Groundwater Management and Allocation Models: A Review

Abstract

Effective groundwater management and allocation are essential from economic and social points of view due to increasing high-quality water demands. This study presents a review and bibliometric analysis of the popular techniques in groundwater management and allocation models, which have not yet been captured in the literature, as our knowledge allows. To this extent, the literature on this state-of-the-art is categorized based on four primary sectors intervening in efficient groundwater management. The first sector discusses the simulation and surrogate models as the central groundwater predictive models, wherein quantitative and qualitative groundwater models are scrutinized. The second section is dedicated to applying different classic and smart optimization models, followed by a summary of state-of-the-art works on applying accurate and heuristic optimization models in groundwater management. Third, uncertainty analysis techniques in conjunction with groundwater modeling are studied as analytical tools, approximation methods, and simulation methods to identify the most exciting subject fields. The fourth section reviews decision-making models coupled with groundwater models as multi-criteria decision-making, social choice, and game-theory models. Finally, a summary of this review and goals for future studies are presented. Additionally, several new ideas are recognized, advising scholars to find critical gaps in the field.

1. Introduction

Groundwater management techniques play a prominent role in water and environmental engineering due to the increasing demand for optimal high-quality water management. Over the past decades, systematic attitude, computational tools, novel optimization, simulation models, and their applications in groundwater planning and management have developed rapidly. Groundwater systems may be analyzed by determining their components and exploring their relationships. These components comprise hydrologic models, including qualitative and quantitative management of groundwater resources, technological and economic factors, and decision-making processes. The groundwater systems would become more complicated when the uncertainties which are effective in resource allocation are considered. Therefore, researching “management of groundwater resources” to investigate the groundwater systems and the methods developed can benefit effective and successful groundwater management and allocation strategies.

Groundwater predictive models are a significant part of groundwater management and modeling since they provide informative knowledge about the behavior of groundwater flow systems [1,2,3,4]. Moreover, optimization models are essential management tools in groundwater management to obtain the optimal values of the targeted function, leading to effective groundwater management strategies [5,6,7]. Furthermore, most groundwater management and allocation models face conflicting objectives, in which decision-making models are necessary [8,9]. However, groundwater predictive and optimization models coupled with decision-making approaches usually have some deviation from actual values since there are considerable uncertainties in simulating the natural hydrological events in groundwater models [10,11]. Therefore, reviewing the groundwater management and allocation models from four aspects of predictive modeling, management alternative, uncertainty analysis, and decision-making can be considered significant in environmental and water engineering (Figure 1).

The present study aims to investigate the most critical research in the optimal allocation and management of groundwater resources, as presented in Figure 1 and Figure 2. Over 200 papers from 1992 to 2021 are examined, categorized, and compared, with most studies taking place in Asia, Africa, North America, and Europe, respectively. These investigations have been studied considering the structure of applied models, including predictive modeling comprising of qualitative and quantitative simulations, management alternatives such as the method of solving optimization problems, uncertainty analysis, and decision-making (how to address the stakeholder’s desires), as well as integrating surface water models with groundwater models (Figure 2).

The remainder of this study is structured as follows. Section 2 provides an overview of the predictive modeling for groundwater management and allocation. Section 3 presents the management alternatives related to groundwater resources. Section 4 discusses the uncertainty analysis to manage groundwater resources suitably, followed by Section 5, defining the decision-making procedures within the groundwater modeling systems. Finally, concluding remarks and suggestions for future studies are noted in Section 6 and Section 7.

2. Predictive Modeling

Groundwater models help understand the behavior of groundwater flow systems via observation and prediction, which plays a significant role in long-term groundwater resource management. Therefore, groundwater modeling (of any kind) is the central part of most groundwater protection, development, and improvement projects. The groundwater models have primarily three goals:

• Predicting and forecasting natural and artificial changes in the aquifer; Forecasting is exclusive to deterministic models with high certainty. However, prediction is commonly utilized for probability models,
• Investigating the plan by elaborating the system based on different hypotheses about the nature and dynamics of descriptive groundwater models, which have not been intrinsically designed as a forecasting tool,
• Producing a hypothetical system to investigate groundwater flow principles with general or specific models for training as a part of a computer code advancement.

The implementation of groundwater models has been captured in several studies over the last five decades [1,2,3,4,22,23,24,25,26,27,28]. These studies showed that predictive groundwater models were successful when each groundwater model’s limitations were recognized. Therefore, reviewing predictive groundwater models can be considered a substantial achievement in water and hydrological engineering. The literature is scrutinized to evaluate these models, mainly in simulation and surrogate models (Figure 3), elaborated in the following sub-sections.

2.1. Simulation Model

Modern simulation models tend to be computationally intensive since they accurately show knowledge about real-world systems [33,52,53]. Recently, many studies have employed quantitative and qualitative simulations to determine hydraulic parameters, hydrochemical characteristics, and aquifer management. To this extent, for quantitative and qualitative simulations, most studies in the literature have used MODFLOW and MT3D (MT3DMS), respectively [8,11,13,15,46,47,48,49,54,55,56,57,58,59,60,61,62,63,64]. In order to investigate the groundwater quantitative and qualitative simulation models, 80 articles are surveyed in this study. This survey includes time steps, studied regions, types of aquifers, applied software, and the methods used in different stages of groundwater management in various types of aquifers as confined, unconfined, karstic, and sandy.

