Specific yield for a two-dimensional flow

Transcription

1 WATER RESOURCES RESEARCH, VOL. 36, NO. 6, PAGES , JUNE 2000 Specfc yeld fr a tw-dmensnal flw Peter Trtscher, W. Wayne Read, 2 and Phlp Bradbrdge Abstract. We nvestgate the systematc secular spatal varatn f specfc yeld. As a vehcle fr ths analyss we cnsder a canncal uncnfned aqufer cnsstng f a prus zne whse crss sectn s a smple lng rectangle. The hydraulc cnductvty n the unsaturated zne s mdeled by the quas-lnear apprxmatn. We fnd that lcally the specfc yeld may be strngly nfluenced by the water table depth and mldly dependent n the recharge rate f that rate s hgh. Fr the smple gemetry cnsdered, a lateral cmpnent f flw has been fund t have an nsgnfcant effect n the lcal specfc yeld and that a mdel that assumes lcally purely vertcal flw t the gven phreatc surface prvdes a mre-than-adequatestmate f the specfc yeld. Fr the verall yeld f an aqufer we fnd that the smplest mdel, wheren the flw thrugh the sl s neglected,.e., the mdel wth statc water and hrzntal phreatc surface, prvdes a reasnable ndcatn f the actual specfc yeld fr mst nfltratn rates and aqufer dmensns. Hwever, f the nfltratn rate s hgh r the aqufer s partcularly lng, then the yeld btaned frm an assumed purely vertcal flw, presuppsng that the phreatc depth s accurately knwn, gves an excellent estmate f the actual specfc yeld. 1. Intrductn The characterstc msture release curve and the depth t the water table are the mst mprtant factrs n determnng Estmatng the strage capacty and sustanable yeld f un- the vlume f water held by the aqufer. It s relatvely smple cnfned aqufers s f fundamental cncern t nhabtants f t shw frm ne-dmensnal studes that the specfc yeld s ard and semard regns. Fr each aqufer the water flw mderately dependent upn the rate f water mvng thrugh regme s determned by a cmplex nteractn amng the surface and subsurface recharge-dscharge dstrbutn, the aqufer bundares, and the sl characterstcs. Central t the sl [Chlds, 1960; Gardner, 1958]. Hwever, t s nt knwn f a lateral water flw cmpnent, whch s typcal n tw- and three-dmensnal flws, bears sgnfcance upn the lcal spequantfcatn f avalable water s the cncept f "specfc cfc yeld and hence n the ttal vlume f water avalable. yeld." It has been defned as "... the vlume f water that an uncnfned aqufer releases frm strage per unt surface area f aqufer per unt declne n the water table" [Freeze and Cherry, 1979, p. 62]. Ths s an mplctly lcal defntn, wth The answer t ths questn has nw becme accessble snce we have develped exact seres slutns fr saturatedunsaturated flw n tw dmensns [Trtscher et al., 1998]. In rder t nvestgate the sgnfcance f a tw-dmensnal flw the specfc yeld fr any chsen clumn f sl dependent upn the spatal varablty f specfc yeld, we emply a caupn, amng ther thngs, the lcal water flw, water table depth, and sl hetergenety [Stewart, 1962; Gllham, 1984; Everett et al., 1984; Rekerk, 1989; Fetter, 1994]. In practce, hwever, spatal varatn f specfc yeld has rarely been cnsdered. Where the water table lwers by ne unt f depth, the specfc yeld s the area f the regn between the tw relevant water cntent-depth curves, as depcted gemetrcally n Fgure 2.23 f Freeze and Cherry [1979]. In nenncal uncnfned aqufer cnsstng f a prus zne whse crss sectn s a smple lng rectangle. The permeable regn verlays an mpermeable (r nearly mpermeable) base and s bunded by vertcal mpermeable dkes. In ths explratry mdel, half f the sl surface s subjected t a unfrm nfltratn rate, wth the remander f the sl surface dschargng by evapratn. Ths gemetry yelds a physcally meanngful and practcal recharge-dscharge prfle wth a manageable dmensnal zer-flux slutns these curves are dentfable as number f parameters whle retanng the essental character msture release curves, but here they are mre general water cntent prfles. In sectn 3 we fnd t cnvenent further generalze the defntn f specfc yeld t mnus the rate f change f water cntent depth wth respect t water table depth. Ths defntn des nt depend n the smewhat arbtrary chce f unt length n a ntnal "unt declne n the water table," but t wuld agree wth the prevus defntn f a very small unt f length were used. f tw-dmensnal flw. We specfcally nvestgate the nfluence f recharge rate, depth t the water table, and aqufer length fr a representatve sl. Prevusly, numercal methds have been requred t slve saturated-unsaturated flw prblems wth cmplex bundary gemetres and hghly varable sl cnductvtes [Bear and Verrujt, 1987; Zaradny, 1993]. Hwever, t s nt wdely knwn that n arbtrary, rregularly shaped dmans, lnear bundary value prblems can be slved by separatn f varables and by Schl f Mathematcs and Appled Statstcs, Unversty f Wl- cnsequent expansns usng nnrthgnal bases fr functn lngng, Wllngng, New Suth Wales, Australa. spaces. Wth sme smple precedents n acustcs by Raylegh 2Department f Mathematcs and Statstcs, James Ck Unversty, [1945] and n saturated flw by Pwers et al. [1967], ths methd Twnsvlle, Queensland, Australa. was appled by Read and Vlker [1993] and Read [1993] t slve Cpyrght 2000 by the Amercan Gephyscal Unn. Paper number 1999WR /00/1999WR Laplace's equatn fr saturated flw and by Read and Bradbrdge [1996] fr the quas-lnear unsaturated flw f Gardner [1958] and Phlp [1969]. 1393

