CHAPTER 4 RESULTS AND DISCUSSION
4.1 Parameter Estimation Using the PEM Based on the SA
4.1.1 Estimation of leaky aquifer parameters
Table 1 lists the observed drawdown data obtained from a test with three monitoring wells reported in Cooper [1963] and cited by Lohman [1972] for parameter identification using the three-parameter model. The r for observation wells 1, 2, and 3 are, respectively, 30.48 m, 152.4 m, and 304.8 m. The Q and total pumping time are 5450.98 m3/day and 1000 minutes. In the four-parameter model, the time-drawdown data is taken from Sridharan et al. [1987] and listed in Table 2. The r is 29.0 m and the Q is 136.26 m3/day.
Three-parameter model
The upper and lower bounds for parameters estimated by SA when analyzing field data using the three-parameter model are 3000 and 0 m2/day for parameter T, 10-3 and 10-5 for the parameter S, and 5 and 0 for the parameter L, respectively. The control parameters of SA, initial temperature, reduction factor Rt, and number of algorithm iteration, are 10, 0.75, and 100, respectively. The choice of the initial temperature is generally case by case.
Nevertheless, Kirkpatrick et al. [1983] gave a guideline for setting the initial temperature that
solution is worse than the current solution, should be larger than 80% initially. This criterion has the merit of avoiding the situation that the current solution is trapped in a local optimum at early search. The process of SA will be terminated if the absolute differences between the two successive values of the optimal objective function are all within less than 10-6 through four iterations. The results determined from SA are compared with those obtained from the EKF and NLN method [Yeh and Huang, 2005] and listed in Table 3. The estimated T for drawdown data obtained from those three wells by the proposed methods, EKF, and NLN range from 1200 to 1300 m2/day. The estimated S ranges from 9.7×10-5 to 1.0×10-4 and the estimated leakage factor L ranges from 0.05 to 0.51. These results indicate that the aquifer of the study site is relatively homogenous. Figure 4 shows the observed drawdowns measured from those three wells and the estimated drawdowns generated by Hantush and Jacob’s model with those parameters obtained from the SA. Apparently, the estimated drawdowns quite suitably fit the pumping test data, as indicated in the figure.
Four-parameter model
In the data analysis of the four-parameter model, the upper and lower bounds for parameters T, S, L, and Ψ are 3000 to 0 m2/day, 10-3 to 10-5, 5 to 0, and 10-3 to 10-5 respectively. The initial temperature, reduction factor, and number of algorithm iteration are
the same as previous section. The results of the estimation for parameters T, S, L, and ψ are listed in Table 4. The estimated parameters obtained from SA almost agree with those obtained from NLN [Yeh and Huang, 2005], as indicated in Table 4. Figure 5 shows the observed drawdowns and the estimated drawdowns determined by Neuman and Witherspoon’s model and those parameters obtained from SA.
4.1.2 Estimation of unconfined aquifer parameters
The upper and lower bounds of Kr are respectively 10-2 and 10-4 (m/sec), of Kz are respectively 10-3 and 10-5 (m/sec), of S are respectively 5×10-3 and 10-5, and of Sy are respectively 3× 10-1 and 10-2 for an unconfined aquifer when applying SA. The initial temperature, reduction factor Rt, and number of algorithm iteration of SA are 10, 0.75, and 100, respectively. The other control parameters of SA are similar to those of the leaky aquifer case given in the previous section.
Table 5 lists the analyzed results and the estimated errors from the graphical methods such as the Neuman type-curve method and Neuman’s semilogarithmic method [Batu, 1998].
In addition, the estimated parameters and related errors resulting from the computer methods such as NLN, EKF [Leng and Yeh, 2003], and the present methods are also listed in Table 5.
The estimated parameters obtained by the SA are: 2.23×10-3 m/s for Kr; 1.67×10-5 m/s for Kz; 1.31× 10-3 for S; and 3.83× 10-2 for Sy, respectively. The estimated errors by SA are
parameters of SA give a better fit to the observed drawdown data. Figure 6 displays the estimated drawdown and the pumping test data in the unconfined aquifer. This figure also indicates that the proposed methods can optimally search the parameters of the unconfined aquifer. Clearly, these estimated results and related errors demonstrate that the proposed methods are much superior to the graphical methods and give the results with the same degree of accuracy when compared with those of NLN and EKF.
4.1.3 The sensitivity analysis of SA’s control parameters
The use of control parameters in SA, such as the initial guess value and the temperature reduction factor Rt, may affect the results of the parameter estimation. For demonstrating robustness and reliability of SA in parameter identification, this study presents two sensitivity analyses of the control parameters in SA for the parameter estimation when analyzing pumping-test data from Sridharan [1987].
Table 6 lists the aquifer parameters with a different reduction factor Rt. The estimated parameter T ranges from 23.34 to 23.36 (m2/day), the parameter S ranges from 1.64×10-4 to 1.65×10-4, the parameter L is 0.13 and keeps the same in different Rt cases. The results of estimated aquitard storage coefficient Ψ range from 8.60×10-4 to 9.59×10-4 with slight variation. The means of T, S, L, and Ψ are 23.4 (m2/day), 1.64×10-4, 0.13, and 9.29×10-4, respectively, which is very close to those estimated by various methods as shown in Table 4.
The standard deviations of T, S, L, and are 6.90×10-3, 3.78×10-7, 0, and 3.93×10-5
respectively, which are very small when compared with their mean values, indicating that the identified results are independent of Rt values. In other words, the influence of choosing various values of Rt on the results of the parameter estimation is negligible.
If the initial guess values are far away from the target parameters, gradient-type methods for solving the nonlinear least-square equations might give divergent results. This is the major disadvantage of employing the NLN method in solving nonlinear least equations.
Therefore, different initial guess values for SA are chosen to examine the performance of SA in parameter estimation. Table 7 displays the estimated parameters with fourteen different combinations of initial guesses. The estimated parameters are almost identical, even if the initial guesses are different from several orders of magnitude. These results indicate that SA can not only successfully estimate the aquifer parameters but also give a consistent estimation when using different temperature reduction factor and initial guesses.