First International Conference on Saltwater Intrusion and Coastal — Monitoring, Modeling, and Management. Essaouira, Morocco, April 23–25, 2001

Optimal Management in Saltwater-Intruded Coastal Aquifers By Simple Genetic Algorithm

M.K. Benhachmi1, D. Ouazar1 , A. Naji2, A.H-D. Cheng3, K. EL Harrouni4 1 Ecole Mohammadia d'Ingénieurs, Morocco

2 Faculté des Sciences et Techniques de Tanger, Morocco 3 University of Delaware,USA

4Ecole Nationale d’Architecture, LabHAUT, Morocco

Abstract

In this paper, a management model with an economic objective of coastal aquifers threatened by saltwater intrusion, is developed. To provide decision makers with several alternative policies, a management objective such as maximization of the net benefits, that may be obtained from water use, is considered in the presented management model . Optimal management decisions aim to maximize the net benefit from allocation of the supplies, while minimizing both interference effects and the invasion of saltwater front to the . For problems involving multiple pumping wells, a Simple Genetic Algorithm based on selection, mutation, and crossover operators, and conventional binary string is used as optimization tool to search for optimal solution. The steady-state model simulates groundwater flow on a horizontal plane using an analytical function developed by Strack. The Strack Pumping Well Solution, i.e. sharp-interface approach, has been incorporated into a simple genetic algorithm to solve this management problem . The example of hypothetical unconfined is presented to demonstrate that analytical solutions are most useful for engineers to conduct feasibility study and preliminary design. The results show that genetic algorithm can effectively and efficiently be used to obtain nearly global solutions to this groundwater management problem. The computational effort needed to determine the optimal solution increases with complexity of the problem.

1 INTRODUCTION : Ground water aquifers are an important resource in coastal regions because he serve as major sources for freshwater supply in many countries around the world, especially in arid and semi-arid zones. In many coastal areas, high rates of urbanization and increased agriculture have arises the demand for groundwater (Bear & Cheng, 1999). Several wells have been drilled to supply increasing water demand .The increase in water withdrawals from the wells have caused unacceptable drawdowns and deterioration of the quality of water pumped by some of the wells. A set of well-established withdrawal and management policies is necessary to achieve more efficient management and operation of this aquifers. Groundwater management is posed as the maximization of the net benefit obtained from water use subject to constraints of no intrusion of saltwater front to the wells, and pumping capacity limits restrictions.

Initial efforts to support and improve the development and operation of groundwater systems by simulation and optimization techniques were made in the early 1970s. Since then, various types of groundwater management models have been proposed and successfully applied to real–world aquifer systems. Many reviews on the types of groundwater management models and their applications are made by [Gorelick, 1983], and [Yeh, 1986]. The management models applications in saltwater intrusion, is relatively recent, [Cummings and McFarland, 1974], [Shamir, et al., 1984], [Willis and Finney, 1988], [Barlow, Wagner and Belitz, 1996], [Emch and Yeh, 1998], [El Harrouni et al., 1998],[Das and Datta, 1999 ], [Cheng et al , 1999], and [Loaiciga and Leipnik, 2000]. Most of these problems have been investigated in a more complex setting involving various management objectives. Concerning saltwater intrusion into wells, it is often addressed in an indirect manner such as constraining drawdown at control points, or minimizing the intruded saltwater volume. These studies were conducted to maintain aquifer levels and prevent the saltwater intrusion so that undesirable economic consequences and legal violations are prevented. In this paper, we deal the development and application of an operational groundwater management model for a coastal aquifer system . The preliminary development here, however, is deterministic in order to focus on the modeling procedures. Optimal management decisions aim to maximize the net benefit from allocation of the groundwater supplies , while minimizing well interference effects and the invasion of saltwater front to the wells . The optimal decisions of the model define the expected water targets for the principal agricultural , municipal and industrial demand centers and their greatest net economic benefit. For simplicity and feasibility demonstration purposes only the sharp-interface saltwater intrusion model is used, particularly; the single-potential formulation of Strack [1976] is adopted for solving boundary value problems. A Simple Genetic Algorithm (SGA) [Goldberg, 1989] [Michalewicz; 1992] is used for optimization purposes. This biological-evolution-based search algorithm is used in place of a conventional gradient method because there exist discontinuities and multiple local optima in the solution space. The gradient methods generally fail under these conditions. The SGA is based on three specialized genetic operators: selection, crossover, and mutation and uses a conventional binary string to represent the design variables, i.e. the pumping rates.