2 1394 TRITSCHER ET AL.: QUASI-STEADY SPECIFIC YIELD steady nfltratn-evapratn /I.,,,,,_unsaturated zne,,. D. '' ' ' ' ' ' ' ' ß ' ',,.I,,,,,,, '..'..'-- --'.,',',','/ ',',',, F--- at au ted// z/de;- Fgure 1. Schematc dagram f the sl prfle and water recharge-dscharge znes. Recently, Trtscher et al. [1998] have used the analytcal seres methd t slve the steady quas-lnear saturatedunsaturated seepage flw prblem fr prus dmans f rregular shape. The seepage prblem was mdeled as a varatnal prblem t determne the pstn f the saturatedunsaturated nterface. A seres slutn fr the ntegrand f the penalty functnal was derved, whch n turn allwed a smple drect numercal methd t be appled t acheve an ptmum lcatn. By sme relatvely mnr mdfcatns t the frmulatn f the seepage prblem, we may derve a seres-type slutn fr ur flw dman wheren the phreatc surface n lnger ntersects the surface seepage face. Fr prescrbed-flux bundary cndtns the steady slutn fr water cntent s unque nly after specfyng anther parameter such as the ttal water cntent. Ths nnunqueness f the slutn enables us t nvestgate the relatnshp between the ttal water cntent and water table depth, purely frm steady state slutns. The specfc yeld can n fact be unquely determned, as a functn f ttal water cntent. The seres apprach has several advantages that are useful fr ur study. In partcular, the explct dependence f the functnal n the pstn f the phreatc surface yelds an accurate lcatn fr the water table, whch s essental t ur analyss. The ther advantages, as n the seepage flw prblem, are that t affrds a realstc descrptn f the water dstrbutn n bth saturated and unsaturated znes; the frmulatn s well defned, algrthmc, and reprducble; n spatal dscretzatns are necessary; and glbal slutn errrs are readly estmated frm maxmum prncples. 2. Mdel Descrptn A schematc dagram f the sl hrzn used n these analyses s gven n Fgure 1. A layer f permeable sl verlays an mpervus base materal wth vertcal dkes at the ends AF and CE. The sl surface AC and basement FE are hrzntal E the sl surface BC (frmx, = L,/2 tx, = L,) s subject t mstly unfrm evapratn. Fr a small length where the water supply changes frm unfrm nfltratn t unfrm evapratn, we ft a snusdal curve t smth the transtn. Ths s ntrduced t smth the phreatc surface s that cal- culatn tme s reduced. At the sl surface the vlumetrc water flux takes the frm s that the crss sectn f the flw regn ABCEF s a rectangle. The flw regn has thckness D, and length L,, whch are measured n the vertcal and hrzntal drectns, respectvely. At the left vertex n the sl surface we fx the rgn f a sutable (x,, z,) crdnate system, wth z, pstve vertcally dwnward. The equatns fr the sl surface and mper- Ths yelds the famlar quas-lnear apprxmatn f Gardner [1958] and Phlp [1969], whch has been fund t be sutable fr a wde varety f sl types [Pullan, 1990]. The cnstant h (=h,/l,) s the dmensnless bubblng pressure s that we may ncrprate a tensn-saturated zne and a s the meable base are gven by z, = 0 and z, = D,, respectvely. dmensnlessrptve number, whch, n terms f dmensnal Steady and essentally unfrm nfltratn ccurs alng the unts, s the rat f the gemetrc length scale L, t the sl surface AB frm x, - 0 t x, = L,/2, whle the rest f ntrnsc srptve length a -l: r,0, 0 -< x, -< 0.4L, r,(x,) = r, cs(5rrx,/l,), 0.4L, <x, -< 0.6L, -r,0, 0.6L, <x, -< L, When the sl s suffcently mst, the rate f evapratn s gverned by atmspherc cndtns. We have assumed that the atmspherc cndtns are unfrm ver the evapratn regn and that the atmspherc demand s near t the rate that balances the vlume f water suppled by the nfltratn regn. Ths s a reasnable apprxmatn untl the sl s very near dry and sl-water transprt s the rate-determnng prcess [Phlp, 1957; Gardner and Hllel, 1962]. Hwever, fr ths case the lcal specfc yeld s smply near the maxmum value. At ths pnt, we ntrduce dmensnless varables, usng the length L, f the regn and the saturated hydraulc cnductvty K,. The nndmensnal lengths and varables satsfy the fllwng relatnshps: D, r, r,0 -- r r D L, K,0 K,0 Nte that D s the aspect rat f the prus regn Gvernng Equatn We assume a hmgeneus, strpc aqufer, wth a sl that dsplays neglgble hysteress n the ptental energy-water cntent relatnshp and that the flw s gverned by Darcy's law. Then fr steady saturated-unsaturated flw, the flw equatn may be expressed as (l) (2) V. (K(0) VH) = 0, (3) where K(O) (= K,(O)/K,) s the dmensnless hydraulc cnductvty, H ( = H,/L, ) s the dmensnless ttal hydraulc head, 0 (x, z) s the vlumetrc msture cntent, and V s the gradent peratr [Bear and Ferrujt, 1987]. In saturated-unsaturated flw, K s a hghly nnlnear func- tn f vlumetrc msture cntent. Here we assume K s a cnstant functn fr the saturated zne and an expnental functn f the pressure head h(x, z) (= H + z) fr the unsaturated zne: K = ea(h+h), 1, h >-h0 < -h' (4)

