A novel approach is developed based on simulated annealing (SA) integrated with aquifer drawdown models to identify aquifer parameters of leaky and unconfined aquifer systems. In the leaky aquifer system, Hantush and Jacob’s analytical model [1955] is chosen to combine with SA to optimally determine the aquifer transmissivity T, storage coefficient S, and leakage coefficient L. Except these three parameters, Neuman and Witherspoon’s model [1969] is used to estimate the additional parameter ψ which describes the effect of the aquitard storage. Three sets of drawdown data given by Cooper [1963] and the drawdown data given by Sridharan [1987] were chosen for data analyses. The aquifer parameters obtained from SA suitably agree with those obtained from NLN or EKF coupled with Hantush and Jacob’s model or Neuman and Witherspoon’s model when analyzing those available drawdown data.
Likewise, the Neuman solution [1975] can also be employed with SA to estimate the horizontal and vertical hydraulic conductivities Kr and Kz, storage coefficient S, and specific yield Sy for an unconfined aquifer if the assumptions of Neuman model are satisfied. Two sets of pumping test data in the confined aquifers and one set in the unconfined aquifer are utilized to demonstrate the application of the proposed method in parameter identification.
The results show that the present method can determine the aquifer parameters with very good
accuracy. The identified results and related estimated errors indicate that the proposed method is superior to the graphical methods and gives results with the same degree of accuracy when compared with those of NLN and EKF.
The analyzed results based on SA with various control parameters are compared and discussed. The results indicate that the temperature reduction factor does not seem to affect the results of the parameter estimation. In addition, the estimated results are almost identical for various initial guesses which is different from several orders of magnitude. This fact shows that SA has a wide range of initial guess values and is a significant advantage over the NLN and EKF approaches. These analyses demonstrate that the proposed method is robust and reliable even if the user is not experienced in using SA.
The sensitivity analysis is used to investigate the influence period of aquifer parameters in both leaky and unconfined aquifers. The influences of parameters L and Sy on the drawdown are shown to have time lag in response to pumping in the leaky and unconfined aquifers, respectively. An on-line parameter estimation model is applied to estimate the parameters based on the data obtained from hypothetical and field pumping tests for both leaky and unconfined aquifers. The results indicate that the on-line estimation can be terminated when the estimated parameters are stabilized and their corresponding normalized sensitivities start to response to the pumping. In the hypothetical cases, the termination time of the on-line estimation is consistent with the influence period of the parameter which has
longest time lag from the beginning of the pumping. This fact indicates that the on-line estimation can be terminated if all identified parameters tend to be stabilized, i.e., the drawdown already reacts to the affect of aquifer parameters. In the field cases, the results indicate that the on-line parameter estimation model can save 90% pumping time in the leaky aquifer and 77% pumping time in the unconfined aquifer. Note that the small fluctuation in the estimated parameters at the late period of pumping and a longer on-line estimation time than that of the hypothetical case occur. These results may be mainly caused by aquifer heterogeneity and/or measurement errors in the observed drawdown data. Finally, different values of the specific yield and distance between pumping well and observation well do not significantly affect the influence period of specific yield during the pumping. These results may provide a useful reference for on-line aquifer parameter estimation.
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Table 1 Time-drawdown data of three observations wells [Cooper, 1963, p. 31]
Drawdown at observation well (m)
