Introduction to groundwater flow modeling: finite difference methods. Tyson Strand
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1 Introducton to groundwater flow odelng: fnte dfference etods Tson Strand
2 Darc s law contnut and te groundwater flow equaton Fundaentals of fnte dfference etods 3 FD soluton of Laplace s equaton 4 FD soluton of Posson s equaton 5 Transent flow
3 Darc s law contnut and te groundwater flow equaton q d q q d q
4 d q q d q q d d 0 Evertng tat goes n ust coe out q d d q d q d q 0 dd 0 After dvdng b te volue area let d and d go to zero q q 0 Contnut
5 Darc s law: sotropc edu K q K q z K q z Scalar for K k z K ˆ ˆ ˆ Vector for
6 Darc s law contnut n D K q K q 0 q q 0 K K K 0 Laplace s equaton
7 Te groundwater flow equaton t S W z K z K K s zz Sources and snks Transent flow ter Darc s law contnut
8 Splfng assuptons Isotropc edu: K K K K zz K W t W ft Stead-state flow Posson s equaton W f Stead-state flow no sources/snks Laplace s equaton S W K s 0
9 Fundaentals of fnte dfference etods Dscretzaton of space Dscretzaton of contnuous quanttes Dscretzaton of te Te frst spatal dervatve Te second spatal dervatve Boundar condtons and ntal condtons Solvng te proble
10 Dsctretzaton of space L N d - d - d
11 Dscretzaton of quanttes Eac spatal locaton as assocated wt t a ead value and a vector darc flu - q q -
12 Dscretzaton of te t n nt nt n3t n4t n5t t t t t 3 t 4 t 5 Increasng te
13 Te frst spatal dervatve - - Upstrea dfference Central dfference
14 Te second spatal dervatve: step - NOTE: Ts s essentall a central dfference approaton of te frst dervatve evaluated at te d-postons -
15 Te second spatal dervatve: step Te second dervatve s ust te dervatve of te dervatve
16 Boundar condtons Wat are boundar condtons and w are te necessar? Consder te sple proble: 0. We can solve ts drectl b separatng varables and ntegratng: C
17 0. C Tere are an nfnte nuber of solutons to ts equaton. Te are referred to as a fal of curves We ust specf a value of at a known poston n order to solve for C.
18 For a second spatal dervatve two boundar condtons ust be specfed. Tere are 3 tpes of bc s tat we can appl Head s specfed at a boundar - Called Drclet condtons Flow frst dervatve of ead s specfed at a boundar - Called Neuann condtons 3 Soe cobnaton of and - Called ed condtons
19 Intal condtons Equvalent to boundar condtons ecept tat te boundar s now teporal nstead of spatal Tpcall te state of te sste.e. ead values are specfed at te t 0 K K z K zz z W S s t Queston: How an boundar and ntal condtons Would be needed to solve te above governng equaton?
20 Solvng te proble Set of dfferental equatons Mateatcal odel Fnte Dfference Fnte Eleent Set of algebrac equatons dscrete odel Calculus tecnques Iteratve or drect etods Analtcal soluton Not alwas possble Copare If possble Approate soluton Feld observatons After Wang & Anderson 98
21 Recall ow we arrved at Laplace s equaton K q K q 0 q q 0 K K K 0 Laplace s equaton We are focusng on teratve etods
22 Solvng Laplace s equaton D 0 0
23 Let [ ] 4
24 [ ] 4 Set equal to zero 0 4 Te above equaton s te basc fnte dfference soluton to Laplace s equaton
25 Now we rearrange te prevous equaton so tat we can pleent t nto our regular grd Solve for 4
26 Now wat do we do? Specf boundar condtons Guess at ntal values for ead at all locatons Begn teratons
27 Wat does t ean to terate? Gven: Do: Wle: We know te boundar condtons and ave ntal guess values for eadposton Update ead values at ever locaton based on prevous equaton ncludng boundares f applcable Convergence crtera are not et
28 Convergence Copare obtaned ead values at a gven teraton to tose at te prevous teraton Contnue to terate as long as ead values contnue to cange wtn soe preset lt 3 Test te soluton to be descrbed
29 Testng a coputatonal odel Coparson to oter analtcal results/feld data valdaton? Cangng te grd spacng Cangng te convergence crteron Mass balance ceck
30 Valdaton/verfcaton Analtcal soluton s known uncoon Coparson to feld data 3 Test on sple stuatons for wc or are known 4 Ceck odel predctons
31 Possble odel probles Stablt Robustness
32 Saple proble: Laplace s equaton
33 Lecture Posson s equaton Dgresson: Inflow outflow and sgn conventons Fnte dfference for for Posson s equaton Eaple progras solvng Posson s equaton Transent flow Dgresson: Storage paraeters Fnte dfference for for transent gw flow equaton eplct etods & stablt Eaple transent flow progra Iplct teratve etods Eaple transent flow progra full plct
34 Solvng Posson s equaton T R K W Wat s W? K W b R W Were b s te tckness of te aqufer z denson
35 Inflow outflow and sgn conventons Tnk about te agntudes of q out and q n Wat does t ean n ters of sources/snks? 0 > 0
36 Sgn conventons < 0 < 0 Tere ust be recarge or soe oter source at te REV stead-state assupton > 0 Tere ust be dscarge or soe oter snk at te REV stead-state assupton
