Development of a Combined Quantity and Quality Model for Optimal Management of Unsteady Groundwater Flow Fields

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1 University of Arkansas, Fayetteville Technical Reports Arkansas Water Resources Center Development of a Combined Quantity and Quality Model for Optimal Management of Unsteady Groundwater Flow Fields R. C. Peralta University of Arkansas, Fayetteville J. Solaimanian University of Arkansas, Fayetteville C. L. Griffis Unversity of Arkansas, Fayetteville Follow this and additional works at: Part of the Fresh Water Studies Commons, Hydrology Commons, Soil Science Commons, and the Water Resource Management Commons Recommended Citation Peralta, R. C.; Solaimanian, J.; and Griffis, C. L Development of a Combined Quantity and Quality Model for Optimal Management of Unsteady Groundwater Flow Fields. Arkansas Water Resources Center, Fayetteville, AR. PUB This Technical Report is brought to you for free and open access by the Arkansas Water Resources Center at ScholarWorks@UARK. It has been accepted for inclusion in Technical Reports by an authorized administrator of ScholarWorks@UARK. For more information, please contact scholar@uark.edu, ccmiddle@uark.edu.

2 DEVELOPMMENT OF A COMBINED QUANTITY AND QUALITY MODEL FOR OPTIMAL MANAGEMENT OF UNSTEADY GROUNDWATER FLOW FIELDS R.C. Peralta and J. Solaimanian Department of Agricultural Engineering C.L. Griffis Department of Agronomy University of Arkansas Fayetteville, AR Publication No. 134 June, 1988 Technical Completion Report Research Project G Arkansas Water Resources Research Center University of Arkansas Fayetteville, Arkansas AWRRC Arkansas W ater Resources Research Center Prepared fo r United States D epartm ent o f th e Interior

3 DEVELOPMENT OF A COMBINED QUANTITY AND QUALITY MODEL FOR OPTIMAL MANAGEMENT OF UNSTEADY GROUNDWATER FLOW FIELDS R. C. Peralta and J. Solaimanian Department o f A g ric u ltu ra l Engineering C. L. G r iffis Department of Agronomy U n iversity of Arkansas F a y e tte v ille, AR Research Project Technical Completion Report P roject G The research on which th is report is based was financed in part by the United States Department of the In te rio r as authorized by the Water Research and Development Act of 1978 (P.L ). Arkansas Water Resources Research Center U n ive rsity o f Arkansas 113 Ozark Hall F a y e tte v ille, Arkansas P ublication No. 134 June, 1988 Contents o f th is p u b lica tio n do not necessarily re fle c t the views and p o lic ie s o f the U.S. Department of the In te r io r, nor does mention of trade names or commercial products c o n s titu te th e ir endorsement or recommendation fo r use by the U.S. Government. The U n ive rsity of Arkansas, in compliance w ith federal and state laws and the regulations governing a ffirm a tiv e action and nondiscrim ination, does not discrim inate in the recruitm ent, admission and employment of students, fa c u lty and s ta ff in the operation o f any of it s educational programs and a c tiv itie s as defined by law. Accordingly, nothing in th is p u b lica tio n should be viewed as d ire c tly or in d ir e c tly expressing any lim ita tio n, s p e c ific a tio n or d iscrim in a tio n as to race, re lig io n, co lo r or national o rig in ; or to handicap, age, sex, or status as a disabled Vietnam-era veteran, except as provided by law. In q u irie s concerning th is p o licy may be directed to the A ffirm a tiv e Action O ffic e r.

4 ABSTRACT DEVELOPMENT OF A COMBINED QUANTITY AND QUALITY MODEL FOR OPTIMAL MANAGEMENT OF UNSTEADY GROUNDWATER FLOW FIELDS Presented are a lte r n a tiv e techniques fo r including conservative solute tra n sp o rt w ith in computer models fo r optim izing groundwater e xtra ctio n rates. Unsteady two-dimensional flow and dispersed conservative solute tra n sp o rt are assumed. Comparisons are made o f the p r a c tic a lity o f in clud ing modified forms o f im p lic it and e x p lic it f in i t e d iffe re n ce solute tran sport equations w ith in optim iza tion models. These equations can be ca lib ra te d and subsequently used w ith in a MODCON procedure. The MODCON m odelling procedure consists o f an integrated series o f fiv e optim iza tion or sim ulation modules. The procedure is applicable fo r e ith e r an e n tire aq u ife r system or fo r a subsystem o f a la rg e r system. The f i r s t module, A, computes p h ysica lly fe a sib le recharge rates across the boundaries o f the modelled subsystem. Module B computes optimal e xtra ctio n rates w ithout considering groundwater q u a lity. Module C uses method o f c h a ra c te ris tic s sim ulation to compute solute tra n sp o rt th a t would re s u lt from implementing the pumping strategy o f model B. Module D uses lin e a r goal programming and nonlinear solute transport equations to c a lib ra te lin e a r c o e ffic ie n ts. I t attempts to duplicate the solute transport predicted by module C. C a lib ra tio n is performed because coarsely d iscre tize d im p lic it or e x p lic it solute transport equations may not be as accurate as the method of c h a ra c te ris tic s. Module E includes appropriate ca lib ra te d equations o f module D as well as the flow equations o f module B. I t computes an optimal pumping (e xtra ctio n or recharge) strategy th a t can s a tis fy futu re groundwater contaminant concentration c r it e r ia. Testing o f the v a lid ity o f th is optimal pumping strategy is subsequently accomplished using module C. I f necessary, one may cycle through modules C, D and E u n til convergence is o b ta in e d --u n til concentrations re s u ltin g from implementing the strategy o f E are demonstrated to be acceptable. R. C. Peralta, J. Solaimanian, C. L. G r iffis Completion Report to the U. S. Department o f the In te rio r, Reston, VA, October 1988 Keywords - - Groundwater Management/Optimization/Groundwater Qual ity/w ater Q uality Management/Modeling i

5 TABLE OF CONTENTS Page Abstract... i L is t o f F i g u r e s... i i i Acknowledgements... iv Introdu ction... 1 A. Purpose and Objectives... 1 B. Related Research and A c tiv itie s... 2 Methods and Procedures... 4 A. Modelling Methodology Overview... 4 and Functions B. Model Development P rincipal Findings and S ignificance A. Testing o f E x p lic it Solute Transport Form o f MODCON B. Comparison of E x p lic it and Im p lic it Solute Transport Versions o f MODCON C o n c lu s io n s L ite ra tu re Cited i i

