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- 1. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935 Elitist Genetic Algorithm For Design Of Aquifer Reclamation Using Multiple Well System S.M.V. Sharief* T.I.Eldho ** A.K.Rastogi ***Professor, Department of Mechanical Engineering, Ramachandra College of Engineering, Eluru, Andhra Pradesh.** Professor, Department of Civil Engineering, Indian Institute of Technology, Bombay. Mumbai-400076, India.ABSTRACT Groundwater contamination is one of the I. INTRODUCTIONmost serious environmental problems in many Worlds growing industrialization andparts of the world today. For the remediation of planned irrigation of large agricultural fields havedissolved phase contaminant in groundwater, the caused deterioration of groundwater quality. Withpump and treat method is found to be a the growing recognition of the importance ofsuccessful remediation technique. Developing an groundwater resources, efforts are increasing toefficient and robust methods for identifying the prevent, reduce, and abate groundwater pollution. Asmost cost effective ways to solve groundwater a result, during the last decade, much attention haspollution remediation problems using pump and been focused on remediation of contaminatedtreat method is very important because of the aquifers. Pump and treat is one of the establishedlarge construction and operating costs involved. techniques for restoring the contaminated aquifers.In this study an attempt has been made to use The technique involves locating adequate number ofevolutionary approach (Elitist genetic algorithm pumping wells in a polluted aquifer where(EGA) and finite element model (FEM)) to solve contaminants are removed with the pumped outthe non-linear and very complex pump and treat groundwater. Various treatment technologies aregroundwater pollution remediation problem. The used for the removal of contaminants and cleanupmodel couples elitist genetic algorithm, a global strategies. Depending on the site conditions, somesearch technique, with FEM for coupled flow and times the treated water is injected back in to thesolute transport model to optimize the pump and aquifer. Many researchers found that the combinedtreat groundwater pollution problem. This paper use of simulation and optimization techniques can bepresents a simulation optimization model to a powerful tool in designing and planning strategiesobtain optimal pumping rates to cleanup a for the optimal management of groundwaterconfined aquifer. A coupled FEM model has been remediation by pump and treat method. Differentdeveloped for flow and solute transport optimization techniques have been employed in thesimulation, which is embedded with the elitist groundwater remediation design involving lineargenetic algorithm to assess the optimal pumping programming [1], nonlinear programming [2-4] andpattern for different scenarios for the dynamic programming [5] methods. Application ofremediation of contaminated groundwater. Using optimization techniques can identify site specificthe FEM-EGA model, the optimal pumping remediation plans that are substantially lesspattern for the abstraction wells is obtained to expensive than those identified by the conventionalminimize the total lift costs of groundwater along trial and error methods. Mixed integer programmingwith the treatment cost. The coupled FEM-EGA was also used to optimal design of air strippingmodel has been applied for the decontamination treatment process [4].of a hypothetical confined aquifer to demonstratethe effectiveness of the proposed technique. The Bear and Sun [6] developed design formodel is applied to determine the minimum optimal remediation by pump and treat. Thepumping rates needed to remediate an existing objective was to minimize the total cost includingcontaminant plume. The study found that an fixed and operating costs by pumping and injectionoptimal pumping policy for aquifer rates at five potential wells. Two–level hierarchicaldecontamination using pump and treat method optimization model was used to optimization ofcan be established by applying the present FEM- pumping /injection rates[6]. At the basic level, wellEGA model. locations and pumping and injection rates are sought so as to maximize mass removal of contaminants. AtKeywords: Pump and treat, Finite element method, the upper level, the number of wells for pumping /Simulation–optimization, Aquifer reclamation, injection is optimized, so as to minimize the cost,Elitist genetic algorithm. taking maximum contaminant level as a constraint. Recently the applications of combinatorial search algorithms, such as genetic algorithms (gas) have been used in the groundwater management to 924 | P a g e
- 2. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935identify the cost effective solutions to pump and study develops an optimal planning model fortreat groundwater remediation design. Genetic cleanup of contaminated aquifer using pump andalgorithms are heuristic programming methods treats method. This study integrates the elitistcapable of locating near global solutions for genetic algorithm and finite element model to solvecomplex problems [7]. A number of researchers the highly nonlinear groundwater remediationhave applied genetic algorithms in the recent past to systems problem. The proposed model considers thesolve groundwater pollution remediation problems operating cost of pumping wells and subsequent[8-19]. treatment of pumped out contaminated groundwater. Ritzel and Eheart [8] applied GA to solve a Minimizing the total cost to meet the water qualitymultiple objective groundwater pollution constraints, and the model computes the optimalcontainment problem. They used the response matrix pumping rate to restore the aquifer.approach. Vector