Table Of ContentAN INTRODUCTION TO SOLUTE TRANSPORT IN
HETEROGENEOUS GEOLOGIC MEDIA
Overthepastseveraldecades,analysesofsolutemigrationinaquifershavewidelyadopted
theclassicaladvection–dispersionequation.However,misunderstandingsoveradvection–
dispersionconcepts,theirrelationshipwiththescalesofheterogeneity,ourobservationand
interest, and their ensemble mean nature have created furious debates aboutthe concepts’
validity.Thisbookprovidesaunifiedandcomprehensiveoverviewandlucidexplanations
ofthestochasticnatureofsolutetransportprocessesatdifferentscales.Italsopresentstools
foranalyzingsolutetransportanditsuncertaintytomeetourneedsatdifferentscales.Easy-
to-understand physical explanations without complex mathematics make this book an
invaluable resource for students, researchers, and professionals performing groundwater
quality evaluations, management, and remediation.
tian-chyi “jim” yeh is a professor in the Department of Hydrology and Atmospheric
SciencesattheUniversityofArizona.Heisaninternationallyrenownedleaderinstochas-
tic/numerical analysis and laboratory/field investigations of flow and solute transport in
variablysaturatedheterogeneousgeologicmedia.HewastheleadauthorofFlowthrough
Heterogeneous Geologic Media (Cambridge University Press, 2015) and pioneered the
new generation ofaquifercharacterization technology: hydraulictomography.
yanhuidongisanassociateprofessorintheInstituteofGeologyandGeophysics,Chinese
AcademyofSciences.Hespecializesinfluidflow,reactivetransport,andporosity/permea-
bility evolution in porous and fractured geological media with application to the manage-
ment of groundwater resources, the disposal of high-level radioactive wastes, and the
environmentalimpactassessmentofhydrocarbonextractionactivities.
shujun ye is a professor in the School of Earth Sciences and Engineering at Nanjing
University.Shehasmorethan20yearsofteachingandresearchexperienceinthenumerical
simulationofgroundwaterflow,multiphaseflow,masstransport,landsubsidence,andEarth
fissures.Sheisthevice-chairoftheUNESCOLandSubsidenceInternationalInitiativeanda
memberoftheGroundwaterTechnicalCommitteeoftheAmericanGeophysicalUnion.She
isthecoauthorofHydrogeology:StrategyoftheDisciplinesDevelopmentinChina(2021).
Published online by Cambridge University Press
Published online by Cambridge University Press
AN INTRODUCTION TO SOLUTE
TRANSPORT IN HETEROGENEOUS
GEOLOGIC MEDIA
TIAN-CHYI “JIM” YEH
UniversityofArizona
YANHUI DONG
ChineseAcademyofSciences
SHUJUN YE
NanjingUniversity
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Informationonthistitle:www.cambridge.org/9781316511183
DOI:10.1017/9781009049511
©Tian-ChyiJimYeh,YanhuiDong,andShujunYe2023
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Firstpublished2023
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Yeh,T.-C.(Tian-Chyi),author.|Dong,Yanhui,author.|Ye,Shujun,author.
Title:Anintroductiontosolutetransportinheterogeneousgeologicmedia/Tian-ChyiJimYeh,
YanhuiDong,ShujunYe.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2023.|Includesbibliographical
referencesandindex.
Identifiers:LCCN2022029342(print)|LCCN2022029343(ebook)|ISBN9781316511183(hardback)|
ISBN9781009049511(epub)
Subjects:LCSH:Waterquality–Mathematicalmodels.|Water–Analysis–Mathematicalmodels.|
Soils–Solutemovement–Mathematicalmodels.|Transporttheory–Mathematicalmodels.|
Diffusioninhydrology.|Reaction-diffusionequations–Numericalsolutions.|BISAC:SCIENCE/
EarthSciences/Hydrology
Classification:LCCTD380.Y442022 (print)|LCCTD380 (ebook)|DDC628.1/61–dc23/eng/20220811
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Contents
Preface page ix
1 Fundamental Concepts 1
1.1 Introduction 1
1.2 Linear First-Order Nonhomogeneous Ordinary
Differential Equation 1
1.3 Random Variable 2
1.4 Stochastic Process or Random Field 8
1.5 Stochastic Fluid-Scale Continuum 24
1.6 Summary 32
2 Well-Mixed Models for Surface Water Quality Analysis 33
2.1 Introduction 33
2.2 Well-Mixed Systems 33
2.3 Mathematical Model for Water Quality of Surface
Water Reservoir 33
2.4 Analytical Solutions 41
2.5 An Application to Water Quality Analysis 54
2.6 Homework 55
3 Well-Mixed Models for Subsurface Water Quality Analysis 60
3.1 Introduction 60
3.2 Water Balance Equation 60
3.3 Application to the Recharge Estimation 65
3.4 Water Quality Formulation 69
3.5 Well-Mixed Models with Reactive Chemicals 77
3.6 Sensitivity and Uncertainty Analysis 84
3.7 Applications and Uncertainty Analysis 87
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vi Contents
3.8 Final Remarks 89
3.9 Homework 90
4 Molecular Diffusion 93
4.1 Introduction 93
4.2 Molecular Diffusion and Fick’s Law 93
4.3 Diffusion Equation 98
4.4 Brownian Motion, Random Walk, Fickian Regime, and
Ensemble Mean 100
4.5 Analytical Solutions of the Diffusion Equation 108
4.6 Advection–Diffusion Process 113
4.7 Remarks 121
4.8 Homework 122
5 Numerical Methods for Advection–Diffusion Equations 123
5.1 Introduction 123
5.2 Finite Difference Approximation 123
5.3 Method of Characteristics (MOC) 138
