**Author details**

Igor Arambasic, Javier Casajus Quiros and Ivana Raos *ETSI Telecomunicacion, Universidad Politecnica de Madrid, Spain*

26 Will-be-set-by-IN-TECH 156 Simulated Annealing – Single and Multiple Objective Problems **Chapter 0**

## **6. References**

[1] Arambasic, I. [2008]. *RF Front-End Non-Linear Coupling Cancellation*, PhD thesis, Escuela Técnica Superior de Ingenieros de Telecomunicación.

**Application of Simulated Annealing in Water**

**Chapter 8**

**Resources Management: Optimal Solution of**

Estimating various characteristics of an unknown groundwater pollutant source can be formulated as an optimization problem using linked simulation-optimization. Meta-heuristics based optimization algorithms such as Simulated Annealing (SA), Genetic Algorithm (GA), Tabu Search etc. are now being accepted as reliable, faster and simpler ways to solve this optimization problem. In this chapter we discuss the suitability of a variant of traditional Simulated Annealing (SA) known as the Adaptive Simulated Annealing (ASA) in solving unknown groundwater pollutant source characterization problem. Growing anthropogenic activities and improper management of their impacts on groundwater quality has resulted in widespread contamination of groundwater worldwide. Coupled with ever increasing water demand leading to increased reliance on groundwater, it has resulted in a widespread recognition of public health risk posed by contaminated groundwater. This has triggered massive efforts for better management of groundwater quality in general and remediation of contaminated aquifers in particular. The sources of contamination in groundwater are often hidden and inaccessible. Characteristics of these pollutant sources such as their location, periods of activity and contaminant release history are often unknown. Groundwater contaminant source identification problem aims at estimating various characteristics of an unknown groundwater pollutant source using measured contaminant concentrations at a number of monitoring locations over a period of time. It has been widely accepted that for any remediation strategy to work efficiently, it is very important to know the pollutant source characteristic. A detailed account of different categories of source identification problems and various approaches to solve them has been presented in Pinder

> ©2012 Datta and Jha, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 Datta and Jha, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Characterization Problem and Monitoring**

**Groundwater Contamination Source**

**Network Design Problems**

Additional information is available at the end of the chapter

cited.

Manish Jha and Bithin Datta

http://dx.doi.org/10.5772/45871

**1. Introduction**

[26].


**Application of Simulated Annealing in Water Resources Management: Optimal Solution of Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems**

Manish Jha and Bithin Datta

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/45871

**1. Introduction**

26 Will-be-set-by-IN-TECH

[1] Arambasic, I. [2008]. *RF Front-End Non-Linear Coupling Cancellation*, PhD thesis, Escuela

[2] Arambasic, I., Quiros, F. J. C. & Raos, I. [2007a]. Efficient rf front-end non-linear multi antenna coupling cancellation techniques, *PIMRC 2007. IEEE 18th International*

[4] Drago, G., Manella, A., Nervi, M., Repetto, M. & Secondo, G. [1992]. A combined strategy for optimization in non linear magnetic problems using simulated annealing and search

[5] Finnerty, S. & Sen, S. [2004]. Simulated annealing based classification, *Sixth International Conference on Tools with Artificial Intelligence, 1994. Proceedings.* (Nov): 824 – 827.

[7] Ingber, L. [1989]. Very fast simulated re-annealing, *Lester Ingber Papers 89vfsr*, Lester

[8] Ingber, L. [1993]. Simulated annealing: Practice versus theory, *Mathl. Comput. Modelling*

[9] Ingber, L. [1996]. Adaptive simulated annealing (asa): Lessons learned, *Lester Ingber Papers 96as*, Lester Ingber. available at http://ideas.repec.org/p/lei/ingber/96as.html. [10] Ingber, L. & Rosen, B. [1992]. Genetic algorithms and very fast simulated reannealing: a

[11] Johnson, D. S., Aragon, C. R., McGeoch, L. & Schevon, C. [1989]. Optimization by simulated annealing: An experimental evaluation, *Operations Research* pp. 865–892. [12] Kirkpatric, S., Galett, C. D. & Vecchi, M. P. [1983]. Optimisation by simulated annealing,

[13] Li, Y., Yao, J. & Yao, D. [2002]. An efficient composite simulated annealing algorithm for global optimization, *IEEE 2002 International Conference on Communications, Circuits and*