6 LIST OF FIGURES 1. Flowchart o f module functions in MODCON S ig n ific a n t ch a ra c te ris tic s o f MODCON modules Cell g rid notation system fo r f in it e diffe re n ce e q u a t io n s Page 4. Assumed i n i t i a l potentiom etric surface in hypothetical study area, in m above sea level Assumed i n i t i a l NaCl concentrations in groundwater, in ppm Concentrations that w ill re s u lt a fte r 25 years w ithout pumping, in p p m Optimal groundwater e xtra ctio n (+) and in je c tio n (-) rates computed by two versions o f module E, in 103 m3 y r -1 : a) e x p lic it form o f solute transport equation b) im p lic it form of solute transport equation Concentrations that w ill re s u lt a fte r 25 years of pumping at optimal rates computed by e x p lic it version o f module E, in ppm Change in potentiom etric surface elevations by year 25 caused by optimal pumping computed in f i r s t ite ra tio n using e x p lic it version of module E, in m Heads th a t w ill re s u lt a fte r 25 years o f pumping at optimal rates computed by e x p lic it version of module E, in m above sea l e v e l i i i

7 ACKNOWLEDGEMENTS A uthors are g ra te fu l to Alex Meeraus o f GAMS Development C o rp o ra tio n f o r h is a id and advice in using GAMS/MINOS. We very much a p p re c ia te B i l l Ashmore o f U niv. o f Arkansas Computing S ervices f o r developing the GAMSOP p o st-p ro ce ssin g ro u tin e. We a lso a p p re c ia te the advice o f R osalinda C a n tille r and her in te r a c tio n w ith B i l l Ashmore in the development o f GAMSOP. A uthors are g ra te fu l f o r the fin a n c ia l support o f the U. S. Department o f the I n t e r io r, G eological Survey, which provided the p rim a ry funds f o r t h is stu d y. The supplem entary fin a n c ia l support p ro v id e d by the A g r ic u ltu r a l and I r r ig a t io n Engineering Department and the C ooperative Extension S e rvice, Utah S tate U n iv e r s it y and th e A g r ic u lt u r a l E ngineering Department, U n iv e rs ity o f Arkansas, F a y e tte v ille, is also a p p re cia te d. iv

8 INTRODUCTION A. Purpose and Obj e c tiv e s A ssu ring th e s u s ta in a b le a v a ila b ilit y o f groundwater fo r w a ter users re q u ire s c o n s id e ra tio n o f both q u a n tita tiv e and q u a lita t iv e is s u e s. Numerous computer models have been re p o rte d f o r o p tim iz in g th e q u a n tita tiv e use o f groundw ater. Far fewer models in c lu d e s o lu te tra n s p o rt. Most o f those are a p p ro p ria te f o r in je c tio n o f contam inated w a te r, because in th a t case the mass f lu x ra te a t in je c tio n w e lls can be considered as a s in g le v a ria b le. O p tim iz in g groundw ater e x tra c tio n ra te s i f groundwater q u a lity is unknown is le s s fre q u e n tly done (except through g ra d ie n t-c o n tro l m ethods). T h is re lu c ta n c e re s u lts from the fa c t th a t s o lu te tra n s p o rt equations are n o n lin e a r. The use o f n o n lin e a r c o n s tra in ts in o p tim iz a tio n models re s u lts in lo c a lly ra th e r than g lo b a lly optim al s o lu tio n s. For some management o b je c tiv e s, lo c a l o p tim a lity is a cce p ta b le. This re p o rt discusses such a sce n a rio. The p rim a ry purpose o f t h is re p o rt is to dem onstrate how a lin k e d sequence o f fiv e o p tim iz a tio n, c a lib r a tio n and s im u la tio n models (modules) can be used to develop optim al groundwater management s tra te g ie s th a t a p p ro p ria te ly consid e r groundwater q u a lity. To accom plish t h is we present an enhanced versio n o f the MODCON m ethodology presented by P e ra lta e t al (1987). The f i r s t 1

9 objective is to demonstrate the p r a c tic a lity o f using ca lib ra te d e x p lic it solute transport equations w ith in MODCON to solve a hypothetical problem. The second objective is to compare e x p lic it and im p lic it versions o f MODCON. B. Related Research and A c tiv itie s G o re lic k (1983) provides a review o f methods fo r representing solute transport w ith in optim ization models. Each method has lim ita tio n s. Several researchers, including G orelick (1984) have used nonlinear constraints to represent solute tra n sp o rt. This is done because both e xtra ctio n rates and the concentration o f the extracted water are unknown. A weakness of using nonlinear constraints is the d if f ic u lt y in assuring global o p tim a lity o f the computed solutions. A second approach to managing groundwater concentrations is to use gradient control or v e lo c ity influence c o e ffic ie n ts (C olarullo et a l., 1984; G orelick and L e fk o ff, 1985). This approach may be unnecessarily r e s tr ic tiv e i f some contaminant movement (in addition to that caused by dispersion) is acceptable. Using predetermined lim its on acceptable hydraulic gradients may also be somewhat im practical i f the region of contamination is large. A th ird method u tiliz e s influence c o e ffic ie n ts that describe the e ffe c t of a change in potentiom etric head on steady-state 2

10 concentractions (Datta and Peralta, 1986). This approach is overly r e s tr ic tiv e since steady-state concentrations do not usually occur ra p id ly. I t is also cumbersome and im practical i f concentrations must be managed at m u ltip le lo ca tio n s. Another influence c o e ffic ie n t approach is described by Louie et a l., (1984). I t does not include detailed sim ulation of tra n sp o rt processes and may be im practical i f concentrations must be managed at numerous lo cations sim ultaneously. Peralta et al (1987) demonstrate use o f lin e a r mass tra n sp o rt equations w ith in two-dimensional models fo r optim izing groundwater management. These equations u t iliz e lin e a r coeffic ie n ts c a lib ra te d to approximately represent the solute tra n s port th a t is predicted using method o f c h a ra c te ris tic s (MOC) s i m ulation. They use a MODCON procedure consisting o f fiv e linked modules. Their computed optimal pumping strategy does cause an acceptable reduction in fu tu re concentrations at ta rg e t c e lls. However, the accuracy o f the solute transport equations was u n sa tisfa cto ry. One would expect a re p e titiv e cycle o f c a lib ra tio n, o p tim iza tion, sim ulation, c a lib ra tio n, e tc., to cause concentrations predicted by the optim ization model to converge to those predicted by the sim ulation model. This did not occur. This paper reports te s tin g performed using s ig n ific a n tly modified MODCON modules. Described changes re s u lt in enhanced 3