evaluated genetic algorithm andPareto genetic algorithms have been used for In the present problem, the geneticmaximization of reliability and minimization of algorithm is chosen as optimization tool. GAs arecosts. The Pareto algorithm was shown to be adaptive methods which may be used to solve searchsuperior to the vector evaluated genetic algorithm. and optimization problems. GA is robust iterativeGenetic algorithms have been used to determine the search methods that are based on Darwin evolutionyield from a homogeneous, isotropic, unconfined and survival of the fittest [7, 31]. The fact that GAsaquifer and also, to determine the minimum cost of have a number of advantages over mathematicalpump and treat remediation system to remove the programming techniques traditionally used forcontaminant plume from the aquifer using air optimization, including (1) decision variables arestripping treatment technology [9]. represented as a discrete set of possible values, (2) the ability to find near global optimal solutions , and Wand and Zheng [10] developed a new (3) the generation of a range of good solutions insimulation optimization model developed for the addition to one leading solution. Consequently, GAsoptimal design of groundwater remediation systems is used as the optimization technique for theunder a variety of field conditions. The model has proposed research. The GA technique andbeen applied to a typical two-dimensional pump and mechanism selected as a part of proposed approachtreat example with one and three management are binary coding of the decision variables, rouletteperiods and also to a large-scale three-dimensional wheel selection, uniform crossover, bitwise mutationfield problem to determine the minimum pumping and elitism technique. In the present study, aneeded to contain an existing contaminant plume simulation-optimization model has been developed[10]. Progressive Genetic algorithms were used to by embedding the features of the finite elementoptimize the remediation design, while defining the method (FEM) with elitist genetic algorithm (EGA)locations and pumping and injection rates of the for the assessment of optimal pumping rate forspecified wells as continuous variables [20]. The aquifer remediation using pump and treat method.proposed model considers only the operating cost of The developed model is applied to a hypotheticalpumping and injection. But the pumping and problem to study the effectiveness of the proposedinjection rates are not time varying. Real-coded technique and it can be used to solve similar realgenetic algorithm are also applied for groundwater field problems.bioremediation[21]. II. GROUNDWATER FLOW AND MASS Integrated Genetic algorithm and TRANSPORT SIMULATIONconstrained differential dynamic programming were The FEM formulation for flow and soluteused to optimize the total remediation cost[20]. But transport followed by genetic algorithm approach isthe decision variables only involve determining time considered in this study. The governing equationvarying pumping rates from extraction wells and describing the flow in a two dimensionaltheir locations. FEM is well established as a heterogeneous confined aquifer can be written assimulation tool for groundwater flow and solute [26]transport [22-26]. h h h Many of the research works revealed that Tx x y Ty y S t Q w ( x x i )(y y i ) q x the high cost of groundwater pollution remediation can be substantially reduced by simulation (1)optimization techniques. Most of the groundwater Subject to the initial condition ofpollution remediation optimization problems are h(x, y,0) = h 0 ( x, y) x, y (2)non-convex, discrete and discontinuous in nature and the boundary conditionsand GA has been found to be very suitable to solve h (x, y, t ) = H (x, y, t ) x, y 1 (3)these problems. GA uses the random search schemesinspired by biological evolution [7]. The present 925 | P a g e
- 3. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935 h g 2 = concentration gradient normal to theT = q (x, y, t ) x, y 2 (4) n boundary 2 ; n = unit normal vector.where h (x, y, t ) = piezometric head (m); T(x, y) =transimissivity (m2/d); S = storage coefficient; x, y For numerical analysis, the finite element= horizontal space variables (m); Q w = source or method is selected because of its relative flexibilitysink function ( Q w source, Q w = Sink) (m3/d/m2); to discretize a prototype system which represents boundaries accurately. The finite element methodt = time in days; = the flow region; = (FEM) approximates the governing partial differential equation by integral approach. In theboundary region ( 1 2 ); = normal n present study, two-dimensional linear triangularderivative; h 0 ( x, y) = initial head in the flow elements are used for spatial discritization anddomain (m); H(x, y, t ) = known head value of the Galerkin approach [29] is used for finite element approximation. In the FEM, the head variable inboundary head (m) and q(x, y, t ) = known inflow equation (1) is initially approximated asrate ( m3/d/m). NP The governing partial differential equation h ( x , y, t ) h L 1 L ( t ) N L ( x , y) (9)describing the physical processes of convection andhydrodynamic dispersion is basically obtained from where h L = unknown head; N L = known the principle of conservation of mass [27]. The basis function at node L ; h ( x, y, t ) = trial solution;partial differential equation for transport of totaldissolved solids (TDS) or a single chemical NP = total number of nodes in the problem domain. A set of simultaneous equations is obtained whenconstituent in groundwater, considering advectiondispersion and fluid sources/sinks can be given residuals weighted by each of the basis function are forced to be zero and integrated over the