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Optimal groundwater remediationdesign of pump and treat systems via asimulation–optimization approach andfirefly algorithmMohammad Javad Kazemzadeh-Parsia, Farhang Daneshmandbc,Mohammad Amin Ahmadfarda, Jan Adamowskib & Richard Martelda Department of Mechanical Engineering, Shiraz Branch, IslamicAzad University, Shiraz, Iranb Department of Bioresource Engineering, Faculty of Agriculturaland Environmental Sciences, McGill University, Ste. Anne deBellevue, Canadac Department of Mechanical Engineering, McGill University,Montreal, Canadad Institut National de la Recherche Scientifique (Centre Eau, Terreet Environnement), Université du Québec, Québec, CanadaPublished online: 09 Jan 2014.

To cite this article: Mohammad Javad Kazemzadeh-Parsi, Farhang Daneshmand, Mohammad AminAhmadfard, Jan Adamowski & Richard Martel (2015) Optimal groundwater remediation design ofpump and treat systems via a simulation–optimization approach and firefly algorithm, EngineeringOptimization, 47:1, 1-17, DOI: 10.1080/0305215X.2013.858138

To link to this article: http://dx.doi.org/10.1080/0305215X.2013.858138

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Engineering Optimization, 2015Vol. 47, No. 1, 1–17, http://dx.doi.org/10.1080/0305215X.2013.858138

Optimal groundwater remediation design of pump and treatsystems via a simulation–optimization approach

and firefly algorithm

Mohammad Javad Kazemzadeh-Parsia, Farhang Daneshmandb,c∗ , Mohammad AminAhmadfarda, Jan Adamowskib and Richard Marteld

aDepartment of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran;bDepartment of Bioresource Engineering, Faculty of Agricultural and Environmental Sciences, McGill

University, Ste. Anne de Bellevue, Canada; cDepartment of Mechanical Engineering, McGill University,Montreal, Canada; dInstitut National de la Recherche Scientifique (Centre Eau, Terre et Environnement),

Université du Québec, Québec, Canada

(Received 6 October 2012; accepted 1 October 2013)

In the present study, an optimization approach based on the firefly algorithm (FA) is combined with afinite element simulation method (FEM) to determine the optimum design of pump and treat remediationsystems. Three multi-objective functions in which pumping rate and clean-up time are design variables areconsidered and the proposed FA-FEM model is used to minimize operating costs, total pumping volumesand total pumping rates in three scenarios while meeting water quality requirements. The groundwaterlift and contaminant concentration are also minimized through the optimization process. The obtainedresults show the applicability of the FA in conjunction with the FEM for the optimal design of groundwaterremediation systems. The performance of the FA is also compared with the genetic algorithm (GA) andthe FA is found to have a better convergence rate than the GA.

Keywords: groundwater; multi-objective optimization; firefly algorithm; finite element method; remedi-ation design

1. Introduction

The protection and improvement of groundwater quality are becoming increasingly importantowing to agricultural intensification, population growth and climate change (Chang, Chu, andHsiao 2007, 2011). However, the inaccessibility of groundwater results in field experimentsthat are costly, time consuming and difficult, if not impossible, to conduct. During the pastfew decades, considerable research has been focused on developing an effective simulation–optimization approach for the remediation of contaminated aquifers. In general, the ‘pump andtreat’ method, as a conventional technique (Chu and Chang 2009a, 2009b; Guan and Aral 1999;Chang, Shoemaker, and Liu 1992), is often combined with an optimization method to determinethe most appropriate design based on proper selection of cost, time and other related parameters.In recent years, a large number of methods has been proposed by researchers (e.g. Zheng andWang 2002; Guan and Aral 2004; Chu and Chang 2009a). However, these methods have a varietyof restrictions owing to the complexity of the problem, its highly nonlinear nature and design

