Visual display of the Optimization of ground-water withdrawal in the lower Fox River communities, Wisconsin


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,


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
                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 ,


s2 = d2, 1+ d2,2 = R2, 1oQ1+ R2,2 *Q2 , and (10)

s3 = d3,I+d3,2 = R3,1*Ql+R3,2*Q2 


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


Hm = HU-S2 = Hu-(R2,1 .Q1 +R2,2*Q2)  and (13)
  2    2       2


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).



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 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