Effectiveness of a Vertical Barrier against Intrusion of Flood Plain Infiltrated Water into an Aquifer

Transcription

1 Hydrlgy Days 007 Effecveness f a Vercal Barrer agans Inrusn f Fld Plan Inflraed Waer n an Aqufer Cnza Mracapll Unversy f Appled Scences Nrhwesern Swzerland, Insue f Cvl Engneerng cnza.mracapll@fhnw.ch Absrac. In hs paper he prcess f unsauraed aqufer recharge frm a rver channel and s fld plan laerally cnfned wh fully penerang barrers s nvesgaed. The same basc mehdlgy descrbed prevusly by he auhrs [Mrel Seyux e al., 1988] fr he cases f rechargng areas n a hmgeneus medum s appled here n case f heergenees due dammng. An apprxmae slun s baned by machng w ne-dmensnal flws, a vercal and a hrznal ne. The frmulan leads an negr-dfferenal uan, whch can be slved numercally. The resuls shw he effecveness f he barrer and s dependency upn prjec parameers. 1. Inrducn Grundwaer barrers are fen called dfferen srucures, lke subsuace dams and srage dams. They can be made f arfcal maeral, such as cncree, r f mre r less mpermeable naural maeral lke sand r sl. Dammng grund waer fr cnservan purpses s ceranly n a new cncep. Grundwaer dams were cnsruced n Sardna n Rman mes and dammng f grund waer was pracsed by ancen cvlsans n Nrh Afrca. In he las cenury hey have been develped and appled n many pars f he wrld, nably n ard regn as a mehd f vercmng waer srage. Mre recenly grundwaer barrers have been cnsdered fr rver managemen purpses. In he las few years her use fr aqufer precn n rver resran prjecs has gven hs mehd renewed aenn. The use f grundwaer barrers s f ncreasng neres snce hey allw he srage f he nflran waer belw he rver bed and drascally reduce he laeral spreadng f he perclang waer. Thus he pssbly f neracn beween he suace waer and he laeral aqufer usde he barrers s defnely reduced.. Am f he research Am f he presen sudy s develp a cncepual mdel and a mahemacal cde fr he develpmen f he waer able rse under sanary and unsauraed recharge cndns belw he fld plan f a rver n case he laeral spreadng f he mund s lmed by he presence f a barrer made f sl f knwn characerscs. The scpe f praccal neres fr engneerng prjecs s defne he effecveness f he barrer and s dependency upn prjec parameers. 3. Descrpn f he prblem The aqufer s assumed be hmgeneus, ansrpc, uncnfned and has a fne deph. The gemery f he prblem s shwn n Fg.1. In he case f lw and medum waer levels, he rver flws whn s curren bed, whch shws a smaller permeably due he clggng prcess. In hs case he Hydrlgy Days

2 Effecveness f Vercal Barrer agans Inrusn f Fld Plan Inflraed Waer n an Aqufer recharge rae belw he rver channel s neglgble and he waer able level s hrznal. The grundwaer level s nally he same a bh sdes f he barrer. Fgure 1. The gemery f he prblem In he case f hgh waer levels, he rver channel flls up s fld plan, whch has a hgher permeably han he clgged layer n he curren bed. Thus s realsc assume ha he recharge rae belw he fld plan s sgnfcan, bu sll n bg enugh ensure sauraed cndns. The unsauraed descendng frn reaches he waer able and a hs me he refleced frn rses fllng he empy pre space n he unsauraed aqufer belw he fld plan. In hs case he rsng mund s alms fla and can be apprxmaed as an upward hrznal frn. The hydraulc head drp whn he barrer s he dfference beween he level f he refleced frn and he waer able level n he exernal sde f he barrer. The waer able level laerally keeps decreasng and a nfny reaches cndns a res. 4. Basc apprach Clearly he flw phenmenn fllwng he arrval f he perclan flux a he nal lcan f he waer able s w-dmensnal n naure. Fllwng a general mehdlgy prevusly appled n sream-aqufer neracn mdellng and aqufer recharge [Mrel-Seyux H.J. and C. Mracapll, 1988] he wdmensnal naure f he flw prcess s apprxmaed by he machng (lnkng and cuplng) f a dmensnal vercal flw prcess and an hrznal ne (Fg.). In hs manner he cmplexy due he hgher dmensns s avded. The jusfcan fr he apprxman s ha, excep n he barrer, where a relevan 15

