2-D Groundwater Flow Through A Confined Aquifer

Transcription

1 -D Groudwater Flow Troug A Coied Aquier Gordo Wybur ad Ajoy Vase Pooa College May 5 t, 006 Abstract We attepted to odel te groudwater low i a -D coied aquier uder dieret coditios usig te iite dierece etod. We used Matlab or all our coputatios. We solved te Laplace equatio uerically to odel te steady state low troug a aquier wit separate regios o dieret coductivities. We solved te tie depedet low case to predict te beavior o te aquier uder certai iitial lows we te low was tured o. We also attepted to solve te tie idepedet low case or a aquier tat as a coductivity tat varies cotiuously. Itroductio A aquier is a layer o pereable aterial tat ca trasit water udergroud. Aquiers cael groudwater to wells. Te coductivity o a aquier is a easure o ow ast water ca low troug it. A sigle aquier is coprised o dieret aterials, eablig te local water low rate to vary i dieret regios o te aquier. Coo aquier aterials iclude sad, gravel, sadstoe, liestoe ad doloite. Aquiers all ito two ai categories ucoied ad coied. A ucoied aquier as pereable aterial extedig ro te lad surace dow to te botto o te aquier. It ca be recarged by water seepig dow ro te surace or by lateral groudwater low. A ideal coied aquier is oe tat is bouded above ad below by a copletely ipereable layer o aterial. Tus te oly way i wic te aquier ca be recarged is by lateral groudwater low. Te water i a coied aquier is uder pressure. Aquiers are a vital source o reswater to ua beigs i rural areas ad eve etire cities. Te Edwards aquier i Texas provides eoug water to sustai 1.5 illio people. Groudwater low odels or coied aquiers ca predict te pressure i dieret regios o te aquier. Tis ioratio ca be used to select te ost appropriate spot or diggig a well. Overexploitatio o aquiers leads to depletio. Aquier depletio is a growig proble. Metods o artiicial recarge are beig developed i order to address te issue. Groudwater low odels tat predict water low i a aquier or dieret etry low coditios will be useul or ipleetig artiicial recarge etods. 1

2 Equatios used: We wated to odel groudwater low troug te ollowig -D aquier: Ipereable regio Ilow y Pereable regio x Ipereable regio Figure 1. Diagra o aquier. Te PDE equatio tat describes -D groudwater low i a coied aquier is: 1 x y T t (1) Here is te ydraulic ead ad T is te costat coductivity o te aquier ediu. Tis equatio assues tat te aquier is 100% coied, i.e. tere are o leaks ro te top ad te botto. Tis equatio is derived usig ass ad eergy coservatio or a cotrol volue i te aquier. Te ydraulic ead is deied to be te total ecaical eergy per uit weigt o te water. I groudwater low, te ydraulic ead equals te su o te elevatio ead ad te pressure ead. Tis is describes ateatically by te ollowig equatio: z P g Equatio (1) ca be solved or te tie idepedet case, wic is tataout to solvig Laplace s equatio. For te tie-idepedet case were te coductivity varies cotiuously, te low is described by te ollowig equatio: ( T T T x, y) 0 (3) x y y y x x ()

3 Oce te ead as bee coputed, te orizotal ad vertical copoets o groudwater low ca be oud usig Darcy s law: q y q x T x T y (4a) (4b) Boudary coditios: Sice tere is o low across te top or te botto o te aquier, q y = 0. Fro (3b), tereore, = 0 at all poits alog te ceilig ad te loor o te aquier. We assue y tat te icoig low to te aquier ad te ead at te begiig o te aquier are kow. Te outgoig low depeds o te iteractio o te water wit te local regios i te aquier, so we leave tis boudary coditio uspeciied. Solutio strategy: We decided to use te iite dierece etod to look at te kid o low odels tat could be created wit tis relatively easy etod. We approaced tis odel i te ollowig steps: 1. Solve Laplace s equatio or te steady state low. Te solutio o tis equatio is idepedet o te local coductivity o dieret regios i te aquier. Te coductivity is cosidered we coputig te velocity ield o te groudwater i te aquier.. Solve equatio (1) or dieret iitial lows to observe ow te low varies i tie. 3. Solve equatio () to observe te low we te coductivity o te aterial i te aquier varies cotiuously, i.e. a case were te packig o te gravel i te aquier varied liearly alog its legt. Assuptios ade: 1. Te aquier is rectagular i sape. Te aquier is ideally coied, wic eas tat te loor ad te ceilig are absolutely ipereable. I oral coditios, tere is soe leakage o water ito te aquier ro te top ad out troug te botto 3. Te iitial ead ad velocity o te water are kow. 4. I te steady state case, we assue tat te low does ot cage sigiicatly wit tie. Tis is a good assuptio i cases were te slope or positio o te water table does ot cage. 3