∗Corresponding author. Email: farhang.daneshmand@mcgill.ca

© 2013 Taylor & Francis

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variable noise. To overcome these difficulties, metaheuristic optimization algorithms with lesscomputational error and faster convergence have been proposed (Yang 2008, 2010; Talbi 2009).In such techniques, an attempt is made to reproduce social behaviours or natural phenomena toconduct optimization. Biological evolution, animal behaviours and thermal annealing are typicalexamples of such techniques (Talbi 2009; Chaiead, Aungkulanon, and Luangpaiboon 2011). Bio-logically inspired algorithms such as the genetic algorithm (GA) (Wang and Zheng 1997; Zhengand Wang 2002; Bayer and Finkel 2004; Guan and Aral 2004; Mondal, Eldho, and Rao 2010), par-ticle swarm optimization (PSO) (Sepehri, Daneshmand, and Jafarpour 2012; Meenal and Eldho2012; Mategaonkar and Eldho 2012) and ant colony optimization (ACO) (Dorigo, Maniezzo, andColorni 1996), and nature-inspired algorithms such as simulated annealing (SA) (Kirtrick, Gelatt,and Vecchi 1983; Dougherty and Marryott 1991), big bang–big crunch (BB-BC) (Erol and Eksin2006) and charged system search (CSS) (Kaveh and Talatahari 2010) are common examples ofmetaheuristic algorithms.

The firefly algorithm (FA), a metaheuristic algorithm, has been used in various practical appli-cations (Yang 2009; Basu and Mahanti 2011; Yang, Hosseini, and Gandomi 2012; Senapati andDash 2012; Rao 1996). Its inspiration is the same as the artificial bee colony (ABC) (Chaiead,Aungkulanon, and Luangpaiboon 2011), the bees algorithm (Bees) techniques and PSO (Sepehri,Daneshmand, and Jafarpour 2012). However, FA has advantages over some conventional algo-rithms (Lukasik and Zak 2009). For example, it has superior performance in problems with largenoise rates compared to the Bees algorithm and so it is more appropriate for exploiting a searchspace (Chaiead, Aungkulanon, and Luangpaiboon 2011). Moreover, it results in superior conver-gence and better results compared with PSO (Basu and Mahanti 2011; Chaiead,Aungkulanon, andLuangpaiboon 2011; Gandomi,Yang, andAlavi 2011; Gandomi,Yang, Talatahari, andAlavi 2012;Parpinelli 2011; Zaman and AbdulMatin 2012), quantum particle swarm optimization (QPSO)(Horng and Liou 2011), ABC (Basu and Mahanti 2011), SA (Basu and Mahanti 2011), networkmanagement system (NMS) (Basu and Mahanti 2011), harmony search (HS) (Miguel and Miguel2012) and exhaustive research (Horng and Liou 2011). It has also been shown to be superior toGAs (Gandomi, Yang, and Alavi 2011; Gandomi et al. 2012) because fireflies aggregate moreclosely around each optimum without ‘jumping around’. In addition, it outperforms the randomsearch method (Gandomi, Yang, and Alavi 2011) through the fine tuning of parameters.

In the present study, an attempt is made to solve some typical pump and treat problemsvia a simulation–optimization approach. To study the effectiveness of the proposed approach,a hypothetical contaminated groundwater study area and three different scenarios with variousmulti-objective functions are analysed. In these scenarios, (1) cost; (2) total pumping rate, averagegroundwater lift and average remaining contaminant concentration; and (3) total pumping volume,average contaminant concentration, average groundwater lift and clean-up time are minimized.Four constraints—concentration, head, minimum (maximum) and total pumping rates—are con-sidered in the mathematical formulation. The results obtained from the proposed approach showthe effect of remediation time, pumping rate, the contaminant concentration and average heightof groundwater on cost and pumping volume of groundwater. The performance of the fireflyoptimization algorithm is also compared with the GA.