entire[24,28] as follows: domain. Applying the finite element formulation for C C C R = D xx D yy equation (1) gives t x x x y h h h Vx C Vy C c`w W C0 Q C Tx Ty Q w q S N L (x, y)dxdy 0 x x y y t x y nb (10) (5) Equation (10) can further be written as thewhere Vx and Vy = seepage velocity in x and y summation of individual elements as e e e he N L ˆ e ˆ N e ˆe T y h L dxdy S h N L dxdy principal axis direction; D xx , D yy = components Tx e e x x y y e t of dispersion coefficient tensor[L2T-1]; C = Q N dxdy q N dxdyDissolved concentration [ML-3]; = porosity(dimensionless); w = elemental recharge rate with w e L e L e esolute concentration c ; n = elemental porosity; b (11)= aquifer thickness under the element; R = N i Retardation factor: C 0 = concentration of solute ininjected water into aquifer; W = rate of water where Ne L N j .The details of interpolation injected per unit time per unit aquifer volume (T -1) ; N k Q = volume rate of water extracted per unit aquifer function for linear triangular elements are givenvolume. among others by [29].The initial condition for the transport problem is For an element eq(11) can be written in matrix form C(x, y,0) f (x, y) (6) as G h P ht F and the boundary conditions are e e e e I e C(x, y, t ) g1 (x, y) 1 (7) I (12) C ( x , y, t ) g 2 (8) where I = i, j, k are three nodes of nwhere 1 = boundary sections of the flow region ; triangular elements and G, P, F are the element matrices known as conductance , storage matricesf = a given function in ; g 1 = given function or recharge vectors respectively.along 1 , which is known solute concentration. Summation of elemental matrix equation (12) for all the elements lying within the flow region gives the global matrix as 926 | P a g e
- 4. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935G h L P h L F III. ELITIST GA OPTIMIZATION (13) MODEL FORMULATION t Genetic algorithm is a Global searchApplying the implicit finite difference scheme for procedure based on the mechanics of natural h selection and natural genetics, which combines anthe term in time domain for equation (13) artificial survival of the fittest with genetic operators t abstracted from the nature [7]. In this paper, elitistgives genetic algorithm is used as optimization tool. The h t t h t EGA procedure consists of three main operators:Gh L t t P F (14) selection, crossover and mutation Elitism ensures t that the fitness of the best solution in a population doesn‟t deteriorate as generation advances Genetic The subscripts t and t t represent the algorithm converge the global optimal solution ofgroundwater head values at present and next time some functions in the presence of elitism In order tosteps. By rearranging the terms of equation (14), the preserve and use previously best solutions in thegeneral form of the equation can be given as[25] subsequent generations an elite preserving operator[P] t [G] ht t = is often recommended [30]. In addition to overall increase in performance, there is another advantage[P] (1 ) t [G] ht + t (1 ) Ft + Ft t of using such elitism. In an elitist GA, the statistics of the population best solutions cannot degrade with (15)where t = time step size; ht and ht t are generations. There exists a number of ways to introduce elitism. In the present study, the best %groundwater head vector at the time t and of the population from the current population is t t respectively ; = Relaxation factor which directly copied to next generation. The rest (100 - )depends on the type of finite difference scheme % of the new population is created by usual geneticused. In the present study Crank-Nicholson scheme operations applied on the entire current populationwith = 0.5 is used. These global matrices can be including the chosen % elite members. In this wayconstructed using the element matrices of different the best solutions of the current population is notshapes based on discritization of the domain. The set only get passed from one generation to another, butof linear simultaneous equations represent by eq (15) they also participate with other members of theare solved to obtain the groundwater head population in creating other population members.distribution at all nodal points of the flow domain The general scheme for the elitist GA is explainedusing Choleski‟s method for the given initial and below [31].boundary conditions, recharge and pumping rates,transsmissivity, and storage coefficient values. After 1. The implementation of GA starts with a randomlygetting the nodal head values, the time step is generated set of decision variables. The initialincremented and the solution proceeds in the same population is generated randomly with a randommanner by updating the time matrices and seed satisfying the bounds and sensitivityrecomputing the nodal head values. From the requirement of the decision variables. Theobtained nodal head values, the velocity vectors in population is formed by certain number of encoded x and y directions can be calculated using Darcy‟s strings in which each string has values of „0‟s andlaw. For solving transport simulation, applying „1‟s, each of which represents a possible solution.Gelarkin‟s FEM to solve partial differential equation The total length of the string (i.e., the number of(5), final set of implicit equations can be given as binary digits) associated with each string is the total(similar to eq 15) number of binary digits used to represent each [S] t [D] Ct t = substring. The length of the substring is usually [S] (1 ) t [D] Ct + t (1 ) Ft + Ft t determined according to the desired solution accuracy. (16) 2. The fitness of each string in the population iswhere S = sorption matrix; D = advection– evaluated based on the objective function anddispersion matrix; F = flux matrix; C = nodal constraints. Various constraints are checked duringconcentration column matrix and the subscripts t the objective function evaluation stage. The amountand t+ t indicates the concentration at present and of any constraint violation is multiplied by weightforward time step respectively. Solution of eq (16) factor and added to the fitness value as a penalty.with the initial and boundary conditions gives the The multiplier is added to scale up the penalty, sosolute concentration distribution in the flow region that it will have a significant, but not excessivefor various time steps. impact on the objective function. 3. This stage selects an interim population of strings representing possible solutions via a stochastic selection process. The selected better fitness strings 927 | P a g e