[14] Mendonca, P. & Caloba, L. [1997]. New simulated annealing algorithms, *Proceedings of 1997 IEEE International Symposium on Circuits and Systems, 1997. ISCAS '97.* 3: 1668 – 1671. [15] Renyuan, T., Shiyou, Y., Yan, L., Geng, W. & Tiemin, M. [1996]. Combined strategy of improved simulated annealing and genetic algorithm for inverse problem, *IEEE*

[16] Rosen, B. [1992]. Function optimization based on advanced simulated annealing,

[17] Szu, H. H. & Hartley, R. L. [1987]. Nonconvex optimization by fast simulated annealing,

[18] van Laarhoven, P. J. M. & Aarts, E. [1987]. *Simulated Annealing: Theory and Applications*,

*Symposium on Personal, Indoor and Mobile Radio Communications, 2007* pp. 1 – 5. [3] Arambasic, I., Quiros, F. J. C. & Raos, I. [2007b]. Improvement of multiple antennas diversity systems through receiver nonlinear coupling cancellation, *2007. International*

Técnica Superior de Ingenieros de Telecomunicación.

*Waveform Diversity and Design Conference* pp. 102 – 106.

techniques, *IEEE Transactions on Magnetics* 28(2): 1541–1544.

[6] Forrest, S. [1996]. Genetic algorithms, *Computing Surveys* 28: 77–80.

comparison, *Mathematical and Computer Modelling* pp. 87–100.

*Systems and West Sino Expositions* 21: 1165– 1169.

*Workshop on Physics and Computation, 1992* pp. 289–293.

*Transactions on Magnetics* 32: 1326 – 1329.

*Proceedings of the IEEE* 75(11): 1538 – 1540.

Ingber. available at http://ideas.repec.org/p/lei/ingber/89vfsr.html.

**6. References**

18(11): 29–57.

D. Reidel.

*Science* 220(4598): 621–680.

Estimating various characteristics of an unknown groundwater pollutant source can be formulated as an optimization problem using linked simulation-optimization. Meta-heuristics based optimization algorithms such as Simulated Annealing (SA), Genetic Algorithm (GA), Tabu Search etc. are now being accepted as reliable, faster and simpler ways to solve this optimization problem. In this chapter we discuss the suitability of a variant of traditional Simulated Annealing (SA) known as the Adaptive Simulated Annealing (ASA) in solving unknown groundwater pollutant source characterization problem. Growing anthropogenic activities and improper management of their impacts on groundwater quality has resulted in widespread contamination of groundwater worldwide. Coupled with ever increasing water demand leading to increased reliance on groundwater, it has resulted in a widespread recognition of public health risk posed by contaminated groundwater. This has triggered massive efforts for better management of groundwater quality in general and remediation of contaminated aquifers in particular. The sources of contamination in groundwater are often hidden and inaccessible. Characteristics of these pollutant sources such as their location, periods of activity and contaminant release history are often unknown. Groundwater contaminant source identification problem aims at estimating various characteristics of an unknown groundwater pollutant source using measured contaminant concentrations at a number of monitoring locations over a period of time. It has been widely accepted that for any remediation strategy to work efficiently, it is very important to know the pollutant source characteristic. A detailed account of different categories of source identification problems and various approaches to solve them has been presented in Pinder [26].

©2012 Datta and Jha, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Datta and Jha, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Will-be-set-by-IN-TECH 158 Simulated Annealing – Single and Multiple Objective Problems Application of Simulated Annealing in Water Resources Management: Optimal Solution of Groundwater Contamination Source Characterization Problem and Monitoring

Typically, groundwater contamination is first detected by one or more arbitrarily located well or monitoring location. Unknown Groundwater source identification problem specifically attempts to ascertain the following source characteristics:

Network Design Problems 3

Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems

Application of Simulated Annealing in Water Resources Management: Optimal Solution of

159

optimization algorithm then calculates the value of objective function by calculating the difference in contaminant concentrations estimated by simulation model and actual observed values at the same monitoring locations over a period of time. Over a number of iterations, the optimization algorithm minimizes the objective function value. Most prominent approaches in this category are linear programming with response matrix approach [9], nonlinear optimization with embedding technique [21–23], artificial neural network approach [28–30], constrained robust least square approach [31, 32], classical optimization based approach [5–7],

