Ey25924935

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. INTRODUCTION
most serious environmental problems in many World's growing industrialization and
parts of the world today. For the remediation of planned irrigation of large agricultural fields have
dissolved phase contaminant in groundwater, the caused deterioration of groundwater quality. With
pump and treat method is found to be a the growing recognition of the importance of
successful remediation technique. Developing an groundwater resources, efforts are increasing to
efficient and robust methods for identifying the prevent, reduce, and abate groundwater pollution. As
most cost effective ways to solve groundwater a result, during the last decade, much attention has
pollution remediation problems using pump and been focused on remediation of contaminated
treat method is very important because of the aquifers. Pump and treat is one of the established
large 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 of
evolutionary 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 out
the non-linear and very complex pump and treat groundwater. Various treatment technologies are
groundwater pollution remediation problem. The used for the removal of contaminants and cleanup
model couples elitist genetic algorithm, a global strategies. Depending on the site conditions, some
search technique, with FEM for coupled flow and times the treated water is injected back in to the
solute transport model to optimize the pump and aquifer. Many researchers found that the combined
treat groundwater pollution problem. This paper use of simulation and optimization techniques can be
presents a simulation optimization model to a powerful tool in designing and planning strategies
obtain optimal pumping rates to cleanup a for the optimal management of groundwater
confined aquifer. A coupled FEM model has been remediation by pump and treat method. Different
developed for flow and solute transport optimization techniques have been employed in the
simulation, which is embedded with the elitist groundwater remediation design involving linear
genetic algorithm to assess the optimal pumping programming [1], nonlinear programming [2-4] and
pattern for different scenarios for the dynamic programming [5] methods. Application of
remediation of contaminated groundwater. Using optimization techniques can identify site specific
the FEM-EGA model, the optimal pumping remediation plans that are substantially less
pattern for the abstraction wells is obtained to expensive than those identified by the conventional
minimize the total lift costs of groundwater along trial and error methods. Mixed integer programming
with the treatment cost. The coupled FEM-EGA was also used to optimal design of air stripping
model has been applied for the decontamination treatment process [4].
of a hypothetical confined aquifer to demonstrate
the effectiveness of the proposed technique. The Bear and Sun [6] developed design for
model is applied to determine the minimum optimal remediation by pump and treat. The
pumping rates needed to remediate an existing objective was to minimize the total cost including
contaminant plume. The study found that an fixed and operating costs by pumping and injection
optimal pumping policy for aquifer rates at five potential wells. Two–level hierarchical
decontamination using pump and treat method optimization model was used to optimization of
can be established by applying the present FEM- pumping /injection rates[6]. At the basic level, well
EGA model. locations and pumping and injection rates are sought
so as to maximize mass removal of contaminants. At
Keywords: 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