3 TRITSCHER ET AL.' QUASI-STEADY SPECIFIC YIELD 1395 '"E =02-0 ' -:3- = E 0.0- I I I I 0 I Ig f pressure head (cm) -5- I I I I 0 I Ig f pressure head (cm) Fgure 2. Sl hydraulc prpertes fr slt lam GE3 (frm Fgure 7 f van Genuchten [1980]). Crcles, sld lnes, and dashed lne ndcate bserved, van Genuchten relatnshp, and quas-lnear ft (a, = cm -1, frm Phlp [1984]), respectvely. In ur dervatn, = œ,,,. (5) f the slutn we wll cnsder the less restrctve case where we have a lwer bundary and sl surface f arbtrary gemetry, as ths may be acheved wth mnmal addtnal effrt. The rectangular dman s a specal case f the arbtrary cnfguratn. The bundary value prblem takes n a partcularly smple frm f we frmulate the prblem usng a dmensnless stream functn, ½(x, z), whch quantfes the mass flux n the flw dman. We relate the stream functn t the ttal hydraulc head by 00 OH 00 OH... g() --. (6) Ox - K(O) Oz Oz Ox There wll be n mass flux acrss the mpermeable basement. Cnsequently, the stream functn n ths bundary wll be a cnstant, whch we chse as zer, wthut lss f generalty. That s, ½(0, z) = ½(x, ft'(x)) = ½(1, z) = 0. (7) and we mnmze F subject t the cnstrant that the ttal hydraulc head alng the bubblng-pressure surface s the neg- Here we have defned z = fb(x) as the functn specfyng the atve f the elevatn plus the bubblng pressure: depth f the basement. We assume the sl surface s subject vertcal nfltratn H(x, *l(x)) = -*l(x) - h. (11) and evapratn at a rate whch s a relatvely general functn Snce we can cnstruct explct seres slutns fr ½(x, z) and r*(x) such that J' r*(x) dx = 0. Subsequently, the stream H(x, z), t s assumed that ½ and H satsfy all ther gvernng functn alng ths bundary s gven by equatns and bundary cndtns stated earler. As n the seepage prblem [Trtscher et al., 1998], the func- ½(x, ft(x)) = R(x) = - r*(x) dx, (8) tnal (10) represents the rt-mean-squarerrr n stream 0 X functn cmpared wth ur target value R(x). In ur varatnal prblem ths functnal s t be mnmzed ver the where z = ft(x) s the functn specfyng the elevatn f the range f allwed tral functns fr the bubblng-pressure sursl surface. Fr example, ur partcular case gven by (1) face z = r/(x). We then slve the varatnal prblem by a yelds drect numercal scheme such as an Euler methd r Rtz's ½(x, 0) = = r(sn(5wx)/(5w) - 0.4), r(x- 1), 0_<x_< <x -< <x_<l (9) 2.2. A Varatnal Frmulatn Nrmally, the bundary value prblem (3)-(8) wuld be slved numercally fr the ptental H wthut explct reference t saturated r unsaturated znes. The lcatn f the phreatc surface, h(x, z) = 0, wuld be btaned by nversn technques. Hwever, we use a mre drect apprach by psng a functnal whch ncrprates the phreatc surfacexplctly. Actually, we defne the surface where the pressure head s equal t the bubblng pressure h (x, z) = -h and then derve the lcatn f the phreatc surface. Hwever, ften the bubblng pressure s small s the tensn-saturated zne may be absrbed nt the unsaturated zne. Let us dente the bubblng-pressure surface h(x, z) = -h as z = rt(x). We specfy the flux bundary cndtn ver the unsaturated sl surface (8) n terms f the functnal F(,l(x)) = [½(x, ft(x)) - R(x)] 2 dx, (10) methd [El'sgl'ts, 1961] after dervng an explct frm fr the stream functn, ½(x, z). We refrmulate the prblem (3)-(8) t ncrprate the phreatc surface explctly. Wth an admssble frm fr r/(x) the flw dman may be dvded nt a saturated zne tls =

4 ._ 1396 TRITSCHER ET AL.: QUASI-STEADY SPECIFIC YIELD E, 10 c 0._ -10,l.-- I I I I ' I I ' _ ß I n the unsaturated zne. Wth ur assumed expnental relatnshp between K and h, the Krchhff transfrmatn t = K dh = K/a (14) h n (13) yelds the well-knwn lnear equatn >'0.15._ 0.10 n 0.05 Z : (x' z) , 5 I I I 0 15 I 20 I 25 _. 0 -]L...,..., '-,..,_.,,,,, ' ' ' ß '., ß -! - I -r---r-,--.1! ' ' ' ' "";... r'-'r-.a... * ' ' ' ' "t- ' - ' - -' - / /l""r'"r'"r'"r'-'r' % _.... y,. "'"'; '-,,.-.,,., 1,..,, J //.=.,=.!! ' I,..... j..,. -,,... 2 HI', u'";'" '-'a.*.-,-.-,.-.,...,, ';"-r...,.-.r-... /L..,......,_,,, ß...,.-.-,...,...a,. J/, ;"'c'"'... q.-. ""' ','" "r.- r '5, _ ' >' I I I I 0.10 n _.. I I I I I 2-3-,,, 'j"''"'"t... "k'... ß... }......,....,... '.:':r.::c'".c" F')"" K'"œ'" r... 'r'-q.-.,.-.'.-.'.-. ', '" " U' '" "" "'... '... ß ,.-....a...a I Fgure 3. Graphs f the lcal specfc yeld fr an aqufer that s almst full r almst empty. Sld, dtted, and dashed lnes ndcate actualcal specfc yeld, lcal specfc yeld fr assumed purely vertcal flw, and quas-statc lcal specfc yeld, respectvely. Fr cmparsn we dsplay the nfltratn- evapratn rate and the flw regmes. Shwn are nrmalzed streamlnes and msture cntent cntur curves. Dashed curves dente the unsaturated zne. The msture cntent dvsns are cm3/cm 3 unts. The phreatc surface s the lwermst msture cntent cntur curve and s shwn sld. 12 >'0.15, I 15 20, 215 {(x, z) ' 0 -< x -< 1 andfb _> z --> r/} and an unsaturated znelu = {(x,z)' 0-<x-< 1 andr/_>z_>f'}.wththe abve parttn, equatn (3) yelds V2H = 0 (x, z) 6 ls (12) fr the gvernng equatn n the saturated zne and yelds the steady state Rchards' equatn ak - :0 (x,z)l:u V. (KVh) z 10 ' 2'0 30 ' 4'0 50 ' 60 ' 70 ' Fgure 4. Lcal specfc yeld fr an ncrease n aqufer depth r an ncrease n aqufer length. The prprtn f sl under nfltratn s held cnstant at 50%. Otherwse, the pctral representatn s the same as n Fgure 3.

