Derived Operating Rules for Allocating Recharges and Withdrawals among Unconnected Aquifers

Transcription

1 Derved Operatng Rules for Allocatng Recharges and Wthdrawals among Unconnected Aqufers Davd E. Rosenberg 1 and Jay R. Lund, M.ASCE 2 Abstract: Sx balancng rules are derved to nform short-term drawdown and recharge of water n multple, unconnected aqufers. Management objectves are: 1 mnmzng costs; 2 maxmzng duraton of operaton; and 3 maxmzng accessblty as a tradeoff between maxmzng nstantaneous wthdrawal rate and the duraton to sustan wthdrawals. Engneerng optmzaton formulatons use ether a specfed target delvery rate for wthdrawals or avalable surface water supply to recharge. Aqufers are modeled as separate, sngle-celled basns wth lumped parameters representng key physcal, nsttutonal, and fnancal characterstcs. Each formulaton s solved analytcally for the case where constrants are nonbndng. Solutons are explaned as operatng rules. Two examples confrm the analytcal solutons. The results show how cost characterstcs, fracton of recharged water avalable for wthdrawal fractonal recovery, ntal storage, maxmum recharge and pumpng rates, and uncertantes regardng the future avalablty of water for extracton nfluence recharge and wthdrawal decsons. DOI: /ASCE :125 CE Database subject headngs: Aqufers; Ground-water management; Water dscharge; Water storage; Pumps; Optmzaton. Introducton Water storage for many water supply systems s movng underground. In Calforna, major urban water provders now contract wth nearby and dstant rrgaton and water management dstrcts that overle large aqufers Puldo-Velázquez et al Although these aqufers can meet urban water demands for several years duraton, they often are far from the urban areas and requre extensve water exchanges for delvery. Conjunctve use creates elaborate engneerng problems for water supply and drought response. Even when surplus surface water supples or target delveres are specfed or recommended by economc analyss, the regonal water provder s often challenged wth how best to dstrbute recharges or extractons among the multple unconnected aqufers gven vared physcal and non-physcal characterstcs Fg. 1. The spatal aqufer balancng problems presented n Fg. 1 are remnscent of operatng surface water reservors confgured n parallel Bower et al. 1966; Lund and Guzman 1999; Sand However, managng multple unconnected aqufers dffers n several respects. Frst, aqufer managers can often regulate nflow and wthdrawal through choce of recharge and pumpng facltes 1 Doctoral Student, Dept. of Cvl and Envronmental Engneerng, Unv. of Calforna, Davs, One Shelds Ave., Davs, CA correspondng author. E-mal: derosenberg@ucdavs.edu 2 Professor, Dept. of Cvl and Envronmental Engneerng, Unv. of Calforna, Davs, One Shelds Ave., Davs, CA E-mal: jrlund@ucdavs.edu Note. Dscusson open untl June 1, Separate dscussons must be submtted for ndvdual papers. To extend the closng date by one month, a wrtten request must be fled wth the ASCE Managng Edtor. The manuscrpt for ths paper was submtted for revew and possble publcaton on October 11, 2004; approved on May 26, Ths paper s part of the Journal of Water Resources Plannng and Management, Vol. 132, No. 1, January 1, ASCE, ISSN /2006/ /$ and volumes. Such nflows are generally more certan than reservor nflows. Durng droughts, demand s relatvely constant, and natural recharge s lkely small or trval. Second, aqufer storage often s reflled and drawn down over several years or decades rather than seasons when antcpatng or respondng to droughts. Thrd, recharge and extracton capacty characterstcs, storage losses, and legal, nsttutonal, and other nonphyscal characterstcs of aqufers may constran aqufer operatons. These condtons apply where: 1 drawdowns are small as compared to the saturated thckness of the aqufers; 2 geologc formatons confnng layers or lenses hydraulcally solate aqufers; 3 large dstances separate the aqufers; or 4 the hydraulc response tme s much longer than the plannng horzon so that management for one aqufer does not affect other aqufers. These assumptons reduce stochastc conjunctve use problems Knapp and Olson 1995; Maddock 1974; Phlbrck and Ktands 1998; Provencher and Burt 1994; Rechard 1995 nto steady, determnstc problems that can use lumped aqufer parameters representng physcal and accountng losses; storage, recharge, and extracton capactes; water qualty; cost; and future avalablty to wthdraw water to specfy near optmal engneerng management rules. Ths paper derves operatng rules that allocate steady recharge and wthdrawal for multple ndependent aqufers wth vared hydrogeologcal, fnancal, and nsttutonal characterstcs. We present sx optmzaton formulatons that represent management objectves for: 1 fnancal performance; 2 duraton of operaton; and 3 accessblty as a tradeoff between the nstantaneous wthdrawal rate and the duraton to sustan wthdrawals. We derve analytcal solutons for each case where constrants do not bnd and nterpret the solutons as operatng rules. Two examples verfy the analytcal solutons, extend them, and show ther lmtatons. Because target wthdrawal or recharge quanttes are specfed exogenously, the paper focuses on spatal rules for short-term operaton. Rules for temporal, economc, and multobjectve, dynamc aqufer management are mportant areas for further work. JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006 / 25