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identify the cost effective solutions to pump and study develops an optimal planning model for
treat groundwater remediation design. Genetic cleanup of contaminated aquifer using pump and
algorithms are heuristic programming methods treats method. This study integrates the elitist
capable of locating near global solutions for genetic algorithm and finite element model to solve
complex problems [7]. A number of researchers the highly nonlinear groundwater remediation
have applied genetic algorithms in the recent past to systems problem. The proposed model considers the
solve 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 quality
multiple objective groundwater pollution constraints, and the model computes the optimal
containment problem. They used the response matrix pumping rate to restore the aquifer.
approach. Vector evaluated genetic algorithm and
Pareto genetic algorithms have been used for In the present problem, the genetic
maximization of reliability and minimization of algorithm is chosen as optimization tool. GAs are
costs. The Pareto algorithm was shown to be adaptive methods which may be used to solve search
superior to the vector evaluated genetic algorithm. and optimization problems. GA is robust iterative
Genetic algorithms have been used to determine the search methods that are based on Darwin evolution
yield from a homogeneous, isotropic, unconfined and survival of the fittest [7, 31]. The fact that GAs
aquifer and also, to determine the minimum cost of have a number of advantages over mathematical
pump and treat remediation system to remove the programming techniques traditionally used for
contaminant plume from the aquifer using air optimization, including (1) decision variables are
stripping 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 in
simulation optimization model developed for the addition to one leading solution. Consequently, GAs
optimal design of groundwater remediation systems is used as the optimization technique for the
under a variety of field conditions. The model has proposed research. The GA technique and
been applied to a typical two-dimensional pump and mechanism selected as a part of proposed approach
treat example with one and three management are binary coding of the decision variables, roulette
periods and also to a large-scale three-dimensional wheel selection, uniform crossover, bitwise mutation
field problem to determine the minimum pumping and elitism technique. In the present study, a
needed 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 element
optimize 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 for
specified 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 hypothetical
pumping and injection. But the pumping and problem to study the effectiveness of the proposed
injection rates are not time varying. Real-coded technique and it can be used to solve similar real
genetic algorithm are also applied for groundwater field problems.
bioremediation[21].
II. GROUNDWATER FLOW AND MASS
Integrated Genetic algorithm and TRANSPORT SIMULATION
constrained differential dynamic programming were The FEM formulation for flow and solute
used to optimize the total remediation cost[20]. But transport followed by genetic algorithm approach is
the decision variables only involve determining time considered in this study. The governing equation
varying pumping rates from extraction wells and describing the flow in a two dimensional
their locations. FEM is well established as a heterogeneous confined aquifer can be written as
simulation 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 of
pollution 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 conditions
and 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 schemes
inspired by biological evolution [7]. The present

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h g 2 = concentration gradient normal to the
T = 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 flexibility
sink function ( Q w source, Q w = Sink) (m3/d/m2); to discretize a prototype system which represents
boundaries accurately. The finite element method
t = time in days;  = the flow region;  = (FEM) approximates the governing partial
 differential equation by integral approach. In the
boundary region ( 1   2   ); = normal
n present study, two-dimensional linear triangular
derivative; h 0 ( x, y) = initial head in the flow elements are used for spatial discritization and
domain (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 in
boundary head (m) and q(x, y, t ) = known inflow equation (1) is initially approximated as
rate ( 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 and
hydrodynamic 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 total
dissolved solids (TDS) or a single chemical
NP = total number of nodes in the problem domain.
A set of simultaneous equations is obtained when
constituent in groundwater, considering advection
dispersion 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 the
where 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 dxdy
Dissolved concentration [ML-3];  = porosity
(dimensionless); w = elemental recharge rate with  w
e
L
e
L
' e e
solute concentration c ; n = elemental porosity; b (11)
= aquifer thickness under the element; R =
N i 
Retardation factor: C 0 = concentration of solute in
injected 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 given
volume. 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
n
where 1 = boundary sections of the flow region  ; triangular elements and G, P, F are the element
matrices known as conductance , storage matrices
f = 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