5 ._ -, TRITSCHER ET AL.: QUASI-STEADY SPECIFIC YIELD 1397 The dmensnless dependent varable/, cmmnly referred t as the matrc flux ptental, s related t the stream functn by [Raats, 1970] Ox = - zz + al O = Ox' (!6) We specfy the bundary cndtns acrss the bubblngpressure surface dvdng the unsaturated and tensnsaturated znes. The ttal hydraulc head must equal the elevatn plus the bubblng pressure alng the bubblng-pressure surface: H(x, *l(x)) = -*l(x) - h. (17) , >' _ ' 0.08 _ ' 2 5 Hence ur cnstrant (11) s satsfed. Addtnally, the sl water cntent tends tward saturatn t yeld a cnstant matrc flux ptental: /x(x, r (x))= 1/a. (18) Fnally, we requre cntnuty f stream functn: lm ½(x, z)= lm ½(x, z). (19) Z- }- Z- /+ These bundary cndtns wth ur specfcatn f n flw acrss the basement (equatn(7)) cmplete the varatnal frmulatn. We nte that the cntnuty f stream functn (19) and the cntnuty f ptental (17)-(18) are suffcent guarantee cntnuty f the Darcan flux vectr acrss the bubblng-pressure surface [Trtscher et al., 1998]. We have transfrmed the nnlnear bundary value prblem (3)-(8) t a frm that prvdes lnear gvernng equatns and bundary cndtns fr each zne f the aqufer. By a Furer seres technque prevusly appled t Laplace's equatn by Read and Vlker [1993] and Read [1993] and t quas-lnear flw by Read and Bradbrdge [1996], a seres frm f the slutn may be btaned by classcal separatn f varables. We present an utlne f the slutn n the appendx Nnunqueness f the Phreatc Surface _.. 0 'q --!-4-:,. ', ', ',...,'"'-4";'": ,-" /I! % % \ '- ',... '-,, _'"/"-.,.,.&,, m ". "3'" E 11 \ \ ',,'"',... :... *' 2 ' ''' ' ; ; ' 4._e >' 0.12._ Calculatns have shwn that fr each specfc bundary value prblem a famly f phreatc surfaces exst, each wth ts Fgure 5. Graphs f lcal specfc yeld fr an ncrease n the wn stream functn slutn. We fnd that there s suffcent nfltratn rate. Shwn are lcal specfc yeld fr the cases freedm n the slutn t allw ne pnt n the phreatc where the aqufer dmensns are the same as n Fgure 3 and surface t be specfed. Alternatvely, we may specfy the ttal fr a case when the aqufer depth s greater. Otherwse, the water cntent, whch, because f ur assumed nnhysteretc pctral representatn s the same as n Fgure 3. sl, s n a ne-t-ne crrespndence wth the lcatn f the phreatc surface. Ths may be cmpared wth purely uns turated flw, wheren the matrc flux ptental s nt unquely specfc yeld n the aqufer gemetry and n recharge cnddetermned and that nfntely many cmpletely unsaturated tns. Heurstcally, the extreme ends f the range f sl types msture dstrbutns exst [Read and Bradbrdge, 1996]. wll yeld relatvely smple results. Fr sandy sls the band Analgus t the case f saturated-unsaturated flw, addwhere the sl msture cntent s hghly varyng s relatvely tnal specfcatn f the msture cntent at ne pnt n the narrw, and typcally the sl surface s dry. The specfc yeld dman s suffcent guarantee unqueness [Read and Bradn ths case becmes essentally that fr a deep water table and brdge, 1996]. Ths nnunqueness permts a quas-steady flw hence assumes a value near the maxmum, regardless f the t be used fr the analyss f specfc yeld fr tw-dmensnal flw regme. Cnversely, an extremely fne clay can supprt flw. Fr fxed and balanced flux bundary cndtns we may nly very lw recharge rates befre becmng cmpletely satcmpare the change n water table depth wth the change n urated. Hence, n mst recharge stuatns the aqufer s practhe ttal water cntent as the slutn changes frm ne steady tcally full and the specfc yeld must be near zer. state t anther. Fr ur medum sl we chse a slt lam (GE3) frm Resenauer [1963]. Ths s chsen because the saturated cn- 3. Applcatn ductvty s apprxmately n the mddle range at 4.96 cm/d. We gve a detaled analyss fr a medum texture sl snce Addtnally, the assumed cnductvty functn (4) yelds a ths wll gve a far ndcatn f the degree f dependence f clse ft t the expermentally determned data. Graphs f the _ 0.00,, e ,-.-,....,....-.m.....-, "' '-,,,,.. ß -...",-,... '-.-.-;g--,- '...,-.,l,-.... T......,, ß, ß,,-:::.';.-;... :::::::::::::::::::::::::: I 2