11 sim ulation o f solute tra n sp o rt. Although lin e a r c o e ffic ie n ts are s t i l l u tiliz e d, nonlinear solute transport equations are used as constrain ts. As a re s u lt, computed stra te g ie s are lo c a lly, not necessarily g lo b a lly, optim al. Another enhancement to the previously reported MODCON procedure is the coding o f MOC solute transport using GAMS/MINOS, perm itting more rapid in te ra ctio n between modules. In the previously reported MODCON, MOC sim ulation was accomplished using an external FORTRAN sim ulation model. METHODS AND PROCEDURES A. Modelling Methodology Overview and Functions We assume: 1) an unconfined is o tro p ic heterogeneous aquifer in which the change in water le vels w ith time w ill cause in s ig n ific a n t change in tra n s m is s iv ity (although anisotropic hydraulic cond uctivity can be re a d ily considered, is o tro p ic c o n d u c tiv ity is used here), 2) two-dimensional unsteady groundwater flow, 3) two-dimensional solute transport and in s ig n ific a n t v e rtic a l density gradients, 4) conservative dispersed contaminant, and 5) groundwater extra ctio n rates that are unchanging w ith time during the planning period (This requires fewer variables than would be needed i f pumping varies with time. Subject to computer memory and optim ization algorithm lim ita tio n s, pumping rates can be permitted to vary w ith tim e.) 4

12 The purpose o f the proposed model is to develop v o lu m e trica lly optimal groundwater extraction stra tegie s that assure acceptable fu tu re groundwater q u a lity. We wish concentrations predicted by the optim ization model to be as accurate as possible, but recognize th a t optim ization models cannot generally use as fin e a d is c re tiz a tio n in time or space as sim ulation models. Therefore, i t is desirable to be able to improve the accuracy o f the transport predicted by the optim ization models. For th is reason, the MODCON procedure includes c a lib ra tio n o f optim ization module solute transport equations w ith respect to solute transport predicted via a more d e ta ile d sim ulation module. Furthermore, sim ulation, c a lib ra tio n and optim ization are performed c y c lic a lly u n til sa tis fa c to ry s im ila r it y e x is ts between c o n c e n tra tio n s predicted by optim ization and sim ulation modules. The MODCON procedure, outlined in Figure 1, consists o f four optim iza tio n /sim u la tio n modules (A,B,D,E) and a sim ulation module (C). A ll modules are w ritte n using GAMS/MINOS (Kendrick and Meeraus, 1985; Murtagh and Saunders, 1983). Components A, B and E incorporate the two-dimensional lin e a rize d Boussinesq equation to model groundwater flow. Module C u tiliz e s method o f chara cte ristic s solute transport sim ulation. The function o f module C can also be accomplished using an external sim ulation model. 5

13 Aquifer parameters, i n i t i a l conditions, bounds on v a ria b le A Compute SS boundary flu x e s Compute o p tim a l US s tra te g y B wo/ c o n s id e rin g w ater q u a lity Compute c o n c e n tra tio n s re s u ltin g from optim al stra te g y STOP a c c e p ta b le? Calibrate c o e ff. fo r solute tra n s p o rt equations E Compute m o d ifie d o p tim a l s tra te g y w hile considering water q u a lity 1. Flowchart o f module fun ction s in MODCON. 6

14 (Konikow and Bredehoeft, 1978). Modules D and E incorporate im p lic it or e x p lic it solute transport equations and lin e a r influence c o e ffic ie n ts. The functions o f each part o f MODCON are discussed below. Figure 2 illu s tra te s the most important c h a ra c te ris tic s o f each module. The use o f module A is o p tio n a l. I t performs steady-state flow sim ulation and weighted lin e a r goal programming (LGP) optim ization to determine acceptable boundary flu x rates fo r the study area. This function is important when i t is im practical to model an e n tire aquifer system. I t aids developing a pumping strategy fo r only a portion o f the aquifer (a subsystem) in such a way as to prevent d isru p tio n o f flow outside th a t subsystem. To do th is, an assumption th a t must be v a lid is th a t aquifer stim u li outside the system during the management period w ill maintain the regional flow patterns th a t e x is t at the beginning o f the era (t= 0 ), as long as pumping w ith in the subsystem does not induce more groundwater flow in to the subsystem than occurred i n i t i a l l y. The recharge fluxes computed fo r boundary c e lls by module A can be used as upper lim its on recharge in subsequent optim ization modules. As w ritte n, module B uses unsteady flow sim ulation and weighted LGP optim ization to compute a pumping strategy that w ill 7

15 Module Module Objective Type; Output Constraints (c :) and bounds (b :) A Linear goal-programming (LGP) O ptim ization; Steady boundary fluxes {Q* } th a t best maintain in it ia l heads. c: c: b: 2D steady flow LGP fo r head on head B LGP O ptim ization; Pumping strategy {Q*} th a t best a tta in s ta rg e t subsystem heads {Ht k} by time k. Predicted heads {H* } c: c: b: b: 2D unsteady flow LGP fo r head on head on pumping C Nonlinear MOC solute transport Sim ulation; Future concentrations {Cck} re s u ltin g from {Q* }. Sim ulation o f: 2D unsteady flow 2D advect-dispersion D LGP O ptim ization; Calibrated c o e ffic ie n ts {Fp},{F r },{F d} so {CkD}={Cck} c: c: b: 2D advect-dispersion LGP fo r conc. on c o e ffic ie n ts E LGP O ptim ization; Modified pumping strategy {Q* } th a t best a tta in s ta rg e t heads and achieves satis fa c to ry concentrations {CEk} c: c: c: c: b: b: 2D unsteady flow LGP fo r head 2D advect-dispersion LGP fo r conc. on head on pumping Figure 2. S ig n ific a n t c h a ra cte ristics o f MODCON modul es 8

16 cause fu tu re potentiom etric heads to be as close to ta rg e t heads as possible. These ta rg e t heads can be the heads th a t e x is t at t=0. They may also be the fu tu re desirable heads computed by other optim ization models, such as models th a t maximize groundwater e xtra ctio n or the economic b e n e fit from groundwater use. A lte rn a tiv e ly, the e x is tin g objective function o f module B can be replaced w ith a function representing those goals d ire c tly. Module C uses nonlinear solute tra nsport sim ulation to compute fu tu re c o n c e n tra tio n s th a t w ill r e s u lt from imple mentation o f the pumping strategy computed by module B. I f fu tu re concentrations w ill be unacceptable in some loca tions, the pumping strategy w ill need to be m odified. To accomplish strategy m o d ifica tio n, solute tra nsport must be appropriately included in a model s im ila r to module B. This is u ltim a te ly achieved in module E, a fte r invoking module D. Module D uses LGP to c a lib ra te two-dimensional im p lic it or e x p lic it solute transport equations so th a t they can re p lic a te concentrations predicted by module C. When module D uses im p lic it equations, or e x p lic it equations with more than one time step, its solute transport equations are nonlinear. The use of nonlinear constraints is acceptable when one is g ra te fu l simply to have a v a lid strategy and global o p tim a lity o f the solution is 9