2.1.1. Quantitative Modeling

Generally, groundwater quantitative simulation models describe groundwater resources and their management. More than 100 papers, detailed in the supplementary, are examined for quantitative and qualitative models, time steps, and case studies.

Table 1. Limitation of groundwater models.

Model Type	Limitations
Analog model	Careful construction of the model is required because the flow rate varies with cube width [65]. Temperature is another factor that must be considered [65]. Application restrictions involve nonlinear conditions of variable transmissivity in unconfined aquifers and two-fluid flow issues [65,66]. Limited applications exist in reducing groundwater in the construction industry [67].
Analytical model	An exact analytical solution can be outweighed by errors introduced by simplifying assumptions of a complex field environment [68,69]. Transport issues are convoluted and cumbersome [68]. It is restricted to steady and uniform flow problems [68]. Initial boundary conditions are relatively simple; To fit the model, hydrogeological boundary conditions must be idealized [68]. Applying the analytical model requires professional discretion and field experience [68]. It is appropriate for resolving groundwater issues involving small portions of aquifer systems or small area extent [70,71]. It cannot deal with spatial/temporal variations in the groundwater system [70].
Porous media model	The capillary rise in such models is significantly more significant than in the field [66]. The water table is difficult to visualize and identify [65,66]. It is time-intensive and prohibitively expensive [66].

gives an overview of more recent references.

A quantitative evaluation of groundwater resources is customized to three main objectives [72]:

• Estimating the groundwater recharge, discharge, and storage locally [29,34,73,74],
• Evaluating the whole managerial threats on water resources [41,60,64,75],
• Investigating the full effects of developing groundwater using the existing and predicted water [76].

These objectives can be evaluated regarding two aspects:

• The part related to groundwater flow,
• The part related to contamination and its relevant reactions.

The quantitative groundwater models’ objectives can be achieved effectively by exploring the quantitative analysis from three main points of view: (1) Mathematical and numerical modeling, (2) Optimization model, and (3) Surface water parameters (Table S1, Supplementary Materials).

Conjunctive with Mathematical and Numerical Models

There have been several studies on different aspects of mathematical models applications in quantitative analysis of groundwater systems, such as sensitivity analysis, calibration, verification, and validation, among which, the presentation of an organized method for calibrating groundwater flow models under stable conditions [38], and automatic calibration of groundwater flow models [77], are the primarily cited studies. Automatic calibration methods in groundwater models such as PEST have been accelerated. Manual calibration remains an effective method, but automatic calibration suffers from computational bottlenecks. Advances in developing practical algorithms for searching and improving processing capabilities are the key drivers behind recent advances in automatic calibration.

Numerical models are substantial quantitative groundwater simulation models, mainly MODFLOW models in the literature, because of their ease of use, user-friendliness, and adaptability [30]. However, numerical models such as MODFLOW cannot model the drawdowns near the wells that have not entirely permeated in the saturation layers with adequate accuracy, which causes issues on a local scale. In order to improve this shortcoming, other effectual numerical models have been proposed in the literature, including WatFlow [50], ISOQUAD [29], FEFLOW [31], Finite element heat and mass transfer [33], analytic element method [34], GMS [35], ParFlow [36], and SGMP [37]. Finally, the GIS approach has been adopted, accompanied by groundwater numerical models such as the MODFLOW model, to present helpful groundwater management and modeling [32].

Conjunctive with Surface Water Model

Surface water models have been used in the literature to determine the recharge in aquifer simulations and evapotranspiration estimations as the main parameters in aquifer simulation. In most previous studies, these values for the model have been defined as a percentage of average precipitation of the region for recharge and the recorded evaporation values. However, these values are not independent of surface hydrological processes and are associated with precipitation, surface runoff, and flow characteristics in the unsaturated part of the soil [78]. A comprehensive simulation of hydrological processes increases the accuracy of estimating evapotranspiration and recharge values in opposition to each other. As a result, quantitative information about the recharge and evapotranspiration values are among the main required inputs in modeling groundwater flow. To fill this knowledge gap, the Soil and Water Assessment Tool (SWAT) model is adopted to calculate these values concerning other hydrological processes and present them to each HRU (Table S2, Supplementary Materials). Therefore, the recharge and evapotranspiration values in each HRU were allocated individually with higher certainty to the corresponding cells in the groundwater model after modeling the surface hydrological processes and adapting them to the region.

The SWAT model simulates aquifer-river interaction and estimates flow rates in hydrologically connected regions. This model can also account for changes in groundwater operation conditions such as pumping values, extraction patterns, and land use, which affect groundwater resource return rates that MODFLOW cannot estimate. In addition, the SWAT surface water model can estimate recharge value in new operational conditions. Thus, water engineers can improve hybrid groundwater-surface water modeling by reviewing articles on surface water and groundwater integration for management, primarily based on SWAT and MODFLOW.