The optimization model is applied to hypothetical unconfined coastal aquifer. the results demonstrate that the nontraditional algorithm , the simple genetic algorithm, is presented as an alternative solution scheme to provide a robust, global solution for determining optimal or near-optimal pumping rates and their greatest net benefit .

2 STRACK BASED PUMPING SOLUTION .

As a first step toward the understanding of the fundamental hydraulic issues of saltwater intrusion, a number of approximations that are common in groundwater and saltwater intrusion studies have been made. First, the sharp interface model, rather than the miscible transport model, is used. Second, the Dupuit-Forchheimer hydraulic assumption is employed to vertically integrate the flow equation, reducing it from three-dimensional geometry to two-dimensional. Third, the aquifer storativity is ignored such that the governing equation becomes time independent. Finally, the Ghyben-Herzberg assumption of stagnant saltwater is utilized to interpret the interface location. Based on the above assumptions, Strack [1976] derived a single potential theory such that a single governing equation could be applies to both the saltwater and the freshwater zones. Figures 1(a) and (b) give a definition sketch in the vertical cross-section of a confined and an unconfined aquifer, respectively. Distinction has been made between two zones---a freshwater only zone (zone 1), and a freshwater-saltwater coexisting zone (zone 2). Strack [1976] demonstrated that for a homogeneous aquifer of constant thickness, a potential Ф which is continuous across the two zones, can be defined:

for confined aquifer : (s −1)B 2 Φ = Bh + − sBd zone 1 f 2 1 = [h + (s −1)B − sd]2 zone 2 (1) 2(s −1) f

for unconfined aquifer : 1 Φ = [h 2 − sd 2 ] 2 f zone 1

s = (h − d) 2 zone 2 (2) 2(s −1) f

In the above hf is the freshwater piezometric head, d is the elevation of mean sea level above the datum, B is the confined aquifer thickness, see Figure 1. We also note that

ρ s = s (3) ρ f is the saltwater and freshwater density ratio, and ρs and ρf are respectively the saltwater and freshwater density.

We note these functions and their first derivatives are continuous across the zonal interface. The potential defined in (1} and (2) satisfies the Laplace equation

∆Φ=0 (4) in the horizontal (xy) plane. The problem is solved as a one-zone problem with appropriate boundary conditions. Once the problem is solved by analytical or numerical means, the interface location xi (see Figure fig1) is evaluated as:

2φ ξ = + d − B...... For.confined..aquifer s −1 2φ (5) ξ = ...... For.unconfined..aquifer s(s −1)

Figure 1: Definition sketch of saltwater intrusion in (a) a confined aquifer , and (b) an unconfined aquifer .

The toe of saltwater wedge (see Figure fig1) is located atξ =d. From (4), this means that the toe is located at where Φ takes these values: s −1 Φ = B 2 For confined aquifer toe 2 s(s −1) Φ = d 2 For uncofined aquifer (6) toe 2 For both the confined and unconfined aquifer , once the solution is found , the location of the toe can be tracked using the above equations. In our problem , we consider a two –dimensional geometry of semi infinite coastal plain bounded by a straight coastline figure 2 :