17 not c r i t i c a l. Module E includes the objective function and unsteady volum etric sim ulation o f module B, as well as the ca lib ra te d lin e a r solute transport equations o f module D. I t develops a modified pumping strategy th a t considers groundwater q u a lity constra in ts. Because o f the re la tiv e ly coarse d is c re tiz a tio n used in module E, one should v e rify the concentrations predicted by th a t module. Module C, or an external sim ulation model, is used fo r th a t purpose. Figure 1 shows th a t ite r a tio n through modules D, E and C is continued u n til concentrations predicted by module E are acceptably close to those predicted by module C, or its su b s titu te. B. Model Development For a n c e ll subsystem, the generic objective function fo r modules, A, B, D and E can be expressed as a va ria tio n o f th a t shown by Yazdanian and P eralta (1986). minimize y = + - +c +c - c -c ( W ){ D } + ( W ){ D } + g ( W ){ D } + g ( W ){ D } where ( W ) = a l x n vector o f weighting fa c to rs fo r head,..[1] 10

18 (dim ensionless) + { D } and { D } are n x 1 column vectors o f nonegative over- and under-achievement variables fo r fin a l heads, (L) g = dummy fa c to r to convert concentration in to head, (L/ppm) +c -c ( W ) and ( W ) are 1 x n vectors o f weighting factors applied to those fin a l concentrations th a t exceed or are less than ta rg e t concentrations, re spective ly, (dimensionless) +c -c { D } and { D } are n x 1 column vectors o f nonegative over- and under-achievement variables fo r fin a l con centrations, (ppm) Modules A and B use only those portions o f equation [1] that contain weights and achievement variables fo r head. Module D uses only weights and achievement varia bles fo r concentration. Module E uses the f u l l equation and weights and achievement variables fo r both heads and concentrations. Optimal solutions fo r modules A and B are constrained subject to the fo llo w in g, simply described fo r e ith e r steady- state flow (module A) or unsteady flow (module B). Equation [2] is a m atrix representation o f the im p lic it fin ite -d iffe re n c e two- 11

19 dimensional lin e a rize d flow equation. In the follo w in g equations, fo r k time steps, vectors o f magnitude n become n x k. L * * U {Q } { Q } = { B } - [ A ] { H } { Q }..[2 ] L * U { H} { H } { H }..[3] * + - t { H } - { D } + { D } = { H }..[4 ] 0.0 { D+ }, {D - }..[5 ] where L U { Q } and { Q } = n x 1 column vectors o f lower and upper bounds, re sp e ctive ly, on pumping (or recharge), (L3T-1 ) { Q* } = n x 1 column ve cto r o f optimal net steady pumping (or recharge) ra tes, where discharge is positive-va lu ed, (L3T-1 ) { B } = n x 1 vector de scrib in g the change in storage w ith tim e, (L3T-1). { B } is a zero ve cto r fo r stea dy-state flow. [ A ] = n x n symmetric banded m atrix o f aquifer properties, (l2t-1) * { H } = n x 1 column vector o f optimal fin a l or interm ediate heads, depending on the number o f time steps, (L) 12

20 L U { H } and { H } = n x 1 column vectors o f lower and upper bounds on head, (L) { Ht } = ta rg e t heads, (L) Note th a t the o b jective function considers a ll c e lls, not merely in te rn a l c e lls. In th is example, through equations [2,3 ], boundary c e lls are treated as va ria b le head/restrained flu x boundary conditions, ra th e r than as cla ssica l constant head (D iric h le t) or constant flu x (Neumann). The use o f weights o f large magnitude fo r boundary c e lls e ffe c tiv e ly forces heads to c lo s e ly approximate desired values. Module C is a sim ulation model o f two-dimensional advection and dispersion o f a conservative contaminant. I t performs no o p tim izatio n. I t is a GAMS representation o f the FORTRAN method o f c h a ra c te ris tic s code developed by Konikow and Bredehoeft (1978). Here fiv e p a rtic le s are used i n i t i a l l y in each c e ll. As previously stated, the function o f module D is to c a lib ra te im p lic it or e x p lic it f in it e diffe rence advective solute tra n sp o rt equations so th a t they w ill p redict the same concentrations as the p o te n tia lly more accurate module C. The change in concentration due to disp ersion as computed by module C is used d ire c tly in modules D and E. Thus neither D nor E need to include the nonlinear dispersion equations. The MODCON ite r a tio n 13

21 procedure serves to cause convergence between the change in concentration caused by dispersion assumed in modules D and E, and the values computed by module C. Module D uses the la t t e r h a lf o f obje ctive function [1] subject to constrain ts and bounds [6-9 ] mentioned below. The o b jective function o f module D usually applies the same weight to both over- or under-achievement variables fo r concentration. In p ra ctical a p plicatio n i t has been useful to also include in the o b jective function the sum o f a ll f r and f d c o e ffic ie n ts (defined below). These are included to make th e ir values be as small as p ra c tic a l. This forces them to be zero when no contaminant needs to be moved between c e lls. To maintain consistency in units while implementing th is a r tific e, these c o e ficie n ts must be m u ltip lie d by one linear u n it. In subsequent discussion, variables or constants used to describe values fo r individ ual c e lls are shown using lower-case le tte rs, as opposed to the upper-case notation used fo r vectors. For b re v ity, d e fin itio n s o f lower-case terms are omitted i f d e fin itio n s have already been provided fo r analagous vectors. +c -c { CDk } - { D ) + { D } = { CCk }..[6] +c -c 0.0 { D }, { D }..[7] 14

22 0. 0 { FP } { FpU }..[8] r d 0.0 { F }, { F } { FrU }..[9] where { CDk } and { CCk } are n x 1 vectors o f the concentrations predicted fo r the end o f the fin a l time time step, using modules D and C, re spective ly, (ppm) { Fp ), { Fr ) and { Fd } are n x 1 vectors o f c o e ffic ie n ts being applied to solute transport due to pumping in a c e ll and due to advection across the right-hand face and the down-side face of th a t c e ll respectively (dimensionless) { FpU ) and { FrU } are upper bounds on the c o e ffic ie n ts applied to solute movement e ith e r due to extraction by pumping, or due to advection, (dimensionless) W ithin module D, concentration at a c e ll located in row i and column j at the end o f time step k is computed using the g rid system shown in Figure 3. For a c e ll (i,j), midpoint terms with d sup erscrip ts ( f, t and v) apply to the boundary between c e ll 15