time(min) 1 2 3
0.2 0.536 0.003 0.000
0.5 0.838 0.043 0.000
1 1.094 0.137 0.006
2 1.298 0.284 0.043
5 1.609 0.536 0.168
10 1.798 0.713 0.302
20 1.972 0.869 0.445
50 2.109 1.009 0.594
100 2.167 1.067 0.640
200 2.195 1.070 0.643
500 2.198 1.073 0.643
1000 2.198 1.073 0.643
Table 2 Time-drawdown data [Sridharan, 1987, p. 170]
time(min) Drawdown(m)
5 0.30
28 0.95
41 1.10
60 1.25
75 1.34
250 1.75
500 1.90
700 1.95
970 1.98
1000 1.99
1200 1.99
Note: Q = 136.26 m /3 day, r = 29.00 m
Table 3 Comparison of results from three-parameter model when using SA, EKF, and NLN to analyze Cooper’s data [Cooper, 1963]
Estimated parameters Errors
Observation
well T (m2/day) S (×10−4) L ME (×10−4) SEE (×10−3) SA
1 1239.4 0.98 0.05 3.50 13.30
2 1243.4 0.97 0.25 -0.39 5.73
3 1221.2 1.01 0.51 -1.54 3.43
EKF
1 1257.9 0.91 0.05 -6.53 19.90
2 1311.4 0.93 0.23 37.20 8.62
3 1228.0 1.00 0.51 -2.44 4.09
NLN
1 1239.1 0.98 0.05 -1.10 13.30
2 1242.1 0.98 0.25 4.98 5.69
3 1215.2 0.97 0.51 -1.90 3.72
Table 4 The estimated parameters and estimated errors when using SA, EKF, and NLN to analyze Sridharan’s data [Sridharan et al., 1987]
Estimated parameters Errors
Method
T (m2/day) S (×10−4) L Ψ (×10−4) ME (×10−3)
SEE (×10−2)
SA 23.4 1.64 0.13 9.04 -1.81 1.02
EKF 22.6 1.73 0.14 3.16 1.49 1.36
NLN 23.3 1.65 0.13 7.04 -1.78 1.00
Table 5 Comparison of results when applying graphical methods, NLN, EKF, and SA methods to analyze the pumping test data obtained from an unconfined aquifer
Estimated parameters Errors
Method Kr×10-3 (m/s)
Kz×10-5 (m/s)
S×10-3 Sy×10-2 ME×10-3 (m)
SEE×10-3 (m) Graphical methods
Neuman type curve 2.40 1.62 1.46 5.73 32.90 34.59 Neuman semilograithmic 2.40 1.62 1.87 2.13 14.23 14.96
Computer methods
NLN 2.22 1.68 1.31 3.85 0.28 8.06
EKF 2.25 1.56 0.97 4.10 1.68 8.36
SA 2.23 1.67 1.31 3.83 0.31 8.06
Table 6 Estimated parameters using different temperature reduction factor
Estimated parameters
Temperature reduction factor Rt
T (m2/day) S (×10−4) L Ψ (×10−4)
0.90 23.36 1.64 0.13 9.06
0.80 23.35 1.64 0.13 9.58
0.75 23.36 1.64 0.13 9.04
0.70 23.35 1.64 0.13 8.60
0.60 23.35 1.64 0.13 9.57
0.50 23.34 1.65 0.13 9.59
0.30 23.35 1.64 0.13 9.58
Mean 23.35 1.64×10−4 0.13 9.29
Standard deviation 6.90×10-3 3.78×10-7 0.00 3.93×10-5
Table 7 Comparison of the results in leaky aquifer considering storage effect when using different initial guesses
Initial guess Estimated parameters T
Table 8 The synthetic drawdown data for the leaky aquifer No Time (min) Drawdown (m)
1 0.017 0.013 2 0.050 0.099 3 0.100 0.203 4 0.250 0.380 5 0.750 0.621 6 1.000 0.687 7 1.500 0.781 8 2.000 0.847 9 2.500 0.899 10 3.500 0.977 11 4.500 1.035 12 5.000 1.059 13 10.000 1.215 14 20.000 1.365 15 50.000 1.539 16 100.000 1.640 17 200.000 1.702 18 500.000 1.728 19 700.000 1.730 20 1000.000 1.730 Q = 3000 m3/day, r = 30 m
Table 9 The synthetic drawdown data set 1 for the unconfined aquifer
Table 10 The synthetic drawdown data set 2 for the unconfined aquifer
Table 11 Number of observations used in the synthetic data analysis and the estimated parameters for a leaky aquifer
Estimated parameters Number of
observations Time (min)
T (m2/day) S×10-4 L×10-2
3 0.10 1000.53 1.00 1.12
4 0.25 1000.32 1.00 2.52
5 0.75 1000.52 1.00 2.74
6 1.00 999.93 1.00 3.06
7 1.50 1000.02 1.00 3.00
8 2.00 999.96 1.00 3.03
9 2.50 999.98 1.00 3.01
10 3.50 999.99 1.00 3.00
11 4.50 999.99 1.00 3.01
12 5.00 999.95 1.00 3.01
13 10.00 1000.06 1.00 3.00
14 20.00 1000.02 1.00 3.00
15 50.00 1000.01 1.00 3.00
16 100.00 1000.02 1.00 3.00
17 200.00 1000.02 1.00 3.00
18 500.00 1000.04 1.00 3.00
19 700.00 1000.06 1.00 3.00
20 1000.00 1000.05 1.00 3.00
20 0.10 1000.83 1.00 1.25
Target values: T = 1000 (m2/day), S = 10-4, and L = 3×10-2
Table 12 The field time-drawdown data and the estimated parameters for a leaky aquifer using different number of observations
Number of Observations Time (min) Drawdown (m)
1 0.2 0.536
Estimated results using on-line PEM Estimated values Number of Observations
T (m2/day) S×10−4 L×10−2
Table 13 Number of observations used in the data analysis and the estimated parameters based on the synthetic data set 1
Estimated parameters
Number of
observations Time (s)
Kr (m/s)×10−3 Kz (m/s)×10−4 S×10−4 Sy×10−1
