Applied Flow and Solute Transport Modeling in Aquifers: Fundamental Principles and Analytical and Numerical Methods

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ISBN: 9780849335747 | 0849335744
Cover: Hardcover
Copyright: 7/12/2005

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Over recent years, important contributions on the topic of solving various aquifer problems have been presented in numerous papers and reports. The scattered and wide-ranging nature of this information has made finding solutions and best practices difficult. Comprehensive and self-contained, Applied Flow and Solute Transport Modeling in Aquifers compiles the scattered literature on the topic into a single-source reference of the most up-to-date information in the field.Based on Dr. Batu's 20 years of practical experience tackling aquifer problems in a myriad of settings, the book addresses essentially all currently applied aquifer flow and contaminant transport solutions, combines theory with practical applications, covers both analytical and numerical solutions, and includes solutions to real world contaminant transport modeling scenarios. Batu approaches the subject from the practicing consultant's point of view and elucidates the difficulties real world professionals have faced in solving aquifer flow and contamination problems.The author simplifies the necessary theoretical background as much as possible and provides all derivational details of the theoretical background as worked examples. He uses this method to explore how the derivations were generated for those who need to know while allowing others to easily skip them and still benefit and learn from the practical applications of the mathematical approaches. Containing 51 tables and 323 figures, the book covers both the breadth and the depth of currently applied aquifer flow and contaminant transport modeling solutions.

Chapter 1 Introduction

1.1 Importance of Flow and Solute Transport Modeling in Aquifers

(1)

1.2 Purpose of this Book

(1)

1.3 Scope and Organization

(2)

1.4 How to use this Book

(2)

Chapter 2 Fundamental Principles of Solute Transport in Aquifers

2.1 Introduction

(1)

2.2 Concepts for Concentration and its Units

(4)

2.2.1 Volume-Averaged (Resident) Concentration

(3)

2.2.1.1 Solute and Solvent

(1)

2.2.1.2 Definitions for Concentration

(3)

Mass Concentration

(1)

Mass Fraction

(1)

Dilution

(1)

2.2.2 Flux-Averaged Concentration

(1)

2.2.3 Distinctions between Volume- and Flux-Averaged Concentrations

(1)

2.3 Physical Mechanism of Solute Transport in Saturated Porous Media without Sorption and Decay: One-Dimensional Case

(6)

2.3.1 Diffusive Transport

(4)

2.3.1.1 Physical Explanation of Molecular Diffusion

(1)

2.3.1.2 Quantification of Molecular Diffusion: Fick's First Law

(1)

Fick's First Law for Pure Water

(1)

Fick's Modified First Law for Porous Media

(1)

2.3.1.3 The Effective Molecular Diffusion Coefficient D

(2)

Effective Molecular Diffusion Coefficients in Pure Liquids

(1)

Effective Molecular Diffusion Coefficients in Porous Media

(1)

Importance of Diffusive Solute Transport

(1)

2.3.2 Convective (Advective) Transport

(1)

2.3.3 Dispersive Transport

(1)

2.3.4 Hydrodynamic Dispersion: One-Dimensional Case

(2)

2.3.4.1 The Effects of Convection and Dispersion

(1)

2.4 Generalization of the Convective-Dispersive Solute Flux Equations without Sorption and Decay for Saturated Porous Media

(3)

2.4.1 Three-Dimensional Convective-Dispersive Flux Components for General Case: Anisotropic Dispersion

(1)

2.4.1.1 Tensorial Forms of the Convective-Dispersive Flux Components

(1)

2.4.1.2 Convective-Dispersive Flux Components in Cartesian Coordinates

(1)

2.4.2 Three-Dimensional Convective-Dispersive Flux Components for Special Cases

(2)

2.4.2.1 The Principal Axes of the Dispersion Tensor can be Defined: the Porous Medium is Homogeneous and Anisotropic

(1)

2.4.2.2 The Principal Axes of the Dispersion Tensor Can be Defined: the Porous Medium Is Homogeneous and Isotropic and Has a Unidirectional Flow

(1)

2.5 Differential Equations for Solute Transport in Saturated Porous Media without Sorption and Decay

(2)

2.5.1 General Case: Homogeneous and Anisotropic Dispersion

(1)

2.5.2 Homogeneous and Isotropic Dispersion

(1)

2.5.3 Homogeneous and Isotropic Medium with Uniform Flow

(1)

2.5.3.1 Three-Dimensional Case

(1)

2.5.3.2 Two-Dimensional Case

(1)

2.5.3.3 One-Dimensional Case

(1)

2.6 Differential Equations for Solute Transport in Saturated Porous Media Under Sorption and Decay Conditions

(10)

2.6.1 Solute Transport Equations under Sorption Conditions

(5)

2.6.1.1 One-Dimensional Solute Transport Equation with Sorption and Desorption

(1)

Definition of Sorption and Desorption and their Incorporation into the One-Dimensional Equation

(1)

2.6.1.2 Freundlich Isotherm

(1)

2.6.1.3 Linear Sorption Isotherm (N = 1) and Distribution Coefficient: The Kai Approach

(1)

2.6.1.4 Retardation Factor Rd

(1)

2.6.1.5 Representative Parameter Values in the Retardation Factor (Rd) Equation

(1)

2.6.1.6 Two- and Three-Dimensional Solute Transport Equations with Sorption and Desorption

(1)

2.6.2 Solute Transport Equations under Degradation Conditions

(5)

2.6.2.1 Radioactive Decay

(2)

Radioactive Decay Equation

(1)

Decay Constant and Half-Life

(2)

2.6.2.2 Biodegradation

(4)

Pesticide Biodegradation in Soils

(3)

2.7 Generalized Solute Transport Differential Equations for Saturated Porous Media

(1)

2.7.1 General Case: Homogeneous and Anisotropic Dispersion

(1)

2.7.2 Homogeneous and Isotropic Dispersion

(1)

2.7.3 Homogeneous and Isotropic Medium with Uniform Flow Field

(1)

2.7.3.1 Three-Dimensional Case

(1)

2.7.3.2 Two-Dimensional Case

(1)

2.7.3.3 One-Dimensional Case

(1)

2.8 Solute Transport Differential Equations for Unsaturated Porous Media Under Uniform Flow Conditions

(2)

