By John B. Czarnecki, Brian R. Clark, and Thomas B. Reed

U.S. GEOLOGICAL SURVEY

Water–Resources Investigations Report 03-4230

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The Mississippi River Valley alluvial aquifer is a water-bearing assemblage of gravels and sands that underlies about 32,000 square miles of Missouri, Kentucky,
Tennessee, Mississippi, Louisiana, and Arkansas.
Because of the heavy demands placed on the aquifer, several large cones of depression over 100 feet deep have formed in the potentiometric surface, resulting
in lower well yields and degraded water quality in some areas. A ground-water flow model of the alluvial aquifer was previously developed for an area covering 14,104 square miles, extending northeast from the Arkansas River into the northeast corner of Arkansas and parts of southeastern Missouri. The flow model showed that continued ground-water withdrawals at rates commensurate with those of 1997 could not be sustained indefinitely without causing water levels to decline below half the original saturated thickness of the aquifer.

To develop estimates of withdrawal rates that could be sustained in compliance with the constraints of critical ground-water area designation, conjunctive-use optimization modeling was applied to the flow model of the alluvial aquifer in northeastern Arkansas. Ground-water withdrawal rates form the basis for estimates
of sustainable yield from the alluvial aquifer and from rivers specified within the alluvial aquifer model. A management problem was formulated as one of maximizing
the sustainable yield from all ground-water and surface-water withdrawal cells within limits imposed by plausible withdrawal rates, and within specified constraints involving hydraulic head and streamflow. Steady-state flow conditions were selected because the maximized withdrawals are intended to represent sustainable
yield of the system (a rate that can be maintained
indefinitely).

Within the optimization model, 11 rivers are specified. Surface-water diversion rates that occurred in 2000 were subtracted from specified overland flow at the appropriate river cells. Included in these diversions
were the planned diversions of 63,339,248 ft3/d for the Bayou Meto project area and 55,078,367 ft3/d for the Grand Prairie project area, which factor in an additional 30 and 40 percent transmission loss, respectively.
Streamflow constraints were specified at all 1,165 river cells based on average 7-day minimum flows for 10 years. Sustainable yield for all rivers ranged from 0 (Current, Little Red, and Bayou Meto Rivers) to almost 5 billion cubic feet per day for the Arkansas River. Total sustainable yield from all rivers combined was 12.8 billion cubic feet per day, which represents a substantial source for supplementing ground water to meet the total water demand.

Sustainable-yield estimates are affected by the allowable upper limit on withdrawals from wells specified
in the optimization model. Ground-water withdrawal
rates were allowed to vary as much as 200 percent of the withdrawal rate in 1997. As the overall upper limit on withdrawals is increased, the sustainable yield generally increases. Tests with the optimization model show that without limits on pumping, wells adjacent
to sources of water would have optimized withdrawal
rates that were orders of magnitude larger than rates corresponding to those of 1997. The sustainable yield from ground water for the entire study area while setting the maximum upper limit as the amount withdrawn
in 1997 is 360 million cubic feet per day, which is only about 57 percent of the amount withdrawn in 1997 (635.6 million cubic feet per day). Optimal sustainable
yields from within the Bayou Meto irrigation project area and within the Grand Prairie irrigation project area are 18.1 and 9.1 million cubic feet per day, respectively, assuming a maximum allowable withdrawal
rate equal to 1997 rates. These values of sustainable
yield represent 35 and 30 percent respectively of the amount pumped from these project areas in 1997.

Unmet demand (defined as the difference between the optimized withdrawal rate or sustainable yield, and the anticipated demand) was calculated using different demand rates based on multiples of the
1997 withdrawal rate. Assuming that demand is the 1997 withdrawal rate, and that sustainable-yield estimates
are those obtained using upper limits of withdrawal
rates of 100-, 150-, and 200-percent of 1997 withdrawal rates, then the resulting unmet demand for the entire model area is 275.5, 190.9, and 110 million cubic feet per day, respectively. Whereas, if the demand is specified as 100-, 150-, and 200-percent of the 1997 withdrawal rate, and the sustainable-yield estimates remain the same, then the resulting unmet demand for the entire model area is 275.5, 508.8, and 745.8 million cubic feet per day, respectively. These unmet demands for ground water could be obtained from large sustainable
surface-water withdrawals.

**TABLE OF CONTENTS**

- Abstract
- Introduction
- Purpose and Scope
- Previous Studies
- Acknowledgments
- Study Area
- Irrigation Project Areas

- Conjunctive-Use Optimization
- Flow Model
- Optimization Model
- Problem Formulation
- Objective Function
- Hydraulic-Head Constraints
- Streamflow Constraints
- Ground-Water Withdrawal Limits
- Surface-Water Withdrawal Limits
- Predevelopment Recharge From Non-River Sources
- Wells Used in Optimization Model
- Streamflow

- Optimization Results
- Sustainable Yield
- Unmet Demand
- Optimal Simulated Hydraulic-Head Altitude
- Nonlinear Effects
- Limitations

- Problem Formulation
- Summary
- References Cited

- Location of study and modeled area
- Irrigation project areas within the model area
- Potentiometric surface within the alluvial aquifer Spring, 1998
- Flow chart of optimization modeling process
- Map showing location of hydraulic-head constraint points and thickness of aquifer below hydraulic-head constraint
- Graph showing hypothetical variation in recharge to the alluvial aquifer from non-river sourcesas a function of time
**Maps showing:**- Location of streams within model showing cells and rates at which water could be with drawn and still meet constraints within optimization model
- Ratio of optimal ground-water withdrawal calculated by the optimization model to the amount withdrawn in 1997 for withdrawal limits at each well set to 100 percent of the1997 withdrawal rate
- Ratio of optimal ground-water withdrawal calculated by the optimization model to the amount withdrawn in 1997 for withdrawal limits at each well set to 150 percent of the 1997 withdrawal rate
- Ratio of optimal ground-water withdrawal calculated by the optimization model to the amount withdrawn in 1997 for withdrawal limits at each well set to 200 percent of the1997 withdrawal rate
- Simulated hydraulic head at steady state using (A) 1997 withdrawal rates; and (B) sustainable yield
- Graph showing cumulative percentage of model cells with values less than or equal to thedifference between simulated hydraulic head and the altitude corresponding to half the aquifer thickness
- Map showing difference between simulated hydraulic head and altitude of half the aquiferthickness

- Characteristics of the flow model
- Rivers, streamflows, and streamflow constraints
- Sustainable yield and unmet demand for different upper limits on withdrawals and different demandrates
- Optimized total streamflow withdrawals from the optimization model
- Comparison of sustainable yield obtained for hydraulic-head constraints everywhere in the modelarea and only in the Bayou Meto and Grand Prairie irrigation project areas
- Sustainable yield from ground water and unmet demand relative to 1997 withdrawal rates in sixirrigation Project areas and the remainder of the model area

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