Fifteen articles used surface water software to estimate recharge value and temporal and spatial distribution along with groundwater models (Table S2, Supplementary Materials) because using groundwater models alone to predict future conditions under different management scenarios usually yields unreliable results. MODFLOW cannot account for recharge rate variations caused by precipitation and river infiltration, land use, irrigation, and agricultural activities on the surface domain [79]. Several attempts have been made to solve the problems noted above by combining SWAT and MODFLOW (e.g., [45,74,78,80,81,82,83,84,85,86,87,88,89], SWAT and MT3D [60,61], MT3DMS, and MODFLOW [8,11,15,54], SWAT, MT3DMS, and MODFLOW [8,11,15,54], and GMS and Visual Basic [35] to model SWGW interaction. SWAT surface water model and MODFLOW groundwater have been the most effective combination in the literature to increase simulation accuracy. With this combination, the groundwater model can predict and manage more accurately.

The SWAT-MODFLOW has been used extensively in the United States [80,83,90,91], Italy [54], Canada [86], Korea [78,81], Spain [61]; India [60]; China [80]; Denmark [87], Iran [8,11,15,92], and Japan [88]. These studies showed that the simultaneous application of surface water models such as SWAT and groundwater models such as MODFLOW has been able to cover the semi-distributive and distributive limitations of these two models individually to identify regional hydraulic conditions.

2.1.2. Groundwater Quality Simulation Model

Water quality models investigate water quality variables’ local and timing variations and groundwater resources’ physical, chemical, and biological properties. In addition, water quality models sometimes estimate groundwater contamination leakage. Many researchers have studied qualitative groundwater modeling recently. The most popular software is MT3D, which can be linked to MODFLOW (

Table 2. Classification of different quantitative simulation groundwater models.

Simulation Model
Quantitative Model	Time Step	Reference	Case Study
Analytic Element Method	Annual	[34]	Dore-France
FEFLOW	Monthly	[31]	Zhangye, China
Finite Element Heat and Mass Transfer	Annual	[33]	Yucca Flat, USA
GMS	Annual	[35]	Maraghe plain, Iran
GMS-MODFLOW 2000	Monthly	[76]	Nakuru district, Nairobi
ISOQUAD	3 Months	[29]	Hypothetical
MODFLOW	Annual	[93]	Hypothetical
		[39]	Island
		[40]	Upper San Pedro River Basin, southeastern Arizona, Mexico
		[94]	Cosumnes River in California
		[73]	The southern part of Tehran, Iran
		[17]	Ismarida plain, northeastern Greece
		[78]	Musim-cheon Basin in Korea
		[81]	Mihocheon watershed, south Korea
		[45]	Yellow River Basin, Inner Mongolia, China
		[95]	Ølgod, Jutland, Denmark
		[42]	Izmir, Turkey
		[96]	NCP, China
		[97]	East Owienat, Egypt
		[44]	Izmir, Turkey
		[98]	Uromieh, Iran
		[99]	Rafsanjan plain, Iran
		[100]	aquifer-Muscat, Oman
MODFLOW and ANN	Monthly	[101]	Najafabad plain in westcentral, Iran
MODFLOW and UCODE	Annual	[102]	Glacial-till plain on the Jutland peninsula in western Denmark
MODFLOW-96	Monthly	[103]	Hill Country, USA
ParFlow	Monthly	[36]	Klamath River, California, USA
WatFlow	Annual	[50]	Oro Moraine in Canada

and Table S3, Supplementary Materials). The qualitative analysis of groundwater models can be reviewed from two primary views: (1) Optimization model and (2) Combination with the surface water model.

The optimization models used for qualitative analysis of groundwater models can be classified into two groups of classic models, such as Bundle-Trust [46], Monte Carlo Simple Genetic algorithm, and Noisy Genetic algorithm [16], and intelligent algorithms, such as GA [47], NSGA-II [48], MINOS [49], and MOPSO [8].

2.2. Surrogate Model

Surrogate models have been developed to become a more economical substitute for “original” groundwater simulation models while providing an approximate analysis of the original model. The primary motivation for developing surrogate modeling strategies is better application of the limited existing computational budgets. It was reported that surrogate modeling was mentioned in six highly cited meta-modeling articles to highlight the potential advantages of these models [104]. The main characteristics impacting the surrogate models are:

Analytical type of model that should be added by the surrogate model—search or sampling; Search analysis means single or multi-objective optimization (management and calibration) or uncertainty-based/Bayesian model calibration procedure; Other modeling analyses refer to sampling analyses,

Computational budget limitation; This explains how many original model evaluations can build a surrogate model and ultimately perform the model analysis of interest. In applications where a surrogate model is used repeatedly after initial construction (i.e., optimizing real-time operational decisions), the available time for each usage can be highly important,

Explanatory variables numbers; In general, with the increase in the number of explanatory variables, surrogate modeling is less advantageous and even infeasible,

Single-output versus multi-output surrogates; This causes a fundamental difference in environmental simulation modeling, where the model’s output typically varies in time and space. Single-output surrogates are usually common when the original model output of interest is a function calculated from many model outputs (such as calibration error metric),

Availability of original model developers/experts; Some surrogate modeling techniques require experts (lower fidelity modeling). They can provide valuable insights into the importance of surrogate modeling errors over original modeling [105].