2. Mathematical formulation

The governing equation of a two-dimensional, steady-state, isotropic, homogeneous confinedaquifer is given as follows (Mondal, Eldho, and Rao 2010):

∂2h

∂x2+ ∂2h

∂y2−

∑Qiδ(x − xi)

K= 0 (1)

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Engineering Optimization 3

The boundary conditions for the problem can be written as

h = h̄ on prescribed head boundary (2)

∂h

∂n= q̄

Kon prescribed flux boundary (3)

where K is the hydraulic conductivity (LT−1); ∂/∂n is the normal derivative; Qi is the pumpingrate of the ith well; h(x, y) is the piezometric head (L), and h̄ and q̄ are the known head (L)and flux (L2T−1) of the specific boundaries, respectively. Equations (2) and (3) express Dirichletand Neumann boundary conditions, respectively. After solving the above governing differentialequation with given boundary conditions, the groundwater head distribution and the velocitycomponents in the x and y directions can be calculated at all nodal points of the flow domainusing Darcy’s law.

In this study, the contaminant transport is considered transient and in an anisotropic domain.Therefore, the partial differential equation of a solute in the groundwater flow is given as follows(Batu 1984, 2006):

R∂C

∂t= ∂

∂x

(Dxx

∂C

∂x+ Dxy

∂C

∂y

)+ ∂

∂y

(Dxy

∂C

∂x+ Dyy

∂C

∂y

)− Vx

∂C

∂x− Vy

∂C

∂y

−∑

QiCδ(x − xi) (4)

where C is the solute concentration (ML−3), Vx and Vy are the seepage velocity components, t istime, and δ is the Dirac delta function. R = 1 + ρKd/θ is the dimensionless retardation factor inwhich ρ is the bulk density (ML−3), Kd is the distribution coefficient (M−1L3) and θ is the aquiferdimensionless porosity. D is the hydrodynamic dispersion tensor (L2T−1) with the followingcomponents:

Dxx = αLV 2x + αT V 2

y

|V| + D0 (5)

Dyy = αLV 2y + αT V 2

x

|V| + D0 (6)

Dxy = (αL − αT )Vx + Vy

|V| + D0 (7)

where Vx and Vy are the velocity components in the x and y directions, respectively, |V| is theEuclidian norm of the velocity vector, αL and αT are the longitudinal and transverse dispersivitycoefficients, respectively, and D0 is the molecular diffusion coefficient. The contamination trans-port is assumed without sorption or generation. The initial condition for the problem is given by

C(x, y, 0) = g (8)

The boundary conditions used in this study are also given as

C(x, y, t) = C̄(x, y, t) (9)

on the prescribed concentration boundary and(Dxx

∂C

∂x+ Dxy

∂C

∂y

)nx +

(Dxy

∂C

∂x+ Dyy

∂C

∂y

)ny = f̄ (x, y, t) (x, y) ∈ � (10)

on the prescribed flux boundary. � is the flow domain, g is the initially known concentrationdistribution of the domain, C̄ and f̄ are known functions on prescribed boundaries, and nx and ny

are components of the unit outward normal vector to the given boundary.

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3. Simulation–optimization approach

In this article a simulation model is applied with an optimization technique for determiningthe optimal design of a groundwater remediation system while satisfying the constraints. Thesimulation model is based on the finite element method (FEM) while the optimization techniqueis based on the FA. State variables, which are hydraulic head and concentrations, are determinedby FEM and the results for pumping rates and remediation time are used in the FA as designvariables. These variables are updated based on their corresponding objective function valuesby the firefly optimization algorithm (FA). The simulation model uses the updated variables assubsequent input values and determines the new state variables for the next step. This procedureruns iteratively until the specified convergence criterion is satisfied.