2.8.1 Unsaturated Porous Media Flow Equations under Uniform Flow Conditions

(1)

2.8.2 Solute Transport Differential Equations for Unsaturated Porous Media under Uniform Flow Conditions

(1)

2.9 Volume- and Flux-Averaged Concentrations Under Uniform Flow Conditions

(1)

2.9.1 Relationship between Volume- and Flux-Averaged Concentrations under Uniform Flow Conditions

(1)

2.9.2 Differential Equation for Flux-Averaged Concentrations under Uniform Flow Conditions

(1)

2.9.2.1 Invariance of C, and C1 Based Solute Transport Differential Equations

(1)

2.10 Initial and Boundary Conditions

(4)

2.10.1 Introduction

(1)

2.10.2 Initial Conditions

(1)

2.10.3 Types of Boundary Conditions

(3)

2.10.3.1 Constant Concentration Boundary

(1)

2.10.3.2 Flux Boundary

(1)

Continuous Flux Boundary Condition

(1)

Instantaneous Flux Boundary Condition

(1)

2.10.3.3 No-Solute Flux Boundary

(1)

2.11 Equations for the Mechanical Dispersion Coefficients for Saturated Porous Media

(9)

2.11.1 Laboratory Experiments - Lead Equations

(3)

2.11.1.1 Results from One-Dimensional Experiments

(1)

2.11.1.2 Results from Two-Dimensional Experiments

(2)

2.11.2 Generalized Equations with Theory and Postulation

(7)

2.11.2.1 Fundamental Relationships for Dispersion Coefficients

(1)

2.11.2.2 Mechanical Dispersion Coefficients

(1)

2.11.2.3 Dispersivities of the Medium

(1)

2.11.2.4 Mechanical Dispersion Coefficients: General Case

(1)

2.11.2.5 A Special Case for Mechanical Dispersion Coefficients: the Average Velocity (v) Coincides with the x-Coordinate Axis

(2)

Chapter 3 Analytical Solute Transport Modeling in Aquifers

3.1 Introduction

(1)

3.2 Deterministic vs. Stochastic Modeling Approaches

(1)

3.3 Some Selected Analytical Solute Transport Models and Fundamental Modeling Concepts

(165)

3.3.1 Introduction

(1)

3.3.1.1 Advantages of Analytical Solute Transport Models

(1)

3.3.2 First-Type Source Analytical Models

(69)

3.3.2.1 One-Dimensional Analytical Solute Transport Models for the First-Type Sources

(15)

Problem Statement and Assumptions

(1)

Governing Differential Equation

(1)

Initial and Boundary Conditions

(1)

Unsteady-State Solution for Concentration Distribution

(5)

Unsteady-State Convective-Dispersive Flux Component

(1)

Steady-State Solution for Concentration Distribution

(1)

Steady-State Convective-Dispersive Flux Component

(1)

Applications of the One-Dimensional Analytical Models

(7)

3.3.2.2 Two-Dimensional Analytical Solute Transport Models for the First-Type Strip Sources

(27)

Problem Statement and Assumptions

(1)

Governing Differential Equation

(1)

Initial and Boundary Conditions

(2)

Unsteady-State General Solution for Concentration Distribution

(5)

Unsteady-State Special Solution for a Single-Strip Source

(4)

Unsteady-State Special Solutions for Other Sources

(3)

Steady-State General Solution for Concentration Distribution

(3)

Steady-State Special Solution for a Single-Strip Source

(1)

Steady-State Special Solutions for Other Sources

(1)

Evaluation of the Models

(2)

Application of the Two-Dimensional Analytical Solute Transport Models to Ground Water Flow Systems

(7)

3.3.2.3 Three-Dimensional Analytical Solute Transport Models for the First-Type Rectangular Sources

(27)

Problem Statement and Assumptions

(2)

Governing Differential Equation

(1)

Initial and Boundary Conditions

(1)

Unsteady-State General Solution for Concentration Distribution

(8)

Unsteady-State Special Solution for a Single Rectangular Source

105

(4)

Unsteady-State Special Solutions for Other Sources

109

(2)

Steady-State General Solution for Concentration Distribution

111

(2)

Steady-State Special Solution for a Single Rectangular Source

113

(1)

Steady-State Special Solutions for Other Sources

114

(1)

Evaluation of the Models

114

(3)

Application of the Three-Dimensional Analytical Solute Transport Models to Ground Water Flow Systems

117

(3)

3.3.3 Third-Type Source Analytical Solute Transport Models

120

(57)

3.3.3.1 One-Dimensional Analytical Solute Transport Models for the Third-Type Sources

120

(11)

Problem Statement and Assumptions

122

(1)

Governing Differential Equation

122

(1)

Initial and Boundary Conditions

122

(2)

Unsteady-State Solution for Concentration Distribution

124

(1)

Unsteady-State Convective-Dispersive Flux Component

125

(1)

Steady-State Solution for Concentration Distribution

125

(1)

Steady-State Convective-Dispersive Flux

126

(1)

Evaluation of the Models

126

(1)

Applications of the One-Dimensional Analytical Solute Transport Models

126

(3)

Comparative Evaluation of the First- and Third-Type Source Conditions

129

(2)

Relationship between the First- and Third-Type Source Models

131

(1)

3.3.3.2 Two-Dimensional Analytical Solute Transport Models for the Third-Type Strip Sources

131

(26)

Problem Statement and Assumptions

131

(1)

Governing Differential Equation

132

(1)

Initial and Boundary Conditions

132

(1)

Unsteady-State General Solution for Concentration Distribution

133

(3)

Unsteady-State Special Solution for a Single-Strip Source

136

(2)

Unsteady-State Special Solutions for Other Sources

138

(3)

Steady-State General Solution for Concentration Distribution

141

(1)

Steady-State Special Solution for a Single-Strip Source

142

(1)

Steady-State Solutions for Other Sources

143

(1)

Evaluation of the Models

144

(4)

Application of the Two-Dimensional Analytical Models to Ground Water Flow Systems

148

(2)

Comparative Evaluation of the First- and Third-Type Source Models

150

(7)

3.3.3.3 Three-Dimensional Analytical Solute Transport Models for the Third-Type Rectangular Sources

157

(20)

