DAVID EZECHIEL ROSENBERG. B.S.E. (Cornell University) M.S. (University of California, Davis) 2003 THESIS. MASTER OF SCIENCE in

Transcription

1 Derved Operatng Rules for Storage and Recovery n Multple, Unconnected Aqufers By DAVID EZECHIEL ROSENBERG B.S.E. (Cornell Unversty) 1998 M.S. (Unversty of Calforna, Davs) 2003 THESIS Submtted n partal satsfacton of the requrements for the degree of MASTER OF SCIENCE n Internatonal Agrcultural Development n the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Jay R. Lund Rchard E. Howtt Beth A. Faber Commttee n Charge

2 Acknowledgements Tm Blar ntroduced the aqufer wthdrawal problem to Dr. Jay Lund. Dr. Lund presented wthdrawal problem formulaton sketches at a research group meetng durng the summer of At the tme, the problem seemed nterestng, but confounded wth another topc I was pursung as IAD master s thess materal that nvolved cooperatve management of the shared water resources n the Jordan Rver Basn. To me, recharge problems seemed lke an analogous and natural extenson to the wthdrawal problems. In September 2002, I gave the problem formulatons exploratory musngs on successve 24-hour plane rdes between San Francsco and Santago, Chle. My frst set of thanks go to Grupo Taca arlnes and Emmanuelle for provdng the space and envronment to focus ths energy. Fall 2002 dd not seem lke the optmal tme to study cooperatve water management n the Jordan Rver. Whle attendng a free concert n Golden Gate Park, Nna nspred the dea to nstead use aqufers n Jordan as an example to demonstrate the theoretcal formulatons. However, by December, wth no response from aqufer managers n Jordan but the theoretcal formulatons most fnshed, I was left conjurng two, smple, numercal examples to demonstrate potental solutons. I thank Jay for hs tme-and-agan efforts to revew and dscuss all drafts of the problem formulatons and wrte-up. Also for encouragng short-duraton solutons to potentally long task of thess work. Dr. Howtt suggested chance constrants as an elegant and smple way to address the ssue of future avalablty. As always, Beth provded useful suggestons to help clarfy and better communcate the formulatons and wrte-up. --

3 Thanks agan to Daman, La, Aron, and my parents for provdng support, love, encouragement, celebraton, and dstractons through the two-year process. Support provded for a second Master s thess! Can I conjure more for a dssertaton? And fnally, thanks BR for provdng wonderful ncentve late n process to fnsh the pen-ultmate draft. Road-trppng north to Ashland and the Calforna coast was blssful reward. You have been a wonderful, new presence that I hope can extend for longer duraton. --

4 Contents Abstract Introducton Management to Maxmze Net Expected Economc Performance A. Maxmze the beneft of wthdrawals...4 2B. Maxmze the expected economc value of recharge Optmzng Duraton of Aqufer Operatons A. Maxmze duraton of wthdrawals...9 Mnmze Duraton of Recharge B. Mnmze duraton to recharge a small volume of water C. Mnmze duraton to fll all aqufers Maxmzng Accessblty Model formulaton Analytcal solutons Maxmze nstantaneous wthdrawal rate (W R ) Maxmze duraton of wthdrawal (D max ) Tradeoff between solutons Example Applcatons Example #1 (Sngle objectve programs) Example #2 (Mult-objectve accessblty) Conclusons Appendx A. Dervaton of Lnear Programs to Optmze Duraton of Steady Water Supply Appendx B. Solutons to Accessblty Program wth bndng constrants on future avalablty Appendx C. Defnton of Terms References v-

5 Lst of Fgures Fgure 1: Water System Balancng for Multple, Unconnected Aqufers 2 Fgure 2. Shadow prces to mprove avalablty of recharges for future wthdrawal (Program 2B) 24 Fgure 3. Tppng pont between solutons that maxmze total wthdrawal rate and duraton of wthdrawal 26 Fgure 4. Accessblty program objectve functon value versus tradeoff coeffcent for 4 solutons 29 Fgure 5. Tradeoff between duraton and nstantaneous wthdrawal rate by varyng aqufer avalabltes 30 Lst of Tables Table 1. Aqufer characterstcs 20 Table 2. Addtonal parameter values for extracton and recharge problems n Example #1 21 Table 3. Aqufer balancng solutons for economc and duraton-based objectves 22 Table 4. Economc calculatons 22 Table 5. Parameter values for Example #2 25 Table 6. Two numercal solutons to accessblty program n Example #2 27 -v-

6 1 Derved Operatng Rules for Storage and Recovery n Multple, Unconnected Aqufers Abstract Sx optmzaton formulatons and balancng rules are presented to nform management for the drawdown and recharge of storage n multple, unconnected aqufers. Program objectves relate to () economc performance, () duraton of operaton, and () accessblty as a tradeoff between maxmzng nstantaneous wthdrawal rate and the duraton that wthdrawals can be sustaned. Aqufers are modeled as separate, sngle-celled basns wth lumped parameters representng key physcal, nsttutonal, and economc characterstcs. Recharges and wthdrawals are treated separately for the frst two program objectves and sequentally for the thrd objectve. Each program s solved analytcally wth resultng operatng rules for the case where constrants are non-bndng. Solutons are also demonstrated for two smple numercal examples. Results show how economc characterstcs, fracton of recharged water avalable for wthdrawal (fractonal recovery), ntal storage, maxmum recharge and pumpng rates gude optmal recharge and wthdrawal decsons. Uncertantes regardng future avalablty of water for extracton also nfluence decsons. Where hgher relablty s desred, managers should recharge and pump aqufers wth hgher expected avalabltes. However, ncreased relablty wll decrease ether economc performance or duraton that wthdrawals can be sustaned. Lastly, to maxmze flexblty and accessblty, managers should preferentally recharge aqufer(s) wth both large maxmum pumpng capactes and hgh fractonal recoveres. 1. Introducton Water storage for many water supply systems s movng underground. As surface and ground waters have come to be more conjunctvely managed, more elaborate problems n aqufer management for water supply and drought response arse. In Calforna, major urban areas n Southern Calforna (Metropoltan Water Dstrct) and the San Francsco Bay regon (Santa Clara Valley Water Agency) now contract wth rrgaton and water management dstrcts that overle large aqufers (Puldo-Velazquez 2002). These aqufers can provde water storage to meet demands for several years duraton. However, they are often far from the urban water demand area and requre extensve water exchanges to

