# Optimal management of groundwater pumping of the cache critical groundwater area, Arkansas

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## Abstract

A simulation model for part of the Mississippi River Valley alluvial aquifer in the Cache area, Arkansas, was coupled with an optimization model to determine maximum optimal pumping from irrigation wells in the areas where cones of depression exist. Groundwater Vistas and Groundwater Management software were used for simulation and optimization model, respectively. The Cache area was designated as a critical groundwater area in 2009 due to the decline in its water level to below 50 % of the saturated thickness of the aquifer. The optimization model was formulated with the objective of maximizing water production from wells subjected to minimum head constraints and drawdown constraints, while limiting groundwater withdrawals to a maximum of 100 and 200 % of the rate pumped in 2010. Four different sets of managed wells were tested in Scenarios 1, 3 (938 wells) and Scenarios 2, 4 (3870 wells). The optimal pumping rates from groundwater in the case of minimum head constraints were 0.59 and 2.43 Mm^{3}/d for Scenarios 1 and 2, respectively. In the case of maximum pumping constraints of the managed wells specified as 200 % of the pumping rate of 2010, the optimal pumping rates from groundwater in the case of minimum head constraints were 0.88 and 3.28 Mm^{3}/d for Scenarios 3 and 4, respectively. The average optimal pumping increased by 6–49 % in the case of the maximum pumping constraint specified as 200 % of the pumping rate of the year 2010.

## Keywords

Groundwater flow Modeling Groundwater vistas Optimization Groundwater management## Introduction

The first effort to optimize groundwater withdrawals from the alluvial aquifer was conducted by Peralta and others (Peralta R et al. 1985) who estimated future groundwater availability in the Grand Prairie area. Peralta et al. (1991b) developed deterministic distributed parameter Simulation/Optimization computer models for each of the response matrix (RM) and embedding (EM) approaches for computing maximum steady-state groundwater extraction strategies in a hypothetical confined aquifer. Two scenarios were tested in which recharge into the area was unconstrained and restricted. Both EM and RM models computed the same optimal pumping strategies. As a result, if heads must be restricted at many locations, or there are many pumping cells, the embedding approach is preferred to the response matrix approach for computing sustained yield groundwater pumping strategies for confined aquifers. Peralta et al. (1995) presented conjunctive water-use strategies for the alluvial aquifer in northeastern Arkansas using the sequential steady-state embedding approach for the five decades of 1990–2039. The effect of the aquifer parameter on the conjunctive use strategy was tested by changing hydraulic conductivity and river conductance. Changing hydraulic conductivity by ±20 % caused about 3 % change in total provided water. Similarly, changing river conductance caused a 3 % change in total provided water.

Barlow et al. (2003) developed conjunctive use management models for estimating sustainable yield from surface water and groundwater for two alluvial valley stream aquifer systems that were representative of those of the northeastern United States. The first was a hypothetical system for a typical stream aquifer system and the second was a stream aquifer system in central Rhode Island. Outcome from management model showed that current withdrawal from aquifer increased by 50 % by adjusting current withdrawal schedules, adjusting the number and arrangement of wells in the supply well network, or increased stream flow depletion in the Annaquatucket and Pettaquamscutt rivers.

Czarnecki J et al. (2003a) developed a conjunctive use optimization model based on Reed’s model (Reed 2003) for the alluvial aquifer in northeastern Arkansas. The strategy for solving the problem was formulated to maximize the sustainable yield from all groundwater and surface water withdrawal rates within specified constraints involving hydraulic head and stream flow. Czarnecki J (2006) evaluated the effects of varying groundwater level constraints and river withdrawal specifications of the conjunctive use optimization model using four different management scenarios for the model described in Czarnecki J et al. (2003a). Average differences in sustainable yield of groundwater in Scenarios 2, 3, and 4 in comparison to Scenario 1 showed an increase of 6.74, 6.82, and 13.24 %, respectively. Czarnecki J (2008) utilized the existing conjunctive optimization model described in Czarnecki J (2006) to evaluate the effect of varying (1) the upper limit of groundwater withdrawal rates by reducing them to 75, 50, and 25 % of the 1997 groundwater withdrawal rates; (2) the stream flow constraint associated with the White River representing minimum specified flow rates, and (3) the specified stage of the White River. The results indicated that, as the upper limit is reduced, the distribution of cells with sustainable yields greater than zero increases.