4 Fiite dierece grid I order to copute te ead at all poits i te aquier, te irst step was to discretize te doai ito a grid o total eigt N ad legt M. d/dy = 0 = 0 d/dx = q 0 d/dy = 0 Figure. Fiite dierece grid represetig te aquier doai. Te distace betwee eac poit is, suc tat te locatio o a poit o te grid is described by x y ; ; 0.. M 0.. N Te ext step was to discretize te PDE ad orgaize te resultig set o equatios ito a liear syste. Te discretizatio proceeded accordig to te locatio o te poits i te aquier. Table 1 suarizes te discretizatio orulas used or te solutio to te Laplace equatio ad te ial appearace o te liear syste i eac case. We used secod order dierece orulas to discretize tis PDE. 4

5 All poits excludig boudary coditios Aquier loor ad ceilig Rigt boudary o te aquier Top ad botto rigt corer o te aquier x- copoet Secod order cetral dierece Secod order cetral dierece Secod order backwards dierece Secod order backwards dierece y-copoet Secod order cetral dierece Secod order orward dierece (discretizatio o te boudary coditio) Secod order cetral dierece Secod order backwards dierece Stecil i Figure Blue Gree Red Yellow Liear syste or Laplace equatio O( ) O( ) O( ) O( ) 0 Table 1. Discretizatio orulas used i te iite dierece grid set up to solve Laplace s equatio. Note: ( x, y ) We relabeled te ead to avoid cousio wit te distace betwee eac poit,. Te superscript is te idex or te y copoet o te low, ad te subscript is te idex or te x-copoet. We ade a vector o te ukows ad a atrix o te coeiciets. It was ot possible to speciy bot te iitial ead ad te iitial velocity i our atrix. We worked aroud tis proble by speciyig te ead or poits i te irst tree colus o te grid. Tis autoatically sets a iitial slope or te ead. A siilar geeral procedure was ollowed i order to discretize te tie depedet PDE i equatio (1). A explicit Euler irst order orward dierece etod was used to project te PDE orward every tie step. Tis orula is o te or: v 1 v t Te solutios were ot expected to be oscillatory i ature, wic ade te Euler etod suitable sice tere was o dager o oversootig te correct aswer. 5

6 For te PDE i equatio (3), we used secod order dierece orulas to represet te irst derivatives i te equatio. We also picked uctios o T wit siple derivatives tat we calculated ad speciied i our atrix. Aalysis: Laplace s Equatio: Te Matlab code worked qualitatively or tis steady state case wit local regios o varyig coductivity. We plotted as a uctio o x ad y i order to observe ow it varied i dieret regios o te aquier. We solved tis steady state equatio or dieret cases. Figure 3. Aquier wit Costat Iput Head. Case 1 (siplest case): Costat ead at te etrace o te aquier wit pockets o dieret coductivity. Our results idicate tat te ead decreases liearly i te orizotal directio. Tis proves tat te code works because decreasig ead is a requireet or te water to low ro let to rigt (see equatio 4a). Furterore, te slope o wit respect to y is zero at te loor ad ceilig o te aquier. Te plot o te low as a uctio o te positio (see Figure 4) also veriies tat our code is qualitatively workig. As we would expect, i regios o a iger coductivity, te low speeds up (idicated by larger arrows i te red portio o te grid). Figure 4. Plot o low as a uctio o positio 6

7 y Our results i te test case allow us to test te code or ore coplex situatios. Figure 5. Aquier wit larger ead i te iddle portio o te etrace Case : Water is artiicially ijected ito te iddle regio o te aquier (peraps troug a large pipe): Tese coditios ea tat tere is iger ead i te iddle o te aquier ta at te top or te botto. Oce agai, te results atc wat we would aturally expect. Te water spreads out ro te regio o iger ead to lower ead, so it lows away ro te cetral regio o te aquier (see Figure 6). Te iger pressure i te ceter orces soe o te water to low backwards i te aquier. I case tis is a udesirable side eect, geologists ca use tis result to decide tat tis particular recarge etod sould ot be eployed x Figure 6. Plot o low or te above case. 7