3.1. Finite element method

The governing partial differential equations and natural boundary conditions are converted intothe integral weak form using the weighted residual method. The discrete form of the groundwaterflow in Equation (1) is obtained using the Galerkin method as follows:

Kh = Q (11a)

(K)ij =∫

�e

(∂Ni

∂x

∂Nj

∂x+ ∂Ni

∂y

∂Nj

∂y

)d�e (11b)

(R)i =∫

�

q̄

KNid� (11c)

where K is the hydraulic conductivity matrix, R is the flux vector, and h is the nodal head vector.Similarly, the discrete form of contaminant transport given in Equation (4) is obtained as follows:

R∂C∂t

= (Kdis + Kadv)C + S (12a)

(R)ij =∫

�e

RNiNjd�e (12b)

(Kdis)ij =∫

�e

(∂Ni

∂x

(Dxx

∂Nj

∂x+ Dxy

∂Nj

∂y

)+ ∂Ni

∂y

(Dxy

∂Nj

∂x+ Dyy

∂Nj

∂y

))d�e (12c)

(Kadv)ij =∫

�e

(vxNi

∂Nj

∂x+ vyNi

∂Nj

∂y

)d�e (12d)

where C is the nodal concentration vector, and Kdis and Kadv are the coefficient matrices fordispersion and advection, respectively. Applying the Crank–Nicolson algorithm over the timedomain gives the following equation:

(R − ωt(Kdis + Kadv))Ct+t = (R − (1 − ω)t(Kdis + Kadv))Ct

+ t((1 − ω)R + ωRt+t) (13)

The solution procedure begins by specifying the initial value of C. Equation (13) then gives theconcentration and the procedure continues for the next time step.

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Engineering Optimization 5

3.2. Firefly algorithm

In this study the FA is used as the optimization tool to identify the global optimum of the problem.As the FA mimics the social behaviour of fireflies, it has three rules:

(1) All fireflies are unisex and so they will be attracted to other fireflies regardless of their sex.(2) Attractiveness in firefly communications is proportional to brightness and as a consequence

it decreases by increasing the distance between fireflies. Therefore, a less bright firefly willmove towards a brighter firefly.

(3) The analytical form of the objective function and firefly brightness are somehow related toeach other.

The attractiveness β of a firefly is defined by (Yang, Hosseini, and Gandomi 2012):

β(r) = β0e−γ r2(14)

where β0 is the attractiveness at r = 0. The distance between any two fireflies i and j at xi and xj isdefined as rij = ‖xi − xj‖. The movement of a firefly i attracted to a brighter firefly j is determinedby (Yang, Hosseini, and Gandomi 2012):

xt+1i = xt

i + β0e−γ r2(xt

j − xti) + αεi (15)

where the last two terms are attraction and randomization terms, respectively. In Equation (15),α ∈ (0, 1)is the randomization parameter and εi is a vector of random numbers. In general, thebest ranges of α and β0 are 0.4–0.9 and 0.5–1, respectively (Yang, Hosseini, and Gandomi 2012).The parameter γ , which characterizes the variation of attractiveness, can theoretically vary from0 to a large number. However, an appropriate range in most applications is (0.1, 10). Moreover,it has been found in some recent studies that the best population size is 25–50 fireflies (Yang,Hosseini, and Gandomi 2012).

Owing to the oscillatory behaviour of FA, as the optimum design is approached, therandomization of parameter α is reduced by a geometric progression reduction scheme:

α = α0θt (16)

where 0 < θ < 1 is the reduction factor of randomization. By this strategy α is reduced from 0.9to 1E-6 and it improves the solution quality.

The purpose of optimization in this work is to find the global minimum of the objective function.Therefore, the brightness of fireflies is defined and optimized in such a way that they immi-grate to locations with minimum objective function values. The coupled simulation–optimizationalgorithm attempts to determine optimal design variables to minimize operating costs, total pump-ing rates and total pumping volumes in three scenarios. The groundwater lift and contaminantconcentration are also minimized through the optimization process. Thus, three multi-objectivefunctions are analysed in which clean-up time and pumping rate are considered as design variables.These design variables will then be updated by the FA based on their corresponding objectivefunction values. It should be noted that objective function values of those fireflies that do notsatisfy inequality constraints will be much higher than for other fireflies, and the next generationof fireflies will not tend to immigrate to such locations. In general, this procedure is implementedby consecutive parameter adjustments and design variable (firefly location) improvements.