Problem Statement and Assumptions

157

(1)

Governing Differential Equation

158

(1)

Initial and Boundary Conditions

158

(1)

Unsteady-State Solution for Concentration Distribution

159

(3)

Unsteady-State-Special Solution for a Single Rectangular Source

162

(2)

Unsteady-State Special Solutions for Other Sources

164

(2)

Steady-State General Solution for Concentration Distribution

166

(3)

Steady-State Special Solution for a Single Rectangular Source

169

(1)

Steady-State Special Solutions for Other Sources

170

(1)

Evaluation of the Models

171

(5)

Application of the Three-Dimensional Analytical Models to Ground Water Flow Systems

176

(1)

Comparative Evaluation of the First- and Third-Type Source Models

177

(1)

3.3.4 Instantaneous Source Analytical Solute Transport Models

177

(39)

3.3.4.1 One-Dimensional Analytical Solute Transport Models for Instantaneous Sources

177

(4)

Problem Statement and Assumptions

177

(1)

Governing Differential Equation

178

(1)

Initial and Boundary Conditions

178

(1)

Solution for Concentration Distribution

179

(1)

Convective-Dispersive Flux Component

179

(1)

Evaluation of the Model

180

(1)

Applications of the One-Dimensional Analytical Solute Transport Model

180

(1)

3.3.4.2 Two-Dimensional Analytical Solute Transport Models for Instantaneous Strip Sources

181

(13)

Problem Statement and Assumptions

182

(1)

Governing Differential Equation

182

(1)

Initial and Boundary Conditions

182

(1)

Solution for Concentration Distribution

183

(2)

Special Solution for a Single Instantaneous Strip Source

185

(3)

Special Solutions for Other Sources

188

(1)

Evaluation of the Models

189

(1)

Application of the Two-Dimensional Analytical Models to Ground Water Flow Systems

189

(5)

3.3.4.3 Three-Dimensional Analytical Solute Transport Models for Instantaneous Rectangular Sources

194

(15)

Problem Statement and Assumptions

194

(1)

Governing Differential Equation

195

(1)

Initial and Boundary Conditions

196

(1)

Solution for Concentration Distribution

196

(4)

Special Solution for a Single Rectangular Source

200

(2)

Special Solutions for Other Sources

202

(3)

Evaluation of the Models

205

(3)

Application of the Three-Dimensional Analytical Models to Ground Water Flow Systems

208

(1)

3.3.4.4 Three-Dimensional Analytical Model For Instantaneous Parallelepiped Sources

209

(7)

Problem Statement and Assumptions

209

(1)

Governing Differential Equations

209

(1)

Initial and Boundary Conditions

210

(1)

Solution

211

(1)

Special Solutions

212

(3)

Chapter 4 Numerical Flow and Solute Transport Modeling in Aquifers

4.1 Introduction

215

(1)

4.2 Numerical Flow and Solute Transport Modeling Approaches

216

(1)

4.2.1 Reasons for the Popularity of the Finite-Difference Method

216

(1)

4.2.2 Historical Background of the Finite-Difference Method

216

(1)

4.3 Finite-Difference Modeling Approaches for Ground Water Flow

216

(88)

4.3.1 Fundamental of Flow in Aquifers

216

(21)

4.3.1.1 Hydraulic Head and Drawdown

217

(1)

Hydraulic Head

217

(1)

Drawdown

218

(1)

4.3.1.2 Porosity and Effective Porosity

218

(1)

4.3.1.3 Discharge Velocity and Ground Water Velocity

219

(2)

4.3.1.4 Darcy's Law

221

(4)

Darcy Experiment

223

(1)

Validity of Darcy's Law

224

(1)

4.3.1.5 Hydraulic Conductivity, Transmissivity, and Permeability

225

(4)

Hydraulic Conductivity

225

(1)

Transmissivity

226

(1)

Permeability

226

(3)

4.3.1.6 Classification of Aquifers with Respect to Hydraulic Conductivity

229

(4)

Basic Definitions

230

(1)

Classifications of Aquifers

231

(2)

4.3.1.7 Storage Coefficient, Specific Yield, and Specific Retention

233

(4)

Storage Coefficient

233

(1)

Specific Yield

234

(3)

Specific Retention

237

(1)

4.3.2 Equations of Motion in Aquifers: Generalization of Darcy's Law

237

(3)

4.3.2.1 Equations of Motion: Principal Directions of Anisotropy Coincide with the Directions of Coordinate Axes

237

(3)

Darcy Velocities in Two- and Three-Dimensional Cartesian Coordinates

237

(2)

Darcy Velocities in Two- and Three-Dimensional Cylindrical Coordinates

239

(1)

4.3.2.2 Equations of Motion: Principal Directions of Anisotropy Do Not Coincide with the Directions of Coordinate Axes

240

(1)

4.3.3 Principal Hydraulic Conductivities

240

(5)

4.3.3.1 Principal Hydraulic Conductivities and Transmissivities for Two-Dimensional Case

241

(2)

4.3.3.2 Mohr's Circle for Principal Hydraulic Conductivities and Transmissivities for Two-Dimensional Case

243

(2)

4.3.4 Directional Hydraulic Conductivities

245

(3)

4.3.4.1 Directional Hydraulic Conductivities and Transmissivities for Two-Dimensional Cases

245

(2)

Hydraulic Conductivity and Transmissivity Ellipses

247

(1)

4.3.4.2 Directional Hydraulic Conductivities for Three-Dimensional Cases

247

(1)

Hydraulic Conductivity Ellipsoid

248

(1)

4.3.5 Derivation of the Governing Differential Equation in Rectangular Coordinates

248

(3)

4.3.5.1 General Case: The Principal Directions of Anisotropy Are Not the Coordinate Axes

248

(1)

Three-Dimensional Flow Equation

248

(1)

Two-Dimensional Flow Equation under Special Conditions

249

(1)

4.3.5.2 Special Cases: The Principal Directions of Anisotropy Are the Coordinate Axes

249

(2)

Homogeneous and Isotropic Aquifer

250

(1)

Homogeneous and Anisotropic Aquifer

250

(1)

Heterogeneous and Isotropic Aquifer

250

(1)