23 Figure 3. Cell g rid notation system fo r f in it e difference equations. 16

24 ( i, j ) and cell ( i+ l, j). Midpoint terms with r superscripts apply to the boundary between cell ( i, j ) and cell ( i, j+ 1). Because the same f coefficients apply to cells on both sides of a boundary, mass balance is maintained. The same amount of contaminant leaves through a boundary as enters the cell on the other side of the boundary. Equation 10 is the im p lic it finite-difference equation that is used for advective solute transport. ( I t can be converted into an e x p lic it form by using ck-1 to compute the t terms in Equations [11-15].) For all active cells i, j in the subsystem: Δ t q i,j,k ci,j,k = ci,j,k-l ( 1 - f pi,j ) + d STi,j Δ x 2 i, j, k - fri,j t ri,j,k+ f ri,j-l tri,j-l,k - f di,j t di,j,k+ f di-l,j tdi-l,j,k where qi,j,k = pumping or recharge in cell i, j, time step k, (L3T-1) Δ t S = length of time step, (T) = effective porosity, (dimensionless) 17

25 T i, j = saturated thickness in c e ll i,j, (L) Δ x= length o f a c e ll side, (L) k d i, j, k = change in concentration in c e ll i, j due to dispersion, as computed by module C, based on the concentrations e xistin g at the beginning o f time step k, (ppm) r d t and t = change in concentration in c e ll i, j caused by i, j i, j Δ Δ across a c e ll boundary, (ppm) r a t r t = - v c i, j, k Δ x i, j, k i,j+ 1 /2, k r Δ t r t = v c i,j -1,k Δ x i,j -1,k i, j-1 /2,k..[11]..[12] d Δ t d t = - v c i, j, k Δ x i, j, k i+ 1 /2, j, k d a t d t = v c i - l, j, k a x i -1,j, k i -1 /2,j, k r K v = ( h - h ) i, j, k S i, j, k i, j + l, k..[13] [14]..[15] = v e lo c ity o f solute movement between c e ll i,j r and c e ll i, j + l, (L /T ). Since v denotes to d the r ig h t, v denotes down in a plan view. 18

26 c i, j, k + c i, j + l, k..[1 6 ] i j + l / 2, k 2 = midpoint concentration between c e ll i, j and c e ll i, j + l, (ppm) Expressions analagous to equation [15] e x is t fo r the other three v e lo c itie s shown in equations [12-14]. S im ila rly, there are equations analagous to [16] fo r the other three midpoint concentrations shown in equations [12-14]. In module D, heads used to compute v e lo c itie s in equations [11-14] are known fo r a ll time steps, having been computed e a rlie r in module B. The concentrations used in equations [11-14] are unknown varia bles. The f c o e ffic ie n ts are also variables whose values are optimized in the module. Thus, im p lic it advective tra nsport equations are nonlinear. An e x p lic it tra nsport form ulation is lin e a r fo r a single time step, but is nonlinear fo r m u ltip le time steps. Module E u tiliz e s o b jective function [1 ], constraint equations [2-5] fo r unsteady flow sim ulation and equations [7-9,17] fo r solute transport sim ulation. 19

27 E +c -c t { C } - { D ) + { D } = { C } k k..[17] where E {C } = a n x l vector o f the fin a l concentrations k re s u ltin g at the end o f time step k from optimal pumping in module E, (ppm) t {C } = a n x l vector o f ta rg e t concentrations, i.e. the k upper lim it on acceptable concentrations re sultin g at the end o f the planning period, (ppm). These ta rg e t concentrations are predetermined by management agency. Equation [17] is a s o ft co n stra in t in th a t i t is possible to exceed the fin a l concentrations. In p ra ctice, using a large W+c and re la tiv e ly small values o f W- c and W in the objective function causes target concentrations to be attained i f i t is p h ysica lly feasible to do so. In Module E, pumping values and future concentrations and heads are unknown variables. Even though the f c o e ffic ie n ts are known from Module D, transport constraints are nonlinear. Dispersion is s t i l l treated as a known value, having la s t been computed in module C. The e rro r in the assumed transport due to dispersion is corrected through the process o f ite ra tin g through modules C, D and E. 20

28 An a lte rn a tiv e to using the five-module MODCON approach is to use Module E by it s e lf. In th a t case, a ll f c o e ffic ie n ts have values o f 1.0 and im p lic it or e x p lic it solute transport sim ulation is used. A Crank-Nicolson form ulation might also be used. That approach is p ra c tic a l caused by crude d is c re tiz a tio n. i f one can accept the e rro r I t permits one to forego the process o f ite ra tin g through modules C, D and E. In that case, Module A would probably be used only to determine lim its on boundary recharge ra te s. 21

29 PRINCIPAL FINDINGS AND SIGNIFICANCE A. Testing o f E x p lic it Solute Transport Form o f MODCON A hypothetical unstressed steady-state system is assumed (Figure 4). Flow assumptions are as mentioned previously. An e ffe c tiv e porosity o f 0.3 and a tra n sm issivity o f 1,092 m2/day (11,750 f t 2/day) are assumed. In th is te s t contaminant movement occurs only by advection ( d is p e rs iv ity equals ze ro). E ffe c tiv e weights o f 1 are assumed fo r head over- and underachievement variables in modules A, B and E. Weights o f 1 are used fo r concentration over- and under-achievement variables in module D. In module E weights o f 1,000 and 1 are used fo r concentration over- and under-achievement variables re spective ly. Thus module E attempts to insure th a t concentrations do not exceed ta rg e t concentrations. I f necessary, the same emphasis can be achieved in module D by increasing the magnitude o f the achievement variables fo r concentration. A ll f c o e ffic ie n ts are bounded to be between 0.0 and 10.0 in value. Lower bounds on recharge and discharge are zero. Upper bounds are a large enough value th a t they never are re s tric tiv e. A ll constant-head c e lls are permitted to e ith e r discharge or accept recharge, depending on what the model prefers. Discharge is also permitted at a ll in te rn a l c e lls, but recharge can occur only at c e lls ( 9,4 ), (9,5) and (9,6 ). P otentiom etric heads 22

