MODFLOW and More 2003: Understanding through Modeling Conference Proceedings, Poeter, Zheng, Hill & Doherty

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1 MODFLOW and Moe 2003: Undestandng toug Modelng onfeence Poceedngs, Poete, Zeng, Hll & Doety MODFLOW: A Fnte-Dffeence Goundwate Flow Model o an Integated Fnte- Dffeence Goundwate Flow Model? Dave M. Roeo 1 and Toas Maddoc, III 2 1 Balleau Goundwate, Inc., oeod@balleau.co, Albuqueque, NM, USA 2 Te Unvesty of Azona, addoc@w.azona.edu, AZ, USA ABSTRAT Te U.S. Geologcal Suvey odula goundwate flow odel (MODFLOW) by McDonald and Habaug s egaded as an pleentaton of a fnte-dffeence nuecal scee appled to te govenng goundwate flow equaton. Howeve, a copason of MODFLOW s dscetzed fo of te flow equaton wt tat deved by an ntegated fnte-dffeence (IFD) tecnque eveals tat MODFLOW pleents an IFD nuecal scee wtn te confnes of a fnte-dffeence gd. An IFD nuecal scee neent n MODFLOW enables no odfcatons to be ade to te etod n wc te odel eads and pepaes data to enable te constucton of a gd wt a oe coplcated geoety tan tat of a fnte dffeence gd. Adaptng MODFLOW n ts fason enables sulaton of flow toug a cuvlnea gd constucted wt tapezodal saped cells as well as ectangula fnte dffeence gd cells. Te odfcatons gve MODFLOW te capacty to explot te potental of ts nuecal scee by addng vesatlty to te spatal dscetzaton of te flow doan wtout coposng ts odelng capablty o ntoducng new solves. Te teoy nvolved n llustatng MODFLOW s non-genealzed IFD nuecal scee s outlned and a test poble s pesented tat vefes te etod wt an adapted veson of MODFLOW. INTRODUTION MODFLOW (McDonald and Habaug, 1988) s one of te ost wdely used goundwate flow odels n te felds of consultng and eseac. Te autos of MODFLOW desgned t so tat new capabltes could be added to te odel stuctue. Te Genealzed Fnte-Dffeence Pacage (Habaug, 1992) pesents a etod tat eoves soe of te assuptons tat ae neent n fnte-dffeence gd constucton. Te dscusson een explans wy eovng tese assuptons s possble and extends te genealzaton. Ts pape llustates tat MODFLOW s nuecal scee s a non-genealzed ntegated fntedffeence (IFD) etod ate tan a fnte-dffeence etod. Ts s deonstated by outlnng te dscetzaton pocess nvolved wen an ntegated fnte-dffeence etod s used. Te equaton tat esults fo te IFD dscetzaton pocess s ten copaed wt te equaton solved by MODFLOW and found to be dentcal. A non-genealzed IFD etod s explaned. Mno odfcatons to MODFLOW s souce code ae llustated to enable flow sulatons toug a cuvlnea gd. Fnally, a test poble s pesented tat vefes te etod wt an adapted veson of MODFLOW. IFD NUMERIAL FORMULATION Te IFD nuecal foulaton, een, follows tat of Naasan (Naasan, 1976). onsde te govenng patal dffeental equaton fo goundwate flow K q Ss. (1) t Spatally ntegatng (1) ove a sall fnte subegon V of te flow egon gves K dv qv SsV. (2) V t Now te dvegence teoe s appled to te ntegal of te net outflux due to te ead gadent n Equaton 2. Te dvegence teoe states 116