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4. Numerical implementation

The main goal of the present study is to evaluate the applicability of the recently proposed FA in thedesign of remediation systems. To the authors’ knowledge, the application of this optimizationalgorithm has not been considered so far in the design of remediation systems. To check theperformance of the algorithm and study the performance using FA, the numerical examples wereselected to be the same as those previously found in the literature (Li, Chen, and Pepper 2003;Sharief et al. 2012; Chang, Chu, and Hsiao 2011; Singh and Chakrabarty 2011). To determinethe simulation–optimization performance, three scenarios in which a hypothetical, homogeneousconfined groundwater aquifer is polluted by a substance are considered. Figure 1 shows theflow direction, dimensions and orientation of the considered domain. All reported results in thefollowing subsections are calculated based on regular quadrilateral finite elements. It should alsobe mentioned that the total number of nodes and elements is 861 and 800, respectively.

4.1. Example 1: Minimization of pumping and remediation cost

In this case, the contaminant is assumed to be dispersed for 10 years. No hydraulic flow and zeroconcentration gradient are considered on the upside and downside boundaries of the aquifer. Aconstant concentration of 0 g/m3 is assumed for left and right boundaries. Dirichlet conditionsare assigned on the left (hl = 60 m) and right (hr = 57 m) boundaries, forming groundwater fromleft to right. Table 1 shows the simulation parameter values considered in this example. The initialplume after removal of the pollution source is shown in Figure 2. As can be observed in Figure 2,the contaminant source, which is in a hexagonal shape, has a peak concentration of 100 g/m3.

4.1.1. Case study 1: Using one well for remediation

A well is assumed for remediation at x = 45, y = 45. The location of the pumping well is selectedin such a way that the concentration at the pumping location would be highest at the beginningof the remediation procedure. The well location and remediation time are presumed fixed inthis example and only the pumping rate is optimized. Since remediation time is considered as aconstraint, the problem is solved for different times.

Figure 1. A hypothetical field area.

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Engineering Optimization 7

Table 1. Aquifer properties and simulation parameters.

Parameter Value

Hydraulic conductivity 25.92e−1 (m/day)Porosity 0.25Longitudinal dispersivity (αL) 6.855 (m)Transverse dispersivity (αT ) 1.371 (m)Media bulk density (ρ) 1.7e6(g/m3)Distribution coefficient for linear sorption (Kd) 1.58e−7(m3/g)Diffusion coefficient 0 (m2/s)Ground level at well 70 m

Figure 2. Initial plume for Example 1, Case 1: left head = 60 m, right head hr = 57 m; dimensions are in metres,concentrations are in g/m3.

The model attempts to minimize the total cost of remediation (composed of pumping andtreatment) while meeting water quality constraints. The cost function is formulated as follows

min f = a1

N∑i=1

Qitγ (hgli − hi) + a2

N∑i=1

Qit (17)

where f is the remediation cost, Qi denotes the pumping rate of well i (in this case only one pumpis considered), hi denotes hydraulic head, hgl

i is the distance between the ground surface and thelower datum of the aquifer at well i, hgl

i − hi represents drawdown at pumping well i, a1 and a2

are pumping and treatment cost coefficients, respectively, t is the pumping duration, and γ is thewater unit weight. Constraints on the problem are specified as

hmin ≤ hi i = 1, . . . , NP (18)

Ci ≤ Cmax i = 1, . . . , NP (19)

Qmin ≤ Qi ≤ Qmax i = 1, . . . , N (20)

N∑i=1

Qi ≤ Qtot (21)

where N is the number of pumps considered for remediation, NP is the number of all nodepoints including pump points, and Cmax is the maximum allowable concentration in the regionor monitoring wells. Equation (19) ensures that the water quality standard will be met anywhereat the end of the planning period. hmin represents minimum allowable groundwater head, Qtot isthe minimum allowable treatment capacity, and Qmin and Qmax are related to the minimum and

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Table 2. Model parameters associated with the constraints and cost coefficients.