- 5. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935are arranged randomly to form mating pool on which which include concentration constraints, constraintfurther operations like crossover mutation are for extraction rates and also constraints for nodalperformed. Reproduction is implemented in this head distribution.study using the Roulette wheel approach [7]. In In the present study, the objective is to minimize theroulette wheel selection, the string with the higher total cost of groundwater lift and the treatment costfitness value represents a large range in the for a fixed remediation period to achieve a desiredcumulative probability values and it has higher level of cleanup standards. The appropriateprobability of being selected for reproduction. objective function for this remediation is consideredTherefore, the algorithm can converge to a set of as, Qi t h chromosomes with higher fitness values. NP NP4. Crossover involves random coupling of the newly f = a1 gl hi + a 2 Qi t (17) ireproduced strings, and each pair of strings partially i 1 i 1exchanges information. Crossover aims to exchange where, f = objective function for minimization ofgene information so as to produce new offspringstrings that preserve the best material from the the system cost; Q i = pumping rate from the well;parent strings. The end result of crossover is the h iglcreation of a new population which has the same = pump efficiency; = ground level at well i;size as the old population, and which consists of h i = piezometric head at well i; a 1 = cost coefficientmostly child strings whose parents no longer existand a minority of parents luckily enough to enter the for total energy unit in kWhr (INR 20.0/kWhr); a 2 =new population unaltered. cost coefficient for treatment (INR 10.0 /m3); t =5. Mutation restores lost or unexplored genetic duration of pumping; = unit weight of water;material to the population to prevent the GA from glprematurely converging to a local minimum. (h h i ) i = groundwater lift at well i and INR =Mutation is performed by flipping the values of „0‟ Indian national Rupees (1US$ = INR 47/-and „1‟ of the bit that is selected. A new population approximately); and NP number of pumping wells.is formed after mutation. The new population isevaluated according to the fitness, i.e., objective Constraints to the problem are specified as Subjectfunction. to6. The best % of the population from the current Ci Cpopulation is directly copied to next generation. The i 1...... Nrest (100- ) % of the new population is created by h m in h i h m axusual genetic operations applied on the entire current ; i 1...... Npopulation including the chosen % elite members. Qi, m in Qi Qi, m ax7. The next step is to choose the appropriate i 1...... NPstopping criterion. The stopping criterion is based on where, i is the node number; N is the total number the change of either objective function or optimized of nodes of the flow domain;. C is specified limitparameters. If the user defined stopping criterion is Qmet or when the maximum allowed number of of concentration in the region; and i = Pumpinggenerations is reached, the procedure stops, rate at well at i. Decision variables for the pump andotherwise it goes back to reproduction stage to treat remediation system include the pumping ratesperform another cycle of reproduction, crossover, for the wells and the state variables are the hydraulicand mutation until a stopping criterion met. head and solute concentration which are dependent variables in the groundwater flow and transport equations. For the remediation design, simulationIV. INTEGRATION OF FEM-ELITISTGA model has to update the state variables, which areSIMULATION–OPTIMIZATION passed on to the optimization model to select theAPPROACH optimal values for the decision variables. The flow To clean up the contaminated aquifer by chart for the developed FEM-EGA model is shownpump and treat, pumping rates are to be optimized to in Fig.1.achieve the minimum specified concentration levelswithin the system after a particular period. Here a V. OPTIMAL REMEDIATION SYSTEMcoupled FEM-EGA model is developed to find out DESIGN WITH FEM-EGA MODEL – Athe optimal pumping rates in the pump and treat CASE STUDYmethod. A binary coded genetic algorithm is A hypothetical problem of solute massembedded with the flow and transport FEM model transport in a confined aquifer is considered fordescribed earlier, where pumping rates are defined remediation of contaminated groundwater. Theas continuous decision variables. In the optimization aquifer is assumed to be homogeneous, anisotropicformulation, three constraints were considered, and flow is considered to be two-dimensional. When 928 | P a g e