2 MODFLOW and Moe 2003: Undestandng toug Modelng onfeence Poceedngs, Poete, Zeng, Hll & Doety F dv V F n d Applyng Equaton 3 to te fst te n Equaton 2 yelds K n d qv SsV t. (3). (4) Equaton 4 s te ntegal foulaton of Equaton 1, wc can be dscetzed to obtan an IFD nuecal appoxaton. onsde ow Equaton 4 can be dscetzed. Fgue 1 sows a two-densonal, fvesded cell n a genealzed IFD gd. Assung te cell n Fgue 1 as a unt dept nto te page and expessng te scala poecton of n te n decton as n / n, ten te two-densonal dscetzed fo of Equaton 4 fo node = 6 s wtten as 5 n1 n L t 1 K n An qv SsV, (5) n K n = ydaulc conductvty between nodes and n [LT -1 ], A n = aea toug wc flow occus [L 2 ], L n = dstance between nodes and n [L], q = souce o sn extenal to aqufe at node [T -1 ], V = volue of cell enclosng node [L 3 ], Ss = specfc stoage of cell enclosng node [L 3 ], = ead at node at te step [L], t = te step cosen fo teatve pocedue [T]. n = 4 n = 3 = 6 n = 5 n = 2 n = 1 Node = 6 as fve sdes esultng n a suface ntegal dscetzed nto a su wt fve tes. Lewse, fo te geneal case, Equaton 5 would be wtten fo evey cell (node) n te gd and te nube of tes n eac su would coespond to te nube of sdes on te assocated cell. Ts aes t possble to sulate flow toug cells wt vayng nubes of sdes n two densons. If conductance s expessed as te poduct of ydaulc conductvty and te aea toug wc flow occus dvded by te lengt of flow, ten t can be descbed as = KA/L and Equaton 5 can be wtten as Fgue 1. Fve-sded cell n a genealzed IFD gd c n1 n n q V Ss V t 1, (6) c = nube of cell faces aound node. onductance can also be descbed n tes of tansssvty. In ts case, = Tw/L w s te wdt of te cell pependcula to te decton of flow. A notable featue of Equaton 6 ust be ade clea. Snce te dvegence teoe conveted te volue ntegal of te net outflux due to a ead gadent n Equaton 2 to a suface ntegal of te scala poecton of te ead gadent noal to a suface enclosng a sall fnte subegon of te flow doan, a constant was neted tat affects IFD gd constucton. Ideally, te ntefaces between eleents sould be pependcula to te lne onng any two nodal ponts and sould ntesect tat lne at ts dpont; altoug ts deal stuaton ay be dffcult to aceve n pactce, t sould be appoxated as closely as possble (Naasan, 1976). Ts ltaton s dscussed below. MODFLOW s DISRETIZATION PROESS Rate tan devng a fnte-dffeence analog to Equaton 1, te autos of MODFLOW pesent an altenatve appoac splfyng te ateatcs and explanng te coputatonal pocedue n tes of fala pyscal concepts egadng te flow syste (McDonald and Habaug, 1988). Essentally, te 117

3 MODFLOW and Moe 2003: Undestandng toug Modelng onfeence Poceedngs, Poete, Zeng, Hll & Doety dscetzaton pocess followed n MODFLOW s to dscetze Dacy s law fo flow toug te sx faces of a tee-densonal, bloc-centeed cell. Ten, fo te sae cell, a souce/sn te wc accounts fo extenal flow ates s deved. Fnally, te contnuty equaton tat states te su of all flows nto and out of te cell ust be equal to te ate of cange n stoage wtn te cell s appled. Usng te sae subscpt notaton as Equaton 5, te esultng equaton s expessed as 6 1 n( n ) P Q SsV n1 t t 1. (7) Fo detals on te actual dscetzaton pocess, te eade s efeed to te MODFLOW anual (McDonald and Habaug, 1988). Te second and td tes n Equaton 7 ae te su of a ead dependent and a non-ead dependent souce o sn te extenal to te aqufe tat can be expessed as q V P Q, (8) q = souce o sn extenal to aqufe at node [T -1 ], V = volue of cell enclosng node [L 3 ]. obnng Equaton 8 wt 7 poduces Equaton 6, ence, te dscetzed equaton solved by MODFLOW pleents an IFD nuecal scee. Howeve, MODFLOW pleents te scee wtn cells tat ae lted to fou sdes n two densons wc ceates a non-genealzed scee. It s appaent tat MODFLOW utlzes an ntegated fnte-dffeence nuecal scee n tat te dscetzed equaton tat MODFLOW solves s dentcal to te equaton we ave at wen pleentng te IFD dscetzaton pocess. MODFLOW pleents ts scee, oweve, toug a flow doan epesented wt a fnte-dffeence gd. Fgue 1 llustates te capacty of an IFD etod to sulate flow toug cells wt sapes oe coplcated tan ectangles. Hence, n ts pesent fo, MODFLOW as te potental to sulate flow toug a doan copsed of sapes ote tan ectangles. ENABLING MODFLOW TO UTILIZE ITS IFD METHOD MODFLOW was desgned to sulate flow toug a fnte-dffeence gd wt ntnscally ectangula cells. Te pocess of devng a nuecal soluton s constaned by te sape of te gd cells n two aeas: te calculaton of aea fo eac cell and te calculaton of conductance between cells. Bot of tese calculatons assue cell wdts ae constant along any ow o colun. MODFLOW s solves ae also equpped to solve fo te ead n a cell wt only fou sdes n two densons. Howeve, no odfcatons to te souce code of MODFLOW enable te cell aeas and conductances between cells to be calculated wtn cells tat fo tapezods n two densons. Ts enables MODFLOW to sulate flow toug a cuvlnea gd, snce appoxatng te cuvlneaty wt stagt-lne segents esults n a gd coposed of tapezodal cells as sown n Fgue 2. Alteng te gd n ts fason gves te code te capacty to explot te benefts neent n ts nuecal scee. How MODFLOW s code would ave to be adusted to calculate te Fgue 2. Exaple of cuvlnea gd aea of eac cell and te conductance between cells s gven below. Wt geogapcal nfoaton systes (GIS), obtanng te aea fo eac cell wtn a gd s a outne pocess. A GIS can ceate o pot a odel gd, calculate te aea of eac cell, and etun te cell aeas n te fo of a two-densonal aay tat MODFLOW can be adapted to ead. In ts case, te aea would be used n te calculaton of stoage capacty, vetcal conductance, voluetc ecage and evapotanspaton ates wtn te odel. Modfcaton of te conductance calculaton entals usng MODFLOW s exstng equaton n ts uneduced fo. Pesently, MODFLOW calculates equvalent conductance between two cells as 118