Transversely Isotropic Aquifer

250

(1)

Heterogeneous and Anisotropic Aquifer

250

(1)

4.3.6 Derivation of the Governing Differential Equation in Cylindrical Coordinates

251

(2)

4.3.6.1 Differential Equation in Three Dimensions for a Transversely Isotropic Aquifer Case

251

(2)

4.3.6.2 Differential Equation in Two Dimensions

253

(1)

4.3.7 Differential Equations Under Steady-State Conditions

253

(1)

4.3.8 Governing Differential Equation for the Finite-Difference Modeling Approach

254

(1)

4.3.9 Finite-Difference Grid and Discretization Convention

255

(2)

4.3.9.1 Grid Coordinates and Cartesian Coordinates

255

(1)

4.3.9.2 Block- and Point-Centered Grid Discretizations

256

(1)

4.3.9.3 Discretization with Time

257

(1)

4.3.10 Derivation of the Finite-Difference Equation

257

(9)

4.3.10.1 Flow Rates into Cell (i, j, k) From the Six Adjacent Cells

258

(1)

4.3.10.2 Flow Rates into the Cell From Outside the Aquifer

259

(2)

4.3.10.3 Whole Continuity Equation

261

(1)

4.3.10.4 Approximation for Head Change with Respect to Time

261

(2)

4.3.10.5 Transient Simulation

263

(1)

4.3.10.6 Determination of Head Distribution

263

(1)

4.3.10.7 Final Form of the Finite-Difference Equation

264

(1)

4.3.10.8 Approximate Solution and Truncation Error

264

(2)

4.3.11 Boundary and Initial Conditions

266

(2)

4.3.11.1 Boundary Conditions

266

(2)

Constant Head Boundary

266

(1)

Specified Head Boundary

266

(1)

Specified Flux Boundary

266

(1)

No-Flow Boundary

267

(1)

Seepage Face Boundary

267

(1)

Head-Dependent Flux Boundary

267

(1)

4.3.11.2 Initial Conditions

268

(1)

4.3.12 Finite-Difference Grid Discretization of Space

268

(5)

4.3.12.1 Finite-Difference Grid System Orientation

268

(1)

4.3.12.2 Vertical Grid Discretization

268

(3)

Vertical Discretization of Permeable Materials

268

(1)

Vertical Discretization of Less Permeable Materials (Aquitards)

269

(2)

4.3.12.3 Horizontal Grid Discretization

271

(2)

The Rule of Thumb for Adjacent Grid Sizing

271

(1)

Peclet Number Requirement for Grid Sizing

271

(1)

Practical Considerations for the Design of Grids

272

(1)

4.3.13 Discretization of Time

273

(4)

4.3.13.1 Input Parameters for Time in the MODFLOW Family Codes

273

(2)

4.3.13.2 Time Step Selection

275

(2)

Expressions for Selection of Time Step

275

(1)

Trial-and-Error Runs for Selection of Time Steps

276

(1)

4.3.14 The Conductance Components of the Finite-Difference Equations in the Modflow Family Codes

277

(5)

4.3.14.1 Basic Conductance Equations

277

(3)

Definition of Conductance

277

(1)

Equivalent Conductance

277

(3)

4.3.14.2 Horizontal Conductance

280

(1)

4.3.14.3 Vertical Conductance

281

(1)

Vertical Hydraulic Conductivity and Vertical Grid Spacing Parameter (V cont)

281

(1)

4.3.15 Implementation of the Boundary Conditions

282

(19)

4.3.15.1 Constant Head Boundaries

282

(1)

4.3.15.2 Constant Flux Boundaries

283

(5)

Boundary Conditions at Wells

284

(1)

Areally Distributed Recharge Boundaries

285

(3)

No-Flow Boundaries

288

(1)

4.3.15.3 Head-Dependent Flux Boundaries

288

(2)

4.3.15.4 River Boundaries

290

(5)

River-Aquifer Interrelationships

292

(1)

Leakage through a Reach of a Riverbed

293

(1)

The Relationship between River Leakage and Head in the Cell

294

(1)

Required Data for MODFLOW

294

(1)

4.3.15.5 Drain Boundries

295

(3)

MODFLOW Drain Boundary Approach

295

(2)

Required Data for MODFLOW

297

(1)

4.3.15.6 Evapotranspiration Boundaries

298

(3)

Definitions

298

(1)

Representation of Evapotranspiration in a Ground Water Flow Model

299

(1)

MODFLOW Evapotranspiration Modeling Approach

300

(1)

4.3.16 Solution Techniques for Finite-Difference Equations for Flow

301

(3)

4.3.16.1 Matrix-form Equations

301

(1)

4.3.16.2 Direct Methods

302

(1)

4.3.16.3 Iterative Methods

303

(1)

Strongly Implicit Procedure

303

(1)

Slice-Successive Overrelaxation Procedure

303

(1)

Preconditioned Conjugate-Gradient 2 (PCG2) Procedure

303

(1)

4.4 Finite-Difference Modeling Approach for Solute Transport

304

(6)

4.4.1 Governing Equations

304

(2)

4.4.1.1 Governing Differential Equation

304

(1)

4.4.1.2 Hydrodynamic Dispersion Coefficients

305

(1)

4.4.1.3 Expression for the Retardation Factor

306

(1)

4.4.1.4 Expression for the Degradation Constant

306

(1)

4.4.2 Finite-Difference Equation

306

(2)

4.4.3 Initial and Boundary Conditions

308

(1)

4.4.4 Discretization of Space

308

(1)

4.4.5 Discretization of Time

308

(1)

4.4.6 Solution Techniques for Finite-Difference Equations for Solute Transport

309

(1)

4.4.6.1 Slice-Successive Overrelaxation (SSOR) Procedure

309

(1)

4.4.6.2 Orthomin Procedure

310

(1)

4.5 Application of Numerical Flow and Solute Transport Models

310

(11)

4.5.1 Introduction

310

(1)

4.5.2 Ground Water Flow Model Application

311

(5)

4.5.2.1 Model Conceptualization and Dimensionality

311

(2)

Conceptual Model Development

311

(1)

Selection of Model Dimensionality

311

(1)

Selection of Computer Code(s)