6 1398 TRITSCHER ET AL.: QUASI-STEADY SPECIFIC YIELD = >' I mean phreatc depth (m) Fgure 6. The regnal specfc yeld fr varus aqufer dmensns and nfltratn rates related t mean phreatc depth. Fr cmparsn, the dash-dtted curve s the yeld btaned by assumng the water s statc and the phreatc surface s hrzntal. Symbls are translated as fllws: plus sgns, r, = 15 cm/yr, D, = 2.5 m,l, = 25 m; sld crcles, r, - 15 cm/yr, D, - 5 m,l, = 25 m; squares, r, = 15 cm/yr, D, = 12.5 m, L, = 25 m; trangles, r, - 15 cm/yr, D, = 5 m, L, = 75 m; sld damnds, r, cm/yr, D, = 5 m,l, = 12.5 m; pen damnds, r, = 110 cm/yr, D, - 5 m,l, = 25 m; and pen crcles, r, = 110 cm/yr, D, = 12.5 m, L, = 25 m. msture cntent and hydraulc cnductvty, wth ur quaslnear ft, are shwn n Fgure 2. Because f the nnunqueness f the slutn fr the water cntent, we are free t specfy the depth (x) f the water table at ne lcatn, fr example, at x - 0, wthn sme allwable range f values. The bundary cndtns and flw equatns wll then unquely determne the water cntent 0 (x, z) and the depth f the phreatc surface (x) at all ther lcatns. At a gven value x f the hrzntal crdnate, the ttal water vlume per unt crss-sectnal area s,(x) = O(x,, z,) dz,. Ntnally, the specfc yeld s regarded as the depth f water (- A, ) remved frm a sl when the water table lwers by a unt length AT,. Hwever, the chce f length unt s smewhat arbtrary. A natural unambguus defntn f lcal dmensnlesspecfc yeld s the lmt f (- A, )/A,, d, d SY(x) = d q, d q where -,a, and =,a,. In practce, we need t evaluate ths dervatve numercally by cmparng the apprxmate phreatc surface lcatns n dfferent slutns. These numercal evaluatns have sme errrs, and ths, alng wth the small errrs n water table lcatn, may explan the small rpples evdent n the functn SY(x) f Fgures 3 and 4. Fgure 3 shws the specfc yeld fr a water table that s just under the sl surface when the aqufer s almst full, and separately fr a water table that s near the basement when the aqufer s almst empty. We cmpare these yelds wth thse predcted by smpler mdels n rder t gauge the effect f any lateral cmpnent f water flw. The smplest mdel s t assume that the water n the clumn abve the phreatc surface s statc. Ths s equvalent neglectng varatns n pressure dstrbutn due t water mvement, and the lcal specfc yeld s btanable drectly frm the msture release curve. Ths s a glbally ne-dmensnal mdel wth unfrm zer water flux, beng the average f the flux at the surface f the twdmensnal regn. Wth ths smplfcatn the relatve errr frm the actual specfc yeld s less than 13%. The errr tends t ncrease as the water table falls. T avd any ambguty, we cmmenthat n ths mdel and n any ther smpler mdel, we stll calculate the pstn f the phreatc surface by the full saturated-unsaturated flw mdel; t s nly n the specfc yeld calculatn that we make smplfyng apprxmatns. The next smplest mdel s t assume that the water mvement abve the phreatc s purely vertcal. Ths s a lcally ne-dmensnal mdel n whch the nnzer unfrm vertcal flux agrees wth that mpsed at the surface f the twdmensnal regn. In effect, ths allws fr a smple dependence n the nfltratn and evapratn rate. Inspectn f the lcal specfc yeld graphs n Fgure 3 shws that there s neglgble dfference frm the actual specfc yeld. Ths ndcates that the lcal pressure dstrbutn fr ur tw-

7 TRITSCHER ET AL.: QUASI-STEADY SPECIFIC YIELD 1399 dmensnal flw s nt far frm the lcal dstrbutn fr purely vertcal flw, a stuatn that seems reasnable, snce t ß, rase water aganst gravty typcally requres a greater pressure = dfference than s needed t mve the water hrzntally. In Fgure 4 we shw the effect f ncreasng the aqufer n e depth r ncreasng the aqufer length. Increasng the depth f ß ; the aqufer tends t flatten the water table, and the graph f the specfc yeld shws ths effect. Hwever, an ncrease n the ß.2- (a) aqufer length causes a larger varatn n the depth t the. - I I I I water table, and we have a crrespndng change n the lcal specfc yeld curve. Our smpler mdels stll prvde gd mean phreatc depth (m) estmates f the actual lcal specfc yeld. We cmment that ne-dmensnal studes have shwn that fr any prescrbed evapratn rate there s a maxmum depth f the water table after whch that rate f evapratn can n lnger be man- >'0.06 taned [Gardner, 1958; Phlp, 1969]. Physcally, the cnductvty rate near the surface becmes s lw that n amunt f 004 Q. ß suctn can drve the water at the requested flw rate. In ur frmulatn we have assumed a cnstant evapratn rate t smplfy the prblem. Hence there s a maxmum depth fr 0.02 whch ur slutn s vald. Fr example, the phreatc surface (b) presented fr the deep aqufer f Fgure 4 s near the max I I I I I I mum depth beynd whch ur frmulatn s n lnger vald Fnally, we ncrease the nfltratn rate. Fgure 5 shws mean phreatc depth (m) graphs f lcal specfc yeld fr the same aqufer dmensns as n Fgure 3 r fr the deep aqufer f Fgure 4. The nfltratn Fgure 7. A cmparsn f regnal specfc yeld wth smrate s ncreased t near the maxmum that can be sustaned fr pler mdels. Sld, dtted, dashed, and dash-dtted lnes dente actual regnal specfc yeld; yeld calculated frm assumed purely vertcal flw; yeld calculated frm quas-statc lcal specfc yeld; and yeld btaned wth n water flw, the water s assumed statc, and the phreatc s hrzntal, respectvely. Fr ur cmparsn we use aqufer cndtns fr whch