrusion ( ) for the value of the dimensionless ratio = 0. 1 were the greatest results and t he results for the value of the dimensionless ratio = 0. 5 w ere the least results However, the PAGE 187 187 results for the dimensionless ratio 0.3 and approaching = 10.0 indicate that (1) t he difference in the results of the extent of saltwater intrusion ( ) w ere more significant than the results for the dimensionless ratio = 0.1 and approaching = 0.3; (2) each of the extent of saltwater intrusion ( ) results was approximately the same for the dimensionless ratio 1.0; an d (3) t he results of the extent of saltwater intrusion ( ) for the value of the dimensionless ratio = 0.5 were the greatest results and the results for the value of the dimensionless ratio = 0.1 were the least results Fourth, for both the uncoupl ed and coupled solutions [Figure 7 60 (A) and (B), respectively] t he reverse point of the pattern of the extent of saltwater intrusion ( ) results occurred at the dimensionless ratio = 0. 3 w hen the dimensionless ratio 1.0 for both the uncoupled and coupled solutions, the extent of the saltwater intrusions ( ) was approximately the same because the effects of combinations of increasing the dimensionless ratio and the dimensionless ratio that is the freshwater advectiv e flux ( ), the density driven buoyancy flux ( ) and the effect of the molecular diffusion ( ), were balanced by the combined effects of internal free convection caused by density dependent flow processes from the transport due to forced conve ction caused by external force ( i.e., the saltwater hydrostatic heads at the downgradient boundary). Thus, the dimensionless ratio = 0.3 on the saltwater intrusions ( ) results is considered as a balance point between the fluxes and convections of the five curve family for both the uncoupled and coupled solution s. Moreover, it is also considered as a separation point of the five uncoupled solution curve family. It is represented by equivalent freshwater heads. PAGE 188 188 W hen considering the degree of saltwater recirculation ( ) results ( Figure 7 61), a s the dimensionless ratio which was the effect of the molecular diffusion ( ) relative to the ratio of freshwater advective flux ( ), was increased from 0.1 to 0.5 for the uncoupled and coupled solutions and both set s of the investi gations, the following occurred. First, f or both the uncoupled and coupled solutions, the degree of saltwater recirculation ( ) was significantly decreased as the dimensionless ratio was increased from 0.1 to 10.0 Also, when the dimensionless ratio 1.0, the degree of saltwater recirculation ( ) was negligible f or both the uncoupled and coupled solutions since the large amount of freshwater advective flux ( ) dominated the flow in the domain Second, f or the uncoupled solutions [ Figure 7 6 1 (A)], the degree of saltwater recirculation ( ) did not obtain any effect from increasing the dimensionless ratio value. The results of were still the same results for each result of the dimensionless ratio Third, f or the coupled solution s [Figure 7 61(B)], w hen the dimensionless ratio = 0.1 approach ed = 1 .0 and the dimensionless ratio increased from 0.1 to 0.5 : (1) t he results of the degree of saltwater recirculation ( ) presented slightly different values for each result of the dimensionless ratio ; (2) unstable results of the degree of saltwater recirculation ( ) for the dimensionless ratio = 0.1 might occur from the calculation when there were the large e ffect s of both molecular diffusion ( ) and density driven buoyancy flux ( ) compar ed to the small amount of freshwater advective flux ( ) ; and (3) the results of the degree of saltwater recirculation ( ) for PAGE 189 189 increasing dimensionless ratio from 0. 2 to 0.5 demonstrated the stabilit y of the decrease in the value of the results Based on the g rid Pclet number ( ) and the LST output files, all solutions demonstrated accuracy and stability of numerical calculation. To obtain the most accuracy, the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) should be determined in a co upled variable density flow and transport code for a groundwater model which was the same assumption as the Henry constant dispersion problem Therefore, r esults of the extent o f saltwater intrusion ( ) and the degree of saltwater recirculation ( ) from investigations indicate that the dimensional analysis can be applied to determine the solutions of the Henry constant dispersion problem and (1964) problem by considering three dimensionless ratios as ind ependent variables, namely the aspect ratio the dimensionless ratio and the dimensionless ratio and t he extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) are treated as dependent d imensionless variables PAGE 190 190 CHAPTER 8 HENRY VELOCITY DEPENDENT DISPERSION P ROBLEM Benchmark Problem s for the Henry Velocity dependent Dispersion Problem In this problem the use of SEAWAT and Groundwater Vistas 5 in this investigation was verified by comparing to results previously obtained by Sgol (1993) and Abarca et al. (2007) as benchmark problems. Sgol (1993) reviewed the background from basic to advanced solutions of the (1964) analytical solution and presented (1964) problem Abarca et al. (20 07) used the saturated unsaturated transport (SUTRA) finite element code (Voss and Provost 2002) to solve (1964) problem which w as compared to the 50% isochlor s from Sgol (1993) work. The works of Sgol ( 1993) and Abarca et al. (2007) presented results of the solution to (1964) problem which used = 0.263, = 0.1 and = 2.0 as shown in Figure 8 1 (A) and (B), respectively A B Figure 8 1 The work s of A) Sgol (1993) and B) Abarca et al. (2007) illustrate isochlors for the solutions of problem Additionally, Abarca et al. (2007) also introduced two solutions of reference s (1964) problem Their modifications were the adjustments of hydraulic conductivities and dispersivities values of the medium to be representative of an anisotropic medium PAGE 191 191 T hey performed solutions of two reference cases as shown in Figure 8 2 (A) and (B), respectively A B Figure 8 2 Abarca et al. (2007) isochlors for the solutions of the reference cases namely A) diffusive case ( constant dispersion case ) and B) dispersive case ( velocity dependent dispersion case) using the equivalent freshwater head and the varied concentration of seawater at the coastal boundary Notice of Sgol (199 3 ) and Abarca et al. (2007) There were important topics of Sgol (1993) and Abarca et al. (2007) which should be considered: 1. Henry (1964) applied the constant concentration of seawater at the coastal boundary with the following homogeneous boundary conditions to the governing equations in Figure 8 3 to obtain the solution, as shown in Figure 8 4 (Henry 1964; Simpson and Clement 2004) : at = 0, 1; at = 0, ; at = 0; and at = 0; and at = 1. at = where: is the ratio of the fluid density to a reference freshwater density; is the non dimensional concentration; and is non dimensional stream function. Actually, to reflect on the assumption of each case of Abarca et al. (2007), the diffusive case should be arca et al. (2007) should be called the 3 ) and Simpson and Clement (2004). PAGE 192 192 Figure 8 3 (1964) problem using the constant concentration of seawater at the coastal boundary (Simpson and Clement 2004) Figure 8 4 Isochlors of Henry idealized mathematical model using the constant concentration of seawater at the coastal boundary (Henry 1964) 2. Sgol (1993) (1964) analytical solution and presented results of (1964) problem using the constant concentration of seawater at the coastal boundary as shown in Figure 8 5. She did not clearly indicate which approach (the coupled or the uncoupled approach ) was used to determine the (1964) problem ; 3. Sgol (1993) noted that she personal ly communicated with Clifford I. Voss in 1992, and then she received the s teady state concentration distribution from the finite element solution b y using a constant scalar value of hydrodynamic PAGE 193 193 dispersion ( ) = 1.62925344 m 2 /day and the varied concentration of seawater at the coastal boundary as shown in Figure 8 6 which correspond ed to the work of Voss and Souza (1987) as shown in Figure 8 7; Figure 8 5 (1964) analytical solution for the dimensionless ratio = 0.1 using the constant concentration of seawater at the coastal boundary revised by Sgol (1993) Figure 8 6 The s teady state concentration distribution from the finite element numerical solution using a constant scalar value of hydrodynamic dispersion ( ) and the varied concentration of seawater at the coastal boundary by Sgol personal communicated with Clifford I. Voss in 1992 Sgol (1993) Figure 8 6 is the same figure as Figure 8 1 ( A) PAGE 194 194 Figure 8 7 T he work of Voss and Souza (1987) 0.5 isochlors with numerical solutions using a constant scalar value of hydro dynamic dispersion ( ) = 18.8 571 x10 6 m 2 /s and the varied concentration of seawater at the coastal boundary The result from = 6.6 x10 6 m 2 /s numerical isochlors (dotted) is also shown (Sgol 1993; Voss and Souza 1987) 4. Simpson and Clement (2004) re ca (1964) semi analytical solution for the dimensionless ratios = 0.263, = 0.1 and the aspect ratio = 2.0 with results obtained numerically using a Galerkin finite element numerical solution and compared their results to numerical results for the original (or standard) Henry problem, as shown in Figure 8 8 ; 5. Abarca et al. (2007) (1964) analytical solution by SUTRA and (1964) problem using the varied concentration of seawater at the coastal boundary as shown in Figure 8 9 which was different from (1964) solution and the reproduced (1964) solutions, as shown in Figure 8 4 to 8 5 and 8 8; 6. Abarca e t al. (2007) (1964) problem to be diffusive case ( constant dispersion case ) and dispersive case ( velocity dependent dispersion case) and solve d them by using the equivalent freshwater head and the varied Figure 8 9 is the same figure as Figure 8 1 (B ) PAGE 195 195 concentration of seawater at the coastal boundary as shown in Figure 8 10 (A) and (B), respectively; Figure 8 8 The results of the standard Henry problem from Simpson and Clement (2004) present the comparison of coupled numerical and semi analytical results for = 0.263, = 0.1 and = 2.0 and the constant concentration of seawater at the coastal boundary The uncoupled 50% numerical isochlors (dashed) is also shown. Figure 8 9 The work of Abarca et al. (2007) illustrates isochlors for the solution of (1964) problem Fi gure 8 10 is the same figure as Figure 8 2. PAGE 196 196 A B Figure 8 10 Abarca et al. (2007) isochlors for the solutions of the reference cases namely A) diffusive case ( c onstant dispersion case ) and B) dispersive case ( velocity dependent dispersion case) using the equivalent freshwater head and the varied concentration of seawater at the coastal boundary 7. Abarca et al. (2007) mentioned both the uncoupled and coupled approaches, including equivalent freshwater heads, but they did not clearly indicate which solutions (the coupled or the uncoupled solutions) were used to obtain solutions of (1964) problem and two reference cases as shown in Figure 8 9 and 8 10, respectively; 8. Povich et al. (2013) developed a model us ing a continuum approach to describe saturated flow and transport in a porous medium where the bulk fluid mass and momentum balance eq uations and the solute mass balance equation were coupled through density and viscosity equations of state. Their model setup scheme is presented as shown in Figure 8 11. They presented the reproduced work of Abarca et al. (2007) reference cases, namely the diffusive case (constant dispersion case) and the dispersive case (velocity dependent dispersion case) as shown in Figure 8 12 (A) and (B), respectively; and 9. The comparison of Povich et al. (2013) work, as shown in Figure 8 12 (A) and (B), to Abarca et al. (2007) reference cases as shown in Figure 8 10 (A) and (B), respectively, showed a good agreement of the isochlors. For (1964) problem of Sgol (1993) and Abarca et al. (2007) works including Abarca et al. (2007) reference cases all topics as shown above presented some conflicts between the approaches to solve problems and the ways to describe the approaches PAGE 197 197 Figure 8 11 The numerical model setup to reproduce Abarca et al. (2007) Henry problem and reference cases using the bulk fluid mass and momentum balance equations and the solute mass balance equation were coupled through density and viscosity equations of state in the domain (Povich et al. 2013) A B Figure 8 12 The work of Povich et al. (2013) presented isochlor s for coupled solutions of Abarca et al. (2007) reference cases namely A) diffusive case and B) dispersive case. A few obvious topics about the condition at the coastal boundary were considered, namely: F irstly, the solutions were obtained by specifying a constant concentration of seawater at the coastal boundary, which were different from the results obtained by the varied concentration of seawater at the coastal boundary; and Lastly, the solutions of Sg ol (1993) and Abarca et al. (2007) including Abarca et al. (2007) references case, i.e., diffusive case ( constant dispersion case ) and dispersive PAGE 198 198 case ( velocity dependent dispersion case), should be the coupled solutions with the varied concentration of seawater at the coastal b oundary. Aquifer Parameters for Sgol (1993) and Abarca et al. (2007) Aq uifer p arameters to reproduce the coupled solution s of the (1964) problem in the work s of Sgol (1993) and Abarca et al. (2007) including the coupled solution s of Abarca et al. (2007) r eference cases are described as follow s: Aquifer parameters for the c oupled s olution s of the Henry (1964) p roblem for Sgol (199 3 ) and Abarca et al. (2007) To solve the H (1964) problem of Sgol (1993) and Abarca et al. (2007) t he domain was assigned to be an isotropic, homogeneous, confined aquifer with the hydraulic conductivit y ( ) = 864 m/day and the porosity ( ) = 0.35 The dispersivities ( and ) were set equal to zero The molecular diffusion coefficient ( ) was assigned to represent all the effects of a constant dispersion coefficient ( ) and set equal to 1.62925 m 2 /day, which represented the constant dispersion coefficient in (1964) problem The freshwater inflow per unit width ( (1964) problem was equal to 5.702 m 2 /day. Aquifer parameters for the c oupled s olution s of Abarca et al. (2007) reference cases To obtain the results of the reference case s of Abarca et al. (2007) i.e., the diffusive case ( constant dispersion case) and the dispersive case ( velocity dependent dispersion case), t he domain was assigned to be an anisotropic, homogeneous, confined aquifer The problem parameters used for SEAWAT Version 4 model of Abarca et al. (2007) reference cases are listed in Table 8 1 (Abarca et al. 2007; Povich et al. 2013) PAGE 199 199 Table 8 1 Aquifer parameters used for Abarca et al. (2007) reference cases with values of parameters from Povich et al. (2013) Parameter Diffusive case Dispersive case Unit Density contrast parameter ( ) 0.025 0.025 Porosity ( ) 0.35 0.35 Dimensionless ratio 0.3214 0.3214 Horizontal dimension of the domain ( ) 2.0 2.0 m Vertical dimension of the domain ( ) 1.0 1.0 m Horizontal (longitudinal) dispersivity ( ) 0.0 0.1 m Transverse vertical dispersivity ( ) 0.0 0.1 m Vertical dispersivity ( = 0.1 ) 0.0 0.01 m Horizontal hydraulic conductivity ( ) 1 074.816 1 074.816 m/day Transverse vertical hydraulic conductivity ( ) 1 074.816 1 074.816 m/day Vertical hydraulic conductivity ( = 0.66 ) 709.379 709.379 m/day M olecular diffusion coefficient ( ) 1.62925 0.0 m 2 /day F reshwater inflow per unit width ( ) 5.702 5.702 m 2 /day Note: 1. ch c ase of Abarca et al. (2007); and 2. Although the transverse horizontal dispersivity ( ) was assigned equal to the longitudinal dispersivity ( ), and the transverse horizontal hydraulic conductivity value ( ) were assigned equal to the longitudinal hydraulic conductivity ( ), the investigations were still accurate since the inve stigations were considered in a two dimensional cross section model. Methods and Schemes for Sgol (1993) and Abarca et al. (2007) Solutions were obtained by running SEAWAT Version 4 (Langevin et al. 2008) to steady state The f low and transport components in SEAWAT were solved for both the uncoupled and coupled sol utions with identical solution procedures. The f low model was solved using the Pre Conditioned Conjugate Gradient (PCG2) package The head change criterion for convergence (HCLOSE) and residual criterion for convergence (RCLOSE) were set equal to 1x10 8 m and 1 m 3 /day, respectively The third order total variation diminishing (TVD) scheme was used to solve the advection term of the solute transport equation The lengths of transport time PAGE 200 200 steps were calculated using a specified Courant number ( ) of 0. 1 in order to obtain accurate solutions (Zheng and Wang 1999) The remaining terms (dispersion, source/sink mixing and reaction) were solved using the iterative implicit Generalized Conjugate Gradient (GCG) solver with the Modified Incomplete Cholesky (MIC) pre conditioner. The convergence criterion in terms of relative concentration (CCLOSE) was set equal to 1x10 6 The transport time step size within each time step of the flow solution (DT0) was assigned a value of 1x10 4 days Variable Density Flow package options were applied with the first transport time step (FIRSTDT) = 1x10 4 days. Each solution was run to steady state in the transient mode for a 2 day simulation period, which required 50,000 timesteps. The output files, i.e., the LST files, were checked to ensure that steady state was reached. The coupled flow and transport option was assigned to represent density dependent flow and transport in a coastal aquifer for the coupled solution. SEAWAT Version 4 Results for the Works of Sgol (1993) and Abarca et al. (2007) In th (1964) problem of Sgol (1993) and Abarca et al. (2007) works using the regular coupled methods and schemes with SEAWAT Version 4 the domain wa s saturated by freshwater as shown in Figure 8 1 3 Moreover, h ori zontal velocity vectors indicate that the direction of freshwater inflows go directly from the upgradient to downgradient boundaries This meant that the freshwater moved directly through the domain to the sea and SEAWAT could not The blue color in Figure 8 11 represents the freshwater. PAGE 201 201 calculate the saltwater intrusion in the domain. Th ese result s contrast with the work s of Sgol (1993) and Abarca et al. (2007) Figure 8 1 3 (1964) problem of Sgol (1993) and Abarca et al. (2007) works using the internally calculate equivalent freshwater heads and the varied concentration at the downgradien t condition. These results indicate that the regular coupled methods and schemes in SEWAT Version 4 c an (1964) problem of Sgol (1993) and Abarca et al. (2007) works to obtain the coupled solution. Thus, the regular coupled methods and schemes need to be adjusted in order to obtain the results After investigat ing it was determined that errors in results might come from downgradient conditions and parameters that controlled the calculation of SEAWAT Version 4 with the variable density flow (VDF) package for the regular coupled methods and schemes suc h as: 1. The varied concentration option at the downgradient boundary, which c an adjust the concentration of TDS equal to 0.0 kg/m 3 and change the sea water in numerical calculation becoming the freshwater; 2. The freshwater i s controlled by the fixed flow rate for each solution which c an a ffect the concentration at the downgradient boundary since the saltwater was PAGE 202 202 controlled by equivalent freshwater heads, which are internally calculate d in SEAWAT Version 4, and could not build the recirculation of seawate r ( ) at the downgradient boundary. The fixed flowrate of freshwater could dominate flow in the domain and dilute the concentration of seawater at the downgradient boundary: 3. The co nvergence criterion in terms of relative concentration (CCLOSE ) of t he Generalized Conjugate Gradient package solver is used to solve the matrix of the transport equation for implicit solution schemes in MT3DMS. The GCG solver has two iteration loops, an inner loop and an outer loop. Within the inner loop, the solver wi ll continue to iterate toward the solution until the user specified convergence criterion is satisfied or the user specified maximum number of inner iterations allowed is reached (Zheng and Wang 1999) The inner loop of calculatio n would continue to iterate on the transport equation until the results reached the convergence criterion; 4. The convergence criterion for convergence between flow and transport (DNSCRIT ) in SEAWAT Version 4, if the maximum fluid density difference betwe en two consecutive implicit coupling iterations is not less than DNSCRIT, the program will continue to iterate on the flow and transport equations, or will terminate if the flow and transport coupling procedure (NSWTCPL § ) is reached (Langevin et al. 2008) The calculation to solve on the flow and transport CCLOSE is the convergence criterion in terms of relative concentration; a real value between 10 4 and 10 6 is generally adequate (Zheng and Wang 1999) If the Generalized Conjugate Gradient package ( GCG ) is selected, dispersion, sink/source, and reaction terms are solved implicitly without any stability constraints. For the advection term, the user has the option to select any of the solution schemes available, including the standard finite difference method using either the upstream or the central in space weighting, the particle tracking based Eulerian Lagrangian methods, and the third order TVD method. The finite difference method can be fully impl icit without any stability constraint to limit transport step sizes, but the particle tracking based Eulerian Lagrangian methods and the third order TVD method still have time step constraints associated with particle tracking and TVD methodology (Zheng an d Wang 1999) DNSCRIT is a user specified density value To update the flow field if the density changes by more than 10 percent of the freshwater seawater range, users would specify a value of 2.5 kg/m 3 for DNSCRIT (Langevin et al. 2008) § NSWTCPL is a flag used to determine the flow and transport coupling procedure (Langevin et al. 2008) PAGE 203 203 equations by iterating continue s until the maximum fluid density difference between two consecutive implicit coupling iterations is le ss than DNSCRIT, which i s the concentration of seawater at the downgradient boundary = 0.0 kg/m 3 i.e., freshwater, because of the effect of varied concentration option; and 5. The combinations of coupled approach as shown above are not suitable for calcul ating the co (1964) problem of Sgol (1993) and A barca et al. (2007) works. Revised Methods and Schemes for Problem of Sgol (1993) and Abarca et al. (2007) W orks After investigating several approaches, a procedure was developed that could perform the c (1964) prob lem of Sgol (1993) and Abarca et al. (2007) works including Abarca et al. (2007) reference cases to obtain results which w ere cons istent with their results, namely: 1. The manually input equivalent freshwater heads at the downgradient conditions as shown in Equation 8 1 (Guo and Langevin 2002) : 8 1 where: = equivalent freshwater head, ; and = elevation of given point above or below a datum (sea level), i.e., = 0 at sea level. 2. The initial concentration condition specified at a concentration of TDS equal to 35 kg/m 3 ; and 3. Using the varied concentration at the boundary condition PAGE 204 204 SEAWAT Version 4 Results Problem of Sgol (1993) and Abarca et al. (2007) Works, including Abarca et al. (2007) Reference Cases The coupled solutions of th problem o f Sgol (1993) and Abarca et al. (2007) works By using the (1964) problem of Sgol (1993) and Abarca et al. (2007) works SEAWAT V ersion 4 model reproduce d the results, as shown in Figure 8 1 4 and Appendix B, which were in quite good agreement with the work s of Sgol (1993) and Abarca et al. (2007) A B Figure 8 14 SEAWAT Version 4 isochlor s for the coupled solution s of Henry problem using the manually input equivalent freshwater heads and the varied concentration at the downgradient condition A) Sgol (1993) and B) Abarca et al. (2007) The coupled solutions of Abarca et al. (2007) reference cases For the two reference cases of Abarca et al. ( 2007) i.e., the diffusive case (constant dispersion case) and the dispersive case (velocity dependent dispersion case) S EAWAT Version 4 model reproduced the coupled solutions b y using the revised approach which were in favorable agreement with the wo rk of Abarca et al. (2007) as shown in Figure 8 15 and Appendix B. PAGE 205 205 A B Figure 8 15 SEAWAT Version 4 isochlor s for the coupled solutions of Abarca et al. (2007) reference cases, using the manually input equivalent freshwater heads and the varied concentration at the downgradient conditions A) diffusive case ( constant dispersion case) and B) dispersive case ( velocity dependent dispersion case). Henry Velocity dependent Dispersion Problem Methods and Scheme s for the H enry Velocity dependent Dispersion Problem Sgol (1993) and Abarca et al. (2007) (1964) Problem of Sgol (1993) and Abarca et al. (2007) method s and the schemes were verified and revised to solve the benchma rk problems, i.e., the works of Sgol (1993 ) and Abarca et al. (2007) However, t he methods and schemes to solve the Henry velocity dependent dispersion problem still used the same methods and schemes as presented in topics Sgol (1993) and Abarca et al. (2007) also used the boundary condi tions for the coupled solution in the regular way, i.e. using the internally calculate d equivalent freshwater heads and specifying a constant concentration of seawater at the coastal boundary. After investigating, each solution was run in the transient mode for the 2 day simulation period which required 20,000 to 300,000 timesteps depending on increasing of the value s of t he dispersivity ratio ( ) and the hydraulic conductivity ratio ( ) PAGE 206 206 The uncoupled flow and transport option in SEAWAT was us ed to represent constant density flow and transport for the uncoupled solutions and the coupled flow and transport option was us ed to represent density dependent flow and transport for the coupled solutions. Input Parameters for the Henry Velocity dependent Dispersion Problem The r esults of Henry velocity dependent dispersion i nvestigations were represented in terms of the extent of saltwater intrusion and the degree of recirculation of seawater over a broad range of the dimensionless ratio Based on the values of t he dispersivity ratio ( ) and the hydraulic conductivity ratio ( ) results of Henry velocity dependent dispersion problem can be categorized into three sets, namely: 1. The dispersivity ratio = = 0.01 : 1.1 The hydraulic conductivity ratio = = 0.01 ; 1.2 The hydraulic conductivity ratio = = 0.10; and 1.3 The hydraulic conductivity ratio = = 1.00. 2. The dispersivity ratio = = 0.10: 2.1 The hydraulic conductivity ratio = = 0.01 ; 2.2 The hydraulic conductivity ratio = = 0.10; and 2.3 The hydraulic conductivity ratio = = 1.00. 3. The dispersivity ratio = = 1.00: 3.1 The hydraulic conductivity ratio = = 0.01 ; 3.2 The hydraulic conductivity ratio = = 0.10; and 3.3 The hydraulic conductivity ratio = = 1.00. PAGE 207 207 The extent of saltwater intrusion ( ) and the degree of recirculation of seawater ( namely the aspect ratio the dimensionless ratio = the dimensionless ratio = the dispersivity ratio = and the hydraulic conductivity ratio = The dimensionless variables were assigned to determine quantitatively the response of saltwater intrusion and the recirculation of seawater as shown below. 1. The aspect ratio = was equal to 2 .0 since the medium size was defined as a constant domain (1964) problem. 2. There were three main cases of the dispersivity ratio = namely 0.01, 0.10 and 1.00: a) Based on the work of Gelhar et al. (1992) t he longitudinal dispersivity ( ) which is the dispersivity of the medium parallel to groundwater flow, was equal to one tenth of the overall length scale in the horizontal (longitudinal) flow direction which was equal to 0.2 m ; b) The transverse horizontal dispersivity ( ) was assigned equal to the longitudinal dispersivity ( ); c) The transverse vertical dispersivity ( ) was assigned equal to 0.002, 0.02 and 0.2 so as to obtain the dispersivity ratio s ( ) = 0.01, 0.10 and 1.00 respectively ; and d) For e ach case of the dispersivity ratio ( ) there were three case s of the hydraulic conductivity ratio = namely 0.01, 0.10 and 1.00 3. There were three main cases of the hydraulic conductivity ratio = namely 0.01, 0.10 and 1.00 for each case of the dispersivity ratio ( ) : PAGE 208 208 a) The longitudinal hydraulic conductivity ( ) i.e., the hydraulic conductivity of the medium parallel to groundwater flow and the transverse horizontal hydraulic conductivity ( ), were assigned equal to 864 m/day; and b) The t ransverse vertical hydraulic conductivity ( ) w as assigned equal to 8.64, 86.4 and 864 m/day so as to obtain the hydraulic conductivity ratio s ( ) = 0.01, 0.10 and 1.00 respectively 4. Th ere could be a question regarding the accuracy of the transverse horizontal hydraulic conductivity value ( ) and the transverse horizontal dispersivity value ( ). The investigations were still accurate since the investigations were considered in the plane, and there was only one row in the direction. 5. The dimensionless ratio was equal to : a) T he freshwater inflow per unit width and depth ( ) was calculat ed by vary ing the t ransverse vertical hydraulic conductivity ( ) ; and b) The dimensionless ratio was varied from 0.1 to 10.0 for each of investigations which required that the freshwater inflow per unit width and depth ( ) extend from 0.00108 to 10.8 m/day 6. All of the effects of the hydrodynamic disper sion ( ) were re pre sented by the dispersion tensor. 