30 J N Km / I / Constant-head Cell Variable-head Cell / Potentiometric surface contour Figure 4. Assumed in it ia l potentiom etric surface in hypothetical study area, in m above sea le v e l. 23

31 can change as long as they do not exceed the ground surface ele vation. They never approach th a t lim it in tested s itu a tio n s. Figure 5 shows the in it ia l s a lin ity concentrations th a t are assumed. Figure 6 shows the concentrations th a t w ill re s u lt a fte r 25 years o f steady-state flow, assuming no addition of contaminant to the system. Values shown in Figure 6 are computed using two 12.5-year time steps. Concentrations predicted using tw e n ty-five one-year time steps or a single 25-year step are w ith in 10 ppm o f the displayed values. Accordingly, MODCON modules discussed below use two 12.5 year time steps. To reduce computational requirements, optimal pumping is steady in time. Head response to pumping is tra n s ie n t. Assume th a t a management agency wishes to assure th a t 25- year concentrations in ta rg e t c e lls ( 9,5 ), (9,6) and (9,7) do not exceed 200 ppm. In Figures 5 and 6 we see th a t i n i t i a l concentrations in those c e lls are 375 ppm and 25-year concentrations w ithout management are 390 ppm. C learly some e xtra ctio n or in je c tio n o f water to the aquifer is needed to achieve the management o b je ctive. In the i n it ia l ite ra tio n o f the MODCON modules, the e x p lic it form o f module E computes the optimal pumping values shown in Figure 7a. Note th a t to ta l discharge and recharge rates via w ells are 927 and m3y r -1 (751 and 743 a c -ft y r -1 ) re spective ly. 24

32 J I Figure 5. Assumed i n i t i a l NaCl concentrations in groundwater, in ppm. 25

33 J I Figure 6. Concentrations th a t w ill re s u lt a fte r 25 years without pumping, in ppm. 26

34 a) J I b) J I Figure 7. Optimal groundwater extraction (+) and in je c tio n (-) rates computed by two versions o f module E in 103m3/yr: a) e x p lic it form o f solute transport equation, b) im p lic it solute transport equation. 27

35 Since the model attempts to d isru p t regional heads as l i t t l e as possible, to ta l discharge and recharge rates are very s im ila r in magnitude. Module E also predicts th a t as a re s u lt o f th a t pumping, concentrations o f p recisely 200 ppm w ill be attained by year 25 in the target c e lls. Subsequent reuse o f module C shows the concentrations that would more probably occur (Figure 8 ). Note concentrations o f 195, 205 and 195 in c e lls (9,4 ), (9,5) and (9,6) re spective ly. Assuming that a concentration o f 205 ppm is close enough to 200 to be acceptable, the flow chart o f Figure 2 indicates th a t no more ite ra tio n s are necessary. For demonstration purposes however, modules D, E and C are run again and provide the follo w in g re s u lts. A fte r re c a lib ra tin g the f c o e ffic ie n ts in module D to emulate the concentrations projected by the second use o f module C, module E computes a new pumping strategy. Again module E expects th is strategy to cause 200 ppm concentrations in the three ta rg e t c e lls. In the new strategy to ta l discharge and recharge by w ells is 965 and m3 y r - 1 (782 and 783 a c -ft y r -1 ) re spective ly. These represent 4 to 5 percent increases from the re su lts o f the previous ite ra tio n. According to the module C M0C model, concentrations that would re s u lt from implementing the revised pumping strategy are 191, 219 and 202 ppm fo r c e lls (9,4 ), (9,5) and (9,6 ). For these target c e lls, th is 28

36 J I Figure 8. Concentrations th a t w ill re s u lt a fte r 25 years of pumping at optimal rates computed by e x p lic it version o f module E, in ppm. 29

37 re s u lt is less accurate than th a t predicted in the f i r s t ite ra tio n. When a ll contaminated c e lls are considered however, ite ra tio n seems to improve the o ve rall accuracy o f the presented lin e a r c o e ffic ie n t sim ulation scheme. The sum o f the absolute values o f a ll differences between fu tu re concentrations predicted by f i r s t ite ra tio n use o f the e x p lic it module E and subsequent use o f MOC sim ulation is 600 ppm (average o f 12 ppm per contaminated c e ll). On the other hand the sum o f a ll p o s itiv e and negative-valued differences in concentration computed by E and C, (E - C) is only 24 ppm. Analagous sum o f absolute-valued differences computed using the second ite ra tio n re s u lts from module E is 535 (average o f 11 ppm per contaminated c e ll). In th is case the sum o f concentration differences between E and C is - 57 ppm. As long as there is d iffe rence between the heads used to c a lib ra te module D and the heads computed by subsequent optim ization in module E, one expects some e rro r in concentrations predicted by module E. A fte r a ll, modules D and E use e x p lic it or im p lic it representations of the p a rtia l d iffe re n tia l expression o f solute tra nsport, while module C uses a method o f c h a ra c te ris tic s p a rtic le tracking method. The f c o e ffic ie n ts in modules D and E permit advective transport to be increased or decreased to match th a t predicted by MOC sim ulation (or any other so rt o f predictio n used in module C). For example in modules D and E, unless the f c o e ffic ie n t describing advective 30

38 tra nsport equals zero, i f there is contaminant in a c e ll, some o f that contaminant w ill move to an adjacent c e ll i f water flows to th a t c e ll from the contaminated c e ll. On the other hand, in a MOC model, contaminant w ill appear in the down-gradient c e ll only i f c h a ra c te ris tic p a rtic le s tra vel fa r enough to cross the c e ll boundary. I f p a rtic le s do not traverse c e ll boundaries, then a MOC model w ill p re d ict no concentration in the down-gradient c e ll. Through use of c o e ffic ie n ts, modules D and E can match MOC-predicted concentrations. The tested scenario provides a more rigorous te s t o f the model than would be imposed i f there is o r ig in a lly no contamination in the target c e lls and i f no contaminant should enter them. In th a t case, the model merely needs to pump in such a way as to prevent m igration due to advection. This can be accomplished by causing heads in the target c e lls to be greater than those in surrounding c e lls. That simple approach is commonly used in management models th a t do not incorporate solute tra n sp o rt equations. The simple hydraulic gradient control approach is inadequate i f some contaminant is acceptable in ta rg e t c e lls and i f contaminant in i t i a l l y e xists in those c e lls. In th is case, the MODCON procedure ensures that fin a l concentration is acceptable. I t causes the development of hydraulic gradients th a t simultaneously lim it the inflow of contaminant w hile flush ing e xistin g contaminant out o f the target c e lls. Figure 9 shows the changes in p o ten tiom e tric surface elevations 31