2 pror recharges are sunk costs lterally! and not consdered. The cost mnmzaton objectve s expressed by the mathematcal formulaton: Mnmze c W 2 Subject to: Wthdrawals lmted by maxmum pumpng rates, p max volume tme 1 W p max, 3 Wthdrawal rates for a predetermned and relatvely short duraton, t tme, lmted by ntal, operatonal storages, S o volume Fg. 1. Water system balancng for n unconnected aqufers consderng steady surface water supply Q S, steady target delvery rate W T, and aqufer characterstcs such as ntal storage S o and unflled storage capacty K : a wthdrawal extracton problem; and b recharge problem Optmzng Fnancal Performance Fnancal performance management nvolves maxmzng the beneft from extractng water or mnmzng the actual and nsttutonal transactonal costs assocated wth rechargng, pumpng, treatng, conveyng, and delverng water to end users. Wth a downward-slopng demand curve, v (w) $ volume 1, an upward-slopng supply curve, c (w) $ volume 1, and decsons on wthdrawal rate for each aqufer, W volume tme 1, the fnancal objectve s Maxmze v W c W dw 1 W Formulatons are posed, separately, for wthdrawals and recharges and are derved as follows. Mnmze Cost of Wthdrawals For wthdrawals, Eq. 1 reduces to a lnear cost mnmzaton problem when: 1 the amount of water to be delvered from aqufer wthdrawals wll meet a small porton of a sngle large drought demand.e., the regonal water utlty uses aqufer wthdrawals n conjuncton wth other water delveres, urban demand management, and water transfers; and 2 operatonal storages are small as compared to overall aqufer storages.e., wthdrawals cause small drawdowns. Wth these condtons, pumpng costs are proportonal to the pumpng lft, and the end-use value of extractng water becomes fxed and the same for all aqufers. The objectve s to dentfy wthdrawal rates from each aqufer, W volume tme 1, that mnmze the cost of usng groundwater. The total cost of wthdrawal ncludes the per-unt costs, c $ volume 1 of extractng, treatng, conveyng, and securng the rght to access and use the aqufer. Cost characterstcs wll lkely dffer among aqufers, because aqufers can dffer n hydraulc pumpng lfts, extracted water qualty, treatment requrements, and conveyance dstances. Intal storages are gven so W t S o, 4 No negatve wthdrawals, W 0,. Ths lnear program LP s solved by the general wthdrawal rule: Unless lmted by pumpng rates or storage, wthdraw water n order of ncreasng cost, c. Take water from aqufers wth the smallest costs. Ths strategy makes water wthdrawals more costly as a wthdrawal program s sustaned, for example, n response to a drought. However, over a populaton of droughts of uncertan lengths, the rule wll generally mnmze the cost of drought response. Maxmze Expected Fnancal Value of Recharge For rechargng, Eq. 1 s modfed and expanded to nclude a dscountng factor, 1+r t untless, wth an nterest rate, r untless, that relates recharge costs, rc $ volume 1, borne n the present wth use benefts, u $ volume 1, and other costs, c as prevously, n the future, t years, when water s extracted, conveyed, treated, delvered, and used: Maxmze 1+r t u c rc Q 5 Here, recharge decsons to each aqufer, Q volume, should maxmze the value expected from extractng water at a specfed, future tme t. Because ths tme s the same for each aqufer, the dscount factor s assumed to be constant across aqufers and wll often not affect the short-term allocaton of recharges. Therefore, the beneft and cost terms can be aggregated nto a sngle, constant, dscounted, unt net value of storng water n each aqufer, v $ volume 1 : v 1+r t u c rc 6 As n the cost-mnmzng wthdrawal problem, use benefts, u, wll be the same for each aqufer when the future wthdrawals wll meet a small porton of a sngle, large drought demand. The fractonal recovery term untless n Eq. 5 descrbes losses as a fxed fracton of the recharge amount and covers accountng and physcal losses. Recoverablty wll nfluence the volume of water that can be later extracted and delvered. wll be 1 for aqufers where groundwater flows away from the recharge ste. could equal 1 for recharge by n-leu exchanges, but may be less wth nsttutonal accountng losses. The recovery term can also be a put-take rato or rent on aqufer storage mposed by regulators or overlyng landowners. Eq. 5 omts a scarcty rent on recharged water because the regonal authorty s pror economc analyss has specfed the total quantty of water to be recharged see Eq. 8. The problem s to 26 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006