4 MODFLOW and Moe 2003: Undestandng toug Modelng onfeence Poceedngs, Poete, Zeng, Hll & Doety TT w 2, (9) T L TL = equvalent conductance between cells and [L 2 T -1 ], T = tansssvty of cell [L 2 T -1 ], L = lengt of flow wtn cell [L], w = wdt of cell noal to decton of flow n cell o [L]. Howeve, Equaton 9 was educed by te assupton tat te wdt of cells and s equal. If te wdts wee not equal, Equaton 9 would tae te fo 2T wt w T w L T w L. (10) Replacng Equaton 9 wt 10 s necessay to enable MODFLOW to calculate flow toug a cuvlnea gd. Fnally, a etod fo calculatng L and w ust be developed. Te splest etod s to adapt te odel to ead te x and y locatons of gd vetces. Te aveage lengt of flow wtn any cell n te ow and colun dectons can ten be calculated. Upon copleton of tese odfcatons, MODFLOW becoes capable of sulatng flow toug a cuvlnea gd. Te ntefaces between eleents sould be pependcula to te lne onng any two nodal ponts and sould ntesect tat lne at ts dpont. Ts deal stuaton can only be aceved f actual cuved lnes wee to be used n te gd constucted fo sulatng flow. If Equaton 10 s used to calculate te equvalent conductance between nodes, ten te way te deal lengt, s appoxated s sown n Fgue, 3. It s also evdent fo Fgue 3 tat te su of alf te aveage,+1,+1 lengt of flow n cell, and alf te IDEAL aveage lengt of flow n cell,+1 s an appoxaton of te actual cuved lengt tat poves as te,+2 APPROXIMATE,+2 spatal dscetzaton s efned. Spatal dscetzaton of a cuve always ples suc an Fgue 3. Appoxated lengt of flow between cells appoxaton. Souce code contanng te odfcatons descbed above s avalable fee of cage fo te auto at VERIFIATION OF MODEL MODIFIATIONS A test poble s pesented to cec te valdty of te odel obtaned fo te afoeentoned code odfcatons. A gd wt adal syety s eadly constucted wt tapezodal saped cells. Fgue 4 sows a plan vew of te odel gd fo te poble. Te poble s specfed oogeneous and sotopc wt a gadent of ove a adus of 15 feet. An analytcal soluton to te steady-state adal flow poble exsts and s gven as (an, 1967) 2 1 log 2 log 1 ; 1 2, (11) 2 log 1 119

5 MODFLOW and Moe 2003: Undestandng toug Modelng onfeence Poceedngs, Poete, Zeng, Hll & Doety 1 = ead at adus 1, 2 = ead at adus 2. Fgue 5 llustates a plot of densonless ead vs. densonless adus solved ove te flow doan and copaes te esult wt te analytcal soluton. 20 DIRIHLET BOUNDARY 15 RADIUS Fgue 4. uvlnea gd wt adal syety Fgue 5. opason of analytcal and nuecal solutons SUMMARY MODFLOW s a wdely used goundwate flow odel tat pleents a non-genealzed IFD nuecal scee wtn te confnes of a fnte-dffeence gd. One of te benefts assocated wt an IFD scee s ts ablty to sulate flow toug a gd wt geoety oe coplcated tan tat of a fnte-dffeence gd. Mno odfcatons can be ade to MODFLOW s souce code to enable flow sulatons toug a cuvlnea gd constucted wt tapezodal saped cells. Te odfcatons antan copatblty wt MODFLOW s standad pacages and ncease te vesatlty of gd constucton. REFERENES an, J., Te Mateatcs of Dffuson, outaulds Lted Reseac Laboatoy, Madenead, Oxfod Unvesty Pess, Habaug, A.W., A Genealzed Fnte-Dffeence Foulaton fo te U.S. Geologcal Suvey Modula Tee-Densonal Fnte-Dffeence Gound-Wate Flow Model, U.S. Geologcal Suvey, Open-Fle Repot McDonald, M.G., Habaug, A.W., A Modula Tee-Densonal Fnte-Dffeence Gound-Wate Flow Model, U.S. Geologcal Suvey, Open-Fle Repot , Boo 6, apte A1. Naasan, T.N., Wtespoon, P.A., An Integated Fnte Dffeence Metod fo Analyzng Flud Flow n Poous Meda, Wate Resouces Reseac, 12(1), Neuan, S.P., Pesonal councaton/lectue notes, Te Unvesty of Azona, Depatent of Hydology and Wate Resouces. Roeo, D., IFD: An Integated Fnte-Dffeence Pacage fo Use wt MODFLOW, Te Unvesty of Azona, Depatent of Hydology and Wate Resouces. 120