toe

coastline

Qq x well xw A pumping well with discharge Qw is located at a distance xw from the . There are also exists a uniform freshwater outflow of rate q. the aquifer can be either confined or unconfined .Solution of this problem can be found by the method of images for multiples pumping wells and is given by Strack( 1976) : q n Q (x − x ) 2 + (y − y ) 2  Φ = x + w ln w w (7) ∑  2 2  K w=1 4πK  (x + xw ) + (y − yw ) 

where ( xw, yw ) are well coordinates, Qw is the pumping rate of well i.,and K is the hydraulic conductivity. The toe location xtoe can be solved from : q n Q (x − x )2 + (y − y )2  Φ = x + w ln toe w toe w (8) toe toe ∑  2 2  K w=1 4πK  (xtoe + xw ) + (ytoe − yw )  1

Figure 2: A pumping well near coastline

3 GENETIC ALGORITHM: Nowadays, GAs are recognized as powerful search algorithms and offer nice alternative to conventional optimization technique. They are based on the mechanics of natural selection and natural genetics [Goldberg, 1989]. A genetic algorithm allows a population composed of many individuals to evolve under specified selection rules to a state that maximizes the ‘’fitness’’(i.e. minimizes the objective function). The method was developed by J. Holland [1975] and finally popularized by one of his students, David Goldberg, who was able to solve a difficult problem involving the control of gas-pipeline transmission for his dissertation [Goldberg, 1989]. The genetic algorithm begins, like any other optimization, by defining the optimization parameters , the objective function. It ends like other optimization algorithms too, by testing for convergence. In between, however, this algorithm is very different from other optimization algorithms. A path through the components of the genetic algorithm is shown as a flow chart in Figure 3.The GAs consists of three main operations: selection , crossover, and mutation. The GAs used in our simulations can be described by the following steps

1. An initial population of ‘’strings’’ which represent realizations of the set of decision variables, is randomly generated. A binary encoding is used, in which each string has values of 0s and 1s for their alleles (the values at particular locations on the string). 2. The ‘’fitness’’ of search string in the population is evaluated, based on the objective function and constraints. The strings are then mapped to a ranking fitness value in which the strings are sorted linearly by order of decreasing evaluation value. Selection of the strings that continue onto the mating pool is done by the roulette wheel weighting selection method . Roulette wheel weighting selection assigns probabilities to the strings in the mating pool according to their fitness. A string with lowest fitness value has the greatest probability of mating , while the strings with the highest fitness value has the lowest probability of mating . A random number determines which string is selected. Additionally , an elitist strategy is implemented , where the string with the best fitness evaluation is always copied to the new population. 3. The strings selected are then randomly assigned a mating partner from within the mating pool, and two random crossover locations on the strings pairs are determined . Mating between two strings occurs with the probability Pcross. If the strings pairs were determined to mate, a two point crossover then occurs in which information is exchanged between two parent strings as the crossover locations. This results in the formation of children strings. The population size remains constant through the generations, or iterations, by replacing the parent strings with the new children strings. If no crossover between parent strings take place, then these strings are copied to new population. 4. To prevent premature convergence to a local optima, some alleles on the strings are randomly muted so that allele values of 0 are changed to 1 and vice versa. This occurs with the probability Pmut. 5. This process of selection, crossover, and mutation (Steps 2-4) is repeated for many generations until a stopping criterion is met. As the maximum value of the fitness value is not known only a maximum number of iterations is specified to stop the process and the best strings within these iterations can be selected as the optimum solution of the problem.

For most real-world problems, these pseudo-optimal solutions are still much better than could be obtained using less robust methods. The GA must have some control parameters such as population size , (n)-usually 20 to 100, and probabilities for applying genetic operators, e.g. crossover probability (Pcross )- usually 0.7 to 1 , and mutation probability (Pmut ) – usually 0.01 to 0.05.

In the pumping management problem , the design variables are the pumping rates. The pumping rates to be optimized are encoded as binary string within the well capacity constraints. Each population will contain strings to represent all design variables. GAs are ideally suited for unconstrained optimization problems . As the present problem is a constrained one, it is necessary to transform it into an unconstrained problem.