3 engneer the fnancally optmal spatal allocaton of recharges. The problem s fully specfed wth objectve Eq. 5 subject to constrant Eqs. 7 10: Storage capacty on each aqufer, K volume Q K, 7 Total recharges lmted by surface water supply, Q S volume Q Q S 8 Recharges for perod t tme lmted by maxmum recharge rates, r max volume tme 1 Q r max t, 9 The probablty that a fracton untless of total recharges are avalable for future wthdrawal must exceed the target relablty untless P r a Q Q 10 No negatve recharges, Q 0,. The random varable, a untless, ncorporates nsttutonal and physcal rsks for future aqufer wthdrawals. Recharged water may be unavalable later for extracton due to unforeseen regulatory, legal, or water qualty concerns, or lack of avalable capacty to convey the water. Aqufers governed by dfferent enttes and wth dfferent physcal-chemcal characterstcs are lkely to dffer n these rsks. When the dstrbuton of a s known, Eq. 10 can be reduced to a determnstc constrant Tung 1986; Wagner For example, when a takes the Gaussan dstrbuton wth expected avalablty ā untless, standard devaton of that avalablty untless, and standard normal varate Z untless for probablty, Eq. 10 becomes ā Z Q 0 11 The unflled storage capacty, K, s readly calculated from the unsaturated vod space n aqufer Rosenberg Alternatvely, regulators, agences, and local landowners can stpulate unflled capacty wth agreements or by legal precedent. Ths problem s also solved as an LP. The followng general recharge allocaton rule results: Recharge aqufers n order of v, unless lmted by recharge or storage capacty or future avalablty. Recharge water frst to basns wth the hghest dscounted net fnancal value and fracton of recoverable water. As the water avalable to recharge ncreases, the margnal value of storng the water wll decrease. As hgh-valued and large fractonal recovery aqufers fll, lower-valued and less desrable aqufers reman for use. Optmzng Duraton of Aqufer Operatons Optmzng the tme to fll or empty aqufer storage s a second objectve for managng a portfolo of aqufers. Duraton becomes a relevant operatonal objectve when ether the surface water supply avalable for recharge or the target delvery rate from wthdrawals s known or desred. For blendng, regulatory, or operatonal reasons, we assume steady wthdrawal or recharge rates. Formulatons for the recharge and extracton problems follow. Maxmze Duraton of Wthdrawals The objectve s to fnd the steady wthdrawal rates, W volume tme 1, to maxmze the duraton to sustan a specfed, steady, total target delvery rate. Ths objectve may be mportant to sustan operatons through a drought. The nonlnear mathematcal program s Maxmze WD max 12 Subject to: Wthdrawals lmted by maxmum pumpng rates, p max volume tme 1 W p max, 13 Total wthdrawals must meet or exceed a target delvery rate, W T volume tme 1 W W T 14 Wthdrawal duraton, WD tme, for aqufer defned by ntal storage n aqufer, S o volume WD S o /W, 15 Defnton of maxmum feasble duraton for wthdrawal program, WD max tme WD max WD, 16 No negatve wthdrawals, W 0,. Ths nonlnear program balances wthdrawals across all aqufers. When the nonnegatvty and pumpng capacty constrants do not bnd, the program can be solved analytcally for a general balancng rule. Under ths condton, the set of optmal, duratonmaxmzng steady wthdrawals W wll exhaust all aqufers at the same tme, so WD max WD S o /W,. Rearrangng gves S o W W T S o or W S o W T S o or W W T S o S o 17 Ths rule shows that the duraton-maxmzng wthdrawal from aqufer s proportonal to the fracton of the total system water ntally stored n aqufer. Inversng the wthdrawal duraton 1/WD transforms the problem nto a lnear program Rosenberg Mnmze Duraton of Recharge Here, the objectve s to fnd recharges that mnmze the duraton to 1 recharge a specfc, total quantty of water, or 2 fll all aqufers. The former objectve should apply when the amount of surface water s small as compared to unflled aqufer storage. The later objectve apples when avalable surface water s sgnfcantly more than aqufer storage capacty. These two problems are formulated separately. Mnmze duraton to recharge a small volume of water: The objectve functon s Mnmze RD mn Subject to: Storage capacty avalable n each aqufer, K volume 18 Q K, 19 JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006 / 27

