2.1 Groundwater flow and transport simulation
Darcy’s law can be written as (Konikow et al., 1996)
3 the effective porosity (dimensionless), is the hydraulic conductivity tensor of the porous media [L/T], h is the hydraulic head [L], and xi are the Cartesian coordinates. Combining Darcy’s law with the continuity equation, the three-dimensional groundwater flow equation can be expressed as (Konikow et al., 1996) unit volume (positive for inflow and negative for outflow [1/T]). Equation (2) can be used to predict the hydraulic head distribution for the groundwater flow field.
The governing equation for three-dimensional solute transport in groundwater can be written as (Konikow et al., 1996)
( ) ( )
⎟⎟− ′ =0 , =1 ,2 ,3the dispersion coefficient [L2/T], and C′ is the concentration of the source or sink fluid [M/L3]. The average linear velocity can be determined by equation (1).
The computer model MODFLOW-GWT developed by the United States Geological Survey (USGS) and developed based on equations (2) and (3) can be used to simulate the groundwater flow and contaminant transport simultaneously. This model combined the modular three-dimensional finite-difference ground-water flow model, MODFLOW-2000, (Harbaugh et al. 2000) and the three-dimensional method-of-characteristics solute-transport model (MOC3D) (Konikow et al. 1996) to simulate groundwater flow field and spatial and temporal plume distribution, respectively.
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i2.2 Simulated annealing
The concept of SA is based on an analogy to crystallization process of the physical annealing from a high temperature state. Annealing is a physical process of heating up a solid to a very high temperature and then slowly cooling the solid down until it crystallizes. If the temperature is cooled properly, a most stable crystalline structure of the rock will be gained with the system reaching a minimum energy state.
The set of solution space looks like the different crystalline structures and the optimal solution is equivalent to the most stable crystalline structure.
In the SA, the Metropolis mechanism is employed to determine the acceptance of
adjacent solution. The Metropolis mechanism has a property to let the SA having the ability to accept the bad solution, preventing the SA from having the same defect as the descent method. Figure 1 is the flowchart of the SA algorithm (Pham and Karaboga, 2000). Yeh et al. (2007a) gave more detailed introduction on the algorithm of SA. The SA been successfully applied to various types of problem such as the THM forecast (Lin and Yeh, 2005), aquifer parameter estimation (e.g., Yeh and Chen, 2007b; Huang and Yeh, 2007c), pipe wall surface reaction rate (Yeh et al., 2008), and pumping source information (Lin and Yeh, 2008).
2.3 Tabu search
Glover (1986) proposed the two main concepts of TS: memory and learning.
The objective of tabu is through interdicted some attributes and improved the search more efficient and accurate. Through memory and learning, the TS is able to have more intensification and diversification in algorithm. Memory means to memorize the passed by solutions and to avoid the repetition of evaluations. During the process of learning, the prior result is memorized to influence the result of next experiment. A better result may encourage the next trial to increase the accuracy of the obtained solution. Then through the learning result, the following search can focus on the better solutions but not wasting time on worse solutions. According to these two ideas, TS utilizes the tabu list and aspiration criterion to interdict or to
encourage some trial solutions during the iterative process. The utility of the tabu list is to memorize some lately evaluated trial solutions. The goal of the aspiration criteria is to release some of the solutions memorized in the tabu list to avoid the iteration cycling and may finally trap solutions in a local optimum. Figure 2 illustrates the flowchart of the TS algorithm. The TS been successfully applied to identify optimal parameter structure (Zheng and Wang, 1996) and spatial pattern of groundwater pumping rates (Tung and Chou, 2004).
2.4 Ordinal optimization
Recently, the OOA has been applied to many areas in terms of simulation-based complex optimization problem. The OOA has two major tenets: ordinal comparison and goal softening procedures. The ordinal comparison procedure is to see the relative relationship between each solution because it is much easier to find better solutions. The goal softening procedure is to determine a reliable and good enough solution instead of directly evaluating the optimal solution in a complex optimization model. The purpose of goal softening procedure is to reduce the consumption time on computer calculation and to obtain the optimum solution from the feasible solution space. To get the top proportion solutions is much easier than to find out the best one. Lau and Ho (1997) showed that the OOA ensures that top 5% solutions can be regard as good enough solutions and have very high probability (≧0.95) to be
reliable.
According to the OOA, all the possible trials are estimated coarsely and ranked quickly. The solution domain is divided to several different parts, the possible optimum solution located in which sub-domain might be effortlessly to recognize.
The optimum solution can then be easily to obtain while all the calculation efforts are focused in searching the possible sub-domain. Therefore, a crude model should first be employed to estimate and rank the solution, and then the good solutions can be differentiated from the bad solutions. Then, the goal softening procedure is focused on the top proportion solutions to determine the optimum solution. Accordingly, the simulation time can be reduced effectively. The OOA been successfully applied to power system planning and operation (Guan et al., 2001; Lin et al., 2004), the electricity network planning (Liu et al., 2006) and the wafer testing (Lin and Horng, 2006) and so on.
2.5 Roulette wheel
The roulette wheel selection method is an important part of GA. The key concept of GA is survival of the fittest by natural selection. Better solutions have good objective function values and thus the areas occupied on roulette wheel are larger in proportion and their corresponding solutions will be selected with greater probability. During the process of iteration, the ones that hope good solutions can
constantly be selected. Strengthen and calculate in good solution nearby, will have a very high chance to find out the global optimal solution. Through this method, much time can be saved to avoid evaluating the bad solutions.
2.6 SATSO-GWT model
A new model called SATSO-GWT is developed based on SATS-GWT and OOA.
The objective function value in SATSO-GWT is to minimize the sum of square errors between the simulated concentration and observed concentration and could be defined as where nm is the total number of monitoring wells, np is the number of observed concentration measured in a monitoring well, Cij,sim is the simulated concentration at jth terminated time period in ith monitoring well, Cij,obs is the observed concentration sampled at jth terminated time period in ith monitoring well. The value nm×np is generally greater than the number of unknowns (Yeh et al., 2007a). Equation (4) is used to calculate the objective function value of the trial solution generated by the approach.
Figure 3 shows the flowchart of SATSO-GWT while Figures 4 and 5 show the flowchart of the TS process and OOA, respectively. TS and SA are used to generate the candidate location and NS trial solutions for the release period and concentration,
respectively. The objective function value is then calculated based on the sampled concentrations and simulated concentrations generated based on those source location and the release periods and concentrations. Each candidate location is regarded as one sub-domain and the OOA is utilized to choose the best 5% sub-domains. The best combination of the source location and the release periods and concentrations, i.e., the least objective function value, is recorded at each sub-domain. Totally, NT locations are generated by TS at each temperature; therefore, NT sets of best combination are obtained. As the number of generated combinations reaches total candidate locations about 3 times for several temperature levels, the top 5% best sub-domains, can be sifted. After obtaining the top 5% best sub-domains, the roulette wheel method is applied and the best combination regarding source release information has more opportunity to be chosen when decreasing the temperature. In reality, the real source location falls in the best combinations. The algorithm is terminated when the objective function values are less than 10-6 four times successively. Finally, the latest updated solution, including the estimated location and the release concentrations and time periods, is considered as the final solution.