39 I J Figure 9. Change in potentiom etric surface elevation by year 25 caused by optimal pumping computed in f i r s t ite ra tio n using e x p lic it version o f module E, in meters. 32

40 th a t re s u lt from the optimal e xtra ctio n and in je c tio n strategy. Figure 10 shows th a t although the gradient is changed i t is not reversed to the extent th a t contaminant from up-gradient no longer enters the ta rg e t c e lls. The MODCON strategy slows contaminant entry to the ta rg e t c e lls and hastens contaminant e x it. This is seen by observing head differences between c e lls immediately above and below the ta rg e t c e lls. Note that i n i t i a l l y there is a uniform 2 -fo o t drop in head per mile (per row). By year 25, the drop between rows 8 and 9 fo r columns 4, 5 and 6 has decreased to 1.123, and fe e t re s p e c tiv e ly. These correspond to gradient and contaminant v e lo c ity decreases o f 44, 46 and 44 percent re spective ly. By the same time the head drop between rows 9 and 10 has increased to 2.877, 2.9 and fe e t, corresponding to v e lo c ity increases o f 43, 45 and 43 percent re sp e ctive ly. Although the presented methodology seems to adequately achieve desired concentrations, one may question whether computed strategies are optimal from other perspectives. In th is example, one part o f the objective function o f module E attempts to maintain in it ia l heads to the extent possible. Thus, th a t function includes the sum of differences between 25 year heads re s u ltin g from the optimal strategy and those heads th a t would re s u lt from no pumping. I t also contains the weighted concentration achievement v a ria b le s. For th is 33

41 J N Km Km Constant-head Cell I * Variable-head Cell Potentiometric surface contour Figure 10. Heads th a t w ill re s u lt a fte r 25 years o f pumping at optimal rates computed by e x p lic it version of module E, in m above sea le v e l. 34

42 m u ltio b je c tiv e problem, weights are used to emphasize one objective versus another. When objectives are in c o n flic t optimal strategies lie on the pareto optimum and enhancing attainment o f one objective can only be accomplished by harming attainment o f the other. In such a case i t is common practice to compute tra d e -o ff functions fo r selected stra te g ie s. These describe the change in one objective caused by an incremental change in the other o b je c tiv e. There is a p a rtic u la r advantage to using head achievement variables in the obje ctive function. As mentioned previously, the target heads may be those developed by a ta rg e t design management model before invoking MODCON. They may be steady-state heads or tra n sie n t heads computed as being optimal fo r the end o f the planning period. They may be developed by models w ith o b jective functions th a t maximize economic or other b e n e fits. By seeking to maintain those heads to the extent possible, module E seeks to d is ru p t the previously developed optimal stra tegie s as l i t t l e as possible, while s a tis fy in g q u a lita tiv e goals. A simple example o f th is decomposition process occurs i f the prelim inary optim ization model computes a pumping strategy that maximizes volum etric extraction o f groundwater. Associated with that strategy are the heads th a t w ill re s u lt at the end o f the planning period. By using those heads as its ta rg e t heads, MODCON seeks to d isru p t the vo lu m e trica lly optimal strategy as l i t t l e as possible, w hile achieving ta rg e t concentrations. 35

43 One may ask why recharge should be used and ta rg e t concentrations should not be achieved using only e xtra ctio n. I f only discharge wells are used, there is a much greater disru p tio n o f the regional flow than i f both discharge and recharge are used. This somewhat negates the benefits o f using the ta rg e t head achievement component o f the objective function o f module E. Furthermore, when using inte rnal discharge but no recharge, concentrations predicted by module E in a prelim inary ite ra tio n are much less accurate than those presented and discussed above. In th is case Module C demonstrates th a t implementing the optimal e xtra ctio n strategy re s u lts in concentrations o f about 300 ppm in the ta rg e t c e lls. This occurs because as the differences in head between those assumed by modules C and D and those computed by E increase, p re d ic tiv e e rro r also increases. J u s tific a tio n fo r using recharge only at ta rg e t c e lls is found by performing prelim inary optim izations using MODCON. I f le f t free to recharge or discharge at a ll in te rn a l c e lls, module E computes to ta l discharge and recharge o f 1,175 and 1, m3 y r -1 (952 and 987 acf t y r -1 ). Although most o f the recharge occurs at the target c e lls, some occurs up to two rows away. Since the objective function does not attempt to minimize to ta l pumping, the model does not o f it s e lf consider the p ra ctical aspects o f having to recharge at many d iffe re n t loca tions. For management purposes, concentrating recharge at target c e lls is most reasonable. In a d d itio n, to ta l pumping is less when 36

44 concentrated. B. Comparison o f E x p lic it and Im p lic it Solute Transport Versions o f MODCON For the problem described above, optimal annual pumping computed by an im p lic it so lu te -tra n sp o rt version o f MODCON is shown in Figure 7b. Discharge and recharge through w e lls each to ta l about 1, m3 y r - (940 and 938 a c -ft y r -1 re s p e c tiv e ly ). These rates are about 125 percent o f the respective pumping rates computed by the e x p lic it model. Above we describe e rro r as the sum o f the absolute-valued or real differences between concentrations predicted by modules E and C. Total absolute valued e rro r o f the im p lic it model is 125 percent o f th a t of the e x p lic it model (750 versus 600 ppm). Total real-valued e rro r is almost ten times th a t o f the e x p lic it model (-250 versus +24 ppm). In a d d itio n, e rro r in concentrations computed fo r the ta rg e t c e lls in the f i r s t ite ra tio n is s lig h tly greater fo r the im p lic it than the e x p lic it version. Module C projected concentrations o f 205, 211 and 205 ppm in c e lls (9,4 ), (9,5) and (9,6) re spective ly, when using the optimal stra te g y from the im p lic it model. Since the im p lic it module E predicted concentrations o f 200 ppm in a ll three c e lls, i t underestimated by a to ta l o f 21 ppm. The e x p lic it module E on the other hand overestimated by a to ta l o f 5 ppm. In the described scenario, a manager would g e n e ra lly p re fe r th a t his model overestimate 37

