Entropy-based groundwater quality monitoring network design using a simulation–optimization approach by coupling genetic algorithm, MODFLOW and MT3DMS

Extractable water resources are classified as groundwater and surface water and have a cycle of interactions. The operating conditions of water resources over time affect groundwater resources management. Operating parameters such as aquifer extraction to develop the agricultural system, urban expansion, and population growth can change in a region (Shourian & Davoudi 2017; Ghaseminejad & Shourian 2019). The addition of a water resource, e.g., a reservoir, to the aquifer system to supply water demands may lead to a different management approach to the aquifer, for surface and groundwater resources are interdependent. In fact, a change in one of the two resources will quantitatively and qualitatively change the other one (Ehtiat et al. 2018). The efficient management of groundwater systems would require reliable and valid data which could be collected through quantitative and qualitative groundwater monitoring (Khader & McKee 2014). These data are collected from a monitoring network, which consists of a set of observational wells distributed randomly across the region (Ayvaz & Elçi 2018).

Monitoring network optimization is a decision-making procedure for obtaining the optimal combination of the wells. Optimization is conducted when monitoring stations are distributed inefficiently and/or insufficiently and could not reliably measure the relevant parameters or when an excessively large number of wells are employed and report unnecessarily high quantities of data. It can also lead to a cost-inefficient monitoring network. In the following, some studies carried out in this field are included: Ning & Chang (2005) reported that the position and distribution optimization of stations in a monitoring network would require empirical data, intuition, and expert knowledge. Wu et al. (2005) proposed the removal of unnecessary samples to reduce the sampling cost in groundwater quality monitoring through a simulation–optimization approach. They employed a hybrid MODFLOW–MT3DMS model to simulate the groundwater flow and implemented the genetic algorithm (GA) to optimize the monitoring network. An optimal monitoring network should provide sufficient rather than excessive data. In other words, monitoring stations should not be distributed; therefore, sufficient data are obtained in some regions, whereas excessive data are provided in other regions (Yeh et al. 2006).

A monitoring network design could be optimized by using a variety of techniques. Several criteria can be employed to select an effective optimization technique. The available water yield and dataset size/type are the most important criteria. A review of the literature indicates that the modeling of the flow and mineral transport can be utilized to design a monitoring network. In such approaches, the groundwater level and quality data are collected by identifying the hydrological system and aquifer contamination distribution. In this regard, Bashi-Azghadi & Kerachian (2010) adopted the MODFLOW and MT3D models to identify the groundwater flow and qualitative aquifer behavior in groundwater quality monitoring network optimization. Luo et al. (2016) used MODFLOW and MT3DMS in order to identify groundwater quality and trend in the aquifer. Shafiei et al. (2013) defined an optimal network as a network in which the distribution of stations allowed for accurately estimating the stations with no observational data. Monitoring network optimization is implemented by increasing, reducing, relocating, or redesigning stations. The station increase, relocation, and redesign approaches are employed more often for rain gauge or hydrometric networks than for groundwater monitoring networks (Akbarzadeh et al. 2016).

In the present research, it is aimed to develop a model to optimize the number of groundwater monitoring wells. Such a model should relatively consider the behavior determinants of the aquifer system. To find the optimal network for monitoring a groundwater resource system, it is necessary to integrate the simulation models with an optimization routine. Hence, the predefined objectives are required for the design of the monitoring networks. In this paper, a new approach based on the concept of information agglomeration is developed to obtain the optimal monitoring network and characterize the entropy criteria in locations with a high number of multivariate data. For this purpose, the soil and water assessment tool (SWAT) is utilized to simulate the surface water resources and estimate the spatiotemporal distributions of the nitrate flow and water percolation into the aquifer. The MODFLOW model is employed to simulate the groundwater resources and obtain the groundwater level and other aquifers' responses to the hydrologic variations. Moreover, MT3DMS is used to estimate the nitrate content in the desired locations. The optimization of monitoring wells is formulated using the entropy theory by the GA in MATLAB to optimize the number of nitrate measurement wells.

The research methodology consists of two main steps:

• 1. Surface water modeling: Firstly, a surface water model is developed in SWAT. The model is calibrated and validated using SWAT-CUP to provide an accurate representation of water percolation and nitrate recharge rates to the aquifer. These rates are extracted from the outputs of the SWAT model.

• 2. Groundwater simulation and quality monitoring: After surface water modeling, the aquifer is simulated quantitatively and qualitatively using MODFLOW and MT3DMS models, based on previously obtained recharge and hydrological data. The nitrate content in 19 nitrate monitoring wells from October 2010 to September 2018 is determined through MT3DMS simulations.

Additionally, a GA routine is developed based on entropy theory to exclude insignificant monitoring wells from the existing network. The significant wells with higher entropy values are reported as the optimal groundwater quality monitoring network.