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G h L  P   h L   F
III. ELITIST GA OPTIMIZATION
    (13) MODEL FORMULATION
 t  Genetic algorithm is a Global search
Applying the implicit finite difference scheme for procedure based on the mechanics of natural
h selection and natural genetics, which combines an
the term in time domain for equation (13) artificial survival of the fittest with genetic operators
t abstracted from the nature [7]. In this paper, elitist
gives genetic algorithm is used as optimization tool. The
h t  t  h t  EGA procedure consists of three main operators:
Gh 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 of
groundwater head values at present and next time some functions in the presence of elitism In order to
steps. By rearranging the terms of equation (14), the preserve and use previously best solutions in the
general form of the equation can be given as[25] subsequent generations an elite preserving operator
[P]   t [G] ht  t = is often recommended [30]. In addition to overall
increase in performance, there is another advantage
[P]  (1  ) t [G] ht + t (1  ) Ft +  Ft 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; ht and ht 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 genetic
used. In the present study Crank-Nicholson scheme operations applied on the entire current population
with  = 0.5 is used. These global matrices can be including the chosen % elite members. In this way
constructed using the element matrices of different the best solutions of the current population is not
shapes based on discritization of the domain. The set only get passed from one generation to another, but
of linear simultaneous equations represent by eq (15) they also participate with other members of the
are 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 explained
using 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 randomly
getting the nodal head values, the time step is generated set of decision variables. The initial
incremented and the solution proceeds in the same population is generated randomly with a random
manner by updating the time matrices and seed satisfying the bounds and sensitivity
recomputing the nodal head values. From the requirement of the decision variables. The
obtained 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 and
law. 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] Ct t = substring. The length of the substring is usually
[S]  (1  ) t [D] Ct + t (1  ) Ft +  Ft t determined according to the desired solution
accuracy.
(16) 2. The fitness of each string in the population is
where S = sorption matrix; D = advection– evaluated based on the objective function and
dispersion matrix; F = flux matrix; C = nodal constraints. Various constraints are checked during
concentration column matrix and the subscripts t the objective function evaluation stage. The amount
and t+ t indicates the concentration at present and of any constraint violation is multiplied by weight
forward 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, so
solute concentration distribution in the flow region that it will have a significant, but not excessive
for 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

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are arranged randomly to form mating pool on which which include concentration constraints, constraint
further operations like crossover mutation are for extraction rates and also constraints for nodal
performed. Reproduction is implemented in this head distribution.
study using the Roulette wheel approach [7]. In In the present study, the objective is to minimize the
roulette wheel selection, the string with the higher total cost of groundwater lift and the treatment cost
fitness value represents a large range in the for a fixed remediation period to achieve a desired
cumulative probability values and it has higher level of cleanup standards. The appropriate
probability of being selected for reproduction. objective function for this remediation is considered
Therefore, the algorithm can converge to a set of as,
Qi  t
h 
chromosomes with higher fitness values. NP NP
4. Crossover involves random coupling of the newly f = a1  gl
 hi + a 2  Qi t (17)

i
reproduced strings, and each pair of strings partially i 1 i 1
exchanges information. Crossover aims to exchange where, f = objective function for minimization of
gene information so as to produce new offspring
strings 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 igl
creation 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 coefficient
mostly child strings whose parents no longer exist
and 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 gl
prematurely 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 is
evaluated according to the fitness, i.e., objective
Constraints to the problem are specified as Subject
function.
to
6. The best % of the population from the current
Ci  C
population is directly copied to next generation. The i  1......
N
rest (100-  ) % of the new population is created by h m in  h i  h m ax
usual genetic operations applied on the entire current ; i  1......
N
population including the chosen % elite members. Qi, m in  Qi  Qi, m ax
7. The next step is to choose the appropriate
i  1......
NP
stopping 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 limit
parameters. If the user defined stopping criterion is
Q
met or when the maximum allowed number of of concentration in the region; and i = Pumping
generations is reached, the procedure stops, rate at well at i. Decision variables for the pump and
otherwise it goes back to reproduction stage to treat remediation system include the pumping rates
perform another cycle of reproduction, crossover, for the wells and the state variables are the hydraulic
and 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, simulation
IV. INTEGRATION OF FEM-ELITISTGA model has to update the state variables, which are
SIMULATION–OPTIMIZATION passed on to the optimization model to select the
APPROACH optimal values for the decision variables. The flow
To clean up the contaminated aquifer by chart for the developed FEM-EGA model is shown
pump and treat, pumping rates are to be optimized to in Fig.1.
achieve the minimum specified concentration levels
within the system after a particular period. Here a V. OPTIMAL REMEDIATION SYSTEM
coupled FEM-EGA model is developed to find out DESIGN WITH FEM-EGA MODEL – A
the optimal pumping rates in the pump and treat CASE STUDY
method. A binary coded genetic algorithm is A hypothetical problem of solute mass
embedded with the flow and transport FEM model transport in a confined aquifer is considered for
described earlier, where pumping rates are defined remediation of contaminated groundwater. The
as continuous decision variables. In the optimization aquifer is assumed to be homogeneous, anisotropic
formulation, three constraints were considered, and flow is considered to be two-dimensional. When