4 Total recharges must equal surface water supply, Q S volume Q S Q 20 No negatve recharge duratons, RD tme RD 0, 21 Recharges lmted by maxmum recharge rates, r max volume tme 1 Q /RD r max, 22 Defnton of program recharge duraton, RD mn tme RD mn RD, 23 No negatve recharges, Q 0, The determnstc constrant on future avalablty Eq. 11. When the nonnegatvty, storage capacty, and future avalablty constrants do not bnd, the program can be solved analytcally for a general balancng rule. Under ths condton, the set of optmal, duraton-mnmzng steady recharges Q wll be related to the largest allowable recharge rate of each aqufer, so RD max RD Q /r max,. Substtutng gves: Q r max Q S or Q r max r max Q S r max or Q Q S r max r max 24 Ths rule s that the duraton-mnmzng recharge to aqufer should be proportonal to the fracton of the total recharge rate capacty aqufer can handle. Mnmze duraton to fll all aqufers: The objectve s Mnmze FD mn 25 Subject to: Fll duraton for aqufer, FD tme, s defned by aqufer storage capacty, K volume; fractonal recovery, fracton; and recharge rate, R volume tme 1 FD K 26 R Total recharges are less than steady surface water avalable each perod, R S volume tme 1 R R S 27 Recharges are lmted by maxmum recharge rates, r max volume tme 1 R r max, 28 Defnton of program fll duraton, FD mn tme FD mn FD 29 No negatve recharge rates, R 0,. The fll duraton for each aqufer s a functon of the fractonal recovery Eq. 26 and assumes that losses occur as recharges are made. Ths assumpton should hold when unflled capacty s large, recharge rates are small, and expected duratons are long. Wth nonbndng recharge constrants, optmal steady recharges R should make all aqufers fll at the same tme, FD mn FD K / R. Substtutng gves K R R S K or R R S K / K or R R S K K 30 To mnmze the duraton to fll all aqufers, recharge more water nto aqufers wth larger unflled capactes or smaller fractonal recoveres,.e., aqufers that are most empty or wth the least effcent recharge. Lower fractonal recoveres wll lengthen the fll duraton. Note that the fractonal recovery terms drop out when they are equal across all aqufers. Inversng the fll duraton 1/FD transforms the problem nto a lnear program Rosenberg Maxmzng Accessblty When fllng groundwater storage capacty n wet years, an agency often s unsure about the future demands for water. The agency may want to optmze flexblty to delver water from a portfolo of groundwater storages at ether hgh wthdrawal rates or for a long duraton. A formulaton s presented to smultaneously address the recharge and wthdrawal portons of the problem. Two analytcal solutons are derved and the tradeoff between them s presented. Model Formulaton The bobjectve maxmzes the total wthdrawal rate, W R volume tme 1 plus the duraton of wthdrawals, D max tme weghted by a tradeoff factor, d volume tme 2 Maxmze W R + d D max 31 Subject to: Recharges, Q volume, for a specfed, short perod t tme are lmted by maxmum recharge rates, r max volume tme 1 Q r max t, 32 Maxmum pumpng capactes, p max volume tme 1 lmt wthdrawals, W volume tme 1 W p max, 33 Aqufer duraton, D tme, s defned by ntal storage, S o volume, and fractonal recovery, untless D S o + Q, 34 W Recharges are lmted by remanng storage capacty, K volume Q K, 35 Total recharges are lmted by surface water supply, Q S volume Q S Q 36 Defnton of program wthdrawal duraton, D max tme D max D, 37 Expected wthdrawal rate, W R volume tme 1, s defned by random varables representng future avalablty, a untless 28 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006

5 Table 1. Aqufer Characterstcs for Example No. 1 Physcal Fnancal Insttutonal Aqufer Unflled storage capacty, K Mm 3 Maxmum pumpng rate, p max Mm 3 /mon Maxmum recharge rate, r max Mm 3 /mon Fractonal recovery, fracton Recharge cost, rc $m 3 Use cost, c $m 3 Use value, u $m 3 Mean expected avalablty, ā fracton Standard devaton of avalablty, fracton A B C D W R a W 38 Expected wthdrawal rate must meet target delvery rate, W T volume tme 1, wth target relablty fracton P r W R W T 39 No negatve recharges or wthdrawals, Q 0,W 0,. Future avalablty s a functon of the wthdrawal, Eq. 38, rather than recharge as n Eq. 10. Eqs. 38 and 39 are reduced to an equvalent determnstc form as shown prevously ā Z W W T 40 In ths nonlnear program, both recharge volumes Q and wthdrawal rates W are decson varables. The recharge perod s fxed to tme t, whle the wthdrawal perod D max s assumed to start after recharge s completed. Solvng the nonlnear program determnes the wthdrawal duraton. Selectng a small value for the tradeoff factor d 0d1 yelds recharge and extracton operatons that maxmze nstantaneous wthdrawal capacty, gvng slght preference to operatons that lengthen the duraton to sustan wthdrawals. Conversely, selectng d1 yelds operatons that maxmze the duraton to sustan wthdrawals, gvng slght preference to operatons that ncrease the rate to wthdraw water. Analytcal Solutons Analytcal solutons are derved for cases where the coeffcent d s ether large or small and the nonnegatvty, lmted recharge rate, lmted extracton rate, aqufer storage capacty, and future avalablty constrants do not bnd. Maxmze Instantaneous Wthdrawal Rate W R. When the value of d s small, an analytcal soluton can be derved to maxmze the expected nstantaneous wthdrawal rate W R. Frst, ncrease aqufer wthdrawals to ther maxmum pumpng rates W p max,. 