3 Mracapll, C. hydraulc head drp happens alng a shr flw pah, n he res f he aqufer he vercal velcy s neglgble and Dupu assumpns sll hld. The majr dffculy n he prcedure s he develpmen f he machng cndn whch leads ulmaely an negral uan frmulan. A dscrezed frm f he negral uan can be slved easly prvde numercal answers a specfc prblem. CONCEPTUAL MODEL Fgure. Cncepual Mdel 5. Mahemacal frmulan f he prblem The mahemacal frmulan presened n hs paper s n he ms general because s based n several assumpns whch are relaed he gemery and he physcs f he prblem, namely: he wdh f he rver channel s much smaller han he wdh f he fld plan he hckness f he barrer s much less han he wdh f he rver bed he barrers are lcaed a he exernal lms f he fld plan 16

4 Effecveness f Vercal Barrer agans Inrusn f Fld Plan Inflraed Waer n an Aqufer he gemery f he prblem has an axs f symmery n he cenre f he rver bed he bm f he aqufer s hrznal he nal waer able prfle s hrznal everywhere (cndn a res he nflran rae belw he fld plan s less han he vercal hydraulc cnducvy due he clgged layer a he bm f he rver bed, he nflran rae belw he rver channel s neglgble he descendng flux belw he fld plan s unsauraed nflran n he unsauraed zne ccurs under seady cndns In hs suan s realsc assume ha when he unsauraed frn hs he waer able, he nflran flux wll spl n w pars, ne refleced (fllng upward he resdual pre space sll avalable afer he passage f he descendng unsauraed frn and ne ransmed usde he barrer (n he aqufer n verlan by he recharge area). In Fg.1 he psn f he waer able a me s ndcaed. Fr reasns f symmery nly ne half f he sysem s cnsdered and dsplayed. The prfle s apprxmaed beween he w barrers by a plaeau a a dsance z abve he nal waer able level and by a prfle h ( x, laerally usde he barrers. The rgn fr he x axs s he exernal sde f he barrer. Whn he barrer he waer able s apprxmaed by a sragh lne cnnecng he aqufer elevans a he w sdes, namely z and h ( 0,. Thus he hydraulc head drp n he barrer s z ( h(0,. In he fllwng mahemacal frmulan he change f he waer cnen n he barrer due he develpmen f he mund s n cnsdered. I s assumed ha hs quany s neglgble cmpared he change f waer srage n he neghburng par f he aqufer Deermnan f he mund hegh h ( x, The evlun f he mund h ( x, hegh s gverned by he lnear frm f he ne-dmensnal Bussnesq uan: h = K H e h x T h = x h = x (1) where s me, K H s fully sauraed hydraulc cnducvy n he hrznal drecn, e s sauraed hckness, s effecve prsy, x s abscssa n he hrznal drecn, T s ransmssvy (r, mre precsely, hrznal ransmssvy) and s aqufer dffusvy. In he parcular case when he laeral flux a he bundary s cnsan and f value 1 and fr an nally fla waer able, he slun f he bundary value prblem fr. (1) s well knwn [e.g. Carslaw, H.S. and Y.C. Yaeger, 1959] and s: K h, q ( x, = ( T 1 / exp( x 1 / / 4 ) x (4 ) 1 / x ec (4 ) 1 / () 17