In recent past, heuristic global search approaches such as Genetic Algorithm [12], Harmony Search, Tabu Search, Ant-Colony Optimization, Simulated Annealing[19] have developed rapidly and have been applied to a wide range of optimization problems. One of the major reasons for their popularity is the fact that these optimization methods do not easily get trapped in the local optima, thereby maximizing the probability of achieving a global optimal solution. Genetic algorithm (GA) and its variants, in particular, have been widely applied for solving unknown pollutant source identification. [11, 24, 27]. Genetic Algorithms are computational optimization algorithms that simulate the laws of natural genetics and natural

Apart from GA or its variants, Simulated Annealing has also been used in solving inverse problems in groundwater management. Simulated annealing is inspired by the physical process of annealing in metallurgy which involves heating and controlled cooling of a material to reduce defects in crystal structure. The atoms are excited by heat and they become agitated while getting into higher energy states. The slow cooling allows a better chance for these atoms to achieve lower energy states than the ones they started with. In simulated annealing, a current solution may be replaced by a random "neighborhood" solution chosen with a probability that depends on the difference between corresponding function values and on a global parameter T (called temperature) that is gradually decreased in the process. Implementations of simulated annealing has been relatively limited because the traditional simulated annealing algorithm is reported to converge slower compared with GA or its variants. However, faster variants of simulated annealing have been developed and one of the most promising variants in terms of convergence speed is Adaptive Simulated Annealing (ASA) [14, 15]. The ASA code was first developed in 1987 as Very Fast Simulated Re-annealing (VFSR) [13]. Ingber & Rosen [16] showed that VFSR is at least an order of magnitude superior to Genetic Algorithms in convergence speed and is more likely to find the global optima

Linked simulation-optimization based approaches are computationally intensive as the simulation model has to be run many thousands of times before an acceptable solution is produced. This has been a deterrent to any desktop based implementation of the simulation-optimization approach. Faster convergence can reduce the computational burden significantly and thereby enhance the possibility of a desktop based implementation of linked simulation-optimization approach. This paper investigates the applicability of ASA to unknown groundwater contaminant source release history reconstruction problem and compares its performance to genetic algorithm based solution. The performance evaluation of competing simulation-optimization approaches are based on a realistic scenario of missing measurement data, where contaminant concentration measurements are available a few years after the sources have ceased to exist. Apart from the convergence speed, the two algorithms are compared for their ability to produce accurate source release histories with moderately

genetic algorithm based approach [1, 24, 27] etc.

selection and use it to search for the optimal solution.

during a time limited search.


Source type is often obvious. In some cases, information on groundwater contaminant source location may be available from preliminary investigations. If an exhaustive record of pollutant inventory and industrial activities of the area is available, it may be possible to infer start time. Release history of the source, however, is difficult to ascertain as the source is not physically accessible for measurements and hence it is unlikely that any accurate temporal record of contaminant fluxes released from the source exists.

Release history reconstruction problem is one of the most widely studied groundwater source identification problems. Ascertaining release history of the contaminant sources from available contaminant concentration measurements is an inverse problem as it requires solving groundwater flow and transport equations backwards in time and space. The process of solving this inverse problem is essentially the process of finding various unknown characteristics of source using observed information about the transport media and the effects caused by the source. In such circumstances, a solution cannot be guaranteed, especially when observed information is sparse. Even if the solution exists, it may not be unique. This is because different combinations of various source characteristics can produce the same effect at a monitoring location. Moreover, the solution of this problem is highly sensitive to measurement errors either in the observation data or model parameters and hence this problem has been classified as an ill-posed inverse problem[33] . When this inverse problem has to be solved by using inaccurate values of media parameters such as hydraulic conductivity and porosity and contaminant concentration observed at arbitrarily placed monitoring wells, it becomes even more challenging to obtain a reliable solution.