4 CONSTRAINTS-HANDLING METHOD : ADDITIVE PENALTY METHOD : A general constrained optimization problem has the form

maxCost = f (x) (9) x subject to

g m (x) ≤ bm , m = 1,…….M (10)

f (x) ; 0, ∀x (11) where f(x)= objective function; x= set of decision variables; gm(x) = mth constraint; bm = constant constraint upper bound; and M = total number of constraints.

START

Generate initial population and evaluate each chromosome; set gen=1 (gen=generation)

selection

crossover

gen = gen+1 mutation

evaluate newly formed chromosomes

is gen < max. gen ? Yes (max. gen is defined by user)

No

STOP

Figure 3: Flow chart of binary genetic algorithm

GAs typically reformulate the system in (9)-(11) into an unconstrained optimization problem By incorporating constraints into the objective function. In the traditional approach , which will be referred to as the additive penalty method, a penalty function that is proportional to the total violation on each of the constraints may be added to the objective function. For the additive penalty method , the objective function becomes

Cost = f (x) −α(x) (12)

The penalty function associated with the set of M constraints α(x ) can be described as M α(x) = ∑ wmvm2 (13) m=1 where the squared constraint violation vm2 is defined as 2 vm2 = [max(0, g m (x) − bm )] (14) and wm= weight applied to the violation on constraint gm(x).The weight wm can be constant or vary with generation and may be the same for all M constraints or differ with each constraint [Goldberg, 1989]. By adding a penalty function α(x ) to the original cost (objective function), a string that violates the constraints is less fit because it has a higher total cost and less likely to survive into future generations. A feasible string with no violations on the constraints will have an α(x ) value of 0. It is expected that a feasible string will be more fit than string with low costs but large constraint violations, although this is clearly dependent on the relative magnitudes of the weights wm and the objective function. The Additive penalty method require the analyst to set the value(s)of the penalty weights wm. In general , if the weights are set too low, infeasible solutions may be found; if the weights are set to high, convergence may be slow and feasible but suboptimal policies may result. If the “optimal’’ policy found by the GAs using this initial weight is infeasible(i.e., does not satisfy all the constraints), then the weight is increased by 5-10 times . The process of applying different initial weights values is repeated until the optimal solution found becomes feasible.

5 FORMULATION OF OPTIMIZATION PROBLEM :

The coastal aquifer management model is developed for sustainable water withdrawal from the aquifer for beneficial uses. Due to application of spatially distributed pumping strategy , the intrusion takes place, and the salinity of the pumped water varies with the magnitude and location of pumping in the two-dimensional space domain of the aquifer. The management model is developed considering the plausible of the objective function. The model as formulated here incorporate explicitly the economic values. Explicit consideration of economic values requires the definition of cost and benefit functions. This aspect is included in the model. The objective is to obtain an optimal pumping policy by maximizing the net benefit, subject to constraints of no intrusion of saltwater front to the wells, and pumping capacity limits restrictions. The solution of the model determines the optimal sustainable spatial distribution of the pumping for beneficial uses from a specified set of potential locations in the two dimensional space, while the saltwater intrusion will not contaminated wells . The pumping costs are assumed to be directly proportional to the product of the pumping rate and the total lift at each well [ Nelson and Busch, 1968] , [Kwanyuen and Fontane, 1998].

The mathematical expression of the objective function of this model can be written as [Haimes,1977]:

m n 1 Maximize Z = (B Q − C Q (L − h )) (15) ∑ ∑ n p i P i i i i k =1 (1+ r)

with respect to Qi and other design variables where Z is the objective function, subject to the constraints :