In the review article, surrogate modeling and its methods in water resources were discussed in detail [106]. In order to find more information about these models, it is recommended to refer to [107]. In most surrogate models, they were linked to optimization algorithms (i.e., generalized likelihood uncertainty estimation [53,108], ACUARS [109], and Markov Chain Monte Carlo [110]. Moreover, response surface modeling and lower-fidelity modeling are among Surrogate models.

The Surrogate models have been addressed in some studies while using other groundwater analysis techniques. Some studies that have utilized MODFLOW to identify the hydrological system and estimate the volume of water and water level changes are [17,76,96,103]. A numerical model called WatFlow has been used to simulate groundwater flow, employing steady-state calibration for long-term simulation [50]. Then, simulations have been employed in unstable situations caused by unregulated groundwater withdrawal, simulating the Zhangye basin in China using EFLOW [31]. Finally, the MODFLOW and GIS were combined to investigate the outcomes of the critical scenario for unstable circumstances [51].

3. Management Alternative

Optimization models can help develop beneficial groundwater management strategies by finding the target function’s maximum and minimum values under constraints. Figure 4 shows groundwater optimization models. This figure shows that optimization models can be accurate or heuristic. Advanced optimization models for quantitative and qualitative management strategies are:

Nonlinear chance-constrained groundwater management modeling,

Simulated annealing,

Sharp interface model.

These optimization models are classified into two categories regarding the type of solution space: continuous and discrete response space. First, these response spaces are utilized to develop simulation–optimization models for groundwater management. Then, the continuous response space includes non-convex and convex problems associated with nonlinear and linear issues. In addition, discrete response space categorizes as a discrete input, logical, and combinational.

Simulation–optimization (S–O) models are increasingly being used in 36% of the surveyed articles in groundwater management to build dependable optimum groundwater management systems, with the benefit of providing a wealth of information about the interplay between social and economic factors and water resources [5,6,111,112,113,114]. S–O models optimize well interval, dewatering region design, water level prediction, pumping, supply, and demand timing (Tables S4 and S5, Supplementary Materials). In water resources engineering, simulation-based optimization models can consider important factors other than economic efficiency. This advantage is that strategies and policies include performance sustainability indicators such as justice and environmental sustainability in addition to economic efficiency criteria. These hybrid S–O models have assessed groundwater management issues:

Drawdown and reduction of the withdrawal capacity from the aquifer,

Simultaneous application of surface and groundwater resources,

The problem of salt and freshwater interference,

Groundwater remediation,

Prediction of the future behavior of the aquifer under different operating conditions,

Justice in meeting the water requirement

The S–O models are generally based on interpolation methods, response matrices, embedding methods, and metamodels, particularly metamodels and embedding models. Different optimization approaches have been used in conjunction with groundwater simulation models, wherein NSGA-II is used nine times as the highest applicable optimization approach in the reviewed studies based on groundwater management techniques. The second rank is allocated to the GA approach, which is repeated seven times in the studies of this review. Then, other optimization approaches are used only in one study of this review, proving that the GA and NSGA-II are the most practical optimization algorithms in groundwater management. S–Os can use several optimization algorithms to find the best design and extraction options if the problem is appropriately defined and formulated, but the computational speed and computer memory limits limit them. This study reviews groundwater modeling optimization model literature. Interpolation methods, response matrices, embedding methods, and metamodels, particularly metamodels and embedding models, form the foundation of the S–O models.

3.1. Accurate Optimization

Mathematical models such as linear programming (LP), nonlinear programming (NLP), mixed-integer programming (MIP), LP based on optimal control theory, constrained differential dynamic programming, combinatorial optimization (CO), and multi-objective programming are used to optimize groundwater systems accurately [115]. Groundwater models were optimized using two main procedures. The first approach identifies the aquifer behavior as linear or nonlinear, and a mathematical model calculates the withdrawal-drawdown relationship (simulation). Due to its convenience, many researchers optimize the cyclic storage system using this method. Second, conjunctive simulation–optimization models optimize aquifers. After calibrating and determining the aquifer’s hydrodynamic parameters, objective functions, and constraints, the simulation and optimization models are coupled. This approach is more accurate than the first because the simulation model is rerun for any change in the decision variables vector during optimization.

3.2. Heuristic Optimization

Evolution, survival, life, and progress in nature, living creatures, and animals often inspire evolutionary algorithms [116]. Excellent management balances the element of demand and supply. Since many water resource problems are nonlinear, linking a robust optimizer model as evolutionary algorithms helps. Evolutionary methods find the optimal point without the cost function derivative. They also avoid the local minimum trap better than gradient-based methods. Depending on the application, these methods may be slower than gradient-based methods for finding an optimal local solution. Groundwater management features:

Hydrogeological and socio-economic conditions of the system,

Regulatory arrangements,

Regulatory procurement,

Costs and benefits of management activities.