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the remediation process commences, it is reasonably dynamic conditions operative on the system is
assumed that the source of pollutants entering in to shown in Fig. 3. Pump and treat method is applied
the 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 through
are as follows: The aquifer is discretised into finite the polluted pond and recharge well). From the
element mesh with a grid spacing of 200 m  200 existing three abstraction wells, the contaminants in
m. Thickness of the confined aquifer = 25 m; Size of the system are pumped out for the treatment. Based
the 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 of
in the area of 0.6 km2 (between node numbers 19 to pumping wells has been changed, so that the
42) at the rate of 0.00012 m/d; porosity = 0.2; pumping wells are situated in the highly polluted
longitudinal dispersivity = 150 m; transverse area. Therefore for the aquifer remediation the
dispersivity = 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 observation
The boundary conditions for the flow model are well which is located at nodes 33 and 52 are
constant head on east (70 m) and west sides (75 m), considered for pumping out the contaminated water
and no flow on the north and south sides. For for treatment on the ground surface.
transport model, the boundary conditions are zero
concentration on west and concentration gradient is The FEM simulation and EGA optimization
zero on north and south directions. Since the area of techniques have been used for obtaining an optimal
the aquifer is relatively small (1.8 km2) the boundary pumping pattern to cleanup the contaminated
for 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 to
triangular elements consisting of 60 nodes and 90 calculate the objective function associated with the
elements as shown in Fig. 2. The groundwater flow possible design, and to evaluate whether various
and contaminant spread in the aquifer system is hydraulic and concentration constraints are met. In
simulated using FEM to obtain nodal heads, velocity the present work, the compliance requirement for the
vectors and nodal concentration distribution in the contaminants concentration level is considered to be
aquifer domain. Three production wells extracting lower than 750 ppm everywhere in the aquifer at the
water at the rate of 500, 600 and 250 m 3/d end of 3960 days. The objective of the present study
respectively are considered in the flow region. These is to obtain optimal pumping rates for the wells to
wells 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 for
3
/d is located at node 21. Three observation wells are treatment for the specified time period. Three
located in the aquifer domain at nodes 44, 47, and 52 pumping wells scenario is considered for this
respectively to monitor the concentration levels remediation period to cleanup the aquifer for a fixed
during the remediation period. The problem involves cleanup period of 3960 days. The optimal pumping
seepage from a polluted pond into the underlying pattern (least cost) for the chosen scenario has been
aquifer. The bottom of the pond is assumed to be obtained by the coupled simulation optimization
sufficiently close to the groundwater surface so that model.
seepage underneath the pond occurs in a saturated
region. The recharge rate (0.005 m/d) from the pond Three scenarios have been considered in
controls the amount of solute introduced into the this study to cleanup the contaminated aquifer for a
system with a certain velocity. In this problem, the fixed cleanup period of 3960 days. Three pumping
contaminant is seeping into the flow field from the wells, two wells and one well respectively are
polluted pond (4000 ppm TDS) and is also added by considered for this remediation period to treat these
a 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 coupled
uncontaminated (C  0) . This study examines the simulation optimization model.
coupled flow and solute transport problem with
these input sources. Pumping wells are also active VI. TUNING OF GENETIC ALGORITHM
for a simulation period of 10000 days during which PARAMETERS
period the aquifer continues to get contaminated. At Important genetic algorithm parameters
the 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 criterion
concentration levels in the aquifer domain before the affect the EGA performance. Hence appropriate
remediation begins. Concentration distribution at values have to be assigned for these parameters
the end of 10000 days of simulation period for these before they are used in the optimization problem.