45 fu tu re concentrations, ra th e r than underestimate them. In the performed comparison, c o e ffic ie n ts computed by the e x p lic it module D p re d ict future concentrations b e tte r than those computed by an im p lic it module--at least when heads and pumping values change from those assumed in module D. For the e x p lic it approach, values o f f p and f r computed by module D are 1.0 and 0.0 re spective ly. For th is approach f d is e ith e r 0.0 (fo r c e lls without in it ia l contamination) or between and (average o f 1.226). For the im p lic it approach the values are 1.0, 0.0 and e ith e r 0.0 or between and (average o f 1.181) re spective ly. One expects the im p lic it c o e ffic ie n ts to be s lig h tly sm aller than the e x p lic it c o e ffic ie n ts because the im p lic it equations u tiliz e concentrations at the end o f each time step to predict advective tra nsport, w hile the e x p lic it equations u tiliz e concentrations at the beginning o f each time step. Because o f the flow fie ld, concentrations are increasing in more rows than they are decreasing (eight out o f ten rows th a t have contaminated c e lls by year 25). Therefore there is s lig h tly less need fo r c o e ffic ie n ts in the im p lic it model to increase tra nsport. I t is also useful to compare e x p lic it and im p lic it models when including dispersion in MODCON. Assuming longitudinal and transverse d is p e rs iv itie s o f 200 f t, both models were run fo r the same problem in i t i a l l y posed. The e x p lic it model computed to ta l discharge and recharge by w e lls as 884 and m3 y r -1 (716 and 748 a c - ft y r -1 ) 38

46 re sp e ctive ly. This s lig h t decrease from pumping rates computed previously possibly re su lts because the contaminant moves a l i t t l e fa rth e r in the same time period even i f gradients are unchanged. The im p lic it model pumps about the same as previously, discharging 1, m3 y r -1 (945 a c -ft y r -1 ) and recharging 1, m3 y r - 1 (940 a c -ft y r -1 ). The accuracy o f concentrations predicted by e x p lic it and im p lic it models using dispersion are s im ila r. Both versions o f module E predicted concentrations o f 200 ppm in the three target c e lls. When te s tin g the optimal stra teg y computed by the e x p lic it model, module C predicted concentrations o f 207, 209 and 207 ppm from le f t to rig h t in those c e lls. When te stin g the strategy developed by im p lic it model, module C predicted concentrations o f 207, 212 and 207 ppm. For unexplained reasons, n e ith e r e x p lic it nor im p lic it versions of Module E can compute optimal solutions i f a ll f c o e ffic ie n ts are assigned values o f 1.0. This precludes the use o f an uncalibrated Module E and means th a t i t functioned s a tis fa c to rily only when used as pa rt o f the MODCON procedure. The e x p lic it form o f MODCON requires less computer processing time than does the im p lic it version. In th is re port, a ll processing is accomplished using an IBM 4381 mainframe computer running under VM/CMS. Average CPU time required to run a ll fiv e modules in the e x p lic it version fo r the above problem is 91.5 minutes. The im p lic it 39

47 version required about 127 minutes, 139 percent o f the e x p lic it requirement. Assuming th a t the module D c a lib ra tio n process elim inates the p o s s ib ility o f numerical in s ta b ility in the e x p lic it approach, the e x p lic it version o f MODCON seems p referable to the im p lic it versio n. CONCLUSIONS A methodology fo r sim ulating conservative solute transport w ith in computer models fo r optim izing groundwater management is presented. The technique allows the achievement o f ta rg e t groundwater contaminant concentrations w ith in groundwater use stra tegie s th a t may optimize attainment o f volum etric, economic or other p o licy objectives. The technique d iffe rs from the more common approach of preventing contaminant m igration by absolutely re s tric tin g advective movement. The presented method is fle x ib le in th a t advective contaminant movement may be perm itted toward and through concentration control c e lls. This is especially valuable i f some contamination already e xists w ith in a control c e ll, or i f preventing contaminant movement through such c e lls may be economically or te c h n ic a lly im p ra c tic a l. The technique u tiliz e s a fiv e module approach consisting o f four optim ization modules and a single sim ulation module. The f i r s t two modules optimize volum etric management and do not consider groundwater q u a lity constra in ts. They u tiliz e the embedding approach fo r representing steady or tra n s ie n t groundwater flo w. The th ir d module 40

48 simulates advective/dispersive solute transport using the method of c h a ra c te ris tic s. I t performs no optim izatio n. The fo u rth module uses optim ization to compute lin e a r c o e ffic ie n ts th a t best c a lib ra te e x p lic it or im p lic it transport diffe rence equations. The f i f t h module combines unsteady flow and calib ra te d solute transport equations to develop a volum etric strategy th a t achieves target concentrations as much as possible. Comparisons performed fo r a hypothetical system show th a t an e x p lic it form o f ca lib ra te d solute transport equation requires s ig n ific a n tly less computer processing time than an im p lic it form ulation. In a d dition, probably because o f the c a lib ra tio n process, the e x p lic it form y ie ld s answers th a t are at le a st as accurate as those from an im p lic it representation. 41

49 LITERATURE CITED C o la ru llo, S. J., M. Heidari and T. Maddock Id e n tific a tio n o f an optimal groundwater management strategy in a contaminated aquifer. Water Resources B u lle tin. v. 20, no. 5, pp Datta, B. and R. C. Peralta Optimal m odification o f regional potentiom etric surface design fo r groundwater contaminant containment. Transactions o f the ASAE. v. 29, no. 6. G orelick, S. M A review o f d is trib u te d parameter groundwater management modeling methods. Water Resources Research. v. 19, no. 2, pp G orelick, S. M., C. I. Voss, P. E. G i l l, W. Murray, M. A. Saunders and M. H. W right A quifer reclamation design: The use o f contaminant transport sim ulation combined w ith nonlinear programming. Water Resources Research. v. 20, no. 4, pp G orelick, S. M. and L. J. L e fk o ff Design and cost analysis o f rapid aquifer re sto ra tio n systems using flow sim ulation and quadratic programming. Groundwater. v. 24, no. 6, pp Kendrick, D. and A. Meeraus GAMS An in tro d u c tio n. The World Bank. Washington, D. C. Konikow, L. F. and J. D. Bredehoeft Computer model o f two-dimensional solsute transport and dispersion in groundwater. Techniques o f water resources in v e s tig a tio n s o f the United States Geological Survey. Book 7. Chapter C2. 91 pp. Louie, W. F., W. Yeh and N. S. Hsu M u ltio b je ctive water resources management planning. Journal o f Water Resources Planning and Management. ASCE. v. 110, no. 1, pp Murtagh, B. A. and M. A. Saunders MINOS 5.0 user s guide. Technical Report SOL 83-20, Stanford U n iv e rs ity, C a lifo rn ia. 42

50 Yazdanian A. and R. C. P era lta Sustained y ie ld groundwater planning by goal programming. Groundwater. v. 24, no. 2, pp