3.1 Groundwater Flow and Transport
The contaminant transport in the groundwater is a complicate process. The transport process may include advection, dispersion, diffusion, adsorption, and biodegradation. The groundwater flow field should be known before simulating the contaminant transport process. The three-dimensional groundwater flow equation may be expressed as (Konikow et al. 1996)
t W
storage, t is time, W is the volumetric flux per unit volume (positive for inflow and negative for outflow), and xi are the Cartesian coordinates. The three-dimensional contaminant transport in groundwater may be written as (Konikow et al. 1996)( ) ( ) − ′ =
0 velocity of groundwater flow, Di is dispersion coefficient in x, y, and z directions, andC′ is concentration of the source fluid. Equation (1) is used to predict the head distribution for a flow field. The average groundwater flow velocity in equation (2)
can be determined by the Darcy’s law, i.e., (Konikow et al. 1996)
ε h V
iK
ij∇
−
=
(3)The temporal and spatial concentration distribution of a contaminant being released at a specified point can be simulated by Equation (2). The computer model MODFLOW-GWT developed by the United State Geology Survey (USGS) can be used to simulate the groundwater flow and contaminant transport simultaneously.
This model consists of a three-dimensional method-of-characteristics solute transport model (MOC3D) (Konikow et al. 1996) and the modular three-dimensional finite-difference ground-water flow model (MODFLOW-2000) (Harbaugh et al.
2000).
3.2 Simulated Annealing
The algorithm of SA is based on an analogy to the physical annealing process.
Annealing is a physical process of heating up a solid and then cooling the solid down slowly until it crystallizes. Physically, when a rock is heated, the activity of molecules in the rock is increased with the temperature. Then the temperature is slowly decreased to let molecules form crystalline structures. A most stable crystalline structure of the rock only formed when the rock is properly cooled. If the cooling is carried too fast, an irregularities structure may be obtained and the system
does not reach the minimum energy state.
At the beginning of SA, a higher temperature is given and an initial guess x is required to evaluate the objective function value f(x). An upper bound and a lower bound are given and the region between upper bound and lower bound can be defined as the solution domain. All the trial solutions are generated within the solution domain. For any point x, a neighborhood trial solution x’ is randomly generated and its objective function value is denoted as f(x’). The neighborhood trial solution x’ is given as:
r RD
x
x ' = + ( 2 *
1− 1 ) *
(4) where RD1 is a random number between zero and one generated from a uniform distribution and r is the step length vector. For point x, the next neighborhood trial solution x’ is selected from x-r to x+r. Note that the trial solution must be generated between the upper bound and lower bound. For solving the minimization problem, if f(x’) is smaller than f(x), then x’ is accepted and the current optimal solution x is replaced by x’. If f(x’) is not smaller than f(x), then the Metropolis’ criterion is used to test the acceptability of the trial solution. The Metropolis’ criterion provides a mechanism to accept the inferior solutions and the acceptance of the inferior solutions avoids the trial solution trapped in local. For solving minimization problem, the Metropolis’ criterion is given as (Pham and Karaboga 2000)
where PSA
is the acceptance probability of the trial solution x’ and Te is current
temperature. A random number RD2 ranging between zero and one is generated from a uniform distribution. If RD2is smaller than P
SA, the trial solution x’ is accepted and called ascent move. Otherwise, keep on generating the trial solutions from the current solution. After a serial of trial solutions are generated, the current temperature is decreased by a constant, called temperature reduction factor, and the prior steps are repeated continually. Note that the acceptance probability of the ascent move decreases with the temperature going down. The algorithm is terminated when the stopping criteria are satisfied.3.3 Tabu Search
Learning and memory are the main concepts of TS proposed by Glover (1986).
During the process of learning, the prior result is memorized to influence the next experiment. A worse result may cause the next experiment to be canceled and a better result may encourage the next trial. 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 purpose of tabu list memorizes some lately evaluated trial solutions. The intention of aspiration criteria release the solutions
memorized in tabu list to avoid the solutions trapping in local optimum.
The iterative process of TS contains four components: initial guess, candidate solution and movement, tabu list, and aspiration (Tung and Chou 2004). At the beginning, an initial guess for the unknown variables is considered as the current solution (CUS) and the guess values are used to calculate the objective value. Let this objective value as the global optimal objective value (GOOV). Next, several adjacent candidate solutions (CASs) are generated in the neighborhood of the current solution and their objective values are also evaluated. For the minimization problem, when the best objective value (BOV) is less than GOOV, then the aspiration criterion is applied to remove the best CAS from the tabu list (TL) if it is in the list. At the same time, the CUS is moved to the TL. In addition, the best CAS becomes the new CUS and the BOV becomes new GOOV. For BOV > GOOV, the next best CAS will be selected if the best CAS is in the TL; otherwise, the best CAS becomes the new CUS. The procedures are repeated continually by generating other adjacent CASs from the neighborhood of the new CUS if the stopping criterion is not satisfied.
Figure 1 shows the flowchart of tabu search.