7. The value of the dimensionless ratio was assigned equal to zero because all of the effects of molecular diffusion coefficient ( ) w ere neglected. Summation of dimensionless variables used in the Henry velocity dependent dispersion problem are shown in Table 8 2, and details are shown in Appendix B. PAGE 209 209 Table 8 2 Summation of dimensionless ratios used for the Henry velocity dependent dispersion problem. Set and (m) and m m m/day m/day m 2 /day 0.01 8.64 0.1 to 10.0 0.00108 to 0.108 1 2.0 0.01 0.2 0.002 0.1 0 864 86.4 0.1 to 10.0 0.01080 to 1.080 1.0 0 864 0.1 to 10.0 0.10800 to 10.800 0.01 8.64 0.1 to 10.0 0.00108 to 0.108 2 2.0 0.1 0 0.2 0.02 0.1 0 864 86.4 0.1 to 10.0 0.01080 to 1.080 1.0 0 864 0.1 to 10.0 0.10800 to 10.800 0.01 8.64 0.1 to 10.0 0.00108 to 0.108 3 2.0 1.0 0 0.2 0.2 0.1 0 864 86.4 0.1 to 10.0 0.01080 to 1.080 1.0 0 864 0.1 to 10.0 0.10800 to 10.800 Note: 1. Density contrast parameter = 0.025; 2. Porosity ( ) = 0.35; 3. Horizontal dimension of the domain ( ) = 2.0 m and vertical dimension of the domain ( ) = 1.0 m which gave the aspect ratio = 2.0; 4. and = Longitudinal and transverse horizontal hydraulic conductivity, respectively; 5. = Transverse vertical hydraulic conductivity; 6. and = Longitudinal and transverse horizontal dispersivity, respectively; 7. = Transverse vertical dispersivity; and 8. Although the transverse horizontal dispersivity ( ) was assigned equal to the longitudinal dispersivity ( ), and the transverse h orizontal hydraulic conductivity value ( ) were assigned equal to the longitudinal hydraulic conductivity ( ), the investigations were still accurate since the inves tigations were considered in a two dimensional cross section model. PAGE 210 210 Result s of Henry Velocity dependent Dispersion Investigations Comparison of the 0.5 Isochlor Results between Uncoupled and Coupled Solutions In a ll cases of the Henry velocity dependent dispersion problem, the toe of saltwater intrusion interface ( ) was determined by measuring the landward extent of the = 0.5 concentration contour, i.e., the 0.5 isochlors, along the base of the aquifer T he 0.5 isochlors of the dimensionless ratio s 0.1, 1.0 and 10.0 were used to represent the degree of the extent of saltwater intrusion. A c omparison of the 0.5 isochlors between uncoupled and coupled of the investigations for the Henry velocity dependent dispersion problem w as obtained by considering the dispersivity ratio ( ) and the hydraulic conduc tivity ratio ( ) over a broad range of the dimensionless ratio 1. Based on the value of dispersivity ratio ( ) : 1.1 Dispersivity ratio ( ) = 0.01 and hydraulic conductivity ratio s ( ) = 0.01, 0.10 and 1.00 as shown in Figures 8 16 through 8 21; 1.2 Dispersivity ratio ( ) = 0.10 and hydraulic conductivity ratio s ( ) = 0.01, 0.10 and 1.00 as shown in Figure s 8 22 through 8 27; and 1.3 Dispersivity ratio ( ) = 1.00 and hydraulic conductivity ratio s ( ) = 0.01, 0.10 and 1.00 as sh own in Figure s 8 28 through 8 33 2. Based on the value of hydraulic conductivity ratio ( ) : 2.1 Hydraulic conductivity r atio ( ) = 0.01 and dispersivity ratio s ( ) = 0.01, 0.10 and 1.00 as shown in Figures 8 34 through 8 36; 2.2 Hydraulic conductivity r atio ( ) = 0.10 and dispersivity ratio s ( ) = 0.01, 0.10 and 1.00 as shown in Figure s 8 37 through 8 39; and 2.3 Hydraulic conductivity r atio ( ) = 1.00 and dispersivity ratio s ( ) = 0.01, 0.10 and 1.00 as shown in Figur e s 8 40 through 8 42 PAGE 211 211 Results for the Henry Velocity dependent Dispersion Problem: Dispersivity Ratio ( ) = 0.01 ; and Hydraulic Conductivity Ratio s ( ) = 0.01, 0.10 and 1.00 1. Comparison of the 0.5 isochlor results of = 0.1, 1.0 and 10.0 with the constant value of dispersivity ratio ( ) = 0.01 where = 0.01, 0.10 and 1.00. A B Figure 8 16 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 0.01 A B Figure 8 17 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 0.10. A B Figure 8 18 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 1.00. PAGE 212 212 2. Comparison of the 0.5 isochlor results of the constant value of dispersivity ratio ( ) = 0.01 where = 0.1, 1.0 and 10.0 and = 0.01, 0.10 and 1.00. A B Figure 8 19 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 0.1 and = 0.01, 0.10 and 1.00. A B Figure 8 20 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 1.0 and = 0.01, 0.10 and 1.00. A B Fig ure 8 21 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 10.0 and = 0.01, 0.10 and 1.00. PAGE 213 213 Results for the Henry Velocity dependent Dispersion Problem: Dispersivity Ratio ( ) = 0.10; and Hydraulic Conductivity Ratio s ( ) = 0.01, 0.10 and 1.00 1. Comparison of the 0.5 isochlor results of = 0.1, 1.0 and 10.0 with the constant value of dispersivity ratio ( ) = 0.10 where = 0.01, 0.10 and 1.00. A B Figure 8 22 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.10 and = 0.01 A B Figure 8 23 Comparison of the pattern of the 0.5 is ochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.10 and = 0.10. A B Figure 8 24 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.10 and = 1.00. PAGE 214 214 2. Comparison of the 0.5 isochlor results of the constant value of dispersivity ratio ( ) = 0.10 where = 0.1, 1.0 and 10.0 and = 0.01, 0.10 and 1.00. A B Figu re 8 25 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 0.1 and = 0.01, 0.10 and 1.00. A B Figure 8 26 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 1.0 and = 0.01, 0.10 and 1.00. A B Figure 8 27 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 10.0 and = 0.01, 0.10 and 1.00. PAGE 215 215 Results for the Henry Velocity dependent Dispersion Problem: Dispersivity Ratio ( ) = 1.00; and Hydraulic Conductivity Ratio s ( ) = 0.01, 0.10 and 1.00 1. Comparison of the 0.5 isochlor results of = 0.1, 1.0 and 10.0 with the constant value of dispersivity ratio ( ) = 1.00 where = 0.01, 0.10 and 1.00. A B Figure 8 28 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1 1.0 and 10.0 in case of = 1.00 and = 0.01 A B Figure 8 29 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 1.00 and = 0.10. A B Figure 8 30 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 1.00 and = 1.00. PAGE 216 216 2. Comparison of the 0.5 isochlor results of the constant value of dispersivity ratio ( ) = 1.00 where = 0.1, 1.0 and 10.0, and = 0.01, 0.10 and 1.00. A B Figu re 8 31 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 0.1 and = 0.01, 0.10 and 1.00. A B Figure 8 32 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 1.0 and = 0.01, 0.1 0 and 1.00. A B Figure 8 33 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 10.0 and = 0.01, 0.10 and 1.00. PAGE 217 217 Results for the Henry Velocity dependent Dispersion Problem: Hydraulic Conductivity Ratio ( ) = 0.01 where = 0.1, 1.0 and 10.0 ; and Dispersivity Ratio s ( ) = 0.01, 0.10 and 1.00 A B Figure 8 34 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 0.1 and = 0.01, 0.10 and 1.00. A B Figure 8 35 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 1.0 and = 0.01, 0.10 and 1.00. A B Figure 8 36 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.01 in case of = 10.0 and = 0.01, 0.10 and 1.00. PAGE 218 218 Results for the Henry Velocity dependent Dispersion Problem: Hydraulic Conductivity Ratio ( ) = 0.10 where = 0.1, 1.0 and 10.0; and Dispersivity Ratio s ( ) = 0.01, 0.10 and 1.00 A B Figure 8 37 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 0.1 and = 0.01, 0.10 and 1.00. A B Figure 8 38 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 0.10 in case of = 1.0 and = 0.01, 0.10 and 1.00. A B Figure 8 39 Comparison of the pattern of the 0.5 isochlors of A) uncou pled B) coupled solutions for = 0.10 in case of = 10.0 and = 0.01, 0.10 and 1.00. PAGE 219 219 Results for the Henry Velocity dependent Dispersion Problem: Hydraulic Conductivity Ratio ( ) = 1.00 where = 0.1, 1.0 and 10.0 ; and Dispersivity Ra tio s ( ) = 0.01, 0.10 and 1.00 A B Figure 8 40 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 0.1 and = 0.01, 0.10 and 1.00. A B Figure 8 41 Comparison of the pattern of the 0.5 isochlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 1.0 and = 0.01, 0.10 and 1.00. A B Figure 8 42 Comparison of the pattern of the 0.5 iso chlors of A) uncoupled B) coupled solutions for = 1.00 in case of = 10.0 and = 0.01, 0.10 and 1.00. PAGE 220 220 Comparison of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) between Uncoupled and Coupled Solutions In a ll cases of the Henry velocity dependent dispersion problem the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) versus the dimensionless ratio from 0.1 to 10.0 were used to determine quantitatively the extent of saltwater intrusion. For both the uncoupled and coupled solutions the dimensionless ratio was increased from 0.1 to 10.0, the extent of saltwater intrusion ( ) decreased Also, the dimensionless ratio was increased from 0.1 to 10.0, the reci rculation of seawater ( ) considerably decreased 1. Based on the value of dispersivity ratio ( ) : 1.1 Dispersivity r atio ( ) = 0.01 and h ydraulic conductivity r atio s ( ) = 0.01, 0.10 and 1.00 as shown in Figures 8 43 through 8 51 ; 1.2 Dispersi vity ratio ( ) = 0.10 and hydraulic conductivity ratio s ( ) = 0.01, 0.10 and 1.00 as shown in Figures 8 52 through 8 60 ; and 1.3 Dispersivity ratio ( ) = 1.00 and hydraulic conductivity ratio s ( ) = 0.01, 0.10 and 1.00 as shown in Figures 8 61 through 8 69 2. Based on the value of hydraulic conductivity ratio ( ): 2.1 Hydraulic c onductivity r atio ( ) = 0.01 and d ispersivity r atio s ( ) = 0.01, 0.10 and 1.00 as shown in Figure s 8 70 through 8 75; 2.2 Hydraulic c onductivity r atio ( ) = 0. 1 0 and d ispersivity r atio s ( ) = 0.01, 0.10 and 1.00 as shown in Figure s 8 76 through 8 81; and 2.3 Hydraulic c onductivity r atio ( ) = 1.00 and d ispersivity r atio s ( ) = 0.01, 0.10 and 1.00 as shown in Figure s 8 82 through 8 87 PAGE 221 221 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Dispersion Pr oblem when D ispersivity R atio ( ) = 0. 0 1 and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 A B Figure 8 43 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01. A B Figure 8 44 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.10. PAGE 222 222 A B Figure 8 45 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 1.00. A B Figure 8 46 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for un coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. PAGE 223 223 A B Figure 8 47 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. Some results were affected by the medium size, especially the extent of saltwater intrusion ( ) in case of = 0.01 for the uncoupled and coupled solutions, as shown in Figure 8 46 (A) and 8 47 (A) namely : T he dimensionless ratio < 5.0 when = 0.01; T he dimensionless ratio < 2 .0 when = 0.10; and T he dimensionless ratio < 0.2 when = 1.00 (only the coupled solution s). Some results reached the lower bound limit of the extent of saltwater intrusion ( ) which was approximately 0.1 0, since the freshwater inflow per unit width ( ) dominated the flow domain. The degree of saltwater recirculation ( ) for the uncoupled and coupled solutions was not affected by the domain size, since the third order total variation diminishing (TVD) scheme used in SEAWAT is mass conservative PAGE 224 224 Figure 8 48 E xtent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 225 225 Figure 8 49 E xtent of saltwater intrusion ( ) for coupled solutions versus the dimensi onless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 226 226 Figure 8 50 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 227 227 Figure 8 51 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 228 228 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Dispersion Pr oblem when D ispersivity R atio ( ) = 0.10 and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 A B Figure 8 52 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01. A B Figure 8 53 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.10. PAGE 229 229 A B Figure 8 54 Results of A) extent of saltwater intrusion ( ) and B) recirculation of s eawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 1.00. A B Figure 8 55 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for un coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00. PAGE 230 230 A B Figure 8 56 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00. Some results were affected by the medium size, especially the extent of saltwater intrusion ( ) in case of = 0.10 for the uncoupled and coupled solutions, as shown in Figure 8 46 (A) and 8 47 (A), namely : T he dimensionless ratio < 4 .0 when = 0.01; T he dimensionless ratio < 1.0 when = 0.10; and T he dimensionless ratio < 0.2 when = 1.00 (only the coupled solution s). Some results reached the lower bound limit of the extent of saltwater intrusion ( ) which was approximately 0.1 0, since the freshwater inflow per unit width ( ) domina ted the flow domain. The degree of saltwater recirculation ( ) for the uncoupled and coupled solutions was not affected by the domain size, since the third order total variation diminishing (TVD) scheme used in SEAWAT is mass conservative PAGE 231 231 Figure 8 57 E xtent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 1 0 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity v ectors for the second set of the Henry constant dispersion problem PAGE 232 232 Figure 8 58 E xtent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 1 0 and = 0.01, 0.10 and 1.00 and is ochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 233 233 Figure 8 59 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 1 0 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 234 234 Figure 8 60 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 1 0 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 235 235 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Dispersion Pr oblem when D ispersivity R atio ( ) = 1.00 and Hydraulic Conductivity Ratios ( ) = 0.01, 0.10 and 1.00 A B Figure 8 61 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01. A B Figure 8 62 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.10. PAGE 236 236 A B Figure 8 63 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled sol utions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 1.00. A B Figure 8 64 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for un coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00. PAGE 237 237 A B Figure 8 65 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00. Some results were affected by the medium size, especially the extent of saltwater intrusion ( ) in case of = 1.00 for the uncoupled and coupled solutions, as shown in Figure 8 46 (A) and 8 47 (A), namely : T he dimensionless ratio < 2 .0 when = 0.01; and T he dimensionless ratio < 0.7 when = 0.10 Some results reached the lower bound limit of the exten t of saltwater intrusion ( ) which was approximately 0.1 0, since the freshwater inflow per unit width ( ) dominated the flow domain. The degree of saltwater recirculation ( ) for the uncoupled and coupled solutions was not affected by the domain size, since the third order total variation diminishing (TVD) scheme used in SEAWAT is mass conservative PAGE 238 238 Figure 8 66 E xtent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 239 239 Figure 8 67 E xtent of saltwater intr usion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 240 240 Figure 8 68 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concent ration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 241 241 Figure 8 69 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 242 242 Results of Extent of Saltwater Intrusion ( ) and Recirculat ion of Seawater ( ) for the Henry Velocity dependent Dispersion Problem when Hydraulic Conductivity Ratio ( ) = 0.01 and Dispersivity Ratio s ( ) = 0.01, 0.10 and 1.00 A B Figure 8 70 Results of A) extent of saltwater intr usion ( ) and B) recirculation of seawater ( ) for un coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. A B Figure 8 71 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. PAGE 243 243 Figure 8 72 E xtent of saltwater in trusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the H enry constant dispersion problem PAGE 244 244 Figure 8 73 E xtent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 245 245 Figure 8 74 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 246 246 Figure 8 75 The recirculation of seawater ( ) for coupled solut ions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 247 247 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Dispersion Problem when Hydraulic Conductivity Ratio ( ) = 0.10 and Dispersivity Ratio s ( ) = 0.01, 0.10 and 1.00 A B Figure 8 76 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for un coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 a nd 1.00. A B Figure 8 77 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.10 and = 0.01, 0.10 and 1.00. PAGE 248 248 Figure 8 78 E xtent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.1 0 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 249 249 Figure 8 79 E xtent of saltwater intrusion ( ) fo r coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0 10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 250 250 Figure 8 80 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 251 251 F igure 8 81 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0. 10 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 252 252 Results of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) for the Henry Velocity dependent Disper sion Problem when Hydraulic Conductivity Ratio ( ) = 1.00 and Dispersivity Ratio s ( ) = 0.01, 0.10 and 1.00 A B Figure 8 82 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for un cou pled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00. A B Figure 8 83 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00. PAGE 253 253 Figure 8 84 E xtent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 254 254 Figure 8 85 E xtent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the H enry constant dispersion problem PAGE 255 255 Figure 8 86 The recirculation of seawater ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with con centration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 256 256 Figure 8 87 The recirculation of seawater ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 1.00 and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors for the second set of the Henry constant dispersion problem PAGE 257 257 Discussion and Conclusions for the Henry Velocity dependent Dispersion Problem The results of the numerical experiments indicate that t he two dimensionless dependent variables, that is, the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ), are f unction s independent dimensionless ratios : the aspect ratio the dimensionless ratio the dimensionless ratio the dispersivity ratio and the hydraulic conductivity ratio Below are the results for the extent of saltwater intrusion and the degree of saltwater recirculation. When comparing the results at the same value of the dispersivity ratio ( ) the 0.5 i sochlor distribution s indicate that some of them are affected by the bo undary due to the domain size as shown in Figures 8 16 through 8 18. For example, = 0.1 for both the uncoupled and coupled solutions when = 0.01 and = 0.01 (Figure 8 16) or when = 0. 01 and = 0. 10 (Figure 8 17). However, = 0.1 only for the coupled solutions when = 0. 01 and = 1.00 (Figure 8 18) When comparing the results at the same value of the dimensionless ratio the 0.5 i sochlor distribution s indicate that some of them are affected by the boundary effect due to the domain size as shown in Figures 8 19 through 8 21. For example, = 0.01 and = 0.1 for both the uncoupled and coupled solutions when = 0.01 and 0.10 (Figure 8 19), whereas = 0.01 and = 1.0 only for the coupled solution when = 0.01 (Figure 8 20). However, t he 0.5 i sochlor distribution s indicate that they are unaffected by the boundary effect due to the domain size when = 0.01 and = 10.0 (Figure 8 21). PAGE 258 258 When comparing the results at the same value of the hydraulic conductivity ratio ( ), the 0.5 isochlor distribution s indicate that some of them are affected by the boundary effect due to the domain size F or example = 0.1 for both the uncoupled and coupled solutions when = 0.01 and = 0.01, 0.10, an d 1.00 (Figure 8 34) or when = 0.10 and = 0.01, 0.10, and 1.00 (Figure 8 37), whereas = 0.1 only for t he coupled solutions when = 1.0 and = 0.01 and 0.10 [ Figure 8 40 ( B) ] Although the domain size dominated some 0.5 i sochlor distribution results some of them still present good results for the dimensionless parameters as shown in Figures 8 18, 8 24, and 8 30. F or example, = 0.1, 1.0 and 10.0 for the uncoupled solutions when = 0.01 and = 1.00 ( Figure 8 1 8) or when = 0.10 and = 1.00 ( Figure 8 24), whereas, = 0.1, 1.0 and 10.0 for both the uncoupled and coupled solutions when = 1 .0 0, and = 1.00 ( Figure 8 30) When comparing the results at the same value of dimensionless ratio ( the 0.5 i sochlor distribution shows the effect of hydraulic conductivity ratio ( ) The increasing value of the hydraulic conductivity ratio ( ) ma kes the influence of freshwater inflow increase (i.e., the 0.5 isochlors move seaward). For example, = 1.0 for both the uncoupled and coupled solutions when = 0.01 and = 0.01, 0.10 and 1.00 ( Figure 8 20), when = 0.10 and = 0.01, 0.10 and 1.00 ( Figure 8 26) and when = 1.00 and = 0.01, 0.10 and 1.00 ( Figure 8 32) When comparing the results at the same value of dimensionless ratio ( ) the 0.5 i sochlor distribution show s the effect of the dispersivity ratio ( ) which has the same result as increasing the value of the hydraulic conductivity ratio ( ) The i ncreasing value of the dispersivity ratio ( ) also mak e s the influence of freshwater inflow increase PAGE 259 259 ( i.e., the 0.5 isochlors move seaward ). For example, = 1.0 for both the uncoupled and coupled solutions when = 0.01 and = 0.01, 0.10 an d 1.00 ( Figure 8 35) or when = 0.1 and = 0.01, 0.10 and 1.00 ( Figure 8 38) However, when = 1.0 and = 0.01, 0.10 and 1.00, the results are not satisf actory due to the effect of domain size ( Figure 8 40) The influence of the hydraulic conductivity ratio ( ) o n the 0.5 i sochlor distribution results is greater than the influence of the dispersivity ratio ( ) when comparing the 0.5 i sochlor distribution results of = 1.0 and = 0.01, 0.10, and 1.00 to the results of = 0.01, 0.10, and 1.00. Although the effect of medium size dominates some results, some of the results are still good enough to interpret When the dimensionless ratio ( which is the ratio of freshwater inflow per unit width ( ) relative to the density driven vertical buoyancy flux ( ), i s increased from 0.1 to 10.0 for both the uncoupled and coupled solutions and the value of the dispersivity ratio ( ) is held constant the results demonstrate that t he extent of saltwater intrus ion ( ) and t he degree of saltwater recirculation ( ) are significantly decreased Significant differences can occur between the uncoupled and coupled solutions as shown in Figure s 8 43 to 8 51 for = 0.01, and = 0.01, 0.10 and 1.00; Figur e s 8 52 to 8 60 for = 0.10, and = 0.01, 0.10 and 1.00; and Figure s 8 61 to 8 69 for = 1.00, and = 0.01, 0.10 and 1.00. At larger values of hydraulic conductivity ratio ( ) when the value of dispersivity ratio ( ) held constant t he difference in values of the extent of saltwater intrusion ( ) is decreased until there i s a slight difference in values for the uncoupled solutions PAGE 260 260 (Figures 8 70, 8 76 and 8 81) and for the coupled solutions (Figures 8 71, 8 77 and 8 83). For the uncoupled solutions, the degree of saltwater recirculation ( ) does not indicate the difference in values when the value of the dispersivity ratio ( ) i s varied and the value of the hydraulic conductivity ratio ( ) is held constant [ Figures 8 70 ( B ) 8 76 ( B ) and 8 82 ( B )]. F or the coupled solutions, the degree of saltwater recirculation ( ) indicate s a slight difference in values when the value of the dispersivity ratio ( ) i s varied and the value of the hydraulic conductivity ratio ( ) held constant [ Figures 8 71 ( B ) 8 77 ( B ) and 8 83 ( B) ] The degree of saltwater recirculation ( ) i s un affected by the domain size since the third order total variation diminishing (TVD) scheme used in SEAWAT is mass conservative On the Courant numbe r ( ) and the LST output files all of the solutions are accurate and stab le numerical calculations. To obtain the most accuracy, the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) at the coastal boundary should be determined in a coupled variable density flow and transport model. There is no clear conclusion that the uncoupled solution can yield results without a difference in values from the coupled solutions since some results are affected by the boundary du e to the domain size However, at larger values of the dimensionless ratio the uncoupled solutions yield ed approximately the same results as the coupled solutions for both the extent of saltwater intrusion ( ) and the degree of saltwater recirc ulation ( ) PAGE 261 261 CHAPTER 9 FIELD SCALE P ROBLEM Benchmark Problem for the Field s cale Problem In this problem the use of SEAWAT and Groundwater Vistas 5 in this investigation was verified by comparing the results to results previously obtained by Motz and Sedighi (2013) They presented results of the e xtent of saltwater intrusion ( ) and the recirculation of seawater ( ) for both the uncoupled and coupled simulations for the field scale problem as shown in Figur e 9 1 in which the dimensionless ratio = ranged from 0.1 to 10.0 This problem was based on data from the upper part of the Floridan aquifer system at Hilton Head Island in South Carolina, U.S.A. described by Bush (1 988), in which = 0.8 (Motz and Sedighi 2009 b ) A B Figur e 9 1 The work of Motz and Sedighi (2013) A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for both uncoupled and coupled solutions. PAGE 262 262 Aquifer Parameters for Motz and Sedighi (2013) To reproduce the work of Motz and Sedighi (2013) the domain was assigned to be an an isotropic, homogeneous confined aqu ifer Aquifer parameters used for the numerical simulation of the field scale problem are shown in Table 9 1 Table 9 1 Aquifer parameters used for the field scale problem to reproduce the work of Motz and Sedighi (20 13) Parameter Value Unit Porosity ( ) 0.30 Longitudinal dimension of the domain ( ) 12,750 m Longitudinal discretization ( ) 250 m Transverse vertical dimension of the domain ( ) 200 m Transverse vertical discretization ( ) 5 m Transverse horizontal dimension of the domain ( ) 250 m Transverse horizontal discretization ( ) 250 m Longitudinal dispersivity ( ) 125 m Transverse horizontal dispersivity ( ) 125 m Transverse vertical dispersivity ( ) 1.25 m Longitudinal hydraulic conductivity ( ) 50 m/day Transverse horizontal hydraulic conductivity ( ) 50 m/day Transverse vertical hydraulic conductivity ( ) 0.5 m/day Freshwater inflow per unit width ( ) for specific flux boundary condition 0.25 25.0 m 2 /day Note: Although the transverse horizontal dispersivity ( ) was assigned equal to the longitudinal dispersivity ( ), and the transverse horizontal hydraulic conductivity value ( ) were assigned equal to the longitudinal hydraulic conductivity ( ), the investigations were still accurate since the investigations were considered in a two dimensional cross section model Methods and Schemes for Motz and Sedighi (2013) Solutions were obtained by running SEAWAT Version 4 (Langevin et al. 2008) to steady state The f low and transport components in SEAWAT were solved for both the uncoupled and coupled solutions with identical solution procedures. The f low model was solved using the Pre Conditioned Conjugate Gradient (PCG2) package The head change criterion for convergence (HCLOSE) and residual PAGE 263 263 criterion for convergence (RCLOSE) were set at 1x10 9 m and 1 m 3 /d ay respectively. The advective term of th e solute transport equation was solved using the third order total variation diminishing (TVD) scheme The lengths of the transport time steps were calculated using a specified Courant number ( ) of 1.0 in order to obtain accurate solutions (Zheng and Wang 1999). The remaining terms (dispersion, source/sink mixing and reaction) were solved using the iterative implicit Generalized Conjugate Gradient (GCG) solver with the Modified Incomplete Cholesky (MIC) pre conditioner The convergence criterion in te rms of relativ e concentration (CCLOSE) was set equal to 1x10 6 The transport time step size within each time step of the flow solution (DT0) was assigned a value of 10 days Variable Density Flow package options were applied with the first transport time step (FIRSTDT) = 1x10 2 days. Each solution w as run in the transient mode for a 4x10 6 day simulation period which required 4 x10 5 timesteps The o utput file s, i.e., the LST files, w ere checked to ensure that steady state was reached. SEAWAT Version 4 Results for Motz and Sedighi (2013) The results of the benchmark problem for saltwater intrusion ( ) and the recirculation of seawater ( ) were compared to Motz and Sedighi (2013) as shown in Figure 9 2. The results obtained with SEAWAT Version 4 with the TVD scheme were generally in good agreement with the work of Motz and Sedighi (2013) At small values of such as 0.1 and 0.2, the extent of saltwater intrusion ( ) was a ffected by the domain size. PAGE 264 264 A B Figur e 9 2 Results of SEAWAT Version 4 with TVD scheme for the field scale problem A) e xtent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for both uncoupled and coupled sim ulations comparing to the work of Motz and Sedighi (2013) Field s cale P roblem Methods and Schemes Solutions were obtained by running SEAWAT Version 4 (Langevin et al. 2008) to steady state The f low and transport components in SEAWAT were solved for both the uncoupled and coupled solutions with identical solution procedures. The f low model was solved using the Pre Conditioned Conjugate Gradient (PCG2) package The head change criterion for convergence (HCLOSE) and residual criterion for convergence (RCLOSE) were set at 1x10 9 m and 1 m 3 /day, respectively. The third order total variation diminishing (TVD) scheme was used to solve the advection term of the solute transport equation The lengths of transport time steps were calculated using a specified Courant number ( ) of 0.1 in order to obtain accurate solutions (Zheng and Wang 1999). PAGE 265 265 The remaining terms (dispersion, source/sink mixing and reaction) were solv ed using the iterative implicit Generalized Conjugate Gradient (GCG) solver with the Modified Incomplete Cholesky (MIC) pre conditioner The convergence criterion in terms of relative concentration (CCLOSE) was set equal to 1x10 6 The transport time ste p size within each time step of the flow solution (DT0) was assigned a value of 0.5 days. Variable Density Flow package options were applied with the first transport time step (FIRSTDT) = 1x10 3 days Each solution was run in the transient mode for a 3x10 6 day simulation period, which required 6x10 6 timestep s. The o utput file s, i.e., the LST files, w ere checked to ensure that steady state was reached. The uncoupled flow and transport option in SEAWAT was used to represent constant density flow and transp ort for the uncoupled solutions, and the coupled flow and transport option was us ed to represent density dependent flow and transport for the coupled solutions. Input Parameters The r esults of the field scale i nvestigations were represented in terms of the extent of saltwater intrusion and the degree of recirculation of seawater over a broad range of the dimensionless ratio Based on t he value of hydraulic conductivity ratio ( ) results of the field scale problem can be categorized into three sets namely: 1. The hydraulic conductivity ratio = = 0.01 ; 2. The hydraulic conductivity ratio = = 0.1 ; and 3. The hydraulic conductivity ratio = = 1.0 The value of the dispersivity ratio = was held as a constant value equal to 0.01 for all three sets. PAGE 266 266 T he extent of saltwater intrusion ( ) and the degree of recirculation of seawater ( ) were considered to be dependent functions of ve independent dimensionless ratios namely the aspect ratio = the dimensionless ratio = the dimensionless ratio = the dispersivity ratio = and the hydraulic conductivity ratio = For both the uncoupled and coupled solutions, input parame ters were divided into two groups, namely a quifer parameters and dimensionless variables, which are shown in Table 9 2 and 9 3, respectively. Table 9 2 Aquifer parameters used in the field scale investigations. Parameter Value Unit Porosity ( ) 0.30 Total length of the domain 20,250 m Longitudinal dimension of the domain ( ) 20,000 m Longitudinal discretization ( ) 250 m Transverse horizontal dimension of the domain ( ) 250 m Transverse horizontal discretization ( ) 250 m Transverse vertical dimension of the domain ( ) 200 m Transverse vertical discretization ( ) 5 m Longitudinal dispersivity ( ) 125 m Transverse horizontal dispersivity ( ) 125 m Transverse vertical dispersivity ( ) 1.25 m Longitudinal hydraulic conductivity ( ) 50 m/day Transverse horizontal hydraulic conductivity ( ) 50 m/day Transverse vertical hydraulic conductivity ( ) 0.5, 5.0 and 50 m/day Freshwater inflow per unit wi d th ( ) for specific flux boundary condition 0.25 2,500 m 2 /day Note: Although the transverse horizontal dispersivity ( ) was assigned equal to the longitudinal dispersivity ( ), and the transverse horizontal hydraulic conductivity value ( ) were assigned equal to the longitudinal hydraulic conductivity ( ), the investigations were still accurate since the inve stigations were considered in a two dimensional cross section model. PAGE 267 267 The dimensionless variables were assigned to determine quantitatively how saltwater intrusion an d the recircul ation of seawater were related as follow s : 1. Since the medium size was defined as a constant domain dimension, the aspect ratio = was equal to 100 ; 2. The dispersivity ratio = was equal to 0.01 based on the longitudinal dispersivity ( ) and the transverse vertical dispersivity ( work ; 3. T here were three case of the hydraulic conductivity = namely 0.01, 0.10 and 1.00: a) The longitudina l hydraulic conductivity ( ) and the transverse vertical hydraulic conductivity ( ) were equal to 50 m/day; and b) The transverse vertical hydraulic conductivity ( ) was assigned values equal to 0.5 5.0 and 50 so as to obtain hydraulic conductivi ty ratio s ( ) = 0.01, 0.10 and 1.00 respectively 4. The dimensionless ratio was equal to : a) T he freshwater inflow per unit width and depth ( ) were calculat ed by varyin g the transverse v ertical hydraulic conductivity ( ) ; and b) The dimensionless ratio was varied from 0.1 to 10.0 for each of investigations That extended the freshwater inflow per unit width and depth ( ) from 62.5 to 6. 