Methods proposed in the past to solve this ill-posed inverse problem can be broadly classified as optimization approaches, analytical solutions, deterministic direct methods and probabilistic and geo-statistical simulation approaches. A detailed review of these methodologies can be found in Atmadja & Bagtzoglou [2]; Michalak & Kitanidis [25]; Bagtzoglou & Atmadja [3] and Sun et al. [31, 32]. The most effective of all suggested methods seems to be those based on optimization or probabilistic and geo-statistical simulation. Of the optimization methods, linked simulation-optimization approaches have been established as one of the most efficient methods. In this approach, a numerical groundwater flow and transport simulation model is linked to the optimization model. All the linked simulation-optimization approaches aim at solving a minimization problem with an objective function representing the difference in measured concentration and simulated concentration at various monitoring locations. The optimization model generates candidate solutions for various source characteristics. This is used as an input for the simulation model to generate estimated contaminant concentration observations at designated monitoring locations. The optimization algorithm then calculates the value of objective function by calculating the difference in contaminant concentrations estimated by simulation model and actual observed values at the same monitoring locations over a period of time. Over a number of iterations, the optimization algorithm minimizes the objective function value. Most prominent approaches in this category are linear programming with response matrix approach [9], nonlinear optimization with embedding technique [21–23], artificial neural network approach [28–30], constrained robust least square approach [31, 32], classical optimization based approach [5–7], genetic algorithm based approach [1, 24, 27] etc.

2 Will-be-set-by-IN-TECH

Typically, groundwater contamination is first detected by one or more arbitrarily located well or monitoring location. Unknown Groundwater source identification problem specifically

5. Contaminant flux released as a function of time elapsed since start time (release history)

Source type is often obvious. In some cases, information on groundwater contaminant source location may be available from preliminary investigations. If an exhaustive record of pollutant inventory and industrial activities of the area is available, it may be possible to infer start time. Release history of the source, however, is difficult to ascertain as the source is not physically accessible for measurements and hence it is unlikely that any accurate temporal record of

Release history reconstruction problem is one of the most widely studied groundwater source identification problems. Ascertaining release history of the contaminant sources from available contaminant concentration measurements is an inverse problem as it requires solving groundwater flow and transport equations backwards in time and space. The process of solving this inverse problem is essentially the process of finding various unknown characteristics of source using observed information about the transport media and the effects caused by the source. In such circumstances, a solution cannot be guaranteed, especially when observed information is sparse. Even if the solution exists, it may not be unique. This is because different combinations of various source characteristics can produce the same effect at a monitoring location. Moreover, the solution of this problem is highly sensitive to measurement errors either in the observation data or model parameters and hence this problem has been classified as an ill-posed inverse problem[33] . When this inverse problem has to be solved by using inaccurate values of media parameters such as hydraulic conductivity and porosity and contaminant concentration observed at arbitrarily placed monitoring wells, it becomes even more challenging to obtain a reliable solution.

Methods proposed in the past to solve this ill-posed inverse problem can be broadly classified as optimization approaches, analytical solutions, deterministic direct methods and probabilistic and geo-statistical simulation approaches. A detailed review of these methodologies can be found in Atmadja & Bagtzoglou [2]; Michalak & Kitanidis [25]; Bagtzoglou & Atmadja [3] and Sun et al. [31, 32]. The most effective of all suggested methods seems to be those based on optimization or probabilistic and geo-statistical simulation. Of the optimization methods, linked simulation-optimization approaches have been established as one of the most efficient methods. In this approach, a numerical groundwater flow and transport simulation model is linked to the optimization model. All the linked simulation-optimization approaches aim at solving a minimization problem with an objective function representing the difference in measured concentration and simulated concentration at various monitoring locations. The optimization model generates candidate solutions for various source characteristics. This is used as an input for the simulation model to generate estimated contaminant concentration observations at designated monitoring locations. The

attempts to ascertain the following source characteristics:

3. Release pattern (slug, continuous, intermittent etc.)

contaminant fluxes released from the source exists.

4. Point of time when the source first became active (start time)

1. Source Type (point, areal etc.)

2. Spatial location and extent of the source

In recent past, heuristic global search approaches such as Genetic Algorithm [12], Harmony Search, Tabu Search, Ant-Colony Optimization, Simulated Annealing[19] have developed rapidly and have been applied to a wide range of optimization problems. One of the major reasons for their popularity is the fact that these optimization methods do not easily get trapped in the local optima, thereby maximizing the probability of achieving a global optimal solution. Genetic algorithm (GA) and its variants, in particular, have been widely applied for solving unknown pollutant source identification. [11, 24, 27]. Genetic Algorithms are computational optimization algorithms that simulate the laws of natural genetics and natural selection and use it to search for the optimal solution.