The management aspects are satisfied : i) Toe location constraints : X < X i=1,m (16) toei i ii) The pratical pumping capacity limits should not be exceeded :

min max Qi ≤ Qi ≤ Qi i=1, m (17)

i In the above Qi is the discharge rate of well i, X w is the distance of well i from the i min max coast, X toe is the toe location from the coast in front of well i, Qi and Qi are respectively the minimum and maximum discharge of a well constrained by equipment capacity, hi is the hydraulic head at well point i , Li initial lift at well i, B p is the benefit per unit supply of water at well point i ,C p is the cost of pumping a unit volume per unit head at well point i ; r = interest rate; n= periods management ; and m is the number of wells. For existing wells at fixed locations, the design variable is the pumping rate Qi . For new wells to be installed, the design variables can include the number of wells, their discharges, and well locations in terms x- and y-coordinates. The main constraint of the problem is the free from invasion of the wells. Other constraints may include the maximum and minimum pumping rate of each well. To allow a genetic algorithm to be used, the constrained problem in (15)-(17) is first transformed into an unconstrained one. In applying the additive penalty method , the penalty function associated with toe locations constraint violations is determined by combining the toe locations constraint ( 16) with (13) and (14). For example, to ensure no saltwater intrusion, a benefit function can be defined as :

2 m n  m  1 1− X toe  F = B Q − C Q ()L − h − w   j   (20) ∑ ∑∑n p i p i i i xtoe     i k==1 (1+ r)  j 1 X    wi   The toe locations constraint violation weight wxtoe was constant with generation number .If the constraints are violated , we assume that the benefit function was defined as: m n 1 m n 1 F = B Q − C Q L − h − w B Q max − C Q max L − h ∑ ∑∑∑n p i p i ()i i xtoe n p i p i ()i i i k===1 (1+ r) i 11k (1+ r) min max The constraints Qi ≤ Qi ≤ Qi in ( 17 ) is not included in ( 15 ) because is automatically satisfied in genetic algorithm by properly selecting the search space of Qi. . The function F is then maximized in a unconstrained optimization procedure.

6 APPLICATION : The proposed optimization model , is applied to a specific hypothetical unconfined coastal aquifer .The management model solutions is useful for establishing the potential applicability of the proposed model . However, the evaluations of these solutions are no doubt limited in scope, because : 1) the study area is hypothetical , and 2) uncertainties in parameter estimates, boundary conditions, and imposed stresses are not incorporated For this problem, the numerical solution is required and two search processes are involved. In the first step, the governing equation (4) with a given pumping pattern is solved to give the solution in potential. The saltwater toe is located at where the potential takes a specific value. This search is conducted using the Gauss-Marquardt method (Sun, 1994; Naji et al., 1998). Since the potential is continuous in the problem domain, this gradient method is quite efficient. Once the toe location is found and the state of invasion or non-invasion is identified, another search is conducted in the wells discharges space to find the optimal pumping rates. This search is discontinuous and the GAs described in the preceding section is used for the optimization. Due to the two levels of search, the cost of computation is generally large. 3 We examine an unconfined aquifer with K=70m/day,q=1m /day/m,d=20m, ρs=1.025 3 3 g/cm , and ρf =1 g/ cm . Figure 4 gives an aerial view of the coast and the locations of 8 pumping wells. The well coordinates are shown in column (2) and (3) in Table1 . For each well we assume that there exist a lower bound Qlow and an upper bound Qup for pumping rate. The upper bound is limited by equipment and operational conditions. A constant Qup =1500 m3/day is initially chosen for all wells. We then check the critical pumping rates of each well assuming that the well exists alone. Since there is no reason that a well can pump more than 3 the critical rate Qc, the smaller of Qc and 1500 m /day is used as the upper bound, which is listed in column (4) of Table 1. From economic considerations, there exists a minimum pumping rate below which the well is no longer cost-effective to operate . The value of 3 Qlow=150 m /day is used for all wells and is represent the minimum demand of the location , as shown in column (5). Thus, the minimum supply needed is Qlow multiplicated by the number of wells. With a uniform benefit rate of 0,16 $ per m3 of water and a pumping cost of 0,0024 $ per m3 per m lift of water ( Gupta et al , 1996), the total optimum water withdrawals are given in Figure 4. In illustrative example the locations in table 1 identified with numbers 1,8 represent the possible pumping locations :