37 W K R T You need to be careful as to ow tngs are defned W MUST be defned dscarge snk postve It s ore ntutve to defne a recarge source functon R tat s postve for recarge sources and negatve for dscarge snks Ts wll also elp gve us an ntutve understandng of te transent gw flow equaton
38 Back to Posson s equaton T R Proceedng as we dd for Laplace s equaton [ ] T R 4 Solvng for T R 4
39 Solvng Posson s equaton nuercall Bascall we can proceed eactl as we dd for Laplace s equaton usng te prevous fnte dfference approaton for Defne boundar condtons Set ntal guess values Iterate Ceck results
40 Saple proble: Posson s equaton wt unfor recarge/dscarge
41 Saple proble: Posson s equaton wt pont source recarge/dscarge
42 Solvng transent flow probles b t R t S K s We ve learned ow to take care of evertng ecept te te dervatve t t t Forward dfference approaton t t t Backward dfference approaton
43 Central dfference approaton t t t BAD NEWS Te central dfference approaton s uncondtonall unstable for te dervatve Ts etod sould NOT be used to estate te frst te dervatve
44 Storage paraeters S storage coeffcent or storatvt Densonless Measures te volue of water epelled absorbed per unt surface area per unt ead cange S s specfc storage S/b Unts [/L] also called elastc storage coeff. Measures te water volue per unt aqufer volue tat s epelled stored due to copressblt atr & water per unt ead cange Used for confned unts or te saturated parts of an unconfned unt b aqufer tckness for confned unts use b s tckness of te saturated regon for unconfned sstes
45 S specfc eld Densonless Specfc eld Used to descrbe unconfned sstes Rato of water volue tat drans fro a saturated regon of aqufer due to gravt forces per unt aqufer volue Generall several orders of agntude larger tan b s S s ecept n ver fne graned sstes Relatonsp between S S s and S S S b s S s Fetter 994
46 Storage paraeters concl. In unconfned sstes te specfc storage can generall be neglected Ecept n ver fne graned eda In confned sstes onl te specfc storage s pertnent so long as ead values rean above te upper confnng unt In general for coputatonal odels t s best to use te storage coeffcent storatvt Defntons of storage paraeters are not alwas consstent n lterature be careful
47 Transent flow So let s consder te transent flow equaton n te for S R t T t T Assue te left and sze s negatve out > n: ten b te conventon we ve set eter recarge s postve or te te rate of cange of storage s negatve rgt?
48 T t R t T S Let s wrte te fnte dfference approaton for ts equaton T R t T S Ts s called te eplct representaton snce all spatal ters are evaluated at te old te
49 Let and solve for te ead at at te nde [ ] S t T S R t S t T 4 Te eplct equaton s eas to solve: Gven a set of ntal condtons & bc s Te ead at all locatons at te nde can be calculated drectl fro te ead values at te prevous te nde Notce tere s no teraton once ou ave an ntal condton t s a sple atter to step forward n te b an aount t
50 Stablt of te eplct etod One ust be careful to keep te te steps sall enoug suc tat te eplct etod reans stable If te te step s too large fluctuatons wll develop n te ead as a functon of te wc wll be aplfed as te odel contnues to step troug te Te stablt crtera are: Tt < 0.5 S Tt < 0.5 S -D -D
51 Eaple transent proble: Eplct soluton D Intal Fnal We are nterested n wat appens n between te ntal and fnal states Te ead at te rgt boundar drops fro blue to red at te t 0 Te red potentoetrc surface s te fnal state
52 Iplct etods Notce n te eplct forulaton tat we ave evaluated all spatal dervatves at te old te t Is tat te best? Wat f we evaluated spatal dervatves at te new te t? Wat f we evaluated spatal dervatves alfwa between?
53 α α Were α can var between 0 full eplct and full plct A slar epresson can be wrtten for te second dervatve wt respect to We splf te algebra b defnng te followng and lettng 4 ~
54 [ ] [ ] ~ 4 ~ 4 α α Put ts approaton nto te gw flow equaton [ ] [ ] T R t T S ~ 4 ~ 4 α α Rearrange [ ] T R t T S t T S 4 ~ 4 ~ 4 α α α
55 Solvng te prevous equaton for te new ead value at [ ] T R t T S t T S 4 ~ 4 ~ 4 α α α Ts s te fundaental plct fnte dfference approaton α 0 Full eplct α ½ Crank-Ncolson α Full plct Te above epresson can be solved b teratve or drect etods We wll contnue to focus on teratve etods Te advantage of plct etods s STABILITY
56 To solve teratvel: - Sweep troug te lattce updatng te ead value at te new te b te above forula - Contnue to sweep troug te lattce updatng as long as te new ead values keep cangng - Stop wen new ead values do not cange wtn preset lts - update as ou go along: called Gauss-Sedel teraton - Repeat for te net te step
57 Full plct D saple proble t T S t T S ~ ~ Were s now defned b ~
58 References H. F. Wang and M. P. Anderson 98. Introducton to Groundwater Modelng Acadec Press Inc. C. W. Fetter 994. Appled Hdrogeolog Macllan College Publsng Copan. F. W. Scwartz and H. Zang 003. Fundaentals of Ground Water Wle publsng.
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