41 Second, confgure recharges so the wthdrawal rates are maxmally sustaned, equalzng wthdrawal duratons for all aqufers D max D S o + Q W S o + Q W 42 Here, the astersk superscrpt represents the optmal value of a decson varable. Substtutng Eq. 41 nto Eq. 42 and rearrangng gves: p max p max S o + Q S o + Q,. 43 Equaton 43 s a set of ndependent equatons that can be solved smultaneously for Q. The soluton suggests rechargng water n aqufer so that the rato of pumpng capacty of aqufer to total pumpng capacty for all the aqufers equals the rato of water recoverable for extracton from aqufer to the total water recoverable from extracton from all aqufers. To maxmze the nstantaneous wthdrawal rate, the rule suggests rechargng more water nto aqufers wth hghest pumpng capactes, lower ntal storages, and lower fractonal recoveres.e., hgher losses. Losses are borne to equalze ratos of recoverable water and to maxmze the capacty for but not necessarly the duraton of subsequent wthdrawal. When fractonal recoveres and expected avalabltes are dentcal across aqufers, Eq. 43 reduces to the Metropoltan Water Dstrct of Southern Calforna s MWD aqufer allocaton rule. The MWD rule equalzes the rato of pumpng capacty to total water storage n each aqufer, p max / p max S o +Q / S o +Q, Tm Blar, personal communcaton, Maxmze Duraton of Wthdrawal D max. A second analytcal soluton apples where the tradeoff coeffcent d s large. To maxmze duraton of steady wthdrawals, all basns should empty at the same tme, so D max D S o + Q W S o + Q W 44 Recharge and extracton decsons are taken sequentally. Frst, wthout knowng the duraton-maxmzng wthdrawal rates for each aqufer W, we observe that the duraton wll be largest when the sum of the wthdrawals s smallest. Therefore, mnmze wthdrawals subject to constrant Eq. 40 on the wthdrawal target W T determned exogenously. Ths substtuton gves: D max S o + Q W T 45 Second, recognze that duraton s maxmzed when the term Q s maxmzed. Q represents recharged water recoverable for extracton. To maxmze the recoverable amount, recharge nto aqufers wth the hghest fractonal recoveres. The duraton-maxmzng recharge rule s: Recharge aqufers n order of, unless lmted by recharge or storage capactes. JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006 / 29

6 Table 2. Addtonal Parameter Values for Example No. 1 Parameter Value Wthdrawal problem Target water delvery rate, W T,Mm 3 /mon 25 Intal storages, S o,af K Recharge Problem Water avalable to recharge, Q S,Mm 3 7 Recharge perod, t, mon 1 Intal storages, S o,mm 3 0 Steady water avalable to recharge, R S,Mm 3 /mon 7 Dscount factor, b, fracton Target avalablty for wthdrawal,, fracton 0.85 Target relablty,, fracton 0.9 Standard normal devate for relablty, Z, fracton Thrd, wth Q known, solve Eq. 45 for the duratonmaxmzng, steady wthdrawal rates for each aqufer. Because the ntal storages and addtonal storage generated from recharge were determned n step 2, Eq. 45 takes the same form as the soluton for Eq. 17. Rearrangng and solvng Eq. 45 for the duraton-maxmzng, steady wthdrawal rate gves W W T S o + Q S o + Q 46 Ths rule makes the duraton-maxmzng wthdrawal from aqufer proportonal to the fracton of total recoverable water stored n aqufer. The rules for wthdrawal Eq. 46 and recharge precedng paragraph represent sequental solutons for recharge followed by wthdrawal decsons. These solutons are smlar to those for maxmzng the wthdrawal duraton Eq. 17. Tradeoff between Solutons The two analytcal solutons frame a tradeoff between wthdrawal duraton and rate. Operatons to maxmze the nstantaneous wthdrawal rate ncur hgher water losses that dmnsh the duraton over whch wthdrawals can be sustaned, and vce versa. The tradeoff wll be most apparent when one group of aqufers has hgh pumpng capactes but low fractonal recoveres whle a second group of aqufers has low pumpng capactes but hgh fractonal recoveres. Solvng the nonlnear program for a range of values for d can also llustrate the tradeoff. Example Applcatons The frst example verfes solutons for the fve derved sngleobjectve operaton rules. A second example demonstrates solutons for the bobjectve accessblty formulaton. Examples were set up n Excel and solved wth the Solver add-n. Example No. 1 Sngle-Objectve Programs We select a portfolo of four aqufers wth dfferent physcal, nsttutonal, and fnancal characterstcs Table 1. Aqufer A has hgh use costs, whle aqufer D has low use costs. We assume aqufers have smlar water qualtes and use values and that extracted water s delvered to a sngle locaton. The parameter values fall wthn ranges of values the Metropoltan Water Dstrct of Southern Calforna and the Santa Clara Valley Water Dstrct commonly use for aqufer storage Puldo-Velázquez et al However, values do not represent specfc aqufers. Table 2 summarzes addtonal parameter values for recharge and extracton. All aqufers are assumed to start full for the wthdrawal problems S o K and start