5.4 Finite Element Approach 143
6 Shear Flow Dispersion 157
6.1 Introduction 157
6.2 Concept of Hydrodynamic Dispersion 157
6.3 G. I. Taylor’s Analysis of Shear Flow Dispersion 163
6.4 Solute Spreading Without Molecular Diffusion 170
6.5 Conclusion 172
7 Solute Transport in Soil Columns 174
7.1 Introduction 174
7.2 Advection and Dispersion Concepts in Solute Transport in
Porous Media 174
7.3 Mathematical Formulation of Solute Transport in Porous Media 180
7.4 Dispersion in Porous Media at Darcy Scale 183
7.5 Mathematical Models for Dispersive Flux 185
7.6 Advection–Dispersion Equation for Solute Transport in
Porous Media 185
7.7 Properties of Dispersion Coefficients 186
7.8 The Equation for Solution Transport in Laboratory-Scale
Porous Media 188
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Contents vii
7.9 Non-Fickian Solute Transport 189
7.10 Reactive Solute Transport 198
7.11 Homework 201
8 Parameter Estimation 202
8.1 Introduction 202
8.2 Estimation of Parameters Using Analytical Solution 202
8.3 Spatial Moments 209
8.4 Temporal Moments 213
8.5 Method of Moments for Multidimensional Plumes 217
8.6 Least Square Nonlinear Regression Approach 219
8.7 Remarks 220
8.8 Homework 221
9 Solute Transport in Field-Scale Aquifers 222
9.1 Introduction 222
9.2 Stochastic Representation of Heterogeneity 223
9.3 Classification of Random Fields 231
9.4 Field Example 235
9.5 Highly Parameterized Heterogeneous Model (HPHM) 240
9.6 Zonation Conceptual Model 243
9.7 Ensemble Mean Based on Monte Carlo Simulation (MCS) 243
9.8 Equivalent Homogeneous Media (EHM) Approach 244
9.9 Homework 253
10 Field-Scale Solute Transport Experiments under Natural Gradient 254
10.1 Introduction 254
10.2 Borden Site, Canada 254
10.3 Macrodispersion Experiment (MADE) Site, Mississippi 266
10.4 Summary and Discussions 269
10.5 Lessons Learned 273
11 Forced Gradient Field-Scale Tracer Experiments 275
11.1 Introduction 275
11.2 May 1990 Two-Well Tracer Experiment 276
11.3 May–September 1992 Field Experiments and Analysis 283
11.4 August–September 1992 Injection Experiments (Validation) 294
11.5 Lessons Learned 294
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viii Contents
12 High-Resolution Characterization (Tomographic Surveys): Hydraulic
Tomography and Geophysics ERT Surveys 298
12.1 The Need for Highly Parameterized Heterogeneous Models 298
12.2 Hydraulic Tomography (HT) 298
12.3 HT Field Examples 303
12.4 Electrical Resistivity Tomography (ERT) 311
12.5 Benefits of High-Resolution Characterization 314
12.6 Predictions of Transport with Different Degrees of
K Resolutions 322
12.7 Importance of Large-Scale Flow Variations 325
12.8 Illustrative MC Simulation 332
12.9 Summaries of the Theme of the Book 335
12.10 Homework 337
References 338
Index 348
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Preface
Environmental fluids migrating at multiple scales and affected by heterogeneity
and factors of various scales is a well-known fact. However, there is widespread
and profound confusion in the hydrology literature, textbooks, and community
about advection–dispersion concepts; the relationship between heterogeneity and
dispersion; the stochastic nature of velocity at different scales; and model,
observation, and heterogeneity scale issues. Countless publications on anomalous
dispersion phenomena and inventions of non-Fickian models, and textbooks’
omission of the fundamental scale issues testify to this perplexity. Expressly, the
hydrologycommunitypoorlyunderstandstheensemblemeannatureofFick’slaw,
Darcy’s law, the advection–dispersion equation (ADE), and their validity relative
to the observation, model and heterogeneity scales. Inadequate and insufficient
explanations of theories and mathematical analyses in the literature and textbooks
arelikelytoblame.Thehydrologycommunityandeducationdeserveaunifiedand
comprehensible articulation of these theories and concepts.
Scales of heterogeneity, models, observations, and our interests dictate the
analysisofsolutetransportingeologicmedia.Textbooksandreferencebookshave
overlooked this fact and omitted the stochastic concepts (such as the volume
average,ensemblemean,andergodicity)embeddedinallsolutetransporttheories.
This book first introduces stochastics, volume average, ensemble average, and
well-mixed concepts. It then elucidates the solute transport theories and principles
frommoleculartofluid-dynamicscalesandsoilcolumntofieldscalesbasedonthe
stochastic concepts. Below are summaries of each chapter of this book.
Chapter 1, “Fundamental Concepts,” provides a review of basic differential
equations and statistics. It then introduces the spatial and temporal stochastic
process, volume average, and ensemble-average concepts behind the continuum,
concentration, mixing concepts, and scale issues in our observations and theories.
Chapter 2, “Well-Mixed Models for Surface Water Quality Analysis,”
presents classical lumped or well-mixed water quality models in surface-water
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