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Show The error in this linearization is negligible for n Y d-( K.- K. 1)> K„ , h ( 2.9) +- d iv i l+ l' n+ 1 v ' i= l The expression ( 1/ a) is analogous to the definition of a storage coefficient for a confined aquifer. The representation of equation ( 2.7) with dependent variable G, is also similar to that for a confined aquifer with a dependent variable < J> ( the piezometric head). Within the limits of the approximation, the stratified aquifer flow problem may now be treated using linear superposition and the response matrix optimization approach. The practical procedure that can be followed for such applications may follow that of Yeh ( 1982). Since the left hand side of equation 2.9 is a constant for any given situation, the limiting value of h for which the Girinski potential linearization for the stratified aquifer flow problem is valid, can be readily evaluated. Aquifer flow simulations may still be performed using a three dimensional finite difference or finite element model with head h as the dependent variable. Then using equation ( 2.5) a mapping may be established between G and h and conversely between h and G for every node of interest. The response matrix ( Green's function) for unit pumping at a specified well may then be evaluated in terms of drawdowns in G. The optimization formulation may then be similarly formulated in terms of G, rather than h. 2.2 General Optimization Formulations for Groundwater management Ground water management models are developed to aid planners in a number of areas where ground water is of concern. Two of these areas are water supply and site dewatering. The basic flow equations used to develop response matrices for these models are the same but objectives and constraints for the management models may differ. Objectives and Constraints: The objective of most optimization models is to minimize cost. Dewatering costs include discounted, capital and operation costs of the well system. In the case of water supply, surface water and imported water sources are sometimes required; these incur additional costs. When adding up costs, the life of the project must be considered. Included in the cost of the well system will be set- up or fixed costs for short term projects, otherwise operational costs dominate. The fixed costs account for cost of well installation. Fixed costs are generally excluded from water supply models due to the long term nature of water supply projects. Hydraulic head constraints are common to both models: Dewatering models set maximum head limits; water supply models require maintenance of minimum levels of hydraulic head. In either case a response matrix can be developed for aquifer system response, measured by the Girinski potential, to pumping stress. Limits on pumping, set by pump capacities, are included in both models. Water supply model constraint sets must also account for reservoir storage limits, water rights, water demand, and stream- aquifer interaction, if conjunctive use of surface and groundwater is considered. Formulations of optimization models for water supply and dewatering models, emphasizing ground water management, are presented next. 14 |