empty for the recharge problems S o 0. Wthdrawals should meet the target delvery rate of 25 mllon cubc meters per month Mm 3 /mon; 20,000 acre-feet per month 20 kaf/mon. 7 Mm 3 6 kaf of surface water s avalable for recharge. We assume an nterest rate of 5% over a plannng horzon of fve years to calculate a dscount factor b Furthermore, we requre 85% of recharged water be avalable for wthdrawal wth 90% relablty. Numercal solutons to the fve sngle-objectve models verfy the analytcal solutons derved prevously Table 3. To mnmze the cost of wthdrawals column 2, aqufers D and C were pumped. These aqufers have the lowest and second-lowest extractve costs Table 1, column 6. Pumpng capacty for aqufer D lmted wthdrawal to 19 Mm 3 mon 1 15 kaf/mon. Remanng delveres were met from aqufer C. When the objectve was to maxmze the duraton of meetng target wthdrawals Table 3, column 3, water was wthdrawn from each aqufer proportonal to the ntal storage n each aqufer. Each wthdrawal was sustaned for 100 months. When the objectve was to maxmze the expected value of recharge, water was recharged nto both aqufers B and D. These aqufers have the hghest and second-hghest dscounted net fnancal value of recoverable water Table 4, column 4. Recharge capacty for aqufer B lmted recharge to 3.7 Mm 3 /mon 3 kaf/mon. Excess surface water was recharged to aqufer D. To mnmze the duraton to recharge 7 Mm 3 6 kaf, recharge each aqufer n proporton to each aqufer s recharge capacty Table 3, column 5. Recharges were sustaned for 0.38 Table 3. Aqufer Balancng Solutons for Fve Sngle Objectve Model Formulatons Wthdrawal problems Aqufer Mnmze cost, W, Mm 3 /mon Maxmze duraton of wthdrawal, W Mm 3 /mon Maxmze expected value of recharge, Q Mm 3 Recharge problems Mnmze duraton to recharge small volume, Q Mm 3 Mnmze duraton to fll all aqufers, R Mm 3 /mon A B C D / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006

7 Table 4. Calculated Net Fnancal Values of Water $ m 3 Aqufer Net extractve use value, u c Dscounted net fnancal value, v,b u c rc Dscounted net fnancal value of recoverable water, v A B C D months n all aqufers to fully recharge the 7 Mm 3 6 kaf. To mnmze the duraton to fll all aqufers wth a supply of 7Mm 3 /mon 6 kaf/mon, recharge each aqufer Table 3, column 6 n proporton to the water needed to fll each aqufer space avalable/fractonal recovery. Aqufer D took the most water 987 Mm 3 /0.921,073 Mm 3, whle aqufer B took the least 247 Mm 3 / Mm months were requred to fll all aqufers and reflects large unflled capactes. Example No. 2 Multobjectve Accessblty We demonstrate solutons to maxmze accessblty usng the aqufer portfolo from example no. 1. However, several parameter values were changed so recharge and wthdrawal decsons could be examned smultaneously and the avalablty and recharge constrants were not ntally bndng Table 5. The accessblty program was solved 20 tmes for values of tradeoff coeffcent d rangng from 0.1 to 100 Mm 3 /mon 2. For all values of d, solutons converged to one of two solutons. A tppng pont between the two solutons was seen at d1.5 Mm 3 mon 2 Fg. 2. The corner soluton that maxmzed the nstantaneous wthdrawal rate Table 6, columns 2 and 3 verfed the analytcal soluton derved for that case Eqs. 41 and 43. The corner soluton that maxmzed duraton of wthdrawals Table 6, columns 4 and 5 resembles the analytcal soluton Eq. 46. However, aqufer B s also recharged because the pumpng rate for aqufer A was constraned. When the tradeoff coeffcent was less than 1.5 Mm 3 /mon 2, water was wthdrawn from each aqufer at maxmum pumpng rates Table 6, column 2. Water was recharged to aqufers A, C, and D n proporton to the pumpng rates and ntal storage Table 6, column 3. No water was recharged to aqufer B because t had the smallest pumpng rate. Aqufer B s preexstng storage Fg. 2. Tppng pont between accessblty solutons that maxmze total nstantaneous wthdrawal rate and duraton of wthdrawals could sustan ts maxmum pumpng rate longer than the other aqufers 16.7 perods. Recharges to aqufers A, C, and D allowed the program to sustan the maxmum wthdrawal rate of 44 Mm 3 /mon 36 kaf/mon for 15.5 months. For tradeoff coeffcent values larger than 1.5 Mm 3 /mon 2, recharge was lmted to aqufers A and B Table 6, column 4 aqufers wth the largest and second-largest fractonal recoveres. Wthout lmts on pumpng rates, the nonlnear program soluton would drect all recharge to aqufer A. However, the maxmum pumpng rate for aqufer A constraned wthdrawal to 9Mm 3 /mon 7 kaf/mon, so excess water was recharged to aqufer B. Wthdrawals were then made n proporton to the water stored n each aqufer Table 6, column 4. Aqufer A had the largest wthdrawal rate because t had the most stored water. Aqufer B had the second largest wthdrawal rate. Aqufers C and D had smaller and dentcal wthdrawal rates, because both aqufers started wth 123 Mm kaf of recoverable storage and no recharge was