**4. Discussion of solution results**

The developed methodology was applied to a hypothetical illustrative study area with synthetically generated concentration measurements over space and time. Advantage of using a hypothetical study area lies in the fact that unknown data errors do not distort the performance evaluation of the methodology. This helps in understanding the drawbacks of developed methodology and improving it further.

## **4.1. Study area**

The hypothetical study area is a heterogeneous aquifer measuring 2100m x 1500m x 30m and consisting of three unconfined layers as shown in Figure 2.

The East and west boundaries are constant head boundaries, whereas north and south boundaries are no flow boundaries. There are two sources (S1 and S2) of contamination. S1 is located in the top layer and S2 in middle layer. Five monitoring location (M1 through M5) are located in the first layer as shown in Figure 3. A grid size of 30m x 30m x 10m is used for finite difference based numerical calculation of groundwater flow and transport equations. Transport time step used for MT3DMS is 36.5 days. Other model parameters are listed in Table 1. Only a conservative contaminant is considered. There are two point sources of contaminants. One in the top layer and another one in the middle layer. A time horizon of 16 years is considered. Entire time horizon is divided into 5 different stress periods. The first four stress periods are each 1.5 years long and the final stress period is of 10 years duration. Sources are assumed to be active only in the first four stress periods or in the initial 6 years. Original source fluxes are presented in Table 2. It is assumed that groundwater contamination is detected at five different locations in the study area at the

10 Will-be-set-by-IN-TECH 166 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 11

Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems

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

Parameter Values Length of Study Area (m) 2100 Width of Study Area (m) 1500 Saturated thickness, b(m) 30 Grid spacing in x-direction, Δx (m) 30 Grid spacing in y-direction, Δy (m) 30

Hydraulic conductivity in x-direction, *Kxx* (m/day) 12 Hydraulic conductivity in y-direction, *Kyy* (m/day) 8 Vertical Anisotropy 5 Hydraulic Gradient (m/m) 0.002 Effective porosity, *θ* 0.3 Longitudinal dispersivity, *α<sup>L</sup>* (m) 18 Transverse dispersivity, *α<sup>T</sup>* (m) 4 Initial contaminant concentration (mg/l) 0.00

Sources Layer Row Column Contaminant Flux (g/sec)

Source 1 1 12 15 Stress Period 3 1.5 years 8.000

Source 2 2 38 9 Stress Period 3 1.5 years 6.000

A set of error free observation data is generated. These observations are then used to evaluate the developed linked simulation-optimization methodology based on both GA and ASA. Input parameters used for GA and ASA are presented in Table 3. Every iteration of ASA based method uses one run of the groundwater transport simulation model (MT3DMS) whereas every generation of GA based method uses 100 (population size) runs of the same simulation model. Irrespective of the method, one run of the groundwater transport simulation model takes 4.281 sec to run on a Dell Optiplex®running an Intel®Core™2 Duo Processor at 2.93GHz. The execution time for one transport simulation run is however dependent on the computing platform. In order to keep the comparison independent of computing platform, both the methods were compared based on number of transport simulation runs which is directly proportional to the execution time. Both the methods were used to estimate source release histories using the error free data. In order to verify the convergence of each optimization method, time of run was made practically unconstrained. It was found that eventually both the optimization algorithms were able to achieve an objective function value very close to zero and identified the release history accurately. The objective function

Stress Period 1 1.5 years 6.000 Stress Period 2 1.5 years 4.000 167

Stress Period 4 1.5 years 5.000 Stress Period 5 10 years 0.000 Stress Period 1 1.5 years 7.000 Stress Period 2 1.5 years 9.000

Stress Period 4 1.5 years 7.300 Stress Period 5 10 years 0.000

Grid Spacing in z-direction, Δz (m) 10

**Table 1.** Model Parameters

**Table 2.** Original Source Fluxes

**4.2. Release history estimation with error free data**

**Figure 3.** Top View of Study Area Showing Sources and Monitoring Locations

end of 8 th year, that is two years after the sources had ceased to exist. The observation wells are monitored for a period of 8 years starting from year 9 at an interval of 36.5 days. Observed contaminant concentration measurements at the designated monitoring locations are generated using MT3DMS as transport simulation model followed by perturbation as per Equation 5.

166 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 11 167 Application of Simulated Annealing in Water Resources Management: Optimal Solution of Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems


**Table 1.** Model Parameters

10 Will-be-set-by-IN-TECH

**Figure 2.** Illustrative Study Area

Equation 5.