This study concentrates on the irrigation well optimization only without coupling stream flow optimization. In addition, the management model is limited to the wells located in Poinsett, Cross, St. Francis, Lee, Monroe, and Woodruff counties. Furthermore, different software was used for coupling the simulating model with the optimization model.

- (a)
maintaining groundwater levels at or above specified levels;

- (b)
maintaining the drawdown in water levels at or above minimum specified levels; and

- (c)
limiting groundwater withdrawals to a maximum of 100 and 200 % of the rate pumped in 2010 using transient flow conditions.

## Method

The study and model area is 6,869 km^{2} and is bounded by Crowleys ridge on the east, the Cache River to the west, the Arkansas Stateline to the north, and Lee County to the south (Fig. 1). It includes part of Clay, Greene, Craighead, Cross, Poinsett, St. Francis, Lee, Monroe, Woodruff, and Jackson Counties and is located between latitudes 34°39′1″ to 36°29′53″ North and longitudes 90°10′56″ to 91°23′42″ West. Three rivers are located within the study area: the Cache River, the L’Anguille River, and the Black River. Mean annual precipitation for the years 2000–2010 was approximately 1,219 mm (PRISM Group 2012). The normal annual temperature for the area is about 60 °F (Broom and Lyford 1981; PRISM Group 2012). The dominant land use in the area consists of cultivated crops (almost 90 % of the area) such as rice, corn, soybeans, cotton, sorghum, and wheat (U.S. Department of Agriculture 2010).

^{2}. The simulation period is from 2000 to 2010 with 23 stress periods modeled under transient flow conditions. The model was calibrated using the parameter estimation code (PEST). Additional calibration was achieved using pilot points with regularization and singular value decomposition (SVD-assist). Note the detailed information about the simulation model shown in Table 1.

Detailed summary of the simulation model, in which “residual” refers to the difference between the observed and simulated hydraulic head, RMSE to root mean square error, and NGVD to National Geodetic Vertical Datum

Grid | Boundary conditions | Target types | Wells/statistics | ||||
---|---|---|---|---|---|---|---|

Rows | 294 | Constant heads | 159 | Head | 150 | Analytical wells | 12,078 |

Columns | 149 | Min head constraints | 134 | ||||

260 | |||||||

220 | |||||||

436 | |||||||

Layer | 1 | Max drawdown constraints | 134 | Boundary condition wells | 7,288 | ||

260 | |||||||

220 | |||||||

436 | |||||||

Total Cells | 43,806 | Rivers | 675 | Kx Pilot point | 218 | Mean residual for the entire model simulation | −0.27 m |

Active Cells | 13,742 | ||||||

Total Area | 21,896 km | ||||||

Active Area | 6,869 km | No flow | 30,064 | Storage Pilot point | 218 | RMSE | 1.18 m |

Row Spacing | 707 m | ||||||

Column Spacing | 707 m | Recharge Pilot point | 218 | Range of observations hydraulic head above NGVD 1929 | 48.84 m |

The Groundwater Management (GWM) Process for the United States Geological Survey simulates three a dimensional groundwater model and MODFLOW-2000 and 2005 were used for optimization models. GWM solves several groundwater management formulations, such as linear, nonlinear, and mixed binary linear, using a response matrix approach. The response matrix solution (RMS) package of GWM uses the groundwater flow process of MODFLOW to calculate the change in head at each constraint location that results from a perturbation of a flow rate variable. These changes are used to calculate the response coefficients (Ahlfeld D et al. 2005). Each management formulation consists of an objective function, a set of constraints, and a set of decision variables. Together, these three components define a mathematical model of the management decision making process (Ahlfeld and Mulligan 2000; Kharmah 2007).

### The objective function

*Qw*

_{ n }is a pumping rate from well n of the year 2010; and

*β*

_{ n }is the weight associated with annual groundwater pumping rate

*Qw*

_{ n };

*T*

_{ Qwn }is the total duration of active flow at well site

*n*.