7 2 delver water durng drought. As llustrated n Fgure 1, the water provder may be challenged wth how best to recharge water nto and extract water out of multple aqufers, whle accountng for varous physcal and non-physcal characterstcs among aqufers. W 1 W 2 W n Target Water Demand, W T S o1 S o2 S on Aqufer 1 Aqufer 2 Aqufer n A. Wthdrawal (extracton) problem Surface Water Supply, Q S Q 1 Q 2 K 2 Q n K n S o1 K 1 S o2 S on Aqufer 1 Aqufer 2 Aqufer n B. Recharge problem Fgure 1: Water System Balancng for Multple, Unconnected Aqufers The aqufer balancng problems presented n Fgure 1 are somewhat smlar to the classcal problem of operatng surface water storage reservors confgured n parallel (Bower et al. 1966; Lund 1996; Lund and Guzman 1999; Sand 1984). But managng

8 3 multple, unconnected aqufers dffers n several key regards. Frst, aqufer managers often can regulate aqufer nflow and wthdrawal through choce of recharge and pumpng facltes and volumes. Aqufer operators are subject to less uncertanty n nflow than operators of surface water reservors. Especally n tmes of drought, 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, legal, nsttutonal, and other non-physcal characterstcs of aqufers may constran aqufer operatons. Ths s especally applcable where () drawdowns are small compared to the saturated thckness of the aqufers, () geologc formatons (confnng layers or lenses) hydraulcally solate aqufers, () large dstances separate the aqufers, or (v) the hydraulc response tme s much longer than the plannng horzon so that operatons n one aqufer do not operatonally affect other aqufers. The above assumptons transform a stochastc conjunctve use problem (Phlbrck and Ktands 1998) nto a steady-state one and narrow the scope to a smple case where lumped parameters can represent the dfferent hydro-geologcal, nsttutonal, and economc propertes of multple, unconnected aqufers. Lumped aqufer characterstcs nclude physcal and accountng losses, storage, recharge, and extracton capactes, water qualty, cost, and future avalablty to wthdraw water. Ths thess derves operatng rules to specfy steady recharge to and wthdrawal from multple ndependent aqufers based on lumped ndvdual aqufer characterstcs. Sx optmzaton formulatons are presented for management objectves based on () economc performance, () duraton of operaton, and () water accessblty (as a tradeoff between nstantaneous wthdrawal rate and the duraton that wthdrawals can be sustaned). For each program, analytcal solutons are derved for cases where constrants

9 4 are non-bndng. Analytcal solutons are restated as operatng rules. Solutons are also demonstrated for two smple numercal examples. In the examples, a chance constrant specfyng future avalablty s tghtened to hghlght tradeoffs between relablty and economc performance or the duraton that wthdrawals can be sustaned. The goal of the work s to suggest how aqufer characterstcs should nfluence the optmal recharge and wthdrawal of multple, unconnected aqufers. 2. Management to Maxmze Net Expected Economc Performance A general management objectve s to maxmze the net expected economc value of operatons. Ths valuaton ncorporates the economc value of extracted water, costs assocated wth rechargng, pumpng, treatment, conveyance to end users, and nsttutonal transacton costs assocated wth the above actvtes. Economc management formulatons can be posed, separately, for wthdrawal or recharge problems. Wthdrawal and recharge rules for maxmzng net economc performance are derved as follows. 2A. Maxmze the beneft of wthdrawals The objectve s to dentfy wthdrawal rates from each aqufer that maxmze the beneft of usng groundwater. Intal storages are taken as gven. For wthdrawal, pror recharge costs are sunk (lterally!) and not consdered. The net economc beneft of wthdrawal s the economc use value of wthdrawn water mnus costs of wthdrawng, treatng, conveyng, and securng rght to access and use the aqufer. Because aqufers can dffer n extracted water qualty and use to whch that aqufer s water can be appled, the net use value of water can dffer among aqufers. Hydraulc pumpng lfts, treatment requrements, and conveyance dstances also wll dffer among aqufers. The followng mathematcal program expresses ths net economc objectve:

10 5 ( u c ) Maxmze Z 2 A = W (1) Subject to: No negatve wthdrawals W 0, (2) Wthdrawals lmted by maxmum pumpng rates W p max, (3) Volume of wthdrawals lmted by ntal storages, W t S o, I (4) Wthdrawals must meet the exogenous water demand rate W = WT (5) Where W s the wthdrawal rate decson for aqufer [volume tme -1 ], u s the unt economc use value of water extracted from aqufer [$ volume -1 ], c s the sum of unt costs to extract, pump, treat to the desred qualty standard, convey, and cover other nsttutonal, legal, and transactonal expenses related to ganng access to aqufer [$ volume -1 ], p max s the maxmum wthdrawal rate for aqufer [volume tme -1 ], S o s the amount stored n aqufer avalable for extracton [volume], t s a predetermned, relevant duraton of wthdrawal [tme], and W T s the total target demand rate [volume tme -1 ]. These demand rate and duraton are fxed exogenously. Here we only want to fnd the most proftable allocaton of wthdrawals among aqufers. The assumpton that untpumpng costs are fxed for each aqufer allows the problem to be solved as a lnear program. The followng general wthdraw rule results, Unless lmted by pumpng rates or storage, wthdraw water n order of decreasng net value, u - c. Take water from aqufers wth

