CHAPTER 4 RESULTS AND DISCUSSION
4.3 Parameter Estimation using On-line PEM
Table 11 lists the number of observations (drawdown data) used in the data analysis and the estimated parameters for a hypothetical leaky aquifer case. The estimation process starts with three observations (shown at the first column) since the number of unknown parameter is three. The target values of the parameters T, S and L are 1000 m2/day, 10-4, and 3×10-2, respectively. The parameter estimation indicates that T and S are correctly identified even at
the beginning of the pumping. The results of estimated L using three, four, five, and six observation data points have the same order of magnitude as the target value, and the relative errors of estimated L are 63%, 16%, 8.7%, and 2%, respectively. The parameters are stably identified using more than seven observation data, i.e., after 1.5 minutes. These results indicate that the aquifer parameters are determined when the corresponding normalized sensitivities start to respond to the pumping. Moreover, the temporal curve of estimated L exhibited in Figure 10 shows fluctuation at first few steps and approaches a constant value after about 1.5 minutes. These results imply that the on-line PEM can successfully identify the parameters of leaky aquifer when the estimated L starts to be stabilized. The last row of Table 11 shows the estimated results by analyzing 20 observations during 0.1 minute (i.e., the time interval is setting as 0.005 minute). The estimated T, S, and L are 1000.83 m2/day, 1.00×10-4, and 1.25×10-2, respectively. This result demonstrates that the inaccurate estimate of parameter L is mainly due to the insensitivity of drawdown to the aquifer parameter at early period but not caused by the insufficiency of the observations.
Table 12 displays the field time-drawdown data and the estimated parameters for a leaky aquifer using different number of observations. The time-drawdown data measured from observation wells, as reported in Cooper [1963] and cited by Lohman [1972, p.31, Table 11], are selected for the data analysis. The r is 30.48 m, Q is 5450.98 m2/day, b is 30.48 m, and total pumping time is 1000 minutes (16.67 hours). It is clear that the estimated values of
parameters T and S do not fluctuate drastically when the number of observation using by on-line PEM is larger than 7, i.e., after 20 minutes. The estimated parameters T and S are 1203.80 m2/day and 1.04× 10-4, respectively. Comparing with the estimated parameters calculated based on the total number of observations (1239.28 m2/day for T and 9.80×10-5 for S), the relative errors of parameters T and S are both smaller than 5% when the number of
observation is larger than 7. Similarly, the estimated values of parameter L remain almost the same when the number of the observation utilized by the on-line PEM is larger than 9.
In this case, the on-line estimation can be terminated after 100 minutes. The on-line PEM saves tremendous 90% time and 3407 m3 groundwater resources if compared with total pumping time and pumped water volume required by conventional graphical approaches.
Note that small fluctuation in the estimated parameters at the late period of pumping and a longer parameter estimation time than that of the hypothetical case may be attributed to aquifer heterogeneity and/or measurement errors in the observed drawdowns
The estimation results with different number of observation using on-line PEM for the synthetic unconfined aquifer data set 1 are listed in Table 13. The identification process starts with four observation data points because the number of unknown parameter is four.
The target values of the parameters Kr, Kz, S, and Sy are 1×10-3 m/sec, 1×10-4 m/sec, 1×10-4, and 1×10-1, respectively. This table only lists the results when the number of observations is less than 20 because the estimated parameters are almost the same as the target values when
the number of observation is larger than 20. Figure 8 shows that the normalized sensitivities of parameters Kr, Kz, and S have immediate response to the pumping but the normalized sensitivity of parameter Sy has a time lag in response to the pumping. The identification results also reflect this phenomenon. The estimated Sy ranges from 4.44×10-2 to 2.01×10-1 and the largest relative errors are 101% when using 12 observation data. The identification results of Sy did not approach the target value until the number of observation is over 20, i.e., about 80 seconds. Therefore, the on-line PEM may not obtain accurate results of Sy if the time-drawdown data is too short to cover the response period of Sy. Similar to Figure 10, the curve of estimated Sy versus time displayed in Figure 11 shows dramatic fluctuation in the early period and converges to a constant value after about 80 seconds. Figures 8 and 10 demonstrate that the on-line PEM can successfully identify the aquifer parameters when Sy
just starts to affect the drawdown. Therefore, the on-line estimation based on Neuman’s model can be terminated once the identified parameters become stable.
Similar to Table 13, the identification results for the synthetic data set 2 are listed in Table 14. The target values of the parameters Kr, Kz, S, Sy, and rw are 1×10-3 m/sec, 1×10-4 m/sec, 1×10-4, 1×10-1, and 1 m, respectively. The estimated parameters are all the same as the target values when the number of observation is larger than 30. The parameters Kr, Kz, S, and rw are accurately determined at first few seconds. The estimated Sy ranges from 1.00×10-2 to 2.91×10-1 and did not approach the target value until the pumping time is over
125 seconds. The curve of estimated Sy versus time displayed in Figure 12 also shows dramatic fluctuation in the early period and converges to a constant value after about 125 seconds. Hence, the on-line estimation can be terminated even based on Moench’s model.
Table 15 shows the estimated parameters for the first field pumping test in an unconfined aquifer using different number of observations. The site of Cape Cod, Massachusetts is selected for the study [Moench et al., 2000]. Its aquifer was composed of unconsolidated glacial outwash sediments that were deposited during the recession, 14,000 to 15,000 years before present, of the late Wisconsinan continental ice sheet. The depth of the pumping well was 24.4 m below the land surface. The top and bottom of the screen were located 4.0 and 18.3 m, respectively, below the initial water table, which was approximately 5.8 m below land surface. The aquifer saturated thickness was about 48.8 m. Well F507-080 was pumped at an average rate 1.21 m3/min for 72 hours. The data set of the observation well F505-032 is selected in this case. The distance between pumping well and observation well is 7.28 m.
From Table 15, the estimated Kr ranges from 2.20 ×10-4 m/sec to 1.97×10-3 m/sec, the estimated Kz ranges from 1.0×10-6 m/sec to 2.25×10-4 m/sec, the estimated S ranges from 3.45×10-3 to 7.29×10-3, and the estimated Sy ranges from 0.016 to 0.3. It can be found that the ranges of estimated Kr and S are small as compared with those of the Kz and Sy. This phenomenon may attribute to the fact that the parameters Kr and S have influence on the drawdown as the pumping starts and thus can be estimated using only few observations.
Oppositely, the influence periods of parameters Kz and Sy have time lags after the start of pumping and the estimated results fluctuate significantly at the early period of the pumping.
Note that the estimated parameter Sy keeps the largest value (0.3) at early pumping period then dramatically decreases to small value (0.016) after 20 minutes (18 observations). This result implies that Sy does not affect the estimation for other parameters before that time, i.e., the variation of parameter Sy does not significantly change the estimation result. Figure 13 displays the estimated Sy versus pumping time (different number of observations). In addition, the value of Sy versus logarithmic time is also shown in the upper part of the figure.
The estimated Sy keep almost constant before 20 minutes and decreases to a small value.
Then the estimated Sy gradually increases and becomes flatly after 1000 minutes implying that the on-line estimation can be terminated at that time. In this case, the on-line PEM can save 77% pumping time if the test is terminated and 4041.4 m3 groundwater resources if compared with total pumping time and pumped water volume required by conventional graphical approaches. Note that the gradual increasing of the estimated parameters at the late period and a longer parameter estimation time than that of the hypothetical case also occur in this real unconfined case.