*β*

_{ n }sets to the dimensionless values of 1.0.

### Model constraints

#### Hydraulic Head constraints

*i, j, k*at the end of the stress period

*t*were set at 50 % of the predevelopment saturated thickness of the alluvial aquifer. This is to prevent the head from dropping below the 50 % of initial saturated thickness. Minimum head constraints were calculated by taking half of the saturated thickness in each cell and adding the bottom elevation to it, to accommodate the ASWCC critical groundwater area criteria that water levels within the alluvial aquifer should remain above half the original saturated thickness of the aquifer. Four different sets of constraints were chosen: two set (134, 260) constraints for scenario 1 and 3 as shown in Figs. 2 and 3, and two set (220, 436) constraints for scenario 2 and 4 as shown in Figs. 4 and 5. The distribution of 134, 260 constraints was in the areas enclosed of the northern and southern cones every 3 cells horizontally and every 6, 3 cells vertically for 134 and 260, respectively. The distribution of 220, 436 constraints was in the areas enclosed in Poinsett, Cross, St. Francis, Lee, Monroe, and part of Woodruff Counties every 6, 3 cells horizontally for 220 and 436, respectively, and every 6 cells vertically. The constraints value ranged between 30.03 and 54.56 m.

#### Drawdown constraints

*i, j, k*at the end of stress period

*t*, \( (h_{i,j,k,t} )^{^\circ } \), and the head calculated at the location at the end of the stress period after implementation of the optimal management strategy,\( h_{i,j,k,t} \):

*i, j, k*at the end of stress period

*t.*The optional value of each drawdown constraint was 2 m, which was specified at the same location of the minimum head constraints.

#### Groundwater withdrawal constraints

### Decision variables and management scenarios

Details of management scenarios, in which Hmin is the minimum head constraints and Dd is drawdown constraints

Scenarios | Sub-scenarios | Constraints no. and type | Decision variables no. (managed wells) | Pumping constraint |
---|---|---|---|---|

1 | 1A | 134-Hmin | 938 | 100 % of the pumping rate of 2010 |

1B | 134-Dd | |||

1C | 260-Hmin | |||

1D | 260-Dd | |||

2 | 2A | 220-Hmin | 3,870 | |

2B | 220-Dd | |||

2C | 436-Hmin | |||

2D | 436-Dd | |||

3 | 3A | 134-Hmin | 938 | 200 % of the pumping rate of 2010 |

3B | 134-Dd | |||

3C | 260-Hmin | |||

3D | 260-Dd | |||

4 | 4A | 220-Hmin | 3,870 | |

4B | 220-Dd | |||

4C | 436-Hmin | |||

4D | 436-Dd |

### Optimization formulation

The optimization model was formulated as a sequential linear programing (SLP) problem to address the aquifer type (unconfined). SLP solves a series of linear programing subproblems, each formulated using the response matrix technique. In each iteration of the method, the response coefficients are calculated using a unit rate added to the current managed flow rates; the current managed rates are derived from the previous iteration’s optimal withdrawal rates. The sequential process is continued until convergence is achieved. Two convergence criteria were specified to control the end of the SLP process. The first criteria, *ε*_{1} (ranged 0.01–0.55), requires that the change in flow rate variable values from the prior iteration to the current iteration be less than a fraction of the magnitude of the flow rate variables at the current iteration. The second criteria, *ε*_{2} (0.001), requires that the change in objective function value be less than a specified fraction of the magnitude of the objective function value. For most scenarios, three to four SLP iterations were required to meet these two criteria.

## Results

### A- Maximum pumping constraints for the managed wells equal to 100 % of the pumping rate of the year 2010

^{3}/d and 2.43 Mm

^{3}/d for Scenario 1 and Scenario 2, respectively, which are only about 41.66 and 60.24 % of the amount withdrawn in 2010, respectively; 1.01 and 3.1 Mm

^{3}/d in the case of drawdown constraints for Scenario 1 and Scenario 2, respectively, which are only about 71.42, and 76.87 % of the amount withdrawn in 2010, respectively. Figures 8, 9, 10, and 11 show the location of operating and shutdown wells with the optimal pumping rate.