3.2.1. Combination with Accurate Methods

Formulating the new objective function for optimal compromise is the central optimization pillar. Heuristic optimization models have several drawbacks: only one answer, unverifiable objective functions, and problem-solving only in convex search spaces [115]. Recently, intelligent methods have been used with accurate models to address these issues, with the NSGA-II algorithm being the most popular. Despite their popularity, these methods have drawbacks in finding Pareto optimal solutions. The answer takes a long time because these algorithms are linked to groundwater simulations such as MODFLOW. Finally, many of these methods are sensitive to Pareto front continuity and shape.

3.2.2. Metaheuristic Models

The metamodel balances quality and cost in most practical optimization problems. Eliminating the optimality condition is economically necessary because CO problems require many computations. Thus, the metamodel needs algorithms to balance computation and near-optimality. These algorithms need adjustable parameters to balance solution and analysis.

Table 3. Classification of different qualitative simulation groundwater models.

Simulation Model
Groundwater Quality Simulation Model	Time Step	Reference	Case Study
MT3D	Annual	[59]	Hypothetical
	Monthly	[46]	Na’aman Aquifer, Western Galilee, Israel
		[16]	Hypothetical
		[56]	Balasore coastal basin, India
		[48]	Tehran aquifer, Iran
		[57]	Tehran aquifer, Iran
		[58]	Hypothetical
		[60]	Upper Yamuna watershed, India
		[61]	Jucar river basin, Spain
		[64]	Nile Delta, Egypt
MT3DMS	Annual	[49]	Hypothetical
	Daily	[55]	Hypothetical
	Monthly	[54]	Bonello watershed, Italy
		[63]	Najaf Abad plain, Iran
		[12]	Kavar-Maharloo aquifer, Iran
		[11]	Isfehan-Barkhoar, Iran
		[13]	Bad-Khaledabad, Iran
		[8]	Isfehan-Barkhoar, Iran
		[15]	Isfehan-Barkhoar, Iran
Parallel Groundwater Transport and Remediation Codes	Daily	[117]	Hypothetical
SAHYSMOD	Seasonal	[56]	Haryana State, India
SEAWAT	Annual	[118]	Ras Sudr, Egypt

shows groundwater optimization models using single-objective metaheuristic optimization algorithms. This table shows that most groundwater modeling studies use a genetic algorithm (GA).

3.2.3. Response Structure Models

Response matrices have been used instead of full groundwater models in groundwater management (e.g., [119]), in which similar methods are commonly used to produce response surface surrogates. For example, [120,121] suggested methods to develop “reduced models” of high-dimensional groundwater models based on proper orthogonal decomposition. [122,123] used such reduced models as surrogates for original groundwater models for inversion of the model (calibration). Moreover, [124] used a response matrix as a “reduced model,” which replaced a groundwater model to resolve the optimization problem in groundwater management. However, it is noteworthy that in all studies for optimization objectives, reduced models (i.e., lower-fidelity models) behave as if they have high-fidelity representations of the basic real-world systems and replace the original models completely (i.e., high-fidelity models) after development. In other words, the discrepancies between the high- and low-fidelity models are ignored in the analysis.

Optimization algorithms based on response surface surrogates such as EGO [125], GMSRBF, MLMSRBF [126], and the Gutmann method [127], and also uncertainty analysis algorithms such as ACUARS [109] and Markov chain Monte Carlo [110] have all been created to evade the limitations of the computational budget associated with computationally intensive simulation models.

3.3. Optimization Techniques Coupled with Surrogate Modeling

Surrogate modeling can be helpful only when the simulation model is computationally intensive. It justifies the transferring cost to the second level of abstraction (reduced model fidelity), which generally decreases the accuracy of the analyses [104,128]. Among the studies in which surrogate models are linked to optimization, and uncertainty algorithms, [52,106,109,126,129,130,131] can be named. Moreover, [132,133,134] used the Monte Carlo sampling algorithm to speed up the study. Furthermore, [12] used a neural network as a surrogate. Although few surrogate modeling studies of water resources resolve the discrepancies between the response surfaces of the low-fidelity surrogate and the original model, [135]) used “coarse-grid models” (“upscaled models”) in the underground water model (high fidelity) to accelerate Markov chain Monte Carlo calculations for uncertainty determination. In order to avoid the approximated posterior distribution, a linear correction function created discrepancies before Markov chain Monte Carlo testing.

4. Uncertainty Analysis

Since flow equations can partially simulate natural hydrological processes in a groundwater model, and observation data under the region’s hydrological conditions are inaccurate; the groundwater simulation model results usually deviate from actual values [9,10]. As a result, groundwater modeling and decision-making have usually been accompanied by uncertainty analysis as an essential series of technical factors to account for uncertainties resulting from the lack of undersurface system information or natural changeability in system processes and region conditions [136] (

Table 4. Classification of different single-objective metaheuristic groundwater optimization models.