Apart from GA or its variants, Simulated Annealing has also been used in solving inverse problems in groundwater management. Simulated annealing is inspired by the physical process of annealing in metallurgy which involves heating and controlled cooling of a material to reduce defects in crystal structure. The atoms are excited by heat and they become agitated while getting into higher energy states. The slow cooling allows a better chance for these atoms to achieve lower energy states than the ones they started with. In simulated annealing, a current solution may be replaced by a random "neighborhood" solution chosen with a probability that depends on the difference between corresponding function values and on a global parameter T (called temperature) that is gradually decreased in the process. Implementations of simulated annealing has been relatively limited because the traditional simulated annealing algorithm is reported to converge slower compared with GA or its variants. However, faster variants of simulated annealing have been developed and one of the most promising variants in terms of convergence speed is Adaptive Simulated Annealing (ASA) [14, 15]. The ASA code was first developed in 1987 as Very Fast Simulated Re-annealing (VFSR) [13]. Ingber & Rosen [16] showed that VFSR is at least an order of magnitude superior to Genetic Algorithms in convergence speed and is more likely to find the global optima during a time limited search.

Linked simulation-optimization based approaches are computationally intensive as the simulation model has to be run many thousands of times before an acceptable solution is produced. This has been a deterrent to any desktop based implementation of the simulation-optimization approach. Faster convergence can reduce the computational burden significantly and thereby enhance the possibility of a desktop based implementation of linked simulation-optimization approach. This paper investigates the applicability of ASA to unknown groundwater contaminant source release history reconstruction problem and compares its performance to genetic algorithm based solution. The performance evaluation of competing simulation-optimization approaches are based on a realistic scenario of missing measurement data, where contaminant concentration measurements are available a few years after the sources have ceased to exist. Apart from the convergence speed, the two algorithms are compared for their ability to produce accurate source release histories with moderately erroneous data and with uncertainty in estimation of hydro-geological parameters. One of the most important factors that affects the execution time and accuracy of solutions generated by linked simulation-optimization approaches is the choice of observation locations. Poorly chosen contaminant observation locations often produce misleading results and hence it becomes important that after the initial estimation of the contaminant sources, a monitoring network is designed and implemented. In this study we use a monitoring network designed specifically to enhance the efficiency of source identification. However, a detailed discussion of the methodology used for monitoring network design is beyond the scope of this book.

Network Design Problems 5

Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems

Application of Simulated Annealing in Water Resources Management: Optimal Solution of

161

**Figure 1.** Schematic Representation of Linked Simulation-Optimization Model using SA

*qs* is the volumetric flux of water per unit volume of aquifer representing sources

*Dij* is the hydrodynamic dispersion coefficient, *L*2*T*−1; *ϑ<sup>i</sup>* is the seepage or linear pore water velocity, *LT*−1;

*Cs* is the concentration of the sources or sinks, *ML*−3;

(positive) and sinks (negative), *T*−1;

## **2. Methodology**

The linked simulation-optimization approach consists of two parts. An optimization algorithm generates the candidate solutions corresponding to various unknown groundwater source characteristics. The candidate solutions are used as input in the numerical groundwater transport simulation model to generate the concentration of contaminant in the study area. The generated concentration at designated monitoring locations is matched to the observed values of contaminant concentrations at various time intervals at the same locations. The difference between simulated and observed concentration is used to calculate the objective function value which is utilized by the optimization algorithm to improve the candidate solution. The process continues until an optimal solution is obtained. A detailed schematic representation of this process of using SA as the optimization algorithm in a linked simulation-optimization model is presented in Figure 1. The classical simulated annealing (SA) algorithm has many associated guiding parameters such as the initial parameter temperature, annealing schedule, acceptance probability function, goal function etc. Effective application of the classical simulated annealing to a particular optimization problem normally involves a lot of trials and adjustments to achieve ideal values for all or most of these parameters. ASA, which is a variant of classical SA, helps overcome this difficulty to a certain extent by automating the adjustments of parameters controlling temperature schedule and random step selection thereby making the algorithm less sensitive to user defined parameters compared with classical SA. This additional ability of ASA combined with inherent ability of classical SA to find the global optimal solution even when multiple local optimums exists, makes it a natural choice for solving the groundwater pollutant source identification problem.