11 6 the largest, postve dfferences between use value and costs. Because hgh-valued water s used frst, water wthdrawals become ncreasngly costly (lower net value) as a wthdrawal program s sustaned, for example n response to a drought. Over a populaton of droughts of uncertan lengths, ths rule wll generally maxmze the expected net beneft of drought response. 2B. Maxmze the expected economc value of recharge For rechargng, the objectve s to dentfy volumes to recharge aqufers that wll maxmze the net, expected economc value of wthdrawng water at a future date. Equaton 1 s modfed slghtly to consder the cost of recharge borne n the present, and that benefts and other assocated costs enumerated for the wthdrawal problem occur n the future. Also, recoverablty and future avalablty of recharged water must be explctly consdered snce these characterstcs wll lkely nfluence the amount of water that can be later extracted and used. Ths problem formulaton s: Z = v λ Q Maxmze 2B (6) Subject to: No negatve recharges Q 0, (7) Storage capacty on each aqufer Q K, (8) Total recharges lmted by surface water supply Q QS (9) Recharges lmted by maxmum recharge rates Q r max t, (10) A fracton β of total recharges must be avalable for future wthdrawal wth target relablty α

12 7 P r a Q β Q α (11) Where Q s the decson on the amount to recharge nto aqufer [volume], λ s the expected fracton of recharge recoverable durng extracton [untless], K s the unflled, remanng storage capacty of basn [volume], r max s the maxmum recharge rate for basn [volume tme -1 ], a s a random varable representng future avalablty to extract water from aqufer [fracton], β s the fracton of recharged water whch s desred to be recovered n the future [untless], and α s the target relablty that the water should be avalable [fracton]. t (duraton) and S o (ntal storage) are as defned prevously. v s the dscounted, unt net economc value of storng water n aqufer [$ volume -1 ] and can be calculated as: ( ) v = b u c rc (12) Where, u and c are values and costs as specfed prevously, rc s unt cost to recharge aqufer [$ volume -1 ], and b s a factor that relates the value of money when water s recharged to the future when water s extracted, conveyed, treated, and sold [untless]. Snce t s typcal for wthdrawal from dfferent aqufers to occur at smlar tmes, the parameter b s assumed to be constant across aqufers. As such, b does not affect the dstrbuton of recharges and v can be treated as a constant. The fractonal recovery term λ lumps accountng and physcal losses nto a sngle factor and expresses losses as a fxed fracton of the recharge amount. λ wll be < 1 for aqufers where groundwater flow drects recharge water away from the recharge ste. λ wll lkely equal 1 for recharge by n-leu exchanges, but may be less f nsttutonal accountng losses are sgnfcant. The nsttutonal losses can also be though of a put / take rato representng rent on groundwater storage.

13 8 The random varable a allows the nsttutonal and physcal rsks of each aqufer to be represented explctly n the formulaton (equaton 11). Recharged water mght not be avalable later for extracton due to unforeseen regulatory, legal, or water qualty concerns, or lack of avalable conveyance capacty to move 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, the chance constrant can be reduced to a determnstc constrant (Tung 1986; Wagner 1969). For example, when a takes the Gaussan dstrbuton, equaton (11) becomes [( a Z σ β ) Q ] 0 α (13) where ā s the expected avalablty of aqufer [fracton], σ s the standard devaton of that avalablty, and Z α s the standard normal varate for probablty α. The unflled storage capacty, K, s lmted by the unsaturated vod space n aqufer, typcally below the root zone of overlyng vegetaton. K can be readly calculated from the aqufer porosty parameter usng: z= e xyus,, K (14) ρ, xyzdz dy dx, x, y z= e, xy, ls Where x and y are the longtudnal and lattudnal domans of the recharge area for aqufer, ρ,xyz s the porosty of aqufer at locaton x,y,z, and e xy us and e xy ls are, respectvely, the upper (.e., ground surface) and lower (.e., groundwater) surface elevatons of the unsaturated depth profle avalable for storng water at locaton x,y n aqufer [length]. The unflled avalable storage capacty also can be lmted by nsttutonal arrangements made wth local or regonal aqufer owners or regulators. Ths problem also can be solved as a lnear program. The followng general recharge allocaton rule results, Recharge aqufers n order of v λ, unless lmted by recharge or

14 9 storage capacty or future avalablty. Water s recharged frst to basns wth the hghest dscounted net economc value and fracton of recoverable water. As the amount of 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. 3. Optmzng Duraton of Aqufer Operatons Optmzng the tme to fll or empty aqufer storage s another objectve for managng a portfolo of aqufers. Duraton can be descrbed as the recharge perod or the duraton of steady aqufer wthdrawal. Objectves and constrants to optmze these duratons are specfc to recharge and extracton problems as follows. For blendng, regulatory, or operatonal reasons, we assume steady wthdrawal or recharge rates. 3A. Maxmze duraton of wthdrawals For the wthdrawal problem, the objectve s to fnd the steady wthdrawals that maxmze the duraton over whch a specfed steady target demand rate can be met. Ths mght represent the duraton of a drought over whch the portfolo can supply a gven delvery rate. The followng non-lnear mathematcal program results: Maxmze WD max (15) Subject to: No negatve wthdrawals W 0, (16) Wthdrawals lmted by maxmum pumpng rates W p max, (17) Wthdrawals must meet or exceed a target demand rate W WT (18)

15 10 Defntons of wthdrawal duraton for each aqufer WD = S o /W, (19) Defnton of maxmum feasble duraton for program WD max WD, (20) Where WD max s the duraton to mantan the wthdrawal target [tme] and WD s the wthdrawal duraton from aqufer [tme]. W (wthdrawal rate), p max (maxmum pumpng rate), S o (ntal storage), and W T (target demand rate) are as defned prevously. Snce each aqufer wthdrawal contrbutes to the target wthdrawal rate, the duraton for the program of wthdrawals wll be constraned by the aqufer wth the smallest duraton of wthdrawal. Ths nonlnear program balances wthdrawals across all aqufers. When the non-negatvty 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, duraton-maxmzng steady wthdrawals (W ) wll exhaust all aqufers at the same tme, so WD max = WD = S o /W,. Algebracally, ths gves the followng wthdrawal rule: S W S o o = or WT W S = o W S o T or W So = (21) W S T o Ths rule says that the duraton-maxmzng wthdrawal from aqufer should be proportonal to the fracton of the total system water ntally stored n aqufer. Program 3A can be transformed nto a lnear program by takng the nverse of the decson varable (1/W ) as derved n Appendx A.