25 x10 5 m /day. 5. All of the effects of the hydrodynamic dispersion ( ) were re presented by the longitudinal dispersivity ( ) the transverse vertical dispersivity ( ) and the dispersion tensor; 6. The value of the dimensionless ratio was equal to zero because all of the effects of molecular diffusion coefficient ( ) w er e considered negligible PAGE 268 268 A s ummation of the dimensionless variables used in the field scale problem is shown in Table 9 3, and details are shown in Appendix C. Table 9 3 Summation of dimensionless ratios used for the field scale problem. and (m) and M m m/day m/day m 2 /day 0.01 0.50 0.1 to 10.0 0.25 to 25 100 0.01 125 1.25 0.1 0 50 5.0 0 0.1 to 10.0 2.50 to 250 1.0 0 50.0 0 0.1 to 10.0 25.00 to 2,500 Result s of the Field s cale Problem In a ll cases of the field scale problem, the toe of the saltwater intrusion interface ( ) was determined by measuring the landward extent of the = 0.5 concentration contour, i.e., the 0.5 isochlors, along the base of the aquifer. For both the uncoupled and coupled solutions the 0.5 isochlors of the dimensionless ratio s 0.1, 1.0 and 10.0 were used to represent the degree of the extent of saltwater intrusion Comparison s of the 0 .5 isochlors between the investigations for the field scale problem were obtained by considering the dispersivity ratio ( ) and the hydraulic conductivity ratio ( ) over a broad range of the dimensionless ratio based on the value of the hydraul ic c onductivity ratio ( ), as shown in Figure s 9 3 through 9 8 Comparison of the 0.5 isochlor results of = 0.1, 1.0 and 10.0 with the dispersivity ratio ( ) = 0.01 where = 0.01, 0.10 and 1.00. The results indicate that the landward extent of saltwater intrusion ( the 0.5 isochlors) decreased significantly as the dimensionless ratio was increased from 0.1 to 10.0 at the same values of the dispersivity ratio ( ) and the hydraulic conductivity ratio ( ) for both the uncoupled and cou pled solutions, as shown in Figures 9 3 t o 9 5 PAGE 269 269 A B Figure 9 3 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 0.01 A B Figure 9 4 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 0.1 A B Figure 9 5 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 0.1, 1.0 and 10.0 in case of = 0.01 and = 1.0 PAGE 270 270 Comparison of the 0.5 isochlor results of the dispersivity ratio ( ) = 0.01 where = 0.1, 1 .0 and 10.0 and = 0.01, 0.10 and 1.00. The results indicated that the landward extent of saltwater intrusion ( 0.5 isochlors) decreased significantly as the hydraulic conductivity ratio ( ) was increased from 0. 0 1 to 0.1 at the same values of the dimensionless ratio and the dispersivity ratio ( ) for both the uncoupled and coupled solutions, as shown in Figures 9 6 through 9 8 A B Figure 9 6 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) c oupled solutions for = 0.1 in case of = 0.01 and = 0.01, 0.10 and 1.00. A B Figure 9 7 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 1.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. PAGE 271 271 A B Figure 9 8 Comparison of the pattern of the 0.5 isochlors of A) uncoupled and B) coupled solutions for = 10.0 in case of = 0.01 and = 0.01, 0.10 and 1.00. Comparison of the Extent of Saltwater Intrusion ( ) and the Recirculation of Seawater ( ) between Uncoupled and Coupled Solutions In a ll cases of the field scale problem the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) versus the dimensionless ratio from 0.1 to 10.0 were used to determine quantitatively the degree of saltwater intrusion. For both the uncoupled and coupled solutions as the dimensionless ratio was increased from 0.1 to 10.0, the extent of salt water intrusion ( ) decreased significantly. Also, as the dimensionless ratio was increased from 0.1 to 10.0, the recirculation of seawater ( ) also decreased significantly based on the value of dispersivity ratio ( ), as shown in Figur es 9 9 through 9 13 and Appendix C Some results reached the lower bound limit of the extent of saltwater intrusion ( ) which was approximately 0. 70, since the freshwater inflow per unit width ( ) dominated the flow domain. The degree of saltwater r ecirculation ( ) for the uncoupled and coupled solutions was not affected by the domain size, since the third order total variation diminishing (TVD) scheme used in SEAWAT is mass conservative PAGE 272 272 A B Figur e 9 9 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.01. A B Figur e 9 10 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 0.1. PAGE 273 273 A B Figur e 9 11 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01 and = 1.0. A B Figur e 9 12 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for un coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00. PAGE 274 274 A B Figur e 9 13 Results of A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for coupled solution s versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00. Figures 9 14 through 9 17 also presented the investigation results of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) of the field scale problem. The upgradient inland boundary on the right hand side of the domain was specifi ed as the freshwater inflow (represented by the blue color). The downgradient coastal boundary on the left hand side of the domain was assigned as seawater (represented by the red color). PAGE 275 275 Figure 9 14 E xtent of saltwater intrusion ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors PAGE 276 276 Figure 9 15 E xtent of saltwater intrusion ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors PAGE 277 277 Figure 9 16 D egree of saltwater recirculation ( ) for uncoupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and velocity vectors PAGE 278 278 Figure 9 17 D egree of saltwater recirculation ( ) for coupled solutions versus the dimensionless ratio from 0.1 to 10.0 in case of = 0.01, and = 0.01, 0.10 and 1.00 and isochlors from 0.1 to 0.9 with concentration color floods and vel ocity vectors PAGE 279 279 Comparison of Extent of Saltwater Intrusion ( ) and Recirculation of Seawater ( ) when Decreasing the V alue of Hydraulic C onductivity The Verification of the Consistency of Results for the Saltwater Numerical Model in case of = 0.01 To verify the consistency of results for the saltwater numerical model of the field scale problem, a set of the dimensionless ratio values was sele cted as a regular set and a set of the dimensionless ratio values was obtained by d ecreasing the v alue of h ydraulic c onductivity by a factor of two as a v erified s et The results of the verified set were performed and compared to the results of regul ar set. The r egular and verified sets for the field scale problem in case of = 0.01 The s altwater numerical model was verified to present the consistency of the results wh en decreasing t he v alue of h ydraulic c onductivity by a factor of two, but the h ydraulic c onductivity ratio = was still maintained at a constant value equal to 0.01. Also t he value of the dispersivity ratio = was held as a constant value equal to 0.01 Aquifer parameters and dimensionless ratios in case o f = 0.01 Six of the dimensionless ratio values were select ed to be the regular set of investigation, namely = 0.1, 0.2, 0.5, 1.0, 4.0 and 5.0, which w ere the values of and equal to 50 m/day and equal to 0.5 m/day. For both the uncoupled and coupled solutions the a quifer p arameters and the d imensionless r atios for the r egular s et in case of = 0.01 are shown in Table 9 4. To obtain the verified set, the value of hydraulic conductivity from six of the dimensionless r atio values was decreased by a factor of two. That affected t he value of the dimensionless ratio which was increased twice by comparing of the regular PAGE 280 280 set. Thus, t he verified set consist ed of = 0.2, 0.4, 1.0, 2.0, 8.0 and 10.0, which w ere the values of and equal to 25 m/day and equal to 0.25 m/day For both the uncoupled and coupled solutions the a quifer p arameters and the d imensionless r atios for the v erified s ets in case of = 0.01 are shown in Table 9 5. Results of the regular and the verified s et s in case of = 0.01 The isochlor results of the regular set with concentration color floods and velocity vectors when th e hydraulic conductivity ratio ( ) = 0.01, the values of and = 50 m/day and = 0.5 m/day are shown in Table 9 6 and 9 7 and also in Figures 9 18 to 9 20 for the un coupled and coupled solutions, respectively The verified results were compared to the regular set of investigation, namely = 0.2, 0.4, 1.0, 2.0 8.0 and 10.0 since the comparison was based on the same value of the dimensionless ratio The verified results with concentration color floods and velocity vectors for the field scale problem when the hydraulic conductivity ratio ( ) = 0.01 are shown in Table 9 6 and 9 7 and also in Figures 9 21 to 9 23 for the un coupled and coupled solutions, respectively Upon compari ng the results between the regular and t he verified sets, it was indicate d that there was no significant difference in either the patterns or values of results in the case of the hydraulic conductivity ratio ( ) = 0.01. PAGE 281 281 Table 9 4 A quifer parameters and dimensionless ratios for six selected regular cases to verify the consistency of results for = 0.01. and and m/day m/day m/day m/day m/day m/day 0.1 50 0.5 0 0.01 0.25 0.1 62.5 125 1.25 0.01 0.2 50 0.5 0 0.01 0.50 0.2 125.0 125 1.25 0.01 0.5 50 0.5 0 0.01 1.25 0.5 312.5 125 1.25 0.01 1.0 50 0.5 0 0.01 2.50 1.0 625.0 125 1.25 0.01 4.0 50 0.5 0 0.01 10.00 4.0 2,500.0 125 1.25 0.01 5.0 50 0.5 0 0.01 12.50 5.0 3,125.0 125 1.25 0.01 Table 9 5 A quifer parameters and dimensionless ratios for six verified cases to verify the consistency of results when decreasing t he v alue of h ydraulic c onductivity by a factor of two for = 0.01. and and m/day m/day m/day m/day m/day m/day 0.2 25 0.25 0.01 0.25 0.2 62.5 125 1.25 0.01 0.4 25 0.25 0.01 0.50 0.4 125.0 125 1.25 0.01 1.0 25 0.25 0.01 1.25 1.0 312.5 125 1.25 0.01 2.0 25 0.25 0.01 2.50 2.0 625.0 125 1.25 0.01 8.0 25 0.25 0.01 10.00 8.0 2,500.0 125 1.25 0.01 10.0 25 0.25 0.01 12.50 10.0 3,125.0 125 1.25 0.01 PAGE 282 282 Table 9 6 C omparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing t he v alue of h ydraulic c onductivity by a factor of two and the hydraulic conductivity ratio ( ) = 0.01 for uncoupled solutions. The regular set The verified set and and m/day m/day m/day m m/day m/day m/day m 0.2 50 0.5 0 0.50 2,616.686 13.08343 15.13136 25 0.25 0.25 2,616.686 13.08343 15.13136 0.4 50 0.5 0 1.00 2,182.773 10.91387 7.32537 25 0.25 0.50 2,182.773 10.91387 7.32537 1.0 50 0.5 0 2.50 1,587.128 7.93564 2.65174 25 0.25 1.25 1,587.128 7.93564 2.65174 2.0 50 0.5 0 5.00 1,156.551 5.78276 1.11378 25 0.25 2.50 1,156.551 5.78276 1.11378 8.0 50 0.5 0 20.00 374.173 1.87086 0.07463 25 0.25 10.00 374.173 1.87086 0.02440 10.0 50 0.5 0 25.00 312.090 1.56045 0.03225 25 0.25 12.50 312.090 1.56045 0.01113 Table 9 7 C omparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing t he v alue of h ydraulic c onductivity by a factor of two and the hydraulic conductivity ratio ( ) = 0.01 for coupled solutions. The regular set The verified set and and m/day m/day m/day m m/day m/day m/day m 0.2 50 0.5 0 0.50 11,951.872 59.75936 1.74782 25 0.25 0.25 11,641.667 58.20833 1.80258 0.4 50 0.5 0 1.00 7,893.304 39.46652 1.06653 25 0.25 0.50 7,891.827 39.45913 1.06675 1.0 50 0.5 0 2.50 4,301.639 21.50820 0.49754 25 0.25 1.25 4,301.639 21.50820 0.49754 2.0 50 0.5 0 5.00 2,427.870 12.13935 0.24514 25 0.25 2.50 2,427.870 12.13935 0.24514 8.0 50 0.5 0 20.00 461.319 2.30659 0.02440 25 0.25 10.00 461.319 2.30659 0.02440 10.0 50 0.5 0 25.00 325.227 1.62614 0.01113 25 0.25 12.50 325.231 1.62615 0.01113 PAGE 283 283 A B Figur e 9 18 R esults of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0. 2 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.01 when comparing between Figure 9 18 and Figu re 9 21. PAGE 284 284 A B Figur e 9 19 R esults of the regular set A) uncoupl ed and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.01 when comparing between Figure 9 19 and Figure 9 22. PAGE 285 285 A B Figur e 9 20 R esults of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.01 when comparing between Figure 9 20 and Figure 9 23. PAGE 286 286 A B Figur e 9 21 R esults of the verified set A) uncoupled and B) coupled solutions with conc entration color floods and velocity vectors when = 0. 2 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.01 when comparing between Figure 9 18 and Figure 9 21. PAGE 287 287 A B Figur e 9 22 R esults of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.01 when comparing between Figure 9 19 and Figure 9 22. PAGE 288 288 A B Figur e 9 23 R esults of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.01 * There was no significant d ifference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.01 when comparing between Figure 9 20 and Figure 9 23. PAGE 289 289 The Verification of the Consistency of Results for the Saltwater Numerical Model in case of = 0.10 To verify the consistency of results for the saltwater numerical model of the field scale problem, a set of the dimensionless ratio values was sele cted as a regular set and a set of the dimensionless ratio values was obtained by d ecreasing the v alue of h ydraulic c onductivity by a factor of two as a v erified s et The results of the verified set were performed and compared to the results of regul ar set. The regular and verified sets f r om the field scale problem in case of = 0.10 The s altwater numerical model was verified to present the consistency of the results wh en decreasing t he v alue of h ydraulic c onductivity by a factor of two, but the h ydraulic c onductivity ratio = was still maintained at a constant value equal to 0.10. Also t he value of the dispersivity ratio = was held as a constant value equal to 0. 01. Aquifer parameters and dimensionless ratios in case o f = 0.10 Six of the dimensionless ratio values were select ed to be the regular set of investigation, namely = 0.1, 0.2, 0.5, 1.0, 4.0 and 5.0, which were the values of and equal to 50 m/day and equal to 5.0 m/day. For both t he uncoupled and coupled solutions the a quifer p arameters and the d imensionless r atios for the r egular s et in case of = 0. 10 are shown in Table 9 8. To obtain the verified set, the value of hydraulic conductivity from six of the dimensionless ratio values was decreased by a factor of two. That affect ed the value of the dimensionless ratio which was increased twice by comparing of the regular set. Thus, t he verified set consist ed of = 0.2, 0.4, 1.0, 2.0, 8.0 and 10.0, which were the values of and equal to 25 m/day and equal to 2.5 m/day For both the PAGE 290 290 uncoupled and coupled solutions the a quifer p arameters and the d imensionless r atios for the v erified s ets in case of = 0. 10 are shown in Table 9 9 Results of the regular and the verified s et s in case of = 0.10 The isochlor results of the regular set with concentration color floods and velocity vectors when th e hydraulic conductivity ratio ( ) = 0.10, the values of and = 50 m/day and = 5.0 m/day are shown in Table 9 10 and 9 11 and also in Figures 9 24 to 9 26 for the un coupled and coupled solutions, respectively The verified results were compared to the regular set of investigation, namely = 0.2, 0.4, 1.0, 2.0, 8.0 and 10.0 sinc e the comparison was based on the same value of dimensionless ratio The verified results with concentration color floods and velocity vectors for the field scale problem when the hydraulic conductivity ratio ( ) = 0. 10 are shown in Table 9 10 and 9 11 and also in Figures 9 27 to 9 29 for the un coupled and coupled solutions, respectively Upon compari ng the results between the regular and t he verified sets, it was indicate d that there was no significant difference in either the patterns or values of results in the case of the hydraulic conductivity ratio ( ) = 0.10. PAGE 291 291 Table 9 8 A quifer parameters and dimensionless ratios for six selected regular cases to verify the consistency of results for = 0.10. and and m/day m/day m/day m/day m/day m/day 0.1 50 5.0 0.10 2.5 0.1 625 125 1.25 0.01 0.2 50 5.0 0.10 5.0 0.2 1,250 125 1.25 0.01 0.5 50 5.0 0.10 12.5 0.5 3,125 125 1.25 0.01 1.0 50 5.0 0.10 25.0 1.0 6,250 125 1.25 0.01 4.0 50 5.0 0.10 100.0 4.0 25,000 125 1.25 0.01 5.0 50 5.0 0.10 125.0 5.0 31,250 125 1.25 0.01 Table 9 9 A quifer parameters and dimensionless ratios for six verified cases to verify the consistency of results when decreasing t he v alue of h ydraulic c onductivity by a factor of two for = 0.10. and and m/day m/day m/day m/day m/day m/day 0.2 25 2.5 0.10 2.5 0.2 625 125 1.25 0.01 0.4 25 2.5 0.10 5.0 0.4 1,250 125 1.25 0.01 1.0 25 2.5 0.10 12.5 1.0 3,125 125 1.25 0.01 2.0 25 2.5 0.10 25.0 2.0 6,250 125 1.25 0.01 8.0 25 2.5 0.10 100.0 8.0 25,000 125 1.25 0.01 10.0 25 2.5 0.10 125.0 10.0 31,250 125 1.25 0.01 PAGE 292 292 Table 9 10 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing t he v alue of h ydraulic c onductivity by a factor of two and the hydraulic conductivity ratio ( ) = 0.10 for uncoupled solutions. The regular set The verified set and and m/day m/day m/day m m/day m/day m/day m 0.2 50 5.0 2.5 630.581 3.15291 2.93317 25 2.5 2.5 630.581 3.15291 2.93317 0.4 50 5.0 5.0 481.505 2.40752 1.24690 25 2.5 5.0 481.505 2.40752 1.24690 1.0 50 5.0 12.5 350.528 1.75264 0.28109 25 2.5 12.5 350.528 1.75264 0.28109 2.0 50 5.0 25.0 275.785 1.37893 0.03143 25 2.5 25.0 275.784 1.37892 0.03143 8.0 50 5.0 100.0 147.044 0.73522 0.00000 25 2.5 100.0 146.950 0.73475 0.00000 10.0 50 5.0 125.0 146.346 0.73173 0.00000 25 2.5 125.0 146.194 0.73097 0.00000 Table 9 11 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing t he v alue of h ydraulic c onductivity by a factor of two and the hydraulic conductivity ratio ( ) = 0.1 0 for coupled solutions. The regular set The verified set and and m/day m/day m/day m m/day m/day m/day m 0.2 50 5.0 2.5 2,471.706 12.35853 0.36104 25 2.5 2.5 2,471.706 12.35853 0.36106 0.4 50 5.0 5.0 1,331.526 6.65763 0.19004 25 2.5 5.0 1,331.526 6.65763 0.19005 1.0 50 5.0 12.5 483.333 2.41667 0.06756 25 2.5 12.5 483.333 2.41667 0.06756 2.0 50 5.0 25.0 212.999 1.06500 0.00793 25 2.5 25.0 212.999 1.06500 0.00793 8.0 50 5.0 100.0 146.507 0.73254 0.00000 25 2.5 100.0 146.404 0.73202 0.00000 10.0 50 5.0 125.0 145.892 0.72946 0.00000 25 2.5 125.0 145.736 0.72868 0.00000 PAGE 293 293 A B Figur e 9 24 R esults of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0. 2 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.1 0 when comparing between Figure 9 24 and Figure 9 27. PAGE 294 294 A B Figur e 9 25 R esults of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.1 0 when comparing between Figure 9 25 and Figure 9 28. PAGE 295 295 A B Figur e 9 26 R esults of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.1 0 when comparing between Figure 9 26 and Figure 9 29. PAGE 296 296 A B Figure 9 27 R esults of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1 0 * There was no significant difference in patterns of results in the case of the hydraulic conduc tivity ratio ( ) = 0.1 0 when comparing between Figure 9 24 and Figure 9 27. PAGE 297 297 A B Figure 9 2 8 R esults of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1 0 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.1 0 when comparing between Figure 9 25 and Figure 9 28. PAGE 298 298 A B Figure 9 2 9 R esults of the veri fied set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 0.1 0 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 0.1 0 when comparing between Figure 9 26 and Figure 9 29. PAGE 299 299 The Verification of the Consistency of R esults for the Saltwater Numerical Model in case of = 1.00 To verify the consistency of results for the saltwater numerical model of the field scale problem, a set of the dimensionless ratio values was selected as a regular set and a set of the dimensionless ratio values was obtained by d ecreasing the v alue of h ydraulic c onductivity by a factor of two as a v erified s et The results of the verified set were performed and compared to the results of regular set. The regular and verified sets for the field scale problem in case of = 1.00 Saltwater numerical model was verified to present the consistency of the results wh en decreasing t he v alue of h ydraulic c onductivity by a factor of two but the h ydraulic c onductivity ratio = was still maintained at a constant value equal to 1.00. Also t he value of the dispersivity ratio = was held as a constant value equal to 0. 01. Aquifer parameters and dimensionless ratios in case of = 1.00 Six of the dimensionless ratio values were select ed to be the regular set of investigation, namely = 0.1, 0.2, 0.5, 1.0, 4.0 and 5.0, which were the values of and equal to 50 m/day and equal to 50 m/d ay. For both solutions, the aquifer parameters and the dimensionless ratios for the regular set in case of = 1.00 are shown in Table 9 12. To obtain the verified set, the value of hydraulic conductivity from six of the dimensionless ratio values was decreased by a factor of two. That affected the value of the dimensionless ratio which was increased twice by comparing of the regular set. Thus, t he verified set consist ed of = 0.2, 0.4, 1.0, 2.0, 8.0 and 10.0, which were the values of and equal to 25 m/day and equal to 25 m/day For both PAGE 300 300 solutions the a quifer p arameters and the d imensionless r atios for the v erified s ets in case of = 1.00 are shown in Table 9 13 Results of the regular and the verified s et s in case of = 1.00 The isochlor results of the regular set with concentration color floods and velocity vectors when th e hydraulic conductivity ratio ( ) = 1.00, the values of and = 50 m/day and = 5 0 m/day are shown in Table 9 1 0 and 9 11 and also in Figures 9 30 to 9 32 for the uncoupled and coupled solutions, respectively. The verified results were compared to the regular set of investigation, namely = 0.2, 0.4, 1.0, 2.0, 8.0 and 10.0 since the comparison was based on the same value of dimensionless ratio The verified results with concentration color floods and velocity vectors for the field scale problem when the hydraulic conductivity ratio ( ) = 1.00 are shown in Table 9 1 0 and 9 11 and also in Figures 9 33 to 9 35 for the uncoupled and coupled solutions, respectively. Upon compari ng the results between the regular and the verified sets, it was indicate d that there was no significant difference in either the patterns or values of results in the case of the hydraulic conductivity ratio ( ) = 1.00. PAGE 301 301 Table 9 12 A quifer parameters and dimensionless ratios for six selected regular cases to verify the consistency of results for = 1.00. and and m/day m/day m/day m/day m/day m/day 0.1 50 50 1.00 25 0.1 6,250 125 1.25 0.01 0.2 50 50 1.00 50 0.2 12,500 125 1.25 0.01 0.5 50 50 1.00 125 0.5 31,250 125 1.25 0.01 1.0 50 50 1.00 250 1.0 62,500 125 1.25 0.01 4.0 50 50 1.00 1,000 4.0 250,000 125 1.25 0.01 5.0 50 50 1.00 1,250 5.0 312,500 125 1.25 0.01 Table 9 1 3 A quifer parameters and dimensionless ratios for six verified cases to verify the consistency of results when decreasing t he v alue of h ydraulic c onductivity by a factor of two for = 1.00. and and m/day m/day m/day m/day m/day m/day 0.2 25 25 1.00 25 0.2 6,250 125 1.25 0.01 0.4 25 25 1.00 50 0.4 12,500 125 1.25 0.01 1.0 25 25 1.00 125 1.0 31,250 125 1.25 0.01 2.0 25 25 1.00 250 2.0 62,500 125 1.25 0.01 8.0 25 25 1.00 1,000 8.0 250,000 125 1.25 0.01 10.0 25 25 1.00 1,250 10.0 312,500 125 1.25 0.01 PAGE 302 302 Table 9 1 4 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing t he v alue of h ydraulic c onductivity by a factor of two and the hydraulic conductivity ratio ( ) = 1.00 for u ncoupled solutions. The regular set The verified set and and m/day m/day m/day m m/day m/day m/day m 0.2 50 50 25 272.767 1.36384 0.10433 25 25 25 272.896 1.36448 0.10433 0.4 50 50 50 144.576 0.72288 0.00000 25 25 50 144.614 0.72307 0.00000 1.0 50 50 125 140.716 0.70358 0.00000 25 25 125 140.607 0.70304 0.00000 2.0 50 50 250 144.323 0.72161 0.00000 25 25 250 143.981 0.71990 0.00000 8.0 50 50 1,000 147.049 0.73524 0.00000 25 25 1,000 145.358 0.72679 0.00000 10.0 50 50 1,250 147.356 0.73678 0.00000 25 25 1,250 145.736 0.72868 0.00000 Table 9 1 5 Comparison of extent of saltwater intrusion ( ) and recirculation of seawater ( ) when decreasing t he v alue of h ydraulic c onductivity by a factor of two and the hydraulic conductivity ratio ( ) = 1.00 for coupled solutions. The regular set The verified set and and m/day m/day m/day m m/day m/day m/day m 0.2 50 50 25 277.390 1.38695 0.02758 25 25 25 277.390 1.38695 0.02758 0.4 50 50 50 155.230 0.77615 0.00000 25 25 50 155.235 0.77618 0.00000 1.0 50 50 125 147.351 0.73676 0.00000 25 25 125 147.227 0.73614 0.00000 2.0 50 50 250 145.726 0.72863 0.00000 25 25 250 145.373 0.72686 0.00000 8.0 50 50 1,000 147.039 0.73520 0.00000 25 25 1,000 145.344 0.72672 0.00000 10.0 50 50 1,250 147.565 0.73782 0.00000 25 25 1,250 145.707 0.72854 0.00000 PAGE 303 303 A B Figur e 9 30 R esults of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0. 2 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00 * There was no significant difference in patterns of results in the case of the hyd raulic conductivity ratio ( ) = 1.00 when comparing between Figure 9 30 and Figure 9 33. PAGE 304 304 A B Figur e 9 31 R esults of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 1.00 when comparing between Figure 9 31 and Figure 9 34. PAGE 305 305 A B Figur e 9 32 R esults of the regular set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.00 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 1.00 when comparing between Figure 9 32 and Figure 9 35. PAGE 306 306 A B Figure 9 33 R esults of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.0 0 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 1.00 when comparing between Figure 9 30 and Figure 9 33. PAGE 307 307 A B Figure 9 3 4 R esults of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 1.0 0 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 1.00 when comparing between Fig ure 9 31 and Figure 9 34. PAGE 308 308 A B Figure 9 3 5 R esults of the verified set A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 for the field scale problem for the hydraulic conductivity ratio ( ) = 1. 