#### **2.1. Governing equations**

The three-dimensional transport of contaminants in groundwater can be represented by the following partial differential equation [17]

$$\frac{\partial \mathbb{C}}{\partial t} = D\_{i\bar{j}} \frac{\partial}{\partial \mathbf{x}\_{\bar{i}}} \left( \frac{\partial \mathbb{C}}{\partial \mathbf{x}\_{\bar{j}}} \right) - \frac{\partial}{\partial \mathbf{x}\_{\bar{i}}} \left( \theta\_{\bar{i}} \mathbb{C} \right) + \frac{q\_{\bar{s}}}{\theta} \mathbb{C}\_{\bar{s}} + \sum\_{k=1}^{N} R\_{k} \tag{1}$$

Where

C is the concentration of contaminants dissolved in groundwater,*ML*−3; t is time, T;

*xi*, *xj* is the distance along the respective Cartesian coordinate axis, L;

160 Simulated Annealing – Single and Multiple Objective Problems Application of Simulated Annealing in Water Resources Management: Optimal Solution of Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems 5 161 Application of Simulated Annealing in Water Resources Management: Optimal Solution of Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems

4 Will-be-set-by-IN-TECH

One of the most important factors that affects the execution time and accuracy of solutions generated by linked simulation-optimization approaches is the choice of observation locations. Poorly chosen contaminant observation locations often produce misleading results and hence it becomes important that after the initial estimation of the contaminant sources, a monitoring network is designed and implemented. In this study we use a monitoring network designed specifically to enhance the efficiency of source identification. However, a detailed discussion of the methodology used for monitoring network design is beyond the scope of

The linked simulation-optimization approach consists of two parts. An optimization algorithm generates the candidate solutions corresponding to various unknown groundwater source characteristics. The candidate solutions are used as input in the numerical groundwater transport simulation model to generate the concentration of contaminant in the study area. The generated concentration at designated monitoring locations is matched to the observed values of contaminant concentrations at various time intervals at the same locations. The difference between simulated and observed concentration is used to calculate the objective function value which is utilized by the optimization algorithm to improve the candidate solution. The process continues until an optimal solution is obtained. A detailed schematic representation of this process of using SA as the optimization algorithm in a linked simulation-optimization model is presented in Figure 1. The classical simulated annealing (SA) algorithm has many associated guiding parameters such as the initial parameter temperature, annealing schedule, acceptance probability function, goal function etc. Effective application of the classical simulated annealing to a particular optimization problem normally involves a lot of trials and adjustments to achieve ideal values for all or most of these parameters. ASA, which is a variant of classical SA, helps overcome this difficulty to a certain extent by automating the adjustments of parameters controlling temperature schedule and random step selection thereby making the algorithm less sensitive to user defined parameters compared with classical SA. This additional ability of ASA combined with inherent ability of classical SA to find the global optimal solution even when multiple local optimums exists, makes it a natural choice for solving the groundwater pollutant source identification problem.

The three-dimensional transport of contaminants in groundwater can be represented by the

(*ϑiC*) <sup>+</sup> *qs*

*<sup>θ</sup> Cs* <sup>+</sup>

*N* ∑ *k*=1

*Rk* (1)

erroneous data and with uncertainty in estimation of hydro-geological parameters.

this book.

**2. Methodology**

**2.1. Governing equations**

Where

t is time, T;

following partial differential equation [17]

*∂C <sup>∂</sup><sup>t</sup>* <sup>=</sup> *Dij*

*∂ ∂xi*  *∂C ∂xj*  <sup>−</sup> *<sup>∂</sup> ∂xi*

C is the concentration of contaminants dissolved in groundwater,*ML*−3;

*xi*, *xj* is the distance along the respective Cartesian coordinate axis, L;

*Dij* is the hydrodynamic dispersion coefficient, *L*2*T*−1;

*ϑ<sup>i</sup>* is the seepage or linear pore water velocity, *LT*−1;

*qs* is the volumetric flux of water per unit volume of aquifer representing sources (positive) and sinks (negative), *T*−1;

*Cs* is the concentration of the sources or sinks, *ML*−3;

*θ* is the porosity of the porous medium, dimensionless;

*N* is the number of chemical species considered;

∑*<sup>N</sup> <sup>k</sup>*=<sup>1</sup> *Rk* is the chemical reaction term for each of the N species considered, *ML*−3*T*−1.