16 11 Mnmze Duraton of Recharge For the recharge problem, the objectve s to fnd the recharges that mnmze the duraton to recharge a specfc quantty of water (3B) or fll all aqufers (3C). The former objectve should apply when the amount of surface water s small compared to unflled aqufer storage. The later objectve apples when avalable surface water s sgnfcantly more than aqufer storage capacty. Formulatons for each problem are: 3B. Mnmze duraton to recharge a small volume of water Mnmze RD mn (22) Subject to: No negatve recharges Q 0, I (23) Storage capacty avalable n each aqufer Q K, I (24) Total recharges must equal surface water supply Q S = Q (25) No negatve recharge duratons RD 0, I (26) Recharges lmted by maxmum recharge rates Q /D r max, (27) Defnton of program duraton RD mn RD, (28) A fracton β of total recharges must be avalable for future wthdrawal wth at least relablty α

17 12 [( a Z σ β ) Q ] 0 α (29) Where RD s the recharge duraton for aqufer [tme], RD mn s the overall recharge duraton for the program [tme], and S o (ntal storage), Q (recharge volume), K (unflled capacty), Q S (avalable surface water), r max I (maxmum recharge rate), ā (future expected avalablty), σ (standard devaton of future avalablty), α (target relablty), and β (target fracton of recovery) are as defned prevously. When the non-negatvty, 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,. Algebracally, ths gves the followng wthdrawal rule: Q r max = Q r S max or Q = r max r Q S max or Q Q S = r max r max (30) Ths rule says that the duraton-mnmzng recharge to aqufer should be proportonal to the fracton of the total recharge rate capacty aqufer can handle. 3C. Mnmze duraton to fll all aqufers Mnmze FD mn (31) Subject to: No negatve recharges R 0, (32) Defnton of fll duratons for each aqufer FD K =, (33) λ R

18 13 Total recharges less than steady surface water avalable n each perod R RS (34) Recharges lmted by maxmum recharge rates R r max, (35) Defnton of program fll duraton FD mn FD, (36) Where FD s the recharge duraton to fll aqufer [tme], FD mn s the program duraton to completely fll all aqufers [tme], R s the rate at whch to recharge water nto aqufer n each tme perod [volume tme -1 ], and R S s the steady surface water avalable for recharge n each perod [volume tme -1 ]. λ (fractonal recovery), K,(unflled capacty), and r max (maxmum recharge rate) are as defned prevously. The fll duraton for each aqufer s a functon of the fractonal recovery (equaton 33). Ths statement assumes that losses occur as recharges are made. Ths assumpton should hold when unflled capacty s large, recharge rates are small, and long duratons are expected. If aqufer recharge constrants do not bnd, the steady recharge allocaton (R ) wll equalze duratons across aqufers, so that all aqufers fll at the same tme FD mn = FD = (K )/(λ R ). Ths gves the followng recharge allocaton rule: K λ K λ = or R RS R = R S K / λ K λ or R R S = K λ K λ (37)

19 14 To mnmze the duraton of recharge across aqufers, more water should be recharged nto aqufers wth larger unflled capactes or smaller fractonal recoveres,.e., aqufers that are most empty and wth the least effcent recharge. Lower fractonal recoveres wll lengthen the duraton to fll all aqufers. The term λ drops out when expected fractonal recoveres of rechargng water are equal across all aqufers or the fll perod s short. 4. Maxmzng Accessblty When fllng groundwater storage capacty n wet years, an agency often s unsure about the future demands for nstantaneous wthdraw or the duraton over whch aqufer wthdrawals must persst. The agency may want to optmze flexblty to take water from a portfolo of groundwater storages at hgh wthdrawal rates or sustan wthdrawals over 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. 4.1 Model formulaton The multple objectve problem s to maxmze the total nstantaneous wthdrawal rate plus a weghted duraton of wthdrawals: Maxmze W R + d D max (38) Subject to: No negatve recharges Q 0, (39) No negatve wthdrawals W 0, (40) Recharges lmted by maxmum recharge rates Q r max t, (41)

20 15 Wthdrawals lmted by maxmum pumpng capactes W p, (42) max Defnton of aqufer duraton D So + λ Q =, W Recharges lmted by remanng storage capacty (43) Q K (44) Recharges lmted by surface water supply Q S Q (45) Defnton of program wthdrawal duraton D D, max Defnton of total expected wthdrawal rate (46) W R a W = (47) Total expected wthdrawal rate must meet target demand rate wth target relablty α r [ W W ] α P (48) R T Where W R s the total expected rate of water wthdrawal from all aqufers [volume tme - 1 ], D s the duraton that wthdrawal W can be sustaned n aqufer [tme], D max s the maxmum duraton of tme the operaton program can be sustaned [tme], d s a userselected factor weghtng the relatve mportance of the two components of the objectve [volume tme -2 ], and S o (ntal storage), λ (fractonal recovery), a (future avalablty), Q (recharge volume), W (wthdrawal rate), W T (target demand rate), K (unflled storage capacty), Q S (avalable surface water), r max (maxmum recharge rate), p max (maxmum pumpng rate), α (target relablty), and t (perod) are as defned prevously.

