I   I


           WEST    ,=                                                                                              EAST
N                                                 <
                 A                        0         Z                                    Z
    CZ           : < -                              3 : o                                 o <                      A '
0         1,000 -
		

					





(fig. 4). The Precambrian crystalline rock is assumed to
form the base of the active ground-water flow system,
because it is virtually impermeable.
     In the upper aquifer, precipitation recharges
ground water in topographically high areas and move-
ment is toward discharge areas such as streams and
lakes in nearby, topographically low areas. Recharge to
the sandstone aquifer occurs mainly to the west of the
Lower Fox River Valley, where the Maquoketa-Sin-
nipee confining unit is absent and the sandstone aquifer
is in good hydrologic connection to the upper aquifer.
Ground-water movement in the sandstone aquifer prior
to development was generally west to east. Since
development, the direction of ground-water movement
is towards the Lower Fox River Valley near the Central
Brown County and the Fox Cities pumping centers.


Water Use and Description of Wells

     The CBCWC and ECWRPC provided estimates of
municipal water use for the year 2030. These estimates
were needed for the purpose of comparison of opti-
mized to non-optimized solutions. In the Central
Brown County area, a 240 percent increase for the
period 1990 to 2030 is projected, from 7.32 to 24.7
Mgal/d (million gallons per day). In the Fox Cities
Heart-of-the-Valley communities, a 41 percent
increase for the period 1990 to 2030 is projected, from
3.9 to 5.5 Mgal/d. For the Fox Cities Western Towns, a
110 percent increase for the period 1990 to 2030 is pro-
jected, from 1.7 to 3.6 Mgal/d. Water use in the Fond
du Lac area, in the southern portion of the model, was
assumed to remain fixed at 1990 rates.
    Numerous high-capacity wells in the Central
Brown County area and Fox Cities area have had a
regional effect on water levels in the sandstone aquifer.
Wells withdrawing water from the upper aquifer are
typically shallow domestic wells with low pumping
rates of about 5 to 10 gal/min (gallons per minute).
Such wells typically affect water levels only locally in
the upper aquifer and can therefore be ignored. Wells
withdrawing water from the sandstone aquifer are typ-
ically deep, high-capacity municipal, industrial, and
commercial wells that pump about 500 to 1,000 gall
mmi. Pumping rates for 1990 and 2030, along with
descriptions of the high-capacity wells included in the
ground-water model, have been compiled and are
included in the Appendix.


OPTIMIZATION MODELING

     Optimization modeling is a general class of prob-
lems in which an objective function is either minimized
or maximized subject to a series of constraints. The
objective function and constraints are expressed as
known mathematical functions of the variables of inter-
est, termed decision variables. There are several classes
of optimization models, depending in part on the form
of the objective function and constraints. These include
linear programming (linear objective function and con-
straints with continuous decision variables), integer
programming (linear objective function and constraints
with integer decision variables), mixed integer pro-
gramming (linear objective function and constraints
with integer and continuous decision variables), and
nonlinear programming (nonlinear objective function
and decision variables). Most introductory texts on
operations research describe the classes of optimization
models (for example, Gue and Thomas, 1968).
    Ground-water optimization involves applying
optimization modeling to problems of ground-water
flow. A review of ground-water optimization tech-
niques is given elsewhere (Gorelick, 1983). In most
cases, linear programming has been applied to prob-
lems of ground-water flow due to its ability to handle
large numbers of decision variables and constraints and
the relative speed of the solution technique.
    Specification of the objective function is a crucial
step in optimization modeling. The objective function
should represent the overall goal of the optimization.
Typically the objective function is written using well
pumping rates as the decision variables. Objective
functions can range from a simple summation of pump-
ing rates (for example, maximize total withdrawal) to a
detailed function involving pumping rates and water
levels (for example, minimize total cost).
    The constraints impose limits on the decision vari-
ables and are very important in ensuring a realistic opti-
mal solution. The constraints can vary from simple
limits to more complex expressions. Examples of sim-
ple limits include upper bounds on pumping rates,
lower bounds on water levels, and upper bounds on
drawdowns. Examples of more complex limits include
upper bounds on horizontal gradients and upper and
lower bounds on flow velocity or direction.
    Several approaches have been devised for repre-
senting the ground-water flow system as a linear sys-
tem, but the most common approach is to use a
response matrix to represent the response of the aquifer


OPTIMIZATION MODELING  7