WATER RESOURCES MANAGEMENT IN THE WATERSHED KORONIA USING FUZZY LINEAR PROGRAMMING

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1 Proceedngs of the 13 th Internatonal Conference on Envronmental Scence and Technology Athens, Greece, 5-7 September 2013 WATER RESOURCES MANAGEMENT IN THE WATERSHED KORONIA USING FUZZY LINEAR PROGRAMMING PANAGIOTA GKINIDI *, CHRISTOS TZIMOPOULOS *, STAVROS ΥANNOPOULOS* AND CHRISTOS EVANGELIDES* * Laboratory of Hydraulcs and Envronmental Management, Arstotle Unversty of Thessalon, GREECE gotgn@yahoo.gr, evan@vergna.eng.auth.gr. EXTENDED ABSTRACT A lnear programmng model s appled to the management of Korona aqufer near the cty of Thessalon. The process of smulaton of the aqufer and the water mass balance was obtaned usng the Modflow model of the G.S.U.S. The optmal management of the aqufer has been carred out utlzng the Smplex method for the soluton of the lnear programmng model. Our prncpal am was the determnaton of the best water supply rate n some of the exstng wells n the area, ensurng the approprate hydraulc heads n them, wth the lowest possble overall cost. Furthermore the nvestgaton of optmal management of the aqufer has been carred out by the use of lnear programmng (Zmmerman, 1996) wth conventonal and fuzzy numbers. Key words: aqufer, water mass balance, lnear programmng model, fuzzy lnear programmng. 1. INTRODUCTION Water s an extremely valuable resource for every lvng organsm. However the nterventon of man, the growth of the world populaton, the change of lvng habts and the technologcal development, coupled wth the fact of a gven quantty of water avalable, generate two contradctory trends. Reducton of per capta avalable water resources and contnuous ncrease n per capta requrements. Ratonal management s very mportant durng explotaton of the aquatc resources of aqufers. The formulaton and soluton of management problems s based on the applcaton of a complex model whch ncludes smulaton, management and optmzaton. Ths research was focused towards ths drecton usng optmzed ntegrated management of groundwater for two cases: a) real numbers and b) fuzzy numbers. Fuzzy logc was utlzed because there s often uncertanty n nput data. Fuzzy sets and fuzzy logc s a newly developed area of mathematcs that nterprets stuatons of uncertanty, the development of whch had as ts startng pont the theory of fuzzy sets, (Zadeh, 1965). The objects of ths research are: The development of a model for the aqufer of Korona watershed n order to estmate the water balance of the study area and ts management. For ths purpose Modflow from Geologcal Servce of USA (USGS) was used, whch s the most versatle and wdely used three-dmensonal model for subsurface flow (Chang and Knzelbach, The optmal management of the aqufer, whch was acheved: 1) Wth the method of lnear programmng (Pslovos, 1996, Gnd, 2002) usng real numbers, n order to determne the optmal demand n some of the exstng wells n the area, as well the approprate pezometrc surface n order to sustan the aqufer and eep operaton costs mnmum.

2 2) Wth the ad of lnear programmng usng fuzzy numbers (Zmmerman, 1996), whch examnes the behavor of the aqufer management problems under uncertanty. The man uncertanty concerns: a) the measurement of hydrodynamc parameters (conductvty K, storatvty S, transmsvty T), b) the measurement of pezometrc head h and c) the lmts or the area of the aqufer. 2. SYMMETRIC FUZZY LINEAR PROGRAMMING The fuzzy lnear programmng s defned as follows (Zmmerman, 1996): Fnd the value of x such that: T c.x z0, A,x b, x 0 (2.1) where z 0 s an aspraton level,.e. the level that we wsh to acheve durng optmzaton and A s a matrx. The symbol s the fuzzfed verson of and has the lngustc nterpretaton essentally smaller than or equal to, and respectvely denotes the fuzzfed verson of and has the lngustc nterpretaton essentally greater than or equal to. Equaton (2.1) shows that the fuzzy lnear programmng model s fully symmetrc wth respect to objectve functon and constrants. mn Ths symmetry allows the creaton of generalzed matrces: c z 0 B A και d, so the model taes the followng form: b Fnd the vector x, so that: B x d, x 0. (2.2) μ (x) 1 μ (x)=1-{(bx) -d }/p 0 x=(bx) d d +p Fgure 1. Membershp functon The membershp functon μ( x ) s now ntroduced, whch corresponds to th row of (m+1) rows of model (2.2), and volatons p are accepted of the restrcton lmts b (rght hand sde of the nequalty), so that the membershp functon has the form of Fgure 1. The p are subjectvely chosen constants of admssble volatons of the constrants and the objectve functon from the decson maer. The membershp functon of the fuzzy set «decson» of model (2.2) s μ (x) = mn(μ (x)) D.