21 16 Future avalablty s stated as functon of the wthdrawal (47) rather than recharge (11). Equatons (47) and (48) can be combned and reduced to an equvalent determnstc constrant as shown prevously: [( a Z σ ) W ] WT α (49) Where ā (future expected avalablty), Z α (standard normal devate for target relablty α), and σ (standard devaton of avalablty) are as defned n equaton (13). In the model, both recharge volumes (Q ) and wthdrawal rates (W ) are decson varables, wth the recharge perod fxed to tme t, but the wthdrawal perod (D max ) determned by the program and assumed to start after recharge s completed. If the user selects a small value for d (d << 1), the program wll dentfy recharge and extracton operatons that maxmze nstantaneous wthdrawal capacty gvng slght preference to operatons that lengthen the duraton the wthdrawal can be sustaned. Conversely, for large values of d, the program wll dentfy operatons that maxmze the duraton over whch water can be wthdrawn gvng slght preference to operatons that expand the rate at whch water can be wthdrawn. 4.2 Analytcal solutons Analytcal solutons are derved for cases where the coeffcent d s ether large or small Maxmze nstantaneous wthdrawal rate (W R ) When the non-negatvty, lmted recharge rate, aqufer storage capacty, and future avalablty constrants do not bnd and 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, (50) max

22 17 Second, the program wll confgure recharges so maxmum expected wthdrawal rates can be sustaned for as long as possble, equalzng wthdrawal duratons for all aqufers: ( So + λ Q ) W So + λ Q D max = D = = W Where the astersk superscrpt () represents the optmal values of the decson varables. Substtutng (50) nto (51) and rearrangng gves: (51) p S + λ Q o =, p max ( So + λ Q ) max (52) Equaton (52) s a set of ndependent equatons that can be solved smultaneously for Q. The form of the soluton suggests the decson maker should recharge water n aqufer so that the water recoverable for extracton from aqufer compared to the total water recoverable from extracton (from all aqufers) s proportonal to the pumpng capacty of aqufer compared to the total pumpng capacty of all aqufers. To maxmze the total wthdrawal rate and duraton that rate can be sustaned, the rule suggests that decson makers should recharge more water nto aqufers wth hgher pumpng capactes, lower ntal storages, and lower fractonal recoveres (.e., hgher losses). When fractonal recoveres and expected avalabltes are dentcal across all aqufers, equaton (52) reduces to the Metropoltan Water Dstrct of Southern Calforna s (MWD) rule for allocatng water nto multple aqufers. The MWD rule equalzes the rato of total water storage to pumpng capacty n each aqufer, (S o + Q )/P max = (S oj + Q j )/P max j (Tm Blar, personal communcaton, 2002) Maxmze duraton of wthdrawal (D max ) A second analytcal soluton maxmzes the duraton of wthdrawal (D max ) for the case where the value of d s large and the non-negatvty, lmted extracton rate, and future

23 18 avalablty constrants do not bnd. To maxmze duraton of steady wthdrawals, all basns should empty at the same tme, so ( So + λ Q ) W So + λ Q D max = D = = W (53) 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, the program wll mnmze wthdrawals subject to constrant (49) on wthdrawal target (W T ). Ths allows the substtuton: D max = S o + W T λ Q (54) Duraton wll also be maxmzed when the term Σ λ Q s maxmzed. Ths term represents recharged water recoverable for extracton. To maxmze ths amount, recharge should be nto aqufers wth the hghest fractonal recoveres. The duraton-maxmzng recharge rule s recharge aqufers n order of λ, unless lmted by recharge or storage capactes. Wth Q known, solve (53) for the duraton-maxmzng, steady wthdrawal rates for each aqufer. Because the ntal storages and addtonal storage generated from recharge are now determned, equaton (53) takes the same form as the soluton for problem 3A (equaton 21). Equaton (54) be rearranged and solved for the duraton-maxmzng, steady wthdrawal rate: W = T ( So + λ Q ) ( So + λ Q ) W (55) Ths rule says that the duraton-maxmzng wthdrawal from aqufer should be proportonal to the fracton of total recoverable water stored n aqufer. The rules for

24 19 wthdrawal (equaton 55) and recharge (paragraph above) represent sequental solutons for recharge followed by wthdrawal decsons. These solutons take forms smlar to the soluton for the wthdrawal decson alone (model 3A). 4.3 Tradeoff between solutons The two analytcal solutons frame a tradeoff between wthdrawal duraton and rate. The tradeoff can be stated as follows: hgher water losses ncurred by an operaton plan to maxmze the nstantaneous wthdrawal rate wll dmnsh the duraton over whch those operatons can be sustaned. 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 mathematcal program for a range of values for d can also llustrate the tradeoff. 5. Example Applcatons Two numercal examples demonstrate the sx aqufer balancng rules derved above. The frst example demonstrates solutons for each of the fve sngle-objectve rules. The second example demonstrates solutons for the mult-objectve accessblty formulaton. Examples were and solved wthn Excel worksheets. 5.1 Example #1 (Sngle objectve programs) A portfolo of 4 aqufers was selected wth dfferent physcal, nsttutonal, and economc characterstcs (Table 1). Aqufer A s located farthest from the demand area and Aqufer D was located closest to the demand area. Aqufer D has the largest storage capacty and second-to-smallest fractonal recovery (.e., large losses assocated wth recharge). Aqufer storage capactes range from 200,000 to 800,000 acre-feet (af). Maxmum recharge capactes and pumpng rates vary between 3,000 and 5,000 af/month and 6,000 and 15,000 af/month, respectvely. The fracton of recharge avalable for extracton

25 Aqufer Maxmum storage capacty Maxmum pumpng rate Physcal Maxmum recharge rate Table 1. Aqufer characterstcs Fractonal recovery Mean expected avalablty Insttutonal Standard devaton of expected avalablty Recharge cost Economc Use cost (to pump, treat, convey) Use value K p max r max λ rc c u [af] [af/mo] [af/mo] [fracton] [fracton] [fracton] [$/af] [$/af] [$/af] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) A 400,000 7,000 4, B 200,000 6,000 3, C 600,000 8,000 4, D 800,000 15,000 5, a σ

