A Method for Setting the Artificial Boundary Conditions of Groundwater Model

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1 Open Journal of Geology, 2013, 3, doi: /ojg b012 Published Online April 2013 (hp:// A Mehod for Seing he Arificial Boundary Condiions of Groundwaer Model Yipeng Zhou 1,2, Zhaoli Shen 1, Weijun Shi 2, Jinhui Liu 2, Yajie Liu 2 1 School of Waer Resources and Environmen, China Universiy of Geosciences, Beijing, China 2 Deparmen of Civil and Environmenal Engineering, Eas China Insiue of Technology, Nanchang, China zyp721@163.com, wjshieci@163.com Received 2013 ABSTRACT Numerical simulaion echnology is nowadays an imporan means for groundwaer issues because of is efficiency and economical advanages. Bu in case of naural hydrogeological boundaries are no wihin he ineres area, i may be a big rouble o se boundary condiions of he model arificially wihou enough field invesigaion informaion. This paper inroduced a mehod for solving such problem applying field pumping es and recovery es. The mehod was applied o build an in-siu leaching of uranium model. Resuls showed ha he model boundary condiions can be se saisfacorily, and also he calculaed heads mached he observed daa well in boh wo models. Keywords: Groundwaer; Numerical Model; Arificial Boundary Condiions 1. Inroducion Since he 1960s, wih he developmen of compuer echnology, he mehod of numerical simulaion had been widely used o solve groundwaer flow and solue ranspor problems because of is effeciveness, flexibiliy and relaively economical wih spend, and gradually become an imporan mehod for groundwaer issues [1-7]. However, alhough los of models have been buil in various applicaions, few people care abou he real effecs of hose models in pracices [8]. One of imporan facors influencing he reliabiliy of he groundwaer model is geology and hydrogeology invesigaion; and usually making reasonable undersandings on boundary condiions is a big challenge [9]. Once he boundary condiions are disored o he ruh, i is bound o lead o significan deviaion of he model calibraion parameers from acual values, and hen serious impac on he reliabiliy of he model would no be avoided. Model boundary condiions are usually se according o field invesigaions. When he ineres area is small ha he model boundaries are far away from naural hydrogeological boundaries, arificial model boundary condiions need o be se according o a long-erm observaion of groundwaer a hose boundaries. However, in many cases, he required observaion daa are ofen unavailable; in his dilemma, one alernaive way is o expand model exen so ha he groundwaer can be assumed no o be affeced by human aciviies (such as pumping es) aken placed in he ineres area; hen he boundary condiions of firs ype or of second ype can be se a model s boundaries [9]. Bu his kind of soluion also has shorcomings, one of which is ha o esablish of model hydrogeology configuraion beyond he ineres area wihou supplemenary geology invesigaion informaion may bring unexpeced serious error o he simulaion resuls [10,11]. In his paper, in order o build a flow model of groundwaer and leaching soluion during insiu leaching of uranium process, a mehod has been employed o se arificial model boundaries by combining heoreical calculaion according groundwaer unseady flow heory and he model ieraive calibraion using observaion daa of pumping es and recovery es independenly. 2. Mehods 2.1. Basic Principles The basic principle is using field pumping es and recovery es o calibrae he model parameers and boundary condiions simulaneously. Firs, ge he head drawdown funcion derived from Jacob formula of groundwaer unseady flow a he boundaries locaed wihin he cone depression; Then se iniial heads generalized from he head drawdown funcion of he model boundaries for model building and calibraion; finally, calibrae model parameers and boundary condiions ieraively using he observaion daa of pumping es and recovery es independenly and hen make he resuls fi he facs o he mos degree.