8 Figure 7. Aquier wit Icreasig Head Case 3: Te iitial ead icreases liearly ro top to botto at te etrace o te aquier. Tis siulates te etry o water ro a slopig regio o a aquier ito our orizotally orieted regio. As expected, te water lows towards te botto o te aquier ad curves aroud to low orizotally troug it. Our low plot also idicates tat te water would low aster troug te upper regio o te aquier ta troug te lower regio. It would be iterestig to ceck i tis was true i a actual aquier. 8

9 Tie-depedet solutios (equatio 1): Solvig tis equatio eabled us to observe ow te ead would cage wit tie i cases were te water low was suddely switced o. We cecked tis peoeo i te evet tat te iitial ead was costat. We te low is tured o quickly, te water retaied i te aquier lows out o bot eds util all te water is draied out. Te sae is true or a artiicial recarge case were te water i te iddle regio o te aquier is at a iger pressure ta te water at te top or te botto. Agai, tis beavior is wat we would expect. Figure 8. Movie o Aquier Losig te Artiicial Ijectio 9

10 Cotiuously varyig coductivity: For te case wit a cotiuously varyig coductivity (we tried T = x + y), we ad probles due to te boudary coditios at te loor ad ceilig o te aquier. Te ead at te loor ad te ceilig o te aquier goes to zero very quickly. Tis orces te ead at oter poits i te aquier to zero very quickly as well. We tougt tat te proble lay i te dierece orula tat we were usig. We tried to ceck tis by settig te ead or Figure 9. Aquier wit T= x+y. te two layers surroudig te ceilig ad te loor so tat te derivative was autoatically set to zero, i uc te sae way as we set te iitial etry low velocity. Tis did ot ave ay substatial eect o te outcoe. We rececked te at uerous ties ad ave coe to te coclusio tat it is eiter soe error i te code or a udaetal proble wit our teory o ow tis sould work. Probles wit our code i geeral: Sice we are atteptig to solve a boudary value proble, we would typically set te iitial ead or its derivative o all our boudaries. However, we ave o kowledge o te outgoig groudwater low ad we expect te ead to cage across te aquier, so we caot set te ial boudary coditio i x. Te way we tried to work aroud tis was to speciy te ead ad its derivative or te iitial boudary coditio i x. However, it is ot possible to speciy bot i te sae atrix. Speciyig te ead at te irst tree iitial poits allowed us to idirectly speciy te iitial derivative as well. We tried to ipleet a etod were we solved or te irst set o ukow poits by usig a backwards dierece etod tat took ito accout all tree iitial poits. Our solutio or tis looked really ustable, ad so we resorted to usig te cetral dierece orula. We let i te irst tree kow poits because it akes our atrix o sigular but te iitial derivative o te ead (i.e. te iitial velocity) does ot aect te calculatio o te ead. We also observed tat watever te iitial ead is set to, it will always decrease to zero at te ed. Tis is icorrect because ead is a easure o eergy ad sould ot ecessarily go to zero at te exit poit o te aquier, sice te water sould still possess soe eergy. Te velocity is soeow set so tat te ial ead will go to zero, wic is 10

11 wy te iitial velocity tat we set ever aects te velocity ad te ead calculated by te progra. We tik tat tere is a proble wit ipleetig te at i our etod or tat tere exists a uderlyig part o te at tat we ave ot take ito accout. Coclusio We coclude tat te iite dierece etod ay ot be te best way to odel groudwater low i a aquier sice we ad trouble speciyig te boudary coditios i our coputatio. Qualitatively, our solutios are correct or te siple cases o Laplace s equatio ad siple versios o te tie depedet low. However, we we tried to adapt our code or ore coplex probles were te coductivity o te aquier varied cotiuously, te etod breaks dow. Tere are also parts o our solutio tat we do t really ave a explaatio or, suc as te aer i wic our progra disregards te iitial low velocity. Sortig out tese errors igt result i a workig versio o te progra, but we doubt tat it could be upgraded to solve very coplex cases o te low. Reereces 1. C.W. Fetter, Applied Hydrogeology, Tird Ed., Merill Publisig Copay,1994. Private Couicatio: a. Dr. Ricard Elderki, Mateatics Dept., Pooa College b. Dr. Lida Reie, Geology Dept., Pooa College 11