26 21 ranges between 90 and 96%. All aqufers have the same expected avalablty to wthdraw water (90%), however, avalablty s more certan for aqufers C and D and than for aqufers A and B. Extracton and conveyance costs vary between aqufers. All aqufers are assumed to have smlar water qualtes. Economc use values are dentcal snce extracted water supples a sngle demand locaton. Table 2 summarzes addtonal parameter values related to the recharge and extracton. All aqufers are assumed to start full for the wthdrawal problems (S o = K ) and empty for the recharge problems (S o = 0). Wthdrawals should meet the target demand rate of 20,000 af/month. 6,000 af of surface water s avalable for recharge. I assume an nterest rate of 5% over a plannng horzon of fve years to calculate a dscount factor of b = = ( ) -5. Furthermore, 85% of recharged water was desred to be avalable for wthdrawal wth 90% relablty. Table 2. Addtonal parameter values for extracton and recharge problems n Example #1 Parameter Symbol Value (1) (2) (3) (a) Extracton problem Target water demand rate [af/mon] W T 20,000 Intal storages S o K (b) Recharge problems Water avalable to recharge [af] Q S 6,000 Recharge perod [mon] t 1 Intal storages [af] S o 0 Steady water avalable to recharge [af/mon] R S 6,000 Dscount factor [fracton] b Water avalable for wthdrawal [fracton of amount recharged] β 0.85 Target relablty [fracton] α 0.9 Standard normal devate for relablty α [fracton] Z α The parameter values descrbed n Tables 1 and 2 fall wthn ranges of values common for aqufers and drought water management n the San Joaqun Valley and Southern Calforna. However, the aqufers do not represent specfc aqufers.

27 22 Usng the above example data, the aqufer wthdrawal and recharge programs were solved for the 2 economc and 3 duraton-based objectves (Table 3). Numercal solutons for each objectve matched the analytcal solutons derved prevously. To maxmze the economc value of wthdrawals (column 2), aqufers D and C were pumped. Aqufer D had the hghest extractve value of water (Table 4, column 11); therefore, the full pumpng capacty of 15,000 af/month was utlzed. Remanng demand was met by wthdrawng water from the aqufer wth second hghest extractve value. When the objectve s to maxmze the duraton of meetng target wthdrawals (Table 3, column 3), water was wthdrawn from each aqufer proportonal to the amount of ntal storage n each aqufer. Wthdrawals were sustaned for 100 perods from each aqufer (months). Table 3. Aqufer balancng solutons for economc and duraton-based objectves Aqufer Wthdrawal Problem Recharge Problem Maxmze economc value Maxmze duraton of wthdrawal Maxmze expected value of recharge Mnmze duraton to recharge small volume Mnmze duraton to fll all aqufers (Model 2A) (Model3A) (Model 2B) (Model 3B) (Model 3C) W W [af/mon] [af/mon] [af] [af] [af/mon] (1) (2) (3) (4) (5) (6) A 0 4, ,500 1,153 B 0 2,000 3,000 1, C 5,000 6, ,500 1,845 D 15,000 8,000 3,000 1,875 2,407 Q Q R Aqufer Table 4. Economc calculatons Net extractve use value u - c Dscounted net economc value v = b(u - c ) - rc Dscounted net economc value of recoverable water λ v [$/af] [$/af] [$/af] (1) (2) (3) (4) A B C D For the recharge problems, water s recharged nto both aqufers B and D when the objectve s to maxmze the expected value of recharge. Aqufer B has the hghest

28 23 dscounted net economc value of recoverable water (Table 4, column 4); so, Aqufer B s full recharge capacty of 3,000 af/month s utlzed. Remanng surface water s recharged to aqufer D, the aqufer wth the second hghest dscounted net economc value. To mnmze the duraton to recharge 6,000 af, recharge each aqufer n proporton to each aqufer s recharge capacty (Table 3, column 5). Recharges are sustaned for 0.38 months n all aqufers to fully recharge the 6,000 af. To mnmze the duraton to fll all aqufers wth a supply of 6,000 af/month, recharge each aqufer (Table 3, column 6) n proporton to the water needed to fll each aqufer (space avalable / fractonal recovery). Aqufer D takes the most water (800,000 af / 0.92 = 870,000 af) whle aqufer B takes the least (200,000 af / 0.93 = 215,000 af). Gven the large unflled capactes n all aqufers, 351 months were requred to fll all aqufers. The numercal solutons dscussed above verfy the analytcal solutons. The avalablty constrant for Program 2B was changed to demonstrate how ncreasng desred avalablty could nfluence the net expected value and confguraton of optmal recharges. The parameter β (requred fracton of recharged water to be avalable for extracton) was ncreased sequentally from 0.85 to For each value, the program was re-solved. The shadow value assocated wth the constrant descrbed by equaton (13) was recorded (Fgure 2). As the value of β ncreased from to 0.892, the program decreased recharge to Aqufer B and ncreased recharge to Aqufer D. When β = 0.892, the recharge capacty of Aqufer D was reached. For larger β, the program ncreased recharge to Aqufer C and contnued to decrease recharge to Aqufer B. Aqufers D and C have more narrowly bound expected relabltes than Aqufer B but lower dscounted, net economc values of recoverable water. Fgure 2 shows the stepwse costs ncurred to recharge water nto Aqufers C and D and mprove future avalablty. In ths example, t was not feasble to confgure recharges to make more than 90% of the amount recharged was avalable for extracton.