Parameter Value

Maximum allowable treatment capacity (Qtot) 500 m3/dayMaximum pumping rates for each well (Qmax) 500 m3/dayMinimum pumping rates for each well (Qmin) 0 m3/dayMaximum allowable concentration (Cmax) 1 g/m3

Minimum allowable groundwater head (hmin) 5 mCost coefficient (a1) $2000/(m4/s · t)Cost coefficient (a2) $80, 000/(m3/s · t)Period length (t) 5 daysPeriod number Varied from 73 to 584

maximum pumping rate of each well, respectively. Table 2 illustrates the model parameters usedin this implementation.

To evaluate the performance of the FA, in this case the problem is solved using two optimizationalgorithms: FA and GA. It must be noted that the same number of fireflies and population is selectedin these two methods to enable a reasonable comparison. The FA is executed for 100 iterationsusing 40 fireflies while the same population size is used in GA. Therefore, the total number offunction evaluations is 4000 for both algorithms. Other parameters of FA are selected as: α =0.9, γ = 1, β0 = 0.2. The GA parameters are 0.8 for the crossover rate and 0.01 for the mutation. Insolving this example by GA and before comparing the results, several runs are performed to obtaina better insight into the influence of choosing different values for the GA parameters (crossoverprobability, mutation probability and population size). The parameters reported in the article forGA are actually those values leading to optimal performance (qualitatively) in the optimizationprocedure. A similar procedure is followed for the FA, as the applied parameters are chosenfrom the best choice of parameters. It should be noted that these parameters are also consistentwith those reported by Yang, Hosseini, and Gandomi (2012). For the GA, the population type isselected as the bit string, the population generation is limited to the upper and lower bounds of thepumping rates, the maximum number of generations is selected as the stopping criterion, and the

Figure 3. Convergence history of two cases (with 3 and 6 years remediation time) obtained by the firefly algorithm (FA)and genetic algorithm (GA) methods.

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Engineering Optimization 9

Figure 4. Pumping rate for Example 1, Case 1, obtained by the firefly algorithm (FA) and genetic algorithm (GA).

selection is performed based on roulette function. The convergence graphs for two remediationtimes (3 and 6 years) are plotted in Figure 3. As can be observed, the FA method converges morerapidly to the optimum solution compared with the GA. The problem is also solved for otherremediation times with very similar behaviour. However, the results for other remediation timesare not presented here for brevity.

Optimized pumping rates and corresponding objective functions obtained by FA and GA areplotted in Figures 4 and 5, respectively. As can be seen, the pumping rate decreases gradually withincreasing time. In addition, the cost decreases with time to a specific value but then increases.It can also be seen that costs obtained by FA are slightly less than for GA. It must be noted thatthe remediation time cannot be less than 2 years in this problem because the constraints wouldno longer be satisfied.

In solving this example via the GA, several runs are performed to first obtain a better insightinto the influence of choosing different values for GA parameters. The parameters used in theoptimization are then selected according to those values showing the optimal performance (quali-tatively).A similar procedure is also used for the FA, as the applied parameters are chosen from thebest choice of parameters. It was found that these parameters were consistent with those reportedby Yang, Hosseini, and Gandomi (2012).

4.1.2. Case study 2: Using four wells for remediation

As the second case of Example 1, a confined aquifer with similar characteristics in which thecontaminant is dispersed for 10 years is assumed. The initial plume after removing the sourceis shown in Figure 6. As shown in Figure 6, four wells are assumed in points P1(x = 45,y = 75), P2(x = 75, y = 75), P3(x = 105, y = 75) and P4(x = 135, y = 75). The initial peakconcentration is 120 g/m3 and Cmax is 0.3 g/m3. Boundary conditions, simulation and manage-ment parameters, cost coefficients and other constraints are the same as described in Tables 1 and2.