2 Y. P. ZHOU ET AL Funcions of he Model Boundary Heads During pumping es in he confined aquifer, draw down lead o he formaion of he cone of depression of pressure head. The range of cone expands coninuously wih pumping, and gradually achieves a relaive sable sae. When he whole model is locaed wihin he cone, heads a model boundaries would vary wih ime; so obviously, model boundary condiions need o be se according o head changes. Therefore, he funcion of head variaion mus be go a firs. According o he heory of confined waer s unseady flow owards o fully peneraing well, he variaion of he head drawdown wihin pumping influence scope can be approximaely described by Jacob Formula [12], as in Q 2.25T s ln (1) 2 * 4T r where s is he head drawdown; Q is he pumping rae; T is he coefficien of ransmissibiliy; is he pumping ime; r is he disance o pumping well; μ * is he coefficien of sorage. Thus, he live head can be calculaed via Equaion (2): H (,) r Hr (, ) sr (,) (2) S where H (,) r S is he head a he poin wih he disance of r o he pumping well and a he ime of ; H (, r 0 ) is he iniial head a he poin wih he disance of r o he pumping well; s(,) r is he head drawdown a he poin wih he disance of r o he pumping well and a he ime of. Since in Equaion (2) he head is a coninuiy funcion of ime, i can no o be applied o se model boundary condiions ye; i need o be emporally discreized o n periods, and in each period he head is a consan, hus, he variaion of he head drawdown can be described by Equaion (3): s1, 0 1 s2, 1 2 sr (,) (3)... sn, n 1 n So, he live head anywhere wihin he cone of depression during pumping es can be given by he piecewise consan funcion as Equaion (4): Hr (, 0) s1, 0 1 Hr (, 0) s2, 1 2 Hr (,) S (4)... Hr (, 0) sn, n 1 n Divide he simulaion ime o n sress periods, and in each period se he model boundary head according o he corresponding consan value of each definiion domain 0 of he funcion shown in Equaion (4) Calibraions of he Model Boundary Condiions Afer iniial boundary condiions being se hrough he heoreical calculaion menioned above, run he model o calibrae boundary condiions and parameers ieraively using observaion daa of he pumping es and recovery es independenly. The flow char of he calibraion process is shown in Figure 1. Firs, build groundwaer flow model of he pumping es and se he model iniial boundary condiions for ieraion according o Equaion (4). Then, run he model buil in he former sep o calibrae model parameers and boundary condiions by he observaion daa of he pumping es. If he sandard error of esimae (S.E.E) exceeds 5%, he hydrogeology parameers and boundary condiions would be adjused slighly and hen he calibraion repeas. When he S.E.E is below 5%, he calibraion process goes o he nex sep. Finally, build groundwaer model of he recovery es applying he calibraion resuls of he second sep as iniial sae; run i o calibrae model parameers and boundary condiions again using he observaion daa of he recovery es. The S.E.E of 5% also is applied as calibraion error crierion; if resuls mee he crierion, he calibraion process ends and he model parameers and boundary condiions are fixed; oherwise, reurn o he firs sep, modify he model parameers and boundary condiions and hen he whole process repeas again. 3. Applicaion Example 3.1. Backgrounds and Model Overview The sudy was conduced a he piedmon alluvial slope in he souhern region of he Turpan-Hami basin. The Figure 1. Flow char of he model calibraion process.