**Figure 3.** Top View of Study Area Showing Sources and Monitoring Locations

end of 8 th year, that is two years after the sources had ceased to exist. The observation wells are monitored for a period of 8 years starting from year 9 at an interval of 36.5 days. Observed contaminant concentration measurements at the designated monitoring locations are generated using MT3DMS as transport simulation model followed by perturbation as per


**Table 2.** Original Source Fluxes

#### **4.2. Release history estimation with error free data**

A set of error free observation data is generated. These observations are then used to evaluate the developed linked simulation-optimization methodology based on both GA and ASA. Input parameters used for GA and ASA are presented in Table 3. Every iteration of ASA based method uses one run of the groundwater transport simulation model (MT3DMS) whereas every generation of GA based method uses 100 (population size) runs of the same simulation model. Irrespective of the method, one run of the groundwater transport simulation model takes 4.281 sec to run on a Dell Optiplex®running an Intel®Core™2 Duo Processor at 2.93GHz. The execution time for one transport simulation run is however dependent on the computing platform. In order to keep the comparison independent of computing platform, both the methods were compared based on number of transport simulation runs which is directly proportional to the execution time. Both the methods were used to estimate source release histories using the error free data. In order to verify the convergence of each optimization method, time of run was made practically unconstrained. It was found that eventually both the optimization algorithms were able to achieve an objective function value very close to zero and identified the release history accurately. The objective function

#### 12 Will-be-set-by-IN-TECH 168 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

convergence profile as well as estimated fluxes were plotted at the end of 40,000 simulation runs of the groundwater transport model. Minimum value of objective function achieved is plotted against number of runs of the transport simulation model. The estimated flux values for both the sources in each stress period is also plotted against actual source fluxes. Convergence profile and source flux estimates are shown in Figure 4. Convergence profile Network Design Problems 13

Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems

optimization method using ASA is compared with the one using GA as the optimization algorithm. Parameters used for both the optimization algorithms is presented in Table 3. Unlike the case with error free measurement data, in this case both the methods were used Parameters of ASA Parameters of GA Accepted to generated ratio 1.00E-06 Mutation Strategy: Polynomial Mutation

Cost Precision 1.00E-15 Population size 80

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

Maximum Cost Repeat 5 Total no. of generations 600 Temperature Ratio Scale 1.00E-05 Cross over probability 0.8 Temp. Anneal Scale 100 Mutation probability 0.05

to reconstruct source release histories using the erroneous data with a limit on execution time. In order to make the comparison consistent by ensuring same number of simulation runs in the ASA and GA based methodologies, the number of simulation runs are restricted to 40,000. This restriction was based on the fact that increasing the number of simulation runs even to 80,000 resulted in very little improvement in the objective function value. Minimum value of objective function achieved is averaged over five solutions and is plotted against number of runs of the transport simulation model. The plot is presented in Figure 5. This

plot clearly shows that ASA based method converges much faster in the beginning and the GA based method is able to achieve comparable objective function values only after a much

**Table 3.** Parameters used in Optimization Algorithms

**Figure 5.** Convergence Plot

Variable Boundaries : Rigid

169

**Figure 4.** Convergence Profile and Estimated Release History with Error Free Data

shows that the objective function value for source identification model converges to a value very close to zero with about 5,000 simulation runs. However, further convergence is accelerated when using ASA algorithm. From these results, it can be concluded that the developed methodology is able to achieve optimal solution for an ideal error free scenario which resembles a well-posed problem.

#### **4.3. Release history estimation with erroneous data**

Five sets of erroneous observation data are generated with the formulation described in Equation 5. The value of fraction 'a' is specified as 0.1. These erroneous observations are used to reconstruct the release histories of contaminant sources. Linked simulation optimization method using ASA is compared with the one using GA as the optimization algorithm. Parameters used for both the optimization algorithms is presented in Table 3. Unlike the case with error free measurement data, in this case both the methods were used


**Table 3.** Parameters used in Optimization Algorithms

12 Will-be-set-by-IN-TECH

convergence profile as well as estimated fluxes were plotted at the end of 40,000 simulation runs of the groundwater transport model. Minimum value of objective function achieved is plotted against number of runs of the transport simulation model. The estimated flux values for both the sources in each stress period is also plotted against actual source fluxes. Convergence profile and source flux estimates are shown in Figure 4. Convergence profile

**Figure 4.** Convergence Profile and Estimated Release History with Error Free Data

which resembles a well-posed problem.

