## Model Development A Deterministic Model

Unit response functions describe relationships between state variables of an aquifer system such as drawdown and management decision variables such as pumping. The continuous form of convolution relations between aquifer drawdown and discharge for a linear flow system can be expressed as

where is drawdown at observation point j at time f, bijT_l+l is drawdown at observation point j resulting from a unit impulse of pumping at point /' during time t; q¡ , is unit pumping at well i during time f, and M is the total number of pumping wells under consideration. The time-dependent drawdown response function, 61J (, represents incremental drawdown of each observation point at j at time t resulting from a unit impulse of pumping at each discharging well applied at time t = 0. When the time scale is discretized, Equation (1) can be expressed in an equivalent form as

where Sj „ is drawdown at the jth observation point at the end of the nth period; b¡j k is the response function for the fcth period relating drawdown at the jth observation point to unit pumping at the ith discharging well; and qi n_k+] is pumping at the ith discharging well during the &th period, k < n.

In groundwater management practices, the entire planning horizon is generally divided into operational intervals. An operation policy or management decision may vary from one operational interval to another, but it generally remains the same within each operational interval. As a result, a discrete formulation of the convolution relation, Equation (2), is more practical than the continuous formulation in groundwater management.

The unit response function 6 can be obtained from a distributed parameter groundwater simulation model. However, when hydrogeologic information of an aquifer system is lacking or unavailable, some closed form of analytical solution to an idealized condition can be used to derive the unit response function. In this paper, a stochastic groundwater management model is developed for a confined, homogeneous, and nonuniform aquifer with the following assumptions: (1) The aquifer is nonleaky and infinite in horizontal extent; (2) there is a radial flow pattern; (3) wells fully penetrate the entire thickness of aquifer; and (4) the piezometric head prior to pumping is uniform throughout the entire aquifer. Under these assumptions, the unit response function can be obtained from the well functions:

where i|)IJJc = (l/4)jiTW[3ui j k], in which W\.} is the well function, uiJk = (r*S)/(4Ttky, r is the distance between the pump well and the observation point; S is the storage coefficient; T is the aquifer transmissivity; and tk is the time at the end of the fcth period. The well function for the Theis equation can be written as

In this paper, the Theis equation is used to demonstrate the development of a stochastic groundwater management model. A groundwater contaminant plume management model will be used to demonstrate our development of a stochastic management model. A detailed description of the model is presented in a previous paper by Peralta and Ward .

The problem is to determine the optimal pumpage and pumping pattern over a specified planning horizon such that there are no undesirable consequences. In general, undesirable consequences such as depletion of aquifer and land subsidence can be avoided by properly controlling aquifer drawdown.

Since the response function characterizes an aquifer pumping-drawdown relationship, a groundwater management model can be very easily formulated once the response functions are defined. Without considering the random nature of aquifer properties, the deterministic management model can be stated as follows.

Objective: Minimize the cost of pumping to produce as nearly as possible a horizontal hydraulic gradient within a specified time frame, subject to

1. Upper and lower limits on drawdown

2. Upper and lower limits on pumping

The details of the objective function and constraints are also described in detail in Ref. 1.

Values for transmissivity T and storage coefficient S are normally derived from a pump well test, and as such a test provides in situ values of aquifer parameters averaged over a large and representative aquifer volume, T and S should be treated as random variables. Consequently, the response function 6 and the drawdown constraint are random in nature because they contain random variables T and S.

The objective function equation and the drawdown constraint equations are affected by drawdown. This implies that the objective function and constraints cannot be known with certainty. Thus, it is more appropriate and realistic to examine both probabilistically. In a stochastic environment, we want to specify limitations on allowable risk or required reliability of constraint performance. The reliability we specify for our constraints is actually the confidence limits we are setting for our optimal pumping values. This reliability can be determined based on the confidence of the model user in his measurement of the aquifer parameters. Our expert system (the preprocessor for user input before optimization) uses the confidence in the input and, on the basis of Bayesian theory, produces a confidence factor (reliability p, „) for its for it = 1 for k > 2

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