3 Applyng now optmal decson, (Belman and Zadeh, 1970), the followng relatonshp s obtaned: max μ D max mn ( x 0 μ( ( x ) max mn 1 x0 B x ) d p (2.3) Introducng now a new varable λ= mn μ the above problem s formulated as follows: Max λ, s.t. λp ( Bx) d p 1,...,m 1 For the case of nequaltes of the form.e. x 0. (2.4) ( B.x) d d p ( Bx) d and the problem s confgured smlarly. 3. RESEARCH AREA 3.1. Locaton, the maxmum tolerated volatons are p, The catchment of Mygdona s located n Central Macedona, n the NE sde of the cty of Thessalon, possessng an elongated zone between Galos Rver and Strymonos (Map 1). Map 1. Research area Lae Korona s located about 12 m east-northeast of the cty of Thessalon. It s a freshwater Lae, shaped le an ellpse. The subject of ths research focuses on the management of the aqufer of the Lae and s assocated wth a sgnfcant drop n ts level recently. The negatve water balance recorded n recent years (Knght et al, 1998, Tzmopoulos, 2004) s manly due to nadequate management of water resources and excessve consumpton of rrgaton water.

4 3.2. Aqufer data The aqufer under study les partly beneath Lae Korona and occupes an area of approxmately 232 m 2. Insttute of Geology and Mneral Exploraton (IGME) consders that the bottom of the Lae s mpermeable and therefore there s no communcaton between Lae and groundwater aqufers (Verans and Katrtzoglou, 2001a, b). The aqufer s fed by Korona watershed that has an area of 746 m 2. The exact boundares of the aqufer were determned based on data obtaned from IGME (Map 1). The aqufer parameters for ths study were obtaned from a study conducted from B.R.G.M (B.R.G.M, 1972) and the study of IGME (Verans and Katrtzoglou, 2001a, b). Accordng to IGME, values of transmssvty T ranged from m 2 / s, the values of storatvty S from , and the aqufer s characterzed as heterogeneous. For ths specfc research and the data entry n the aqufer, the regon was dvded n 1024 cells dmensons (500 m) x (500 m) Wells data The man water consumpton n Korona s due to agrculture. In order to estmate the amount of water used from the aqufer, data from a survey conducted by the Natonal Agrcultural Research Foundaton (NAGREF, 2003) was utlzed. For the ntroducton of ths data nto the model each cell was consdered to represent a separate vrtual well. To smplfy the model, the wells whch are placed n adjacent cells are grouped n a sngle cell, wheren the amount of yeld from ths sngle cell s derved from the sum of all correspondng yelds from the same group. After the approprate checs on the pezometrc surface and fnal adjustments of the model, thrty seven (37) wells were used (W 1, W 2,... W 37). Ther locatons are llustrated n Fgure 2. Fgure 2. Dstrbuton of wells n the Korona aqufer after groupng 4. GROUND WATER MODELING The smulaton of the water movement n the aqufer was mplemented usng Modflow. The soluton was obtaned through, Slce-Successve Overrelaxaton - SSOR method