**4.3. Release history estimation with erroneous data**

shows that the objective function value for source identification model converges to a value very close to zero with about 5,000 simulation runs. However, further convergence is accelerated when using ASA algorithm. From these results, it can be concluded that the developed methodology is able to achieve optimal solution for an ideal error free scenario

Five sets of erroneous observation data are generated with the formulation described in Equation 5. The value of fraction 'a' is specified as 0.1. These erroneous observations are used to reconstruct the release histories of contaminant sources. Linked simulation to reconstruct source release histories using the erroneous data with a limit on execution time. In order to make the comparison consistent by ensuring same number of simulation runs in the ASA and GA based methodologies, the number of simulation runs are restricted to 40,000. This restriction was based on the fact that increasing the number of simulation runs even to 80,000 resulted in very little improvement in the objective function value. Minimum value of objective function achieved is averaged over five solutions and is plotted against number of runs of the transport simulation model. The plot is presented in Figure 5. This

**Figure 5.** Convergence Plot

plot clearly shows that ASA based method converges much faster in the beginning and the GA based method is able to achieve comparable objective function values only after a much

#### 14 Will-be-set-by-IN-TECH 170 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

larger simulation runs. Because of the erroneous measurement data this problem may be ill-posed and the solution may not be unique. Therefore, lower objective function values do not always mean accurate reconstruction of the release histories.

Network Design Problems 15

Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems

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

171

Monitoring network design in the context of groundwater quality management essentially means specifying the spatial location of monitoring wells and frequency of sampling. Since this is one of the most cost intensive part of most contaminated groundwater remediation problems, an efficient and cost effective design of monitoring network is essential. Monitoring

Irrespective of the various objectives, the problem of monitoring network design can be formulated as an optimization problem [8, 20]. While designing a monitoring network for estimating unknown groundwater source characteristics, the objective of optimization can be to maximize the reliability of estimated source characteristics or to minimize the total number of monitoring locations in the network or both. Compliance monitoring is aimed at minimizing the area of contamination when the contamination is first detected at monitoring network or maximizing the probability of detection of contaminant in groundwater. Often, only the average values of hydro-geological parameters of the aquifer are known. This results in uncertainty in the modeling results. In order to better characterize an aquifer, spatial distribution of hydro-geological properties should be specified. This objective can be achieved by sampling hydro-geologic parameter at sufficient locations such that the interpolated values can represent actual hydrological parameters accurately. The objective of optimization in this case is to find the minimum number of samples required to accurately represent a population of random hydro-geological parameter values. In all such cases, Adaptive Simulated Annealing can be efficiently used as the tool for optimization. Our attempts to develop classical simulated annealing algorithm for optimal design of a dedicated monitoring network for enhancing the efficiency of source identification was successful to a large extent. However, the mixed integer nature of the decision variables in a monitoring network design problem makes the application of classical simulated annealing algorithm a bit constraining. Adaptive Simulated Annealing is more suitable to solve this monitoring

**5. Application of simulated annealing for monitoring network design**

of groundwater quality may be necessitated by a variety of objectives such as:

2. Compliance monitoring for limiting the effects of groundwater contamination

**6. Suitability and sensitivity of adaptive simulated annealing**

In the application discussed here, Simulated Annealing is utilized for finding the global minimum of a cost function that characterizes large and complex systems such as transport of pollutants in groundwater.Simulated Annealing, as an algorithm, is very efficient in solving non-convex optimization problems by ensuring that it does not always move downhill on a complex non-convex search space and hence avoids getting trapped in local minimum. Simulated annealing also differs significantly from conventional iterative optimization algorithms in that gross features of the final state of the system are seen at higher temperatures whereas the finer details of the state appear at lower temperatures [10]. The fact that simualted annealing ensures a global optimal solution enhances its suitability for solving

1. Unknown groundwater source characterization

3. Better aquifer characterization

network design problems.

4. Hydro-geological parameter estimation

In order to test the effectiveness of the competing methods based on accuracy of solutions produced, reconstructed release histories were compared to the actual release history after every set of 10,000 transport simulation runs. The results are shown in Figure 6. It can be seen that ASA based method is more efficient compared to GA based method after 10,000 and 20,000 simulation runs. However, as the execution time increases further with increase in number of simulation runs, the release histories produced by both methods become similar. This is also confirmed from the calculated values of NAEE presented in Table 4. As the execution time increases, the NAEE of ASA based method appears to increase only slightly. This could be due to statistical variation in the five different solutions and may be attributed to the input data error. Averaging over larger number of solutions may modify this inference. NAEE of GA based method consistently improves. However, the NAEE values obtained using ASA is still better in comparison.

**Figure 6.** Reconstructed Release Histories using the competing methods


**Table 4.** Normalized Absolute Error of Estimation