- 6. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935the remediation process commences, it is reasonably dynamic conditions operative on the system isassumed that the source of pollutants entering in to shown in Fig. 3. Pump and treat method is appliedthe aquifer system is stopped. for the aquifer remediation after the sources entering Important data for the considered aquifer in the system are arrested (i.e., no recharge throughare as follows: The aquifer is discretised into finite the polluted pond and recharge well). From theelement mesh with a grid spacing of 200 m 200 existing three abstraction wells, the contaminants inm. Thickness of the confined aquifer = 25 m; Size of the system are pumped out for the treatment. Basedthe confined aquifer = 1800 m 1000 m; storativity on the concentration distribution at the end of the= 0.0004; aquifer is recharged through some aquitard simulation period in the system, the location ofin the area of 0.6 km2 (between node numbers 19 to pumping wells has been changed, so that the42) at the rate of 0.00012 m/d; porosity = 0.2; pumping wells are situated in the highly pollutedlongitudinal dispersivity = 150 m; transverse area. Therefore for the aquifer remediation thedispersivity = 15 m; rate of seepage from the pond = recharge well at node 21 is converted as a pumping T well (pumping well at node 23, 35 closed).0.005 m/d; Txx = 300 m 2/d, and y y = 200 m2/d. Additionally one pumping well and one observationThe boundary conditions for the flow model are well which is located at nodes 33 and 52 areconstant head on east (70 m) and west sides (75 m), considered for pumping out the contaminated waterand no flow on the north and south sides. For for treatment on the ground surface.transport model, the boundary conditions are zeroconcentration on west and concentration gradient is The FEM simulation and EGA optimizationzero on north and south directions. Since the area of techniques have been used for obtaining an optimalthe aquifer is relatively small (1.8 km2) the boundary pumping pattern to cleanup the contaminatedfor the east direction is kept open. aquifer. The optimization of pump and treat remediation design requires that the flow and solute The flow region is discretised into transport model is run repeatedly, in order totriangular elements consisting of 60 nodes and 90 calculate the objective function associated with theelements as shown in Fig. 2. The groundwater flow possible design, and to evaluate whether variousand contaminant spread in the aquifer system is hydraulic and concentration constraints are met. Insimulated using FEM to obtain nodal heads, velocity the present work, the compliance requirement for thevectors and nodal concentration distribution in the contaminants concentration level is considered to beaquifer domain. Three production wells extracting lower than 750 ppm everywhere in the aquifer at thewater at the rate of 500, 600 and 250 m 3/d end of 3960 days. The objective of the present studyrespectively are considered in the flow region. These is to obtain optimal pumping rates for the wells towells are located at nodes 23, 33 and 35 respectively minimize the total lift cost of groundwater involving(Fig.2). A recharge well with inflow rate of 750 m in definite volume of contaminated groundwater for3 /d is located at node 21. Three observation wells are treatment for the specified time period. Threelocated in the aquifer domain at nodes 44, 47, and 52 pumping wells scenario is considered for thisrespectively to monitor the concentration levels remediation period to cleanup the aquifer for a fixedduring the remediation period. The problem involves cleanup period of 3960 days. The optimal pumpingseepage from a polluted pond into the underlying pattern (least cost) for the chosen scenario has beenaquifer. The bottom of the pond is assumed to be obtained by the coupled simulation optimizationsufficiently close to the groundwater surface so that model.seepage underneath the pond occurs in a saturatedregion. The recharge rate (0.005 m/d) from the pond Three scenarios have been considered incontrols the amount of solute introduced into the this study to cleanup the contaminated aquifer for asystem with a certain velocity. In this problem, the fixed cleanup period of 3960 days. Three pumpingcontaminant is seeping into the flow field from the wells, two wells and one well respectively arepolluted pond (4000 ppm TDS) and is also added by considered for this remediation period to treat thesea recharge well (1000 ppm). Initially the entire scenarios. The optimal pumping pattern (least cost)aquifer domain is assumed to be for three scenarios has been obtained by the coupleduncontaminated (C 0) . This study examines the simulation optimization model.coupled flow and solute transport problem withthese input sources. Pumping wells are also active VI. TUNING OF GENETIC ALGORITHMfor a simulation period of 10000 days during which PARAMETERSperiod the aquifer continues to get contaminated. At Important genetic algorithm parametersthe end of this simulation period the TDS such as size of population, crossover, mutation,concentration distribution is considered as initial number of generations, and termination criterionconcentration levels in the aquifer domain before the affect the EGA performance. Hence appropriateremediation begins. Concentration distribution at values have to be assigned for these parametersthe end of 10000 days of simulation period for these before they are used in the optimization problem. 929 | P a g e