Optimization Approach	Study	Optimization Approach	Study
Simulated Annealing	[137]	Artificial Neural Network (ANN)	[138]
Genetic Algorithm (GA)	[43]	Particle Swarm Optimization	[34]
	[138]		[114]
	[73]		[5]
	[47]		[11]
	[139]		[8]
	[48]	Harmony Search Algorithm	[140]
	[141]		[142]
	[57]	Firefly Algorithm	[6]
	[101]	GAMS	[93]
	[59]	Mixed Integer Non-linear Programming	[40]
	[98]	CMA-ES	[143]
	[99]	Constrained Differential Dynamic Programming-Adaptive-Network-based Fuzzy Inference System	[144]
	[100]	MINOS	[61]
	[63]	Elitist Continuous Ant Colony Optimization	[41]
	[7]	Heuristic Harmony Search	[42]
	[145]	Constrained Differential Dynamic Programming	[29]
	[12]

). Studies show that the Monte Carlo method and Markov Monte Carlo chain have been most used in determining the uncertainty of underground water models (

Table 5. The uncertainty analysis techniques in groundwater modeling.

Reference	Simulation		Uncertainty Technique	Case Study
	Quantitative Modeling	Qualitative Modeling
[146]	No	No	Fuzzy α-cut	Vannetin basin, France
			Monte-Carlo simulation
[147]	MODFLOW	No	generalized likelihood uncertainty estimation	Hypothetical
[148]	BIGFLOW	No	Monte Carlo	Plain of Tadla, Morocco
[149]	No	No	Fuzzy α-cut	Hypothetical
			Monte-Carlo simulation
[55]	MODFLOW	MT3DMS	Spatial bootstrap	Hypothetical
[20]	MODFLOW	No	generalized likelihood uncertainty estimation	Hypothetical
			Bayesian model averaging
[47]	MODFLOW	MT3D	Monte Carlo	Hypothetical
[53]	No	No	Fuzzy-stochastic partial differential equation, Fuzzy partial differential equation, and Stochastic partial differential equation	Hypothetical
[33]	Finite element heat and mass transfer		Null-space Monte Carlo	Yucca Flat, USA
			Markov chain Monte Carlo (Differential Evolution Adaptive Metropolis)
[150]	No	No	Fuzzy-probabilistic	Hanford site, Washington, USA
			Monte Carlo simulation
[151]	No	No	Hybrid propagation	No
[152]	MODFLOW	No	AM- Markov chain Monte Carlo	Hypothetical
[58]	MODFLOW	MT3D	Fuzzy parameterized probabilistic analysis	Hypothetical
[59]	MODFLOW	MT3D	Fuzzy ordinary Kriging	Hypothetical
[117]	No	Parallel groundwater transport and remediation codes	Markov chain Monte Carlo	Hypothetical
[95]	MODFLOW	No	Multiple-point geostatistical method	Ølgod, Jutland, Denmark
			Stanford geostatistical modeling software
[153]	MODFLOW	No	Null-space Monte Carlo	No
[154]	GFLOW		Monte Carlo	Wisconsin, USA
[100]	MODFLOW	No	Monte-Carlo simulation	Muscat, Oman
[14]	MODFLOW	No	generalized likelihood uncertainty estimation	Birjand aquifer, Iran
[12]	MODFLOW	No	Monte Carlo	Tashk-Bakhtegan river basin, Iran
[8]	MODFLOW	MT3DMS	DREAMzs	Isfahan-Barkhoar, Iran
[15]	MODFLOW	MT3DMS	DREAMzs	Isfahan-Barkhoar, Iran

).

The debate over whether to theorize complicated or straightforward models to explain a groundwater system [155,156,157,158,159,160], advances in computational power and growing awareness among researchers to resolve uncertainty in simulated data [161,162,163,164,165] has sparked a growing tendency to postulate alternative conceptualizations [20,102,166,167,168,169,170,171]. Therefore, much effort has been put forward to study the sources producing uncertainty in groundwater or hydrological simulation models, from which different classifications have been carried out by the researchers, four of which are presented as follows.

First, [136] classified the uncertainty sources into three main categories:

Conceptual uncertainty: The first step in modeling is creating a conceptual model from an undersurface system. Considering that decision about the conceptual model is made based on the incomplete data of the system, there would be uncertainty in the conceptualization of the model.

Parametric uncertainty: A model has various parameters which need to be specified. Generally, the insufficient number of data can cause parametric uncertainty.

Random uncertainty: Natural changeability in field conditions, even in a well-calibrated conceptual model, can cause uncertainty in the prediction.

Second, [172] stated that the uncertainty depending on geological characteristics in groundwater modeling originated from two primary sources: the geological structure and the hydraulic parameter values. In a geological structural element, parameter values always show heterogeneity on a local scale, so the uncertainty can be considered or ignored. Therefore, three primary classifications are presented for evaluating uncertainty:

Geological structure: The geological model’s structural elements’ geometrical characteristics have resulted in these uncertainties.

Effective parameters of the model: These uncertainties focus on the effective values of model parameters that describe the changes in hydraulic characteristics.