The SWAT model is an advanced software for hydrological simulation used in ArcGIS, incorporating processes such as evapotranspiration, runoff, and deep percolation. This model plays a crucial role in assessing the interactions between surface and groundwater resources in the Eshtehard plain and examining the impact of land management practices on water quality and quantity in complex watersheds with various land uses and soil types (Ehtiat et al. 2018; Aliyari et al. 2019). Digital elevation models (DEMs) with a resolution of 30 m were utilized to delineate watersheds, slopes, and sub-basin areas. Land-use and soil type data were obtained from the Ministry of Agriculture and the Global Soil Map. The study area was divided into 21 sub-basins and 167 hydrologic response units (HRUs), which represent regions with similar land cover, soil type, and management practices.

Meteorological data, including minimum and maximum temperatures, precipitation, relative humidity, solar radiation, and wind speed, were collected from the Iran Meteorological Organization. Additionally, agricultural management data related to planting seasons, irrigation, fertilizer and pesticide use, and tillage practices were obtained from the Comprehensive Water Updating Research of Iran. These comprehensive settings enable the SWAT model to effectively simulate hydrological processes over the long term.

Crop yield and runoff calibration were conducted utilizing regional crop yield data sourced from the Ministry of Agriculture, along with data from hydrometric stations provided by the Iran Water Resources Management Company. The model was implemented for the period from 2000 to 2018, with the initial 3 years (2000–2003) designated for model warm-up. Sensitivity analysis and calibration of the SWAT model were performed using the SWAT-CUP tool to ensure the reliability of outputs for subsequent analyses.

MODFLOW is a comprehensive numerical model developed by the United States Geological Survey (USGS) for simulating groundwater flow (McDonald & Harbaugh 2003). In this study, the MODFLOW model was integrated with the SWAT outputs to provide an accurate representation of groundwater systems in the Eshtehard plain. This model employs the finite difference method for three-dimensional simulation of groundwater flows in steady and transient states and incorporates Darcy's law to manage mass balance in simulating subsurface flows (Akbari et al. 2022). The aquifer was developed using ground-level point layers, bedrock, and initial water levels obtained through inverse distance weighting (IDW) interpolation. The model was assumed to be uniform and divided into 600 cells. Percolation data from the SWAT model were applied for simulating water movement within the aquifer.

To meet agricultural demands, the model considered 126 extraction wells and 14 observation wells in the Eshtehard plain. Steady-state calibration was focused on hydraulic conductivity and aquifer recharge parameters for October 2010. The calibration process included extracting 20 acceptable solution sets with a Nash–Sutcliffe (NS) coefficient above 0.7, providing a robust framework for groundwater management. Once the groundwater flow model was developed, the MT3DMS model was utilized to simulate nitrate transport as a qualitative parameter. This model is a powerful three-dimensional numerical tool for simulating the transport of dissolved materials under various hydrological conditions and allows for the investigation of complex interactions between multiple contaminants (Zheng et al. 2012; Ehtiat et al. 2018). Utilizing data from the SWAT outputs and integrating land-use maps and sub-basin shape files, the model accurately represents initial nitrate concentrations in 19 observation wells, facilitating comprehensive groundwater quality analysis. In Table 1, the key variables and data sources for modeling processes are presented.

An optimization algorithm seeks to find the optimal values for decision variables. Optimal values are obtained by minimizing or maximizing objective functions with predefined constraints. Classical optimization algorithms are time-consuming and ineffective in solving complex problems. In this study, the GA is adopted using entropy theory for the optimal design of the monitoring network.

In a monitoring problem, the collected dataset grows as the number of observational wells increases. However, the shared data of wells would increase, despite an expensive process. Therefore, there is a trade-off between unique data and shared (repetitive) data in the system as the number of wells increases (Alfonso et al. 2012). Such a trade-off provides decision-makers with good insights into the transfer of data, repetitive and unnecessary data, and the optimal number of monitoring wells.

To examine the relationships between the different well datasets, we utilize Shannon's entropy theory. In this approach, entropy represents the degree of uncertainty in the distribution of the data, while joint correlation calculates the amount of shared information between the wells.

The probability of a specific nitrate value in each well is calculated by dividing the number of occurrences of that specific value by the total number of observations. For instance, Table 2 shows the nitrate values for three wells over 12 months:

The joint correlation between wells is calculated as the sum of individual entropies minus the total joint entropy. The joint entropy is computed by combining the values in the following manner:

• 1. Combining values of wells 1 and 2:

  

  (2)
• 2. Joint entropy for wells 1 and 2:

  

  (3)
• 3. Combining with well 3:

  

  (4)

• 4. The total joint entropy H(X, Y, Z) is then computed.

This result indicates that there are 1.09 units of shared information among the three wells, suggesting a dependency among the well datasets (Alfonso et al. 2012). I(X, Y, Z) denotes the joint correlation between the wells, H is the entropy of a well, and P is the probability distribution of the well data. The terms A and B are intermediate combinations of the well values used to calculate the joint entropy.

In this context, the objective functions are defined as follows:

• - Minimizing overall correlation:

  

  (6)
• - Maximizing joint entropy:

  

  (7)

where represent the data of the wells in the system, P denotes the probability distribution, H is the entropy, and refer to the individual values of the wells.