3 52 Y. P. ZHOU ET AL. ineres confined groundwaer sysem has sable impervious roof and base; he groundwaer flew from souhwes o norheas wih a hydraulic gradien of 0.02, and was mainly recharged indirecly by he Quaernary phreaic waer from he souhern mounainous bedrock fissure. The sudied issue was abou simulaion of groundwaer and leaching soluion flow in he ore-bearing aquifer a an in-siu leaching of uranium sie. There are five wells (Figure 2); he well CK1 was for exracion, and he ress were for leaching soluion injecion. The hydrogeology characerisic of he aquifer wihin he mining scope is as shown in Figure 3. The average hickness of he aquifer is abou 40m; he sable impervious roof and base are mainly of mudsone and sily mudsone (in gray); in he aquifer (in blue) here are four disconinuous inerlayer, one is of sily mudsone (shown in gray) wih he hickness of 1-3 meers, and he hree ohers are of calcareous sandsone wih he hickness of m (in whie). Field pumping es and recovery es were conduced employing well CK1 for pumping, well ZK1 and well ZK3 for observaion, he es resuls are shown in Table 1. Figure 2. The plan view of well disribuion. Figure 3. Cross secion of he aquifer. Table 1 field ess and he hydraulic parameers of he aquifer. Field es Tes ime (min) Pumping rae (m 3 /h) Pumping es Recovery es Coefficien of ransmissibiliy (m 2 /d) coefficien of sorage Based on hose field invesigaions, he concepual model of he groundwaer flow during he pumping es was buil as Equaion (5): H H H ( K xx ) ( Kyy ) ( Kzz ) W x x y y z z H Ss,( x, y, z) D, 0 (5) H( x, y, z, o) H0 ( x, y, z) ( x, y, z) D H ( xyz,,, ) f( xyz,,, ) ( xyz,, ) S S where K xx, K yy and K zz are respecively he conduciviies in x, y and z direcion in he hree-dimension space; H is he confined waer head; W is he flux rae per uni volume, which is used o describe he flow rae of wells; Ss is he specific soraiviy; is he ime; H 0 (x,y,z) is he iniial head a posiion wih he coordinae (x, y, z); H ( xyz,,, ) S is he head a he model boundary. The model area denoed D and boundary S. The iniial head of H 0 (x, y, z) was se according o saic waer level observed a he beginning of he pumping es. Since he modeling area was small, is edges are far away from he naural hydraulic boundaries known, hence arificial boundaries were necessary. A circle surrounding he well CK1 and wih radius of 100 meers was se as he boundary of he model Head Funcions of he Model Boundary In case of he coefficien of ransmissibiliy is 5.7 m 2 /d, he coefficien of sorage is and he pumping rae is 7.2 m 3 /h, he radius of influence of he pumping es is more han 800 meers; clearly, he heads a he pumping es model boundaries which were 100 meers away from he pumping well mus o be varying wih ime. Calculaion according o Jacob formula showed ha head drawdown sared o ake place a he model boundaries afer pumping for 3.65 hours. The drawdown funcion can be derived from Equaion (1), as in 0, s (6) 2.34 ln 2.95, Temporally discreized he coninuiy funcion of he drawdown o a piecewise consan one, as shown in Figure 4. A broken line of discree funcion s () was used

4 Y. P. ZHOU ET AL. 53 o approximaely replace he curve of he original funcion s(), making sure he difference beween he wo adjacen consans of he funcion s () was no greaer han 0.5 m. The funcion values are lised in Table 2. And hen, divided he whole simulaion ime of he pumping es ino 15 periods according o he funcion s (), in each period iniial boundary heads of he model were se as a corresponding consan value, which could be ge from he Equaion (4) by replacing he values of funcion s () lised in Table 2 for he drawdown. Then, he ieraive calibraion process sared Resuls Afer correcing he model parameers and boundary condiions repeaedly via he calibraion processes of he pumping es model and recovery es model, he model parameers and boundary condiions were fixed on. Resuls showed ha he calculaed heads mached he observed daa saisfacorily in boh wo models (Figure 5 and Figure 6); he mean absolue error beween he calculaed heads and observed daa of pumping es simulaion was m, and recovery es simulaion m; boh variances were less han 5%. Furhermore, he coefficien of ransmissibiliy was 5.3 m 2 /d; i was close o he resuls of field ess (5.7 m 2 /d). Figure 5. The resuls of calculaed heads mached o observed heads of he pumping es model. Figure 6. The resuls of calculaed heads mached o observed heads of he recovery es model. Figure 4. The curves of funcion s () and funcion s'(). Table 2. The values of funcion s'(). s' ( ) s' ( ) s' ( ) Conclusions The boundary condiion is one of key facors influencing he reliabiliy of groundwaer model. In case of arificial boundary condiions are needed, hey should be se reasonably using as much field invesigaion daa as we can ge, oherwise, i is prone o cause grea disorion o he ruh and make he model worhless. Sudy resuls show ha he mehod inroduced in his paper can be a feasible choice o se arificial boundary condiions of he groundwaer model. 5. Acknowledgemens Thanks for he fund of he Major Sae Basic Research Developmen Program of China (973 Program) (No. 2012CB723101). REFERENCES [1] H. R. Zhang, Unseady Groundwaer Flow Theory Developmen and Applicaion, Geology Publishing House,

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