Heterogeneity of model parameters on the local scale: These uncertainties are due to the undescribed changes of hydraulic characteristics with each element of the geological structure.

Third, [173] categorized uncertainty methods into three significant groups. The first group includes simple and primitive methods in terms of statistical relationships and their implementation, such as generalized likelihood uncertainty estimation by [174] and Sequential Uncertainty Fitting (SUFI2) by [173,175]. This group reflects the uncertainty of the model output only as the uncertainty caused by the parameter, which cannot demonstrate different uncertainty components, including parameters, model structure, and input data [173]. The second group includes the models investigating the impact of inputs and the model structure by adding the error value (which usually shows the time correlation between the residuals) to the output, such as Markov chain Monte Carlo methods [176] and Autoregressive error models [177]. Finally, the third group includes more complete methods than the other two. These methods try to express the influence of simultaneous error originating from the model inputs and structure by introducing likelihood functions, thus approaching quantifying all the sources of uncertainty. Moreover, the uncertainty analysis methods can be classified into three general categories:

Analytical methods,

Approximation methods,

Simulation methods.

4.1. Analytical Methods

In uncertainty analysis, calculating three statistical characteristics of the mean, variance, and coefficient of variation of the output variable or variables of the mathematical model are of great importance. In order to overcome the uncertainty of groundwater numerical simulations, methodologies based on stochastic and statistical methods are frequently applied. For example, Monte Carlo (MC) method [7], the Neumann expansion method [18], the perturbation method [178], the inversion method [179,180], Kalman filtering method [181,182,183], and Markov chain Monte Carlo method are some examples of such methods [117,176,184]. Metropolis and Metropolis-Hastings are also more efficient sampling algorithms. All of these algorithms are Markov chain Monte Carlo methods. Converging the search space, these optimization methods are based on previous samples. However, the high computation cost may be unaffordable due to groundwater models’ many parameters. Thus, comparing samplers is necessary to pick the best one.

Simplified surrogate models have also been used using various Markov chain Monte Carlo frameworks for the uncertainty analysis of water resource models. For example, Ref. [33] developed a simple groundwater model as a “surrogate model” in a computationally intensive model defined in different parameters. Instead of the original model, they performed auto-calibration and uncertainty experiments on the surrogate model and adjusted the algorithm parameters of the Markov chain Monte Carlo uncertainty analysis method so that the adjusted Markov chain Monte Carlo was used in the expensive original computational model. Finally, analytical methods include derived distributed methods, integral transform techniques, conjunctive simulation, and optimization methods.

4.2. Approximation Methods

In analytical methods, statistical torques are calculated analytically. However, some cases have no analytical solution to the statistical torques for the mathematical model. In such cases, approximate methods are the solution. However, choosing any approximation method depends on the information related to random variables and the mathematical relation between the variables. Therefore, uncertainty analysis methods have been recently applied along with the fuzzy method, which has increased the accuracy of the results.

The approximation methods are the first-order variance estimation method, Rosenblueth’s probabilistic point estimation method, and Harr’s probabilistic point estimation method. These techniques are widely used in analyzing the uncertainty of the mathematical models governing the phenomena. Furthermore, there has been a significant development in research, including the general likelihood uncertainty estimation (generalized likelihood uncertainty estimation) approach [14,20,174,184], the differential requirement evolution adaptive metropolis (Differential Evolution Adaptive Metropolis) algorithm [15], the Bayesian method [185,186], and other approximate methods [187,188,189,190]. Among these methods, [174] confirmed that many time-consuming computations are in the generalized likelihood uncertainty estimation method. Furthermore, they stated that the computational time would decrease with technological advancement and the speed of processing systems. However, considering the complexity of the groundwater model, computations remain; practical sampling algorithms can solve this problem to some extent.

Some uncertainty assessment techniques, such as generalized likelihood uncertainty estimation and SUFI2 based on the Monte Carlo method [191] and Markov chain Monte Carlo and Bayesian model averaging methods [192], reflect all the sources of uncertainty on the parameter. However, these two methods require many simulation model evaluations to estimate significant uncertainty [9]. To fill this knowledge gap, some researchers have used the Differential Evolution Adaptive Metropolis algorithm to assess the uncertainty of hydrological models based on the differential Evolution-Markov chain (DE-MC) method [193,194], which is improved as DREAMzs, in which a small number of parallel chains for posterior sampling are used. Therefore, the DREAMzs algorithm increases the sampling efficiency using fewer function evaluations than Differential Evolution Adaptive Metropolis to converge to the appropriate posterior distribution [195].

4.3. Simulation Methods

Simulation of the models provides a better response to the uncertainty degree in some cases since the numerical simulation of groundwater is the basis of the quantitative analysis of groundwater resources [19]. To this extent, applying these methods seems necessary in some groundwater models.