Similarly, the number and locations of wells and the remediation time are presumed fixedand only pumping rates are optimized as decision variables. The optimization model attemptsto minimize the total cost of remediation while meeting the constraints. The cost function and

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Figure 5. Optimized cost for Example 1, Case 1, obtained by the firefly algorithm (FA) and genetic algorithm (GA).

Figure 6. Initial plume for Example 1, Case 2: left head = 60 m, right head hr = 57 m; dimensions are in metres,concentrations are in g/m3.

constraints are formulated in Equations (17)–(21). In this case, head and concentration data aregathered from 19 monitoring wells.

The total pumping rate and remediation cost figures are shown in Figures 7 and 8, respectively.It can be seen from Figure 8 that cost decreases with time. Figure 9 shows the minimum waterhead of all nodes at different remediation times. As expected, the minimum water head increaseswith time as a result of a decrease in the pumping rate, while the maximum concentration ofcontaminant remains at 0.3 g/m3.

4.2. Example 2: Minimization of total pumping rate, water lift and contaminantconcentration

In the second example, an aquifer with similar dimensions and boundary conditions is considered.As shown in Figure 10, a contaminant source has been dispersed for 20 years in the aquifer, and

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Engineering Optimization 11

Figure 7. Total pumping rate for Example 1, Case 2.

Figure 8. Optimized cost for Example 1, Case 2.

Figure 9. Minimum water head after remediation versus remediation time for Example 1, Case 2.

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Figure 10. Initial plume for Example 2: left head = 60 m, right head hr = 57 m; dimensions are in metres, concentrationsare in g/m3.

is then removed. The initial peak concentration is 120 g/m3 and Cmax is 0.5 g/m3 at all nodes.Because of the symmetry of the problem, six pumps are considered on the centreline of the field.Locations of extraction wells are determined based on groundwater direction, dispersion time andcontaminant source location.

The purpose of this problem is to minimize the summation of pumping rates, average con-taminant concentration and water lift while meeting water quality constraints. The number andlocations of wells and remediation time are presumed fixed and only pumping rates are optimized.The considered objective function is formulated as follows

min h = a3

N∑i=1

Qi + a4∑N

i=1 Ci − a5∑N

i=1 Hi

NP(22)

where h is the objective function, Qi denotes the pumping rate of well i, a3, a4 and a5 are adjustingcoefficients, N is the pump number, NP is the total number of nodes in the area which consistsof well points, and Ci and Hi are contaminant concentration and head at node i, respectively.The aquifer properties, simulation and optimization parameters and constraint coefficients arethe same as in Example 1 and are described in Tables 1 and 2. The purpose of minimizing thetotal pumping rate is to minimize the number and locations of pumping wells. After determiningpumping rates, wells with negligible pumping rates can be eliminated and the solution can beupdated to dominant wells. Figure 11 shows the effect of remediation time by illustrating the totalpumping rates versus remediation time. It can be seen that the pumping rate decreases noticeablywith time.

Figures 12 and 13 illustrate the minimum water head and maximum concentration variationsat different remediation times. It can be observed from Figure 12 that the minimum water headremains steady during the first 6 years of remediation time then starts to increase at remediationtime = 6. In contrast, Figure 13 shows that the maximum contaminant concentration decreaseswith time for the period considered in this example.

4.3. Example 3: Minimization of pumping volume, contaminant concentration, water liftand clean-up time

In the third example, a confined aquifer with similar dimensions and boundaries as describedearlier (Section 4.1.1) is assumed. The dispersion time of the contaminant is 20 years; the initialpeak concentration is 120 g/m3 and Cmax is 0.01 g/m3 at all nodes.As observed earlier, considering

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Figure 11. Total pumping rate of wells versus remediation time for Example 2.

