Application Results And Discussion

The stochastic optimization model was applied to the same hypothetical groundwater contamination problem that was used to analyze the deterministic version. Aquifer parameters (transmissivity and effective porosity), coefficient of variation (CV, ratio of standard deviation to mean), and required solution reliability were varied in consecutive runs.

The simulation component and optimization component were run on an IBM AT with 640K bytes of RAM, a 30 MB internal hard disk a with a floppy disk drive, and math coprocessor.

Physical parameters for model run Id include a transmissivity of 1255 m2/day (13,500 ft2/ day) and an effective porosity of 0.3. The original hydraulic gradient was 0.54%. Maximum and minimum acceptable pumping rates, based on available equipment, are 135 and 0 L/sec. This was based on the performance curve for a pump that can discharge 150 L/sec against 6 m of head at 80% efficiency. The upper limit on head at all injection wells was the ground surface (5.8 m, above the initial water table). This should prevent pressurized injection [9], The lower limit on head at extraction wells prevented such changes in tranmissivity that would invalidate the use of superposition. In general, if the change in transmissivity is less than 10%, the aquifer can be treated as a confined aquifer system.

The stochastic model was applied to the same hypothetical system described for the deterministic model in Ref. 1. Results are shown in Tables 1 and 2 for comparison with the deterministic model (run Id). The coefficients used for this analysis were Wf = 1.0 and c' and c" equal to their original values. Therefore, the results shown for this analysis are for a strongly hydraulic objective function.

The initial pumping (Qa) used in the iterative solution procedure of the stochastic model was the optimal pumping from the deterministic model run. It was found that two iterations brought acceptable agreement (convergence within 5%) between the "estimated" pumping values and the final optimal pumping. The weight factor in the objective function was adjusted for identical runs as described in the previous paper [1], but, as was found then, all weight factors of 1.0 and greater produced the same results. Subsequent tests used a weight factor of 1.0.

Table 1 Effect of Aquifer Parameter Uncertainty on 95% Reliable Optimal Unsteady Pumping Strategy for Hypothetical Problem"

Table 1 Effect of Aquifer Parameter Uncertainty on 95% Reliable Optimal Unsteady Pumping Strategy for Hypothetical Problem"

Id

Is

2s

3s

4s

5s

Pumping (L/sec)

Day 1

96.1

85.8

70.2

51.4

85.3

83.3

2

90.1

76.4

63.4

47.1

74.8

70.9

3

84.9

70.4

59.3

44.7

68.3

63.7

4

80.2

66.3

56.4

43.0

64.0

59.2

5

76.9

63.2

54.2

41.7

60.9

56.2

6

36.9

57.3

52.5

40.7

58.7

54.2

7

0.0

0.0

28.7

40.0

0.0

52.8

8

0.0

0.0

0.0

25.6

23.3

0.0

Avg. pumping

58.1

52.4

48.1

41.8

54.4

55.0

Avg. gradient (%)

0.08

0.079

0.085

0.095

0.098

0.14

Gradient SD

0.058

0.043

0.057

0.062

0.061

0.084

Sum of squared

1.24

1.08

1.30

1.72

1.79

4.99

head diff. (m2)

Obj. function

15.63

13.54

15.66

19.82

21.18

55.53

0 & M costs"

2.31

1.93

1.65

1.32

1.93

Run Transmissivity CV Effective porosity CV

Is

0.2

0.2

2s

0.4

0.2

3s

0.8

0.2

4s

0.2

0.4

5s

0.2

0.8

"Run Id is deterministic model.

bO & M = operations and maintenance.

"Run Id is deterministic model.

bO & M = operations and maintenance.

In all, 10 stochastic optimizations were performed. These used a range of values for the coefficient of variation (CV) for both transmissivity and effective porosity and used two reliabilities (a constant for all wells and all time periods for each run). Figures 2 and 3 graphically depict the pumping strategies developed for the five stochastic model runs made at the 95% reliability level. Figure 2 shows the pumping trends as the uncertainty of transmissivity increases from run Is to 3s as compared to the deterministic run (Id). Figure 3 shows the pumping trends as the uncertainty of effective porosity increases from run Is to runs 4s and 5s. The same general pumping trends are evident for the runs made at the 80% reliability level.

To analyze the predictability of these results we look first at the equation for the stochastic influence coefficient E [Equation (11)] and refer to Figure 1. From a table of standard normal deviates it is known that as reliability [p = F(z)] increases, z (which equals F~'[p]) increases. Therefore, looking at Equation (11) we see that as reliability increases, E for the objective function decreases and E for the drawdown constraint increases. In summary, an increase in the uncertainty of aquifer parameters produces the same result as an increase in reliability— smaller E for the objective function and larger E for the drawdown constraint.

