CHAPTER 11 Pumping Optimization in Saltwater-Intruded Aquifers A.H D. Cheng, M.K. Benhachmi, D. Halhal, D. Ouazar, A. Naji, K. EL Harrouni 1. INTRODUCTION Coastal aquifers serve as major sources for freshwater supply in many countries around the world, especially in arid and semiarid zones. Many coastal areas are heavily urbanized, a fact that makes the need for freshwater even more acute [Bear and Cheng, 1999]. Inappropriate management of coastal aquifers may lead to the intrusion of saltwater into freshwater wells, destroying them as sources of freshwater supply. One of the goals of coastal aquifer management is to maximize freshwater extraction without causing the invasion of saltwater into the wells. A number of management questions can be asked in such considerations. For existing wells, how should the pumping rate be apportioned and regulated so as to achieve the maximum total extraction? For new wells, where should they be located and how much can they pump? How can recharge wells and canals be used to protect pumping wells, and where should they be placed? If recycled water is used in the injection, how can we maximize the recovery percentage? These and other questions may be answered using the mathematical tool of optimization. Efforts to improve the management of groundwater systems by computer simulation and optimization techniques began in the early 1970s [Young and Bredehoe, 1972; Aguado and Remson, 1974]. Since that time, a large number of groundwater management models have been successfully applied; see for example Gorelick [1983], Willis and Yeh [1987], and many other papers published in the Journal of Water Resources Planning and Management, ASCE, and the Water Resources Research. Applications of these models to aquifer situations with the explicit threat of saltwater intrusion in mind, however, are relatively few [Cumming, 1971; Cummings and McFarland, 1974; Shamir et al., 1984; Willis and Finney, 1988; Finney et al., 1992; Hallaji and Yazicigil, 1996; Emch and Yeh, 1998; Nishikawa, © 2004 by CRC Press LLC Coastal Aquifer Management 234 1998; Das and Datta, 1999a, 1999b; Cheng et al., 2000]. In terms of management objectives, some of these studies have addressed relatively complex settings such as mixed use of surface and subsurface water in terms of quantity and quality, water conveyance, distribution network, construction and utility costs, etc. However, saltwater intrusion into wells has been dealt with in simpler and indirect approaches, for example, by constraining drawdown or water quality at a number of control points, or by minimizing the overall intruded saltwater volume in the entire aquifer. The explicit modeling of saltwater encroachment into individual wells resulting in the removal of invaded wells from service is found only in Cheng et al. [2000]. This chapter reviews some of the earlier considerations of pumping optimization in saltwater-intruded aquifers under deterministic conditions, and furthermore, introduces the uncertainty factor into the management problem. The resultant methodology is applied to the case study of the City of Miami Beach in the northeast Spain. 2. DETERMINISTIC SIMULATION MODEL The first step of modeling is to have a physical/mathematical model. Depending on the available data input from the field problem and the desirable outcome of the simulation, models of different levels of complexity, ranging from the sharp-interface model to the density-dependent miscible transport model, can be used [Bear, 1999]. For the method of solution, it can range from simple analytical solutions [Cheng and Ouazar, 1999] to the various finite-element- and finite-difference-based numerical solutions [Sorek and Pinder, 1999]. In principle, any of the above models and methods can be used; in reality, however, the selection of the model is dependent on the tolerable computer CPU time, as both the optimization and the stochastic modeling can be computational time consuming. In our case, the Genetic Algorithm (GA) has been chosen as the optimization tool. Due to the large number of individual simulations needed in the GA, the simulation model needs to be highly efficient in order to stay within a reasonable amount of computation time. For this reason, the sharp interface analytical solution is chosen, which is briefly described in the following. Figures 1(a) and (b) respectively give