MATH MODELING GROUNDWATER SYS
MATH MODELING GROUNDWATER SYS GEGN 483
Colorado School of Mines
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Date Created: 10/05/15
ALL GROUNDWATER HYDROLOGY WORK IS MODELING A Model is a representation of a system Modeling begins when one formulates a concept of a hydrologic system continues with application of for example Darcy39s Law or the Theis equation to the problem and may culminate in a complex numerical simulation MODELS can be used BENEFICIALLY and for DECEPTION GROUND WATER MODELING WHY MODEL To make predictions about a groundwater system s response to a stress To understand the System To design eld studies Use as a thinking tool Characterize the system Governing equation of Ground Water Flow 3K K aimzK ws 6x x6 6y Yay 01 Zaz S t Geometry Material Properties K S T 09 D R 910 Boundary Con ns Head Flux Concentration etc Stresses changing boundary condl Ions Boundary Types necmm HmdCnnuenlmllnn a steal case mcnnslml me mac rm nnslant Head Cnnuenlralnn muld replace mac er In nne snurceislnk ealeulaen head necmed Flux muld he eehzrue muss 1cm nlalnlalns ux n r lnw Streamllnak a sneclal use at speuneu ux mu quot39quotm mu nqmuem Flux could letlane mac EFG source slnk ree Su aue mendIE rhrezmc sumee 1cm axliusls lmsmnn mung mm 39 39 hene 21 round sume um MODELING PROCESS Stat by Ehf39ning Ihs Probsm CmceplLaI MgdeI C us or Bath Unsatisfactory Model Selisfadofy Model Make Racomnantttionlescia ans ALL IMPORTANT MECHANISMS amp PROCESSES MUST BE INCLUDED IN THE MODEL OR RESULTS WILL BE INVALID TYPES OF MODELS CONCEPTUAL PHYSICAL ANALOG EMPIRICAL GRAPHICAL MATHEMATICAL SIMPLE ANALYTICAL COMPLEX NUMERICAL CONCEPTUAL MODEL Geometry Material Properties Boundary Conditions General Flow Patterns System Stresses CONCEPTUAL MODEL FLOW EQUATIONS QUANTITATIVE MODEL OF FLOW SYSTEM CONCEPTUAL MODEL PHYSICAL MODEL Geometry Materials Boundary Conditions ANALOG MODEL Geometry Material Properties Boundary Conditions V Electrical analog model of the ChampaignL Urbana Illinois area groundwater system circa 1960 The top panel is a circuit of resistors and capacitors representing the regional model Measuring the voltage at various locations in the circuit is equivalent to measuring head in the aquifer The middle level includes a local model of a portion of the regional model at both the same scale and twice the scale The lower area includes the controls for imposing current on the model These models are very difficult to calibrate because each 39change of material properties involves removing and resoldering the resistors and capacitors EMPIRICAL MODEL A Mathematical Fit to Data Unrelated to Process Equations eg Manning39s Equation V149R1 3S 2 n where V average velocity in fps R hydraulic radius flow area ft1wetted perimeterft S slope of energy gradient n Manning friction factor GRAPHICAL MODEL FLOW NET Geometry Material Properties Boundary Conditions ANALYTICAL MODEL Closed form algebraic solution Geometry Material Properties Boundary Conditions Recall Dupuit Flow to fixed heads with recharge h hf L Xx x thf h l L W qx 2L Wg x murmur dig 39 2 w 2L 2 Note Recharge is a fixed flux and headgradient will be calculated to accommodate that recharge eg High recharge gtgt High heads M and h2 are fixed heads and flux will be calculated to accommodate those heads eg a high M will shift the divide to the left of the problem domain and produce large influx that joins the recharge and discharges to the right if h2 is very low that influx will be even higher Recall Toth developed a solution for Steady State flow in a 2D section from a stream to a divide he solved the Laplace Equation 2 2 quot agag0 JJJJ L quot39quot39 ax E 2 Eil39rg m Ii quotquot a J 39 gquot quot V I OL ginwatt IIIP 139 r 1quot 16 I a 0 If r v39 fr 3 r HIf Hig boundaries left a h0z0 right a hsz 6X 8X 0 lower 6 11 X0 0 62 upper water table hXzO Z0 ex Z0 tan0LX His result 0 U 000 800 600 400 09 m 2 above 5 n or m c 3 20 l 200 939u 20 0 quot5 s 20000 feet Standard datum 52 b LLJ T quotE gt w m0 HM gt2ltW 39 I 8000 3 9 520000 feel Standard datum a standard datum in feet above Elevation i Fig 3 Twodimensional theoretical potential distributions and ow patterns for different depths to the horizontal impermeable boundary 39 Recall Analytical Transport models eg for 1D slug source cltxVtgtlt y Y 22 3 ant m NUMERICAL MODELING DISCRETIZES THE SYSTEM eg Finite Difference Modeling III W ulna NUMERICAL FLOW MODELING DISCREHZ Write equations of GW Flow between each node Dar 39s Law