5 (Greenwald, 1994), n whch the number of repettons, the acceleraton factor and the convergence crteron are specfed. The results of the smulaton model shows an average drawdown of the aqufer for the year of study of the phenomenon, whch s 1997, equal to 1.42 m, whch agrees well wth that gven from IGME. The drawn water volume was estmated at about m 3, close to the amount gven from the NAGREF (2003). 5. REAL NUMBERS. APPLICATION OF LINEAR PROGRAMMING IN KORONIA AQUIFER. The problem of lnear programmng s formulated usng the objectve functon and the constrants as follows: max s.t. f 37 1 CQ C Q C 37 Q 37 (5.1) 37 j37 Q Q, (5.2) 1 j1 j where Q= dscharge, C= cost coeffcent, = well number, = month 1,2,3..12, ntal dscharge, Q =the optmzed dscharge of well n month. 6. FUZZY NUMBERS. APPLICATION OF FUZZY LINEAR PROGRAMMING IN KORONIA AQUIFER In order to fnd the optmal allocaton of rrgaton supply and the mnmum cost wth fuzzy lnear programmng, equatons (2.1), (2.4) are utlzed. In the present wor, the cases where the objectve functon s volated by percentage p 1 = 15%, 10%, 5%, 2.5% and 1%, the restrctons p 2-39 by the same percentages and all possble combnatons of p 1 and p 2-39 are examned. Thus 25 dfferent combnatons are generated and the effect of each volaton on the fnal result s nvestgated Results The largest varatons n dscharge usng fuzzy lnear programmng are observed n wells W 3, W 16, W 28, W 29, W 32, W 33, W 34, W 35, and W 36. Yelds of wells W 3 and W 16 resulted after mplementaton of lnear programmng for real numbers tae zero values, whle n fuzzy programmng they tae zero values only for cases where the objectve functon s volated by 1%, 2.5% and 5%. Also the yeld of well W 15 taes zero values only n the case of real numbers. The wells W 30, W 31, W 32, W 33, W 34, W 35, W 36 and W 37 cease to operate (no pumpng) when the restrctons are volated by 10% and 15%, whle W 32, W 33 cease to operate n case of volaton of constrants by 5%. The varable λ taes zero value f the restrctons are volated by p 2-39 = 1%, p 2-39 = 2.5% and the objectve functon at p 1 = 1%, p 1 = 2.5%, whle the largest value occurs when p 1 = 15%, p 2-39 = 1%. Table 1 contans analytcally the percentage of the total demand and the cost of the objectve functon for all the combnatons of the volatons. Specfcally, t s observed that n the case of fuzzy lnear programmng, the percentage reducton of the sum of demands follows the correspondng volaton that has been appled, whle the value of the objectve functon (whch represents the operatng costs of pumpng), ncreases n case of volatons by 15%, 10% and reduces for the remanng percentages. When restrctons Q j =the