29 1,200 Shadow value (value to mprove water avalablty, $/%) 1, Requred water avalable for wthdrawal, Beta, (fracton of amount recharged) Fgure 2. Shadow prces to mprove avalablty of recharges for future wthdrawal (Program 2B)

30 Example #2 (Mult-objectve accessblty) The portfolo of 4 aqufers from Example #1 was also used to demonstrate solutons for the tradeoff objectve. However, problem parameter values were changed (Table 5). A surface water supply of 200,000 af was avalable for recharge over a tme perod suffcent to recharge any sngle aqufer. Intal storages were 80,000 af for aqufer A and 100,000 af for aqufers B, C, and D. For each aqufer, expected avalablty (ā ) was ncreased to 1 and standard devaton of avalablty (σ ) was reduced to zero. Changes were made so recharge and wthdrawal decsons could be examned smultaneously and the avalablty and recharge constrants were not bndng ntally. Table 5. Parameter values for Example #2 Parameter Symbol Value (1) (2) (3) Water avalable to recharge [af] Q S 200,000 Recharge perod [mon] t 200 Intal Storage, Aqufer A S o1 80,000 Intal Storages, Aqufer B,C,D S o2,3,4 100,000 Expected avalablty a 1.0 Standard devaton of avalablty σ 0 Target water demand rate [af/mon] W T 20,000 Target relablty [fracton] α 0.5 Standard normal devate for relablty α [fracton] Tradeoff coeffcent [af/mon 2 Z α ] d 10-3 to 10 4 The tradeoff objectve formulaton was solved 20 tmes for dscrete values of the tradeoff coeffcent (d) rangng from 10-3 to 10 5 wthdrawal rate per duraton squared [af/mon 2 ]. For all values of d, solutons converged to one of two solutons. A tppng pont between the two solutons was seen at d = af/mon 2 (Fgure 3). The corner soluton that maxmzed the nstantaneous wthdrawal rate (Table 6, columns 2 and 3) concded wth the analytcal soluton derved for that case (equatons 50 and 52). The

31 26 50, , Wthdrawal rate (left axs) Total Wthdrawal Rate (W T ) [ac-ft/mon] 40,000 35,000 30,000 25,000 20,000 Duraton (rght axs) Duraton (D max ) [mon] 15, , E E E+04 Tradeoff coeffcent (d) [ac-ft/mon 2 ] Fgure 3. Tppng pont between solutons that maxmze total wthdrawal rate and duraton of wthdrawal

32 27 corner soluton that maxmzed duraton of wthdrawals (Table 6, columns 4 and 5) also agreed wth the analytcal soluton (equaton 55). However, recharges were made to both Aqufers A and B because the maxmum pumpng rate for Aqufer A was constranng. Table 6. Two numercal solutons to accessblty program n Example #2 Aqufer Corner soluton that Corner soluton that maxmzes maxmzes duraton wthdrawal rate (d < ) (d > ) W Q W [af/mon] [af] [af/mon] [af] (1) (2) (3) (4) (5) A 7,000 29,603 7, ,381 B 6,000-5,979 75,619 C 8,000 26,564 3,510 - D 15, ,833 3,510 - Q When the tradeoff coeffcent was less than , 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. Based on ts ntal storage, Aqufer B 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 rate of wthdrawal of 36,000 af/month for 15.5 months. When the tradeoff coeffcent was larger than , recharge was lmted to aqufers A and B (Table 6, column 4), aqufers wth the largest fractonal recoveres. The program would have drected all recharge to Aqufer A, but the maxmum pumpng rate for Aqufer A constraned wthdrawal to 7,000 af/month. Therefore, water was also recharged to Aqufer B, the aqufer wth the second largest fractonal recovery. 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

33 28 rates because both aqufers started wth 100,000 af of recoverable storage and no recharge was made to ether aqufer. Total wthdrawals met the target rate of 20,000 af/month. From the recharge and wthdrawal decsons, the program could sustan wthdrawals for 28.5 months. The program 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) agan dentfes the tppng pont (Fgure 4). For values of d much larger or smaller than , 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 ntermedary solutons. The ntermedate solutons n Fgure 4 represent duraton maxmzng solutons acheved when expected avalabltes, ā, were further constraned (to 0.9 and 0.7) and the program was resolved. Lowerng the expected avalablty rases the expected wthdrawal rate requred to meet the target demand. Rasng the wthdrawal rate lowers the duraton. Thus, varyng expected avalabltes n the chance constrant llustrates a tradeoff between the two corner solutons (Fgure 5). Square markers ndcate the corner solutons presented n Table 6 (when the mean expected aqufer avalablty, ā, was 1.0 for all aqufers). Other ponts n Fgure 6 show duratons and total wthdrawal rates when the program was solved for dfferent expected avalabltes (ā = 0.9, , 0.6, and 0.57). Each pont represents a duraton-maxmzng soluton (d > ) where each aqufer was assgned the same mean expected avalablty (ā 1 = ā 2 = ā 3 = ā 4 ).

34 100,000 90,000 80,000 Objectve functon value (Z) 70,000 60,000 50,000 40,000 30,000 20,000 Soluton that maxmzes wthdrawal rate Intermedary Soluton, a = 0.9 Intermedary Soluton, a = 0.7 Soluton that maxmzes duraton of wthdrawals 10, ,000 1,500 2,000 2,500 Tradeoff coeffcent (d) [ac-ft/mon 2 ] Fgure 4. Accessblty program objectve functon value versus tradeoff coeffcent for 4 solutons

35 Wthdrawal rate (af/month) 40,000 35,000 30,000 25,000 20,000 d < , Soluton that maxmzes wthdrawal rate Mean expected avalablty = 1.0 Mean expected avalablty = 0.9 Mean expected avalablty = 0.8 Mean expected avalablty = 0.7 Mean expected avalablty = 0.6 Mean expected avalablty = ,000 d > Solutons that maxmze duraton of wthdrawals 10, Duraton, D max, (months) Fgure 5. Tradeoff between duraton and nstantaneous wthdrawal rate by varyng aqufer avalabltes