4 4 0.997 1.006 1.000 0.612
5 5 1.000 0.999 1.000 0.616
6 6 1.000 0.999 1.000 1.190
7 7 0.997 1.010 1.000 0.444
8 8 1.000 0.998 1.000 1.570
9 9 0.999 1.000 1.000 0.933
10 10 1.000 1.000 1.000 0.972
11 11 0.998 1.000 1.000 0.712
12 12 1.000 0.995 1.000 2.010
13 13 1.000 0.997 1.000 1.140
14 14 1.000 0.998 1.000 1.220
15 15 0.998 0.998 1.000 0.816
16 30 1.000 1.000 1.000 1.040
17 44 0.998 0.997 1.000 0.987
18 58 1.000 1.000 1.000 1.010
19 74 1.000 1.000 1.000 0.993
20 87 1.000 1.000 1.000 1.000
Target Values: Kr = 1×10−3(m/s), Kz =1×10−4(m/s), S = 1×10−4, and Sy = 1×10−1
Table 14 Number of observations used in the data analysis and the estimated parameters based on the synthetic data set 2
Estimated parameters
Table 15 The estimated parameters for an unconfined aquifer (Cape Cod site) using different
(a)
(b)
Figure 1 The sketch of the pumping tests of (a) leaky aquifer and (b) unconfined aquifer
Figure 2 Flowchart of the SA
Initialize
Figure 3 The flowchart of the identification procedure (a) conventional method, (b) present method
0.1 1 10 100 1000 Time(min)
0 0.5 1 1.5 2 2.5
Drawdown(m)
Estimated Drawdown for Well 1 Pumping Test Data for Well 1 Estimated Drawdown for Well 2 Pumping Test Data for Well 2 Estimated Drawdown for Well 3 Pumping Test Data for Well 3
Figure 4 The estimated drawdowns and the pumping test data for the observation wells in the leaky aquifer without considering storage effect
1 10 100 1000 10000 Time(min)
0 0.4 0.8 1.2 1.6 2
Drawdown(m)
Estimated Drawdown Pumping Test Data
Figure 5 The estimated drawdowns and the pumping test data for the observation wells in the leaky aquifer with considering storage effect
1 10 100 1000 10000 100000 1000000 Time (s)
0 0.1 0.2 0.3 0.4 0.5 0.6
Drawdown (m)
Estimated Drawdown Pumping Test Data
Figure 6 The estimated drawdown and the pumping test data obtained from an unconfined aquifer using SA
0.01 0.1 1 10 100 1000 Time (min)
0 0.4 0.8 1.2
Normalized sensitivity Si,t (m)
T S L
Drawdown data
0 0.4 0.8 1.2
Drawdwon(m)
Figure 7 The time-drawdown data and the normalized sensitivities of the leaky aquifer parameters
1 10 100 1000 10000 100000 1000000 Time(s)
0 0.4 0.8 1.2 1.6
Normalized sensitivity Si,t
Kr Kz S Sy
Drawdown data
0 0.4 0.8 1.2 1.6 2
Drawdown(m)
Figure 8 The time-drawdown data and the normalized sensitivities of the unconfined aquifer parameters (Neuman’s model)
0.1 1 10 100 1000 10000 100000 1000000 Time (s)
0 0.2 0.4 0.6
Normalized sensitivity si, t (m)
Kr
Kz S Sy
rw
Drawdown data
0 0.2 0.4 0.6 0.8
Drawdown(m)
0.1 1 10 100 1000 10000 100000 1000000
0 0.04 0.08 0.12 0.16
Figure 9 The normalized sensitivities of the unconfined aquifer parameters (Moench’s model)
0.01 0.1 1 10 100 1000 Time (min)
0.008 0.012 0.016 0.02 0.024 0.028 0.032
L
Figure 10 The estimated L versus time in the leaky aquifer case
0 50 100 150 200 250 Time (s)
0.04 0.08 0.12 0.16 0.2 0.24
Sy
Figure 11 The estimated Sy versus time using the synthetic data set 1
0 50 100 150 200 250 Time (s)
0 0.1 0.2 0.3
Sy
Figure 12 The estimated Sy versus time using the synthetic data set 2
0 1000 2000 3000 4000 5000 Time (min)
0 0.1 0.2 0.3
Sy
0.1 1 10 100 1000 10000
Time (min) 0
0.1 0.2 0.3
Sy
Figure 13 The estimated Sy versus time in the field unconfined aquifer
1 10 100 1000 10000 100000 1000000 Time (s)
0 0.1 0.2 0.3
Sensitivity coefficicent of Sy
r = 10 m Sy = 0.01 Sy = 0.1 Sy = 0.3
Figure 14 The normalized sensitivity of Sy for Sy = 0.01, 0.1, or 0.3 and r = 10 m
1 10 100 1000 10000 100000 1000000 Time (s)
0 0.1 0.2 0.3
Sensitivity coefficicent of Sy
Sy = 0.1 r = 10 m r = 30 m r = 50 m
Figure 15 The normalized sensitivity of Sy for Sy = 0.1 and r = 10, 30, or 50 m
VITA (個人簡歷)
姓 名 黃彥禎 性 別 男
生 日 民國66 年 12 月 25 日
1996-2000 學士，台灣大學造船與海洋工程學系 2000-2002 碩士，交通大學環境工程研究所 2002-2007 交通大學環境工程研究所博士班 學經歷
2006-2007 Visiting Scholar, Center for Research in Water Resources, The University of Texas at Austin, USA
行動電話 0933863499
通訊電話 03-5712121#55528
通訊地址 300 新竹市綠水路 56 號 9 樓之 1
E-mail boyz.ev89g@nctu.edu.tw
PUBLICATION LIST
(A) Journal Papers
1. Yeh, HD, Huang, YC, 2005, Parameter estimation for leaky aquifers using the extended Kalman filter, and considering model and data measurement uncertainties, Journal of Hydrology, 302(1-4), 28-45.