0 0 * There was no significant difference in patterns of results in the case of the hydraulic conductivity ratio ( ) = 1.00 when comparing between Figure 9 32 and Figure 9 35. PAGE 309 309 Discussion and Conclusions for the Field s cale Problem The results of the numerical experiments indicate that t he two dimensionless dependent variables, that is, the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ), are function s ve independent dimensionless ratios : the aspect ratio the dimensionless ratio the dimensionless ratio the dispersivity ratio and the hydraulic conductivity ratio As the dimensionles s ratio ( ) was increased from 0.1 to 10.0, the 0.5 isochlor s decreased significantly (i.e., they move seaward ) as shown in Figures 9 3 to 9 8 In terms of the extent of saltwater intrusion ( ), when comparing the work of Motz and Sedighi (2013) as shown in Figures 9 1 (A), to the results of = 0.01 and = 0.01, as shown in Figures 9 9 (A), it is evident that changing the aspect ratio from = 62.5 in Motz and Sedighi (2013) to = 100 in t he present investigation eliminates the boundary effect due to the domain size. For the recirculation of seawater ( ), when comparing the work of Motz and Sedighi (2013) as shown in Figures 9 1 (B), to the results of = 0.01 and = 0.01, as shown in Figures 9 9 ( B), it is evident that the domain size does not affect the results since the calculation method, that is a third order total variation diminishing (TVD) scheme is mass conservative When considering the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) for the uncoupled and coupled solutions versus the dimensionless ratio from 0.1 to 10.0 for the dispersivity ratio = 0.01 and the hydraulic conductivity ratio = 0.01 0.10, and 1.00, the results demonstrate that, as shown in Figure s 9 9 through 9 17, t he extent of saltwater PAGE 310 310 intrusion ( ) for the uncoupled solutions i s less than the extent of saltwater intrusion ( ) for the coupled solutions. However, the recirc ulation of seawater ( ) for the uncoupled solutions i s greater than the recirculation of seawater ( ) for the coupled solutions The extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) are significantly decreased as the dimensionless ratio i s increased The results of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) also indicate that significant differences can occur between the uncoupled and coupled solutions, especial ly at smaller values of the dimensionless ratio At smaller values of the dimensionless ratio 0.1, where the density driven vertical buoyancy flux ( ) dominates the freshwater advective flux ( ), the extent of saltwater intrusion ( ) obtained from the uncoupled solutions is less than that from the coupled solutions. Conversely, the degree of saltwater recirculation ( ) that is derived from the uncoupled solutions is greater than that from the coupled solutions. At larger values of the dimensionless ratio 10.0, where the freshwater advective flux ( ) dominates the driven vertical buoyancy flux ( ), the extent of saltwater intrusion ( ) simulated for the uncoupled solution becomes equal to that for the coupled s olutions. As can be seen, the degree of saltwater recirculation ( ) for the uncoupled solution asymptotically approaches the corresponding values for the coupled solutions. Results of the extent of saltwater intrusion ( ) and the recirculation of seawater ( ) obtained by numerical experiments for the original and the verified sets, represent the consistency of results as shown in Tables 9 4 to 9 15 and Figures 9 18 to 9 35 F or = 0.01 and = 0.10, the aquifer parameters and the dim ensionless ratios are shown in Table s 9 4 and 9 5 and t he results are shown in Table s 9 6 to 9 7 and in PAGE 311 311 Figure s 9 18 to 9 23. F or = 0.01 and = 0.10, the aquifer parameters and the dimensionless ratios are shown in Table s 9 8 and 9 9 and t he re sults are shown in Table s 9 10 to 9 11 and in Figure s 9 24 to 9 29. F or = 0.01 and = 1.00, the aquifer parameters and the dimensionless ratios are shown in Table s 9 12 and 9 13 and t he results are shown in Table s 9 14 to 9 15 and in Figure s 9 3 0 to 9 35. When th e hydraulic conductivity ratio ( ) = 1.00, t he result s of the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) for the uncoupled and coupled solution s are the lower bound of the investigations. Based on the Courant number ( ) and the LST output files all solut ions present the accuracy and stability of numerical calculation To obtain the most accuracy, the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) at the coastal boundary should be determined in a coupled variable density flow and transport code for a reg ional groundwater model which i s the same assumptions as the field scale problem If an uncoupled variable density flow and transport code is applie d to a re gional groundwater model, which i s the same assumptions as the field scale problem, it is based on these conditions to obtain the accurate results and t o obtain the efficient accuracy of both the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ), for the dispersivity ratio ( ) = 0.01 In this case, t he dimensionless ratio should be greater than 5.0 when the hydraulic conductivity ratio ( ) = 0.01 and should be greater than 1.0 when the hydraulic cond uctivity ratio ( ) = 0.10 The results should be satisf ied with any value of the dimensionless ratio when the hydraulic conductivity ratio ( ) = 1.00 PAGE 312 312 Results from the field scale problem can be applied to estimate the values of the extent of s altwater intrusion ( ) and recirculation of seawater ( ) in the case of a real coastal aquifer since the basic a quifer p arameters would be required to determine the di mensionless r atios and results on the charts. PAGE 313 313 CHAPTER 10 CONCLUSIONS OF INVESTIGATIONS Summary of Contributions In this dissertation, numerical investigations were conducted to simulate three saltwater intrusion problems to illustrate that t he consistency of the results from the three saltwater intrusion problems namely th e Henry constant dispersion problem the Henry velocity dependent dispersion problem and the field scale problem, indicat e that dimensional analysis can be applied to solve the saltwater intrusion problem Th e saltwater intrusion problems with their benc hmark problems were investigated using a numerical code ( i.e. SEAWAT Version 4 ) in two solutions : the uncoupled solution (the less computationally intensive method) and the coupled solution (the more computationally intensive method). The uncoupled solut ions re present constant density flow and transport in a domain, and the coupled solutions re present the density dependent flow and transport in a domain. The investigation wa s undertaken to determine quantitatively how saltwater intrusion and the recircula tion of seawater are associated with the freshwater advective flux ( ), the density driven vertical buoyancy flux ( ), the dispersion coefficient ( ), the molecular diffusion coefficient ( ) the aspect ratio = the dimensionless ratio = the dimensionless ratio = the dispersivity ratio and the hydraulic conductivity ratio The results of the three saltwater intrusion problems are as follows. Including the dimensionless ratio in c ase of a homogeneous and isotropic domain ( the Henry constant dispersion problem ) PAGE 314 314 T he results of the first problem namely the Henry constant dispersion problem and the benchmark problem, that is Simpson and Clement (2004) demonstrate c onstant dispersion in the flow and transport numerical model which is similar to T he dimensionless ratio = 0.3 of the saltwater intrusions ( ) results is considered as a balance point between the fluxes and convections of the five curve family for both the uncoupled and coupled solution s. Moreover, it is also considered as a separation point of the five uncoupled solution curve family. At smaller values of the dimensionless ratio 0.1, there are significant difference s i n the results for both the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) between the uncoupled and coupled solutions At larger values of the dimensionless ratio 10.0, the uncoupled solution will yield approximat ely the same results for both the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) as does the coupled solution T he uncoupled constant density flow and transport code may be an alternative to determine the results a t larger values of the dimensionless ratio instead of a coupled variable density flow and transport code. The results of the second problem namely the Henry velocity dependent dispersion problem and the benchmark problem s that is Sgol (1993) and Abarca et al. (2007) indicate that the results of (1964) problem i n the works of Sgol (1993) and Abarca e t al. (2007) including Abarca et al. (2007) reference cases are t he coupled solutions and present the velocity dependent dispersion with the varied concentration at the downgradient boundary in the flow and transport numerical model However, the Henry velocity dependent dispersion problem i s performed for both the PAGE 315 315 uncoupled and coupled solutions with the constant concentration at the downgradient boundary of the medium based on the hydraulic conductivity ratio ( ) and the dispersiv ity ratio ( ). The work of Gelhar et al. (1992) is applied to determine the dispersivity in the longitudinal direction ( ). The Henry velocity dependent dispersion problem is investigated by using the hydraulic conductivity ratio ( ) = 0.01, 0.1 and 1.00, and also the dispersivity ratio ( ) = 0.01, 0.10, and 1.00. It cannot be concluded from t he results that the uncoupled constant densit y flow and transport code can yield results without difference s in values from the coupled density depende nt flow and transport approach, since some of results are affected by the boundary due to the domain size A t large r value of the dimensionless ratio however, the uncoupled solutions yield approximately the same results as the coupled solutions for both the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ). The results of the third problem namely the field scale problem and the benchmark problem, that is Motz and Sedighi (2013) present the velocity dependent disper sion with the constant concentration at the downgradient boundary based on the data from the upper part of the Floridan aquifer system at Hilton Head Island in South Carolina, U.S.A. (Bush 1988) The field scale problem wa s performed using the varied hydr aulic conductivity ratio ( ) = 0.01, 0.1 and 1.00 with the dispersivity ratio ( ) held at a constant value = 0.01 A t smaller values of the dimensionless ratio 0.1, there are s ignificant difference s of results for both the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) between the uncoupled and coupled solutions At larger values of the dimensionless ratio 10.0, the uncoupled solution s yield approximately the same results for both the extent of salt water intrusion ( ) and the PAGE 316 316 degree of saltwater recirculation ( ) as do the coupled solution s. T he uncoupled constant density flow and transport code may be an alternative to determine the results instead of a coupled variable density flow and trans port code If an uncoupled variable density flow and transport code are applied to a regional groundwater model w ith the same assumptions as in the field scale problem t o obtain sufficient accuracy of both the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ), the conditions would depend on t he dispersivity ratio ( ) being held at a constant value = 0.01 In this case, t he dimensionless ratio should be greater than 5.0 for the hydraulic conductivity ratio ( ) = 0.01; t he dimensionless ratio should be greater than 1.00 for the hydraulic conductivity ratio ( ) = 0.10; and t he results should be satisf ied with any value of the dimensionless ratio for the hydraulic conductivity ratio ( ) = 1.00. R esults from the field scale model can be applied to estimate the values of the extent of saltwater intrusion ( ) and recirculation of seawater ( ) in case of a real coastal aquifer since the basic a quifer p arameters would be required to determine the di mensionless r atios and results on the charts. For three problems, t o obtain the most accuracy, the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) at the coastal boundary should be determined in a coupled variable d ensity flow and transport code for a model. However, the computational costs of the uncoupled solution are less than the costs of the coupled solution. The third order total variation diminishing (TVD) scheme is used to solve a problem by using a small sufficient Courant number ( ). It is un necessary to use a very small Courant number because it needs more computational costs to perform a simulation, and there is no significan t difference in results between using a sufficient ly PAGE 317 317 small Courant number a nd a very small Courant number. However, the Courant number should be less than 1.0 if the problem is advection dominated (Delleur 2006; Zheng and Wang 1999) The accuracy result depends on grid discretization, variables and parameters of numerical method. The use of small grid discretization a ffects other variables and parameters which need to be adjusted to make them suitable for solving the problem. The small grid discretization does not guarante e obtaining a satisfactory result, and it is sometime too challeng ing to determine a suitable set of all variables and parameters for solving the problem. Additionally the small grid discretization also has more computational costs than the l arge grid discretization. Opportunities of Future Work From the summary of contributions, some opportunities for future work are suggested Results obtained from the field scale problem should be verified by actual field scale data to determine the result s and the accuracy of results, i.e., the extent of saltwater intrusion ( ) and the degree of saltwater recirculation ( ) for both the uncoupled and coupled solutions The minimum dimensionless ratio = of the field scale problem i s equal to 0.1 0, which seems to be a small value but in some places the dimensionless ratio = c ould be less than 0.1 0 It should be considered to investigate this for improving the results The dispersivity ratio ( ) and the hydraulic conductivity ratio ( ) of the field scale problem can be extended into the large spans of value s, which is similar to the Henry velocity dependent dispersion problem in order to investigate the results PAGE 318 318 Since the field scale problem assumes a rectang l e to be the shape of the domain to perform the calculations, i t migh t be better to investigate the solutions by using the other shapes of domains based on actual field scale geology. PAGE 319 319 APPENDIX A BENCHMARK AND HENRY CONSTANT DISPERSION PROBLEM S Benchmark Problem Simpson and Clement (2004) was selected as a benchmark problem for the Henry constant dispersion problem since Simpson and Clement (2004) reproduced results of the un coupled and coupled solutions of (1964) that is standard Henry problem and created modified Henry problem by using a Galerkin finite element numeric al solution and a semi analytical solution as shown in Figure A 1 (Simpson and Clement 2004) The standard Henry (1964) problem, except improving the methods and solutions, and variables were similar to Guo and Langevin (200 2 ) The modified Henry problem was almost the same as the standard Henry problem, except the freshwater inflow per unit width ( ). The freshwater inflow per unit width ( ) was reduced by 50% from the standard Henry problem, i.e., from 5.702 to 2.851 m 2 /day which increase d the influence of density dependent effects. Optimal Finite Difference Grid In numerical calculation s the f inite d ifference grid is one of major parameter s to determine a solution. To determine the o ptimal f inite d ifference grid for the Henry constant di spersion p roblem SEAWAT results were verified by comparing to the results obtained by Simpson and Clement (2004) In Figure A 1 ( A) the comparison of n umerical and semi analytical results for the coupled standard Henry saltwater intrusion problem = 0.263, = 0.1 and = 2.0 are presented The uncoupled 50% numerical isochlor s (dashed) was also shown. In Figure A 1 ( B) the comparison of numerical and semi analytical results for the coupled PAGE 320 320 modified Henry saltwater intrusion problem = 0.1315, = 0.2 and = 2.0 which Simpson and Clement (2004) created are presented The uncoupled 50% numerical isochlors (dashed) was also shown in Figure A 1 Langevin and Guo (2006) obtain ed (1964) standard and modified problems by using SEAWAT 2000 as shown in Figure A 2. The result s were in quite good agreement with the work of Simpson and Clement (2004) The comparison might be similar to Motz and Sedighi (2013) Motz and Sedighi (2013) perform ed simulations using SEAWAT Version 4 to obtain the uncoupled and coupled solution (1964) standard and modified problems which compared quite favorably with the work of Simpson and Clement (2004) as shown in Figure A 3 (Motz and Sedighi 2013) Finite difference Grid Optimization To obtain the best discretization for performing the solutions, the three dimensio nal model was discretized into three alternatives, as shown in Table A 1 and Figure A 4. For all alternatives, the total depth was equal to 1.0 m and the width of the row perpen dicular to the direction of flow in the cross section ( ) was arbitrarily set equal to 1.0 m. To simplify the numeri cal model, the finite difference grid and b oundary conditions can be represented in a two dimensional cross section, as shown in Figure A 5. All three alternatives, the last column, i.e., the 21 st 41 st and 81 st was assigned to more precisely locate the saltwater hydrostatic heads at a distance of 2.0 m since the Block Center Flow (BCF) package was applied in calculating. In case of Alternate I 0.1 it wa s similar to the work of Motz and Sedighi (2013) PAGE 321 321 A B Figure A 1 R esults obtained by Simpson and Clement (2004) results A) the standard and B) the modified Henry problems. A B Figure A 2 The work of A) the standard and B) the modified Henry problems. A B Figure A 3 R esults from Motz and Sedighi (2013) work A) the standard and B) the modified Henry problems. PAGE 322 322 Table A 1 Alternative finite difference grids used for for the optimal finite difference grid A lternative Column Row Layer Total dimension No. (m) No. (m) No. (m) X Y Z (m) I 0.1 21 0.100 1 1.0 10 0.100 2.100 1 1 II 0.05 41 0.050 1 1.0 20 0.050 2.050 1 1 III 0.025 81 0.025 1 1.0 40 0.025 2.025 1 1 Figure A 4 Alternative finite (1964) problem. PAGE 323 323 Boundary Conditions and Solu tions F or both the uncoupled and coupled solutions the upgradient inland boundary on the left hand side of the domain was specified as a constant flux boundary with the concentration of TDS equal to zero, representing the freshwater inflow The downgradi ent coastal boundary on the right hand side of the domain was simulated by specifying a concentration boundary condition with a concentration of TDS equal to 35 kg/m 3 representing the concentration of TDS in seawater, as shown in Figures A 4 and A 5. For the uncoupled solutions, the equivalent freshwater heads were calculated over the depth of the cross section at the coastal boundary Conversely f or the coupled solutions, equivalent freshwater heads were c alculated internally in SEAWAT. The boundary cond itions for the Henry constant dispersion problems are shown in Figure A 5. Figure A 5 B oundary conditions (1964) problem. PAGE 324 324 Aquifer Parameters The domain was assigned to be an isotropic, homogeneous, confined aquifer Thus, the horizonta l and vertical hydraulic conductivities ( ) were equal to 864 m/day, and the porosity ( ) was equal to 0.35 The horizontal and vertical dispersivities ( and ) were set equal to zero The molecular diffusion coefficient ( ) was assigned to represent all of the effects of the dispersion coefficient ( ) and set equal to 1.62925 m 2 /day (Gu o and Langevin. 2002) which represented the constant dispersion (1964) problem The freshwater inflow s per unit width ( ) of standard and modified Henry problems were 5.702 and 2.851 m 2 /day, respectively (Simpson and Clement 2004) Methods and Schemes Solutions w ere obtained by running SEAWAT Version 4 (Langevin et al. 2008) to steady state The f low and transport components in SEAWAT were solved for both the uncoupled and coupled solutions with identical solution procedures. The f low model was solved using the Pre Conditioned Conjugate Gradient (PCG2) package The head change criterion for convergence (HCLOSE) and residual criterion for convergence (RCLOSE) were set at 1x10 6 m and 1 m 3 /d ay respectively. The advective term of the solute transport equation was solved using the implicit finite difference with the central in space weighting scheme The remaining terms (dispersion, source/sink mixing and reaction) were solved using the iterative implicit Generalized Conjugate Gradient (GCG) solver with the Modified Incomplete Cholesky (MIC) pre conditioner The convergence criterion in terms of relative concentration (CCLOSE) was set equal to 1x10 6 The transport time step size within each tim e step PAGE 325 325 of the flow solution (DT0) was assigned a value of 1x10 4 days, and a timestep multiplier (TTSMULT) of 1.2 was used to increase subsequent timesteps Variable Density Flow package options were applied with the first transport time step (FIRSTDT) 1x 10 4 days. Each solution was run in the transient mode for the 2 day simulation period required 46 timesteps The o utput file s, i.e., the LST files, w ere checked to ensure that steady state was reached. The uncoupled flow and transport option in SEAWAT wa s us ed to represent constant density flow and transport for the uncoupled solutions, and coupled flow and transport option was us ed to represent density dependent flow and transport for the coupled solutions. Groundwater Vistas Version 5, a user friendly Windows based graphical interface was used to perform the solutions obtained using SEAWAT Solutions of the uncoupled and coupled solutions (1964) standard and modified problems Results of Alternate Finite difference Grids Isochlors and velocity distribution Result s for all alternatives were presented by using the isochlors from 0.1 to 0.9 and velocity distribution in the domain as shown in Figures A 6 through A 11. The location and direction of the velocity vectors where the fluid stagnates and recirculates corre spond well with the shape of the isochlors for both uncoupled and coupled solutions Moreover the velocities near the outflow region are relatively large, as expected since this is a region of convergent flow as Simpson and Clement (2004) mentioned. PAGE 326 326 1. Results of Isochlors and Velocity Distribution for Alternative I 0.1 The isochlors with concentration color floods and velocity distribution in the dom ain for alternative I 0.1 are shown in Figures A 6 to A 7. A B Figure A 6 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative I 0.1 results for standard Henry problem A) uncoupled and B) coupled solution s A B Figure A 7 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative I 0.1 results fo r modified Henry problems A) uncoupled and B) coupled solution s The r esults of isochlors for alternative I 0.1 presented that the isochlors were stable for both uncoupled and coupled solutions of standard and modified Henry problem s PAGE 327 327 2. Results of Isochlors and Velocity Distribution for Alternative II 0.05 The isochlors with concentration color floods and velocity distribution in the domain for alternative I I 0. 05 are shown in Figures A 8 to A 9. A B Figure A 8 Isochlo rs from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative II 0.05 results for standard Henry problem A) uncoupled and B) coupled solution s A B Figure A 9 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative II 0.05 results for modified Henry problems A) uncoupled and B) coupled solution s The r esults of isochlors for alternative I I 0. 05 presented that the isochlors remained stable for both the uncoupled and coupled solutions of the standard and modified Henry problem s PAGE 328 328 3. Results of Isochlors and Velocity Distribution for Alternative III 0.025 Based on the boundary conditions and solu tions as mentioned Methods and Schemes the alternative III 0.025 numerical model could not be calculated to obtain the steady state results due to i nappropriate parameter values with the f inite difference g rids Thus, alternative III 0.025 model need ed to adjust some criteria and parameters so as to perform the steady state results as follow s : a) The convergence criterion in terms of relative concentration (CCLOSE) was set equal to 1x10 4 which was 10 0 times greater than the previous CCLOSE ; b) The transport time step size within each time step of the flow solution (DT0) was assigned a value of 1x10 12 days, which was 1x10 8 ti mes less than the previous DT0; and c) The timestep multiplier (TTSMULT) was assigned a value of 1.2 Each solution was run in the transient mode for the 2 day simulation period for both the uncoupled and coupled solutions which required 147 timesteps The r esults of alternative III 0.025 presented that the i sochlors of standard Henry problem presented that there was so me instability of i sochlors as shown in Figures A 10, and the results of modified Henry problem showed plenty of irregularity and variation of isochlor, as shown in Figures A 11. Therefore, this indicates that alternative III 0.025 could not perform the accurate results for both the uncoupled and coupled solutions of the standard and modified Henry problems PAGE 329 329 A B Figure A 1 0 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative III 0.025 results for standard Henry problem A) uncoupled and B) coupled solution s A B Figure A 1 1 Isochlors from 0.1 to 0.9 with concentration color floods and velocity distribution in the domain of SEAWAT alternative III 0.025 results for modified Henry problem A) uncoupled and B) coupled solution s Instabilities and oscillations All of the results indicated some instabilities and o scillations which is attributed to relatively large velocities near the outflow region (Langevin and Gou 2006). The magnitudes of velocities are presented in terms of contour lines (unit: m/day) as showed in Figure A 12 through A 17. T he velocities near the outflow region ar e relatively large, as expected since this is a region of convergent flow as Simpson and Clement (2004) mentioned PAGE 330 330 1. Results of Velocity Contour Lines for Alternative I 0.1 The results of velocity contour lines in the domain for alternative I 0.1 are shown in Figures A 12 to A 13. A B Figure A 12 The v elocity contour lines (unit: m/day) in the domain from SEAWAT alternative I 0.1 results for standard Henry problem A) uncoupled and B) coupled solution s A B Figure A 13 The v elocity contour lines (unit: m/day) in the domain from SEAWAT alternative I 0.1 results for modified Henry problem A) uncoupled and B) coupled so lution s The r esults of alternative I 0.1 presented velocity contour lines that were stable for both the standard and modified Henry problem. T he velocities near the outflow region are relatively large PAGE 331 331 2. Results of Velocity Contour Lines for Alterna tive II 0.05 The results of velocity contour lines in the domain for alternative II 0.05 are shown in Figures A 14 to A 15. A B Figure A 14 The v elocity contour lines (unit: m/day) in the domain from SEAWAT alternative I I 0. 05 results for standard Henry problem A) uncoupled and B) coupled solution s A B Figure A 15 The v elocity contour lines (unit: m/day) in the domain from SEAWAT alternative I I 0. 05 results for modified Henry problem A) uncoupled and B) coupled solution s The r esults of alternative I I 0. 05 indicate that the velocity contour lines remained stable for both the standard and modified Henry problem. T he velocities near the outflow region are also relatively large The distribution of velocity contour lines of alternative I I 0. 05 is similar to the results of alternative I 0. 10. PAGE 332 332 3. Results of Velocity Contour Lines for Alternative III 0.025 The results of velocity contour lines in the domain for alternative I II 0.0 2 5 are shown in Figures A 16 to A 17. A B Figure A 16 The v elocity contour lines (unit: m/day) in the domain from SEAWAT alternative I II 0. 025 results for standard Henry problem A) uncoupled and B) coupled solution s A B Figure A 17 The v elocity contour lines (unit: m/day) in the domain from SEAWAT alternative I II 0. 025 results for modified Henry problem A) uncoupled and B) coupled solution s For the coupled solutions both the standard and mo dified Henry problem, velocity contour lines presented some instability Especially the velocity contour lines of modified Henry problem for the uncoupled solution were different from the first two a lternative s of the modified Henry problem. PAGE 333 333 Transient 0.5 isochlors movement at the base of the domain The t ransient 0.5 isochlors m ovement at the b ase of the d omain was used to test that the simulation reached steady state, which is the equilibrium position of t he transient 0.5 isochlors T he saltwater intrusion interface ( ) of 0.5 isochlors is used to represent the t ransient 0.5 isochlors m ovement at the b ase of the d omain i.e., the 10 th 20 th and 40 th layer respectively as shown in Figures A 18 through A 20. 1. Results of Transient 0.5 Isoch lors Movement at the Base of the Domain (The 10 th Layer) f or Alternative I 0.1 The results of alternative I 0.1 presented the transient position of the intersection of the 0.5 isochlors with the base of the domain stable for both the standard and modif ied Henry problem, as shown in Figure A 18. A B Figure A 1 8 Transient position of the intersection of the 0.5 isochlors with the base of the domain of SEAWAT alternative I 0.1 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. The transient 0.5 isochlors movement demonstrated that the 0.5 isochlor movement at the initial time period was so fast with the short time steps when compar ed to the late time period. After 0.5 day sim ulation period, the 0.5 isochlor movement finally converged on accurate distances and steady state PAGE 334 334 2. Results of T ransient 0.5 Isochlors M ovement at the B ase of the D omain (The 2 0 th Layer ) for Alternative I I 0. 05 The results of alternative II 0.05 presented the transient position of the intersection of the 0.5 isochlors with the base of the domain remained stable for both the standard and modified Henry problem, as shown in Figure A 1 9 A B Figure A 19 Transient position of the intersection of the 0.5 isochlors with the base of the domain of SEAWAT alternative I I 0. 05 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. The transient 0.5 isochlors movement demonstrated that t he 0.5 isochlor movement at the initial time period was so fast with the short time steps when compar ed to the late time period. After 0.5 day simulation period, the 0.5 isochlor movement finally converged on accurate distances and steady state 3. Result s of Transient 0.5 Isochlors Movement at the Base of the Domain (The 40 th Layer) f or Alternative III 0.025 The results of alternative I II 0.0 2 5 presented the transient position of the intersection of the 0.5 isochlors with the base of the domain un stable for both the standard and modified Henry problem, as shown in Figure A 20 The transient 0.5 isochlors movement demonstrated that the 0.5 isochlor movement at the initial time period was unstable and less stable than the first two alternatives of fi nite difference grids. PAGE 335 335 A B Figure A 2 0 Transient position of the intersection of the 0.5 isochlors with the base of the domain of SEAWAT alternative I II 0. 025 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. Changing of the concentration of total dissolved solids (tds) at the edge of the domain The other way to present that simulations reached the steady was the c hanging of the concentration of total dissolved solids (TDS) at the e dge of the d omain Results for all alternatives were presented by using the c hanging of the TDS at the b ase of the domain as shown in Figures A 21 through A 23. 1. Results of the C oncentration of T otal D issolved S olids (TDS) at the 20 th C olumn an d the 10 th L ayer ( 1.95 m, 0.0 5 m) for Alternative I 0.1 The results of alternative I 0.1 presented t the TDS at the edge of the domain were stable for both the standard and modified Henry problem s as shown in Figure A 21 PAGE 336 336 A B Figure A 21 C hanging of the TDS at the edge of the domain for SEAWAT alternative I 0.1 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. The TDS at the edge of the domain was v aried at the initial time period and increased so fast with the short time steps to 35 kg/m 3 when comparing to the late time period. After 0.05 day simulation period, the TDS finally converged to 35 kg/m 3 and remain steady state. 2. Results of the Concentration of Total Dissolved Solids (TDS) at the 40 th Column and the 20 th Layer ( 1.975 m, 0.025 m) for Alternative II 0.05 The results of alternative I I 0. 