- 7. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935For some parameters several trial runs may beneeded to “tune” these parameters. Each parameter Number of generationswas varied to determine its effect on the system From practical considerations, the selectionwithin reasonable bounds while keeping remaining of maximum generation is governed by the limitsparameters at a standard level. The obtained optimal placed on the execution time and the number ofvalues are used in the simulation optimization model function evaluations. In each generation, thefor these scenarios. objective function is calculated maximum population times, (except in the first generation inRate of mutation which the fitness of the initial population is also In the present study for three scenarios, the calculated) and the various genetic operators are alsomutation rate was varied between 0.1%, 1.0%, and applied several times.10% per bit. Probability of mutation is usuallyassigned a low value between 0.001 and 0.01., VII. RESULTS AND DISCUSSIONsince it is only a secondary genetic operator. Although the specific test cases areMutation is needed to introduce some random synthetic and relatively idealized, the overalldiversity in the optimization process. However if approach is realistic. The installation of pumpingprobability of mutation is too high, there is a wells at appropriate locations may be necessary todanger of losing genetic information too quickly, prevent further spreading of the contaminants toand the optimization process approaches a purely unpolluted areas. At the end of the remediationrandom search period, the concentration levels everywhere in the system are expected to be lower than the specifiedRate of crossover concentration limit of 750 ppm. In all scenarios a Probability of crossover between 0.60 and fixed management period of 3960 days is used.1.0 is used in most genetic algorithm Various design strategies for the remediation periodimplementations. Higher the value of the crossover, are presented in Table.1. The following sectionhigher is the genetic recombination as the algorithm provides the details of the results obtained for theproceeds, ensuring higher exploitation of genetic chosen problem for various scenarios.information. Three wells scenario Population size In this three well scenario, three pumping The optimal value of maximum population wells are considered for pumping out thefor binary coding increases exponentially with the contaminated groundwater. Lower and upperlength of the string [7]. Even for relatively small bounds on the pumping capacity are chosen as 250problems, this would lead to high memory storage m3/d and 1000 m3/d respectively. The pumping ratesrequirements. Results have shown that larger of the three wells are encoded as a 30 bit binarypopulation performs better. This is expected as string with 10 bits for each pumping rate. Thelarger populations represent a large sampling of the optimal GA parameters as discussed earlier sectionsdesign space. are used in this study. The optimal size of the population is chosen as 30. A higher probability ofMinimum reduction in objective function for crossover of 0.85 and lower mutation probability istermination set at 0.001. A converged solution is obtained after The selection of this parameter is problem 128 generations. Fig. 4 shows the final concentrationdependent, and its appropriate value depends on how contours at the end of 3960 days of remediationmuch improvement in objective function value may period based on the optimal pumping rates achievedbe considered significant. The value of this by simulation–optimization model (FEM-EGA). Aparameter is determined by trial and error, and a best value of objective function versus number ofvalue of 3 to 4 usually quite sufficient. Larger values generations for three well scenarios is presented inof this parameter increase the chances of converging Fig.6. The optimal pumping rates for the three wellsto the global optimum, but at the cost of increasing are 300.58, 260.26 and 275.65 m3/d respectively asthe computational time shown in Fig.5. Penalty function Although the three wells scenario performs Specifying a penalty coefficient is a good to cleanup the system to the requiredchallenging task. A high penalty coefficient will concentration level everywhere in the system, asensure that most solutions lie with in the feasible expected it leads to a higher system cost. Also itsolutions space, but can lead to costly conservative needs to be emphasized that an increase in thesystem designs. A low penalty coefficient permits number of pumping wells to pump out thesearching for both feasible and infeasible regions, contaminated groundwater is not the first choice.but can cause convergence to an infeasible system However, if the pumping wells are placed at andesign [30]. appropriate location, a further spreading of the 930 | P a g e
- 8. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935contaminant plume can be minimized. This means scenarios, the concentration levels in the aquiferthat more effective remediation can be achieved by domain is less than that the required valueappropriately locating the pumping well. These everywhere in the system, where as for one wellinvestigations show that an appropriately designed scenario the maximum concentration level ispump and treat system can have a significant effect marginally less than the specified cleanup standardon the decontamination of a polluted aquifer and limit for the system. Further results in Table.1preclude further spreading of contaminant plume. shows that increasing the number of pumping wells is not feasible, since it involves high system cost andTwo wells scenario less performance. Hence the optimal pumping rate During the remediation period, two obtained in this one well scenario is the bestabstraction wells are considered for pumping out the pumping policy for the chosen problem to cleanupcontaminated groundwater. The lower and upper the aquifer with in the specified time period. Thesebounds of the abstraction wells are 250 m 3/d and investigations show that an appropriately designed1000 m3/d respectively. The pumping rates of the pump and treat system can have a significant effecttwo abstraction wells are encoded as a 20 bit