Models and modeling : an introduction for earth and environmental scientists

258 Pages·2016·3.317 MB·English

by  Fairley, Jerry

Models and Modeling
Models and
Modeling
An Introduction for Earth
and Environmental Scientists
Jerry P. Fairley
Thiseditionfirstpublished2017©2017byJohnWiley&Sons,Ltd
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LibraryofCongressCataloging-in-PublicationData
Names:Fairley,JerryP.,author.
Title:Modelsandmodeling:anintroductionforearthandenvironmentalscientists/JerryP.Fairley.
Description:Chichester,UK;Hoboken,NJ:JohnWiley&Sons,2016.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016024807(print)|LCCN2016033087(ebook)|ISBN9781119130369(pbk.)|
ISBN9781119130383(pdf)|ISBN9781119130376(epub)
Subjects:LCSH:Earthsciences–Mathematicalmodels.|Environmentalsciences–Mathematicalmodels.
Classification:LCCQE33.2.M3F352016(print)|LCCQE33.2.M3(ebook)|DDC550.1/5118–dc23
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Contents
Aboutthecompanionwebsite, xi
Introduction, 1
1 Modelingbasics, 4
1.1 Learningtomodel, 4
1.2 Threecardinalrulesofmodeling, 5
1.3 HowcanIevaluatemymodel?, 7
1.4 Conclusions, 8
2 Amodelofexponentialdecay, 9
2.1 Exponentialdecay, 9
2.2 TheBandurragaBasin,Idaho, 10
2.3 Gettingorganized, 10
2.4 Nondimensionalization, 17
2.5 Solvingforθ, 19
2.6 Calibratingthemodeltothedata, 21
2.7 Extendingthemodel, 23
2.8 Anumericalsolutionforexponentialdecay, 26
2.9 Conclusions, 28
2.10 Problems, 29
3 Amodelofwaterquality, 31
3.1 Oasesinthedesert, 31
3.2 Understandingtheproblem, 32
3.3 Modeldevelopment, 32
3.4 Evaluatingthemodel, 37
3.5 Applyingthemodel, 38
3.6 Conclusions, 39
3.7 Problems, 40
4 TheLaplaceequation, 42
4.1 Laplace’sequation, 42
4.2 TheElysianFields, 43
4.3 Modeldevelopment, 44
4.4 Quantifyingtheconceptualmodel, 47
4.5 Nondimensionalization, 48
4.6 Solvingthegoverningequation, 49
4.7 Whatdoesitmean?, 50
vii
viii Contents
4.8 Numericalapproximationofthesecondderivative, 54
4.9 Conclusions, 57
4.10 Problems, 58
5 ThePoissonequation, 62
5.1 Poisson’sequation, 62
5.2 Alcatrazisland, 63
5.3 Understandingtheproblem, 65
5.4 Quantifyingtheconceptualmodel, 74
5.5 Nondimensionalization, 76
5.6 Seekingasolution, 79
5.7 Analternativenondimensionalization, 82
5.8 Conclusions, 84
5.9 Problems, 85
6 Thetransientdiffusionequation, 87
6.1 Thediffusionequation, 87
6.2 TheTwelveLaborsofHercules, 88
6.3 TheAugeanStables, 90
6.4 Carryingouttheplan, 92
6.5 Ananalyticalsolution, 100
6.6 Evaluatingthesolution, 109
6.7 Transientfinitedifferences, 114
6.8 Conclusions, 118
6.9 Problems, 119
7 TheTheisequation, 122
7.1 TheKnightoftheSorrowfulFigure, 122
7.2 Statementoftheproblem, 124
7.3 Thegoverningequation, 125
7.4 Boundaryconditions, 127
7.5 Nondimensionalization, 128
7.6 Solvingthegoverningequation, 132
7.7 Theisandthe“wellfunction”, 134
7.8 Backtothebeginning, 135
7.9 Violatingthemodelassumptions, 138
7.10 Conclusions, 139
7.11 Problems, 140
8 Thetransportequation, 141
8.1 Theadvection–dispersionequation, 141
8.2 Theproblemchild, 143
8.3 TheAugeanStables,revisited, 144
8.4 Definingtheproblem, 144
8.5 Thegoverningequation, 146
Contents ix
8.6 Nondimensionalization, 148
8.7 Analyticalsolutions, 152
8.8 Cauchyconditions, 165
8.9 Retardationanddispersion, 167
8.10 NumericalsolutionoftheADE, 169
8.11 Conclusions, 173
8.12 Problems, 174
9 Heterogeneityandanisotropy, 177
9.1 Understandingtheproblem, 177
9.2 Heterogeneityandtherepresentativeelementalvolume, 179