the statc mdel was apprachng nadequacy: (a) r, = 15 cm/yr, D, = 5 m, L, = 75 m (see Fgure 4 lwermst aqufer fr an example flw regme) and (b) r, = 110 cm/yr, D, = 5 m, L, = 25 m (see Fgure 5 upper aqufer fr an example flw regme). a vald slutn, wth the restrctn that the phreatc des nt ntersect the sl surface. In each case the actual lcal specfc yeld s apprxmated well by the smpler mdel that assumes purely vertcal flw. Hwever, the mdel whch neglects the nfltratn rate s substantally n errr, the relatve errr beng as hgh as 35%. Fr tw-dmensnal flw we have shwn that lcally the specfc yeld may be strngly nfluenced by the water table depth and mldly dependent n the nfltratn rate f the nfltratn rate s hgh. Hwever, t can be reasned that beynd sme very great depth, further changes n water table depth wuld cease t have any apprecable effect n the specfc yeld. Fr the smple gemetry cnsdered, a lateral cmpnent f flw has been fund t have an nsgnfcant effect n the lcal specfc yeld, and the mdel that assumes lcally purely vertcal flw adequately estmates the specfc yeld. Althugh lcal specfc yeld gves an ndcatn f the mvement f the lcal water table as water s remved r added, the verall specfc yeld f the aqufer remans t be determned. T address ths, let us defne the vlume f water released per unt declne n the mean water table depth dvded by the area f the aqufer as a rudmentary measure f the verall specfc yeld f the aqufer, and let us label ths as the regnal specfc yeld. The dvsn by the aqufer area prvdes a nndmen- mdel and cmpare these wth mre sphstcated mdels, namely, the yeld calculated frm assumed purely nedmensnal vertcal flw and the yeld calculated frm quasstatc lcal specfc yeld. The regnal specfc yeld btaned frm the assumed purely vertcal flw mdel gves excellent results. The next clsest s the yeld calculated frm quas-statc lcal specfc yeld. Hwever, the yeld calculated frm quasstatc lcal specfc yeld may nt be much f an mprvement ver the smple statc mdel. It appears that the assumed purely vertcal flw mdel gves an excellent estmate f the actual lcal and regnal specfc yelds. Hwever, perhaps there exst aqufer gemetres where ths n lnger s the case. As the drectn f the vertcal snal unt and allws a cmparsn amng aqufers f dffer- cmpnent f flw has a mderate bearng upn the specfc ng area. yeld, we attempt t frce sme f the flw under the evap- Fgure 6 shws the regnal specfc yeld fr varus aqufer ratng surface t be dwnward. T acheve ths effect, we have dmensns and nfltratn rates as the mean phreatc depth s used a snusdal basement and a hgh nfltratn rate. Fgure lwered. We cmpare these t the yeld btaned by assumng 8 shws the aqufer prfle, flw regme, and lcal and regnal the water t be statc and the phreatc surface t be hrzntal. specfc yelds fr an aqufer that has a mre cmplcated Fr mst nfltratn rates and aqufer dmensns ths smple unsaturated-zne flw pattern than the smple canncal gemdel gves a surprsngly gd estmate f the regnal spe- metry. We have chsen the ampltude f the basement depth cfc yeld. It s nly when there s a large varatn n the water t be near the maxmum allwable fr tlerable bundary errrs. table depth that the errr may be unacceptable. Ths ccurs fr There appears t be sme verestmate n lcal specfc yeld hgh nfltratn rates r lng thn aqufers. In Fgure 7 we take by the assumed purely vertcal flw mdel nly where vertcal the flw regmes that have the mst devatn frm the statc transects cntan a regn f dwnward flw beneath the sur-

8 1400 TRITSCHER ET AL.: QUASI-STEADY SPECIFIC YIELD.12 '.'E 0.08 m I I I I % 0.08-,, m '0 ß 04- c ø ß I I I I I I I mean phreatc depth (m) Fgure 8. Lcal specfc yeld, flw regme, and regnal specfc yeld fr an aqufer gemetry that yelds a mre cmplcated unsaturated-zne flw pattern than the smple canncal gemetry. The chsen basement gemetry frces sme f the flw under the evapratng surface t be dwnward. The legend fr the lcal specfc yeld and flw regme s the same as n Fgure 3. The legend fr the regnal specfc yeld s the same as n Fgure 7. The nfltratn parameter r. s 95 cm/yr. face where evapratn s ccurrng. Hwever, ths effect s barely sgnfcant even when cmpared wth pssble errrs n the specfc yeld calculatn. We cnclude that we have nt yet fund a flw regme fr whch the assumed purely vertcal flw mdel s nt satsfactry as a gd estmatr f the specfc yeld. 4. Cnclusns Fr tw-dmensnal flw we have demnstrated that lcally the specfc yeld may be strngly nfluenced by the water table depth and mldly dependent n the recharge rate f that rate s hgh. Fr the smple gemetry cnsdered, a lateral cmpnent f flw has been fund t have an nsgnfcant effect n the lcal specfc yeld. Fr a hmgeneu sl a mdel that assumes lcally purely vertcal