929 | P a g e

For some parameters several trial runs may be
needed to “tune” these parameters. Each parameter Number of generations
was varied to determine its effect on the system From practical considerations, the selection
within reasonable bounds while keeping remaining of maximum generation is governed by the limits
parameters at a standard level. The obtained optimal placed on the execution time and the number of
values are used in the simulation optimization model function evaluations. In each generation, the
for these scenarios. objective function is calculated maximum
population times, (except in the first generation in
Rate 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 also
mutation rate was varied between 0.1%, 1.0%, and applied several times.
10% per bit. Probability of mutation is usually
assigned a low value between 0.001 and 0.01., VII. RESULTS AND DISCUSSION
since it is only a secondary genetic operator. Although the specific test cases are
Mutation is needed to introduce some random synthetic and relatively idealized, the overall
diversity in the optimization process. However if approach is realistic. The installation of pumping
probability of mutation is too high, there is a wells at appropriate locations may be necessary to
danger of losing genetic information too quickly, prevent further spreading of the contaminants to
and the optimization process approaches a purely unpolluted areas. At the end of the remediation
random search period, the concentration levels everywhere in the
system are expected to be lower than the specified
Rate 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 period
implementations. Higher the value of the crossover, are presented in Table.1. The following section
higher is the genetic recombination as the algorithm provides the details of the results obtained for the
proceeds, 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 the
for binary coding increases exponentially with the contaminated groundwater. Lower and upper
length of the string [7]. Even for relatively small bounds on the pumping capacity are chosen as 250
problems, this would lead to high memory storage m3/d and 1000 m3/d respectively. The pumping rates
requirements. Results have shown that larger of the three wells are encoded as a 30 bit binary
population performs better. This is expected as string with 10 bits for each pumping rate. The
larger populations represent a large sampling of the optimal GA parameters as discussed earlier sections
design space. are used in this study. The optimal size of the
population is chosen as 30. A higher probability of
Minimum reduction in objective function for crossover of 0.85 and lower mutation probability is
termination set at 0.001. A converged solution is obtained after
The selection of this parameter is problem 128 generations. Fig. 4 shows the final concentration
dependent, and its appropriate value depends on how contours at the end of 3960 days of remediation
much improvement in objective function value may period based on the optimal pumping rates achieved
be considered significant. The value of this by simulation–optimization model (FEM-EGA). A
parameter is determined by trial and error, and a best value of objective function versus number of
value of 3 to 4 usually quite sufficient. Larger values generations for three well scenarios is presented in
of this parameter increase the chances of converging Fig.6. The optimal pumping rates for the three wells
to the global optimum, but at the cost of increasing are 300.58, 260.26 and 275.65 m3/d respectively as
the 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 required
challenging task. A high penalty coefficient will concentration level everywhere in the system, as
ensure that most solutions lie with in the feasible expected it leads to a higher system cost. Also it
solutions space, but can lead to costly conservative needs to be emphasized that an increase in the
system designs. A low penalty coefficient permits number of pumping wells to pump out the
searching 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 an
design [30]. appropriate location, a further spreading of the