the definition sketch of a confined and an unconfined aquifer. The aquifers are with homogeneous hydraulic conductivity K and constant thickness B in the confined aquifer case. Distinction has been made between two zones—a freshwater only zone (zone 1), and a freshwater–saltwater coexisting zone (zone 2). Following the © 2004 by CRC Press LLC Pumping Optimization in Saltwater-Intruded Aquifers 235 Figure 1: Definition sketch of saltwater intrusion in (a) a confined aquifer, and (b) an unconfined aquifer. work of Strack [1976], the Dupuit-Forchheimer hydraulic assumption is used to vertically integrate the flow equation, reducing the solution geometry from three-dimensional to two-dimensional (horizontal x-y plane). Steady state is assumed. The Ghyben-Herzberg assumption of stagnant saltwater is utilized to find the saltwater–freshwater interface. With the above common assumptions of groundwater flow, the governing equation for the system is the Laplace equation: 2 0 φ ∇ = (1) where 2 ∇ is the Laplacian operator in two-spatial dimensions (x and y), and the potential φ is defined differently in the two zones © 2004 by CRC Press LLC Coastal Aquifer Management 236 y saltwater invaded zone freshwater zone pumping well inactive well toe (x y ) ii , Q i q coastline sea Figure 2: Pumping wells in a coastal aquifer. 2 2 1 [ ( 1) ] for zone 1 2( 1) 1 [(1) ]for zone 2 2( 1) ff f Bh h s B sd s hsBsd s φ φ == +−− − =+−− − (2) for confined aquifer; and 22 2 1 [ ] for zone 1 2 ( ) for zone 2 2( 1) f f hsd s hd s φ φ =− =− − (3) for unconfined aquifer. We also define s s f ρ ρ = (4) as the saltwater and freshwater density ratio, and other definitions are found in Figure 1. In our problem, we consider a semi-infinite coastal plain bounded by a straight coastline aligned with the y-axis (Figure 2). Multiple pumping x x w Qq toe coastline wellx xw Qq toe coastline well x x w Qq toe coastline wellx xw Qq toe coastline wellx xw Qq toe coastline wellx xw Qq toe coastline wellx xw Qq toe coastline wellx xw Qq toe coastline well © 2004 by CRC Press LLC Pumping Optimization in Saltwater-Intruded Aquifers 237 wells are located in the aquifer with coordinates (, ) ii x y and discharge i Q . There is a uniform freshwater outflow rate q. The aquifer can be confined or unconfined. Solution of the potential φ for this problem can be found by the method of images and has been given by Strack [1976] (see also Cheng and Ouazar, 1999): 22 22 1 ()( ) ln 4 ()( ) n iii i ii Qxxyy q x KK xx yy φ π =   −+− =+   ++−   ∑ (5) With the above solution, the toe location of saltwater wedge toe x is found where the potential takes the value toe φ , 22 22 1 ()() ln 4 ()() toe n toe toe iii toe i ii Qxxyy q x KK xx yy φ π =   −+− =+   ++−   ∑ (6) where 2 1 for confined aquifer 2 toe s B φ − = 2 (1) for unconfined aquifer 2 toe ss d φ − = (7) Since toe φ is some known number evaluated from Eq. (7), Eq. (6) can be solved for toe x for each given y value using a root finding technique. 3. OPTIMIZATION UNDER DETERMINISTIC CONDITIONS The management objective of the coastal pumping operation is to maximize the economic benefit from the pumped water less the utility cost for lifting the water. For simplicity, we assume that the value of water and the utility cost are both linear functions of discharge i Q . The objective is to maximize the benefit function Z with respect to the design variables i Q [Haimes, 1977]: () i Q 1 max Z n ip Pi i i QB C L h =   =−−   ∑ (8) In the above p B is the economic benefit per unit discharge, p C is the cost per unit discharge per unit lift height, i L is the ground elevation at well i, and i h is the water level in well i. It should be remarked that although a relative simple model is used for the right-hand side of Eq. (8), it can be © 2004 by CRC Press LLC Coastal Aquifer Management 238 generalized to a realistic microeconomic model involving supply and demand without complicating the solution process. The pumping operation is subject to some constraints. First, the discharge of each well must stay within the certain limits set by the operation conditions such as the minimum feasible pumping rate, maximum capacity of the pump, restriction on well drawdown, etc. This can be written as min max or 0; for 1, , iii i QQQ Q i n≤≤ = =… (9) We note that the second condition in the above allows the well to be shut down. Second, it is required