311

(2)

4.5.2.2 Model Construction

313

(1)

4.5.2.3 Model Calibration

313

(2)

Definitions for Calibration

313

(1)

Calibration Targets

314

(1)

Evaluation of Calibrated Model Results

314

(1)

4.5.2.4 Application of A Calibrated Model

315

(1)

4.5.2.5 Sensitivity Analysis

316

(1)

4.5.3 Ground Water Solute Transport Model Application

316

(5)

4.5.3.1 Model Conceptualization and Dimensionality

316

(2)

Conceptual Model Development

316

(1)

Selection of Model Dimensionality

316

(2)

Selection of Computer Codes

318

(1)

4.5.3.2 Model Construction

318

(1)

Telescopic Mesh Refinement Approach

318

(1)

Required Data for Solute Transport Model

318

(1)

4.5.3.3 Model Calibration

319

(1)

4.5.3.4 Application of A Calibrated Solute Transport Model

320

(1)

Evaluation of Different Remedial Alternatives

320

(1)

Potential Contaminant Source Identification

321

(1)

Optimizing Ground Water Monitoring Systems

321

(1)

Risk Assessment

321

(1)

4.5.3.5 Sensitivity Analysis

321

(1)

4.6 Model Testing

321

(1)

4.6.1 Model Verification

321

(1)

4.6.2 Model Validation

322

(1)

4.7 Model Misuse

322

(1)

4.7.1 Investigator's (Modeler) Background

322

(1)

4.7.2 Procedural Points

323

(1)

4.7.3 Insufficient Data for Calibration

323

(1)

4.7.4 Blind Faith in Model Results

323

(1)

4.8 Modeling Limitations and Sources of Error

323

(1)

4.8.1 Conceptual Errors

324

(1)

4.8.2 Truncation Errors

324

(1)

4.8.3 Data Errors

324

(1)

4.9 Case Study for Ground Water Flow and Solute Transport Modeling

324

(21)

4.9.1 Introduction

324

(1)

4.9.2 Numerical Ground Water Flow and Solute Transport Modeling Example

324

(21)

Description of Study Area

325

(4)

Ground Water Flow and Solute Transport Models Construction

329

(5)

Calibration of Solute Transport Model

334

(3)

Potential Problems with the Solute Transport Model Calibration

337

(4)

Predictive Capability of the Calibrated Solute Transport Model

341

(4)

Chapter 5 Analytical and Numerical Modeling Approaches for Solute Travel Times and Path Lines in Aquifers

5.1 Introduction

345

(1)

5.2 Path Lines and Travel Times of Solute Under Convective Transport Conditions

345

(29)

5.2.1 Kinematics of a Solute Particle

345

(2)

5.2.2 Analytical Methods for the Determination of Solute Path Lines and Travel Times

347

(27)

5.2.2.1 Travel Time Determination Using Darcy's Equation

347

(3)

5.2.2.2 Travel Time Determination Using Analytical Solutions

350

(10)

General Equations for Travel Time

350

(1)

Travel Times to a Well in a Confined Aquifer

351

(6)

Minimum Travel Time under a Dam

357

(3)

5.2.3 Numerical Methods for Solute Path Lines Determination

360

(17)

5.2.3.1 Introduction

360

(1)

5.2.3.2 Theory of Particle Tracking Based on the Finite-Difference Method

361

(1)

Conservation of Mass Equation

361

(1)

Computation Procedure of Path Lines Under Unsteady-State Flow Conditions by MODPATH

362

(1)

Computation Procedure of Path Lines Under Steady-State Flow Conditions by MODPATH

365

(1)

Modifications for Special Cases

369

(1)

Boundary Conditions and Discharge Points

370

(1)

Examples

372

(2)

5.3 Path Lines of Solute Particles Under Convective-Dispersive Transport Conditions

374

(41)

5.3.1 Introduction

374

(2)

5.3.2 Stream Function Concept for Hydrodynamic Dispersion

376

(1)

5.3.3 Partial Differential Equations for the Stream Function for Hydrodynamic Dispersion

377

(6)

5.3.3.1 General Case

377

(1)

5.3.3.2 Special Case in the Cartesian Coordinates: Uniform Ground water Flow Field

378

(1)

5.3.3.3 Special Case in the Cylindrical Coordinates: Uniform Ground Water Flow Field

379

(2)

5.3.3.4 Properties of the Hydrodynamic Dispersion Stream Function

381

(2)

5.3.4 Some Analytical Solutions for the Hydrodynamic Dispersion Stream Function

383

(33)

5.3.4.1 Potential Solution Methods

383

(1)

5.3.4.2 Two-Dimensional Analytical Solutions in a Semi-infinite Porous Medium

383

(22)

Problem Statement and Assumptions

383

(1)

Governing Differential Equations and Convective-Dispersive Flux Components

384

(1)

Boundary Conditions

384

(2)

Solution for Hydrodynamic Dispersion Stream Function for Multiple Sources

386

(1)

Solution for Concentration Function for Multiple-Strip Sources

387

(2)

Special Solutions for a Single-Strip Source

389

(4)

Evaluation of the Integrals in the Solutions

393

(4)

Application of Strip Source Solutions

397

(8)

5.3.4.3 Two-Dimensional Analytical Solutions for Strip Sources in a Finite Porous Medium

405

(13)

Problem Statement and Assumptions

405

(1)

Governing Differential Equations and Convective-Dispersive Flux Components

405

(1)

Boundary Conditions

406

(1)

General Solution for the Hydrodynamic Dispersion Stream Function

407

(2)

Special Solutions for a Single-Strip Source

409

(6)

Chapter 6 Stochastic Analysis of Flow and Solute Transport in Heterogeneous Aquifers and Determination of Expressions for Dispersivities Using Stochastic Approaches

6.1 Introduction

415

(1)

6.2 Physical Mechanism of Convective-Dispersive Solute Transport in Heterogeneous Porous Media

416

(2)

6.2.1 Mechanical Dispersion Coefficients Based on Laboratory Experiments

416

(1)

6.2.2 Convective-Dispersive Solute Transport Mechanism in Natural Materials

416

(1)

6.2.3 Convective-Diffusive Solute Transport Mechanism in Natural Materials

417

(1)