Conservation of Mass De ne Material Properties Boundary Conditions Initial Condi ions Stresses At each node either H or Q is known the other is unknown n uations amp n unkn solve simultaneously with matrix algebra Result H at each known 0 node 0 at e h wn H node Calibrate Steady State Transie Validate Sen PredIctIons Similar Process for Transport Modeling only Concentration and Flux is unknown NUMERICAL MODELING Finite Element Modeling uuuu hnumlm NUMERICAL MODELING Finite Element Modeling More flexibility in designing Grid Sand and Gravel Sandstone U Lxmesrune Examples of Model Results with links to animations Flow Model Hanford Reservation Waste Water Disposal httpinsidemineseduepoeter 6W23Modelingflow4396avi vzn ism no Flow and Transport Model East Texas Land Fill Plume httpinsidemineseduepoeter 6W23Modelingetex2avi North South cross section through plume W layerZ MODFLOW Block Centered 3D Finite Difference Ground Water Flow Model Developed b McDonald amp Harbaugh at USGS in 1983 enhance many times since then Public Domain Most widely used Saturated Porous Media Flow model Many features available MODFLOW With Packages provided run 39 Output calculation Based on Model Setup MODFLOW lnpm Flles Executable GUI s Facilitate your work Graphical User Interface MODFLOW Boundary Condition Packages El Specified Head Boundaries CHD TimeVariant SpecifiedHead Package FHB Flow and Head Boundary Package El Specified Flux Boundaries FHB Flow and Head Boundary Package FlCH Recharge Package WEL Well Package E HeadDependent FIurr Boundary Packages DAF DAFG DAFLDW with MUDFLUW DHN Drain Package DFlT Drain Fleturn Package ETS Evapotranspiration Segments Package EVT Evapotranspiration Package GHB GeneralH ead Boundary Package LAK Lake Package MNWl MultiNode DrawdownLimited Well Package BES Reservoir Package E HIV Fliver Package SFFl StreamflowFlouting Package STFl Stream Package UZF In alllraled one Flow Package MODFLOW Head dependent Flux Boundary Condition Packages RIVER package For each river reach in each cell MODFLOW requires that the user input Conductance which is all of Darcy39s Law except the head difference for Head Dependent Flux boundaries Q KA dhdl Conductance KAthickness then MODFLOW calculates the flow as Q Conductance dh Conductance of the river bed is calculated as Kv Areathe plan view areaLW thickness MODFLOW Head dependent Flux Boundary Condition Packages RIVER package Q Conductance dh ie CRIV dh Dh is limited to stage bottom of sediment when bottom is above the water table sup42w MODFLOW Head dependent Flux Boundary Condition Packages RAIN package ONLY allows OUTFLOW at d9 on Q Conductance dh ie CD dh if When head is above the associated 3 elevation MODFLOW Head dependent Flux Boundary Condition Packages DRAIN package For each river reach in each cell MODFLOW requires that the user input Conductance which is all of Darcy39s Law except the head difference for Head Dependent Flux boundaries Q KA dhdl Conductance KAthickness then MODFLOW calculates the flow as Q Conductance dh Conductance of the drain is calculated as Kof maturiul om which grudkn39 is calculu39ad Areath ickness Area may be the cylindrical area midway between where the heads used for the gradient are located length of the drain MODFLOW Flux Boundary Condition Packages Recharge package For recharge a rate is specified for each cell MODFLOW calcula39l39es Q ra l39e cell area v Q is specified amp forced in or out unless The cell goes dryquot in which case The modeler may choose To apply if 1390 a lower layer Heads increase as recharge is increased Example alternative recharge distributions MODFLOW Flux Boundary Condifion Packages Well packag Wluen pumping a well in a MODFLOW grid cell Q is specified amp wifludrawn or injecfed unless flue cell goes dryquot Calculafed drawdown represenfs flue average drawdown in flue cel nof flue acfual drawdown in flue we Cell cross section MODFLOW assigns one head value to each cell Nexf we will use MODFLOW and EMS Ground Wafer Modeling System fo gef a feel for ground wafer modeling Intermittent Perennial I Wells TRIEUTARIES TRIEUTARY Locations of Production Wells Where you can adjust withdrawal rate As time allows eXperiment with MODFLOW Using the GMS Ground Water Modeling System GUI Graphical User Interface By Downloading the Example Files and Associated Write up on the class web page for the Modeling Lecture CSM Ground Water Modeling course Mathematical Modeling of Ground Water Systems GEGN 483583 Currently scheduled for Spring 2008 Thursday 14PM