5 Mracapll, C. In. () K h, q ( x, s he mund hegh respnse a un sep excan f laeral flux (als knwn as he un sep respnse f hegh due excan f bundary flux). When he bundary flux vares wh me s knwn frm lnear sysem hery [e.g. Dge, 1973; Mrel-Seyux, 1979; Mrel-Seyux H.J. e al., 1988] ha he respnse h ( x, s relaed he bundary flux q ( (a dscharge per un lengh f srp, hus f dmensn area per me) by he cnvlun uan: q( ) h( x, = q( ) Kh, q ( x, Kh, q ( x; ) d (3) where q (0) s he value f q ( a me zer. In parcular a x = 0 he value f h s: q( ) h(, = h( = q(0) d T (4) T because he value f K h, q ( x, fr x = 0, as deduced frm. (), s: K h q, (, = T 5.. Deermnan f he psn f he refleced frn I s knwn frm he hery f mulphase flw n prus meda [e.g. Mrel-Seyux, 1969, 1973, 1987] ha he velcy f prpagan, V f f a frn f waer cnens s gven by he expressn: vw vw V f = (5) where v w, and v w, are he respecve values f waer velces (Darcy velces) and waer cnens n bh sdes f he frn. Applcan f. (5) he prblem a hand yelds he resul: dz d r( B B ) q( ( )( B B ) ( ) B = (6) where s he unfrm and seady waer cnen n he unsauraed zne under he fld plan (abve he rsng waer able), s he waer cnen a sauran and s he nal waer cnen under he rver channel. In hs case he waer srage n he barrer and s change durng he rse f he waer able are negleced. Ths s jusfed by he fac ha he waer srage under he fld plan ual ( ) z B ( ) z ( B B ) 18

6 Effecveness f Vercal Barrer agans Inrusn f Fld Plan Inflraed Waer n an Aqufer s much bgger han he waer srage n he barrer, whch s ual 1 ( d d )( z h) d where d and d are respecvely he waer cnen a sauran and he nal waer cnen n he barrer and d s he hckness f he barrer Deermnan f he ransmed flux (machng uan) Usng he flw ne apprach [e.g., Mrel Seyux and Mracapll 1988; Mrel- Seyux e al., 1990] he recharge rae q ( can be expressed by Darcy s law as: H q = A (7) LV LH K K V H where A s he average crss-secnal area f he flw ubes, H s he pezmerc head drp, K V and K H are he values f he hrznal and vercal hydraulc cnducves, LV and LH are he average lenghs f he flw ubes n he hrznal and vercal drecns whch carry waer away frm he nflran area he laeral bundary ( x = 0 ) acrss he aqufer underneah he fld plan and acrss he barrer. Thus L V = L V L Vd and L H = L H L Hd where he subscrp and d refer he aqufer and he barrer. The average lengh f he flw lnes s baned by he arhmec mean f he bundary flw lnes, namely: L H = 0. 5B L Hd = d L = 0.5( z ( e) V L Vd = 0.5( z ( h(, Smlarly, he average area f he flw per un lengh s [ B ( e h(, )] A = 0.5 H ( n. (8) s he hydraulc head drp frm he recharge suace alng he flw pahlne he bundary a x=0. 19

7 Mracapll, C. Snce he waer able mund under he fldng plane s apprxmaed wh a hrznal plaeau, he pezmeer head can nly drp whn he barrer frm he value f z ( he value f h ( 0,. Thus H ( = z ( h(0, Subsun f he abve wren uans n.(7) gves [ z ( h(0, )] q( = K ( (8) where K n he nex general case s K = K H ( z ( B e h) h d ) ( z e) B (9) where = If K V / K H and = K d / K H. z remans durng he rse f he waer able much smaller han e (and cnsuenly h ), K = K H ( z r f ( B e) h d ) e B and f ( z h) s much smaller han d (fr nsance fr lle values f ) he expressn fr K smplfes : K = K H ( B e) d e B and n hs case he dependence f K n he me s elmnaed Cuplng f he varus cmpnens The hrznal flw q (, whch spreads laerally u f he barrer s gven by.(8), wh a cnsan K. The expressn fr z ( can be baned frm negran f. (6), wh he resul: z ( = r( B B ) ( )( B B ) ( ) B 1 ( )( B B ) ( ) B q( ) d (10) 0