### B-Maximum pumping constraints for the managed wells equal to 200 % of the pumping rate of the year 2010

^{3}/d for Scenario 3 and Scenario 4, respectively, which are only about 62.21 and 81.32 % of the amount withdrawn in 2010, respectively; 1.07 and 3.45 Mm

^{3}/d in the case of drawdown constraints for Scenario 3 and Scenario 4, respectively, which are only about 75.83, and 85.5 % of the amount withdrawn in 2010, respectively. Figures 12, 13, 14, and 15 show the location of operating and shutdown wells with the optimal pumping rate.

Optimal pumping result for 100 and 200 % of the pumping rate for the year 2010, in which Hmin refers to minimum head constraints and Dd to drawdown constraints

Sub-scenario | Constraints no. and type | No. of managed wells | Total Q2010 (Mm | Total Qoptimal (Mm | No. of shutdown wells | No. of operating wells | %Volume of water store increase |
---|---|---|---|---|---|---|---|

1A | 134-Hmin | 938 | 1.41 | 0.58 | 532 | 406 | 2.4 |

1B | 134-Dd | 938 | 1.41 | 1.02 | 183 | 755 | 0.8 |

1C | 260-Hmin | 938 | 1.41 | 0.60 | 545 | 393 | 2.3 |

1D | 260-Dd | 938 | 1.41 | 1.00 | 161 | 777 | 0.9 |

2A | 220-Hmin | 3,870 | 4.03 | 2.51 | 1,111 | 2,759 | 3.7 |

2B | 220-Dd | 3,870 | 4.03 | 3.11 | 643 | 3,227 | 1.3 |

2C | 436-Hmin | 3,870 | 4.03 | 2.34 | 1,284 | 2,586 | 4.3 |

2D | 436-Dd | 3,870 | 4.03 | 3.08 | 615 | 3,255 | 1.4 |

3A | 134-Hmin | 938 | 1.41 | 1.05 | 551 | 387 | 0.6 |

3B | 134-Dd | 938 | 1.41 | 1.10 | 491 | 447 | 0.5 |

3C | 260-Hmin | 938 | 1.41 | 0.71 | 676 | 262 | 1.5 |

3D | 260-Dd | 938 | 1.41 | 1.04 | 446 | 492 | 0.6 |

4A | 220-Hmin | 3,870 | 4.03 | 3.39 | 1,867 | 2,003 | 0.6 |

4B | 220-Dd | 3,870 | 4.03 | 3.47 | 1,860 | 2,010 | 0.1 |

4C | 436-Hmin | 3,870 | 4.03 | 3.16 | 2,008 | 1,862 | 1.8 |

4D | 436-Dd | 3,870 | 4.03 | 3.42 | 1,758 | 2,112 | 0.4 |

## Conclusions

The Cache area was designated as a critical groundwater area in 2009 due to the decline in its water level to below 50 % of the saturated thickness of the aquifer. Optimization models were constructed to determine maximum optimal pumping of the irrigation wells located in the areas with cones of depression. The results indicated that the number of constraints has no significant effect on the optimal results. The number of shutdown wells increased by 14–188 % in the case of the maximum pumping constraint specified as 200 % of the pumping rate of the year 2010. In addition, the number of shutdown wells was more in the case of minimum head constraints compared with the drawdown constraints. The average optimal pumping increased by 6–49 % in the case of the maximum pumping constraint specified as 200 % of the pumping rate of the year 2010; however, the percent of volume of water stored in the aquifer decreased as compared to the case of the maximum pumping constraint specified as 100 % of the pumping rate of the year 2010. From the results, there is no relation between the optimal pumping rate, cell thickness, and the existence of the constraint in the cells that have been chosen as managed wells. In addition, the same managed well has a different optimal value for different decision variable groups.

## Notes

### Acknowledgments

Mrs. Haveen Rashid is grateful to Mr. Brian Clark from the United States Geological Survey, Arkansas Water Science Center, Fayetteville Field Office for his help and continuous support during this research.

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