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system to withdrawal rates at specified wells. This approach is attractive because it can use complex ground-water flow models to simulate the aquifer response, thus recent advances in ground-water flow modeling are incorporated into the final solution. The response matrix is based on the theory of superposition. For the purpose of illustration, assume there are several wells in the system where the optimal withdrawal rate is to be determined; these are termed managed wells. Further, assume there are various loca- tions in the system where the water level needs to be determined; these are termed control points. The response matrix is determined by operating each of the managed wells in isolation from the other managed wells. If dij equals the drawdown at control point i due to wellj pumping in isolation at rate Q1, and Rij equals the unit response at control point i due to wellj, then it follows that Rii j di, Qj (1) Equation 1 can be rearranged to express drawdown as a function of an individual pumping rate and the unit response factor, thus dij = Rij Qj . (2) Consider a case with two managed wells (j=2) and three control points (i=3). With well 1 pumping in iso- lation at a rate of Q1 (well 2 turned off), equation 2 results in the following drawdowns at the 3 control points: dl, 1 = R11 aQ, (3) d2, I = R2, 1*Q1 ,and (4) d3, I = R3,.1 aQ1 (5) Likewise, with well 2 pumping in isolation at a rate of Q2 (well 1 turned off), equation 2 results in the follow- ing drawdowns at the 3 control points: S2 = dl, 1 +dl,2 = R,1 .Q1+R1,2.Q2 , (9) s2 = d2, 1+ d2,2 = R2, 1oQ1+ R2,2 *Q2 , and (10) s3 = d3,I+d3,2 = R3,1*Ql+R3,2*Q2 (11) If we let Hu equal the water level at control point i when all managed wells are off and let Hm equal the water level at control point i when all managed wells are on, then equations 9-10 can be used to determine the managed heads at the three control points, thus HT' = Hu-s1 = H-(R1, +R 1 11 '1, 9 2 (12) Hm = HU-S2 = Hu-(R2,1 .Q1 +R2,2*Q2) and (13) 2 2 2 (14) Equations 12-14 express the water level at the control points as a linear function of the withdrawals at the managed wells. Thus the response of the flow system can be written as a linear function of the decision vari- ables, and linear programming techniques can be used to determine the optimal solution. The response-matrix approach has been used by numerous investigators to solve a variety of ground- water-management problems. In each case, ground- water-flow simulation models were used to determine the response matrix, which in turn was used in the for- mulation of the optimization problem. In some cases, a simple summation of withdrawal rates at the managed wells is used as the objective function (for example, Heidari, 1982; Danskin and Freckleton, 1992). In other cases, the objective function represents net economic benefit (for example, Bredehoft and Young, 1970; Rei- chard, 1987). (6) (7) d2,2 = R2,2.0 Q2 , and d3,2 = R3,2 *Q2 (8) If both wells are pumping, then by superposition the drawdown at the three control points (s1, S2, and s3) is the sum of the individual drawdowns due to each well pumping in isolation (equations 3-8), thus OPTIMIZATION SIMULATIONS Optimization modeling was used to evaluate sev- eral management alternatives for the Central Brown County area and the Fox Cities area. The MODMAN commercial package (International Groundwater Mod- eling Center, 1996) was coupled with an existing ground-water flow model for the model area to deter- mine optimal withdrawal rates. The UNDO linear-pro- 8 Optimization of Ground-Water Withdrawal in the Lower Fox River Communities, Wisconsin Hnm = Hu-s = Hu-(R3,1QI+R32Q) dl, 2 = RI, 2 Q2 9
gramming package (Schrage, 1991) was used to determine the solutions to the linear program optimiza- tion programs formulated by MODMAN. The optimal simulations were compared to baseline conditions rep- resenting the 2030 projected withdrawals. The ground- water flow model will be described, the management alternatives will be discussed, and baseline conditions will be presented in this section. This section concludes with presentation and discussion of the results. Lower Fox River Basin Ground-Water Model The 3-dimensional finite difference MODFLOW model (McDonald and Harbaugh, 1988) developed in a separate study (Conlon, 1998) was used to simulate the ground-water system in the Lower Fox River Basin in northeastern Wisconsin. In this section, a brief descrip- tion of the model is given; a complete description of model calibration and limitations is presented else- where (Conlon, 1998). The model area (fig. 1) was dis- cretized by use of a finite-difference grid. The extent of the model area was chosen such that: (1) the western boundary includes the western ground-water divide in the sandstone aquifer and the discharge areas of the Wolf River in the west and the upper Fox River in the south; (2) the northern boundary was set to a sufficient distance to minimize the effects of pumping in the Lower Fox River Valley on water levels near the boundary; (3) the eastern boundary incorporates a ground-water discharge divide in Lake Michigan; and (4) the southern boundary includes the area of water withdrawals near the city of Fond du Lac. The grid is rotated 230 east of north to orient the northern and southern boundaries parallel to the primary direction of ground-water flow in the sandstone aquifer. The model grid contains 141 rows and 102 col- umns and two layers: Layer 1 simulates conditions in the upper aquifer, and layer 2 simulates conditions in the sandstone aquifer. The Maquoketa-Sinnipee con- fining unit is not simulated as a model layer, but as a boundary that allows limited vertical flow between the upper aquifer (model layer 1) and the sandstone aquifer (model layer 2). The Precambrian crystalline rock is assumed to be the base of the ground-water system. The upper aquifer is simulated as a water-table aquifer with a combination of no-flow, constant-head, and head-dependent-flux boundaries along the north- ern, western, and southern edges of the model. Con- stant head cells simulate Lake Michigan along the eastern edge of the model. Rivers, streams, and lakes in the upper aquifer are simulated as constant-head or head-dependent-flux cells. The sandstone aquifer is simulated as a convertible model layer, that is, the aquifer is simulated as confined unless water levels in the layer fall below the bottom of the overlying confining unit, in which case the aquifer is simulated as unconfined. The northern, eastern, and western boundaries of the sandstone aquifer are simu- lated as no flow. The southern boundary is simulated as constant head because that location coincides with a mapped ground-water divide in the sandstone aquifer which exists between the Milwaukee metropolitan area and the Fond du Lac area. Wells are included only for the sandstone aquifer and are modeled as being open to the entire thickness of layer 2. Description of Management Alternatives The objective of all of the management alterna- tives is to maximize total ground-water withdrawal. Thus the objective function is the summation of pump- ing rates from the managed wells. General constraints included upper bounds on the pumping rates of individ- ual wells and lower bounds on the water level at model cells containing the managed wells. Maintaining the water level at or above the bottom of the confining unit assures no loss of capacity from a well; however, for some alternatives this constraint was relaxed to increase the amount of water available for withdrawal. As noted previously, there are two main pumping cen- ters of interest in the model area: the Central Brown County area and the Fox Cities area. Because these areas are assigned to separate planning agencies, simu- lations were conducted separately for each area. The main issues in the Central Brown County pumping center include (1) whether the city of Green Bay wells are operated at fixed rates or are managed, and (2) whether potential future wells (growth wells) are installed at two communities (Rockland and Hum- boldt, each with a withdrawal rate of 0.5 Mgal/d). In addition, two alternatives are available for increasing the amount of water available for withdrawal: (1) relax- ing the water-level constraint to a level below the bot- tom of the confining unit, and (2) installing additional wells in outlying areas. Twelve potential well locations were selected for the new wells based primarily on dis- tance from the main cone of depression; the optimiza- tion procedure selects the best 8 locations. Thus four OPTIMIZATION SIMULATIONS 9