05 presented the TDS at the edge of the domain were stable for both the standard and modified Henry problem, as shown in Figure A 22 A B Figure A 2 2 C hanging of the TDS at the edge of the domain for SEAWAT alternative I I 0. 05 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. PAGE 337 337 The TDS at the edge of the domain was varied at the initial time period and increased so fast with the short time steps to 35 kg/m 3 when comparing to the late time period. After 0.05 day simulation period, the TDS finally converged to 35 kg/m 3 and remain steady state. 3. Results of the Concentration of Total Dissolved Solids (TDS) at the 80 th Column and the 40 th Layer ( 1.9875 m, 0.0125 m) for Alternative III 0.025 The results of alternative I II 0. 025 presented t the TDS at the edge of the domain were stable for both the standard and modified Henry problem, as shown in Figure A 23 A B Figure A 2 3 C hanging of the TDS at the edge of the domain for SEAWAT alternative III 0.025 results for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. The TDS at the edge of the domain was varied at the initial time period and increased so fast with the short time steps to 35 kg/m 3 when compa ring to the late time period. After 0.05 day simulation period, the TDS finally converged to 35 kg/m 3 and remain steady state. PAGE 338 338 Compared alternative results to benchmark problems in terms of the 0.5 isochlors 1. Results of Comparison to Benchmark Problem for Alternative I 0.1 Comparison s of the a lternative I 0.1 results to b enchmark p roblem i.e., Simpson and Clement (2004) demonstrate in Figure A 24 by using 0.5 isochlors. A B Figure A 24 Comparison the 0.5 isochlors of SEAWAT alternative I 0.1 results and Simpson and Clement (2004) solutions for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. The a lternative I 0.1 results compared quite favorably with the previous results obtained by Simpson and Clement (2004) as shown in Figure A 1. Minor instabilities occurred near the outflow part of the coastal boundary. 2. Results of Comparison to Benchmark Problem for Alternative II 0. 05 Comparison s of the a lternative I I 0. 05 results to b enchmark p roblem i.e., Simpson and Clement (2004) demonstrate in Figure A 25 by using 0.5 isochlors. The a lternative I I 0. 05 results compared more closely with the results previously obtained by Simpson and Clement (2004) as shown in Figure A 1. Minor instabilities also occurred near the outflow part of the coastal boundary which is similar to the results of a lternative I 0. 1. PAGE 339 339 A B Figure A 2 5 Comparison t he 0.5 isochlors of SEAWAT alternative II 0.05 results and Simpson and Clement (2004) solutions for A) the standard and B) the modified Henry problems both uncoupled and coupled solutions. 3. Results of Comparison to Benchmark Problem for Alternative III 0.025 Comparison s of the a lternative I II 0. 025 results to b enchmark p roblem i.e., Simpson and Clement (2004) demonstrate in Figure A 26 by using 0.5 isochlors. A B Figure A 2 6 Comparison the 0.5 isochlors of SEAWAT alternative III 0.025 results and Simpson and Clement (2004) solutions for A) the standard and B) the modified Henry problems b oth uncoupled and coupled solutions. The a lternative I I I 0. 025 results were in a fairly agreement with the results previously obtaine d by Simpson and Clement (2004) but there was amount of instability and variation in solutions especially modified Henry problem uncoupled solution. Optimal Finite Difference Discretization As illustrated above, when comparing all results to Simpson and Clement (2004) the results from alternative II 0.05 w ere the optimal finite difference discretization to perfor m solutions for the Henry constant dispersion problem. PAGE 340 340 Aquifer Parameters and Dimensionless ratios for the First Set of I nvestigations of the Henry Constant Dispersion Problem Table A 2 Aquifer parameters and dimensionless ratios in case of = 0.1 for the first set of investigations. Molecular diffusion coefficient ( ) m 2 /day m 2 /day 0.1 2.16 0.10 0.617142857 0.2 4.32 0.10 1.234285714 0.3 6.48 0.10 1.851428571 0.4 8.64 0.10 2.468571429 0.5 10.8 0 0.10 3.085714286 0.6 12.96 0.10 3.702857143 0.7 15.12 0.10 4.32 0000000 0.8 17.28 0.10 4.937142857 0.9 19.44 0.10 5.554285714 1 .0 21.6 0 0.10 6.171428571 2 .0 43.2 0 0.10 12.34285714 0 3 .0 64.8 0 0.10 18.51428571 0 4 .0 86.4 0 0.10 24.68571429 0 5 .0 108 .00 0.10 30.85714286 0 6 .0 129.6 0 0.10 37.02857143 0 7 .0 151.2 0 0.10 43.2 00000000 8 .0 172.8 0 0.10 49.37142857 0 9 .0 194.4 0 0.10 55.54285714 0 10 .0 216 .00 0.10 61.71428571 0 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025; 3. Isotropic hydraulic conductivity ( ) = 864 m/day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 341 341 Table A 3 Aquifer parameters and dimensionless ratios in case of = 0.2 for the first set of investigations. Molecular diffusion coefficient ( ) m 2 /day m 2 /day 0.1 2.16 0.20 1.234285714 0.2 4.32 0.20 2.468571429 0.3 6.48 0.20 3.702857143 0.4 8.64 0.20 4.937142857 0.5 10.8 0 0.20 6.171428571 0.6 12.96 0.20 7.405714286 0.7 15.12 0.20 8.64 0000000 0.8 17.28 0.20 9.874285714 0.9 19.44 0.20 11.10857143 0 1 .0 21.6 0 0.20 12.34285714 0 2 .0 43.2 0 0.20 24.68571429 0 3 .0 64.8 0 0.20 37.02857143 0 4 .0 86.4 0 0.20 49.37142857 0 5 .0 108 .00 0.20 61.71428571 0 6 .0 129.6 0 0.20 74.05714286 0 7 .0 151.2 0 0.20 86.4 00000000 8 .0 172.8 0 0.20 98.74285714 0 9 .0 194.4 0 0.20 111.0857143 00 10 .0 216 .00 0.20 123.4285714 00 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025 ; 3. Isotropic hydraulic conductivity ( ) = 864 m/day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 342 342 Table A 4 Aquifer parameters and dimensionless ratios in case of = 0.3 for the first set of investigations. Molecular diffusion coefficient ( ) m 2 /day m 2 /day 0.1 2.16 0.30 1.851428571 0.2 4.32 0.30 3.702857143 0.3 6.48 0.30 5.554285714 0.4 8.64 0.30 7.405714286 0.5 10.8 0 0.30 9.257142857 0.6 12.96 0.30 11.10857143 0 0.7 15.12 0.30 12.96000000 0 0.8 17.28 0.30 14.81142857 0 0.9 19.44 0.30 16.66285714 0 1 .0 21.6 0 0.30 18.51428571 0 2 .0 43.2 0 0.30 37.02857143 0 3 .0 64.8 0 0.30 55.54285714 0 4 .0 86.4 0 0.30 74.05714286 0 5 .0 108 .00 0.30 92.57142857 0 6 .0 129.6 0 0.30 111.0857143 00 7 .0 151.2 0 0.30 129.6000000 00 8 .0 172.8 0 0.30 148.1142857 00 9 .0 194.4 0 0.30 166.6285714 00 10 .0 216 .00 0.30 185.1428571 00 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025; 3. Isotropic hydraulic conductivity ( ) = 864 m/day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 343 343 Table A 5 Aquifer parameters and dimensionless ratios in case of = 0.4 for the first set of investigations. Molecular diffusion coefficient ( ) m 2 /day m 2 /day 0.1 2.16 0.40 2.468571429 0.2 4.32 0.40 4.937142857 0.3 6.48 0.40 7.405714286 0.4 8.64 0.40 9.874285714 0.5 10.8 0 0.40 12.34285714 0 0.6 12.96 0.40 14.81142857 0 0.7 15.12 0.40 17.28000000 0 0.8 17.28 0.40 19.74857143 0 0.9 19.44 0.40 22.21714286 0 1 .0 21.6 0 0.40 24.68571429 0 2 .0 43.2 0 0.40 49.37142857 0 3 .0 64.8 0 0.40 74.05714286 0 4 .0 86.4 0 0.40 98.74285714 0 5 .0 108 .00 0.40 123.4285714 00 6 .0 129.6 0 0.40 148.1142857 00 7 .0 151.2 0 0.40 172.8000000 00 8 .0 172.8 0 0.40 197.4857143 00 9 .0 194.4 0 0.40 222.1714286 00 10 .0 216 .00 0.40 246.8571429 00 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025; 3. Isotropic hydraulic conductivity ( ) = 864 m/day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 344 344 Table A 6 Aquifer parameters and dimensionless ratios in case of = 0.5 for the first set of investigations. Molecular diffusion coefficient ( ) m 2 /day m 2 /day 0.1 2.16 0.50 3.08571428 0.2 4.32 0.50 6.17142857 0.3 6.48 0.50 9.25714285 0.4 8.64 0.50 12.34285714 0.5 10.8 0 0.50 15.42857143 0.6 12.96 0.50 18.51428571 0.7 15.12 0.50 21.6 0000000 0.8 17.28 0.50 24.68571429 0.9 19.44 0.50 27.77142857 1 .0 21.6 0 0.50 30.85714286 2 .0 43.2 0 0.50 61.71428571 3 .0 64.8 0 0.50 92.57142857 4 .0 86.4 0 0.50 123.4285714 0 5 .0 108 .00 0.50 154.2857143 0 6 .0 129.6 0 0.50 185.1428571 0 7 .0 151.2 0 0.50 216 .00000000 8 .0 172.8 0 0.50 246.8571429 0 9 .0 194.4 0 0.50 277.7142857 0 10 .0 216 .00 0.50 308.5714286 0 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025; 3. Isotropic hydraulic conductivity ( ) = 864 m/day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 345 345 Results of the First Set of Investigations of the Henry Constant Dispersion Problem Table A 7 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.1 of the first set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.80940 6 581.032 2,160 0.80940 3.04677 1.45121 1 171.907 2,160 1.45121 0.54255 0.2 0.59070 5 717.032 4,320 0.59070 1.32339 0.81154 1 278.826 4,320 0.81154 0.29602 0.3 0.46471 4 853.032 6,480 0.46471 0.74892 0.54500 1 202.034 6,480 0.54500 0.18550 0.4 0.37772 3 989.032 8,640 0.37772 0.46169 0.40235 1 024.652 8,640 0.40235 0.11859 0.5 0.31436 3 378.887 10,800 0.31436 0.31286 0.31677 811.520 10,800 0.31677 0.07514 0.6 0.26722 2 946.887 12,960 0.26722 0.22738 0.26169 596.682 12,960 0.26169 0.04604 0.7 0.23221 2 514.887 15,120 0.23221 0.16633 0.22453 406.920 15,120 0.22453 0.02691 0.8 0.20586 2 082.887 17,280 0.20586 0.12054 0.19762 252.260 17,280 0.19762 0.01460 0.9 0.18607 1 650.888 19,440 0.18607 0.08492 0.17875 125.035 19,440 0.17875 0.00643 1.0 0.17074 1 218.887 21,600 0.17074 0.05643 0.16401 53.961 21,600 0.16401 0.00250 2.0 0.11062 0.000 43,200 0.11062 0.00000 0.10817 0.000 43,200 0.10817 0.00000 3.0 0.09439 0.000 64,800 0.09439 0.00000 0.09360 0.000 64,800 0.09360 0.00000 4.0 0.08810 0.000 86,400 0.08810 0.00000 0.08783 0.000 86,400 0.08783 0.00000 5.0 0.08445 0.000 108,000 0.08445 0.00000 0.08446 0.000 108,000 0.08446 0.00000 6.0 0.08212 0.000 129,600 0.08212 0.00000 0.08227 0.000 129,600 0.08227 0.00000 7.0 0.08043 0.000 151,200 0.08043 0.00000 0.08072 0.000 151,200 0.08072 0.00000 8.0 0.07922 0.000 172,800 0.07922 0.00000 0.07958 0.000 172,800 0.07958 0.00000 9.0 0.07831 0.000 194,400 0.07831 0.00000 0.07872 0.000 194,400 0.07872 0.00000 10.0 0.07749 0.000 216,000 0.07749 0.00000 0.07802 0.000 216,000 0.07802 0.00000 PAGE 346 346 Table A 8 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.2 of the first set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.80996 6 581.032 2,160 0.80996 3.04677 1.15055 1 342.835 2,160 1.15055 0.62168 0.2 0.59078 5 717.032 4,320 0.59078 1.32339 0.66570 1 340.237 4,320 0.66570 0.31024 0.3 0.46799 4 853.032 6,480 0.46799 0.74892 0.47646 1 109.130 6,480 0.47646 0.17116 0.4 0.38750 3 989.032 8,640 0.38750 0.46169 0.37974 811.603 8,640 0.37974 0.09394 0.5 0.33472 3 378.887 10,800 0.33472 0.31286 0.32302 532.206 10,800 0.32302 0.04928 0.6 0.29795 2 946.887 12,960 0.29795 0.22738 0.28678 300.567 12,960 0.28678 0.02319 0.7 0.27244 2 514.887 15,120 0.27244 0.16633 0.26183 136.413 15,120 0.26183 0.00902 0.8 0.25228 2 082.887 17,280 0.25228 0.12054 0.24369 39.635 17,280 0.24369 0.00229 0.9 0.23798 1 650.888 19,440 0.23798 0.08492 0.23035 0.000 19,440 0.23035 0.00000 1.0 0.22684 1 218.887 21,600 0.22684 0.05643 0.21990 0.000 21,600 0.21990 0.00000 2.0 0.17917 0.000 43,200 0.17917 0.00000 0.17709 0.000 43,200 0.17709 0.00000 3.0 0.16474 0.000 64,800 0.16474 0.00000 0.16398 0.000 64,800 0.16398 0.00000 4.0 0.15754 0.000 86,400 0.15754 0.00000 0.15766 0.000 86,400 0.15766 0.00000 5.0 0.15307 0.000 108,000 0.15307 0.00000 0.15377 0.000 108,000 0.15377 0.00000 6.0 0.15040 0.000 129,600 0.15040 0.00000 0.15122 0.000 129,600 0.15122 0.00000 7.0 0.14872 0.000 151,200 0.14872 0.00000 0.14946 0.000 151,200 0.14946 0.00000 8.0 0.14744 0.000 172,800 0.14744 0.00000 0.14849 0.000 172,800 0.14849 0.00000 9.0 0.14638 0.000 194,400 0.14638 0.00000 0.14774 0.000 194,400 0.14774 0.00000 10.0 0.14553 0.000 216,000 0.14553 0.00000 0.14699 0.000 216,000 0.14699 0.00000 PAGE 347 347 Table A 9 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.3 of the first set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.81075 6,581.032 2,160 0.81075 3.04677 1.01118 1,385.280 2,160.0 1.01118 0.64133 0.2 0.59547 5,717.032 4,320 0.59547 1.32339 0.61493 1,266.994 4,320.0 0.61493 0.29329 0.3 0.47866 4,853.032 6,480 0.47866 0.74892 0.46940 925.817 6,480.0 0.46940 0.14287 0.4 0.41000 3,989.032 8,640 0.41000 0.46169 0.39636 580.097 8,640.0 0.39636 0.06714 0.5 0.36744 3,378.887 10,800 0.36744 0.31286 0.35435 306.731 10,800.0 0.35435 0.02840 0.6 0.33805 2,946.887 12,960 0.33805 0.22738 0.32746 115.636 12,960.0 0.32746 0.00892 0.7 0.31667 2,514.887 15,120 0.31667 0.16633 0.30759 15.917 15,120.0 0.30759 0.00105 0.8 0.30188 2,082.887 17,280 0.30188 0.12054 0.29451 0.000 17,280.0 0.29451 0.00000 0.9 0.29004 1,650.888 19,440 0.29004 0.08492 0.28387 0.000 19,440.0 0.28387 0.00000 1.0 0.28263 1,218.887 21,600 0.28263 0.05643 0.27663 0.000 21,600.0 0.27663 0.00000 2.0 0.24338 0.000 43,200 0.24338 0.00000 0.24137 0.000 43,200.0 0.24137 0.00000 3.0 0.23088 0.000 64,800 0.23088 0.00000 0.23112 0.000 64,800.0 0.23112 0.00000 4.0 0.22571 0.000 86,400 0.22571 0.00000 0.22634 0.000 86,400.0 0.22634 0.00000 5.0 0.22042 0.000 108,000 0.22042 0.00000 0.22338 0.000 108,000.0 0.22338 0.00000 6.0 0.21922 0.000 129,600 0.21922 0.00000 0.22140 0.000 129,600.0 0.22140 0.00000 7.0 0.21714 0.000 151,200 0.21714 0.00000 0.21986 0.000 151,200.0 0.21986 0.00000 8.0 0.21589 0.000 172,800 0.21589 0.00000 0.21871 0.000 172,800.0 0.21871 0.00000 9.0 0.21541 0.000 194,400 0.21541 0.00000 0.21823 0.000 194,400.0 0.21823 0.00000 10.0 0.21487 0.000 216,000 0.21487 0.00000 0.21751 0.000 216,000.0 0.21751 0.00000 PAGE 348 348 Table A 10 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.4 of the first set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.82584 6,581.052 2,160 0.82584 3.04678 0.93377 1,379.656 2,160 0.93377 0.63873 0.2 0.60747 5,717.032 4,320 0.60747 1.32339 0.60323 1,115.863 4,320 0.60323 0.25830 0.3 0.49733 4,853.032 6,480 0.49733 0.74892 0.48561 732.591 6,480 0.48561 0.11305 0.4 0.44314 3,989.052 8,640 0.44314 0.46170 0.42927 388.267 8,640 0.42927 0.04494 0.5 0.40835 3,378.887 10,800 0.40835 0.31286 0.39483 149.974 10,800 0.39483 0.01389 0.6 0.38480 2,946.887 12,960 0.38480 0.22738 0.37500 21.953 12,960 0.37500 0.00169 0.7 0.36796 2,514.887 15,120 0.36796 0.16633 0.35960 0.000 15,120 0.35960 0.00000 0.8 0.35533 2,082.887 17,280 0.35533 0.12054 0.34913 0.000 17,280 0.34913 0.00000 0.9 0.34417 1,650.888 19,440 0.34417 0.08492 0.33455 0.000 19,440 0.33455 0.00000 1.0 0.33810 1,218.887 21,600 0.33810 0.05643 0.33528 0.000 21,600 0.33528 0.00000 2.0 0.30489 0.000 43,200 0.30489 0.00000 0.30724 0.000 43,200 0.30724 0.00000 3.0 0.29587 0.000 64,800 0.29587 0.00000 0.29820 0.000 64,800 0.29820 0.00000 4.0 0.29122 0.000 86,400 0.29122 0.00000 0.29377 0.000 86,400 0.29377 0.00000 5.0 0.28949 0.000 108,000 0.28949 0.00000 0.28964 0.000 108,000 0.28964 0.00000 6.0 0.28524 0.000 129,600 0.28524 0.00000 0.28874 0.000 129,600 0.28874 0.00000 7.0 0.28453 0.000 151,200 0.28453 0.00000 0.28904 0.000 151,200 0.28904 0.00000 8.0 0.28296 0.000 172,800 0.28296 0.00000 0.28819 0.000 172,800 0.28819 0.00000 9.0 0.28342 0.000 194,400 0.28342 0.00000 0.28841 0.000 194,400 0.28841 0.00000 10.0 0.28321 0.000 216,000 0.28321 0.00000 0.28675 0.000 216,000 0.28675 0.00000 PAGE 349 349 Table A 11 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0.5 of the first set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.82836 6,581.032 2,160 0.82836 3.04677 0.89540 1,298.634 2,160 0.89540 0.60122 0.2 0.62011 5,717.032 4,320 0.62011 1.32339 0.61183 980.018 4,320 0.61183 0.22686 0.3 0.52144 4,853.032 6,480 0.52144 0.74892 0.51556 555.699 6,480 0.51556 0.08576 0.4 0.46300 3,989.032 8,640 0.46300 0.46169 0.47230 230.208 8,640 0.47230 0.02664 0.5 0.44440 3,378.887 10,800 0.44440 0.31286 0.44834 44.595 10,800 0.44834 0.00413 0.6 0.43054 2,946.887 12,960 0.43054 0.22738 0.42743 0.000 12,960 0.42743 0.00000 0.7 0.42075 2,514.900 15,120 0.42075 0.16633 0.41610 0.000 15,120 0.41610 0.00000 0.8 0.40720 2,082.887 17,280 0.40720 0.12054 0.40720 0.000 17,280 0.40720 0.00000 0.9 0.40374 1,650.901 19,440 0.40374 0.08492 0.39860 0.000 19,440 0.39860 0.00000 1.0 0.39927 1,218.901 21,600 0.39927 0.05643 0.39480 0.000 21,600 0.39480 0.00000 2.0 0.36827 0.000 43,200 0.36827 0.00000 0.36723 0.000 43,200 0.36723 0.00000 3.0 0.36262 0.000 64,800 0.36262 0.00000 0.36602 0.000 64,800 0.36602 0.00000 4.0 0.35736 0.000 86,400 0.35736 0.00000 0.36141 0.000 86,400 0.36141 0.00000 5.0 0.35619 0.000 108,000 0.35619 0.00000 0.36041 0.000 108,000 0.36041 0.00000 6.0 0.34665 0.000 129,600 0.34665 0.00000 0.35456 0.000 129,600 0.35456 0.00000 7.0 0.34682 0.000 151,200 0.34682 0.00000 0.35586 0.000 151,200 0.35586 0.00000 8.0 0.34806 0.000 172,800 0.34806 0.00000 0.35750 0.000 172,800 0.35750 0.00000 9.0 0.34995 0.000 194,400 0.34995 0.00000 0.35503 0.000 194,400 0.35503 0.00000 10.0 0.35060 0.000 216,000 0.35060 0.00000 0.35798 0.000 216,000 0.35798 0.00000 PAGE 350 350 Aquifer Parameters and Dimensionless ratios for the Second Set of I nvestigations of the Henry Constant Dispersion Problem Table A 1 2 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.1 for the second set of inves tigations Hydraulic Conductivity ( ) m 2 /day m/day 0.1 5.7024 0.1 2 282.1010 0.1 5.7024 0.2 1 143.3380 0. 1 5.7024 0.3 761.5890 0.1 5.7024 0.4 570.9530 0.1 5.7024 0.5 456.6480 0.1 5.7024 0.6 380.4770 0.1 5.7024 0.7 326.0840 0.1 5.7024 0.8 285.2980 0.1 5.7024 0.9 253.5800 0.1 5.7024 1 .0 228.2100 0.1 5.7024 2 .0 114.0760 0.1 5.7024 3 .0 76.0440 0.1 5.7024 4 .0 57.0310 0.1 5.7024 5 .0 45.6230 0.1 5.7024 6 .0 38.0190 0.1 5.7024 7 .0 32.5870 0.1 5.7024 8 .0 28.5130 0.1 5.7024 9 .0 25.3450 0.1 5.7024 10 .0 22.8100 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025; 3. Molecular diffusion coefficient ( ) = 1.62924998 m 2 /day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 351 351 Table A 1 3 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.2 for the second set of inves tigations Hydraulic Conductivity ( ) m 2 /da y m/day 0.2 2.8512 0.1 1 141.0500 0.2 2.8512 0.2 571.6690 0.2 2.8512 0.3 380.7940 0.2 2.8512 0.4 285.4760 0.2 2.8512 0.5 228.3240 0.2 2.8512 0.6 190.2380 0.2 2.8512 0.7 163.0420 0.2 2.8512 0.8 142.6490 0.2 2.8512 0.9 126.7900 0.2 2.8512 1 .0 114.1050 0.2 2.8512 2 .0 57.0380 0.2 2.8512 3 .0 38.0220 0.2 2.8512 4 .0 28.5150 0.2 2.8512 5 .0 22.8110 0.2 2.8512 6 .0 19.0090 0.2 2.8512 7 .0 16.2930 0.2 2.8512 8 .0 14.2560 0.2 2.8512 9 .0 12.6720 0.2 2.8512 10 .0 11.4050 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025; 3. Molecular diffusion coefficient ( ) = 1.62924998 m 2 /day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 352 352 Table A 1 4 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.3 for the second set of inves tigations Hydraulic Conductivity ( ) m 2 /day m/day 0.3 1.9008 0.1 760.7000 0.3 1.9008 0.2 381.1120 0.3 1.9008 0.3 253.8630 0.3 1.9008 0.4 190.3170 0.3 1.9008 0.5 152.2160 0.3 1.9008 0.6 126.8250 0.3 1.9008 0.7 108.6940 0.3 1.9008 0.8 95.0990 0.3 1.9008 0.9 84.5260 0.3 1.9008 1 .0 76.0700 0.3 1.9008 2 .0 38.0250 0.3 1.9008 3 .0 25.3480 0.3 1.9008 4 .0 19.0100 0.3 1.9008 5 .0 15.2070 0.3 1.9008 6 .0 12.6730 0.3 1.9008 7 .0 10.8620 0.3 1.9008 8 .0 9.5040 0.3 1.9008 9 .0 8.4480 0.3 1.9008 10 .0 7.6030 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025; 3. Molecular diffusion coefficient ( ) = 1.62924998 m 2 /day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 353 353 Table A 1 5 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.4 for the second set of inves tigations Hydraulic Conductivity ( ) m 2 /day m/day 0.4 1.4256 0.1 570.5250 0.4 1.4256 0.2 285.8340 0.4 1.4256 0.3 190.3970 0.4 1.4256 0.4 142.7380 0.4 1.4256 0.5 114.1620 0.4 1.4256 0.6 95.1190 0.4 1.4256 0.7 81.5210 0.4 1.4256 0.8 71.3240 0.4 1.4256 0.9 63.3950 0.4 1.4256 1 .0 57.0520 0.4 1.4256 2 .0 28.5190 0.4 1.4256 3 .0 19.0110 0.4 1.4256 4 .0 14.2570 0.4 1.4256 5 .0 11.4050 0.4 1.4256 6 .0 9.5040 0.4 1.4256 7 .0 8.1460 0.4 1.4256 8 .0 7.1280 0.4 1.4256 9 .0 6.3360 0.4 1.4256 10 .0 5.7020 Not e: 1. Porosity ( ) = 0.35. 2. Density contrast parameter ( ) = 0.025. 3. Molecular diffusion coefficient ( ) = 1.62924998 m 2 /day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 354 354 Table A 1 6 Aquifer parameters and dimensionless ratios in case of the dimensionless ratio = 0.5 for the second set of inves tigations Hydraulic Conductivity ( ) m 2 /day m/day 0.5 1.14048 0.1 456.4200 0.5 1.14048 0.2 228.6670 0.5 1.14048 0.3 152.3170 0.5 1.14048 0.4 114.1900 0.5 1.14048 0.5 91.3290 0.5 1.14048 0.6 76.0950 0.5 1.14048 0.7 65.2160 0.5 1.14048 0.8 57.0590 0.5 1.14048 0.9 50.7160 0.5 1.14048 1 .0 45.6420 0.5 1.14048 2 .0 22.8150 0.5 1.14048 3 .0 15.2080 0.5 1.14048 4 .0 11.4060 0.5 1.14048 5 .0 9.1240 0.5 1.14048 6 .0 7.6030 0.5 1.14048 7 .0 6.5170 0.5 1.14048 8 .0 5.7025 0.5 1.14048 9 .0 5.0690 0.5 1.14048 10 .0 4.5620 Not e: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0.025; 3. Molecular diffusion coefficient ( ) = 1.62924998 m 2 /day ; and 4. Vertical dimension of the medium ( ) = 1 .0 m PAGE 355 355 Results of the Second Set of Investigations of the Henry Constant Dispersion Problem Table A 17 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0. 1 of the second set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.81005 17,383.912 5,702 0.81005 3.04874 1.45202 3087.607 5,702.0 1.45202 0.5415 0.2 0.59146 7,570.943 5,702 0.59146 1.32777 0.81340 1692.110 5,702.0 0.81340 0.29676 0.3 0.46518 4,281.792 5,702 0.46518 0.75093 0.54583 1060.347 5,702.0 0.54583 0.18596 0.4 0.37816 2,639.039 5,702 0.37816 0.46283 0.40272 677.822 5,702.0 0.40272 0.11887 0.5 0.31436 1,787.069 5,702 0.31436 0.31341 0.31710 429.921 5,702.0 0.31710 0.07540 0.6 0.26734 1,298.762 5,702 0.26734 0.22777 0.26198 263.176 5,702.0 0.26198 0.04616 0.7 0.23232 950.017 5,702 0.23232 0.16661 0.22466 153.974 5,702.0 0.22466 0.02700 0.8 0.20598 688.587 5,702 0.20598 0.12076 0.19762 83.613 5,702.0 0.19762 0.01466 0.9 0.18624 485.238 5,702 0.18624 0.08510 0.17887 36.809 5,702.0 0.17887 0.00646 1.0 0.17099 322.597 5,702 0.17099 0.05658 0.16409 14.381 5,702.0 0.16409 0.00252 2.0 0.11014 0.000 5,702 0.11014 0.00000 0.10817 0.000 5,702.0 0.10817 0.00000 3.0 0.09427 0.000 5,702 0.09427 0.00000 0.09360 0.000 5,702.0 0.09360 0.00000 4.0 0.08810 0.000 5,702 0.08810 0.00000 0.08781 0.000 5,702.0 0.08781 0.00000 5.0 0.08445 0.000 5,702 0.08445 0.00000 0.08446 0.000 5,702.0 0.08446 0.00000 6.0 0.08212 0.000 5,702 0.08212 0.00000 0.08227 0.000 5,702.0 0.08227 0.00000 7.0 0.08037 0.000 5,702 0.08037 0.00000 0.08072 0.000 5,702.0 0.08072 0.00000 8.0 0.07928 0.000 5,702 0.07928 0.00000 0.07961 0.000 5,702.0 0.07961 0.00000 9.0 0.07825 0.000 5,702 0.07825 0.00000 0.07872 0.000 5,702.0 0.07872 0.00000 10.0 0.07749 0.000 5,702 0.07749 0.00000 0.07799 0.000 5,702.0 0.07799 0.00000 PAGE 356 3 56 Table A 18 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0. 2 of the second set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.80952 8 691.986 2,852 0.80952 3.04768 1.15114 1,773.760 2,852 1.15114 0.62194 0.2 0.59039 3 785.244 2,852 0.59039 1.32722 0.66686 887.972 2,852 0.66686 0.31135 0.3 0.46774 2 140.443 2,852 0.46774 0.75051 0.47694 488.898 2,852 0.47694 0.17142 0.4 0.38686 1 319.163 2,852 0.38686 0.46254 0.38004 268.479 2,852 0.38004 0.09414 0.5 0.33533 893.303 2,852 0.33533 0.31322 0.32321 141.421 2,852 0.32321 0.04959 0.6 0.29670 649.181 2,852 0.29670 0.22762 0.28641 66.983 2,852 0.28641 0.02349 0.7 0.27179 474.809 2,852 0.27179 0.16648 0.26165 26.006 2,852 0.26165 0.00912 0.8 0.25309 344.094 2,852 0.25309 0.12065 0.24341 6.692 2,852 0.24341 0.00235 0.9 0.23812 242.419 2,852 0.23812 0.08500 0.23051 0.000 2,852 0.23051 0.00000 1.0 0.22647 161.131 2,852 0.22647 0.05650 0.21975 0.000 2,852 0.21975 0.00000 2.0 0.17940 0.000 2,852 0.17940 0.00000 0.17682 0.000 2,852 0.17682 0.00000 3.0 0.16444 0.000 2,852 0.16444 0.00000 0.16384 0.000 2,852 0.16384 0.00000 4.0 0.15743 0.000 2,852 0.15743 0.00000 0.15780 0.000 2,852 0.15780 0.00000 5.0 0.15306 0.000 2,852 0.15306 0.00000 0.15337 0.000 2,852 0.15337 0.00000 6.0 0.15000 0.000 2,852 0.15000 0.00000 0.15108 0.000 2,852 0.15108 0.00000 7.0 0.14777 0.000 2,852 0.14777 0.00000 0.14957 0.000 2,852 0.14957 0.00000 8.0 0.14712 0.000 2,852 0.14712 0.00000 0.14839 0.000 2,852 0.14839 0.00000 9.0 0.14586 0.000 2,852 0.14586 0.00000 0.14742 0.000 2,852 0.14742 0.00000 10.0 0.14553 0.000 2,852 0.14553 0.00000 0.14689 0.000 2,852 0.14689 0.00000 PAGE 357 357 Table A 19 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0. 3 of the second set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.81154 5,794.903 1,900 0.81154 3.04995 1.01042 1,223.135 1,900 1.01042 0.64376 0.2 0.59399 2,524.002 1,900 0.59399 1.32842 0.61519 561.993 1,900 0.61519 0.29579 0.3 0.47694 1,427.495 1,900 0.47694 0.75131 0.47143 269.449 1,900 0.47143 0.14182 0.4 0.41039 879.975 1,900 0.41039 0.46314 0.39636 128.643 1,900 0.39636 0.06771 0.5 0.36477 595.843 1,900 0.36477 0.31360 0.35448 55.391 1,900 0.35448 0.02915 0.6 0.33452 433.076 1,900 0.33452 0.22793 0.32746 17.371 1,900 0.32746 0.00914 0.7 0.31721 316.785 1,900 0.31721 0.16673 0.30848 1.968 1,900 0.30848 0.00104 0.8 0.30061 229.656 1,900 0.30061 0.12087 0.29355 0.000 1,900 0.29355 0.00000 0.9 0.29332 161.875 1,900 0.29332 0.08520 0.28340 0.000 1,900 0.28340 0.00000 1.0 0.28000 107.666 1,900 0.28000 0.05667 0.27632 0.000 1,900 0.27632 0.00000 2.0 0.24100 0.000 1,900 0.24100 0.00000 0.24137 0.000 1,900 0.24137 0.00000 3.0 0.23105 0.000 1,900 0.23105 0.00000 0.23085 0.000 1,900 0.23085 0.00000 4.0 0.22394 0.000 1,900 0.22394 0.00000 0.22563 0.000 1,900 0.22563 0.00000 5.0 0.22117 0.000 1,900 0.22117 0.00000 0.22294 0.000 1,900 0.22294 0.00000 6.0 0.21809 0.000 1,900 0.21809 0.00000 0.22068 0.000 1,900 0.22068 0.00000 7.0 0.21690 0.000 1,900 0.21690 0.00000 0.22036 0.000 1,900 0.22036 0.00000 8.0 0.21571 0.000 1,900 0.21571 0.00000 0.21823 0.000 1,900 0.21823 0.00000 9.0 0.21192 0.000 1,900 0.21192 0.00000 0.21727 0.000 1,900 0.21727 0.00000 10.0 0.21068 0.000 1,900 0.21068 0.00000 0.21727 0.000 1,900 0.21727 0.00000 PAGE 358 358 Table A 20 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0. 4 of the second set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.81982 4 345.734 1,426 0.81982 3.04750 0.93503 907.488 1,426 0.93503 0.63639 0.2 0.60102 1 892.579 1,426 0.60102 1.32719 0.60366 373.378 1,426 0.60366 0.26184 0.3 0.47790 1 070.265 1,426 0.47790 0.75054 0.48217 165.687 1,426 0.48217 0.11619 0.4 0.43537 659.582 1,426 0.43537 0.46254 0.42743 65.047 1,426 0.42743 0.04562 0.5 0.39658 446.651 1,426 0.39658 0.31322 0.39218 17.894 1,426 0.39218 0.01255 0.6 0.38302 324.584 1,426 0.38302 0.22762 0.37364 2.528 1,426 0.37364 0.00177 0.7 0.36474 237.411 1,426 0.36474 0.16649 0.35798 0.000 1,426 0.35798 0.00000 0.8 0.35418 172.040 1,426 0.35418 0.12065 0.35339 0.000 1,426 0.35339 0.00000 0.9 0.33891 121.210 1,426 0.33891 0.08500 0.33868 0.000 1,426 0.33868 0.00000 1.0 0.33341 80.546 1,426 0.33341 0.05648 0.33399 0.000 1,426 0.33399 0.00000 2.0 0.30815 0.000 1,426 0.30815 0.00000 0.30855 0.000 1,426 0.30855 0.00000 3.0 0.29178 0.000 1,426 0.29178 0.00000 0.29613 0.000 1,426 0.29613 0.00000 4.0 0.29409 0.000 1,426 0.29409 0.00000 0.29315 0.000 1,426 0.29315 0.00000 5.0 0.29080 0.000 1,426 0.29080 0.00000 0.28851 0.000 1,426 0.28851 0.00000 6.0 0.28215 0.000 1,426 0.28215 0.00000 0.28671 0.000 1,426 0.28671 0.00000 7.0 0.28360 0.000 1,426 0.28360 0.00000 0.28527 0.000 1,426 0.28527 0.00000 8.0 0.27690 0.000 1,426 0.27690 0.00000 0.28679 0.000 1,426 0.28679 0.00000 9.0 0.27810 0.000 1,426 0.27810 0.00000 0.27762 0.000 1,426 0.27762 0.00000 10.0 0.28529 0.000 1,426 0.28529 0.00000 0.27884 0.000 1,426 0.27884 0.00000 PAGE 359 359 Table A 21 E xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled and coupled solutions in case of the dimensionless ratio = 0. 