made to ether aqufer. Total wthdrawals met the target rate of 25 Mm 3 /mon 20 kaf/mon. From the recharges and wthdrawals, the program could sustan wthdrawals for 28.5 months. Fg. 2 shows a dscontnuous tppng pont between the corner solutons, because the objectve functon s lnear wth respect to both the wthdrawal rate and the duraton. Plottng the objectve functon value aganst the tradeoff coeffcent for several dfferent solutons ncludng the two corner solutons presented n Table 6 also dentfes the tppng pont Fg. 3. For values of d much larger or smaller than 1.5 Mm 3 /mon 2, a smooth tradeoff exsts between the corner and ntermedary solutons. However, for values of d near the tppng pont, both corner solutons become superor to the ntermedate solutons. Table 5. Addtonal Parameter Values for Example No. 2 Parameter Value Water avalable to recharge, Q S,Mm Recharge perod, t, mon 200 Intal storage n aqufer A, S o1,mm 3 99 Intal storages n aqufer B, C, D, S o2,3,4,mm Expected avalablty, ā, fracton 1.0 Standard devaton of avalablty,, fracton 0 Target water delvery rate, W T,Mm 3 /mon 25 Target relablty,, fracton 0.5 Standard normal devate for relablty, Z, fracton Tradeoff coeffcent, d, Mm 3 mon Table 6. Two Numercal Solutons to Accessblty Program n Example No. 2 Aqufer Corner soluton that maxmzes wthdrawal rate d1.5 Mm 3 mon 2 Wthdrawals, W Mm 3 /mon Recharges, Q Mm 3 Corner soluton that maxmzes wthdrawal duraton d1.5 Mm 3 mon 2 Wthdrawals, W Mm 3 /mon Recharges, Q Mm 3 A B C D JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006 / 31

8 Fg. 3. Accessblty program objectve functon value plotted versus tradeoff coeffcent for four accessblty solutons; ntermedate solutons represent cases where expected future avalabltes were decreased The ntermedate solutons n Fg. 3 represent duratonmaxmzng solutons when the program was solved wth expected avalabltes, ā, further constraned to 0.9 and 0.7. Lowerng the expected avalablty rases the expected wthdrawal rate requred to meet the target delvery. Rasng the wthdrawal rate lowers the duraton. Thus, varyng expected avalabltes n chance constrant Eq. 40 llustrates a tradeoff between the two corner solutons Fg. 4. Square markers ndcate the corner solutons presented n Table 6 mean expected aqufer avalablty ā1.0 for all aqufers. Other ponts n Fg. 6 show duratons and total wthdrawal rates when the program was solved for dfferent expected avalabltes ā0.9, , 0.6, and Each pont represents a duraton-maxmzng soluton d1.5 Mm 3 mon 2 where each aqufer was assgned the same mean expected avalablty ā 1 ā 2 ā 3 ā 4. As expected avalabltes decreased, larger wthdrawal rates were requred to meet the desred target wthdrawal rate results not shown. Less water was recharged to aqufers A and B and more water was recharged to aqufer D. Water was only recharged to aqufer C when expected avalablty was less than 0.6. Total pumpng rate ncreases wth the largest ncreases n wthdrawals from aqufer D. Aqufer A sustaned a maxmum pumpng rate of 9Mm 3 /mon 7 kaf/mon, and aqufer B reached a maxmum pumpng rate of 7 Mm 3 /mon 6 kaf/mon for avalabltes less than 1.0. As expected avalabltes were decreased, optmal recharges and wthdrawals approached the soluton for maxmzng the wthdrawal rate. No feasble solutons exsted for ā0.55 because the program could not ncrease the total wthdrawal rate above a maxmum pumpng capacty of 44 Mm 3 /mon 36 kaf/mon. Recharges to and wthdrawals from aqufer D were made to ncrease the expected relablty of wthdrawn water. Because aqufer D had a lower fractonal recovery than aqufers A and B, wthdrawals from the aqufer could be sustaned for a shorter tme. Ths relatonshp s represented by the negatvely slopng tradeoff curve n Fg. 4. Despte the tradeoff, recharges to and maxmum pumpng rates from aqufer A were sustaned over all avalabltes, dentfyng aqufers wth large pumpng capactes and hgh fractonal recoveres as the most sutable for wthdrawals when an aqufer manager seeks to maxmze accessblty to stored water as ether duraton or rate of wthdrawal. Conclusons Sx operatng rules were derved to suggest short-term aqufer recharge and wthdrawal decsons to meet fnancal, duraton, and accessblty objectves. The rules are as follows. Fnancal Objectves 1. To mnmze the cost of wthdrawng water, wthdraw water frst from aqufers wth the smallest overall extracton costs. 2. To maxmze the future expected fnancal value, recharge water to the aqufers wth the largest dscounted net fnancal value of recoverable water. Fg. 4. Tradeoff between duraton and nstantaneous wthdrawal rate by varyng aqufer avalabltes 32 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006

