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i=i *=i in which F~1 [p] is a standard normal deviate corresponding to the normal cumulative distribution function of p. A plus sign produces the equation stating that there is a p probability that the actual drawdowns at pumping wells are less than the calculated value Sj „ whereas a minus sign produces the equation stating that there is a 1 — p probability that the actual drawdowns at observations wells are less than the calculated value.

Note that the second term in Equation (10) involves a square root of the variance of drawdown at each observation point, which, in turn, is a quadratic function of unknown decision variables q. The deterministic equivalent of a chance-constrained equation is nonlinear. Standard linear programming codes cannot solve problems with nonlinear constraint equations. However, as suggested by Tung [61, quasi-linearization can be employed to linearize the nonlinear term in Equation (10).

The quasi-linearization results in a linear approximation for the stochastic equivalent to the original deterministic constraint on drawdown:

where EiJk = B,j k + t for the drawdown constraint, £(>/ jt = BiJ k - F~'[p]D,J jt for the objective function, and D is a stochastic term derived during this process.

Checking the signs for the B and D coefficients reveals that the stochastic unit influence coefficient E responds the same whether showing the influence of an injection well or that of an extraction well. At injection wells, both B and D are negative values. Therefore, E is larger in absolute magnitude than the deterministic unit influence coefficient for the drawdown constraint. E is smaller than the deterministic coefficient for the objective function. At extraction wells, both B and D are positive, producing a larger absolute value for £ in the drawdown constraint and a smaller value for the objective function.

To convert the original deterministic model into a stochastic model, replace the drawdown constraint with Equation (10) and use E,j k for b/j k in the objective function. Clearly Ei j k can

AT EXTRACTION WELL

AT EXTRACTION WELL

'CERTAINTY THAT FINAL E.LE.V. AT OBSERVATION

-95X CERTAINTY THAT FINAL ELEV. AT OBSERVATION WELL IS < ELEVATION CALCULATED BY MODEL

AT OBSERVATION WELL'

>.951 CERTAINTY fTHAT FINAL ELEV. f AT PUMP WELL

DOES NOT EXCEED \ GROUND SURFACE

'CERTAINTY THAT FINAL E.LE.V. AT OBSERVATION

-95X CERTAINTY THAT FINAL ELEV. AT OBSERVATION WELL IS < ELEVATION CALCULATED BY MODEL

^-95» CERTAINTY THAT FINAL ELEV. AT PUMP WELL OOES NOT DROP BELOW A CRITICAL LOWER ELEV.

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INITIAL •DIRECTION OF FLOW

WELL IS > ELEV. CALCULATED BY MODEL

Figure 1 Cross section demonstrating sample stochastic constraints on final water levels.

be considered a stochastic unit response function derived from the Theis equation. And it should be noted that the deterministic model actually represents a reliability of 0.50 (when F~'[0.50] = 0).

1. Reliability Determination

There are drawdown terms (for observation wells) in the objective function as well as in the drawdown constraint (for pumping wells). Reliability is treated differently in the two cases. Refer to Figure 1 during the following discussion.

Let's assume a reliability level of 0.95. In a drawdown constraint one wishes to be 95% sure that the change in water level does not exceed the prespecified maximum change (i.e., does not violate predetermined bounds on the head). One uses the standard normal deviate (F~'[p]) corresponding to a reliability of 0.95 for the drawdown constraint (i.e., F~'[.95] = 1.64). The procedure described previously computes a stochastic unit response coefficient for the 95% confidence level. The coefficient is larger than a deterministic coefficient (which corresponds to a 50% confidence level). Since unit pumping causes a greater change in head using the 95% probability influence coefficient, less pumping is feasible before drawdown constraints become tight.

When considering the objective of raising water levels to prevent contaminant movement, one wishes to be 95% confident that head changes equal or exceed calculated values. Therefore, with the objective function one uses the standard normal deviate corresponding to a reliability of 0.05. This produces stochastic influence coefficients that are numerically smaller than 95% of all deterministic influence coefficients. For identical pumping values, the 95% probability change in water levels needed to achieve a horizontal gradient is much greater than that needed using deterministic coefficients. This guarantees that pumping values calculated by the model are equal to or greater than those required by the deterministic model to produce a horizontal gradient.

However, this guarantee also causes the constraint that final heads at downgradient observation wells are greater than the final head at the source to force the objective function value to be larger than an objective function value resulting from only trying to minimize the head differences between the contaminant source and all observation wells. Greater pumping values may actually cause the heads at the downgradient observation wells to "overshoot" the head at the source and produce a reverse gradient. This is demonstrated in Section IV, where the objective function and reverse gradient increase as aquifer parameter uncertainty increases. The "tight" downgradient observation well is the one whose final head is equal to the final head at the source. All other downgradient observation well heads are higher than the source head and therefore produce a larger objective function value.

2. Determination of Aquifer Parameters

Estimation of transmissivity and effective porosity has received much attention in the literature in recent years and was discussed in Section II. From Equations (7) and (8), it is seen that the mean and variance of transmissivity and effective porosity are needed in the stochastic version of the optimization model. Many methods for determining these statistics are described in the literature. Here a Bayesian approach is used to derive the mean and variance for transmissivity and effective porosity.

The Bayesian approach uses a prior (also called unconditional) probability distribution function (pdf) and a likelihood pdf to determine the mean and variance for the aquifer parameters. This mean and variance describe the posterior or conditional pdf used within the stochastic model. The prior pdf is based on knowledge of the aquifer obtained from past experience. This study suggests using aquifer material (soil type) as the basis for the prior pdf. The likelihood pdf is developed from current information (field or lab data) about the aquifer in question.

In this chapter the standard deviation of transmissivity and effective porosity are varied to determine how these changes affect the objective function value. However, in a real situation, one would estimate a mean and variance for these aquifer parameters from a prior pdf and a "likelihood" pdf. The user would select a description of the soil type from a given list. Based on a range of values of transmissivity and effective porosity associated with each soil type (derived from numerous references), a prior pdf mean (X0) and variance (V0) are determined. This determination is made by assuming that the range of values spans three standard deviations each side of the mean (99% confidence interval). With this assumption and assuming a lognormal pdf for transmissivity and a normal pdf for effective porosity, one can compute the mean and standard deviation. If there are no field data values for the problem, the prior pdf becomes the posterior pdf.

If one has field data values, the mean (X) and variance (V) are determined using standard equations for mean and variance of a data population. This mean and variance for the field data values define the likelihood pdf. The mean and variance for transmissivity are calculated using the natural log of all transmissivity values because these log values are known to be normally distributed. The relationship between posterior pdf, the prior pdf, and the likelihood pdf can be expressed as

Posterior distribution « prior distribution x likelihood distribution

The mathematics of multiplying a normally distributed likelihood pdf by a normally distributed prior pdf has been previously derived [8], Assuming that the natural log data values for transmissivity and the data values for effective porosity are normally distributed, the posterior mean, £(.), and posterior variance, var (.), for either parameter are calculated from

and var

The expected value, £(.), and the variance, var(.), for effective porosity are used as the posterior mean and variance. However, because natural log values are used to determine the expected value and variance for transmissivity, these values must be converted back to represent the mean and variance of the actual transmissivity values. Standard equations for the mean and variance of a population that has a log-normal pdf (when the expected value and variance of its natural log values are known) are

Mean = exp

var and

These two equations are used on the assumption that the entire population of values is available. Since the prior pdf uses the knowledge of a large amount of data for each soil type, this assumption is sound.

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