As stated, for the drawdown constraints, increasing reliability or uncertainty of parameters produces a larger influence coefficient. This causes a greater reaction of the potentiometric

Table 2 Effect of Aquifer Parameter Uncertainty on 80% Reliable Optimal Unsteady Pumping Strategy for Hypothetical Problem

Id Is 2s 3s 4s 5s

Pumping (L/sec)

Day 1

96.1

94.6

85.7

69.8

93.2

90.6

2

90.1

86.0

76.7

63.2

85.1

82.0

3

84.9

78.8

71.1

59.2

77.6

74.7

4

80.2

73.9

67.1

56.4

72.4

69.3

5

76.9

70.2

64.1

54.3

68.7

65.6

6

36.9

21.5

44.9

52.7

36.2

63.0

7

0.0

0.0

0.0

20.1

0.0

0.0

8

0.0

0.0

0.0

0.0

0.0

4.9

Avg. pumping

58.1

53.1

51.2

47.0

54.1

56.3

Avg. gradient (%)

0.08

0.067

0.070

0.076

0.076

0.097

Gradient SD

0.058

0.047

0.048

0.050

0.049

0.060

Sum of squared

1.24

.77

.85

1.04

1.01

1.70

head diff. (m2)

Obj. function

15.63

10.37

11.03

12.80

12.89

20.36

0 & M costs

2.31

2.04

1.89

1.62

2.04

2.06

($103)

Figure 2 Daily pumping strategies for increasing uncertainty of transmissivity as compared to the deterministic strategy (95% reliability level).

Figure 3 Daily pumping strategies for increasing uncertainty of effective porosity as compared to the deterministic strategy (95% reliability level).

Days

Figure 3 Daily pumping strategies for increasing uncertainty of effective porosity as compared to the deterministic strategy (95% reliability level).

surface to a unit of pumping. Therefore, this increase allows for less pumping during a unit of time because the upper bound on drawdown is reached more quickly. In the case of a reliability of 0.95 we know that the [0.95] value (1.64) is equal to or larger than 95% of all F~'[p] values; thus the E value for a reliability of 0.95 is equal to or greater than 95% of E values for the same aquifer parameters. This confirms the stochastic constraint that in the field the upper bound on drawdown will not be exceeded 95% of the time. Tables 1 and 2 reflect the trend of increasing reliability or increasing uncertainty of parameters and the resulting decrease in allowable pumping.

Why, then, does the pumping increase for the last time period, or why are there more time periods of pumping as reliability or CV increases? While the large coefficients are causing large head increases at the injection wells (thus restricting the amount of pumping), the small stochastic influence coefficients for the objective function cause a much smaller reaction of the potentiometric surface at the observation wells. Thus, lower pumping values caused by increasing the reliability or uncertainty have even a smaller effect on drawdown at the observation wells. Yet the goal is still to minimize the objective function. To do this, additional pumping periods are needed or more pumping is required during the last time period as reliability or uncertainty increases. This trend is shown in Tables I and 2. The objective function uses the large drawdowns at the pumping wells to calculate pumping costs, thus producing the highest costs. The objective function uses the small drawdowns at the observation wells to determine the difference in head, thus producing a large sum of head differences. Thus we are assured that the objective function value is the largest expected for the given input and that the results in the field will probably not exceed the calculated value.

However, the constraint that downstream heads must be higher than the head at the contaminant source, because it uses the smaller E values for the observation well head calculations, actually causes the hydraulic gradient to "overshoot" horizontal. The smaller E values produced at the 0.05 reliability level for observation well head calculations give us a 95% confidence that the heads are those calculated (using these E values) or greater, thus causing the

Table 3 Summary of Trends Produced by Stochastic Analysis (Hydraulic Objective Function)

Effect on uncertainty

Value affected

Effect on reliability In transmissivity In effective porosity

1. Influence coefficient used with:

Objective function DD constraint

2. Daily pumping

3. Total pumping

4. Gradient (reverse)

5. Objective function value

Decrease

Increase

Decrease

Decrease

Steeper and less smooth

Increase

Large deer. Small deer.

Large incr. Small incr.

Large deer. Small deer.

Large deer. Small incr.

Steeper and less smooth

Small incr.