(1) ( 2) (3) ( 4 ) ( 5 ) ( 6 ) Well xw yw Qmax Qmin Li 3 3 Id ( m ) ( m) (m /day) (m /day) (m)

1 1100 2000 1500 150 11

2 1200 -1000 1500 150 12

3 1700 2400 1500 150 17

4 1100 -2000 1500 150 11

5 1200 400 1500 150 12

6 1800 -2400 1500 150 18

7 1800 -400 1500 150 18

8 1700 1000 1500 150 17

Table 1: Pumping wells input data for the example

Before implementing the GAs, it is necessary to decide on the precision to which the design variables are to be evaluated, as this determines the length of binary code required for each pumping rate. The GAs is then applied using the following parameters : population size = 20, maximum generations = 400. Different values of crossover and mutation probabilities were used and only those obtained for 0.1 for mutation probability and 0.7 for crossover probability are presented. The toe location constraint is handled using the additive penalty method as described previously. The optimal solution given by GAs is shown in figure 4 . This solution is obtained at iteration 21 from 400 fixed genetic iterations. The value of the objective function is approximately 6940,24 $/day .The computational time required for the solution of problem with 9 numbers of optimizations parameters increase than required for the problem with 8 numbers of optimizations parameters .

Figure 4: Saltwater intrusion front for the spatially distributed pumping wells from the unconfined aquifer and the correspondent optimal solution for their net benefit.

6 CONCLUSION: Management model with economic objective function for sustainable use of coastal aquifers is formulated and solved . For the multiple pumping well problem, the analytical solution, e.g. Strack Pumping Well Solution, is most useful for engineers to conduct feasibility study and preliminary design. The optimization technique has been applied to the saltwater problem in such a way to locate either the interface or the toe position of the saltwater that entered into aquifer. The coupling of the analytical solution sharp interface approach with GAs is a very promising tool for use in solving this economic objective optimization problem. One great advantage of GAs over conventional optimization lies in its handling of complex, highly nonlinear problems that more accurately reflect the real world. We have demonstrated the use of a GAs to conduct the search. The search space is huge, but we believe that we have obtained a near-optimal solution. The computational time required for the solution of management model increases with the complexity of the problem

7 REFERENCES: Bear, J. and Cheng, A.H.-D., ``An overview,'' Chap. 1, In” Seawater Intrusion in Coastal Aquifers---Concepts, Methods, and Practices”, eds. J. Bear, A.H.-D. Cheng, S. Sorek , D. Ouazar and I. Herrera, Kluwer, 1-8, 1999.

Cheng, A.H.-D. , Halhal, D. , Naji , A. and Ouazar, D., ``Pumping Optimization in Saltwater –intruded Coastal Aquifers , J. Water Resourc. Research, submitted, 1999

Cheng, A.H.-D. and Ouazar, D., ``Analytical solutions,'' Chap. 6, “Seawater Intrusion in Coastal Aquifers---Concepts, Methods, and Practices”, eds. J. Bear, A.H.-D. Cheng, S. Sorek , D. Ouazar and I. Herrera, Kluwer, 163-191, 1999.

Cummings, R.G. , ``Optimum Exploitation of Groundwater Reserves with Saltwater Intrusion ‘’ , J. Water Resourc. Research ,Vol 7,N° 6 , 1415-1424, 1971.

Cummings, R.G. and McFarland , J. W. , ``Groundwater Management and Salinity Control ‘’ , J. Water Resourc. Research ,Vol 10,N° 5 , 909-915, 1974.

Das Gupta , A. , Nobi , N and Paudyal G.N., ``Ground-Water Management Model for an Extensive Multiaquifer System and an application ,'' J. Ground Water, vol 34,N°2, 349-357, 1996.