flw s mre than adequate as an estmatr fr the specfc yeld. Perhaps a mre mprtant nfluence n specfc yeld wuld be sl hetergenety. Ths nfluence may be nvestgated n the future by technque smlar t thse emplyed here. Fr the verall yeld f an aqufer we fnd that the smplest mdel, where the flw thrugh the sl s neglected,.e., where the water s statc and the phreatc surface s hrzntal, gves a reasnable ndcatn f the actual specfc yeld fr mst nfltratn rates and aqufer dmensns. Hwever, f the nfltratn rate s hgh r the aqufer s partcularly lng, then the yeld btaned frm an assumed purely vertcal flw, presuppsng that the phreatc depth s accurately knwn, gves an excellent estmate f the actual specfc yeld. In ur frmulatn we have emplyed a bundary cndtn f cnstant evapratn rate. Ths n turn has placed a restrctn n the allwable depth f the phreatc surface, and hence ur specfc yeld cvers mst f the range pssble but falls shrt f the theretcal maxmum. It s suggested fr future wrk that a mre realstc radatn-type bundary cndtn fr evapratn at the sl surface be ncrprated nt the slutn. Ths wuld ease the restrctn n the depth f the phreatc surface, whch wuld braden the specfc yeld range avalable. Appendx We present here a seres slutn fr the stream and ptental functns. The prcedure fr the dervatn s dentcal t that f Trtscher et al. [1998]. The slutn fr the stream and ptental may be presented as q,(x, z) = [-A n snh(n rrz) + B n csh(n,rz)] x sn (nrrx)/csh (nrrd) (x, z) Ils e z/2 [-Cn snh (y,z) + D n csh (ynz) ] x sn (nrrx)/csh (ynd), (x, z) Iln H(x, z) -- Aø - n l [An csd (g/qtz) -- B n snh (n,rz)] where x cs (nrrx)/csh (nrrd) (x, z) Ils In (a )/a - h0 - z (x, z) Iln, (A2) I (X, z) = C eø - e ø /2 (l/n,r)[(acn/2- 'YnDn) wth ß snh (7,z) + (7 Cn- adn/2) csh ('¾nZ)] ß cs (nrrx)/csh (7nD), n--- a2/4 + n 2*r2, (A3) (A4) and An, Bn, Cn, D n are the slutn f the fllwng system f lnear equatns. The ceffcents fr the saturated zne are gven by j=l nt- An -- -kn, A 0 -- k + t On t n -': ' (AS) (A6)

9 TRITSCHER ET AL.' QUASI-STEADY SPECIFIC YIELD 1401 Bn-- E k;7qta, (m7) where k h, k?*, and kn are (cnstant) expansn ceffcents. These are gven by the fllwng relatns: snh (n rf b) sn (n rx)/csh(n rd) = kjn q' csh ( rf b) sn ( rx)/csh ( rd), (A8) snh (TnT}) COS (n 'x)/csh (Tn D) = k n a + E ku? csh (T'l]) COS ( 'x)/csh (TnD), (A16) e "'v2= k + kn' csh(tnrl) cs(n'n'x)/csh(tnd), (A17) snh (n r,1) cs (n 'x)/csh (n 'D) =k[n ah "}- E kjff h csh ( "1) cs ( 'x)/csh ( 'D), -*l(x) - h0 = k + kn csh (n '*l) cs(n'n'x)/csh (n'n'd). The ceffcents fr the unsaturated zne are gven by (A9) (A10) e- 'V2/t = k + k2 csh (TnT}) COS (n 'x)/csh (TnD). (A18) Krkham and Pwers [1972] r Read [1993] detal Gram- Schmdt rthgnalzatn, and Read [1993] detals least squares methds t calculate the expansn ceffcents. T prvde reprducblty f the fgures, we detal the numercal mplementatn fr the slutns. As gven by Trtscher et al. [1998], we mnmze an alternatve functnal t reduce errrs frm seres truncatn, as t s pssble that the phreatc s s chsen that the functnal (10) s mnmzed at the expense f the bundary errrs. We avd ths prblem by ncrpratng the bundary errrs explctly, namely, -k - E knq'k;na Tn/(rt qr) = -k C + E E k q'k?t/('n') + k na øzn/(2n'rr) Cn, ) (All) F( (x)) = F( (x)) + Wl l( (x)) -I'- W2 2( (X)) -I- W3 3(T}(X)) -I- W4 4(T}(X)), (A19) where e( q(x)) are rt-mean-square (rms) bundary errrs: { 0L }1/2 :1-- L-1 [½(X, fa)]2 dx, (A20) -k; + kn*a/(2n r) - kq'kn T/(qr ) -- ' 'j"nj r. (j'n') C, j=l -- kn C- TnCn/(t't'rt ) (A12) :2 = L-1 [H(x,,r}) -- ]2 dx, (A21) L e3 = {(L - s(t})) -1 [lm ½(x, z) - lm ½(x, Z)] 2 dx} 1/2, (.1) z---> - z---> + (A22) Dn = kn ' - E k;7 C, (A13) wth the expansn ceffcents ktn q', kn*, kn.av., kn, and kn, gven by snh (TnT}) sn (n 'x)/csh (Tn D) = ktn q' csh (TIT) sn ('n'x)/csh (%D), lm e-' 'v2½(x, z) z-->*l- = k csh (TnT}) sn (n 'x)/csh (TnD), (A14) (A15) 4 = (m- S(T})) -1 [#,(X,'1)-- 1/C ] 2 dx, (A23) and w are weghts. We chse 2 fr the weghts W and calculate the expansn ceffcents requred n the seres slutn prcedure by a least squares methd [Read, 1993]. We mnmze the new functnal by Rtz's methd [El'sgl'ts, 1961] wth cubc splnes fr the bass functns, and we adjust the knt pnts by a Nelder-Mead [1965] mnmzatn scheme. Fve equally spaced splne segments were chsen fr the ndes f the phreatc surface. We specfed 10 seres terms each n the saturated and unsaturated znes, except fr Fgure 8, where 20 seres terms fr the saturated zne were used. Our specfc yelds were calculated by furth-rder fnte dfferences. The abscssae spacng n the fnte dfferences were frm t 0.01 nndmensnal unts. In dmensnal unts ths crre- spnds t a dfference n the water table depth by apprx- 1 mately m.