that saltwater wedge does not invade the pumping wells at ; for all active wells toe ii i xx yy<= (10) where toe i x stands for the toe location in front of well i. Since genetic algorithm can only work with unconstrained problems, it is necessary to convert the constrained problem described by Eqs. (8)-(9) to an unconstrained one. This is accomplished by the adding penalty to the objective function for any violation that takes place: () 2 1 max Z 1 i toe n i ip Pi i ii Q i i x QB C L h rN x =   =−−−−    ∑ (11) where i r are penalty factors, which are empirically selected, and 1 i N = for toe ii x x≥ and 0 i N = for toe ii x x < . We notice that the constraint Eq. (9) is not included in Eq. (11) because it is automatically satisfied by setting the population space in genetic algorithm. 4. GENETIC ALGORITHM Conventional optimization techniques, such as the linear and nonlinear programming, and gradient-based search techniques are not suitable for finding global optimum in space that is discontinuous and contains a large number of local optima, which are the prevalent conditions for the optimization problem defined above. To overcome these difficulties, a genetic algorithm (GA) has been introduced and successfully applied [Cheng et al., 2000]. GA is a probabilistic search based optimization technique that imitates the biological process of evolution [Holland, 1975]. Its application to groundwater problems started in the mid-1990s [McKinney and Lin, 1994; Ritzel et al., 1994; Rogers and Fowla, 1994; Cienlawski et al., 1995], and since that time it has found many applications. (See Ouazar and Cheng [1999] for a review.) © 2004 by CRC Press LLC Pumping Optimization in Saltwater-Intruded Aquifers 239 A brief illustration of the GA solution procedure applied to the current problem is given below. Given the solution space of i Q defined by Eq. (9), we discretize it in order to reduce the number of trial solutions from infinite to a finite set. As an example, if each discharge is constrained between 100 500 i Q≤≤ m³/day, and the desirable accuracy of the solution is 5 m³/day (which is a rather crude resolution), then for each i Q there exist 82 possible discrete values (including the zero pumping rate). If there are 10 wells in the field, then the total number of possible combinations of pumping rate is 10 19 82 1.4 10=× . One of the combinations is the optimal pumping solution we look for. This search space is so huge that if we spend 1 sec of CPU time to conduct a single simulation to check its benefit, it will take 11 410× years to complete the work. The search space of a typical field problem in fact is greater than the above. Hence we must follow some intelligent rules in the search; this is where the GA comes in. GA seeks to represent the search space by binary strings. In the above example, it is sufficient to represent all possible combinations of pumping rate by a 64-bit binary string ( 64 19 21.810=× ). To seed an initial population, a random number generator is used to flip the bits between 0 and 1 to create individuals in the form of 01101…10111 (64 digits long), each one corresponding to a distinct set of pumping rates. Typically a relatively small number of individuals, say 10 to 20, are created to fill a generation. Individuals are then tested for their fitness to survive by running the deterministic simulation as described above. The fitness is determined by the objective function given as the right-hand side of Eq. (11). Once the fitness is determined for each individual in the generation, certain evolutional-based probabilistic rules are applied to breed better offspring. For example, in a simple genetic algorithm (SGA), three rules, selection, crossover, and mutation, are used [Michalewicz, 1992]. First, the selection process decides whether an individual will survive by “throwing a dice” using a probability proportional to the individual’s fitness value. Second, the GA disturbs the resulting population by performing crossover with a probability of c p . In this operation, each binary string (individual) is considered as a chromosome. Segments of chromosome between individuals can be exchanged according to the predetermined probability. Third, to create diversity of the solution, GA further perturbs the population by performing mutation with a probability of m p . In this operation, each bit of the chromosome