5 of the second set of inves tigations Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.80995 1,140 3 476.942 0.80995 3.04995 0.89233 703.727 1,140 0.89233 0.61730 0.2 0.63219 1,140 1 514.435 0.63219 1.32845 0.60781 266.644 1,140 0.60781 0.23390 0.3 0.52876 1,140 856.532 0.52876 0.75134 0.51044 103.064 1,140 0.51044 0.09041 0.4 0.46867 1,140 527.968 0.46867 0.46313 0.46593 33.310 1,140 0.46593 0.02922 0.5 0.42726 1,140 357.488 0.42726 0.31359 0.43250 6.317 1,140 0.43250 0.00554 0.6 0.42878 1,140 259.826 0.42878 0.22792 0.42086 0.000 1,140 0.42086 0.00000 0.7 0.41176 1,140 190.083 0.41176 0.16674 0.41443 0.000 1,140 0.41443 0.00000 0.8 0.40113 1,140 137.791 0.40113 0.12087 0.39837 0.000 1,140 0.39837 0.00000 0.9 0.38390 1,140 97.128 0.38390 0.08520 0.39791 0.000 1,140 0.39791 0.00000 1.0 0.39809 1,140 64.600 0.39809 0.05667 0.39880 0.000 1,140 0.39880 0.00000 2.0 0.36170 1,140 0.000 0.36170 0.00000 0.36167 0.000 1,140 0.36167 0.00000 3.0 0.35450 1,140 0.000 0.35450 0.00000 0.36161 0.000 1,140 0.36161 0.00000 4.0 0.35075 1,140 0.000 0.35075 0.00000 0.35564 0.000 1,140 0.35564 0.00000 5.0 0.34307 1,140 0.000 0.34307 0.00000 0.35445 0.000 1,140 0.35445 0.00000 6.0 0.34378 1,140 0.000 0.34378 0.00000 0.35760 0.000 1,140 0.35760 0.00000 7.0 0.34452 1,140 0.000 0.34452 0.00000 0.35458 0.000 1,140 0.35458 0.00000 8.0 0.34176 1,140 0.000 0.34176 0.00000 0.35835 0.000 1,140 0.35835 0.00000 9.0 0.34368 1,140 0.000 0.34368 0.00000 0.34625 0.000 1,140 0.34625 0.00000 10.0 0.33904 1,140 0.000 0.33904 0.00000 0.35727 0.000 1,140 0.35727 0.00000 PAGE 360 360 Comparison the Values of the E xtent of S altwater I ntrusion ( ) and the Degree of saltwater recirculation ( ) between the First and Second Sets of Investigations for the Henry Constant Dispersion Problem Table A 22 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of investigations i n case of = 0.1 The first set The second set 0.1 0.80940 3.04677 0.81005 3.04874 0.99920 0.99935 0.2 0.59070 1.32339 0.59146 1.32777 0.99872 0.99670 0.3 0.46471 0.74892 0.46518 0.75093 0.99899 0.99732 0.4 0.37772 0.46169 0.37816 0.46283 0.99884 0.99754 0.5 0.31436 0.31286 0.31436 0.31341 1.00000 0.99825 0.6 0.26722 0.22738 0.26734 0.22777 0.99955 0.99829 0.7 0.23221 0.16633 0.23232 0.16661 0.99953 0.99832 0.8 0.20586 0.12054 0.20598 0.12076 0.99942 0.99818 0.9 0.18607 0.08492 0.18624 0.08510 0.99909 0.99788 1.0 0.17074 0.05643 0.17099 0.05658 0.99854 0.99735 2.0 0.11062 0.00000 0.11014 0.00000 1.00436 3.0 0.09439 0.00000 0.09427 0.00000 1.00127 4.0 0.08810 0.00000 0.08810 0.00000 1.00000 5.0 0.08445 0.00000 0.08445 0.00000 1.00000 6.0 0.08212 0.00000 0.08212 0.00000 1.00000 7.0 0.08043 0.00000 0.08037 0.00000 1.00075 8.0 0.07922 0.00000 0.07928 0.00000 0.99924 9.0 0.07831 0.00000 0.07825 0.00000 1.00077 10.0 0.07749 0.00000 0.07749 0.00000 1.00000 PAGE 361 361 Table A 23 Comparison the value s of e xtent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.1 The first set The second set 0.1 1.45121 0.54255 1.45202 0.54150 0.99944 1.00194 0.2 0.81154 0.29602 0.81340 0.29676 0.99771 0.99751 0.3 0.54500 0.18550 0.54583 0.18596 0.99848 0.99753 0.4 0.40235 0.11859 0.40272 0.11887 0.99908 0.99764 0.5 0.31677 0.07514 0.31710 0.07540 0.99896 0.99655 0.6 0.26169 0.04604 0.26198 0.04616 0.99889 0.99740 0.7 0.22453 0.02691 0.22466 0.02700 0.99942 0.99667 0.8 0.19762 0.01460 0.19762 0.01466 1.00000 0.99591 0.9 0.17875 0.00643 0.17887 0.00646 0.99933 0.99536 1.0 0.16401 0.00250 0.16409 0.00252 0.99951 0.99206 2.0 0.10817 0.00000 0.10817 0.00000 1.00000 3.0 0.09360 0.00000 0.09360 0.00000 1.00000 4.0 0.08783 0.00000 0.08781 0.00000 1.00023 5.0 0.08446 0.00000 0.08446 0.00000 1.00000 6.0 0.08227 0.00000 0.08227 0.00000 1.00000 7.0 0.08072 0.00000 0.08072 0.00000 1.00000 8.0 0.07958 0.00000 0.07961 0.00000 0.99962 9.0 0.07872 0.00000 0.07872 0.00000 1.00000 10.0 0.07802 0.00000 0.07799 0.00000 1.00038 PAGE 362 362 Table A 24 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of investigations in case of = 0.2. The first set The second set 0.1 0.80996 3.04677 0.80952 3.04768 1.00054 0.99970 0.2 0.59078 1.32339 0.59039 1.32722 1.00066 0.99711 0.3 0.46799 0.74892 0.46774 0.75051 1.00053 0.99788 0.4 0.38750 0.46169 0.38686 0.46254 1.00165 0.99816 0.5 0.33472 0.31286 0.33533 0.31322 0.99818 0.99885 0.6 0.29795 0.22738 0.29670 0.22762 1.00421 0.99895 0.7 0.27244 0.16633 0.27179 0.16648 1.00239 0.99910 0.8 0.25228 0.12054 0.25309 0.12065 0.99680 0.99909 0.9 0.23798 0.08492 0.23812 0.08500 0.99941 0.99906 1.0 0.22684 0.05643 0.22647 0.05650 1.00163 0.99876 2.0 0.17917 0.00000 0.17940 0.00000 0.99872 3.0 0.16474 0.00000 0.16444 0.00000 1.00182 4.0 0.15754 0.00000 0.15743 0.00000 1.00070 5.0 0.15307 0.00000 0.15306 0.00000 1.00007 6.0 0.15040 0.00000 0.15000 0.00000 1.00267 7.0 0.14872 0.00000 0.14777 0.00000 1.00643 8.0 0.14744 0.00000 0.14712 0.00000 1.00218 9.0 0.14638 0.00000 0.14586 0.00000 1.00357 10.0 0.14553 0.00000 0.14553 0.00000 1.00000 PAGE 363 363 Table A 25 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.2 The first set The second set 0.1 1.15055 0.62168 1.15114 0.62194 0.99949 0.99958 0.2 0.66570 0.31024 0.66686 0.31135 0.99826 0.99643 0.3 0.47646 0.17116 0.47694 0.17142 0.99899 0.99848 0.4 0.37974 0.09394 0.38004 0.09414 0.99921 0.99788 0.5 0.32302 0.04928 0.32321 0.04959 0.99941 0.99375 0.6 0.28678 0.02319 0.28641 0.02349 1.00129 0.98723 0.7 0.26183 0.00902 0.26165 0.00912 1.00069 0.98904 0.8 0.24369 0.00229 0.24341 0.00235 1.00115 0.97447 0.9 0.23035 0.00000 0.23051 0.00000 0.99931 1.0 0.21990 0.00000 0.21975 0.00000 1.00068 2.0 0.17709 0.00000 0.17682 0.00000 1.00153 3.0 0.16398 0.00000 0.16384 0.00000 1.00085 4.0 0.15766 0.00000 0.15780 0.00000 0.99911 5.0 0.15377 0.00000 0.15337 0.00000 1.00261 6.0 0.15122 0.00000 0.15108 0.00000 1.00093 7.0 0.14946 0.00000 0.14957 0.00000 0.99926 8.0 0.14849 0.00000 0.14839 0.00000 1.00067 9.0 0.14774 0.00000 0.14742 0.00000 1.00217 10.0 0.14699 0.00000 0.14689 0.00000 1.00068 PAGE 364 364 Table A 26 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of inves tigations in case of = 0.3 The first set The se cond set 0.1 0.81075 3.04677 0.81154 3.04995 0.99903 0.99896 0.2 0.59547 1.32339 0.59399 1.32842 1.00249 0.99621 0.3 0.47866 0.74892 0.47694 0.75131 1.00361 0.99682 0.4 0.41000 0.46169 0.41039 0.46314 0.99905 0.99687 0.5 0.36744 0.31286 0.36477 0.31360 1.00732 0.99764 0.6 0.33805 0.22738 0.33452 0.22793 1.01055 0.99759 0.7 0.31667 0.16633 0.31721 0.16673 0.99830 0.99760 0.8 0.30188 0.12054 0.30061 0.12087 1.00422 0.99727 0.9 0.29004 0.08492 0.29332 0.08520 0.98882 0.99671 1.0 0.28263 0.05643 0.28000 0.05667 1.00939 0.99576 2.0 0.24338 0.00000 0.24100 0.00000 1.00988 3.0 0.23088 0.00000 0.23105 0.00000 0.99926 4.0 0.22571 0.00000 0.22394 0.00000 1.00790 5.0 0.22042 0.00000 0.22117 0.00000 0.99661 6.0 0.21922 0.00000 0.21809 0.00000 1.00518 7.0 0.21714 0.00000 0.21690 0.00000 1.00111 8.0 0.21589 0.00000 0.21571 0.00000 1.00083 9.0 0.21541 0.00000 0.21192 0.00000 1.01647 10.0 0.21487 0.00000 0.21068 0.00000 1.01989 PAGE 365 365 Table A 27 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.3 The first set The second set 0.1 1.01118 0.64133 1.01042 0.64376 1.00075 0.99623 0.2 0.61493 0.29329 0.61519 0.29579 0.99958 0.99155 0.3 0.46940 0.14287 0.47143 0.14182 0.99569 1.00740 0.4 0.39636 0.06714 0.39636 0.06771 1.00000 0.99158 0.5 0.35435 0.02840 0.35448 0.02915 0.99963 0.97427 0.6 0.32746 0.00892 0.32746 0.00914 1.00000 0.97593 0.7 0.30759 0.00105 0.30848 0.00104 0.99711 1.00962 0.8 0.29451 0.00000 0.29355 0.00000 1.00327 0.9 0.28387 0.00000 0.28340 0.00000 1.00166 1.0 0.27663 0.00000 0.27632 0.00000 1.00112 2.0 0.24137 0.00000 0.24137 0.00000 1.00000 3.0 0.23112 0.00000 0.23085 0.00000 1.00117 4.0 0.22634 0.00000 0.22563 0.00000 1.00315 5.0 0.22338 0.00000 0.22294 0.00000 1.00197 6.0 0.22140 0.00000 0.22068 0.00000 1.00326 7.0 0.21986 0.00000 0.22036 0.00000 0.99773 8.0 0.21871 0.00000 0.21823 0.00000 1.00220 9.0 0.21823 0.00000 0.21727 0.00000 1.00442 10.0 0.21751 0.00000 0.21727 0.00000 1.00110 PAGE 366 366 Table A 28 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of investigations in case of = 0.4 The first set The second set 0.1 0.82584 3.04678 0.81982 3.04750 1.00734 0.99976 0.2 0.60747 1.32339 0.60102 1.32719 1.01073 0.99714 0.3 0.49733 0.74892 0.47790 0.75054 1.04066 0.99784 0.4 0.44314 0.46170 0.43537 0.46254 1.01785 0.99818 0.5 0.40835 0.31286 0.39658 0.31322 1.02968 0.99885 0.6 0.38480 0.22738 0.38302 0.22762 1.00465 0.99895 0.7 0.36796 0.16633 0.36474 0.16649 1.00883 0.99904 0.8 0.35533 0.12054 0.35418 0.12065 1.00325 0.99909 0.9 0.34417 0.08492 0.33891 0.08500 1.01552 0.99906 1.0 0.33810 0.05643 0.33341 0.05648 1.01407 0.99911 2.0 0.30489 0.00000 0.30815 0.00000 0.98942 3.0 0.29587 0.00000 0.29178 0.00000 1.01402 4.0 0.29122 0.00000 0.29409 0.00000 0.99024 5.0 0.28949 0.00000 0.29080 0.00000 0.99550 6.0 0.28524 0.00000 0.28215 0.00000 1.01095 7.0 0.28453 0.00000 0.28360 0.00000 1.00328 8.0 0.28296 0.00000 0.27690 0.00000 1.02189 9.0 0.28342 0.00000 0.27810 0.00000 1.01913 10.0 0.28321 0.00000 0.28529 0.00000 0.99271 PAGE 367 367 Table A 2 9 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.4 The first set The second set 0.1 0.93377 0.63873 0.93503 0.63639 0.99865 1.00368 0.2 0.60323 0.25830 0.60366 0.26184 0.99929 0.98648 0.3 0.48561 0.11305 0.48217 0.11619 1.00713 0.97298 0.4 0.42927 0.04494 0.42743 0.04562 1.00430 0.98509 0.5 0.39483 0.01389 0.39218 0.01255 1.00676 1.10677 0.6 0.37500 0.00169 0.37364 0.00177 1.00364 0.95480 0.7 0.35960 0.00000 0.35798 0.00000 1.00453 0.8 0.34913 0.00000 0.35339 0.00000 0.98795 0.9 0.33455 0.00000 0.33868 0.00000 0.98781 1.0 0.33528 0.00000 0.33399 0.00000 1.00386 2.0 0.30724 0.00000 0.30855 0.00000 0.99575 3.0 0.29820 0.00000 0.29613 0.00000 1.00699 4.0 0.29377 0.00000 0.29315 0.00000 1.00211 5.0 0.28964 0.00000 0.28851 0.00000 1.00392 6.0 0.28874 0.00000 0.28671 0.00000 1.00708 7.0 0.28904 0.00000 0.28527 0.00000 1.01322 8.0 0.28819 0.00000 0.28679 0.00000 1.00488 9.0 0.28841 0.00000 0.27762 0.00000 1.03887 10.0 0.28675 0.00000 0.27884 0.00000 1.02837 PAGE 368 368 Table A 30 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for uncoupled solutions between the first and second sets of investigations in case of = 0.5 The first set The second set 0.1 0.82836 3.04677 0.80995 3.04995 1.02273 0.99896 0.2 0.62011 1.32339 0.63219 1.32845 0.98089 0.99619 0.3 0.52144 0.74892 0.52876 0.75134 0.98616 0.99678 0.4 0.46300 0.46169 0.46867 0.46313 0.98790 0.99689 0.5 0.44440 0.31286 0.42726 0.31359 1.04012 0.99767 0.6 0.43054 0.22738 0.42878 0.22792 1.00410 0.99763 0.7 0.42075 0.16633 0.41176 0.16674 1.02183 0.99754 0.8 0.40720 0.12054 0.40113 0.12087 1.01513 0.99727 0.9 0.40374 0.08492 0.38390 0.08520 1.05168 0.99671 1.0 0.39927 0.05643 0.39809 0.05667 1.00296 0.99576 2.0 0.36827 0.00000 0.36170 0.00000 1.01816 3.0 0.36262 0.00000 0.35450 0.00000 1.02291 4.0 0.35736 0.00000 0.35075 0.00000 1.01885 5.0 0.35619 0.00000 0.34307 0.00000 1.03824 6.0 0.34665 0.00000 0.34378 0.00000 1.00835 7.0 0.34682 0.00000 0.34452 0.00000 1.00668 8.0 0.34806 0.00000 0.34176 0.00000 1.01843 9.0 0.34995 0.00000 0.34368 0.00000 1.01824 10.0 0.35060 0.00000 0.33904 0.00000 1.03410 PAGE 369 369 Table A 31 Comparison the value s of extent of saltwater intrusion ( ) and degree of saltwater recirculation ( ) for coupled solutions between the first and second sets of investigations in case of = 0.5 The first set The second set 0.1 0.89540 0.60122 0.89233 0.61730 1.00344 0.97395 0.2 0.61183 0.22686 0.60781 0.23390 1.00661 0.96990 0.3 0.51556 0.08576 0.51044 0.09041 1.01003 0.94857 0.4 0.47230 0.02664 0.46593 0.02922 1.01367 0.91170 0.5 0.44834 0.00413 0.43250 0.00554 1.03662 0.74549 0.6 0.42743 0.00000 0.42086 0.00000 1.01561 0.7 0.41610 0.00000 0.41443 0.00000 1.00403 0.8 0.40720 0.00000 0.39837 0.00000 1.02217 0.9 0.39860 0.00000 0.39791 0.00000 1.00173 1.0 0.39480 0.00000 0.39880 0.00000 0.98997 2.0 0.36723 0.00000 0.36167 0.00000 1.01537 3.0 0.36602 0.00000 0.36161 0.00000 1.01220 4.0 0.36141 0.00000 0.35564 0.00000 1.01622 5.0 0.36041 0.00000 0.35445 0.00000 1.01681 6.0 0.35456 0.00000 0.35760 0.00000 0.99150 7.0 0.35586 0.00000 0.35458 0.00000 1.00361 8.0 0.35750 0.00000 0.35835 0.00000 0.99763 9.0 0.35503 0.00000 0.34625 0.00000 1.02536 10.0 0.35798 0.00000 0.35727 0.00000 1.00199 PAGE 370 370 Accuracy and Stability Criteria for the Henry Constant Dispersion Problem of the Henry Constant Dispersion Problem Table A 32 Summation of the grid Pclet number of = 0.1 for the first set of inves tigations Case Molecular diffusion coefficient Max. Seepage velocity Grid Pclet number ( ) m 2 /day ( ) m/day ( ) Uncoupled solutions 0.1 0.617142857 97.2591877 7.8798 0.2 1.234285714 97.2591877 3.9399 0.3 1.851428571 97.2591877 2.6266 0.4 2.468571429 97.2591843 1.9699 0.5 3.085714286 97.2591877 1.5760 Coupled solutions 0.1 0.617142857 63.5373133 5.1477 0.2 1.234285714 59.6998368 2.4184 0.3 1.851428571 54.6757851 1.4766 0.4 2.468571429 50.0570502 1.0139 0.5 3.085714286 45.2935696 0.7339 Increasing dimensionless ratio from 0.1 to 0.5 corresponded to the growing of molecular diffusion coefficient ( ) from 0.617142857 to 3.085714286 For the uncoupled solutions, t here was no significant difference in the maximum of seepage velocity ( ) Grid Pclet number of the results were decreased by increasing the molecular diffusion coefficient ( ) from 7.8798 to 1.5760 For the coupled solutions, the maximum of seepage velocity decreased from 63.5373133 to 45.2935696 Grid Pclet number of the results were decreased by increasing of molecular diffusion coefficient ( ) and decreasing of seepage velocity from 5.1477 to 0.7339 In case of = 0.1 and = 0.1, both the uncoupled and coupled solution should be unstable and oscill a tory However, the remaining solutions should PAGE 371 371 be s table because the grid Pclet numbers were less than 4.0 ( Daus et al. 1985 Voss and Souza 1987, Voss and Provost 2002 and Zheng and Bennett 2002). Table A 33 Summation of the grid Pclet number of = 1.0 for the first set of inves tigations Case Molecular diffusion coefficient Max. Seepage velocity Grid Pclet number ( ) m 2 /day ( ) m/day ( ) Uncoupled solutions 0.1 6.171428571 152.802045 1.2380 0.2 12.34285714 152.802045 0.6190 0.3 18.51428571 152.802045 0.4127 0.4 24.68571429 152.802045 0.3095 0.5 30.85714286 152.802052 0.2476 Coupled solutions 0.1 6.171428571 147.227710 1.1928 0.2 12.34285714 127.334513 0.5158 0.3 18.51428571 114.170252 0.3083 0.4 24.68571429 105.277971 0.2132 0.5 30.85714286 99.0635736 0.1605 The s ummation of the grid Pclet number of = 1.0 for the first set of inves tigations were similar trend to the result of = 0.10. However, all of the uncoupled and coupled solutions should be stable because the grid Pclet numbers were less than 4.0 (Voss and Souza 1987, and Voss and Provost 2002) PAGE 372 372 Table A 34 Summation of the grid Pclet number of = 10.0 for the first set of inves tigations Case Molecular diffus ion coefficient Max. Seepage velocity Grid Pclet number ( ) m 2 /day ( ) m/day ( ) Uncoupled solutions 0.1 61.71428571 708.230618 0.5738 0.2 123.4285714 708.230618 0.2869 0.3 185.1428571 708.230618 0.1913 0.4 246.8571429 708.230618 0.1434 0.5 308.5714286 708.230618 0.1148 Coupled solutions 0.1 61.71428571 689.506667 0.5586 0.2 123.4285714 668.277005 0.2707 0.3 185.1428571 655.694689 0.1771 0.4 246.8571429 647.405488 0.1311 0.5 308.5714286 641.290801 0.1039 The s ummation of the grid Pclet number of = 1 0 .0 for the first set of inves tigations were similar trend to the result of = 0.1. All of the uncoupled and coupled solutions should be stable because the grid Pclet numbers were less than 4.0 (Voss and Souza 1987, and Voss and Provost 2002) PAGE 373 373 Table A 35 Summation of the grid Pclet number of the first set of i nvestigations = 0.1 = 1.0 = 10.0 Case Max. Grid Pclet number Max. Grid Pclet number Max. Grid Pclet number ( ) ( ) ( ) Uncoupled solutions 0.1 7.8798 1.2380 0.5738 0.2 3.9399 0.6190 0.2869 0.3 2.6266 0.4127 0.1913 0.4 1.9699 0.3095 0.1434 0.5 1.5760 0.2476 0.1148 Coupled solutions 0.1 5.1477 1.1928 0.5586 0.2 2.4184 0.5158 0.2707 0.3 1.4766 0.3083 0.1771 0.4 1.0139 0.2132 0.1311 0.5 0.7339 0.1605 0.1039 Although i n case of = 0.1 and = 0.1 both the uncoupled and coupled solution s presented the oscill a tory results, they reached steady state condition. However, the remaining solutions should be s table because the grid Pclet numbers were less than 4.0 ( Daus et al. 1985 Voss and Souza 1987, Voss and Provost 2002 and Zheng and Benne tt 2002). The comparison of grid Pclet number showed that the more increasing of dimensionless ratios and were assigned to the numerical model, the more stability result of model would present in the domain. PAGE 374 374 Table A 36 Summation of the grid Pclet number of uncoupled and coupled solutions for the first set of investigations in case of = 0.1. Molecular diffusion coefficient ( ) G rid Pclet number ( ) m 2 /day Uncoupled Coupled 0.1 0.617142857 7.8798 5.1477 0.2 1.234285714 4.1899 3.4588 0.3 1.851428571 2.9599 2.6804 0.4 2.468571429 2.3449 2.2097 0.5 3.085714286 1.9760 1.8922 0.6 3.702857143 1.7300 1.6673 0.7 4.32 0000000 1.5543 1.5008 0.8 4.937142857 1.4225 1.3734 0.9 5.554285714 1.3200 1.2733 1.0 6.171428571 1.2380 1.1928 2.0 12.34285714 0.8690 0.8325 3.0 18.51428571 0.7460 0.7159 4.0 24.68571429 0.6845 0.6588 5.0 30.85714286 0.6476 0.6250 6.0 37.02857143 0.6230 0.6027 7.0 43.2 0000000 0.6054 0.5869 8.0 49.37142857 0.5922 0.5751 9.0 55.54285714 0.5820 0.5659 10.0 61.71428571 0.5738 0.5586 The comparison of grid Pclet number showed that the more increasing of dimensionless ratio was assigned to the numerical model, the more stability result of model would present in the domain. PAGE 375 375 Table A 37 Summation of the grid Pclet number of = 0.1 for the second set of i nvestigations Case Molecular diffusion coefficient Hydraulic conductivity Max. Seepage velocity Grid Pclet number ( ) m 2 /day ( ) m/day ( ) m/day ( ) Uncoupled solutions 0.1 1.629249984 2 282.1010 256.883328 7.8835 0.2 1.629249984 1 141.0500 128.449784 3.9420 0.3 1.629249984 760.7000 85.625860 2.6278 0.4 1.629249984 570.5250 64.221721 1.9709 0.5 1.629249984 456.4200 51.375517 1.5767 Coupled solutions 0.1 1.629249984 2 282.1010 167.712825 5.1469 0.2 1.629249984 1 141.0500 78.841952 2.4196 0.3 1.629249984 760.7000 48.261888 1.4811 0.4 1.629249984 570.5250 32.994918 1.0126 0.5 1.629249984 456.4200 24.249611 0.7442 Increasing dimensionless ratio from 0.1 to 0.5 corresponded to the decreasing of hydraulic conductivity ( ) from 2,282.1010 to 456.4200 The value of molecular was held constant, i.e., 1.629249984 In case of = 0.1 and = 0.1, both the uncoupled and coupled solution s should be unstable and oscillatory However, the remaining solutions were stable because the grid Pclet numbers were less than 4.0 (Daus et al. 1985, Voss and Souza 1987, Voss and Provost 2002, and Zheng and Benne tt 2002). The comparison of grid Pclet number showed that the more increasing of dimensionless ratios and were assigned to the numerical model, the more stability result of model would present in the domain. PAGE 376 376 Table A 38 Summation of the grid Pclet number of = 1.0 for the second set of i nvestigations Case Molecular diffusion coefficient Hydraulic conductivity Max. Seepage velocity Grid Pclet number ( ) m 2 /day ( ) m/day ( ) m/day ( ) Uncoupled solutions 0.1 1.629249984 228.2100 40.350614 1.2383 0.2 1.629249984 114.1050 20.178693 0.6193 0.3 1.629249984 76.0700 13.448301 0.4127 0.4 1.629249984 57.0520 10.089030 0.3096 0.5 1.629249984 45.6420 8.068980 0.2476 Coupled solutions 0.1 1.629249984 228.2100 38.880273 1.1932 0.2 1.629249984 114.1050 16.820719 0.5162 0.3 1.629249984 76.0700 10.050667 0.3084 0.4 1.629249984 57.0520 6.943344 0.2131 0.5 1.629249984 45.6420 5.217720 0.1601 The summation of the grid Pclet number of = 1.0 for the second set of inves tigations were similar to = 0.1 However, all of the uncoupled and coupled solutions should be stable because the grid Pclet numbers were less than 4.0 (Voss and Souza 1987 and Voss and Provost 2002). PAGE 377 377 Table A 39 Summation of the grid Pclet number of = 10.0 for the second set of i nvestigations Case Molecular diffusion coefficient Hydraulic conductivity Max. Seepage velocity Grid Pclet number ( ) m 2 /day ( ) m/day ( ) m/day ( ) Uncoupled solutions 0.1 1.629249984 22.8100 18.696187 0.5738 0.2 1.629249984 11.4050 9.350951 0.2870 0.3 1.629249984 7.6030 6.230122 0.1912 0.4 1.629249984 5.7020 4.675423 0.1435 0.5 1.629249984 4.5620 3.738095 0.1147 Coupled solutions 0.1 1.629249984 22.8100 18.202073 0.5586 0.2 1.629249984 11.4050 8.823865 0.2708 0.3 1.629249984 7.6030 5.768289 0.1770 0.4 1.629249984 5.7020 4.278211 0.1313 0.5 1.629249984 4.5620 3.387439 0.1040 The summation of the grid Pclet number of = 10.0 for the second set of inves tigations were similar to = 0.1. All of the uncoupled and coupled solutions should be stable because the grid Pclet numbers were less than 4.0 (Voss and Souza 1987, and Voss and Provost 2002). PAGE 378 378 Table A 40 Summation of the grid Pclet number of the second set of i nvestigations = 0.1 = 1.0 = 10.0 Case Max. Grid Pclet number Max. Grid Pclet number Max. Grid Pclet number ( ) ( ) ( ) Uncoupled solutions 0.1 7.8835 1.2383 0.5738 0.2 3.9420 0.6193 0.2870 0.3 2.6278 0.4127 0.1912 0.4 1.9709 0.3096 0.1435 0.5 1.5767 0.2476 0.1147 Coupled solutions 0.1 5.1469 1.1932 0.5586 0.2 2.4196 0.5162 0.2708 0.3 1.4811 0.3084 0.1770 0.4 1.0126 0.2131 0.1313 0.5 0.7442 0.1601 0.1040 The comparison of grid Pclet number showed that the more increasing of dimensionless ratios and were assigned to the numerical model, the more stability result s of model would present in the domain PAGE 379 379 Table A 41 Summation of the grid Pclet number of uncoupled and coupled solutions for the second set of investigations in case of = 0.1. Molecular diffusion coefficient ( ) Hydraulic conductivity ( ) G rid Pclet number ( ) m 2 /day m/day Uncoupled Coupled 0.1 1.629249984 2 282.101 7.8835 5.1469 0.2 1.629249984 1 143.338 4.1990 3.4617 0.3 1.629249984 761.589 2.9640 2.6832 0.4 1.629249984 570.953 2.3472 2.2115 0.5 1.629249984 456.648 1.9774 1.8938 0.6 1.629249984 380.477 1.7310 1.6684 0.7 1.629249984 326.084 1.5550 1.5016 0.8 1.629249984 285.298 1.4230 1.3740 0.9 1.629249984 253.580 1.3204 1.2738 1.0 1.629249984 228.210 1.2383 1.1932 2.0 1.629249984 114.076 0.8691 0.8326 3.0 1.629249984 76.044 0.7460 0.7159 4.0 1.629249984 57.031 0.6845 0.6588 5.0 1.629249984 45.623 0.6476 0.6250 6.0 1.629249984 38.019 0.6230 0.6027 7.0 1.629249984 32.587 0.6054 0.5869 8.0 1.629249984 28.513 0.5922 0.5750 9.0 1.629249984 25.345 0.5820 0.5659 10.0 1.629249984 22.810 0.5738 0.5586 When comparing the grid Pclet number of the uncoupled and coupled solutions for the first and second sets of investigations in case of = 0.1 for the Henry constant dispersion problem there was no significant differences between them. PAGE 380 380 APPENDIX B BENCHMARK AND HENRY VELOCITY DE PENDENT DISPERSION PROBLEMS Benchmark Problems Abarca et al. (2007) was selected to be a benchmark problem since they performed simulations which account for the anisotropic hydraulic conductivity of (1964) problem Additionally, Sgol (1993) was also selected to a benchmark problem because Abarca et a l. (2007) (1964) model by comparing the 0.5 isochlors to Sgol (1994) Thus, Sgol (1993) and Abarca et al. (2007) were select to be benchmark problems. Sgol (1993) (1964) anal ytical solution had some errors (1964) solution by using the semi analytical solution Sgol (1993) (1964) problem, as shown in Figur e B 1 (A ) Abarca et al. (2007) reproduced the coupled solution s (1964) problem by using SUTRA which was a model for sat urated unsaturated, variable density ground water flow with solute or energy transport, as shown in Figur e B 1 (B ) Moreover, Abarca et al. (2007) created two new problems namely the diffusive (constant dispersion) and the dispersive reference ca ses which Abarca et al. (2007) assigned the e B 2. PAGE 381 381 A B Figur e B 1 The works of A) Sgol (1993) and B) Abarca et al. (2007) present the coupled solutions of Henry problem A B Figur e B 2 T he coupled solutions for the reference cases A) diffusive case (constant dispersion case) and B) dispersive case ( velocity dependent dispersion case) as Abarca et al. (2007) SEAWAT Version 4 Results of Benchmark Problems SEAWAT Version 4 model presented the coupled solutions, as shown in Figur es B 3 to D 6, which were in good agreement with the work s of Sgol (1993) and Abarca et al. (2007) PAGE 382 382 A B Figur e B 3 The SEAWAT Version 4 results for the coupled solution s of Henry problem using the manually input equivalent freshwater heads and the varied concentration at the downgradient conditions A) Sgol (1993) and B) Abarca et al. (2007) PAGE 383 383 A B Figur e B 4 The SEAWAT Version 4 results for the coupled solution s of Henry problem using the manually input equivalent freshwater heads and the varied concentration at the downgradient conditions in shape of isochlors for A) Sgol (1993) and B) Abarca et al. (2007) with concentration color floods and velocity vectors PAGE 384 384 A B Figur e B 5 The SEAWAT Version 4 results for the coupled solutions of Abarca et al. (2007) reference cases, using the manually input equivalent freshwater heads and the varied concentration at the downgradient conditions A) diffusive case (constant dispers ion case) and B) dispersive case ( velocity dependent dispersion case). PAGE 385 385 A B Figur e B 6 The SEAWAT Version 4 results for the coupled solutions of Abarca et al. (2007) reference cases, using the manually input equi valent freshwater heads and the varied concentration at the downgradient conditions in shape of isochlor s for A) diffusive case (constant dispersion case) and B) dispersive case ( velocity dependent dispersion case) with concentration color floods and velocity vectors PAGE 386 386 Aquifer Parameters and Dimensionless Variables of the Henry Velocity dependent Dispersion Problem Table B 1 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.01 and = 0.01. and and m/day m/day m 2 /day m m 0.1 864 8.64 0.00108 0.2 0.002 0.2 864 8.64 0.00216 0.2 0.002 0.3 864 8.64 0.00324 0.2 0.002 0.4 864 8.64 0.00432 0.2 0.002 0.5 864 8.64 0.00540 0.2 0.002 0.6 864 8.64 0.00648 0.2 0.002 0.7 864 8.64 0.00756 0.2 0.002 0.8 864 8.64 0.00864 0.2 0.002 0.9 864 8.64 0.00972 0.2 0.002 1.0 864 8.64 0.01080 0.2 0.002 2.0 864 8.64 0.02160 0.2 0.002 3.0 864 8.64 0.03240 0.2 0.002 4.0 864 8.64 0.04320 0.2 0.002 5.0 864 8.64 0.05400 0.2 0.002 6.0 864 8.64 0.06480 0.2 0.002 7.0 864 8.64 0.07560 0.2 0.002 8.0 864 8.64 0.08640 0.2 0.002 9.0 864 8.64 0.09720 0.2 0.002 10.0 864 8.64 0.10800 0.2 0.002 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 387 387 Table B 2 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.01 and = 0.10. and and m/day m/day m 2 /day m m 0.1 864 86.4 0.0108 0.2 0.002 0.2 864 86.4 0.0216 0.2 0.002 0.3 864 86.4 0.0324 0.2 0.002 0.4 864 86.4 0.0432 0.2 0.002 0.5 864 86.4 0.0540 0.2 0.002 0.6 864 86.4 0.0648 0.2 0.002 0.7 864 86.4 0.0756 0.2 0.002 0.8 864 86.4 0.0864 0.2 0.002 0.9 864 86.4 0.0972 0.2 0.002 1.0 864 86.4 0.1080 0.2 0.002 2.0 864 86.4 0.2160 0.2 0.002 3.0 864 86.4 0.3240 0.2 0.002 4.0 864 86.4 0.4320 0.2 0.002 5.0 864 86.4 0.5400 0.2 0.002 6.0 864 86.4 0.6480 0.2 0.002 7.0 864 86.4 0.7560 0.2 0.002 8.0 864 86.4 0.8640 0.2 0.002 9.0 864 86.4 0.9720 0.2 0.002 10.0 864 86.4 1.0800 0.2 0.002 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 388 388 Table B 3 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.01 and = 1.00. and and m/day m/day m 2 /day m m 0.1 864 864 0.108 0.2 0.002 0.2 864 864 0.216 0.2 0.002 0.3 864 864 0.324 0.2 0.002 0.4 864 864 0.432 0.2 0.002 0.5 864 864 0.540 0.2 0.002 0.6 864 864 0.648 0.2 0.002 0.7 864 864 0.756 0.2 0.002 0.8 864 864 0.864 0.2 0.002 0.9 864 864 0.972 0.2 0.002 1.0 864 864 1.080 0.2 0.002 2.0 864 864 2.160 0.2 0.002 3.0 864 864 3.240 0.2 0.002 4.0 864 864 4.320 0.2 0.002 5.0 864 864 5.400 0.2 0.002 6.0 864 864 6.480 0.2 0.002 7.0 864 864 7.560 0.2 0.002 8.0 864 864 8.640 0.2 0.002 9.0 864 864 9.720 0.2 0.002 10.0 864 864 10.800 0.2 0.002 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 389 389 Table B 4 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.10 and = 0.01. and and m/day m/day m 2 /day m m 0.1 864 8.640 0.00108 0.2 0.02 0.2 864 8.640 0.00216 0.2 0.02 0.3 864 8.640 0.00324 0.2 0.02 0.4 864 8.640 0.00432 0.2 0.02 0.5 864 8.640 0.00540 0.2 0.02 0.6 864 8.640 0.00648 0.2 0.02 0.7 864 8.640 0.00756 0.2 0.02 0.8 864 8.640 0.00864 0.2 0.02 0.9 864 8.640 0.00972 0.2 0.02 1.0 864 8.640 0.01080 0.2 0.02 2.0 864 8.640 0.02160 0.2 0.02 3.0 864 8.640 0.03240 0.2 0.02 4.0 864 8.640 0.04320 0.2 0.02 5.0 864 8.640 0.05400 0.2 0.02 6.0 864 8.640 0.06480 0.2 0.02 7.0 864 8.640 0.07560 0.2 0.02 8.0 864 8.640 0.08640 0.2 0.02 9.0 864 8.640 0.09720 0.2 0.02 10.0 864 8.640 0.10800 0.2 0.02 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 390 390 Table B 5 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.10 and = 0.10. and and m/day m/day m 2 /day m m 0.1 864 86.4 0.0108 0.2 0.02 0.2 864 86.4 0.0216 0.2 0.02 0.3 864 86.4 0.0324 0.2 0.02 0.4 864 86.4 0.0432 0.2 0.02 0.5 864 86.4 0.0540 0.2 0.02 0.6 864 86.4 0.0648 0.2 0.02 0.7 864 86.4 0.0756 0.2 0.02 0.8 864 86.4 0.0864 0.2 0.02 0.9 864 86.4 0.0972 0.2 0.02 1.0 864 86.4 0.1080 0.2 0.02 2.0 864 86.4 0.2160 0.2 0.02 3.0 864 86.4 0.3240 0.2 0.02 4.0 864 86.4 0.4320 0.2 0.02 5.0 864 86.4 0.5400 0.2 0.02 6.0 864 86.4 0.6480 0.2 0.02 7.0 864 86.4 0.7560 0.2 0.02 8.0 864 86.4 0.8640 0.2 0.02 9.0 864 86.4 0.9720 0.2 0.02 10.0 864 86.4 1.0800 0.2 0.02 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 391 391 Table B 6 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 0.10 and = 1.00. and and m/day m/day m 2 /day m m 0.1 864 864 0.108 0.2 0.02 0.2 864 864 0.216 0.2 0.02 0.3 864 864 0.324 0.2 0.02 0.4 864 864 0.432 0.2 0.02 0.5 864 864 0.540 0.2 0.02 0.6 864 864 0.648 0.2 0.02 0.7 864 864 0.756 0.2 0.02 0.8 864 864 0.864 0.2 0.02 0.9 864 864 0.972 0.2 0.02 1.0 864 864 1.080 0.2 0.02 2.0 864 864 2.160 0.2 0.02 3.0 864 864 3.240 0.2 0.02 4.0 864 864 4.320 0.2 0.02 5.0 864 864 5.400 0.2 0.02 6.0 864 864 6.480 0.2 0.02 7.0 864 864 7.560 0.2 0.02 8.0 864 864 8.640 0.2 0.02 9.0 864 864 9.720 0.2 0.02 10.0 864 864 10.800 0.2 0.02 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 392 392 Table B 7 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 1.00 and = 0.01. and and m/day m/day m 2 /day m m 0.1 864 8.640 0.00108 0.2 0.2 0.2 864 8.640 0.00216 0.2 0.2 0.3 864 8.640 0.00324 0.2 0.2 0.4 864 8.640 0.00432 0.2 0.2 0.5 864 8.640 0.00540 0.2 0.2 0.6 864 8.640 0.00648 0.2 0.2 0.7 864 8.640 0.00756 0.2 0.2 0.8 864 8.640 0.00864 0.2 0.2 0.9 864 8.640 0.00972 0.2 0.2 1.0 864 8.640 0.01080 0.2 0.2 2.0 864 8.640 0.02160 0.2 0.2 3.0 864 8.640 0.03240 0.2 0.2 4.0 864 8.640 0.04320 0.2 0.2 5.0 864 8.640 0.05400 0.2 0.2 6.0 864 8.640 0.06480 0.2 0.2 7.0 864 8.640 0.07560 0.2 0.2 8.0 864 8.640 0.08640 0.2 0.2 9.0 864 8.640 0.09720 0.2 0.2 10.0 864 8.640 0.10800 0.2 0.2 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 393 393 Table B 8 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 1.00 and = 0.10. and and m/day m/day m 2 /day m m 0.1 864 86.4 0.0108 0.2 0.2 0.2 864 86.4 0.0216 0.2 0.2 0.3 864 86.4 0.0324 0.2 0.2 0.4 864 86.4 0.0432 0.2 0.2 0.5 864 86.4 0.0540 0.2 0.2 0.6 864 86.4 0.0648 0.2 0.2 0.7 864 86.4 0.0756 0.2 0.2 0.8 864 86.4 0.0864 0.2 0.2 0.9 864 86.4 0.0972 0.2 0.2 1.0 864 86.4 0.1080 0.2 0.2 2.0 864 86.4 0.2160 0.2 0.2 3.0 864 86.4 0.3240 0.2 0.2 4.0 864 86.4 0.4320 0.2 0.2 5.0 864 86.4 0.5400 0.2 0.2 6.0 864 86.4 0.6480 0.2 0.2 7.0 864 86.4 0.7560 0.2 0.2 8.0 864 86.4 0.8640 0.2 0.2 9.0 864 86.4 0.9720 0.2 0.2 10.0 864 86.4 1.0800 0.2 0.2 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 394 394 Table B 9 Aquifer parameters and dimensionless variables for the Henry velocity dependent dispersion problem in case of = 1.00 and = 1.00. and and m/day m/day m 2 /day m m 0.1 864 864 0.108 0.2 0.2 0.2 864 864 0.216 0.2 0.2 0.3 864 864 0.324 0.2 0.2 0.4 864 864 0.432 0.2 0.2 0.5 864 864 0.540 0.2 0.2 0.6 864 864 0.648 0.2 0.2 0.7 864 864 0.756 0.2 0.2 0.8 864 864 0.864 0.2 0.2 0.9 864 864 0.972 0.2 0.2 1.0 864 864 1.080 0.2 0.2 2.0 864 864 2.160 0.2 0.2 3.0 864 864 3.240 0.2 0.2 4.0 864 864 4.320 0.2 0.2 5.0 864 864 5.400 0.2 0.2 6.0 864 864 6.480 0.2 0.2 7.0 864 864 7.560 0.2 0.2 8.0 864 864 8.640 0.2 0.2 9.0 864 864 9.720 0.2 0.2 10.0 864 864 10.800 0.2 0.2 Note: 1. Porosity ( ) = 0.35; 2. Density contrast parameter ( ) = 0 .025; 3. Vertical dimension of the medium ( ) = 1.0 m ; and 4. Molecular diffusion coefficient ( ) = 0 m 2 /day PAGE 395 395 Results of E xtent of S altwater I ntrusion ( ) and Degree of sal twater recirculation ( ) of the Henry Velocity dependent Dispersion Problem Tabl e B 10 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.01 and = 0.01 Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 1.96286 536.838 21.600 1.96286 24.85361 1.98974 132.954 21.600 1.98974 6.15528 0.2 1.95496 526.038 43.200 1.95496 12.17681 1.96318 127.451 43.200 1.96318 2.95025 0.3 1.92221 515.238 64.800 1.92221 7.95120 1.95304 122.408 64.800 1.95304 1.88901 0.4 1.90870 504.438 86.400 1.90870 5.83840 1.94712 117.467 86.400 1.94712 1.35957 0.5 1.89331 493.638 108.000 1.89331 4.57072 1.94350 112.620 108.000 1.94350 1.04278 0.6 1.86193 482.838 129.600 1.86193 3.72560 1.94313 108.075 129.600 1.94313 0.83391 0.7 1.85435 472.038 151.200 1.85435 3.12194 1.93655 103.602 151.200 1.93655 0.68520 0.8 1.81733 463.015 172.800 1.81733 2.67948 1.91171 99.196 172.800 1.91171 0.57405 0.9 1.80787 454.375 194.400 1.80787 2.33732 1.89668 95.049 194.400 1.89668 0.48894 1.0 1.77717 445.735 216.000 1.77717 2.06359 1.88807 91.039 216.000 1.88807 0.42148 2.0 1.56830 361.145 432.000 1.56830 0.83598 1.79655 61.557 432.000 1.79655 0.14249 3.0 1.36874 295.079 648.000 1.36874 0.45537 1.67468 46.381 648.000 1.67468 0.07158 4.0 1.18590 230.279 864.000 1.18590 0.26653 1.60918 38.122 864.000 1.60918 0.04412 5.0 1.01617 175.155 1,080.000 1.01617 0.16218 1.39526 32.482 1,080.000 1.39526 0.03008 6.0 0.86077 131.181 1,296.000 0.86077 0.10122 1.21178 28.151 1,296.000 1.21178 0.02172 7.0 0.71658 96.410 1,512.000 0.71658 0.06376 1.02765 24.822 1,512.000 1.02765 0.01642 8.0 0.57955 64.812 1,728.000 0.57955 0.03751 0.86313 20.683 1,728.000 0.86313 0.01197 9.0 0.47696 54.012 1,944.000 0.47696 0.02778 0.68265 18.349 1,944.000 0.68265 0.00944 10.0 0.38145 43.212 2,160.000 0.38145 0.02001 0.54872 15.427 2,160.000 0.54872 0.00714 PAGE 396 396 Tabl e B 1 1 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.01 and = 0.10. Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 1.92453 2 395.120 216.00 1.92453 11.08852 1.98858 157.052 216.00 1.98858 0.72709 0.2 1.83081 2 297.407 432.00 1.83081 5.31807 1.96126 158.718 432.00 1.96126 0.36740 0.3 1.74034 2 200.207 648.00 1.74034 3.39538 1.95368 157.667 648.00 1.95368 0.24331 0.4 1.65990 2 103.007 864.00 1.65990 2.43404 1.95278 155.292 864.00 1.95278 0.17974 0.5 1.57309 2 005.807 1,080.00 1.57309 1.85723 1.95364 153.272 1,080.00 1.95364 0.14192 0.6 1.48254 1 908.607 1,296.00 1.48254 1.47269 1.92892 151.143 1,296.00 1.92892 0.11662 0.7 1.41329 1 811.407 1,512.00 1.41329 1.19802 1.90487 151.528 1,512.00 1.90487 0.10022 0.8 1.32976 1 727.437 1,728.00 1.32976 0.99967 1.90448 147.253 1,728.00 1.90448 0.08522 0.9 1.26725 1 651.837 1,944.00 1.26725 0.84971 1.90837 143.388 1,944.00 1.90837 0.07376 1.0 1.19190 1 576.237 2,160.00 1.19190 0.72974 1.85556 140.452 2,160.00 1.85556 0.06502 2.0 0.69277 978.629 4,320.00 0.69277 0.22653 1.25280 100.335 4,320.00 1.25280 0.02323 3.0 0.41174 546.629 6,480.00 0.41174 0.08436 0.62794 62.418 6,480.00 0.62794 0.00963 4.0 0.23957 210.396 8,640.00 0.23957 0.02435 0.30902 27.948 8,640.00 0.30902 0.00323 5.0 0.16074 90.382 10,800.00 0.16074 0.00837 0.13784 0.000 10,800.00 0.13784 0.00000 6.0 0.12792 0.000 12,960.00 0.12792 0.00000 0.13968 0.000 12,960.00 0.13968 0.00000 7.0 0.13699 0.000 15,120.00 0.13699 0.00000 0.14009 0.000 15,120.00 0.14009 0.00000 8.0 0.13889 0.000 17,280.00 0.13889 0.00000 0.14030 0.000 17,280.00 0.14030 0.00000 9.0 0.13953 0.000 19,440.00 0.13953 0.00000 0.14041 0.000 19,440.00 0.14041 0.00000 10.0 0.13987 0.000 21,600.00 0.13987 0.00000 0.14051 0.000 21,600.00 0.14051 0.00000 PAGE 397 397 Tabl e B 1 2 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.01 and = 1.00. Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.81785 6 581.033 2,160.0 0.81785 3.04677 1.94826 310.000 2,160.0 1.94826 0.14352 0.2 0.58839 5 717.033 4,320.0 0.58839 1.32339 1.75460 231.676 4,320.0 1.75460 0.05363 0.3 0.47416 4 853.033 6,480.0 0.47416 0.74892 1.16324 238.602 6,480.0 1.16324 0.03682 0.4 0.38086 3 989.033 8,640.0 0.38086 0.46169 0.81291 233.234 8,640.0 0.81291 0.02699 0.5 0.32479 3 378.888 10,800.0 0.32479 0.31286 0.61839 205.456 10,800.0 0.61839 0.01902 0.6 0.27549 2 946.888 12,960.0 0.27549 0.22738 0.47145 165.237 12,960.0 0.47145 0.01275 0.7 0.23259 2 514.888 15,120.0 0.23259 0.16633 0.33903 134.481 15,120.0 0.33903 0.00889 0.8 0.19110 2 082.888 17,280.0 0.19110 0.12054 0.28597 70.021 17,280.0 0.28597 0.00405 0.9 0.17980 1 650.888 19,440.0 0.17980 0.08492 0.19665 43.416 19,440.0 0.19665 0.00223 1.0 0.14389 1 218.888 21,600.0 0.14389 0.05643 0.14202 14.221 21,600.0 0.14202 0.00066 2.0 0.12190 0.000 43,200.0 0.12190 0.00000 0.14019 0.000 43,200.0 0.14019 0.00000 3.0 0.13798 0.000 64,800.0 0.13798 0.00000 0.14071 0.000 64,800.0 0.14071 0.00000 4.0 0.13974 0.000 86,400.0 0.13974 0.00000 0.14092 0.000 86,400.0 0.14092 0.00000 5.0 0.14036 0.000 108,000.0 0.14036 0.00000 0.14103 0.000 108,000.0 0.14103 0.00000 6.0 0.14069 0.000 129,600.0 0.14069 0.00000 0.14113 0.000 129,600.0 0.14113 0.00000 7.0 0.14081 0.000 151,200.0 0.14081 0.00000 0.14124 0.000 151,200.0 0.14124 0.00000 8.0 0.14092 0.000 172,800.0 0.14092 0.00000 0.14133 0.000 172,800.0 0.14133 0.00000 9.0 0.14103 0.000 194,400.0 0.14103 0.00000 0.14133 0.000 194,400.0 0.14133 0.00000 10.0 0.14111 0.000 216,000.0 0.14111 0.00000 0.14143 0.000 216,000.0 0.14143 0.00000 PAGE 398 398 Tabl e B 1 3 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.10 and = 0. 0 1 Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 1.97303 536.838 21.600 1.97303 24.85361 1.97633 146.850 21.600 1.97633 6.79861 0.2 1.95351 526.038 43.200 1.95351 12.17681 1.94846 141.761 43.200 1.94846 3.28150 0.3 1.91942 515.238 64.800 1.91942 7.95120 1.92818 136.533 64.800 1.92818 2.10699 0.4 1.90420 504.438 86.400 1.90420 5.83840 1.91557 131.705 86.400 1.91557 1.52436 0.5 1.88109 493.638 108.000 1.88109 4.57072 1.90553 126.815 108.000 1.90553 1.17421 0.6 1.85575 482.838 129.600 1.85575 3.72560 1.89356 121.753 129.600 1.89356 0.93945 0.7 1.84295 472.038 151.200 1.84295 3.12194 1.87554 116.489 151.200 1.87554 0.77043 0.8 1.80874 463.015 172.800 1.80874 2.67948 1.85765 111.615 172.800 1.85765 0.64592 0.9 1.80874 454.375 194.400 1.80874 2.33732 1.84283 106.908 194.400 1.84283 0.54994 1.0 1.76457 445.735 216.000 1.76457 2.06359 1.83012 102.157 216.000 1.83012 0.47295 2.0 1.55156 361.145 432.000 1.55156 0.83598 1.67152 73.286 432.000 1.67152 0.16964 3.0 1.33861 295.079 648.000 1.33861 0.45537 1.48077 64.375 648.000 1.48077 0.09934 4.0 1.12978 230.279 864.000 1.12978 0.26653 1.26037 59.416 864.000 1.26037 0.06877 5.0 0.94136 175.155 1,080.000 0.94136 0.16218 1.04171 53.597 1,080.000 1.04171 0.04963 6.0 0.76516 131.181 1,296.000 0.76516 0.10122 0.84316 45.995 1,296.000 0.84316 0.03549 7.0 0.62627 96.410 1,512.000 0.62627 0.06376 0.68441 40.372 1,512.000 0.68441 0.02670 8.0 0.54758 64.812 1,728.000 0.54758 0.03751 0.54974 32.635 1,728.000 0.54974 0.01889 9.0 0.43856 54.012 1,944.000 0.43856 0.02778 0.47789 28.094 1,944.000 0.47789 0.01445 10.0 0.35299 43.212 2,160.000 0.35299 0.02001 0.42407 22.654 2,160.000 0.42407 0.01049 PAGE 399 399 Tabl e B 1 4 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.10 and = 0.10. Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 1.92563 2 395.120 216.00 1.92563 11.08852 # 273.480 216.00 # 1.26611 0.2 1.82969 2 297.407 432.00 1.82969 5.31807 1.96574 310.310 432.00 1.96574 0.71831 0.3 1.73687 2 200.207 648.00 1.73687 3.39538 1.95309 324.598 648.00 1.95309 0.50092 0.4 1.65369 2 103.007 864.00 1.65369 2.43404 1.94481 328.728 864.00 1.94481 0.38047 0.5 1.56712 2 005.807 1,080.00 1.56712 1.85723 1.92143 329.503 1,080.00 1.92143 0.30510 0.6 1.47580 1 908.607 1,296.00 1.47580 1.47269 1.88514 326.490 1,296.00 1.88514 0.25192 0.7 1.40417 1 811.407 1,512.00 1.40417 1.19802 1.86393 321.882 1,512.00 1.86393 0.21288 0.8 1.32219 1 727.437 1,728.00 1.32219 0.99967 1.82093 315.071 1,728.00 1.82093 0.18233 0.9 1.25765 1 651.837 1,944.00 1.25765 0.84971 1.77611 307.176 1,944.00 1.77611 0.15801 1.0 1.18016 1 576.237 2,160.00 1.18016 0.72974 1.71279 298.613 2,160.00 1.71279 0.13825 2.0 0.67865 978.629 4,320.00 0.67865 0.22653 0.96535 194.678 4,320.00 0.96535 0.04506 3.0 0.39653 546.629 6,480.00 0.39653 0.08436 0.50905 102.876 6,480.00 0.50905 0.01588 4.0 0.23270 210.396 8,640.00 0.23270 0.02435 0.27869 41.465 8,640.00 0.27869 0.00480 5.0 0.14918 90.382 10,800.00 0.14918 0.00837 0.13227 0.000 10,800.00 0.13227 0.00000 6.0 0.12431 0.000 12,960.00 0.12431 0.00000 0.13689 0.000 12,960.00 0.13689 0.00000 7.0 0.13326 0.000 15,120.00 0.13326 0.00000 0.13814 0.000 15,120.00 0.13814 0.00000 8.0 0.13587 0.000 17,280.00 0.13587 0.00000 0.13883 0.000 17,280.00 0.13883 0.00000 9.0 0.13718 0.000 19,440.00 0.13718 0.00000 0.13926 0.000 19,440.00 0.13926 0.00000 10.0 0.13793 0.000 21,600.00 0.13793 0.00000 0.13957 0.000 21,600.00 0.13957 0.00000 Note: # This value was greater than 2.0 m, i.e., the length of domain. PAGE 400 400 Tabl e B 1 5 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 0.10 and = 1.00. Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.81642 6 581.033 2,160.0 0.81642 3.04677 1.91949 591.897 2,160.0 1.91949 0.27403 0.2 0.58876 5 717.033 4,320.0 0.58876 1.32339 1.41816 563.321 4,320.0 1.41816 0.13040 0.3 0.47287 4 853.033 6,480.0 0.47287 0.74892 0.94147 523.026 6,480.0 0.94147 0.08071 0.4 0.38080 3 989.033 8,640.0 0.38080 0.46169 0.67582 455.091 8,640.0 0.67582 0.05267 0.5 0.32382 3 378.888 10,800.0 0.32382 0.31286 0.54736 372.877 10,800.0 0.54736 0.03453 0.6 0.27468 2 946.888 12,960.0 0.27468 0.22738 0.40618 288.992 12,960.0 0.40618 0.02230 0.7 0.23215 2 514.888 15,120.0 0.23215 0.16633 0.31578 206.540 15,120.0 0.31578 0.01366 0.8 0.19238 2 082.888 17,280.0 0.19238 0.12054 0.23556 136.605 17,280.0 0.23556 0.00791 0.9 0.17939 1 650.888 19,440.0 0.17939 0.08492 0.18775 68.836 19,440.0 0.18775 0.00354 1.0 0.14518 1 218.888 21,600.0 0.14518 0.05643 0.13684 27.912 21,600.0 0.13684 0.00129 2.0 0.11816 0.000 43,200.0 0.11816 0.00000 0.13803 0.000 43,200.0 0.13803 0.00000 3.0 0.13462 0.000 64,800.0 0.13462 0.00000 0.13947 0.000 64,800.0 0.13947 0.00000 4.0 0.13750 0.000 86,400.0 0.13750 0.00000 0.14009 0.000 86,400.0 0.14009 0.00000 5.0 0.13868 0.000 108,000.0 0.13868 0.00000 0.14049 0.000 108,000.0 0.14049 0.00000 6.0 0.13932 0.000 129,600.0 0.13932 0.00000 0.14071 0.000 129,600.0 0.14071 0.00000 7.0 0.13974 0.000 151,200.0 0.13974 0.00000 0.14081 0.000 151,200.0 0.14081 0.00000 8.0 0.14004 0.000 172,800.0 0.14004 0.00000 0.14092 0.000 172,800.0 0.14092 0.00000 9.0 0.14026 0.000 194,400.0 0.14026 0.00000 0.14103 0.000 194,400.0 0.14103 0.00000 10.0 0.14047 0.000 216,000.0 0.14047 0.00000 0.14113 0.000 216,000.0 0.14113 0.00000 PAGE 401 401 Tabl e B 1 6 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 1.00 and = 0. 0 1 Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 # 536.838 21.600 # 24.85361 1.93506 168.358 21.600 1.93513 7.79435 0.2 1.96728 526.038 43.200 1.96728 12.17681 1.87814 165.404 43.200 1.87814 3.82880 0.3 1.92274 515.238 64.800 1.92274 7.95120 1.83558 161.666 64.800 1.83558 2.49485 0.4 1.88818 504.438 86.400 1.88818 5.83840 1.79903 157.836 86.400 1.79903 1.82681 0.5 1.85328 493.638 108.000 1.85328 4.57072 1.76396 154.489 108.000 1.76396 1.43045 0.6 1.81791 482.838 129.600 1.81791 3.72560 1.73003 150.582 129.600 1.73003 1.16190 0.7 1.78536 472.038 151.200 1.78536 3.12194 1.69700 146.885 151.200 1.69700 0.97146 0.8 1.75258 463.015 172.800 1.75258 2.67948 1.66278 143.469 172.800 1.66278 0.83026 0.9 1.71865 454.375 194.400 1.71865 2.33732 1.62806 139.921 194.400 1.62806 0.71976 1.0 1.68503 445.735 216.000 1.68503 2.06359 1.59281 137.009 216.000 1.59281 0.63430 2.0 1.33221 361.145 432.000 1.33221 0.83598 1.21532 122.884 432.000 1.21532 0.28445 3.0 1.00420 295.079 648.000 1.00420 0.45537 0.89960 113.980 648.000 0.89960 0.17590 4.0 0.76373 230.279 864.000 0.76373 0.26653 0.67695 101.290 864.000 0.67695 0.11723 5.0 0.61071 175.155 1,080.000 0.61071 0.16218 0.53077 88.349 1,080.000 0.53077 0.08180 6.0 0.49868 131.181 1,296.000 0.49868 0.10122 0.44292 73.980 1,296.000 0.44292 0.05708 7.0 0.38940 96.410 1,512.000 0.38940 0.06376 0.33143 62.386 1,512.000 0.33143 0.04126 8.0 0.30917 64.812 1,728.000 0.30917 0.03751 0.26825 48.447 1,728.000 0.26825 0.02804 9.0 0.27698 54.012 1,944.000 0.27698 0.02778 0.26462 40.609 1,944.000 0.26462 0.02089 10.0 0.23731 43.212 2,160.000 0.23731 0.02001 0.26278 31.601 2,160.000 0.26278 0.01463 Note: # This value was greater than 2.0 m, i.e., the length of domain. PAGE 402 402 Tabl e B 1 7 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 1.00 and = 0.10. Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 1.93978 2 395.120 216.00 1.93978 11.08852 1.99686 459.194 216.00 1.99686 2.12590 0.2 1.81903 2 297.407 432.00 1.81903 5.31807 1.89186 560.933 432.00 1.89186 1.29846 0.3 1.71166 2 200.207 648.00 1.71166 3.39538 1.78831 611.423 648.00 1.78831 0.94355 0.4 1.60993 2 103.007 864.00 1.60993 2.43404 1.68783 637.777 864.00 1.68783 0.73817 0.5 1.51340 2 005.807 1,080.00 1.51340 1.85723 1.58703 646.299 1,080.00 1.58703 0.59843 0.6 1.42126 1 908.607 1,296.00 1.42126 1.47269 1.48736 646.322 1,296.00 1.48736 0.49871 0.7 1.33457 1 811.407 1,512.00 1.33457 1.19802 1.39050 637.912 1,512.00 1.39050 0.42190 0.8 1.25441 1 727.437 1,728.00 1.25441 0.99967 1.29837 624.831 1,728.00 1.29837 0.36159 0.9 1.17655 1 651.837 1,944.00 1.17655 0.84971 1.21110 608.660 1,944.00 1.21110 0.31310 1.0 1.10625 1 576.237 2,160.00 1.10625 0.72974 1.12973 587.938 2,160.00 1.12973 0.27219 2.0 0.60757 978.629 4,320.00 0.60757 0.22653 0.57746 367.303 4,320.00 0.57746 0.08502 3.0 0.35103 546.629 6,480.00 0.35103 0.08436 0.31565 186.177 6,480.00 0.31565 0.02873 4.0 0.20390 210.396 8,640.00 0.20390 0.02435 0.19724 73.971 8,640.00 0.19724 0.00856 5.0 0.12830 90.382 10,800.00 0.12830 0.00837 0.11286 0.000 10,800.00 0.11286 0.00000 6.0 0.10911 0.000 12,960.00 0.10911 0.00000 0.12589 0.000 12,960.00 0.12589 0.00000 7.0 0.11831 0.000 15,120.00 0.11831 0.00000 0.13029 0.000 15,120.00 0.13029 0.00000 8.0 0.12374 0.000 17,280.00 0.12374 0.00000 0.13260 0.000 17,280.00 0.13260 0.00000 9.0 0.12701 0.000 19,440.00 0.12701 0.00000 0.13414 0.000 19,440.00 0.13414 0.00000 10.0 0.12927 0.000 21,600.00 0.12927 0.00000 0.13523 0.000 21,600.00 0.13523 0.00000 PAGE 403 403 Tabl e B 1 8 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the Henry velocity dependent dispersion problem in case of = 1.00 and = 1.00. Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 0.81711 6 581.033 2,160.0 0.81711 3.04677 1.41197 1 277.956 2,160.0 1.41197 0.59165 0.2 0.59642 5 717.033 4,320.0 0.59642 1.32339 0.87413 1 293.989 4,320.0 0.87413 0.29953 0.3 0.47650 4 853.033 6,480.0 0.47650 0.74892 0.61492 1 166.247 6,480.0 0.61492 0.17998 0.4 0.38898 3 989.033 8,640.0 0.38898 0.46169 0.46117 1 001.122 8,640.0 0.46117 0.11587 0.5 0.33006 3 378.888 10,800.0 0.33006 0.31286 0.35838 812.841 10,800.0 0.35838 0.07526 0.6 0.28207 2 946.888 12,960.0 0.28207 0.22738 0.28369 629.285 12,960.0 0.28369 0.04856 0.7 0.24139 2 514.888 15,120.0 0.24139 0.16633 0.22635 462.971 15,120.0 0.22635 0.03062 0.8 0.21163 2 082.888 17,280.0 0.21163 0.12054 0.17801 316.630 17,280.0 0.17801 0.01832 0.9 0.18896 1 650.888 19,440.0 0.18896 0.08492 0.13846 188.961 19,440.0 0.13846 0.00972 1.0 0.16831 1 218.888 21,600.0 0.16831 0.05643 0.10967 105.901 21,600.0 0.10967 0.00490 2.0 0.11273 0.000 43,200.0 0.11273 0.00000 0.12662 0.000 43,200.0 0.12662 0.00000 3.0 0.12305 0.000 64,800.0 0.12305 0.00000 0.13305 0.000 64,800.0 0.13305 0.00000 4.0 0.12823 0.000 86,400.0 0.12823 0.00000 0.13562 0.000 86,400.0 0.13562 0.00000 5.0 0.13113 0.000 108,000.0 0.13113 0.00000 0.13697 0.000 108,000.0 0.13697 0.00000 6.0 0.13301 0.000 129,600.0 0.13301 0.00000 0.13779 0.000 129,600.0 0.13779 0.00000 7.0 0.13434 0.000 151,200.0 0.13434 0.00000 0.13597 0.000 151,200.0 0.13597 0.00000 8.0 0.13534 0.000 172,800.0 0.13534 0.00000 0.13894 0.000 172,800.0 0.13894 0.00000 9.0 0.13362 0.000 194,400.0 0.13362 0.00000 0.13923 0.000 194,400.0 0.13923 0.00000 10.0 0.13667 0.000 216,000.0 0.13667 0.00000 0.13947 0.000 216,000.0 0.13947 0.00000 PAGE 404 404 APPENDIX C BENCHMARK AND FIELD SCALE PROBLEMS Benchmark Problem The work of Motz and Sedighi (2013) was selected to be a benchmark for the field scale problem. Motz and Sedighi (2013) presented results for saltwater intrusion and the recirculation of seawater based on data from the upper part of the Floridan aquifer system at Hilton Head Island in South Carolina, U.S.A. (Bush 1988 ) as the study case Their solutions were obtained by running SEAWAT Version 4 to steady state using the implicit finite difference scheme and the generalized conjugate gradient solver. Results are presented as shown in Figur e C 1 A B Figur e C 1 The work of Motz and Sedighi (2013) A) extent of saltwater intrusion ( ) and B) recirculation of seawater ( ) for both uncoupled and coupled solutions. SEAWAT Version 4 Results of Benchmark Problem SEAWAT Version 4 numerical model reproduced solutions of Motz and Sedighi (2013) as shown in Figure C 2 and Tables C 1 to C 2. Additionally, six cases were PAGE 405 405 selected to present isochlors from 0.1 to 0.9 and the velocity distributi on as shown in Figures C 3 to C 5. A B Figur e C 2 Reproduced r esults of SEAWAT A) e xtent of saltwater intrusion and B) recirculation of seawater for both uncoupled and coupled simulations comparing to the work of Motz and Sedighi (2013) A B Figur e C 3 Reproduced results of SEAWAT simulations A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 0.2 of Motz and Sedighi (2013) PAGE 406 406 A B Figur e C 4 Reproduced results of SEAWAT A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 2.0 of Motz and Sedighi (2013) A B Figur e C 5 Reproduced result s of SEAWAT A) uncoupled and B) coupled solutions with concentration color floods and velocity vectors when = 10.0 of Motz and Sedighi (2013) PAGE 407 407 Table C 1 Reproduced results of SEAWAT for uncoupled and coupled solu tions of Motz and Sedighi (2013) Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 3,059.89 1,922,670 62,500 15.299 30.763 12,403.96 170,365 62,500 62.020 2.726 0.2 2,616.65 1,891,300 125,000 13.083 15.130 11,520.03 217,733 125,000 57.600 1.742 0.3 2,356.17 1,860,920 187,500 11.781 9.925 9,462.50 247,011 187,500 47.313 1.317 0.4 2,182.74 1,831,232 250,000 10.914 7.325 7,894.23 266,528 250,000 39.471 1.066 0.5 2,021.38 1,801,654 312,500 10.107 5.765 6,836.01 280,588 312,500 34.180 0.898 0.6 1,923.68 1,771,857 375,000 9.618 4.725 6,066.88 290,632 375,000 30.334 0.775 0.7 1,814.76 1,742,170 437,500 9.074 3.982 5,548.96 298,378 437,500 27.745 0.682 0.8 1,702.36 1,713,485 500,000 8.512 3.427 5,005.49 303,795 500,000 25.027 0.608 0.9 1,663.30 1,685,360 562,500 8.317 2.996 4,621.82 308,021 562,500 23.109 0.548 1.0 1,587.08 1,657,335 625,000 7.935 2.652 4,301.98 310,788 625,000 21.510 0.497 2.0 1,156.45 1,392,148 1,250,000 5.782 1.114 2,428.04 306,273 1,250,000 12.140 0.245 3.0 905.83 1,157,462 1,875,000 4.529 0.617 1,663.21 280,732 1,875,000 8.316 0.150 4.0 706.17 951,256 2,500,000 3.531 0.381 1,236.77 249,956 2,500,000 6.184 0.100 5.0 601.81 772,016 3,125,000 3.009 0.247 926.41 216,061 3,125,000 4.632 0.069 6.0 496.01 618,155 3,750,000 2.480 0.165 721.75 185,149 3,750,000 3.609 0.049 7.0 449.22 485,969 4,375,000 2.246 0.111 580.15 153,090 4,375,000 2.901 0.035 8.0 374.07 373,133 5,000,000 1.870 0.075 460.99 121,995 5,000,000 2.305 0.024 9.0 343.36 279,211 5,625,000 1.717 0.050 383.73 94,844 5,625,000 1.919 0.017 10.0 312.04 201,550 6,250,000 1.560 0.032 325.08 69,483 6,250,000 1.625 0.011 PAGE 408 408 Aquifer Parameters and Dimensionless Variables of the Field s cale Problem Table C 2 Aquifer parameters and dimensionless variables for the field scale problem in case of = 0.01 and = 0.01 and and m/day m/day m 2 /day m m 0.1 50 0.5 0.25 125 1.25 0.2 50 0.5 0.50 125 1.25 0.3 50 0.5 0.75 125 1.25 0.4 50 0.5 1.00 125 1.25 0.5 50 0.5 1.25 125 1.25 0.6 50 0.5 1.50 125 1.25 0.7 50 0.5 1.75 125 1.25 0.8 50 0.5 2.00 125 1.25 0.9 50 0.5 2.25 125 1.25 1.0 50 0.5 2.50 125 1.25 2.0 50 0.5 5.00 125 1.25 3.0 50 0.5 7.50 125 1.25 4.0 50 0.5 10.00 125 1.25 5.0 50 0.5 12.50 125 1.25 6.0 50 0.5 15.00 125 1.25 7.0 50 0.5 17.50 125 1.25 8.0 50 0.5 20.00 125 1.25 9.0 50 0.5 22.50 125 1.25 10.0 50 0.5 25.00 125 1.25 Note: 1. Density contrast parameter ( ) = 0.025; 2. Transverse vertical dimension of the domain ( ) = 250 m; 3. Transverse vertical discretization ( ) = 250 m; 4. Vertical dimension of the domain ( ) = 200 m; and 5. Horizontal discretization ( ) = 5 m. PAGE 409 409 Table C 3 Aquifer parameters and dimensionless variables for the field scale problem in case of = 0.01 and = 0.1 and and m/day m/day m 2 /day m m 0.1 50 5.0 2.50 125 1.25 0.2 50 5.0 5.00 125 1.25 0.3 50 5.0 7.50 125 1.25 0.4 50 5.0 10.00 125 1.25 0.5 50 5.0 12.50 125 1.25 0.6 50 5.0 15.00 125 1.25 0.7 50 5.0 17.50 125 1.25 0.8 50 5.0 20.00 125 1.25 0.9 50 5.0 22.50 125 1.25 1.0 50 5.0 25.00 125 1.25 2.0 50 5.0 50.00 125 1.25 3.0 50 5.0 75.00 125 1.25 4.0 50 5.0 100.00 125 1.25 5.0 50 5.0 125.00 125 1.25 6.0 50 5.0 150.00 125 1.25 7.0 50 5.0 175.00 125 1.25 8.0 50 5.0 200.00 125 1.25 9.0 50 5.0 225.00 125 1.25 10.0 50 5.0 250.00 125 1.25 Note: 1. Density contrast parameter ( ) = 0.025; 2. Transverse vertical dimension of the domain ( ) = 250 m; 3. Transverse vertical discretization ( ) = 250 m; 4. Vertical dimension of the domain ( ) = 200 m; and 5. Horizontal discretization ( ) = 5 m. PAGE 410 410 Table C 4 Aquifer parameters and dimensionless variables for the field scale problem in case of = 0.01 and = 1.0 and and m/day m/day m 2 /day m m 0.1 50 50 25 125 1.25 0.2 50 50 50 125 1.25 0.3 50 50 75 125 1.25 0.4 50 50 100 125 1.25 0.5 50 50 125 125 1.25 0.6 50 50 150 125 1.25 0.7 50 50 175 125 1.25 0.8 50 50 200 125 1.25 0.9 50 50 225 125 1.25 1.0 50 50 250 125 1.25 2.0 50 50 500 125 1.25 3.0 50 50 750 125 1.25 4.0 50 50 1 000 125 1.25 5.0 50 50 1 250 125 1.25 6.0 50 50 1 500 125 1.25 7.0 50 50 1 750 125 1.25 8.0 50 50 2 000 125 1.25 9.0 50 50 2 250 125 1.25 10.0 50 50 2 500 125 1.25 Note: 1. Density contrast parameter ( ) = 0.025; 2. Transverse vertical dimension of the domain ( ) = 250 m; 3. Transverse vertical discretization ( ) = 250 m; 4. Vertical dimension of the domain ( ) = 200 m; and 5. Horizontal discretization ( ) = 5 m. PAGE 411 411 Results of Extent of Saltwater Intrusion ( ) and Degree of Recirculation of Seawater ( ) f or the Field s cale Problem Table C 5 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncouple d and coupled solutions of the field scale problem in case of = 0.01 and = 0.01 Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 3,059.89 1,922,670 62,500 15.299 30.763 16,424.49 179,214 62,500 82.122 2.867 0.2 2,616.69 1,891,420 125,000 13.083 15.131 13,902.06 200,933 93,750 69.510 2.143 0.3 2,356.23 1,861,029 187,500 11.781 9.925 11,951.87 218,477 125,000 59.759 1.748 0.4 2,182.77 1,831,342 250,000 10.914 7.325 9,469.45 247,110 187,500 47.347 1.318 0.5 2,021.38 1,801,654 312,500 10.107 5.765 7,893.30 266,632 250,000 39.467 1.067 0.6 1,923.80 1,771,967 375,000 9.619 4.725 6,835.62 280,716 312,500 34.178 0.898 0.7 1,814.76 1,742,279 437,500 9.074 3.982 6,066.88 290,770 375,000 30.334 0.775 0.8 1,702.39 1,713,585 500,000 8.512 3.427 5,549.24 298,529 437,500 27.746 0.682 0.9 1,663.40 1,685,460 562,500 8.317 2.996 5,005.38 303,955 500,000 25.027 0.608 1.0 1,587.13 1,657,335 625,000 7.936 2.652 4,621.98 308,179 562,500 23.110 0.548 2.0 1,156.55 1,392,231 1,250,000 5.783 1.114 4,301.64 310,961 625,000 21.508 0.498 3.0 905.95 1,157,530 1,875,000 4.530 0.617 2,427.87 306,425 1,250,000 12.139 0.245 4.0 706.28 951,313 2,500,000 3.531 0.381 1,663.21 280,853 1,875,000 8.316 0.150 5.0 601.92 772,067 3,125,000 3.010 0.247 1,236.78 250,023 2,500,000 6.184 0.100 6.0 496.17 618,195 3,750,000 2.481 0.165 926.58 216,099 3,125,000 4.633 0.069 7.0 449.38 486,004 4,375,000 2.247 0.111 721.95 185,166 3,750,000 3.610 0.049 8.0 374.17 373,163 5,000,000 1.871 0.075 580.33 153,107 4,375,000 2.902 0.035 9.0 343.42 279,233 5,625,000 1.717 0.050 461.32 122,004 5,000,000 2.307 0.024 10.0 312.09 201,567 6,250,000 1.560 0.032 384.02 94,906 5,625,000 1.920 0.017 PAGE 412 412 Table C 6 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the field scale problem in case of = 0.01 and = 0.1 Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 772.21 3,961,566 625,000 3.861 6.339 4,262.58 404,721 625,000 21.313 0.648 0.2 630.58 3,666,461 1,250,000 3.153 2.933 2,471.71 451,300 1,250,000 12.359 0.361 0.3 556.62 3,385,211 1,875,000 2.783 1.805 1,738.40 468,690 1,875,000 8.692 0.250 0.4 481.50 3,117,250 2,500,000 2.408 1.247 1,331.53 475,099 2,500,000 6.658 0.190 0.5 482.03 2,860,687 3,125,000 2.410 0.915 1,073.93 476,736 3,125,000 5.370 0.153 0.6 431.20 2,615,358 3,750,000 2.156 0.697 882.11 474,722 3,750,000 4.411 0.127 0.7 385.29 2,381,044 4,375,000 1.926 0.544 731.48 470,892 4,375,000 3.657 0.108 0.8 371.59 2,162,294 5,000,000 1.858 0.432 649.34 456,529 5,000,000 3.247 0.091 0.9 361.34 1,954,353 5,625,000 1.807 0.347 573.37 440,266 5,625,000 2.867 0.078 1.0 350.53 1,756,842 6,250,000 1.753 0.281 483.33 422,247 6,250,000 2.417 0.068 2.0 275.79 392,875 12,500,000 1.379 0.031 213.00 99,137 12,500,000 1.065 0.008 3.0 153.68 0 18,750,000 0.768 0.000 155.06 0 18,750,000 0.775 0.000 4.0 151.05 0 25,000,000 0.755 0.000 150.85 0 25,000,000 0.754 0.000 5.0 149.44 0 31,250,000 0.747 0.000 148.87 0 31,250,000 0.744 0.000 6.0 148.35 0 37,500,000 0.742 0.000 147.73 0 37,500,000 0.739 0.000 7.0 147.58 0 43,750,000 0.738 0.000 147.00 0 43,750,000 0.735 0.000 8.0 147.04 0 50,000,000 0.735 0.000 146.51 0 50,000,000 0.733 0.000 9.0 146.65 0 56,250,000 0.733 0.000 146.15 0 56,250,000 0.731 0.000 10.0 146.35 0 62,500,000 0.732 0.000 145.89 0 62,500,000 0.729 0.000 PAGE 413 413 Table C 7 Results of extent of saltwater intrusion ( ) and recirculation of seawater ( ) for uncoupled and coupled solutions of the field scale problem in case of = 0.01 and = 1.0. Uncoupled simulation Coupled simulation (m) (kg/day) (kg/day) (m) (kg/day) (kg/day) 0.1 310.06 3,172,948 6,250,000 1.550 0.508 502.32 587,407 6,250,000 2.512 0.094 0.2 272.77 1,304,125 12,500,000 1.364 0.104 277.39 344,694 12,500,000 1.387 0.028 0.3 190.96 258,619 18,750,000 0.955 0.014 164.10 13,724 18,750,000 0.821 0.001 0.4 144.58 0 25,000,000 0.723 0.000 155.23 0 25,000,000 0.776 0.000 0.5 137.13 0 31,250,000 0.686 0.000 152.21 0 31,250,000 0.761 0.000 0.6 136.37 0 37,500,000 0.682 0.000 150.41 0 37,500,000 0.752 0.000 0.7 137.31 0 43,750,000 0.687 0.000 149.23 0 43,750,000 0.746 0.000 0.8 138.57 0 50,000,000 0.693 0.000 148.41 0 50,000,000 0.742 0.000 0.9 139.74 0 56,250,000 0.699 0.000 147.80 0 56,250,000 0.739 0.000 1.0 140.72 0 62,500,000 0.704 0.000 147.35 0 62,500,000 0.737 0.000 2.0 144.32 0 125,000,000 0.722 0.000 145.73 0 125,000,000 0.729 0.000 3.0 145.06 0 187,500,000 0.725 0.000 145.54 0 187,500,000 0.728 0.000 4.0 145.48 0 250,000,000 0.727 0.000 145.69 0 250,000,000 0.728 0.000 5.0 145.87 0 312,500,000 0.729 0.000 145.96 0 312,500,000 0.730 0.000 6.0 146.26 0 375,000,000 0.731 0.000 146.29 0 375,000,000 0.731 0.000 7.0 146.65 0 437,500,000 0.733 0.000 146.65 0 437,500,000 0.733 0.000 8.0 147.05 0 500,000,000 0.735 0.000 147.04 0 500,000,000 0.735 0.000 9.0 147.22 0 562,500,000 0.736 0.000 147.44 0 562,500,000 0.737 0.000 10.0 147.36 0 625,000,000 0.737 0.000 147.56 0 625,000,000 0.738 0.000 PAGE 414 414 LIST OF REFERENCES Abarca, E., Carrera, J., Sanchez Vila, X., and Dentz, M. 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PAGE 422 422 BIOGRAPHICAL SKETCH Chanyut Kalakan is from Thailand, which not only is a small country in Southeast Asia but also has a lot of stories, such as a lot of beautiful places to visit, too complicated society problems to understand and too various political issues to believe. He is the son of Mr.Sompong Kalakan and Mrs.Cholao Kalakan. He was born into a warm family in 1975 H is academic career started at Kasetsart University where he received his Bachelor of Engineering in Irrigation Engineering in 1998. At that time, a financ ial cris i s occurred in Thailand, push ing Thailand into the most terrible economic problems in its history, and then the cris i s expanded to become one of the worst Asian financial crises Asian Financial Crisis 1997 om Y um K ung C risis ing to do any businesses and it also was a terrible time in life. He decided to attend Kasetsart University in order to study again and then he achieved his Bachelor of Engineering in Civil Enginee ring in 1999. After that, he started to work with a private company. After working for a couple years, he returned to study again by attending Kasetsart University and earned his Master of Engineering in Water Resources Engineering in 2003, in which his work concentrated on developing a G enetic A lgorithms optimization model for reservoir water management problem s After graduating, he worked with a couple of private companies and he was looking for challenging work. In 2005, he went to the Burapha University in Chonburi, to be a lecturer in the Faculty of Engineering. Although he did not have a Ph.D., the F aculty Executive Board accepted him as a lecturer and then recommended that he pursue a Ph.D. Attempting to do this thing was not only so much challenging work but it also PAGE 423 423 involved so many more barriers than he expected In 2007, it seemed to be his good luck at that moment He was awarded the Royal Thai Government scholarship to purs ue his Ph.D. degree. In 2009, h e received an admission letter from University of Florida which was a great opportunity from Dr Mang Tia and Dr Louis H. Motz. In Fall 2009, he turned into a Ph.D. student unde supervisory at University of Florida. Although studying for the Ph.D. was one of the most important challenges in his life, it pushed him to encounter a lot of tough time and challenges simultaneously. While pursuing his graduate degree at University of Florida, many unexpected things occurred, which was both good and bad He overcame a number of troubles with encourag ement from his family. Finally, after walking through difficult challenges for 5 years, this was the end of his educational journey in Gainesville, Florida He received a Ph.D. in C ivil E ngineering from the University of Florida in Spring 2014 His Ph D dissertation focused on investigat ing s altwater intru sion and determining quantitatively how saltwater intrusion and the recirculation of saltwater can be simulated using numerical groundwater flow and transport models The D epartment of Civil and Coastal Engineering is in the School of Sustainabl e Infrastructure & Environment, which is under the College of Engineering at University of Fl orida PAGE 1 SaltwaterIntrusionandRecirculationofSeawater ataCoastalBoundaryLouisH.Motz,M.ASCE1;andAliSedighi2Abstract: Numericalexperimentswereperformedtoinvestigatesaltwaterintrusionandrecirculationofseawateratacoastalboundary. Afieldscaletwodimensionalcrosssectionwassimulatedinwhichfreshwaterinflowoccurredatanupgradientboundary,andsaltwater inflow,freshwateroutflow,andrecirculatedseawateroutflowoccurredatadowngradientboundary.Theupgradientboundaryisaspecifiedfluxboundarywithazero(i.e.,freshwater)concentration,andthedowngradientboundaryisaspecifiedheadboundarywithaspecified (i.e.,saltwater)concentration.ThisproblemwassolvednumericallyusingSEAWATfortwoconditions;i.e.,firstfortheuncoupledcondition inwhichthefluiddensityisconstant,andthus,theflowandtransportequationsareuncoupledinaconstantdensityflowfield,andthenforthe coupledconditioninwhichthefluiddensityisafunctionoftheconcentrationoftotaldissolvedsolids,andthus,theflowandtransport equationsarecoupledinavariabledensityflowfield.Equivalentfreshwaterheadsarespecifiedatthedownstreamboundaryforboth conditionstoaccountfordensitydifferencesbetweenfreshwaterandsaltwateratthedownstreamboundary.Forbothconditions,itwas determinedthatsaltwaterintrusionandseawaterrecirculationaredecreasedsignificantlyasthedimensionlessratioofthefreshwaterinflow relativetothedensitydrivenbuoyancyflux( az)isincreased.However,theextentofsaltwaterintrusionislessandthedegreeofseawater recirculationisgreaterfortheuncoupledconditionthanforthecoupledconditionatsmallervaluesof az,indicatingthatsignificantdifferencescanoccurbetweenuncoupledandcoupledsimulations.Fortheexperimentsconductedinthisinvestigation,atsmallervaluesof az 0 1 ,wherethedensitydrivenbuoyancyfluxdominatesthefreshwateradvectiveflux,theextentofsaltwaterintrusionislessand thedegreeofsaltwaterrecirculationandpercentofrecirculatedseawateraregreaterfortheuncoupledconditionthanthecoupledcondition. Atlargervaluesof az 10 0 ,wherethefreshwateradvectivefluxdominatesthebuoyancyflux,theextentofsaltwaterintrusionforthe uncoupledconditionbecomesequaltotheextentofsaltwaterintrusionforthecoupledcondition,andthedegreeofsaltwaterrecirculationand percentofrecirculatedseawaterfortheuncoupledconditionasymptoticallyapproachthecorrespondingvaluesforthecoupledcondition. DOI: 10.1061/(ASCE)HE.19435584.0000594 2013AmericanSocietyofCivilEngineers. CEDatabasesubjectheadings: Aquifers;Groundwaterflow;Numericalmodels;Recirculation;Seawater;Saltwaterintrusion; Coastalenvironment. Authorkeywords: Aquifers;Coastalboundary;Groundwaterflow;Numericalmodels;Recirculationofseawater;Saltwaterintrusion; SEAWAT;Submarinegroundwaterdischarge.IntroductionAttheseacoastinacoastalaquifer,freshgroundwaterfrominland terrestrialsourcesdischargestothesea,flowingoverdenserseawaterthatintrudesinland( MotzandSedighi2009 ).Theseawater flowsinlandalongthebaseoftheaquifer,mixeswiththeseawardflowingfreshwater,isrecirculated,anddischargesbacktothe sea.Thisrecirculationofseawaterandthemixwithfreshwater ischaracterizedbyazoneofcontactbetweenthefreshwater andthesaltwater,whichgenerallytakestheformofafinitewidth transitionzonecausedbyhydrodynamicdispersion,acrosswhich thedensityofthemixedwatervariesfromthatoffreshwatertothat ofsaltwater( Bear1979 ).Therecirculatedseawatercaninclude componentsduetowavesetup,tidallydrivenoscillations,and convectioncausedbydensityorthermaldifferencesbetweenthe saltwaterandfreshwater( Taniguchietal.2002 ),andthesumof therecirculatedseawaterandfreshgroundwaterdischargeiscalled submarinegroundwaterdischarge(SGD)( Taniguchietal.2002 ; Smith2004 ; Robinsonetal.2007 ; ParkandAral2008 ).Generally, ithasbecomerecognizedthatinlandterrestrialsourcesalonecannotsustainthemagnitudeofmeasuredSGDand,thus,thatsome partofSGDmustoriginatefromseawater( Younger1996 ; Moore andChurch1996 ).MeasurementsofSGDthathavebeenmadein manypartsoftheworld( Taniguchietal.2002 ; Burnettetal.2006 ) confirmtheimportanceofrecirculatedseawaterasacomponent ofSGD.Forexample,Martinetal.