AGENDA ITEM 11
DATE: March 5, 1998
TO: Jerry L. Maxwell, General Manager
S FROM: Donald J. Polmann, Director of Resource Evaluation and Protection J
SUBJECT: Optimized Regional Operations Plan Status Report
ThIt Auith inter RECOMMENDATION: Receive and file.
DISCUSSION: Staff of the Resource Optimization Department have started development of
the aquifer Unit Response (UR) matrix using the Integrated Surface/Ground Water (ISGW)
simulation model. Staff have been working closely with personnel of SDI Environmental, Inc.,
in updating the model data input, recalibrating the model with recent data (including rainfall
through 1/3/97), and performing rainfall frequency analysis as a basis for operational
simulations. They have also made several changes in the model codes which are needed to
accommodate the process of UR matrix generation. To date, Staff have performed several test
runs to generate samples of the UR matrix; software problems with the model postprocessors
were identified and corrected.
As a part of the optimization model development process and the development of the requisite
objective function, a set of shallow aquifer monitoring wells (in addition to the Floridan
Aquifer regulatory wells) is being selected and reviewed by Resource Evaluation and
Permitting Staff. Upon selection of an appropriate set of monitoring wells, the Optimization
Program will systematically generate the effects of pumpage scenarios on these wells, will
evaluate the performance of each scenario, and will seek an optimal pumping distribution based
on maximized shallow aquifer water levels (as a surrogate for nearby lake and wetland levels).
To generate the UR matrix, Staff will be performing more than 200 ISGW simulation runs. For
each simulation, a single production well will be pulsed by one unit of pumpage (i.e., increase
pumpage by an appropriate unit rate, depending on the well capacity) for one week. The
magnitude of drawdown and the associated recovery (at each of the identified shallow aquifer
monitoring wells and Floridan Aquifer regulatory wells) generated by this pulsed pumpage will
be saved in matrix form to be used by the Optimization Program.
As a continuation of work reported last month, additional simulations have also been
performed to analyze and verify the linearity of aquifer system responses, which is a basic
assumption of the optimization problem as presently being formulated.
As an aid in explaining our development of the Optimization Program, a simplified example
which demonstrates several optimization concepts is provided as an attachment to this Agenda
item.
The proposed timeline reported last month for development and testing of the Optimization
Program remains in effect, leading to preparation of the draft Operations Plan by July 1998.
T:\RESPLAN\PLANNING\LEMEREBOARD\MAR98\OPT RPTZ.DOC
EXAMPLE OPTIMIZATION PROBLEM
INTRODUCTION
Optimization is a descriptive term that refers to a management science called Operations
Research (OR). As its name implies, this involves "research on operations" that rely on
mathematics to represent the management model. OR has been applied extensively to solve
problems of resource allocation in business, industry, the military, and government agencies.
Although OR was introduced to address allocation problems during World War II, its application
has grown most rapidly during the past two decades. Recent advancements are due to the
improvement of solution tools and techniques and the recent revolution of computer processor
speed and expanded memory chips. Scientific literature concerning OR in groundwater
applications can be found back to the early 1970's. Early applications involved only groundwater
quantity issues such as determining yield, optimum pumping distributions, and maximizing
aquifer water levels. More recently, groundwater quality has become a major issue, and more
complex applications have been attempted to minimize the cost of pollution cleanup.
OPTIMIZATION CONCEPT
OR can be used to solve the problem of allocating limited resources among competitive activities
in the best, or optimal, way. This is achieved by repeatedly investigating selected subsets of
activities (decision variables) that lie within the feasible region defined by policy constraints. For
each feasible set, the optimization goal (the objective function) is assessed using the current
values of decision variables and compared with the previous goal. A systematic technique is
applied to spawn a sequence of new solutions until the optimum solution is identified. Insight to
the OR concept can be gained through an example.
A Wellfield Scale Example
Consider a small wellfield of a confined, isotropic, homogeneous aquifer with the steady state
groundwater flowfield defined by the piezometric head contour in Figure 1. There are three
production wells available that have a 0.4 mgd pumping limit (constrained by either the well
capacity or the withdrawal permit). There are four monitoring wells used to regulate the
pumpage such that the drawdown in each well will be no more than 10 feet from the initial head.