is subjected to a small probability of mutation by allowing it to be flipped from 1 to 0 or the other way around. After these steps, a new generation is formed and the evolution continues. The process is terminated © 2004 by CRC Press LLC Coastal Aquifer Management 240 1 2 3 4 5 6 7 89 10 11 12 13 14 15 0 1000 2000 3000 4000 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 y (m) Coastline x (m) pumping well toe Inactive well Figure 3: Pumping wells in a coast and saltwater intrusion front. by a number of criteria, such as no improvement observed in an number of generations, or reaching a pre-determined maximum number of generation. The reader can consult the above-cited references for more detail. 5. EXAMPLE OF DETERMINISTIC OPTIMIZATION This test case was examined in Cheng et al. [2000]. Assume an unconfined aquifer with K = 40 m/day, q = 40 m²/day, d = 15 m, s ρ = 1.025 g/cm³, and f ρ = 1 g/cm³. Figure 3 gives an aerial view of the coast and the locations of 15 pumping wells. The well coordinates are shown in columns (2) and (3) of Table 1. Each well is bounded by a maximum and a minimum well discharge, as indicated in columns (4) and (5). In this optimization problem, only the benefit from the pumped volume is considered, and the utility cost is neglected. The objective function (11) is modified to 2 1 max Z 1 i toe n i iii Q i i x QrN x =  =− −   ∑ (12) © 2004 by CRC Press LLC Pumping Optimization in Saltwater-Intruded Aquifers 241 (1) (2) (3) (4) (5) (6) (7) Well i x i y max i Q min i Q i Q toe i x Id (m) (m) (m 3 /day) (m 3 /day) (m 3 /day) (m) 1 1000 2500 600 150 201 836 2 1700 1100 1300 150 351 1117 3 1500 850 1100 150 0 1257 4 1200 400 800 150 0 1372 5 1700 200 1300 150 150 1514 6 1800 -300 1400 150 0 1344 7 3500 -500 1500 150 1497 1323 8 1600 -800 1200 150 0 1311 9 1600 -1200 1200 150 0 1315 10 1500 -1600 1100 150 0 1332 11 2000 -2000 1500 150 155 1319 12 1000 -2200 600 150 0 1287 13 1600 -2500 1200 150 0 1241 14 3600 -2800 1500 150 1387 1251 15 1400 -3000 1000 150 150 1213 Total 3891 Table 1: Optimal pumping well solution. The GA described earlier is used for optimization. In the first attempt, the optimization was conducted by assuming all 15 wells are in operation. The search space for each well is defined between min i Q and max i Q with increment size of roughly 1 m³/day and also the zero discharge. If a well is invaded, a penalty is imposed with an empirical penalty factor i r to discourage such events. If the well is shut down, 0Q = , the program detects it and no penalty is applied for invasion. This allows the inactive wells to be intruded in order to increase pumping. After three runs of GA with different seeding of initial population, the best solution gives the total discharge of 3,610 m³/day. The optimal solution shows that eight wells are in operation and seven are shut down. The fact that so many wells are shut down is not surprising, as an estimate based on a simple analytical solution [Cheng et al., 2000] shows that the well field is too crowded and some wells can be taken out of action. The program was run on a Pentium 450MHz microcomputer. It was terminated when the maximum number of generations was reached, for about 6 hours of CPU time. Since an near optimal solution may not have been reached, a second search is conducted using a refined strategy. In the second search, only cases with any combinations of seven, eight, and nine wells in © 2004 by CRC Press LLC Coastal Aquifer Management 242 operation are admitted into the search space. Wells not selected do not exist and can be invaded. This strategy much reduces the size of the search space and better solution is obtained. The best solution is a seven-well case as shown in column (6) of Table 1. The toe location in front of the wells is shown in column (7). The total pumping rate is 3,891 m³/day. The saltwater intrusion front is graphically demonstrated in Figure 3, with the well locations marked. We notice that two of the inactive wells, 4 and 12, are intruded by saltwater. 