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Groundwater Modelling L4018/GWM I Groundwater Modelling September 2000 G.H.P. Oude Essink Utrecht University Interfaculty Centre of Hydrology Utrecht Institute of Earth Sciences Department of Geophysics f. 15,- Contents General introduction v I Modelling Protocol 1 1 Introduction 3 1.1 Historical developments towards hydrologic modelling . ...... 3 1.2 Why modelling ? . ................................. 4 1.3 Some drawbacks in modelling . ....................... 5 1.4Definitions..................................... 6 2 Classification of mathematical models 9 2.1Introduction.................................... 9 2.2Basedonthedesignofthemathematicalmodel................. 10 2.3Basedontheprocessesinthehydrologiccycle.................. 15 2.4Basedontheapplicationofthemodel...................... 15 3 Methodology of modelling 19 3.1 Define purpose of the modelling effort . ................ 20 3.2Conceptualisationofamathematicalmodel................... 20 3.3Selectionofthecomputercode.......................... 24 3.4Modeldesign................................... 26 3.4.1 Griddesign................................ 27 3.4.2 Temporaldiscretisation.......................... 30 3.4.3 Boundary conditions . ....................... 31 3.4.4 Initialconditions.............................. 34 3.4.5 Preliminaryselectionofparametersandhydrologicstresses....... 34 3.5Calibration..................................... 35 3.5.1 Evaluatingthecalibration......................... 38 3.5.2 Errorcriterion............................... 39 3.5.3 Firstmodelexecution........................... 42 3.5.4 Sensitivityanalysis............................. 45 3.5.5 Kriging................................... 45 3.6Modelverification................................. 46 3.7Simulation..................................... 47 3.8Presentationofresults............................... 48 3.9Postaudit...................................... 51 3.10Whycanthingsgowrong?............................ 53 i ii 4 Data gathering 55 II Groundwater Modelling 59 5 Introduction 61 5.1Classificationbasedonthedesignofthemodel................. 61 5.1.1 Physicalmodels.............................. 63 5.1.2 Analoguemodels............................. 63 5.1.3 Mathematicalmodels........................... 66 5.2Outline....................................... 68 6 Mathematical description of hydrogeologic processes 69 6.1Fluidflow:equationofmotionandcontinuity.................. 69 6.1.1 Equationofmotion:Darcy’slaw..................... 69 6.1.2 Piezometrichead............................. 72 6.1.3 Hydraulic conductivity and permeability . ........... 73 6.1.4 Density of groundwater . .................. 75 6.1.5 Dynamicviscosity............................. 77 6.1.6 Equationofcontinuity........................... 78 6.1.7 Groundwater flow equation . .................. 79 6.1.8 Equationofstate............................. 81 6.2Solutetransport:advection-dispersionequation................. 82 6.2.1 Equationofsolutetransport....................... 83 6.2.2 Hydrodynamicdispersion......................... 88 6.2.3 Chemicalreactions............................ 91 6.3 Heat transport: conduction-convection equation . ........... 93 7 Solution techniques 95 7.1Introduction.................................... 95 7.2Iterativemethods................................. 95 7.2.1 Taylorseriesdevelopment......................... 95 7.2.2 Laplaceequation............................. 96 7.2.3 Steadystatemethods........................... 97 7.2.4 Non-steadystatemethods........................ 98 7.3Thomasalgorithm................................. 102 7.4Gauss-Jordanelimination............................. 103 7.5Finitedifferencemethod.............................. 105 7.6Finiteelementmethod.............................. 106 7.7Analyticelementmethod............................. 109 7.8Methodofcharacteristics............................. 112 7.9Randomwalkmethod............................... 114 7.10Vorticesmethod.................................. 116 iii 8 Numerical aspects of groundwater models 119 8.1Numericaldispersion............................... 120 8.1.1 Stability analysis of the advection-dispersion equation . ...... 124 8.2 Oscillation . ................................. 127 8.3 Analysis of truncation and oscillation errors . ................ 128 9 Some selected groundwater codes 133 9.1Introduction.................................... 133 9.2MODFLOW.................................... 134 9.2.1 Externalsourcesintoablock:packages................. 139 9.2.2 Layertypes................................ 144 9.2.3 Boundary conditions . ....................... 145 9.2.4 StronglyImplicitProcedurepackage(SIP)................ 146 9.2.5 Slice-SuccessiveOverrelaxationpackage(SSOR)............. 