9 Duraton Objectves 3. To maxmze wthdrawal duraton, wthdraw n proporton to ntal storage. 4. To mnmze the tme to recharge a small quantty of surface water, recharge n proporton to maxmum recharge rate. 5. To mnmze the duraton to fll all aqufers, recharge n proporton to unflled storage capacty weghted by expected water losses. Accessblty Objectve 6. To maxmze flexblty to meet both large future wthdrawal rates and duratons of wthdrawals, preferentally recharge water to aqufers wth both hgh maxmum pumpng capactes and large fractonal recoveres small storage losses. The operatng rules are based on lumped aqufer characterstcs, exogenously determned total recharge or wthdrawal amounts, and represent stuatons where constrants do not bnd. The formulatons were readly extended and solved numercally to nclude constrants for more complex systems such as wthdrawal capactes, recharge capactes, and uncertantes concernng future avalablty of banked water for later wthdrawal. Further extensons mght nclude aqufers operated n conjuncton wth a surface water reservor, multple reservors, and uncertan surface water volumes avalable for recharge. Addtonal modfcatons are requred to derve temporal, economc, dynamc, or broader, multobjectve operatng rules. Acknowledgments Tm Blar ntroduced the aqufer recharge problem. The wrters also thank Rchard Howtt and Beth Faber for ther comments and suggestons. Notaton The followng symbols are used n ths paper: a random varable representng future avalablty to extract water from aqufer, fracton; ā mean expected avalablty of aqufer, fracton; b dscount factor, untless; c sum of unt costs to extract, pump, treat, convey, and cover nsttutonal, legal, and transactonal expenses to gan access to aqufer, $ volume 1 ; D duraton of wthdrawal from aqufer, tme; D max overall duraton of wthdrawal program, tme; d tradeoff objectve coeffcent, volume tme 2 ; FD duraton to fll aqufer, tme; FD mn overall fll duraton for recharge program, tme; aqufer ndex, 1 n; K unflled, remanng storage capacty of aqufer, volume; p max maxmum extracton pumpng capacty for aqufer ; Q decson on amount to recharge nto aqufer, volume; Q optmal amount to recharge to aqufer, volume; R steady recharge rate nto aqufer, volume tme 1 ; R optmal recharge rate nto aqufer, volume tme 1 ; R S steady surface water avalable for recharge n each perod, volume tme 1 ; RD recharge duraton for aqufer, tme; RD mn overall duraton for recharge program, tme; r max maxmum recharge capacty for aqufer, volume tme 1 ; rc unt cost to recharge aqufer, volume tme 1 ; S o ntal storage n aqufer avalable for extracton, volume; t predetermned duraton of wthdrawal/ recharge, tme; u unt use value of water extracted from aqufer, $ volume 1 ; v dscounted, net fnancal value of storng water n aqufer, $ volume 1 ; W decson on wthdrawal rate from aqufer, volume tme 1 ; W optmal wthdrawal rate from aqufer, volume tme 1 ; W R total expected rate of water wthdrawal from all aqufers, volume tme 1 ; W T total target delvery rate, volume tme 1 ; WD wthdrawal duraton from aqufer, tme; WD max overall wthdrawal duraton for program, tme; Z standard normal devate for probablty, untless; relablty that water should be avalable, fracton; requred fracton of recharged water to be avalable n future, untless; expected fracton of recharge that wll be recoverable for extracton, untless; and standard devaton of expected avalablty, fracton. References Bower, B., Hufschmdt, M. M., and Reedy, W. W Operaton procedures: Ther role n the desgn of water-resource systems by smulaton analyss. Desgn of water resource systems, Harvard Unversty Press, Cambrdge, Mass., Knapp, K. C., and Olson, L. J The economcs of conjunctve groundwater-management wth stochastc surface supples. J. Envronmental Economcs and Management, 283, Lund, J. R., and Guzman, J Derved operatng rules for reservors n seres or n parallel. J. Water Resour. Plan. Manage., 1253, Maddock, T The operaton of a stream-aqufer system under stochastc demands. Water Resour. Res., 101, Phlbrck, C. R., and Ktands, P. K Optmal conjunctve-use operatons and plans. Water Resour. Res., 345, Provencher, B., and Burt, O Approxmatng the optmal groundwater pumpng polcy n a multaqufer stochastc conjunctve use settng. Water Resour. Res., 303, Puldo-Velázquez, M., Jenkns, M. W., and Lund, J. R Economc values for conjunctve use and water bankng n Southern JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006 / 33

10 Calforna. Water Resour. Res., 403, W Rechard, E. G Groundwater-surface water management wth stochastc surface-water supples a smulaton optmzaton approach. Water Resour. Res., 3111, Rosenberg, D Derved operatng rules for storage and recovery n multple, unconnected aqufers. MSc thess, Unv. of Calforna, Davs, Calf. Sand, G. M An analytcal nvestgaton of operatng polces for water-supply reservors n parallel. PhD thess, Cornell Unv., Ithaca, N.Y. Tung, Y. K Groundwater management by chance-constraned model. J. Water Resour. Plan. Manage., 1121, Wagner, H. M Prncples of operatons research wth applcaton to manageral decsons, Prentce-Hall, Englewood Clffs, N.J. 34 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ASCE / JANUARY/FEBRUARY 2006