binary on the decontamination of a polluted aquifer andstring with 10 bits for each pumping rate. The preclude further spreading of contaminant plume.optimal size of the population is taken as 30. Ahigher probability of crossover of 0.85 is chosen and VIII. CONCLUSIONSlower mutation probability is set at 0.001. A Here an integrated simulation–optimizationconverged solution is obtained after 30 generations. model is developed for the remediation of a confinedThe progress of the aquifer decontamination for this aquifer that is polluted by a recharge pond andscenario is shown in Fig 6. The optimal pumping injection well operational for a prolonged period ofrates obtained for the two wells are 252.199 m 3/d time. The two-dimensional model consists of threeand 260.263 m3/d respectively. Best values of the scenarios for remediation of contaminated aquifer.objective function versus number of generations for Studies demonstrate the effectiveness and robustnesstwo well scenario is presented in Fig.7. of the simulation optimization (FEM-EGA) model. The problem used a cost based objective function forOne well scenario the optimization technique that took into account the For this case, the length of the management costs of pumping and treatment of contaminatedperiod is same as that of three and two well groundwater for the aquifer system. Optimalscenarios. One pumping well is considered for pumping pattern is obtained for cleanup of thepumping out the contaminated groundwater during aquifer using FEM and EGA for a period of 3960the remediation period. The lower and upper bounds days. Comparing the total extraction and pumpingof the pumping well are kept same as for the above patterns for the three scenarios, it was found that thetwo scenarios. The pumping rate of the one one well scenario reduces the total pumping rate andabstraction well is encoded as a string length of 10 shows the dynamic adaptability of the EGA basedbits. The optimal size of the population is chosen to optimization technique to achieve the cleanupbe 20. A moderate crossover probability of 0.85 is standards. The total remediation cost for the twotaken and lower mutation probability is set at 0.001. well scenarios is relatively less as compared to threeA converged solution is obtained after 25 and one well scenario. This is due to the pumpinggenerations which is relatively less as compared to well locations which are placed in the highlythe three well scenarios. Results indicate that the polluted area, so that removal of contaminant fromoptimal pumping rate for one well for this case is these pumping wells is effective. The total346.77 m3/d. This pumping rate is much lower extraction of contaminated groundwater gives ancompared to other two scenarios. Even for the lower indication of costs associated with pumping andpumping rate the mass remaining in the system (728 treatment costs in the system. Hence the optimalppm) is marginally less than the specified limit (750 pumping rate obtained in this one well scenario isppm). Therefore one well scenario is perhaps the the best pumping policy for the chosen problem tobest option to cleanup the aquifer within the cleanup the aquifer within the specified time period.specified time period. Fig. 8 shows the final The important advantage of using EGA inconcentration contours at the end of 3960 days of remediation design optimization problem is that theremediation period based on the optimal pumping global or near global optimal solutions can be found.rate. Values of objective function versus number of Since there is no need to calculate the derivatives ofgenerations for one well scenario are presented in the objective function which often creates aFig. 9. Figure suggests marginal increase in the numerical instability, the GA based optimizationvalues after 5 generations and minimum at 25 model is robust and stable. However, the studygenerations. found that application of genetic algorithms in remediation design optimization model are The results suggest that, for the remediation computational intensive for tuning of EGAof contaminated aquifer using three and two well parameters. 931 | P a g e
- 9. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935 [13] Smalley,J.B., Minsker, B.S., and GoldbergREFERENCES ,D.E. (2000) . “Risk based insitu bio- [1] Atwood, D.F., and Gorelick, S. M. (1985). remediation design using a noisy genetic Hydraulic gradient control for groundwater algorithm.” Water Resour. Res., 36 (10), contaminant removal,” J. Hydro., 3043-3052. Amsterdam, 76, 85-106. [14] Haiso, C.T., and Chang, L.C. (2002). [2] Ahlfeld, D. P., Mulvey, M .J. Pinder, G.F., “Dynamic optimal groundwater and Wood, E. F. (1988a). “Contaminated management with inclusion of fixed costs.” ground water remediation design using J. Water Resourc. Plng. and Mgmt., 128 simulation, optimization, and sensitivity (1), 57-65. theory 1. Model Development.” Water [15] Maskey, S., Jonoski, A., and Solomatine, Resour. Res., 24(3), 431-441. D. P. (2002). “Ground water remediation [3] Ahlfeld, D. P., Mulvey, J.M., and Pinder, strategy using global optimization G. F. (1988b). “Contaminated ground water algorithms.” J. Water Resour. Plan. remediation design using simulation, Manage. 