The purpose of this example is to find an optimal pumping distribution that minimizes the
environmental impacts (drawdown) and satisfies the water demand of 1.0 mgd while meeting the
permitting constraints. The decision to be made concerns rates of pumpage in each production
well (which we call the decision variables).
The objective of minimizing the environmental impacts can be interpreted mathematically as
maximizing the sum of the water levels in the monitoring wells. In other words, we try to operate
the production wells in such a way that water levels in monitoring wells are at their highest
possible level. The monitored water levels are also viewed as decision variables because they are
used to define the objective function. Concerning pumpage and the water levels, their
relationship is defined physically by the aquifer responses and mathematically by the
groundwater flow equations and/or simulation model. One way to represent this relationship is to
derive a unit pumpagedrawdown matrix (also known as a Unit Response matrix) using a
groundwater analysis or simulation model. Table 1 gives the initial heads (Ho) at each monitoring
well and lists the elements of the Unit Response matrix (UR) for this example problem.
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Figure 1 A simple groundwater optimization example
Table 1 Relationship between drawdown and unit pumpage for a homogeneous aquifer
Monitoring Initial Head, Drawdown, UR (ft) due to one mgd pumpage
Well # Ho (ft) Well #1 Well #2 Well #3
A 100 12.5 12.5 3.125
B 80 10.0 12.5 5.0
C 60 7.5 10.0 10.0
D 40 3.75 5.0 7.5
The aquifer response relationship together with the water demand and permitting constraints
stated above constitute a policy region that contains a number of feasible solutions. The steps for
determining the feasible region are as follows. First, the feasible region on the pumping variable,
q, is determined from the demand and pumping limits. This region is then mapped to the water
level variable using the aquifer response relationship. Permitting conditions on drawdown are
used to define the bounding region on the state space, h. The intersection of both regions on the
state space defines the final feasible region.
Table 2 lists the constraints and their mathematical representations that define a feasible region.
Without using a tool such as OR, one can obtain an optimal solution by explicitly comparing a
large number of possible solutions. However, an OR tool such as the simplex method (a linear
programming (LP) technique) needs to solve only a small portion of the feasible solutions. In
practice, this method systematically selects the next solution to test by moving from one feasible
region's vertex to another, always heading in the optimal direction. Table 3 provides a partial list
of the solutions this method obtained before reaching the optimum.
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Table 2 List of decision variables, constraints, and objective function
Decision variables Notation
Pumpage q = [ql q2, q3]T
Monitoring water level h = [hA, hB, hC, hDIT
Constraint
Water demand Zqi, 1.0
Pumping limits 0.0 < qi 5 0.4, i= 1,2,3
Minimum levels Hoj h < 10, j=A,B,C,D
Aquifer response h = Ho UR.q
Objective function
Minimize environmental impacts Maximize Zhj
Table 3 Partial list of solutions to the example problem
Solution ql q2 q3 hA hB hC hD Obj. Fn. Feasibility
No.
0 1 1 1 100 80 60 40 initial
1 1 1 1 71.87 52.5 32.5 23.75 180.63 infeasible
2 0.4 0.4 0.4 88.75 69 49 33.5 240.25 infeasible
3 0.333 0.4 0.667 90 70.33 50.83 34.75 245.92 feasible
4 0.4 0.333 0.267 90 70.5 51 34.83 246.33 feasible
5 0.4 0.2 0.4 91.25 71.5 51 34.5 248.25 Optimal
Applications in the real world are far more complex than this example. Most of the practical
problems are involved with anisotropic, nonhomogeneous, multilayer aquifers. Use of a
simulation model to obtain relationships of the aquifer response is unavoidable. Furthermore, we
would like to explore the optimal solutions in the transient condition to take advantage of
manageable elements such as dynamic aquifer storage and the seasonal variations of demands and
recharge. With the additional temporal dimension, the optimal pumping operation obtained will
address not only the spatially distributed pumpage but also the pumpage scheduling.
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