10 1402 TRITSCHER ET AL.: QUASI-STEADY SPECIFIC YIELD Acknwledgments. We thank J. H. Knght fr helpful cmments and the annymus referees fr cnstructve crtcsms. We als gratefully acknwledge the fnancal supprt prvded by the Australan Research Cuncl, Department f Emplyment, Educatn, Tranng and Yuth Affars, Canberra, Australa. References Bear, J., and A. Verrujt, Mdelng Grundwater Flw and Pllutn, D. Redel, Nrwell, Mass., Chlds, E. C., The nnsteady state f the water table n draned land, J. Gephys. Res., 65, , El'sgl'ts, L. E., Calculus f Varatns, Pergamn, Tarrytwn, N.Y., Everett, L. G., L. G. Wlsn, and E. W. Hylman, Vadse Zne Mntrng fr Hazardus Waste Stes, Nyes Data Crp., Park Rdge, N.J., Fetter, C. W., Appled Hydrgelgy, p. 376, Prentce-Hall, Englewd Clffs, N.J., Freeze, R. A., and J. A. Cherry, Grundwater, Prentce-Hall, Englewd Clffs, N.J., Gardner, W. R., Sme steady state slutns f the unsaturated msture flw equatn wth applcatn t evapratn frm a water table, Sl Sc., 85, , Gardner, W. R., and D. I. Hllel, The relatn f external evapratve cndtns t the dryng f sls, J. Gephys. Res., 67, , Gllham, R. W., The capllary frnge and ts effect n the water-table respnse, J. Hydrl., 67, , Krkham, D., and W. L. Pwers, Advanced Sl Physcs, Wley- Interscence, New Yrk, Nelder, J. A., and R. Mead, A smplex methd fr functn mnmzatn, Cmput. J., 7, , Phlp, J. R., Evapratn, and msture and heat felds n the sl, J. Meterl., 14, , Phlp, J. R., Thery f nfltratn, Adv. Hydrsc., 5, , Phlp, J. R., Nnunfrm leachng frm nnunfrm steady nfltratn, Sl Sc. Sc. Am. J., 48, , Pwers, W. L., D. Krkham, and G. Snwden, Orthnrmal functn tables and the seepage f steady ran thrugh sl beddng, J. Gephys. Res., 72, , Pullan, A. J., The quas-lnear apprxmatn fr unsaturated prus meda flw, Water Resur. Res., 26, , Raats, P. A. C., Steady nfltratn frm lne surces and furrws, Sl Sc. Sc. Am. Prc., 34, , Raylegh, Lrd (J. W. Strutt), The Thery f Sund, 2nd ed., Dver, Mnela, N.Y., Read, W. W., Seres slutns fr Laplace's equatn wth nnhmgeneus mxed bundary cndtns and rregular bundares, Math. Cmput. Mdell., 17, 9-19, Read, W. W., and P. Bradbrdge, Seres slutns fr steady unsat- urated flw n rregular prus dmans, Transp. Prus Meda, 22, , Read, W. W., and R. E. Vlker, Seres slutns fr steady seepage thrugh hllsdes wth arbtrary flw bundares, Water Resur. Res., 29, , Resenauer, A. E., Methds fr slvng prblems f multdmensnal, partally saturated steady flw n sls, J. Gephys. Res., 68, , Rekerk, H., Influence f slvcultural practces n the hydrlgy f pne flatwds n Flrda, Water Resur. Res., 25, , Stewart, J. W., Water-yeldng ptental f weathered crystallne rcks at the Gerga Nuclear Labratry, n Gelgcal Survey Research, U.S. Gelgcal Survey prfessnal paper, B106-B107, Trtscher, P., W. W. Read, P. Bradbrdge, and J. H. Knght, Steady saturated-unsaturated flw n rregular prus dmans, Preprnt 4/98, Schl f Math. and Appl. Stat., Mchael Brt Lbr., Unv. f Wllngng, Wllngng, N.S.W., Australa, Van Genuchten, M. T., A clsed-frm equatn fr predctng the hydraulc cnductvty f unsaturated sls, Sl Sc. Sc. Am. J., 44, , Zaradny, H., Grundwater Flw n Saturated and Unsaturated Sl, A. A. Blkema, Rtterdam, Netherlands, P. Bradbrdge and P. Trtscher, Schl f Mathematcs and Appled Statstcs, Unversty f Wllngng, N.S.W. 2522, Australa. (phl_bradbrdge@uw.edu.au; peter_trtscher@uw.edu.au) W. W. Read, Department f Mathematcs and Statstcs, James Ck Unversty, Twnsvlle, Queensland 4811, Australa. (wayne.read@jcu.edu.au) (Receved Nvember 2, 1998; revsed Nvember 15, 1999; accepted Nvember 16, 1999.)