9.3 Heterogeneityandeffectiveproperties, 180
9.4 Anisotropyinporousmedia, 187
9.5 Layeredmedia, 188
9.6 Numericalsimulation, 189
9.7 Someadditionalconsiderations, 191
9.8 Conclusions, 192
9.9 Problems, 192
10 Approximation,error,andsensitivity, 195
10.1 Thingswealmostknow, 195
10.2 Approximationusingderivatives, 196
10.3 Improvingourestimates, 197
10.4 Boundingerrors, 199
10.5 Modelsensitivity, 201
10.6 Conclusions, 206
10.7 Problems, 207
11 Acasestudy, 210
11.1 TheBoraxLakeHotSprings, 210
11.2 Studymotivationandconceptualmodel, 212
11.3 Definingtheconceptualmodel, 213
11.4 Modeldevelopment, 215
11.5 Evaluatingthesolution, 224
11.6 Conclusions, 229
11.7 Problems, 230
12 Closingremarks, 233
12.1 Somefinalthoughts, 233
AppendixAAheuristicapproachtonondimensionalization, 236
AppendixBEvaluatingimplicitequations, 238
B.1 Trialanderror, 239
B.2 Thegraphicalmethod, 239
B.3 Iteration, 240
B.4 Newton’smethod, 241
x Contents
AppendixCMatrixsolutionforimplicitalgorithms, 243
C.1 Solutionof1Dequations, 243
C.2 Solutionforhigherdimensionalproblems, 244
C.3 ThetridiagonalmatrixroutineTDMA, 244
Index, 247
About the companion website
Thisbookisaccompaniedbyacompanionwebsite:
www.wiley.com/go/Fairley/Models
Thiswebsiteincludes:
• Powerpointsofallfiguresfromthebookfordownloading
• Solutionstomanyoftheproblemsgiveninthechapters
xi
Introduction
Before serious discussion on any topic can take place, it is usually necessary to
defineone’sterms.Inthepresentinstance,weneedtoaskthefollowingquestion:
“what,exactly,dowemeanwhenwespeakofa‘model’?”Areasonabledefinition
of the term might bea simplified or idealized representation of reality. Although
this is a pretty broad definition, it is perhaps the only one that really embraces
everythingwerefertoas“amodel.”
In order to better understand what is, and what is not, a model, we can
examine some possible examples. Which of the following would be considered
models?
• Amap
• Amodeltrain
• AmodelUnitedNations
• Afashionmodel
• Anatom
Ofcourse,byourdefinitionallofthesethingswouldqualifyasmodels.Students
will sometimes argue that “an atom” is a physical object and, therefore, not a
model.However,everythingweknowaboutatomscomesfromobservingtheir
effects,andourunderstandinganddepictionofatomshaschangedandevolved
over time. In fact, our current understanding of atoms and subatomic particles
is so far outside our everyday experience that our descriptions are entirely
probabilistic.Asaresult,Imaintainthatanydiscussionof“atoms”isnecessarilya
discussionofmodels—simplificationsandidealizationsofthingsthatarebeyond
ourdirectknowledge.
In this book, we will primarily concentrate on a subset of models known as
mathematicalmodels.Amathematicalmodelisakindofmodelthatisformulated
intermsofmathematicalconceptssuchasconstants,variables,functions,deriva-
tives,andsoon.Thenominalgoalofamathematicalmodelistogivequantitative
indicationsofthebehaviorofsomeaspectofasystem.
Note that I said we would primarily be concentrating on mathematical
models. In fact, the word “primarily” glosses over an important fact about the
developmentofmathematicalmodels:everymathematicalmodelisthequantification
ModelsandModeling:AnIntroductionforEarthandEnvironmentalScientists,FirstEdition.JerryP.Fairley.
©2017JohnWiley&Sons,Ltd.Published2017byJohnWiley&Sons,Ltd.
Companionwebsite:www.wiley.com/go/Fairley/Models
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