151 9.3Micro-Fem..................................... 154 9.4MOC3D...................................... 157 9.5MOC,(2D)adaptedfordensitydifferences.................... 158 9.5.1 Theoretical background of the groundwater flow equation . ...... 159 9.5.2 Theoreticalbackgroundofthesolutetransportequation........ 164 Continu¨ıteitsvergelijking: niet stationair 177 References 181 Consultedliterature................................... 181 Interestingtextbooks.................................. 187 Somedistributorsofcomputercodes.......................... 187 Formula sheet 189 Index 191 iv General introduction The development of models has been the direct outcome of the need to integrate our existing theories with all physical and measured data. The key factor in this development is the (still ongoing) breakthrough in computation technologies, because it is the digital computer which is capable to store, to manipulate data and to execute complex calculations beyond the physical ability of man, yet within his mental capacity. In order to avoid that you, as a hydrogeologist-in-spe, will get stranded in the fine art of modelling, these lecture notes1 are written to show you some ropes in the fantastic, tempting, and yet creepy world of groundwater modelling. The presence course, which comes under the ICHU2, is called Groundwater Modelling I. The aim of this course is to gain more insight in the behaviour of groundwater processes, quantitatively as well as qualitatively, by means of numerical modelling. These lecture notes are divided into two parts: I. Modelling protocol in this part, the procedure of modelling hydrologic processes is described. A hydro- geologist should advance this procedure, from coping with a hydrological problem towards solving the problem by means of (numerical) modelling, while skipping all kinds of traps on the track; and II. Groundwater modelling in this part, features of groundwater flow and solute transport are considered and numerical solution techniques of partial differential equations are discussed. Moreover, some groundwater computer codes will be treated. During the computer practicals, attention will be paid to the modelling of standard problems by means of the codes MODFLOW and MOC(3D). Primarily knowledge should comprise basic knowledge on hydrogeologic processes such as the Darcy equation and stationary groundwater flow; as well as basic knowledge on discretisation techniques such as Taylor series and simple numerical solution techniques. For students of the Department of Physi- cal Geography, it is strongly recommended to have followed Groundwaterhydrology (HYDB). For all students, the lectures Hydrological Transport Processes (L3041/KHTP) are also rec- ommended. Gualbert Oude Essink, September 2000 [email protected] 1Parts of these lecture notes are based on the lecture notes Hydrological models (f15D) of the Delft University of Technology (Oude Essink, G.H.P., Rientjes, T. & R.H. Boekelman, 1996). 2The Interfacultair Centrum voor Hydrologie Utrecht (ICHU) is a centre which provides a so-called study- path Hydrologie to students from the Faculty of Earth Sciences and Department of Physical Geography. v vi Groundwater Modelling æ Part I Modelling Protocol 1 Chapter 1 Introduction 1.1 Historical developments towards hydrologic modelling The science of hydrology began with the conceptualisation of the hydrologic cycle. Much of the speculations of the Greek philosophers was scientifically unsound. Nonetheless, some of them correctly described some aspects of the hydrologic cycle. For example, Anaxagoras of Clazomenae (500-428 B.C.) formed a primitive version of the hydrologic cycle (e.g. the sun lifts water from the sea into the atmosphere). Another Greek philosopher, Theophrastus (circa 372-287 B.C.) gave a sound explanation of the formation of precipitation by conden- sation and freezing. Meanwhile, independent thinking occurred in ancient Chinese, Indian and Persian civilizations. During the Renaissance, a gradual change occurred from purely philosophical concepts of hydrology toward observational science, e.g. by Leonardo da Vinci (1452-1519). Hydraulic measurements and experiments flourished during the eighteenth century, when Bernoulli’s equation and Chezy’s formula were discovered. Hydrology advanced more rapidly during the nineteenth century, when Darcy developed his law of porous media flow in 1856 and Manning proposed his open-channel flow formula (1891). However, quantitative hydrology was still immature at the beginning of the twentieth century. Gradually, empiricism was replaced with rational analysis of observed data. For example, Sherman devised the unit hydrograph method to transform effective rainfall to direct runoff (1932) and Gumbel proposed the extreme value law for hydrological studies (1941). Like many sciences, hydrology
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