Das, A. and Datta, B., ``Development of multiobjective management models for coastal aquifers,'' J. Water Resour. Planning Management, ASCE, 125, 76-87, 1999a.

Das, A. and Datta, B., ``Development of management models for sustainable use of coastal aquifers,'' J. Irrigation Drainage Eng., ASCE,125, 112-121, 1999b.

El Harrouni, K., Ouazar, D. and Cheng, A.H.-D., ``Salt /fresh –Water Interface Model and GAs for Parameter Estimation ‘’ Procceding SWIM 15 , Ghent 1998.

Emch, P.G. and Yeh, W.W.G., ``Management model for conjunctive use of coastal surface water and groundwater,'' J. Water Resour. Planning Management, ASCE,124, 129-139, 1998.

Finney, B.A., Samsuhadi and Willis, R., ``Quasi-3-dimensional optimization model of Jakarta Basin,'' J. Water Resour. Planning Management, ASCE, 118, 18-31, 1992.

Goldberg, D.E., Genetic Algorithms in Search, Optimization and Machine Learning, Addison- Wesley,.1989.

Gorelick, S.M., ``A review of distributed parameter groundwater management modeling methods,'' Water Resour. Res., 19, 305-319, 1983.

Haimes, Y.Y., Hierarchical analyses of water resources systems , McGraw-Hill Inc,1977.

Hallaji, K. and Yazicigil, H., ``Optimal management of coastal aquifer in Southern Turkey,'' J. Water Resour. Planning Management, ASCE, 122, 233-244, 1996.

Holland, J., Adaptation in Natural and Artificial Systems, Univ. Michigan Press, Ann Arbor, 1975.

Loaiciga H. A., and Leipnik, R.B., ``Closed –Form For Coastal aquifer Management ,'' J. Water Resour. Planning Management, ASCE, 126, 30-35, 2000

Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs, Springer- Verlag, 1992.

Naji, A., Ouazar, D. and Cheng, A.H.-D., ``Locating saltwater/freshwater interface using nonlinear programming and h-adaptive BEM,'' Eng. Analy. Boundary Elements, 21, 253-259, 1998.

Naji, A., Cheng, A.H.-D. and Ouazar, D., ``BEM solution of stochastic seawater intrusion,'' Eng. Analy. Boundary Elements,23, 529-537, 1999.

Nelson , A. G. and C. P. Busch, ‘’Cost of pumping water in central Arizona,’’ Arizona Agricultural Experiment Station Technical Bulletin No. 182, Tucson, 1968.

Kwanyuen, B. and Fontane, D., ``Heuristic branch and bound method for groundwater development planning,'' J. Water Resour. Planning Management, ASCE, 124, 140-148, 1998.

Ouazar, D. and Cheng, A.H.-D., ``Application of genetic algorithms in water resources,'' Chap. 7, In: Control, ed. K. Katsifarakis, WIT Press, 1999.

Shamir, U., Bear, J. and Gamliel, A., ``Optimal annual operation of a coastal aquifer,'' Water Resour. Res., 20, 435-444, 1984.

Strack, O.D.L., ``A single-potential solution for regional interface problems in coastal aquifers'', Water Resour. Res., 12, 1165-1174, 1976.

Sun, N.-Z., Inverse Problems in Groundwater Modeling, Kluwer, 1994.

Willis, R. and Liu, P., ``Optimization Model For Ground-Water Planning,'' J. Water Resour. Planning Management, ASCE, 110, 333-347, 1984.

Willis, R. and Finney, B.A., ``Planning model for optimal control of saltwater intrusion,'' J. Water Resour. Planning Management, ASCE, 114, 163-178, 1988.

Willis, R. and Yeh, W.W.-G., Groundwater Systems Planning and Management, Prentice- Hall, 1987.

Yeh, W.W.-G, ``Review of Parameter Identification Procedures In Groundwater Hydrology : The inverse Problem,'' Water Resour. Res., 22, 95-108, 1986.