8 Effecveness f Vercal Barrer agans Inrusn f Fld Plan Inflraed Waer n an Aqufer The waer able hegh a x=0, h (0), s baned frm.(4) n case q ( 0) = 0 snce ( 0) = h(0) = 0, yeldng: z q( ) h( = d (11) T Subsun f he expressns fr z and h frm s.(10) and (11) n.(8) leads an negr-dfferenal uan fr he unknwn q (, specfcally: [( )( B B ) ( ) B ] q( K q( ) d K q( ) [( )( B B ) ( ) B ] d T = K r( B B ) whch can be rewren B KB q( ) q( K q( ) d d T (1) = K r( B B ) where B = ( )( B B ) ( ) B Slun f he uan Equan (1) s a lnear negral uan whch can be slved n prncple usng fr example he Laplace ransfrm echnque (e.g. Abdulrazzak and Mrel- Seyux, 1983). Hwever n hs case cmplex negran s rured and successve negrals f specal funcns wuld appear. I s easer slve. (1) numercally by dscrezng n he frm: B n q( n) Ke (1 ) q( ) q( ) q( n) (13) q v= 1 K B T n [ q( v) q( v 1) ] ( n v 1) = = 1 1

9 Mracapll, C. = K ( B B ) rn fr n = 1,,3... N where q (n) s he value f q ( a dscree neger values f a seleced perd f me and N s he al me hrzn f neres. The parameer s analgus he me wegh used n fne dfferences schemes. A value f 1 ndcaes a fully mplc scheme, a value f 0.5 ndcaes a Crank-Nclsn scheme and a value f zer ndcaes an explc scheme. Fr sably reasns s smemes necessary use a value f n he range 0.5 < 1. In hs case, where me s measured frm he mmen when he descendng weng frn hs he waer able, q(0) s zer. In. (13) a parcular negraed dscree kernel s defned as [e.g. Mrel Seyux and Mracapll, 1988]: 1 [ m ] 3/ ( 1) 3/ ( m) = m d = m 3 The dscrezan leads an explc lnear sysem f algebrac uans n he unknwns q ( 1), q(), q( n) he rdnaes f q (n) a dscree nervals f me. The ms general expressn fr lnear sysem f algebrac uans s Aq = B where q and B are vecrs and A s he marx f ceffcens. In hs case A s a lwer rangular marx and he slun f he lnear uans sysem can be baned by frward subsun. The sngle erms are b = K ( B B ) r a = B K e q KB T D 1 a j K B = K j 1 T [ D D ] j where D 3/ [ ( 1) ] 3/ 3 = 6. Resuls The mdel evaluaes he effecveness f a vercal barrer preven cnamnaed nflraed waer frm enerng he clean surrundng aqufer. The vercal barrer reduces he laeral recharge rae n he aqufer. The reducn

10 Effecveness f Vercal Barrer agans Inrusn f Fld Plan Inflraed Waer n an Aqufer depends n he characerscs f he barrer, namely s hckness and s permeably. The mahemacal cde s mplemened and sme numercal resuls are shwn. The evlun f he laeral recharge rae usde he barrer (n belw he fld plan) s shwn fr dfferen values f he hckness f he barrer (Fg.3) and fr dfferen values f he hydraulc cnducvy f he barrer (Fg.4). Inpu parameers are: wdh f he fld plan=40m, wdh f he curren bed=10m, waer cnen f sauran=5%, nal waer cnen=5%, waer cnen f he descendng frn=0%, vercal hydraulc cnducvy f he aqufer=0.01m/hur, hrznal hydraulc cnducvy f he aqufer=0.05m/hurs, recharge rae =0.005m/hur, hckness f he aqufer=0m, number f me seps= E E E-03 Laeral recharge rae (mq/hur) 6.00E E E E-03.00E-03 d=0.5m d=1m d=1.5m 1.00E E Tme (hurs) Fgure 3. Evlun f he laeral recharge rae n he aqufer usde he barrers fr a gven value f he hydraulc cnducvy f he barrer (K d =0.001m/hur) and dfferen values f he hckness f he barrer (d=0.5m, 1m, 1.5m). 3