36 31 As expected avalabltes were decreased, larger wthdrawal rates were requred to meet the desred target wthdrawal rate (Appendx B). Less water was recharged to aqufer A and B and more water 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 7,000 af/month and Aqufer B reached a maxmum pumpng rate of 6,000 af/month for avalabltes less than 1.0. As expected avalabltes were decreased, optmal recharges and wthdrawals approached the soluton for maxmzng the wthdrawal rate. Note that no feasble solutons exsted for ā < 0.55 because the program could not ncrease the total wthdrawal rate above a maxmum pumpng capacty of 36,000 af/month. 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, wthdraws from the aqufer could be sustaned for a shorter tme. Ths s represented by the negatvely slopng tradeoff curve n Fgure 5. 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 takng water from when an aqufer manager seeks to maxmze accessblty to stored water (as ether duraton or rate of wthdrawal). 6. Conclusons Sx operatng rules were derved to suggest optmal aqufer management decsons for three types of objectves based on lumped aqufer characterstcs. The rules are: Economc Objectves 1. To maxmze the economc value of wthdrawng water, wthdraw water from aqufers wth the largest dfferences between use value and extracton costs.

37 32 2. To maxmze the future expected value of water, recharge water to the aqufers wth the largest dscounted net, economc value of recoverable water. Duraton Objectves 3. To maxmze the duraton of wthdrawals, wthdrawal n proporton to ntal storage. 4. To mnmze the duraton 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 or duratons of wthdrawals, preferentally recharge water to aqufers wth both hgh maxmum pumpng capactes and large fractonal recoveres (small storage losses). Operatng rules were demonstrated and verfed for two smple, numercal examples. When uncertantes concernng future avalablty of banked water for later wthdrawal and target relablty were ncorporated nto problem statements, results hghlght cost and performance tradeoffs and changes to recommended allocatons. The operatng rules and examples represent stuatons where constrants were non-bndng. However, the formulatons can readly be extended and solved numercally to nclude constrants for more complex systems. Examples mght nclude multple, unconnected aqufers operated n conjuncton wth a surface water reservor, multple reservors, and uncertan surface water volumes avalable for recharge.

38 33 Appendx A. Dervaton of Lnear Programs to Optmze Duraton of Steady Water Supply Two of the duraton-based programs presented n secton 2 can be transformed nto equvalent lnear programs. The transformaton requres takng the nverse of duraton (1/duraton). The resultng lnear programs are presented for maxmzng the duraton of wthdrawal and mnmzng the duraton to fll all aqufers. A1. Maxmze duraton of wthdrawal (lnear program) The objectve of program 2A was to maxmze the duraton of wthdrawal. When duraton s transformed and nverted (1/duraton), the objectve must lkewse be nverted. Therefore we need to mnmze the nverse duraton to maxmze the duraton (smaller nverse-duratons correspond to larger duratons). The decson varables are stll the wthdrawal rates (W ). The 5 constrants presented for model 2A are ncluded, however, the defntons of nverse duraton for the program (ID mn [tme -1 ]) and for each aqufer (ID [tme -1 ]) must also be nverted. The resultng lnear program s: Mnmze ID mn (A1) Subject to: No negatve wthdrawals W 0, (A2) Wthdrawals lmted by maxmum pumpng rates W p max, (A3) Wthdrawals must meet or exceed a target demand rate W WT (A4) Defntons of nverse duraton for each aqufer wthdrawal ID = W /S o, (A5)

39 34 Defnton of nverse duraton for program ID mn ID, (A6) A3. Mnmze duraton to fll all aqufers (lnear program) The objectve of program 2C was to mnmze the duraton to fll all aqufers. When duraton s transformed and nverted (1/duraton), the objectve must lkewse be nverted. Therefore we need to maxmze the nverse duraton to mnmze the duraton (larger nverse-duratons correspond to smaller duratons). The decson varables are stll the recharge rates (R ). The 5 constrants presented for model 2C are ncluded, however, the defntons of nverse duraton for the program (ID max [tme -1 ]) and for each aqufer (ID [tme -1 ]) must also be nverted. The resultng lnear program s: Max ID max (A7) Subject to: No negatve recharges R 0, (A8) Defnton of nverse duratons to fll each aqufer λ R ID =, ( K S ) o Total recharges less than steady surface water avalable n each perod (A9) R RS (A10) Recharges lmted by maxmum recharge rates R r max, (A11) Defnton of nverse program duraton ID max ID, (A12)

40 Appendx B. Solutons to Accessblty Program wth bndng constrants on future avalablty

41 36 Appendx C. Defnton of Terms λ a Expected fracton of recharge that wll be recoverable for extracton, untless Random varable representng future avalablty to extract water from aqufer, fracton α Relablty that the water should be avalable, fracton ā Mean expected avalablty of aqufer, fracton b Dscount factor, untless β Requred fracton of recharged water to be avalable n the future, 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 Tradeoff objectve coeffcent, volume tme -2 D Duraton of wthdraw from aqufer, tme D max Overall duraton of wthdrawal program, tme e xy ls Lower (.e., groundwater surface) elevaton of unsaturated area avalable to store water at locaton x,y n aqufer, length e xy us Upper (.e., ground surface) elevaton of the unsaturated area avalable to store water at locaton x,y n aqufer, length Aqufer ndex, 1..n FD Duraton to fll aqufer, tme FD mn Overall fll duraton for recharge program, tme K Unflled, remanng storage capacty of aqufer, volume p max Maxmum extracton pumpng capacty for aqufer, Decson on the amount to recharge nto aqufer, volume Q Q Optmal amount to recharge to aqufer, volume rc Unt cost to recharge aqufer, volume tme -1 R Steady recharge rate nto aqufer, volume tme -1 RD Recharge duraton for aqufer, tme Overall duraton for recharge program, tme RD mn R Optmal recharge rate nto aqufer, volume tme -1 r max Maxmum recharge capacty for aqufer, volume tme -1 R S Steady surface water avalable for recharge n each perod, volume tme -1 Porosty of aqufer at locaton x,y,z, fracton ρ,xyz