6.3 Determination of Dispersivities from Solute Plume Spatial Moments

418

(60)

6.3.1 One-Dimensional Convective-Diffusive Solute Transport and Plume Spatial Moments

418

(6)

6.3.1.1 One-Dimensional Analytical Solution of a Tracer Slug

418

(2)

Problem Statement and Assumptions

418

(1)

Governing Differential Equation

419

(1)

Initial and Boundary Conditions

419

(1)

Solution for Concentration Distribution

420

(1)

6.3.1.2 Random Walk: One-Dimensional Form with Only Diffusion

420

(1)

6.3.1.3 Some Statistical Properties of Concentration Distribution

421

(1)

6.3.1.4 Standard Deviation

422

(1)

6.3.1.5 Fundamental Relationship Between σ²x and D*

422

(2)

6.3.2 Three-Dimensional Plume Spatial Moments

424

(2)

6.3.2.1 Zeroth Moment

424

(1)

6.3.2.2 First Moments

424

(1)

6.3.2.3 Second Moments

425

(1)

6.3.3 Mechanical Dispersion Coefficient (Dm) and Dispersivity Tensors for General Case

426

(1)

6.3.4 Dispersivity and Mechanical Dispersion Coefficient Tensors for Uniform Ground Water Flow Case

427

(1)

6.3.5 Important Points For the Application of Spatial Moments

428

(1)

6.3.6 Stochastic Description of Hydraulic Conductivity Variation

428

(6)

6.3.6.1 Definitions For Nonuniformity, Heterogeneity, and Anisotropy

429

(1)

6.3.6.2 Description of Heterogeneity by the Stochastic Process Theory

429

(2)

6.3.6.3 Statistical Parameters for Log-Normal Frequency Distributions of Hydraulic Conductivity

431

(3)

6.3.7 Illustrative Examples for the Determination of Dispersivities from Natural Gradient Experiments Using the Plume Spatial Moments Method

434

(44)

6.3.7.1 Borden Experiment

434

(11)

Description of the Aquifer

434

(1)

Hydraulic Heads of the Aquifer

434

(1)

Horizontal Hydraulic Conductivity

435

(1)

Spatial Moment Estimation Methodology

436

(1)

Solute Plume Migration

436

(1)

Estimation of Mass in Solution

437

(3)

Center of Mass and Estimation of its Horizontal Displacement

440

(1)

Estimation of Vertical Displacement of the Mass

440

(2)

Solute Velocity

442

(1)

Estimation of Spatial Covariances and Dispersivities

442

(3)

6.3.7.2 Twin Lake Experiment

445

(12)

Description of the Aquifer

445

(2)

Hydraulic Heads of the Aquifer

447

(1)

Horizontal and Vertical Hydraulic Conductivities

447

(1)

Radioiodine Plume Characterization

447

(2)

Spatial Moments Analysis

449

(8)

Evaluation of the Observed Plumes with an Analytical Model

457

(1)

6.3.7.3 Cape Cod Experiment

457

(21)

Site Description and Aquifer Characteristics

457

(2)

Design and Operation of the Tracer Test

459

(5)

Observed Migration of the Bromide Tracer Plume

464

(1)

Horizontal Migration

464

(4)

Spatial Moments Analysis

468

(10)

6.4 Determination of Expressions for Dispersivities from Perturbation Analysis Using the Fourier-Stieltjes Integral Representation

478

(103)

6.4.1 Definition of Key Concept and Quantities

478

(3)

6.4.1.1 Realization and Ensemble

478

(1)

6.4.1.2 Expected Value

479

(1)

6.4.1.3 Variance of Joint Distributions and Covariance

479

(1)

6.4.1.4 Stationary Process and a Common Covariance Function

480

(1)

6.4.2 One-Dimensional Stationary Random Process Representation

481

(2)

6.4.2.1 Spectral Representation Theorem

481

(1)

6.4.2.2 Covariance Function for A Zero-Mean Stochastic Process X(t)

482

(1)

6.4.2.3 Important Features of the Covariance and Spectral Density Function Relationships

483

(1)

6.4.3 Three-Dimensional Stationary Random Process Representation

483

(2)

6.4.4 Stochastic Analysis of One-Dimensional Steady-State Flow

485

(11)

6.4.4.1 Perturbed Forms of the Flow Equation

485

(3)

Expressing the Variables in Terms of Mean and Perturbation

485

(1)

Spectral Representation

486

(1)

A Form of the Resistivity Autocovariance

487

(1)

Determination of the Spectrum Sh'h'

487

(1)

Determination of the Autocovariance Function for the Hydraulic Head (Rh'h')

488

(1)

6.4.4.2 Linearized Solution in Terms of Hydraulic Conductivity K

488

(2)

Expressing the Variables in Terms of Mean and Perturbation

488

(1)

Spectral Representation and Determination of the Spectra

489

(1)

Determination of the Autocovariance Function for the Hydraulic Head (Rh'h')

490

(1)

6.4.4.3 Linearized Solution in Terms of ln(K)

490

(3)

Purpose of the Usage of ln(K) and the Flow Equation

490

(1)

Normally Distributed ln(K)

491

(1)

Expressing the Variables in Terms of Mean and Perturbation

491

(1)

Spectral Representation

492

(1)

6.4.4.4 Effective Hydraulic Conductivity for One-Dimensional Case

493

(3)

Approximate Analysis of One-Dimensional Flow

493

(1)

Spectral Representation

494

(2)

6.4.5 Stochastic Analysis of Three-Dimensional Steady-State Flow in Isotropic Porous Media

496

(5)

6.4.5.1 Three-Dimensional Perturbed Flow Equation

496

(1)

6.4.5.2 Spectral Representation

497

(1)

6.4.5.3 Effective Hydraulic Conductivity For Three-Dimensional Case

498

(1)

6.4.5.4 The Hydraulic Head and Loghydraulic Conductivity Processes

499

(2)

Darcy Flux Expressions

499

(1)

Spectral Representations

500

(1)

Usage of Effective Hydraulic Conductivity Expressions

501

(1)

6.4.6 Analysis of Macrodispersion in Three-Dimensionally Heterogeneous Aquifers under Steady Flow Conditions Using Stochastic Solute Transport Theory: Gelhar and Axness' First-Order Analysis Approach