In order to solve this transport equation, linear pore water velocity needs to be known for the study area. Hence, it becomes necessary to first calculate the hydraulic head distribution using a groundwater flow simulation model. The partial differential equation for groundwater flow is given by the following equation:

$$\frac{\partial}{\partial \mathbf{x}} \left( K\_{\mathbf{x}\mathbf{x}} \frac{\partial h}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{x}} \left( K\_{yy} \frac{\partial h}{\partial y} \right) + \frac{\partial}{\partial \mathbf{x}} \left( K\_{zz} \frac{\partial h}{\partial z} \right) + W = S\_{\mathbf{s}} \frac{\partial h}{\partial t} \tag{2}$$

Network Design Problems 7

Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems

Application of Simulated Annealing in Water Resources Management: Optimal Solution of

*iob* = Concentration estimated by the identification model at observation well

*iob*= Observed concentration at well iob and at the end of time period k;

*iob*= Weight corresponding to observation location iob, and the time period k.

*iob* <sup>=</sup> <sup>1</sup> (*cobs<sup>k</sup>*

Where n is a constant, sufficiently large, so that errors at low concentrations do not dominate

Of the various simulated annealing implementations, it is evident in literature that the adaptive simulated annealing algorithm converges faster [16] while maintaining the reliability of results and hence it was preferred over traditional Boltzmann annealing implementation [19]. Its application to the unknown pollutant source identification has been limited but it is potentially a good alternative because its convergence curve is steep, thereby producing better

Currently, the most widely used optimization algorithm for solving groundwater source identification problem using linked simulation-optimization model is Genetic Algorithm and its variants. The effectiveness of ASA in solving this problem is compared against the effectiveness of GA. Genetic algorithms (GAs) are population based search strategies which are popular for many difficult to solve optimization problems including inverse problems. GAs emulate the natural evolutionary process in a population where the fittest survive and reproduce [12]. GA-based search performs well because of its ability to combine aspects of solutions from different parts of the search space. Real coded genetic algorithm was used with a population size of 100, crossover probability of 0.85 and a mutation probability of 0.05.

In order to evaluate the performance of two different optimization algorithms involving comparison of solutions obtained, it is vital to first ensure that only one solution exists. In other words, a unique solution has to be guaranteed. This is possible only under the following

1. The numerical models used for simulation of groundwater flow and transport are able to

2. All the model parameters and concentration measurements are known without any

*iob* <sup>+</sup> *<sup>n</sup>*)<sup>2</sup> (4)

163

Where,

*cest<sup>k</sup>*

*cobs<sup>k</sup>*

The weight *w<sup>k</sup>*

**2.3. Optimization algorithms**

results when execution time is limited.

**3. Performance evaluation**

idealized assumptions [33]:

associated errors.

*wk*

location iob and at the end of time period k.

*iob* can be defined as follows:

nob= Total number of observation wells;

nk = Total number of concentration observation time periods;

*wk*

the solution [18]. It is possible to include other forms of this weight.

The values were chosen based on a series of numerical experiments.

provide exact solution of the governing equations in forward runs.

Where

*Kxx*, *Kyy*, and *Kzz* are the values of hydraulic conductivity (*LT*−1) along the x, y and z co-ordinate axes respectively;

*H* is the potentiometric head (L);

W is the volumetric flux per unit volume representing sources and/or sinks of water (*T*−1);

*Ss* is the specific storage of the porous media (*L*−1); and t is time (T).

The flow equation describes transient groundwater flow in three dimensions in a homogeneous anisotropic medium, provided the principal axes of hydraulic conductivity are aligned with the co-ordinate directions. A computer code called MODFLOW is used to solve this groundwater flow equation. MODFLOW was developed by United States Geological Survey (USGS) and is one of the most popular computer programs being used to simulate groundwater flow today. MODFLOW is based on modular finite-difference method which discretizes the study area into a grid of cells. The potentiometric head is calculated at the center of each cell.