11 Mracapll, C. 7.00E E E-03 Laeral recharge rae (mq/hur) 4.00E E-03.00E-03 kd=0.001m/h kd=0.0001m/h kd= m/h 1.00E E Tme (hurs) Fgure 4. Evlun f he laeral recharge rae n he aqufer usde he barrers fr a gven value f he hckness f he barrer (d=1m) and dfferen values f he hydraulc cnducvy f he barrer (K d = m/hurs, m/hurs, m/hurs). 7. Cnclusn Ths sudy s based n a mehdlgy prevusly appled and already esed n dfferen cases, namely: - sauraed aqufer recharge frm a rver channel n an hmgeneus aqufer Abdulrazzak, M.J. and H.J. Mrel-Seyux, 1983) - unsauraed aqufer recharge n an hmgeneus ansrpc aqufer (Mrel-Seyux H.J. e al., 1990) - unsauraed aqufer recharge frm a crcular spreadng basn n an hmgeneus ansrpc aqufer (Mrel-Seyux, H.J. and C. Mracapll, 1988) In hs sudy he prevus mehdlgy s exended he case f a fld plan laerally cnfned wh fully penerang barrers. The prcedure based n he machng f undrecnal flws, a vercal and a hrznal ne, allws he deermnan f negraed characerscs, such as he verall dscharge hrugh he barrers and he waer sred under he fld plan beween he barrers. These negraed quanes are mpran because hey enable deermne he effecveness f he barrers. Ths knwledge can be very useful, fr nsance, n he plannng and desgn f measures fr grundwaer precn purpses. Ths prcedure has als addnal advanages. Frsly, accuns fr dfferen values fr he specfc yeld n he aqufer belw he rver channel, belw he fldng plane and away frm. In fac he fllable pre space depends n he waer sred n he unsauraed zne abve he rsng waer able. Secndly, can als accun n a smply way fr ansrpy beween he hrznal and he vercal 4

12 Effecveness f Vercal Barrer agans Inrusn f Fld Plan Inflraed Waer n an Aqufer drecns. Ths s a useful feaure snce sgnfcan ansrpy s he rule raher han he excepn. References Abdulrazzak, M. J., and H. J. Mrel-Seyux, Recharge frm an ephemeral sream fllwng weng frn arrval waer able, Waer Resur, Res., 19(1), , 1983 Abramwz, M., and I. A. Segun, Handbk f Mahemacal Funcns, Dver, New Yrk, 197 Carslaw, H. S., and Y. C. Yaeger, Cnducn f Hea n Slds, Oxfrd Unversy Press, New Yrk, 1959 Dge, J. C. I., Lnear hery f hydrlgc sysems, Tech. Bull. 1468, U.S. Dep. f Agrc., Washngn, D.C., 1973 Mrel-Seyux, H. J., Inrducn flw f mmscble lquds n prus meda, n Flw Trugh Prus Meda, eded by R.J.M. de Wes, pp , Academc, San Deg, Calf., 1969 Mrel-Seyux, H. J., Tw-phase flws n prus meda, Adv. Hydrsc., 9, , 1973 Mrel-Seyux, H. J., Cs effecve mehdlgy fr sream acqufer mdelng and use n managemen f large scale sysems, repr, Hydrwer Rep. Dv., Hydrlgy Days Publ., Fr Cllns, Cl., 1979 Mrel-Seyux, H. J., Mul-phase flws n prus meda, n Develpmens n Hydraulc Engneerng, eded by P. Nvak, pp , Elsever Appled Scence, New Yrk, 1987 Mrel-Seyux, H. J., C. Mracapll, and M. J. Abdulrazzak, A reducns physcal apprach unsauraed aqufer recharge, paper presened a Inernanal Sympsum n Ineracn Beween Grund Waer and Suace Waer, In. Assc. fr Hydraul. Res., Ysad, Sweden, May 30-June 3, 1988 Mrel-Seyux, H.J. and C. Mracapll, Predcn f nflran, mund develpmen and aqufer recharge frm a spreadng basn r an nermen sream, Hydrwar Reprs Dvsn, Hydrlgy Days Publcans, Fr Cllns, Cl Mrel-Seyux, H.J., C. Mracapll and M.J. Abdulrazzak, A reducns physcal apprach unsauraed aqufer recharge frm a crcular spreadng basn, Waer Resur. Res., 6(4), ,