501

(63)

6.4.6.1 Objectives and Literature Review

501

(2)

6.4.6.2 Scale Classification

503

(1)

6.4.6.3 Generalization of The Spectral Representation Expressions and First-Order Analysis

504

(1)

6.4.6.4 Description of the Hydraulic Conductivity Variation

504

(1)

6.4.6.5 Correlation Structure of the Random Hydraulic Conductivity Field

505

(5)

Autocovariance Functions for Hydraulic Conductivity Fields

505

(1)

The Exponential Autocovariance

505

(1)

The Modified Exponential Autocovariance

505

(1)

Random Fields that are Not Statistically Homogeneous

506

(1)

Relationship Between the Variogram and Covariance for One-Dimensional Case

506

(1)

Stochastic Characterization of Aquifers

507

(1)

Aquifer Covariance Data From the Literature

507

(2)

Aquifer Covariance Data From Field Tests

509

(1)

6.4.6.6 Macroscopic Dispersive Fluxes

510

(2)

The Effects of Variations of Porosity and Local Dispersion Coefficients

510

(1)

Macroscopic Dispersive Flux Expressions

511

(1)

6.4.6.7 Determination of the General Expression for the Macroscopic Dispersion Tensor Aij

512

(5)

Stochastic Solute Transport Differential Equation: General Case

512

(1)

Unsteady State Deterministic Solute Transport Differential Equation

512

(1)

Steady-State Deterministic Solute Transport Differential Equation

513

(1)

Assumptions and Expressing of the Variables in Terms of a Mean and Perturbation

513

(1)

General Stochastic Differential Equation

514

(1)

Stochastic Solute Transport Differential Equation: Special Case

515

(1)

Spectral Solution and the Macroscopic Dispersion Tensor Aij

516

(1)

6.4.6.8 Principal Macrodispersivities for Two-Dimensional Solute Transport Cases

517

(5)

Two-Dimensional Fickian Dispersive Fluxes

517

(1)

Principal Macrodispersivities

518

(2)

Mohr's Circle for Principal Macrodispersivities for Two-Dimensional Case

520

(1)

Anisotropy is in the x-y Plane and the z and z* Axis Are the Same

520

(1)

Anisotropy Is in the x-z Plane and the z and z* Axis Are the Same

521

(1)

6.4.6.9 Determination of the Macroscopic Dispersion Tensor Au in a Statistically Isotropic Aquifer - Local Dispersivity Isotropic

522

(8)

Statistically Isotropic Media 522 Exponential Autocovariance Function for a Statistically Isotropic Medium

524

(1)

Macroscopic Dispersion Tensor Aij for a Statistically Isotropic Medium

525

(1)

Evaluation of A11

525

(1)

Asymptotic Approach Result for A11

526

(1)

Exact Approach Result for A11

526

(1)

Evaluation of A22 = A33

526

(1)

Discussion of the Results

527

(3)

6.4.6.10 Approximate Determination of Aij for Macroscopic Dispersion in Statistically Isotropic Media Local Dispersivity Anisotropic

530

(2)

Expression for Aij

530

(1)

Evaluation of A11

530

(1)

Evaluation of A22

531

(1)

Special Case: Two-Dimensional Flow

531

(1)

Typical Parameter Values in the A11 and A22 Expressions

531

(1)

6.4.6.11 Determination of Aij for Macroscopic Dispersion in a Statistically Anisotropic Aquifer With Arbitrary Orientation of Stratification

532

(32)

Aquifer Description and General Formulation

532

(1)

Effective Hydraulic Conductivity Tensor

533

(1)

Evaluation of the Term f' h'/ xi

534

(2)

The General Expression for the Fij Integral

536

(1)

The Expressions for g11, g22, and g33

536

(1)

Special Cases for g11 g22, g33, aij, and Fij for λ1 = λ2 (σ = 1)

537

(2)

General Expression for the Effective Hydraulic Conductivity Tensor

539

(1)

Effective Hydraulic Conductivity Tensor for a Special Case: Statistical Isotropy in the x'1, x'2 Plane (λ1 = λ2)

539

(3)

Specific Discharge Spectra

542

(1)

Formulation and General Solution for Macroscopic Dispersion in Statistically Anisotropic Media with Arbitrary Orientation of Stratification

543

(1)

Generalized Fickian Transport Relationship and General Solution

543

(1)

Evaluation of the Integrals

544

(1)

Special Case 1: Isotropy is in the Plane of Stratification with λ1=λ2 and the Mean Flow Inclined at Angle theta with the Bedding

544

(1)

Determination of the Ratio I3/I1 and γ

544

(1)

Components of the Macrodispersivity Tensor

545

(1)

Illustrative Example for the Principal Components of the Macrodispersivity Tensor

545

(4)

Special Case 2: Horizontal Stratification with Horizontal and Vertical Anisotropy

549

(1)

Components of the Macrodispersivity Tensor

550

(2)

Important Features for Case II

552

(2)

Special Case 3: Two-Dimensional Horizontal Flow

554

(2)

Illustrative Example from the Borden Experiment

556

(8)

6.4.7 Analysis of Macrodispersion in Three-Dimensionally Heterogeneous Aquifers under Unsteady Flow Conditions Using Stochastic Solute Transport Theory: The Rehfeldt and Gelhars' First-Order Analysis Approach

564

(17)

6.4.7.1 Introduction

564

(1)

Examination of the Influence of Temporal Variability on Dispersive Mixing

564

(1)

Limitations of the Approach and Assumptions

564

(1)

6.4.7.2 Governing Equations

565

(1)

6.4.7.3 Spectral Solution

566

(1)

6.4.7.4 Development of the Specific Discharge Spectrum and Flow Equation

567

(5)

Decomposing the Hydraulic Gradient into the Sum of Three Parts

567

(1)

Specific Discharge Perturbations Equation

568

(1)

Flow Equation

568

(1)

Spectral Representation for Flow and Flux Equations

569

(3)

6.4.7.5 Hydraulic Gradient Spectrum and Expression Aij (t)

572

(1)

6.4.7.6 Examination of Hydraulic Gradient Variability

572

(4)