Large incr.

reverse gradient. Remembering that the final gradients are always reverse gradients, Tables 1 and 2 show that as reliability or uncertainty increases, the final gradient is larger in the reverse direction. The confidence in the final gradient is further complicated by the fact that the target elevation (normally the head at the contaminant source) is itself stochastic. Therefore, the actual reliability of the final gradient will be something less than the specified value, but that reliability cannot be determined with precision.

Table 3 summarizes the trends that developed as uncertainty of aquifer parameters and reliability were systematically varied. Figures 4 and 5 graphically show the trends in total pumping and the resulting final gradient. Figure 4 shows the five stochastic runs using a reliability of 95% normalized to the deterministic run (Id). Figure 5 shows the five runs normalized using a reliability of 80%. As the coefficient of variation (CV) for tranmissivity increases (runs Is, 2s, and 3s), the influence coefficients for the drawdown constraint increase and those of the objective function decrease. The expected result is a decrease in pumping for each time period (but larger total pumping) and an increase in the final average gradient and objective function value.

Runs Is, 4s, and 5s show the results of increasing the CV for the effective porosity while holding the transmissivity CV constant. The general trend for these runs is the same as those for runs Is, 2s, and 3s. The resulting gradient and objective function for runs 4s and 5s show

Figure 4 Comparisons of final gradient and total pumping for the stochastic runs at the 95% reliability level.
Figure 5 Comparisons of final gradient and total pumping for the stochastic runs at the 80% reliability level.

a sharp increase from run Is. The increased CV produces larger influence coefficients for the drawdown constraint and smaller coefficients for the objective function just as the increased CV for transmissivity does. However, the changes in these coefficients are small compared to those produced by comparable increases in transmissivity CV and cause only small differences in pumping between runs Is, 4s, and 5s. In comparison, the resulting gradient and objective function are much worse than those resulting from comparable tranmissivity changes in runs 2s and 3s.

To explain this difference we look at the difference in sign between the A coefficients, Equation (19), which are affected by changes in transmissivity CV, and the P coefficients, Equation (21), which are affected by changes in effectivity porosity CV. The negative sign with the P coefficient indicates it will affect the optimal strategy in a manner opposite to that of the A coefficient. As the CV of transmissivity is increased, there is a large change in pumping and a small change in gradient and objective function. For the same CV increase in effective porosity, there is a small change in pumping and a large change in gradient and objective function. The two parameters (transmissivity and effective porosity) cause an opposite relationship between pumping and its effect on the objective function and the constraints.

Table 2 displays results of the same variation in the CV of the two parameters, computed using a reliability level of 0.80. As expected, the reduction in reliability increases the optimal pumping values and improves the final gradient and objective function. The smaller reliability produces smaller stochastic unit response coefficients. Resulting strategies and water levels are more similar to those from the deterministic model (reliability = 0.50) than are those developed using a 0.95 reliability.

Strategies for runs 4s at the 95% reliability level and 5s at 80% reliability have no pumping on day 7 and yet require pumping on day 8. This is a definite change in the overall pattern of the stochastically optimal pumping strategies. However, a look at the sensitivity values for the pumping during days 7 and 8 gives an indication that it is not a major change. The sensitivity value (amount the objective function would change with a unit increase in pumping during that day) associated with each pumping value for days 7 and 8 for those two runs is very small. For example, these sensitivities are in the range of 10~4-10~15 as compared to a sensitivity of 0.71.3 for the tight pumping value in most other runs. This indicates that the pumping for day 8 could also be 0 without any significant change in the objective function. Therefore, the zero pumping for day 7 and a pumping value for day 8 of these runs could be zero pumping for both days 7 and 8 without a dramatic change in the overall pattern of the results.

Comparisons to Tung's [6] analysis are difficult to make because his objective function was to maximize pumping, which is not affected by the stochastic influence coefficient. The only constraint was on drawdown. In addition, the Cooper-Jacob equation (which is only appropriate for small values of the Boltzmann variable, u < 0.02) used to derive the stochastic unit influence coefficient shows P to be equal to 0 except for the first time period. However, the general trends Tung speaks of concerning transmissivity apply to is analysis: (1) Pumping increases as reliability or CV decreases and (2) uncertainty of tranmissivity causes a larger change in pumping than does a comparable change in effective porosity. This study indicates that effective porosity has an effect on the drawdown at the observation wells (something Tung considers negligible) and hence has an effect on the objective function value. In addition, the daily pumping increases with decreasing effective porosity CV, but at the same time the total pumping decreases. In summary, the trends shown in this analysis are found in Table 3.

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