The problem can be solved as a multi-objective optimization problem, the output of which is a set of quasi-optimal non-dominated solutions representing a Pareto front. The Pareto front describes the decision-making outcome and indicates how the improvement of one criterion worsens other criteria. In this regard, optimization refers to finding a trade-off between the objectives. The multi-objective genetic algorithm (MOGA) is an efficient multi-objective optimization algorithm as it searches the point space in parallel rather than searching for single points and does not require auxiliary information, e.g., differentiation. The MOGA model was used in this study to implement optimization and entropy theory. NSGA-II is a MOGA model that benefits from simulated binary crossover (SBN) and polynomial mutation. The SBN operation consists of the weighted average calculation of parents, allowing for the use of single-point binary-coded crossover in problems with real codes.

The unsteady model was calibrated on a monthly basis for 8 years from October 2010 to September 2018. In the unsteady model calibration, the surface aquifer recharge was extracted from the calibrated SWAT model and introduced to MODFLOW, calibrating the specific yield of the aquifer. The root mean square error (RMSE) and the NS coefficient of the steady quantitative model were reported as 0.45 and 0.99, respectively. However, the RMSE and NS of the unsteady model were calculated 0.75 and 0.81, respectively. Overall, the obtained results of SWAT-MOFDLOW showed that the proposed model performs well for the watershed simulation.

In the basic state, a 3D reactive transport model was executed with MT3DMS, simulating the rate and transport of nitrate in the aquifer over an 8-year period from October 2010 to September 2018. The model was run without any pollution entering the aquifer, and the results were used to calculate nitrate values in 19 wells on a monthly basis. This provided a baseline to compare against when evaluating the impact of pollutant sources entering the aquifer. The optimization with two objective functions of the entropy theory was performed in MATLAB. To adjust the GA parameters, the model was executed with different parameters' values and the suitable values reported in Table 5 were selected.

The FCM and GA model results were compared using the joint entropy of the selected wells. The joint entropy of the wells selected by FCM was reported 1.52, whereas that of the wells selected by GA was found equal to 1.92 and 3.25 without and with nitrate point source pollution, respectively. Therefore, the FCM model resulted in a lower joint entropy value and thus, represented a lower significance level comparing the GA model results.

In the design of groundwater monitoring networks, a homogeneous distribution of wells is often implemented. However, such an approach would not necessarily lead to an optimal monitoring well distribution due to excessive or insufficient data in the network. Hence, optimization is an effective approach to the design of groundwater monitoring networks. This paper aimed to develop an optimal groundwater monitoring network design based on the current monitoring network. For this purpose, a hydrological model was developed in SWAT. Once the SWAT runoff and crop yield had been calibrated, percolation and nitrate inflow outputs of the SWAT model were used as input to the groundwater simulation model. In fact, a quantitative groundwater simulation was conducted in MODFLOW, whereas qualitative groundwater simulations were performed through MT3DMS. The output nitrate data of the MT2DMS model from October 2010 to September 2018 were employed to optimize the groundwater monitoring network. The optimized monitoring network would have fewer nitrate monitoring wells and, therefore, lower costs. It is notable, excluding some wells, based on optimization models from the monitoring network is temporary, because some areas may become critical in terms of groundwater quality in future and would require more wells. As for an assumed nitrate recharge across a given area, the optimal combination of the wells differed from the combination corresponding to the scenario of no nitrate entrance. All in all, it is suggested groundwater monitoring network should be redesigned periodically (every few years) based on quality measures. It is worth noting that monitoring wells were selected among the existing wells. New locations, especially in areas where there is no monitoring well, can be considered as potential points of monitoring in order to obtain better coverage for the whole aquifer.

In the presented detailed analysis, the outcomes of the GA were compared with other methods used for optimizing the number of monitoring wells. Each method yielded different optimal well configurations, reflecting the unique criteria and algorithms utilized.

Genetic Algorithm (GA): The GA approach identified seven optimal well locations, which provided the best balance between operational costs and the effectiveness of groundwater monitoring. This method effectively minimized the RMSE to 11.43 mg/L, demonstrating its efficiency in achieving accurate predictions with fewer wells.

Other methods: In contrast, the alternative methods, while potentially offering more data points, resulted in higher RMSE values of 13.17, 12.27, and 10.7 mg/L for solutions 2, 3, and 4, respectively. These methods required a larger number of wells (9 and 11) leading to increased operational costs without significant improvements in the accuracy of the results.

Comparative evaluation: The comparative analysis highlights that the GA method, despite having fewer wells, maintains a high level of accuracy, making it the most cost-effective option for groundwater monitoring in the Eshtehard plain. The results suggest that utilizing the optimal number of wells can lead to substantial savings in sampling, transportation, laboratory analysis, and data interpretation.

No funds, grants, or other support were received.

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by H. Farshad. The first draft of the manuscript was written by H. Farshad and edited by M. Javan Salehi and M. Shourian. M. Shourian read and approved the final manuscript.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.