By John B. Czarnecki, Brian R. Clark, and Gregory P. Stanton

U.S. GEOLOGICAL SURVEY

Water–Resources Investigations Report 03-4233

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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 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 3,826 square miles, extending south from the Arkansas River into the southeastern corner of Arkansas, parts of northeastern
Louisiana, and western Mississippi. The flow-model results indicated 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.

Conjunctive-use optimization modeling was applied to the flow model of the alluvial aquifer to develop withdrawal rates that could be sustained relative
to the constraints of critical ground-water area designation.
These 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 conditions were selected because the maximized withdrawals are intended to represent sustainable yield of the system (a rate that can be maintained indefinitely).One point along the Arkansas River and one point along Bayou Bartholomew were specified for obtaining surface-water sustainable-yield estimates within the optimization model. Streamflow constraints were specified at two river cells based on average 7-day low flows with 10-year recurrence intervals.

Sustainable-yield estimates were affected by the allowable upper limit on withdrawals from wells specified
in the optimization model. Withdrawal rates were allowed to increase to 200 percent of the withdrawal rate in 1997. As the overall upper limit is increased, the sustainable yield generally increases. Tests with the optimization model show that without limits on pumping,
wells adjacent to sources of water, such as large rivers, would have optimal withdrawal rates that were orders of magnitude larger than rates corresponding to those of 1997. Specifying an upper withdrawal limit of 100 percent of the 1997 withdrawal rate, the sustainable
yield from ground water for the entire study area is 70.3 million cubic feet per day, which is about 96 percent of the amount withdrawn in 1997 (73.5 million cubic feet per day). If the upper withdrawal limit is increased to 150 percent of the 1997 withdrawal rate, the sustainable yield from ground water for the entire study area is 80.6 million cubic feet per day, which is about 110 percent of the amount withdrawn in 1997. If the upper withdrawal limit is increased to 200 percent of the 1997 withdrawal rate, the sustainable yield from ground water for the entire study area is 110.2 million cubic feet per day, which is about 150 percent of the amount withdrawn in 1997. Total sustainable yield from the Arkansas River and Bayou Bartholomew is about 4,900 million cubic feet per day, or about 6,700 percent of the amount of ground-water withdrawn in 1997. The large, sustainable yields from surface water represent a potential source of water that could supplement
ground water and meet the total water demand.

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 3.3, -7.1, and -36.6 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 3.3, 29.7, and 36.9 million cubic feet per day.

**TABLE OF CONTENTS**

- Abstract
- Introduction
- Purpose and Scope
- Previous Studies
- Acknowledgments
- Study Area
- Hydrogeology

- Conjunctive-Use Optimization
- Flow Model
- Optimization Model
- Problem Formulation
- Objective Function
- Hydraulic-Head Constraints
- Streamflow Constraints
- Ground-Water Withdrawal Limits
- Surface-Water Withdrawal Limits
- 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
- Hydrogeologic sections showing lithology and variation in thickness through 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
- Location of streams within model showing cells and rates at which water could be withdrawnand 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 IV Contents
- 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 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 200 percent of the1997 withdrawal rate
- Simulated hydraulic head at steady state using (A) 1997 withdrawal rates; and(B) sustainable yield
- Difference between simulated hydraulic head and altitude of half the aquifer thickness
- Graph showing cumulative percentage of hydraulic-head constraint points less than or equal to the difference between simulated hydraulic head and the altitude corresponding tohalf the aquifer thickness

- Characteristics of the flow model
- Rivers, streamflows, and streamflow constraints.
- Sustainable yield and unmet demand by county or parish for different upper limits on withdrawalsand different demand rates
- Optimized streamflow withdrawal

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