To solve the three dimensional ground water transport equation, another computer code called MT3DMS is used. This is also a very popular computer program developed by the USGS and uses modular finite-difference just like MODFLOW. The transport simulation model (MT3DMS) utilizes flow field generated by the flow model (MODFLOW) to compute the velocity field used by the transport simulation model. [34]

### **2.2. Formulation of the optimization problem**

It is assumed in this study that information on a set of potential source locations are available. The objective of simulation-optimization method then reduces to regenerating the source release histories at these potential source locations. Spatial and temporal contaminant concentration(C) is known at specific monitoring locations at various point of time. Candidate source fluxes are generated by the optimization algorithm. These values are used for forward transport simulations in MT3DMS. The difference between simulated and observed contaminant concentrations are then used to calculate the objective function. The objective function for this optimization problem is defined as:

$$Minimize \text{F1} = \sum\_{k=1}^{nk} \sum\_{iob=1}^{nob} \left( \text{cost}\_{iob}^k - obs\_{iob}^k \right)^2 . w\_{iob}^k \tag{3}$$

Where,

6 Will-be-set-by-IN-TECH

*<sup>k</sup>*=<sup>1</sup> *Rk* is the chemical reaction term for each of the N species considered, *ML*−3*T*−1.

In order to solve this transport equation, linear pore water velocity needs to be known for the study area. Hence, it becomes necessary to first calculate the hydraulic head distribution using a groundwater flow simulation model. The partial differential equation for groundwater flow

*Kxx*, *Kyy*, and *Kzz* are the values of hydraulic conductivity (*LT*−1) along the x, y and z

W is the volumetric flux per unit volume representing sources and/or sinks of water

The flow equation describes transient groundwater flow in three dimensions in a homogeneous anisotropic medium, provided the principal axes of hydraulic conductivity are aligned with the co-ordinate directions. A computer code called MODFLOW is used to solve this groundwater flow equation. MODFLOW was developed by United States Geological Survey (USGS) and is one of the most popular computer programs being used to simulate groundwater flow today. MODFLOW is based on modular finite-difference method which discretizes the study area into a grid of cells. The potentiometric head is calculated at the

To solve the three dimensional ground water transport equation, another computer code called MT3DMS is used. This is also a very popular computer program developed by the USGS and uses modular finite-difference just like MODFLOW. The transport simulation model (MT3DMS) utilizes flow field generated by the flow model (MODFLOW) to compute

It is assumed in this study that information on a set of potential source locations are available. The objective of simulation-optimization method then reduces to regenerating the source release histories at these potential source locations. Spatial and temporal contaminant concentration(C) is known at specific monitoring locations at various point of time. Candidate source fluxes are generated by the optimization algorithm. These values are used for forward transport simulations in MT3DMS. The difference between simulated and observed contaminant concentrations are then used to calculate the objective function. The objective

+ *W* = *Ss*

*∂h*

*<sup>∂</sup><sup>t</sup>* (2)

*θ* is the porosity of the porous medium, dimensionless; *N* is the number of chemical species considered;

*Ss* is the specific storage of the porous media (*L*−1); and

the velocity field used by the transport simulation model. [34]

**2.2. Formulation of the optimization problem**

function for this optimization problem is defined as:

*MinimizeF*1 =

*nk* ∑ *k*=1

*nob* ∑ *iob*=1

 *cest<sup>k</sup>*

*iob* <sup>−</sup> *cobs<sup>k</sup>*

*iob* 2 .*w<sup>k</sup>*

*iob* (3)

∑*<sup>N</sup>*

Where

(*T*−1);

t is time (T).

center of each cell.

is given by the following equation: *∂ ∂x Kxx ∂h ∂x* + *∂ ∂x Kyy ∂h ∂y* + *∂ ∂x Kzz ∂h ∂z* 

co-ordinate axes respectively; *H* is the potentiometric head (L); *cest<sup>k</sup> iob* = Concentration estimated by the identification model at observation well location iob and at the end of time period k.

nk = Total number of concentration observation time periods;

nob= Total number of observation wells;

*cobs<sup>k</sup> iob*= Observed concentration at well iob and at the end of time period k;

*wk iob*= Weight corresponding to observation location iob, and the time period k.

The weight *w<sup>k</sup> iob* can be defined as follows:

$$w\_{iob}^k = \frac{1}{(cob s\_{iob}^k + n)^2} \tag{4}$$

Where n is a constant, sufficiently large, so that errors at low concentrations do not dominate the solution [18]. It is possible to include other forms of this weight.