Figure 12. Minimum water head after remediation versus time for Example 2.

the remediation time as a fixed constraint often results in unnecessary excessive pumping, andas a consequence it has significant effects on remediation cost as well. Therefore, the purpose ofthis example is to minimize the clean-up time in addition to total pumping rates. The objectivefunction is formulated as Equation (23). It should be mentioned that the groundwater lift andcontaminant concentration are also minimized through the optimization process. As shown inFigure 14, five wells are considered in specific locations and their pumping rates are consideredas design variables.

min g = a3

N∑i=1

Qit + a4∑N

i=1 Ci − a5∑N

i=1 Hi

NP(23)

where g is the objective function and t is the duration of remediation. All other parameters are thesame as in Equation (22).

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Figure 13. Maximum concentration after remediation versus time for Example 2.

Figure 14. Initial plume for Example 3, left head = 60 m and right head hr = 57 m; dimensions are in metres,concentrations are in g/m3.

Table 3. Optimized remediation time, total pumping rate, total volume of extracted water, minimum head,maximum concentration and obtained pumping rates for Example 3.

Pumping rate of wells (m3/day)Min. Max. Total pumping rate Total pumping

Best time head (m) concentration (g/m3) (m3/day) volume (m3) P1 P2 P3 P4 = P5

2170 days 26.69 0.0024 120.94 2.62e+5 14.33 20.77 56.8 14.52

All parameters, coefficients and constraints are similar to those given in Example 1 and Tables 1and 2, except for the maximum period number and the minimum pumping rate Qmin, which areconsidered as 730 and 10 m3/day, respectively.

Table 3 shows the results of the optimization for this example. The results imply that for the casewith a time interval of 10 years and five pumps, the most optimal pump and treat strategy satisfyingthe constraints regarding minimum cost is to use pumps with a pumping rate of 120.94 m3/day for2170 days. To meet this condition, 2.62e+5 m3 of contaminated water should be extracted, which

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reduces the minimum head of the field to 26.69 m. Table 3 also shows the optimized pumping ratesof each specific pump. The result of this remediation strategy reduces the pollutant concentrationto 0.0024 g/m3.

5. Conclusion

Studying the behaviour of groundwater systems in response to natural and human-induced influ-ences is very important because there are many significant stresses on the world’s groundwater,including population growth, agricultural intensification and the growing impact of climatechange. Therefore, new approaches to groundwater modelling are required to develop efficient androbust solutions, strategies and policies to deal with groundwater pollution remediation problemsto protect the quantity and quality of groundwater as well as its contribution to the viability ofecosystems. Groundwater flow and contaminant transport simulation provides useful informationwhich is not otherwise available from a long-term field study. This article focuses on the appli-cability of the FA in solving groundwater remediation problems. In the present study, an attemptwas made to solve typical pump and treat problems using a coupled simulation–optimizationapproach via the FEM and FA. To study the effectiveness of the proposed approach, a multi-objective optimization problem including three scenarios of remediation cost, total pumping rateand total pumping volume was considered in a hypothetical contaminated groundwater studyarea. The minimum contaminant concentration and water lift were also included in the calcu-lations. The results obtained from the proposed approach show the effect of remediation time,pumping rate and constraints on the objective function values. The performance of the FA wasqualitatively compared with the GA, and it was found that the FA had a better convergence ratethan the GA. It can also be concluded that the FA, in conjunction with the FEM, can effectivelybe used to construct a simulation–optimization algorithm for the optimal design of groundwaterremediation systems. However, it should be noted that owing to the random nature of FA and GA,the computational cost of the optimization process may be sensitive to numerical implementationand corresponding parameters in each optimization method. Therefore, it is suggested that morecomprehensive studies be performed to establish reasonable criteria and merits for comparison ofFA and GA as two metaheuristic algorithms. This would be an interesting topic of investigationfor further research.

Funding

This research was partially funded by an NSERC Discovery Grant held by Jan Adamowski.

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