930 | P a g e

contaminant plume can be minimized. This means scenarios, the concentration levels in the aquifer
that more effective remediation can be achieved by domain is less than that the required value
appropriately locating the pumping well. These everywhere in the system, where as for one well
investigations show that an appropriately designed scenario the maximum concentration level is
pump and treat system can have a significant effect marginally less than the specified cleanup standard
on the decontamination of a polluted aquifer and limit for the system. Further results in Table.1
preclude further spreading of contaminant plume. shows that increasing the number of pumping wells
is not feasible, since it involves high system cost and
Two wells scenario less performance. Hence the optimal pumping rate
During the remediation period, two obtained in this one well scenario is the best
abstraction wells are considered for pumping out the pumping policy for the chosen problem to cleanup
contaminated groundwater. The lower and upper the aquifer with in the specified time period. These
bounds of the abstraction wells are 250 m 3/d and investigations show that an appropriately designed
1000 m3/d respectively. The pumping rates of the pump and treat system can have a significant effect
two abstraction wells are encoded as a 20 bit binary on the decontamination of a polluted aquifer and
string with 10 bits for each pumping rate. The preclude further spreading of contaminant plume.
optimal size of the population is taken as 30. A
higher probability of crossover of 0.85 is chosen and VIII. CONCLUSIONS
lower mutation probability is set at 0.001. A Here an integrated simulation–optimization
converged solution is obtained after 30 generations. model is developed for the remediation of a confined
The progress of the aquifer decontamination for this aquifer that is polluted by a recharge pond and
scenario is shown in Fig 6. The optimal pumping injection well operational for a prolonged period of
rates obtained for the two wells are 252.199 m 3/d time. The two-dimensional model consists of three
and 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 robustness
two well scenario is presented in Fig.7. of the simulation optimization (FEM-EGA) model.
The problem used a cost based objective function for
One well scenario the optimization technique that took into account the
For this case, the length of the management costs of pumping and treatment of contaminated
period is same as that of three and two well groundwater for the aquifer system. Optimal
scenarios. One pumping well is considered for pumping pattern is obtained for cleanup of the
pumping out the contaminated groundwater during aquifer using FEM and EGA for a period of 3960
the remediation period. The lower and upper bounds days. Comparing the total extraction and pumping
of the pumping well are kept same as for the above patterns for the three scenarios, it was found that the
two scenarios. The pumping rate of the one one well scenario reduces the total pumping rate and
abstraction well is encoded as a string length of 10 shows the dynamic adaptability of the EGA based
bits. The optimal size of the population is chosen to optimization technique to achieve the cleanup
be 20. A moderate crossover probability of 0.85 is standards. The total remediation cost for the two
taken and lower mutation probability is set at 0.001. well scenarios is relatively less as compared to three
A converged solution is obtained after 25 and one well scenario. This is due to the pumping
generations which is relatively less as compared to well locations which are placed in the highly
the three well scenarios. Results indicate that the polluted area, so that removal of contaminant from
optimal pumping rate for one well for this case is these pumping wells is effective. The total
346.77 m3/d. This pumping rate is much lower extraction of contaminated groundwater gives an
compared to other two scenarios. Even for the lower indication of costs associated with pumping and
pumping rate the mass remaining in the system (728 treatment costs in the system. Hence the optimal
ppm) is marginally less than the specified limit (750 pumping rate obtained in this one well scenario is
ppm). Therefore one well scenario is perhaps the the best pumping policy for the chosen problem to
best 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 in
concentration contours at the end of 3960 days of remediation design optimization problem is that the
remediation 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 of
generations for one well scenario are presented in the objective function which often creates a
Fig. 9. Figure suggests marginal increase in the numerical instability, the GA based optimization
values after 5 generations and minimum at 25 model is robust and stable. However, the study
generations. found that application of genetic algorithms in
remediation design optimization model are
The results suggest that, for the remediation computational intensive for tuning of EGA
of contaminated aquifer using three and two well parameters.

931 | P a g e

[13] Smalley,J.B., Minsker, B.S., and Goldberg
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[4] McKinney, D. C., and Lin, M. D. (1995). remediation design.”J. of Hydroinformatics,
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programming methods for optimal aquifer [17] Espinoza, F. P., Minsker, B. M., and
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[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.
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(1998).”Optimization of Pump and treat Plng. and Mgmt., 131 (1), 67-78.
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contaminated aquifer: multi-stage design (2005). “Groundwater remediation design
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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.
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[27] Freeze, R.A., & Cherry. J. A. (1979).
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0
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[29] Reddy, J. N. (1993). An Introduction to the
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[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


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

800
Distance (m)

600

400

200

0
0 200 400 600 800 1000 1200 1400 1600 1800

Fig.4. Concentration distribution at the end of 3960 Fig.5. Values of objective function versus number
days remediation for three well scenario of generations for three well scenario

934 | P a g e

1000 Vol. 2, Issue 5, September- October 2012, pp.924-935
20000000

System performance in rupees
800 19000000

18000000
Distance (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

800
Distance (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

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