6 are volated by 1%, and the objectve functon s volated by 15% and 10%, the total amount of demand s reduced by 0.6%. Also a decrease of 1.22% s observed for volaton of the objectve functon by 5%, whle when the objectve functon s volated by 1% and 2.5%, the change n the total yeld ranges from 2.8% -3.8%, respectvely. For the volaton of constrants by 2.5%, there s a reducton n the sum of demands, rangng from 1.76% -3.63%. For volaton of the restrctons by 5%, a decrease from 3.11% 4.97% s observed, for 10% rates rangng from 6.37% 9.78%, whle when the volaton s 15%, the total amount of demands s reduced by 12.41%, 13.85% and 14.84%. 7. CONCLUSIONS The mplementaton of the above methods resulted n the followng conclusons: From the calculaton of the water balance n the study area t was found that acceptng vrtual wells correspondng to the consumpton of exstng crops and postonng them n the center of each cell, the smulaton results agree farly well to the actual dmensons of the problem. Ths s verfed by the fact that a) the average drawdown of pezometrc surface s 1.42 m and s very close to the correspondng drop 1.48 m calculated by IGME (Verans and Katrtzoglou, 2001a, b) for year 1997 and b) the pumped volume of rrgaton water was equal to m 3,.e. about as much has been calculated from the NAGREF for the same tme perod. From the results of the lnear programmng for real numbers t follows that yelds n wells W 3, W 15, W 16, W 37 are annhlated, whle there s a decrease to one thrd the ntal yeld values of wells W 12, W 24,,compared to the ntal demands gven as upper lmt. In the remanng wells there s no effect. The wells whose yelds vary are dstrbuted throughout the aqufer. Larger fluctuatons n yelds are observed durng the applcaton of lnear programmng for fuzzy numbers. Specfcally, for the wells whch are manly located n the southern part of the aqufer (W 30, W 31, W 32, W 33, W 34, W 35, W 36 and W 37) zero yelds result, partcularly for volaton of the objectve functon by 10% and 15%, whle for the remanng wells, the yelds are ncreased by 735 m 3 /d18674 m 3 / d compared wth those obtaned from the lnear programmng for real numbers. The above concluson results also for the case where total well yeld s consdered. Specfcally, applyng volatons p 2-39 to the restrctons, the sum of yelds n fuzzy lnear programmng appears reduced compared to the lnear programmng for real numbers. It s worthwhle to note that the fuzzy set process allows volatons of the objectve functon and the constrants, whle n classcal lnear programmng the volaton of any sngle constrant renders the soluton nfeasble. Fnally the decson maer has the possblty to accept more flexble solutons to satsfy needs. REFERENCES 1. Bellman R. and Zadeh L. (1970). Decson-mang n a fuzzy envronment. Management Scence, Vol. 17, No 4, pages B-141-B B.R.G.M.- Bureau de Recherches Geologques et Mneres (1972). Etude Hydrogeologque du Bassn de Mygdona pour l almentaton en eau de la Vlle de Salonque, Greece, 72 RME 011 FE (O.Y.Θ.). 3. Chang W. H. and Knzelbach W. (1993). Pre- and Postprocessors for Smulaton of Flow And Contamnants Transport n Transport n Groundwater System wth MODFLOW, MODPATH and MT3D Verson 3.0, 92 pp. 4. Gnd P. (2002). Management of Aga Varvara aqufer of Drama usng lnear programmng, Msc thess, A.U.T.H., page 298.

7 5. Greenwald R.M. (1994). MODflow MANagement: An optmzaton Module for Modflow. Interanatonal Ground Water Modelng Center, FOS 76 PS, Verson 3.02, Insttute of Ground-water Research and Educaton, Colorado School of Mnes, Colorado. 6. Knght Pesold and Karavoyrs and Partners, Master Plan (1998). Envronmental restoraton of Korona Lae. 7. NAGREF- Natonal Agrcultural Research Foundaton (2003). Korona Lae, Thessalon. 8. Pslovos A. (1996). Optmum management n aqufer studes usng the lnear programmng method. An applcaton to Edomen- Evzones area, Msc thess, A.U.T.H, page Tzmopoulos Ch. (2004). Research program: Restoraton project for Korona Lae, Thessalon, Greece, page Verans N., Katrtzoglou K. (2001a,b). Water mass balance n the hydrologcal basn of Lae Korona, Insttute of Geologcal and Mneral Researches, Thessalon, Greece, pages Zadeh L. Α. (1965). Fuzzy sets. Informaton and control 8(3): Zmmermann H.J. (1996). Fuzzy set theory and ts applcatons. Decson Mang n Fuzzy Envronments, Kluwer Academc Publshers, Boston, Chapter 13, pages

8 Table 1. The percentage of the total demand and the cost of the objectve functon for all the combnatons of the volatons usng fuzzy lnear programmng compared to the results of the lnear programmng for real numbers. Volaton of constrants Volaton of objectve functon 1% 2.5% 5% 10% 15% Objectve functon & constrants 1% 3.8% 5.63% 2.8% 4.25% 1.22% 1.95% 0.69% 0.95% 0.57% 6.05% 2.5% 3.63% 5.61% 2.66% 4.23% 2.26% 2.38% 1.76% 3.47% 1.83% 8.54% 5% 4.7% 5.63% 4.97% 4.21% 3.72% 3.09% 3.90% 5.71% τ 3.11% 7.13% 10% 9.78% 5.53% 7.86% 4.65% 8.79% 1.19% 6.85% 3.62% 6.37% 5.94% 15% 14.84% 1.03% 13.85% 2.91% 12.41% 0.73% 12.41% 0.73% 12.41% 0.70%