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Show number of subdomains or cells. Algebraic equivalents of equation 3.1, are developed for each cell or node of the discretization scheme. Point values of piezometric head are solved for, at discrete times, given appropriate boundary and initial conditions, and stimuli to the aquifer such as pumping and natural recharge rates. The algebraic equations that need to be solved for discrete aquifer simulation may be linear or nonlinear, depending on the boundary conditions, and on whether the aquifer is confined, unconfined or stratified. A large system of equations is usually solved for each time step of the model. The variability of pumping and natural recharge rates over time, and the need for a high degree of accuracy can lead to simulation time steps that do not correspond to the time steps of interest for groundwater management. Thus, if the embedding approach is used for the formulation of the optimization model, the mathematical size of the model may be rather large, and the model may be highly nonlinear. As indicated in section 3.1, the principle of superposition may be used through unit response matrices, even where the governing partial differential equation is nonlinear ( e. g., confined and stratified aquifer systems). The response matrices may be evaluated at the time steps desired for optimization, even if the simulation is carried out over much smaller time steps. This leads to a much more compact and linear optimization formulation. The well sites in the area are first divided into two sets - ( 1) those considered for optimization, and ( 2) others that are treated as background pumping by small users, or in areas of little potential for increased pumping. The response to pumping at a candidate well site is evaluated as follows. A simulation of the aquifer is performed by specifying unit pumping at the candidate site, zero pumping at all other candidate sites. The resulting drawdown at all sites of interest is then recorded for each time step, and entered into the response matrix. The procedure is repeated with each candidate well site to define all the entries of the response matrix. For the optimization model formulated, response matrices to pumping are needed for ( a) incremental drawdown, ( b) incremental flows across water supply agency boundaries, and ( c) incremental flows out of each contaminated area. A three dimensional, finite difference groundwater flow model ( GW3D) developed by Mcdonald and Harbaugh ( 19S4) was used for aquifer simulations. Procedures for the determination of the response matrices and related head and flow computations, are outlined in the next 2 sections, with reference to a finite difference representation of the groundwater flow system. 3.2.3 Response Matrices for Drawdown and Head Computations A two dimensional finite difference grid is superimposed on the area of interest. Sites in the area may then be identified with reference to nodes of the grid as ( ij), where i is a row and j is a column of the grid. The drawdown ajjwt is then defined as the incremental drawdown between time t- 1 and t, at node ( i j) due to unit pumping at well w. The terms ajjwt then constitute the response matrix coefficients for well w. These are recorded only at a subset of the finite difference grid nodes where head changes of flow rates are of interest from the optimization perspective. To reduce computer memory requirements, a region of influence may be defined for each candidate well, and response matrix entries stored only for those sites that are within the region of influence. The region of influence is defined by considering only the sites Rj, where the steady state drawdown due to pumping at site i, is significant. It is advisable to perform the simulation with a pumping rate at site i that is higher than the proposed upper bound at that site ( rather than unit pumping). The response matrix is then recovered by dividing the resulting drawdowns by the pumping rate used. Similarly, a time of influence of each well may also be established. This is defined as the time after 44 |