Hydraulic Gradient Magnitude Variation

572

(4)

6.4.7.7 Examination of Hydraulic Gradient Direction Variability

576

(8)

Determination of the Spectrum Expressions

576

(2)

Components of the Macrodispersivity Tensor

578

(1)

Evaluation of the Macrodispersivity Tensor Components

579

(2)

Chapter 7 Estimation of Dispersivities for Solute Transport in Aquifers

7.1 Introduction

581

(1)

7.2 Laboratory vs. Field-Scale Dispersivities

581

(1)

7.2.1 Laboratory and Field Scales

581

(1)

7.2.2 Dispersivities under Field Conditions

582

(1)

7.3 Approaches for the Determination of Dispersivities

582

(1)

7.3.1 Approach I: Perfectly Layered Aquifer Assumption

582

(1)

7.3.2 Approach II: The Spatial Statistics of the Heterogeneous Velocity Field Is Assumed To Be Given

583

(1)

7.3.3 Approach III: The Statistical Description of the Spatial Variability of Hydraulic Conductivity Is Assumed To Be Given

583

(1)

7.4 Estimation of Dispersivities Using Relationships Between Dispersivities and Field Scale

583

(34)

7.4.1 Introduction

583

(1)

7.4.2 Field Data on Dispersivities

584

(16)

7.4.2.1 Summary of Field Observations

584

(1)

7.4.2.2 Aquifer Characteristics

584

(14)

7.4.2.3 Types of Subsurface Solute Transport Tests

598

(1)

7.4.2.4 Dimensionality and Monitoring

599

(1)

7.4.2.5 Correlation of Longitudinal Dispersivity with Other Quantities

599

(1)

Longitudinal Dispersivity, AL, vs. Scale, s

599

(1)

Longitudinal Dispersivity, AL, vs. Horizontal Hydraulic Conductivity, Kh, and Effective Porosity, φe

600

(1)

7.4.3 Data Evaluation and Methods for Estimation of Dispersivities

600

(12)

7.4.3.1 Evaluation of Data for Dispersivities

601

(1)

7.4.3.2 Criteria for High-Reliability Dispersivities Data

602

(2)

Some Major Problems Regarding Dispersivity Determination

603

(1)

7.4.3.3 Criteria for Low-Reliability Dispersivities Data

604

(1)

7.4.3.4 Results of the Classification for the Longitudinal Dispersivity Data

605

(3)

An Important Feature of the Longitudinal Dispersivity, Data

605

(1)

Reanalysis of Selected Longitudinal Dispersivity, Data

606

(2)

Evaluation of the Longitudinal Dispersivity Values Based on Numerical Simulations

608

(1)

7.4.3.5 Results of the Classification for the Transverse Dispersivities Data

608

(3)

Transverse Horizontal Dispersivity, ATH

609

(1)

Transverse Vertical Dispersivity, ATV

609

(1)

The AL/ATH and AL/ATV Ratios

609

(2)

7.4.3.6 Interpretations of the Dispersivities Data

611

(1)

(1) Clear Trend of Systematic Increase in AL with s

611

(1)

(2) Porous vs. Fractured Media

611

(1)

(3) Status of High-Reliability Points

611

(1)

(4) A Single Universal Line Does Not Seem Reasonable

611

(1)

(5) Explanation Based on the Stochastic Theory

611

(1)

(6) Uncertainty in Estimating Longitudinal Dispersivity, AL

611

(1)

(7) Strong Cautionary About Linear Regression Representation Through the Dispersivity Data

612

(1)

(8) Transverse Dispersivities

612

(1)

(9) Typical Ratios for Dispersivities

612

(1)

7.4.4 Neuman's Interpretation of the Longitudinal Dispersivity Data

612

(3)

7.4.4.1 Theoretical Background

612

(1)

Expression for Asymptotic Longitudinal Dispersivity, AinfintyL

612

(1)

7.4.4.2 Interpretation of Scale Effect and Longitudinal Dispersivity Equations

613

(5)

Neuman's First Interpretation

613

(1)

Neuman's Second Interpretation

613

(1)

Neuman's Conclusions Based on His Interpretations

614

(1)

7.4.5 Usage of the Dispersivities Estimation Procedures

615

(2)

Chapter 8 Determination of Solute Sorption Parameters in Aquifers

8.1 Introduction

617

(1)

8.2 The Kd/Koc Approach for the Determination of Retardation Factor

617

(8)

8.2.1 Schematic Illustration of the Retardation Concept

617

(1)

8.2.2 Characteristics of the Kd/Koc Approach and Related Quantities

618

(3)

8.2.2.1 Relationship between the Distribution Coefficient Kd and Organic Carbon Content foc

619

(1)

8.2.2.2 Organic Carbon Partition Coefficient Koc

619

(1)

8.2.2.3 Octanol-Water Partition Coefficient

620

(1)

8.2.2.4 Solubility in Water

620

(1)

8.2.3 Evaluation of the Kd/Koc Approach with Experiments

621

(4)

8.2.3.1 Comparison of the Experimental and Computed Column Breakthrough Curves

621

(1)

Determination of S = f(C) Relationship

621

(1)

Comparison With Experiments

621

(1)

Normalized Column Breakthrough Curves for Reactive and Nonreactive Solutes

622

(1)

8.2.3.2 Evaluation of the Linear Equilibrium Isotherm Equation with Respect to the Freundlich Isotherm

622

(3)

Comparison of the Linear and Freundlich Isotherm Equations

622

(1)

Comparison of Concentration Profiles Based on Linear and Freundlich Isotherm

623

(2)

8.3 Compiled Data for Koc Values

625

(11)

8.3.1 Direct Data for Koc for Various Chemicals

625

(11)

8.4 Regression Equations for Koc Values

636

(2)

8.4.1 Relationship Between the Koc and Kow

636

(2)

8.4.2 Relationship between Koc and solubility S

638

(1)

8.5 Estimation of Koc from the Regression Equations

638

(7)

8.5.1 Foundations of the Estimation Methods

639

(1)

8.5.2 Available Regression Equations Based on Kow

639

(6)

References

645

(16)

Index

661

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Applied Flow and Solute Transport Modeling in Aquifers: Fundamental Principles and Analytical and Numerical Methods

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