431- 439. optimization, and sensitivity theory 2. [16] Gopalakrishnan, G., MInsker, B.S.,and Analysis of a field site.” Water Resour. Goldberg , D.E. (2003). “Optimal sampling Res., 24(3), 443-452. in a noisy genetic algorithm for risk based [4] McKinney, D. C., and Lin, M. D. (1995). remediation design.”J. of Hydroinformatics, “Approximate mixed-integer non linear 5(1), 11-25 programming methods for optimal aquifer [17] Espinoza, F. P., Minsker, B. M., and remediation design.” Water Resour. Res., Goldberg, D. E. (2005). “Adaptive hybrid 31(3), 731-740. genetic algorithm for ground water [5] Culver, T.B., and Shoemaker, C.A. (1992). remediation design,” J. Water Resour. Plng. “Dynamic optimal control for groundwater and Mgmt., 131 (1), 14-24. remediation with flexible management [18] Horng, J. S., Richard, C., and Peralta, C. periods.” Water Resour. Res., 28(3), 629- (2005). “Optimal in situ bioremediation 641. design by hybrid genetic algorithm – [6] Bear, J., and Sun, Y.W. simulated Annealing,” J. Water Resour. (1998).”Optimization of Pump and treat Plng. and Mgmt., 131 (1), 67-78. (PTI) design for remediation of [19] Chan Hilton, A. B., and Culver, T. B. contaminated aquifer: multi-stage design (2005). “Groundwater remediation design with chance constraints.” J. Contaminant under uncertainty using genetic Hydrology,29(3),225-244. algorithms,” J. Water Resour. Plng. and [7] Goldberg, D. E. (1989). Genetic algorithms Mgmt., 131 (1), 25-34. in search, optimization and machine [20] Guan, J., and Aral, M. M. (1999). “Optimal learning. Addison-Wesley Pub. Company. remediation with well locations and [8] Ritzel, B. J., and Eheart, J. W. (1994). pumping rates selected as continuous “Using genetic algorithms to solve a decision variables.” J. Hydrol., 221, 20-42. multiple objective ground water pollution [21] Yoon, J. H., and Sheomaker, C. A. (2001). containment problem.” Water Resour. Res., “Improved real-coded genetic algorithm for 30(5), 1589-1603. groundwater bioremediation.” J. Comp in [9] McKinney, D. C., and Lin, M. D. (1994). Civil Engg., 15 (3), 224-231. “Genetic algorithm solution of ground [22] Pinder, G. F. (1973). “A Galerkin- finite water management models.” Water Resour. element simulation of ground water Res., 30(6), 1897-1906. contamination on long Island, New York,” [10] Wang, M., and Zheng, C. (1997). “Optimal Water Resour. Res., 9 (6), 1657-1669. remediation policy selection under general [23] Pinder, G. F., and Gray, W. G. (1977). conditions.” Ground Water, 35(5), 757- Finite element simulation in surface and 764. subsurface hydrology. Academic press, [11] Aly, A.H., and Peralta, R.C. (1999). New York. “Comparison of genetic algorithm and [24] Wang, H. and Anderson, M. P. (1982). mathematical programming to design the Introduction to groundwater modeling groundwater cleanup systems.” Water finite difference and finite element methods. Resour. Res., 35(8), 2415-2426. New York: W. H. Freeman and Company. [12] Chan Hilton, A.B., and Culver, T.B. (2000). [25] Istok, J. (1989). Groundwater modeling by “Constraint handling genetic algorithm in the finite element method. Washington: optimal remediation design.” J. Water American Geophysical union. Resour. Plan. Manage. 126(3), 128-137. [26] Bear, J. (1979).”Hydraulics of ground water.” McGra-Hill Publishing, New York. 932 | P a g e
- 10. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935[27] Freeze, R.A., & Cherry. J. A. (1979). Groundwater. New York: Prectice Hal: 0 Englewood Cliffs.[28] Sun, N. 1996. Mathematical modeling of groundwater pollution. New York Springer Coding sel. Publishing House.[29] Reddy, J. N. (1993). An Introduction to the Finite Element Method. New York: Ini. population McGraw-Hill Publishing.[30] Deb, K. (1995). Optimization for engineering design. Prentice Hall of India, Calculate head and New Delhi. concentration using FEM[31] Deb, K. (2001). “Multi-Objective element model for the Optimization Using Evolutionary remediationconstr. Check Algorithms”. John Wiley and sons Ltd., London. constraints Obj.fun. estimate functiomevaluation Sel. % best parents population Select fittest string & mating pool Creation of offspring from crossover Offspring Mutation New population = Best % parents from population + (100 )% of mutated population Population Obj. fun. estimation N Has stop o Criterion met? Y e 1 s 933 | P a g e
- 11. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.924-935 30 36 42 12 18 24 No flow boundary 48 54 6 60 6 90 5 59 Constant 4 58 Constant Head Pond Head 3 57 Head 2 56 2 1 Node 1 55 Numbers 7 13 No flow boundary 43 49 19 25 31 37 Pumping well Recharge well Areal recharge Observation well Fig. 2. Finite element discritization 1000 800 600 400 200 0 0 200 400 600 800 1000 1200 1400 1600 1800 Fig.3. Concentration distribution at the end of 10000 days of simulation (Initial concentration before remediation commences) 1000 800Distance (m) 600 400 200 0 0 200 400 600 800 1000 1200 1400 1600 1800Fig.4. Concentration distribution at the end of 3960 Fig.5. Values of objective function versus numberdays remediation for three well scenario of generations for three well scenario 934 | P a g e
- 12. S.M.V. Sharief, T.I.Eldho, A.K.Rastogi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com 1000 Vol. 2, Issue 5, September- October 2012, pp.924-935 20000000 System performance in rupees 800 19000000 18000000Distance (m) 600 17000000 16000000 400 15000000 200 14000000 0 10 20 30 40 50 No of generations 0 0 200 400 600 800 1000 1200 1400 1600 1800 Fig.7. Values of objective function versus number Fig.6. Concentration distribution at the end of 3960 of generations for two well scenario days remediation for two well scenario 1000 800Distance (m) 600 400 200 0 0 200 400 600 800 1000 1200 1400 1600 1800 Fig.8. Concentration distribution at the end of 3960 Fig.9. Values of objective function versus number days remediation for one well scenario of generations for one well scenario Table.1. Various design strategies for the remediation period Design Pumping rates Total pumping for Evaluated strategies for each well the remediation cost (m3/d) period. (m3) (Rupees) Three 289.58 3300897 25459830 well 264.66 279.32 Two well 252.19 2029302 15384529 260.26 One well 346.77 1373209 10774175 935 | P a g e

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