6. STOCHASTIC SIMULATION MODEL The solution presented above assumes deterministic conditions, i.e., all aquifer data are known with certainty. This is not true in reality as hydrogeological surveys are expensive and time consuming to conduct; hence hydrogeological data are rare. The optimization model needs to take this reality into consideration. The first step of conducting a stochastic optimization is to have a stochastic simulation model. This can be accomplished by applying the second order uncertainty analysis of Cheng and Ouazar [1995] to the deterministic model given as Eq. (6). Based on the approximation of Taylor series, the statistical moments of toe location can be related to the moments of uncertain parameters as [Naji et al., 1998] () 22 22 22 1 , 2 toe toe toe toe qK xx xxqK qK σ σ   ∂∂ =+ +   ∂∂   (13) 22 222 toe toe x qK xx qK σ σσ   ∂∂ =+   ∂∂   (14) where toe x , q , and K are respectively the mean toe location, the mean freshwater outflow rate, and the mean hydraulic conductivity; 2 x σ , 2 q σ , and 2 K σ are respectively the variance of toe location, freshwater outflow rate, and hydraulic conductivity; and ( ) , toe x qK is the toe location evaluated using the mean parameter values. In the above, we have neglected the covariance qK σ by assuming that it is small. The above equations state that in order to obtain the mean toe location and its standard deviation, we first need to calculate the toe location using the mean parameter values, i.e., ( ) , toe x qK . This is obtained from the deterministic solution by solving Eq. (6) using the given q and K values. Next, we need to find the partial derivatives of toe location © 2004 by CRC Press LLC [...]... “Conceptual and mathematical modeling, Chap 5, 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, 127–161, 1999 Bear, J and Cheng, A.H.-D., “An overview,” Chap 1, In: Seawater Intrusion in Coastal Aquifer Concepts, Methods, and Practices, eds J Bear, A.H.-D Cheng, S Sorek, D Ouazar and I Herrera, Kluwer, 1–8,... Intrusion and Coastal Aquifers Monitoring, Modeling, and Management, Merida, Mexico, March 30–April 2, 2003b © 2004 by CRC Press LLC 254 Coastal Aquifer Management Chan, N., “Partial infeasibility method for chance-constrained aquifer management,” J Water Resour Planning Management, ASCE, 120, 70–89, 1994 Charnes, A and Cooper, W.W., “Chance-constrained programming,” Mgmt Sci., 6, 73–79, 1959 Charnes, A and. .. equivalents for optimizing and satisfying under chance constraints,” Oper Res., 11, 18–39, 1963 Cheng, A.H.-D., Halhal, D., Naji, A and Ouazar, D., “Pumping optimization in saltwater-intruded coastal aquifers,” Water Resour Res., 36, 2155–2166, 2000 Cheng, A.H.-D and Ouazar, D., “Theis solution under aquifer parameter uncertainty,” Ground Water, 33, 11 15, 1995 Cheng, A.H.-D and Ouazar, D., “Analytical... 120 113 120 111 120 89 120 81 120 78 120 66 120 67 120 85 120 91 120 70 120 89 120 65 120 25 120 30 120 22 120 19 120 20 120 34 120 20 120 15 120 12 120 13 120 14 120 12 Table 2: Pumping well locations and discharge limits for the Miami Beach aquifer age, corresponding to coastal piedmonts and alluvial fans, and is generally unconfined and single-layered The sediment consists of clay and gravel, and. .. 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Wagner, B.J and Gorelick, S.M., “Optimal groundwater quality management under parameter uncertainty,” Water Resour Res., 23, 116 2 117 4, 1987 Willis, R and Finney, B.A., “Planning model for optimal control of saltwater intrusion,” J Water Resour Planning Management, ASCE, 114 , 333–347, 1988 Willis, R and Yeh, W.W.-G., Groundwater Systems Planning and Management, Prentice-Hall, 1987 Young, R.A and Bredehoe,... front of the three cases, 2, 4, and 5 are shown in Figure 8 for comparison 9 CONCLUSION In this chapter we presented an optimization model for maximizing the benefit of pumping freshwater from a group of coastal wells under the threat of saltwater invasion In view of the real-world situation, the aquifer properties are assumed to be uncertain, and are given in terms of mean values and standard deviations . solutions [Cheng and Ouazar, 1999] to the various finite-element- and finite-difference-based numerical solutions [Sorek and Pinder, 1999]. In principle, any of the above models and methods can. and the evolution continues. The process is terminated © 2004 by CRC Press LLC Coastal Aquifer Management 240 1 2 3 4 5 6 7 89 10 11 12 13 14 15 0 1000 2000 3000 4000 -3 000 -2 500 -2 000 -1 500 -1 000 -5 00 0 500 1000 1500 2000 2500 y. 351 111 7 3 1500 850 110 0 150 0 1257 4 1200 400 800 150 0 1372 5 1700 200 1300 150 150 1514 6 1800 -3 00 1400 150 0 1344 7 3500 -5 00 1500 150 1497 1323 8 1600 -8 00 1200 150 0 1 311 9 1600 -1 200