( 2007 )determinedfromfield measurementsthatthedischargeofrecirculatedseawaterinthe IndianRiverLagooninFloridaismorethantwoordersofmagnitudelargerthangroundwaterdischargefromterrestrialsources.PreviousInvestigationsMuchofthequantitativeanalysisofsaltwaterintrusionandrecirculationofseawateratacoastalboundaryhasbeenconducted intermsoftheHenry( 1964 )problem,whichisasteadystate quasianalyticalsolutionthatdescribestheseawardflowoffreshwater,themixinginthezoneofdispersionbetweenfreshwaterand1AssociateProfessor,Dept.ofCivilandCoastalEngineering,Univ.of Florida,P.O.Box116580,Gainesville,FL32611(correspondingauthor). Email:lmotz@ce.ufl.edu 2AssistantProfessor,Dept.ofCivilEngineering,SharifUniv.ofTechnology,InternationalCampus,KishIsland,Iran.Email:alised@sharif.edu Note.ThismanuscriptwassubmittedonAugust12,2010;approvedon January13,2012;publishedonlineonJanuary16,2012.Discussionperiod openuntilJune1,2013;separatediscussionsmustbesubmittedfor individualpapers.Thispaperispartofthe JournalofHydrologicEngineering ,Vol.18,No.1,January1,2013.ASCE,ISSN10840699/2013/ 11018/$25.00.10 /JOURNALOFHYDROLOGICENGINEERINGASCE/JANUARY2013 PAGE 2 saltwater,andtherecirculationofintrudingsaltwater.Henry( 1964 ) describedhissolutionforanisotropicaquiferintermsoftwodimensionlessparameters, a and b ,whicharetheratiooffreshwater inflowtothebuoyancyfluxandtheratioofthecoefficientof dispersiontothefreshwaterinflow,respectively;andathirddimensionlessparameterthatrepresentstheratioofthehorizontaland verticaldimensionsoftheproblem.PinderandCooper( 1970 ), LeeandCheng( 1974 ),Sgoletal.( 1975 ),Huyakornetal. ( 1987 ),VossandSouza( 1987 ),SengerandFogg( 1990 ),Croucher andO Sullivan( 1995 ),Chengetal.( 1998 ),andTejedoetal. ( 2003 ),amongothers,haveusedHenry soriginalsolutionorrevisionstoHenry ssolutiontocompareorbenchmarktheirsolutions. SimpsonandClement( 2003 2004 )solvedtheHenryproblemin thestandarddensitycoupledmodeandinanewdensityuncoupled mode;theyconcludedthattheHenryproblemhaslimitedusefulnessinbenchmarkingdensitydependentflowmodels,andrecommendeddecreasingthefreshwaterrechargebyafactoroftwoto increasetherelativeimportanceofdensitydependenteffects. Dentzetal.( 2006 )describedHenry sproblemintermsoftwodimensionlessgroups,acouplingparameter, ,whichistheratioof thebuoyancyandviscousforces,andthePcletnumber,whichis theratioofadvectiveanddispersivetransport.ThecouplingparameteristheinverseofHenry s a parameter,andthePcletnumberis theinverseofHenry s b parameter.Dentzetal.( 2006 )conducteda systematicanalysisoftheHenryproblemforafullrangeofthe Pcletnumberfromsmalltolargevalues,includingcoupledand uncoupledsolutions(thelatterarecalledpseudocoupledsolutionsintheirstudy).GoswamiandClement( 2007 )conducted laboratoryscaleexperimentsandnumericalmodelingtodevelop steadystateandtransientsaltwedgedatasetsthatrepresentbenchmarkproblemsfortestingdensitycoupledgroundwaterflowmodels.Theirinvestigationincludedcomparingtheresultsofcoupled anduncoupledsteadystateandtransientSEAWAT( Langevinetal. 2003 )numericalsimulationstoinvestigatedensitycouplingeffects,and,similartoSimpsonandClement( 2003 2004 ),theyproposedreducingthefreshwaterinflowasamorerobustalternative tothetraditionalHenryproblem.Abarcaetal.( 2007 )modified Henry sproblembyaccountingforanisotropichydraulicconductivityandrepresentingthemixingalongthesaltwater freshwater interfacebymeansofvelocitydependentdispersioninsteadofconstantdiffusion.Intheiranalysis,newdimensionlessparametersfor theanisotropyratiosforhydraulicconductivityanddispersivity wereintroducedtosupplementthethreedimensionlessratiosfor viscousandbuoyancyforces,diffusiveandadvectivefluxes(Pclet number),andhorizontalandverticalscalesintheoriginalHenry problem.Solutionsfortheoriginaldiffusivecaseandthemodified dispersivecaseofHenry sproblemwereobtainedforbothdensitycoupledanddensityuncoupledconditions.IncontrasttoSimpson andClement s( 2004 )recommendationtodecreasethefreshwaterdischarge,therebyreducingthe a parameterintheHenryproblem, Abarcaetal.( 2007 )recommendedreducingthePcletnumber, whichistheinverseofthe b parameterintheHenryproblem,asa moresuitablebenchmarkproblem. Saltwaterintrusionandrecirculationofseawaterhavebeeninvestigatedforotherproblemsaswell.Smith( 2004 )usedFEFLOW ( Diersch2002 )andSUTRA( Voss1984 )toinvestigatethecontributionofdensitydrivenconvectiontoSGDinanaquifercross sectionthathadlargerhorizontalandverticaldimensionsthan Henry s1by2mproblem.Itwasdeterminedthatsaltwatercirculation,calculatedastheratioofsaltwaterinflowandfreshwater outflow,couldbeexpressedintermsofthedimensionlessratio ofthebuoyancyandadvectiveforcesandthedimensionlessratio ofthecharacteristicaquiferlengthscaletothecharacteristic dispersionlengthscale,whicharerelatedtoHenry s a and b parameters,respectively.Itwasfoundthatthekeycontrolscould beexpressedintheformofacombinedsingledimensionlessnumberthatincorporatesthecombinedeffectsoffreeconvection, forcedconvection,andhydrodynamicdispersiononconvective overturnwithinthecoastalsaltwedge.Also,Kaleris( 2006 )investigatedSGDforaconfinedaquiferforthreedifferentpermeability distributions(uniform,anisotropic,andheterogeneouslognormal) andforanaquiferinwhichasurfacewaterbodyisembedded. NumericalsimulationswereperformedusingFASTC(2D) ( Holzbecher1998 ).TheportionofrecirculatedseawaterinSGD andthelengthoftherecirculationzonewereinvestigatedinterms offourdimensionlessratios,namely,theratiooftheRaleighand Pcletnumbers(Ra = P )basedonHolzbecher( 1998 2000 ),the Pcletnumber,thehorizontalandverticalpermeabilities,and thehorizontalandverticallengthscales.TheratiooftheRaleigh andPcletnumbers,whichiscalledthemixedconvectionnumber ( Holzbecher2004 ),istheinverseofHenry s a parameter,andthe PcletnumberistheinverseofHenry s b parameter. Inpreviousinvestigationsofthesaltwaterintrusionproblem,a constantfluxoffreshwaterisappliedatanupstream,inlandboundary,whilethecoastalboundaryisopentoabodyofgreaterdensity seawater( SimpsonandClement2004 ).Saltwaterintrudesinland fromthecoastalboundaryuntilanequilibriumisreachedbetween theheavierintrudedseawaterandthelighterrechargingfreshwater. Intheuncoupledsolutions,flowandtransportarecoupledonlyby thesaltwaterboundaryconditionatthecoastalboundary( Dentz etal.2006 ).Intheseuncoupledsolutions,inwhichdensityisconstantintheflowfield,adensefrontofsalinewaterintrudesinland fromthecoastalboundaryintothelessdenseregionsasaresult offorcedconvectioncausedbyexternalforcingattheboundary ( GoswamiandClement2007 ).Inthecoupledsolutions,inwhich densityisafunctionofsalinityintheflowfield,thedensefront intrudesfartherinlandasaresultoffreeconvectiongenerated byinternaldensityvariationsandtheforcedconvectiondueto theexternalforcingatthecoastalboundary( GoswamiandClement 2007 ).Inthecoupledsolutions,thevelocitiesassociatedwiththe heaviersalinewaterintrudefartherinlandthanthevelocitiesforthe uncoupledsolutions( SimpsonandClement2003 ).Thevelocities andresultingrecirculationaresmallerinmagnitudeforthecoupled solutionsthanfortheuncoupledsolutionsbecausetheflowpaths arelongerbetweenthelowerpartsofthecoastalboundarywhere intrusionoccursandtheupperpartsofthecoastalboundarywhere mixingwithfreshwateranddischargefromtheaquiferoccur.ThisInvestigationIntheinvestigationdescribedinthispaper,numericalexperiments wereperformedtoinvestigatesaltwaterintrusionandthecomponentsofSGDconsistingoffreshgroundwaterdischargeandthe recirculationofseawaterduetothedensitydifferencebetweensaltwaterandfreshwater.Theinvestigationwasundertakentoquantitativelydeterminehowsaltwaterintrusionandtherecirculationof seawaterarerelatedtofreshwaterinflowanddensitydrivenbuoyancyfluxoverabroadrangeofvaluesforthedimensionlessratioof freshwaterinflowandbuoyancyflux.Theanalysisofthisproblem wasextendedbeyondtheusual1by2mcrosssectionoftheHenry problembyselectingparametersanddimensionsthatrepresenta morepracticalfieldscalecoastalaquiferandbymorerealistically representingthemixingalongthesaltwater freshwaterinterfaceby meansofvelocitydependentdispersioninsteadofconstantdiffusion.Additionally,theexperimentswereperformedtoinvestigate theaccuracyofusinganuncoupledsolutiontosimulateacoastal aquifer.Innumericalmodelswithlargenumbersofrows,columns,JOURNALOFHYDROLOGICENGINEERINGASCE/JANUARY2013/ 11 PAGE 3 andlayerswithseveralhundredthousandcells,significantdifferencesoccurinruntimesbetweenuncoupledandcoupledsimulations,sometimesontheorderofminutescomparedtohoursfor eachsimulationperiod.Obviously,significanttimesavingsduring calibrationofsuchnumericalmodelscanbeachievedifan uncoupledsimulationisused,butitisimportanttodeterminethe accuracyofthischoice.DesignofNumericalExperimentsThisproblemwassolvednumericallyfortwoconditions,i.e.,first fortheuncoupledconditioninwhichthefluiddensityisconstant andtheflowandtransportequationsareuncoupled,andthenfor thecoupledconditioninwhichthefluiddensityisafunctionofthe concentrationoftotaldissolvedsolids,andthus,theflowand transportequationsarecoupledinthevariabledensityflowfield. Equivalentfreshwaterheadswerespecifiedatthedownstream boundaryforbothconditionstoaccountfordensitydifferences betweenfreshwaterandsaltwateratthedownstreamboundary. ConstantDensityandVariableDensitySolutions ThesolutionswereobtainedbyrunningSEAWATVersion4 ( Langevinetal.2008 )tosteadystateusingtheimplicitfinitedifferencesolutionschemeandthegeneralizedconjugategradient solver.TheuncoupledflowandtransportoptioninSEAWAT wasusedtorepresentconstantdensityflowandtransportin acoastalaquiferfortheuncoupledsolution,andthecoupled flowandtransportoptionwasusedtorepresentdensitydependent flowandtransportinacoastalaquiferforthecoupledcondition. GroundwaterVistas( EnvironmentalSimulations2007 )wasused tocreateinputfiles,runSEAWAT,andanalyzesimulationresults. DiscretizationandBoundaryConditions Atwodimensionalverticalcrosssectionwasusedtorepresenta coastalaquifer(Fig. 1 ).Thecrosssectionwasdiscretizedinto onerow,50columns,and40layers.Thecolumnswereequally spacedwith x 250 mforatotallengthof12,500m.Alllayers werespecifiedasconfinedandwereequallyspacedwith z 5 m foratotaldepthof200m.Thewidthoftherowperpendiculartothe directionofflowinthecrosssectionwasarbitrarilysetto y 250 m.Thelefthandsidecoastalboundarywassimulated byspecifyingaconcentrationboundaryconditionwithaconcentrationoftotaldissolvedsolids(TDS)equalto 35 kg = m3,representingtheconcentrationofTDSinseawater.Intheuncoupled solutions,equivalentfreshwaterheadswerecalculatedoverthe depthofthecoastalboundarywith hs 0 s 1 ; 025 kg = m3, and f 1 ; 000 kg = m3,basedonEq.( 1 )( GuoandLangevin 2002 ): hf sfhs s ffZ (1) where hf= equivalentfreshwaterhead( L ); s= densityofsaltwater(ML 3); f= densityoffreshwater(ML 3); hs= saltwater hydraulichead( L );and Z = elevation( L )aboveorbelowadatum, i.e., Z 0 atsealevel.Inthecoupledsolutions,thisboundaryconditionwascalculatedinternallyinSEAWAT.Forbothuncoupled andcoupledsolutions,theboundaryontherighthandsidewas specifiedasaconstantfluxboundarywiththeconcentrationof TDSequaltozero,representingfreshwaterinflow. AquiferParameters Thehorizontalandverticaldimensionsofthecrosssection,the boundaryconditionsattheseacoast,andthehydraulicconductivities andotheraquiferparameters(Table 1 )werechosensothatthecross sectionwasrepresentativeofafieldscalecoastalaquifer( Motzand Sedighi2009 ).Thehorizontalandverticaldimensions(12,500by 200m)aresimilartothedimensions(12,875by160m)ofa crosssectionalmodeldevelopedbyBush( 1988 )usingSUTRA ( Voss1984 )torepresenttheupperpartoftheFloridanaquifer systematHiltonHeadIslandinSouthCarolina.Theratioof verticaltohorizontalhydraulicconductivity( 0 5 = 50 0 m = day 0 01 )issimilartoBush s( 1988 )ratioofapproximately0.01,and therangeofthefreshwaterinflowforthespecifiedfluxboundary condition( 0 25 25 0 m2= day)spansBush s( 1988 )valueof 1 85 m2= day.Also,theporosity(0.30)isthesameinbothmodels. DimensionlessParameters Thisproblemcanbecharacterizedbyfiveindependentdimensionlessparameters,namely: L d (2) rK KzKx(3) bL Ld (4) Fig.1. Finitedifferencegridusedfornumericalexperiments Table1. AquiferParametersandSpecifiedBoundaryInflowsUsedinthe NumericalSimulations ParameterValue Horizontaldimension( L )12,500m Horizontaldiscretization( x )250m Verticaldimension( d )200m Verticaldiscretization( z )5m Horizontalhydraulicconductivity( KH) 50 m = day Verticalhydraulicconductivity( KV) 0 5 m = day Longitudinaldispersivity( H)125m Verticaldispersivity( V)1.25m Porosity( )0.30 Flowperunitwidth( Q0 f)forspecifiedflux boundarycondition 0 25 25 0 m2= day12 /JOURNALOFHYDROLOGICENGINEERINGASCE/JANUARY2013 PAGE 4 r TL(5) and: az Q0 fKzd (6) where L = horizontaldimensionofthecrosssection( L ); d = verticaldimensionofthecrosssection( L ); Kz= verticalhydraulic conductivity(LT 1); Kx= horizontalhydraulicconductivity (LT 1); L= longitudinaldispersivity( L ); T= transversevertical dispersivity( L ); Q0 f= freshwaterinflowperunitwidthatthe upstreamboundary( L2T 1),and: s f f 0 025 (7) Theratio isanaspectratiooftheoverallhorizontalandvertical lengthscalesofthecrosssection( Henry1964 ); rKaccounts foranisotropyofthehydraulicconductivityoftheaquifer; bLdescribesthescaleofvelocitydependentdispersioncomparedtothe verticaldimensionoftheproblem;and rdescribestheanisotropy oftheverticalandlongitudinaldispersivities( Abarcaetal.2007 ). Thedimensionlessratio azcomparesthefreshwaterinflow Q0 f= d totheverticalbuoyancyflux Kz ,whichisthesameasthedefinitionofthe a parameterusedbyAbarcaetal.( 2007 ).Asdescribed byAbarcaetal.( 2007 ),thisparameteristhesameasHenry s ( 1964 ) a parameterforisotropichydraulicconductivity.Inthe numericalexperimentsdescribedinthispaper,thehorizontal andverticaldimensionsinthecrosssectionwere L 12 ; 500 m and d 200 m,respectively,horizontalhydraulicconductivity was Kx 50 m = day,verticalhydraulicconductivitywas Kz 0 5 m = day,longitudinaldispersivitywas L 125 m,and transverseverticaldispersivitywas T 1 25 m.Thus, L = d 62 5 rK Kz= Kx 0 01 bL L= d 0 01 ,and r T= L 0 01 .AccordingtoSimpsonandClement ( 2004 ),thedimensionlessratio a reflectsthebalanceoftheadvectivefreshwaterinflowandthedensitydrivenbuoyancyflux.Under conditionswhere a 1 ,thefluxesarebalanced.When a << 1 ,the systemisdominatedbythedensitydrivenflux,andwhen a >> 1 thesystemisdominatedbytheadvectiveflux.Fortheanisotropic conditionconsideredinthisproblem(i.e., rK 0 01 ),thefluxes arebalancedwhen az 1 ,thesystemisdominatedbythe densitydrivenfluxwhen az<< 1 ,andthesystemisdominated bytheadvectivefluxwhen az>> 1 .Inthesenumericalexperiments,varyingthefreshwaterinflowattheupgradientboundary from Q0 f 0 25 to 25 0 m2= dayresultedinarangefor azfrom 0.1to10.0,thusresultinginarangeofconditionsoverwhich theaquifersystemwasdominatedbythedensitydrivenfluxatthe smallervaluesof azanddominatedbytheadvectivefluxat thelargervaluesof az.ResultsofNumericalExperimentsSaltwaterintrusionwasdeterminedbymeasuringthelandward extentofthe c = c0 0 5 concentrationcontouralongthebaseof theaquifer.Inboththeuncoupledandcoupledsimulations,as thedimensionlessratio azwasincreasedfrom0.1to10.0,thelandwardextentofsaltwaterintrusiondecreasedsignificantly(Figs. 2 and 3 ).Also,foreachvalueof az,thelandwardextentofsaltwater intrusionwasgenerallygreaterforthecoupledconditionthan thesaltwaterintrusionfortheuncoupledcondition,e.g.,Fig. 4 for az 1 0 Threeratioswerecalculatedtoquantifytheresultsofthesimulations.Theextentofsaltwaterintrusion,whichwasdetermined fromthelocationofthetoeofthe c = c0 0 5 concentration contour,wasexpressedas LD Ltoed (8) where LD= nondimensionalsaltwaterintrusion;and Ltoe= measuredlocationofthe c = c0 0 5 concentrationcontourfrom thecoastalboundary.Thedegreeofrecirculationofseawateratthe downstreamcoastalboundarywascalculatedastheratioofthe saltwatermassinflowatthecoastalboundaryrelativetothefreshwatermassinflowattheupgradientboundaryfrom 020004000600080001000012000 x (meters) 200 150 100 50 0z (meters) a z = 10.00.5 Isochlor a z = 1.0 a z = 0.1 Fig.2. The0.5isochlorsforcoupledsimulationsfor az Q0 f= Kzd = f 0 1 ,1.0,and10.0 020004000600080001000012000x (meters) 200 150 100 50 0z (meters) a z = 10.00.5 Isochlor a z = 1.0 a z = 0.1 Fig.3. The0.5isochlorsforuncoupledsimulationsfor az Q0f= Kzd = f 0 1 ,1.0,and10.0 020004000600080001000012000 x (meters) 200 150 100 50 0z (meters) Coupled Simulation (a z = 1.0) Uncoupled Simulation (a z = 1.0)0.5 Isochlor Fig.4. Comparisonof0.5isochlorsforcoupledanduncoupled simulationsfor az Q0 f= Kzd = f 1 0JOURNALOFHYDROLOGICENGINEERINGASCE/JANUARY2013/ 13 PAGE 5 RD QssQff(9) where RD= nondimensionaldegreeofrecirculation; Qs= saltwater inflowatthecoastalboundary( L3T 1);and Qf= freshwaterinflow attheupgradientboundary( L3T 1).Also,similarly,thepercentof recirculatedseawaterinthemassoutflowatthecoastalboundary wascalculatedfrom PR 100 QssQss Qff (10) where PR= percentofrecirculatedseawater(%). UncoupledSimulations Intheuncoupledsimulations,as azwasincreasedfrom0.1to10.0, LDdecreasedfrom15.2to1.58(Table 2 ;Fig. 5 ).Also,as azwas increasedfrom0.1to10.0, RDdecreasedfrom30.8to0.032,and PRdecreasedfrom96.9to3.12%(Table 2 ;Figs. 6 and 7 ). Table2. SaltwaterIntrusionandRecirculationforUncoupledSimulations Q0 f(m2= day) az Q0 fKzd = fLtoe(m) LD Ltoe=dQss(kg = day) Qff(kg = day) RD Qss= QffPR 100 QssQss Qff 0.250.103,042.715.2 1 92 1066 25 10430.896.9 0.500.202,617.213.1 1 89 1061 25 10515.193.8 0.750.302,355.911.8 1 86 1061 88 1059.9290.8 1.000.402,185.710.9 1 83 1062 50 1057.3288.0 1.250.502,036.710.2 1 80 1063 13 1055.7785.2 2.000.801,701.38.51 1 71 1065 00 1053.4377.4 2.501.001,585.97.93 1 66 1066 25 1052.6572.6 5.002.001,162.95.81 1 39 1061 25 1061.1152.7 7.503.00913.54.57 1 16 1061 88 1060.61738.2 10.004.00701.03.50 9 51 1052 50 1060.38127.6 12.505.00611.93.06 7 72 1053 13 1060.24719.8 15.006.00517.72.59 6 18 1053 75 1060.16514.2 17.507.00450.32.25 4 86 1054 38 1060.11110.0 20.008.00376.21.88 3 73 1055 00 1060.0756.94 22.509.00349.61.75 2 79 1055 63 1060.0504.73 25.0010.00316.01.58 2 02 1056 25 1060.0323.12 Fig.5. Saltwaterintrusionforcoupledanduncoupledsimulations versus az Q0 f= Kzd = f 0 1 to10.0 Fig.6. Recirculationofseawateratthecoastalboundaryforcoupled anduncoupledsimulationsversus az Q0f= Kzd = f 0 1 to10.0 0.1110 0.20.52 5 0 20 40 60 80 100 Coupled Simulations Uncoupled Simulations Fig.7. Percentofrecirculatedseawaterinthemassoutflowatthe coastalboundaryforcoupledanduncoupledsimulationsversus az Q0 f= Kzd = f 0 1 to10.014 /JOURNALOFHYDROLOGICENGINEERINGASCE/JANUARY2013 PAGE 6 CoupledSimulations Inthecoupledsimulations,as azwasincreasedfrom0.1to10.0, LDdecreasedfrom60.4to1.50(Table 3 ;Fig. 5 ).Also,as azwas increasedfrom0.1to10.0, RDdecreasedfrom3.38to0.009,and PRdecreasedfrom77.2to0.85%(Table 3 ;Figs. 6 and 7 ). ComparisonofUncoupledandCoupledSimulations Importantdifferencesbetweentheuncoupledandcoupledsimulationswereobservedovertheinvestigatedrangeofthe azvalues.At thesmallervaluesof az 0 1 ,wherethedensitydrivenbuoyancy fluxdominatesthefreshwateradvectiveflux,theextentofsaltwater intrusionisless(Fig. 5 )andthedegreeofsaltwaterrecirculation andpercentofrecirculatedseawateraregreater(Figs. 6 and 7 )for theuncoupledconditionthanforthecoupledcondition.Atthe largervaluesof az 10 0 ,wherethefreshwateradvectiveflux dominatesthebuoyancyflux,theextentofsaltwaterintrusion fortheuncoupledconditionbecomesequaltotheextentofsaltwaterintrusionforthecoupledcondition(Fig. 5 ),andthedegree ofsaltwaterrecirculationandpercentofrecirculatedseawaterfor theuncoupledconditionasymptoticallyapproachthecorrespondingvaluesforthecoupledcondition(Figs. 6 and 7 ).NumericalErrorsDuetoInstabilitiesand OscillationsDiscretizationandsimulationparameterswerechosentominimize errorsduetoinstabilitiesandoscillationsbyminimizingthehorizontalandverticalgridPcletnumbers.Inthehorizontaldirection, thegridPcletnumber( Px)is Px x L(11) where x = discretization( L )inthehorizontaldirection.Inthe verticaldirection,thegridPcletnumber( Pz)is Pz z T(12) where z = discretization( L )intheverticaldirection.Inthehorizontaldirection,thegridPcletnumber( Px 250 = 125 m 2 0 ) islessthanthemaximumvalueof Px 4 0 recommendedbyVoss andProvost( 2002 )forspatialstabilitytominimizeoscillationsin concentrations.Intheverticaldirection,thegridPcletnumber ( Pz 5 0 = 1 25 m 4 0 )islessthanthemaximumvalueof Pz 10 0 recommendedbyVossandProvost( 2002 )fordiscretization fortransversedispersion.InsomeoftheSEAWATsimulations, minorinstabilitiesoccurredinregionsoflowTDSconcentrations, whichresultedinsomeverysmallnegativeconcentrationsinthese regions,wheretheTDSconcentrationswereessentiallyzero.These instabilitieswereminimal,however,anddidnothaveanyapparent effectonthelocationsofthe c = c0 0 5 concentrationcontoursor themassbalances.DiscussionandConclusionsTheresultsofthenumericalexperimentsdescribedinthispaper indicatethatseawaterintrusionandsaltwaterrecirculationaresignificantlydecreasedastheratiooffreshwaterinflowrelativetothe densitydrivenbuoyancyfluxisincreasedforbothuncoupledand coupledconditions.However,theyalsoindicatethatsignificant differencescanoccurbetweenuncoupledandcoupledconditions; i.e.,theextentofsaltwaterintrusionislessandthedegreeofsaltwaterrecirculationandpercentofrecirculatedseawateraregreater fortheuncoupledconditionthanforthecoupledcondition,particularlyatsmallervaluesof az 0 1 ,wherethedensitydrivenbuoyancyfluxdominatesthefreshwateradvectiveflux.Atlargervalues of az 10 0 ,wherethefreshwateradvectivefluxdominatesthe buoyancyflux,theextentofsaltwaterintrusionfortheuncoupled conditionbecomesequaltotheextentofsaltwaterintrusionforthe coupledcondition,andthedegreeofsaltwaterrecirculationand percentofrecirculatedseawaterfortheuncoupledcondition asymptoticallyapproachthecorrespondingvaluesforthecoupled condition. Inthesesimulations,threedependentdimensionlessparameters, i.e.,theextentofsaltwaterintrusion,thedegreeofsaltwaterrecirculation,andthepercentofrecirculatedseawater[ LD, RD,and PRinEqs.( 8 10 )],wereinvestigatedintermsoftheratiooffreshwater inflowrelativetothebuoyancyflux( az),whichisoneoffiveindependentdimensionlessparametersthatcharacterizethisproblem [ rK, bL, r,and azinEqs.( 2 6 )],alongwiththegridPclet numbers[ Pxand PzinEqs.( 11 and 12 )].Additionalnumerical experimentscouldbedesignedandconductedtotesttherelationshipsofthesedependentparameterstosomeoralloftheotherindependentdimensionlessparameters,tofurtherquantifysaltwater intrusionandrecirculationofseawaterandthedifferencesbetween Table3. SaltwaterIntrusionandRecirculationforCoupledSimulations Q0 f(m2= day) az Q0 fKzd = fLtoe(m) LD Ltoe=dQss(kg = day) Qff(kg = day) RD Qss= QffPR 100 QssQss Qff 0.250.1012,079.160.40 2 11 1056 25 1043.3877.2 0.500.2011,054.955.27 2 40 1051 25 1051.9265.7 0.750.309,375.046.88 2 62 1051 88 1051.4058.3 1.000.407,886.439.43 2 79 1052 50 1051.1252.8 1.250.506,916.734.58 2 93 1053 13 1050.93848.4 2.000.804,989.624.95 3 17 1055 00 1050.63438.8 2.501.004,298.421.49 3 24 1056 25 1050.51934.2 5.002.002,429.112.15 3 18 1051 25 1060.25520.3 7.503.001,670.58.35 2 90 1051 88 1060.15513.4 10.004.001,241.96.21 2 57 1052 50 1060.1039.33 12.505.00863.84.32 2 19 1053 13 1060.0706.55 15.006.00685.83.43 1 81 1053 75 1060.0484.60 17.507.00554.62.77 1 45 1054 38 1060.0333.20 20.008.00418.32.09 1 11 1055 00 1060.0222.17 22.509.00358.21.79 7 93 1045 63 1060.0141.39 25.0010.00299.01.50 5 37 1046 25 1060.0090.85JOURNALOFHYDROLOGICENGINEERINGASCE/JANUARY2013/ 15 PAGE 7 uncoupledandcoupledsimulations.Asanexampleoftheirapplication,theresultsoftheinvestigationdescribedinthispapercould beusedtodeterminewhetheracoupledvariabledensityflowand transportcodeshouldbeselectedforaregionalgroundwatermodel thatincludesacoastalboundary,orwhetheralesscomputationally intenseuncoupledalternativecouldbeselectedthatwouldstillaccuratelysimulatesaltwaterintrusionandrecirculationofseawater atthecoastalboundary.Basedontheresultsofthisinvestigation, anuncoupledconstantdensityflowandtransportcodewouldyield approximatelythesameresultsforsaltwaterintrusionandrecirculationofseawaterasacoupledvariabledensitycodeforvaluesof az 10 0 ,andsomewhatsmallerresultsforintrusionandgreater resultsforrecirculationforvaluesof az 1 0 .Atsmallervaluesof azapproaching0.1,theresultsbetweenanuncoupledanda coupledmodelwouldbesignificantlydifferent.Appendix.BenchmarkProblemsforSEAWATand GroundwaterVistasTheuseofSEAWATandGroundwaterVistasinthisinvestigation wasverifiedbycomparingresultscalculatedaspartofthisinvestigationtoresultspreviouslyobtainedbySimpsonandClement ( 2004 )forHenry s( 1964 )originalandmodifiedproblemsforuncoupledandcoupledsolutions.SimilartoGuoandLangevin ( 2002 ),atwodimensionalverticalcrosssectionwasdiscretized intoonerow,21columns,and10layers(Table 4 ).Columns1 to20wereequallyspacedwith x 0 1 m,andcolumn21at thecoastalboundarywassetequalto0.01mforatotalhorizontal dimensionof2.01m.Allofthelayerswerespecifiedasconfined andequallyspacedwith z 0 1 mforatotaldepthof1.0m.The widthoftherowperpendiculartothedirectionofflowinthecross sectionwasarbitrarilysetequalto y 1 0 m.ThecoastalboundarywassimulatedbyspecifyingaconcentrationboundaryconditionwithaconcentrationofTDSequalto 35 kg = m3.Inthe uncoupledsolutions,equivalentfreshwaterheadswerecalculated overthedepthofthecrosssectionatthecoastalboundarywith hs 0 s 1 ; 025 kg = m3,and f 1 ; 000 kg = m3usingEq.( 1 ). ThisboundaryconditionwascalculatedinternallyinSEAWAT inthecoupledsolutions.Forboththeuncoupledandcoupledsolutions,theupstreamboundarywasspecifiedasaconstantflux boundarywiththeconcentrationofTDSequaltozero,representing freshwaterinflow.Thehorizontalandverticalhydraulicconductivitieswereequalto K 864 m = day,andtheporositywasequalto 0 35 .Thehorizontalandverticaldispersivities( Hand V) weresetequaltozeroandthemoleculardiffusioncoefficient wassetequalto Dm 1 62925 m2= daytorepresenttheconstant dispersioncoefficientinHenry s( 1964 )problem.Theflowperunit widthwas Q0 f 5 702 m2= dayforHenry s( 1964 )originalproblemand Q0 f 2 851 m2= dayforHenry s( 1964 )modified problem. SolutionswereobtainedbyrunningSEAWATVersion4 ( Langevinetal.2008 )tosteadystateusingtheimplicitfinitedifferenceschemewithcentraldifferenceweightingandthegeneralizedconjugategradientsolver.Theuncoupledflowandtransport optioninSEAWATwasusedtorepresentconstantdensityflowand transportfortheuncoupledsolutions,andthecoupledflowand transportoptionwasusedtorepresentdensitydependentflow andtransportforthecoupledsolutions.GroundwaterVistas ( EnvironmentalSimulations2007 )wasusedtocreatetheinput files,runSEAWAT,andanalyzetheresults.SolutionswereobtainedforuncoupledandcoupledconditionsforHenry s( 1964 ) originalandmodifiedproblems(Figs. 8 and 9 ).Asillustrated bythe0.5isochlors,theseresultscomparequitefavorablywith theresultspreviouslyobtainedbySimpsonandClement( 2004 ) forHenry s( 1964 )originalandmodifiedproblemsusinga Galerkinfiniteelementnumericalsolution.Also,theresultsobtainedusingSEAWATforthecoupledconditionforHenry s ( 1964 )originalandmodifiedproblemsarethesameastheresults obtainedbyLangevinandGuo( 2006 )usingSEAWAT2000,comparedtoHenry s( 1964 )semianalyticalsolutionasrecalculatedby SimpsonandClement( 2004 ).Minorinstabilitiesoccurnearthe outflowpartofthecoastalboundaryintheresultsofthisinvestigation(Figs. 8 and 9 ).TheseinstabilitiesaresimilartoresultsobtainedbyLangevinandGou( 2006 )andSimpsonandClement Table4. AquiferParametersandSpecifiedBoundaryInflowsUsedinthe HenryProblem ParameterValue Horizontaldimension2.01m Horizontaldiscretization( x) Columns1to200.1m Column210.01m Verticaldimension1.0m Verticaldiscretization( z)0.1m Hydraulicconductivity(K) 864 m = day Dispersivity( H, V)0 Moleculardiffusion( Dm) 1 62925 m2= day Porosity( )0.35 Flowperunitwidth( Q0 f) Henryproblem 5 702 m2= day ModifiedHenryproblem 2 851 m2= day Fig.8. ComparisonofcoupledanduncoupledSEAWATandSimpson andClementsolutionsforthestandardHenrysaltwaterintrusion problem Fig.9. ComparisonofcoupledanduncoupledSEAWATandSimpson andClementsolutionsforthemodifiedHenrysaltwaterintrusion problem16 /JOURNALOFHYDROLOGICENGINEERINGASCE/JANUARY2013 PAGE 8 ( 2004 ),andareattributedtorelativelylargevelocitiesnearthe outflowregion( LangevinandGou2006 ).AcknowledgmentsFinancialsupportforthisinvestigationwasprovidedinpartbythe St.JohnsRiverWaterManagementDistrict,Palatka,Florida,the FloridaWaterResourcesResearchCenter,andtheU.S.Geological SurveyStateWaterResearchInstituteProgram.Theinformationin thispaperrepresentstheopinionsandconclusionsoftheauthors, anditdoesnotnecessarilyrepresenttheofficialpositionofthe St.JohnsRiverWaterManagementDistrict,theFloridaWater ResourcesResearchCenter,ortheU.S.GeologicalSurvey.AppreciationisexpressedtoDrs.SimpsonandClement,whoprovided copiesoftheoutputfilesfortheirsolutionstotheHenrysaltwater intrusionproblem.NotationThefollowingsymbolsareusedinthispaper: a = Q0 f= Kd = f = Henry s( 1964 ) a parameter; az= Q0 f= Kzd = f ; b = D = Q0 f= Henry s( 1964 ) b parameter; bL= L= d ; D = coefficientofdiffusion( L2T 1); d = verticaldimensionofthecrosssection( L ); hf= equivalentfreshwaterhead( L ); hs= saltwaterhydraulichead( L ); K = isotropichydraulicconductivity(LT 1); Kx= horizontalhydraulicconductivity(LT 1); Kz= verticalhydraulicconductivity(LT 1); L = horizontaldimensionofthecrosssection( L ); LD= Ltoe= d = nondimensionalsaltwaterintrusion; Ltoe= inlandextentofthetoeofthe c = c0 0 5 concentration contour( L ); Px= horizontalgridPcletnumber = x = L; Pz= verticalgridPcletnumber = z = T; PR= 100 Qss= Qss Qff = percentofrecirculated seawater(%); Qf= freshwaterinflow( L3T 1); Qs= saltwaterinflow( L3T 1); Q0 f= freshwaterinflowperunitwidth( L2T 1); RD= Qss= Qff= degreeofseawaterrecirculation; r= T= L; rK= Kz= Kx; Z = elevationaboveorbelowsealevel( L ); L= longitudinaldispersivity( L ); T= transversedispersivity( L ); x = discretizationinhorizontaldirection( L ); z = discretizationinverticaldirection( L ); = ( s f)(ML 3); = s f = f; = L = d ; f= densityoffreshwater(ML 3); s= densityofsaltwater(ML 3);and = porosity.ReferencesAbarca,E.,Carrera,J.,SnchezVila,X.,andDentz,M.(2007). 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