2. Yeh, HD, Huang, YC, 2005, Comment on “Application of BEM with extended Kalman filter to parameter identification of an elastic plate under dynamic loading” by Tanaka, M., T. Matsumoto, and H. Yamamura. Engineering Analysis with Boundary Elements, 29(1) 93-94.
3. Yeh, HD, Lin, YC, Huang, YC, 2007, Parameter Identification for Leaky Aquifers Using Heuristic Approaches, Hydrological processes, 21, 862-872, doi:10.1002/hyp.6274.
4. Huang, YC, Yeh, HD, 2007, The Use of Sensitivity Analysis in On-line Aquifer Parameter Estimation, Journal of Hydrology, 335 (3-4), 406 – 418
doi:10.1016/j.jhydrol.2006.12.007.
5. Huang, YC, Yeh, HD, 2007, Review of the pipeline leak detection methods in the water distribution systems, Journal of Chinese Institute of Civil and Hydraulic Engineering, 34(2), 82-90. (In Chinese).
6. Huang, YC, Yeh, HD, Lin, YC, 2007, A Computer Method Based on Simulated Annealing to Identify Aquifer Parameters Using Pumping Test Data, International Journal for numerical and analytical methods in Geomechanics, 2007. (In press).
7. Chang, YC, Yeh, HD, Huang, YC, 2007, Parameter identification in multi-layer aquifer systems using Simulated Annealing, Hydrogeology Journal. (accepted).
8. Huang, YC, Yeh, HD, 2007, Improvement of parameter identification for a slug test in a well with finite-thickness skin, Journal of Hydrology. (In preparation)
9. Huang, YC, Yeh, HD, Determination of the Specific Yield in unconfined aquifers:
Composite Analysis or not? (In preparation).
10. Wang, CT, Huang, YC, Yeh, HD, The analysis of groundwater sampling data using
討會，國立交通大學，新竹市，論文集(下冊)F181-F185 頁。
10. Huang, YC, Yeh, HD, 2004, Uncertainty and Sensitivity Analyses in Identifying Leaky Aquifer Parameters using Extended Kalman Filter, Western Pacific Geophysics Meeting, AGU, Hawaii, WP60.
11. 黃彥禎、葉弘德，93 年 10 月，含水層參數即時檢定的敏感度分析，九十三年度農
業工程研討會，中國農業工程學會，桃園，論文摘要集258 頁，論文集光碟版
1637-1644 頁。
12. Yeh, HD, Chang, YC, Huang, YC, 2005, Identifying horizontal multi-zone unconfined aquifer parameters using simulated annealing, AOGS 2nd annual meeting, Singapore, 58-HS-A0504.
13. 黃彥禎、葉弘德，94 年 10 月，探討以單口或多口觀測井洩降數據推求自由水層比 出水量參數的差異，九十四年度農業工程研討會，中國農業工程學會，苗栗，論文 摘要集104 頁。
14. 黃彥禎、王智澤、葉弘德、洪慧念、莊士群、邱志昇，94 年 11 月，應用主成分分 析探討地下水污染檢測數據，第三屆土壤與地下水研討會，環境工程學會，桃園，
論文摘要集453 頁。
(C) Others
1. Huang, YC, 2002, Application of extended Kalman filter for identifying leaky aquifer parameters, Master thesis, Institute of Environmental Engineering, National Chiao-tung University, Hsinchu, Taiwan.