Uncertainty in groundwater models involves studying the transmission source, producing uncertainty processes, describing and evaluating the situation and uncertainty characteristics, and studies on controlling and decreasing the uncertainty, prediction, and response for the unknown groundwater environment. Therefore, Monte-Carlo simulation is the main among the recently developed simulation methods. However, one of the limitations of the methods based on Monte-Carlo simulation, as the most applicable investigation and parametric uncertainty analysis techniques, is time-consuming computations.

5. Decision Making

In recent years, conflict resolution and game theory approaches have been used to better model decision-makers’ preferences and those affected by groundwater resource allocation problems in the face of uncertainties because most water resources management decision-making problems involve conflicting goals such as maximizing economic benefits and minimizing environmental impacts. Moreover, in these models, the objective function constraints are usually:

• Reduction of drawdown,
• Reduction of environmental pollution factors,
• Optimal increase in pumping,
• Decreasing water scarcity,
• A balance between water supply and water demand, and justice.

When goals conflict, one goal is achieved only by losing other goals. In this case, decision-makers determine a consensual solution to achieve socially acceptable behavior. Therefore, game theory and conflict resolution methods can help interactions between conflicting objectives (

Table 6. The game theory or conflict resolution approaches in groundwater modeling.

Reference	Simulation		Game Theory or Conflict Resolution	Case Study
	Quantitative Modeling	Qualitative Modeling
[196]	No	No	Nash bargaining scenario	El Paso, Texas and Ciudad Juarez, Mexico
			Nash non-cooperative game
[197]	No	No	Non-cooperative equilibrium	South-central Texas, US
[198]	No	No	Game theory	Guanajuato, Mexico
[199]	No	No	Game-theory model	Hypothetical
[48]	MODFLOW	MT3D	Young conflict–resolution theory	Tehran aquifer, Iran
[57]	MODFLOW	MT3D	Rubinstein’s sequential bargaining theory	Tehran aquifer, Iran
[200]	No	No	Social choice rule	Western La Mancha aquifer, Spain
[99]	MODFLOW	No	MCSGA	Rafsanjan plain, Iran
			NGA
[12]	MODFLOW	MT3DMS	Social choice rule-Fallback Bargaining	Kavar-Maharloo aquifer, Iran
[13]	MODFLOW	MT3DMS	Non-cooperative game theory	Khaledabad, Iran

).

Game theory and conflict resolution models have generally been applied to planned scenarios or optimization outputs (Pareto frontier). In most cases, the results have shown that such models can achieve the goals and satisfy the desires of the conflicting parties well [12,13,15]. However, considering that there are different objectives and different decision-makers with diverse and often conflicting interests in the extraction of groundwater resources, after investigating the problem as a multi-objective problem or as a scenario, using game theory and conflict resolution models can improve the acceptability of optimal extraction policies in terms of decision-makers and system users and enable policies to be implemented. Thus, system decision-makers and groundwater extraction influencers are identified, and their preferences are determined by purpose and constraint. Then, the objectives are ranked by preferences, scenarios, or optimal interaction curve points. Finally, the optimal response considers potential conflicts between decision-makers and system-influenced people.

In order to improve management, game theory examines decision-makers’ behaviors. This theory states that each player maximizes his profit (payoff) while considering how other players’ actions may affect the outcome [201]. Game theory has advantages over quantitative simulation and optimization resource management methods because it can simulate multiple parts of the dispute, account for different problem characteristics, and estimate feasible solutions without quantitative payoff information [202].

Cooperative and non-cooperative games are the two most common types of games. In cooperative games, players are expected to work together to obtain equitable access to a resource. For example, the difficulty of managing groundwater has been treated as a cooperative game [203,204] since it resembles multi-objective and multi-criteria decision-making procedures often employed in water resource management; however, predictions made in the context of cooperative games are sometimes erroneous since they assume that all stakeholders would cooperate to the maximum extent possible [21]. The non-cooperative game models applications in groundwater modeling include non-cooperative equilibrium, non-cooperative game theory [13], Nash [196], Rubinstein sequential bargaining theory [57], and MCSGA and NGA [99] and cooperative game models in groundwater models such as young conflict–resolution theory [48], Social choice rule [12,200], and fallback bargaining [12,15].

6. Summary and Conclusions

Groundwater management and allocation modeling are considered essential practices since they help appropriate groundwater management for each study area, which should be as congruous as possible to the area’s natural conditions so that it can be used effectively and successfully for prediction and management purposes. To this extent, four primary sectors should be studied: groundwater management as predictive groundwater modeling, management alternative, uncertainty analysis, and decision-making.

Predictive groundwater models such as MODFLOW simulate aquifers quickly for effective groundwater management. However, due to the increased decision variables in long-term groundwater planning and management, these effective simulation models have execution time issues. Some articles solved this problem with optimization models. Optimization algorithms are linked to conceptual numerical models such as MODFLOW to create an integrated modular model that covers the model’s weaknesses. Thus, using predictive models and optimizers has optimized aquifer water allocation.

A specific optimization algorithm for groundwater simulation models is not recommended for a specific problem. However, this state-of-the-art shows that the NSGA-II optimization algorithm has been used to overcome simulations’ weaknesses. Due to groundwater system uncertainties, hybrid groundwater optimization–simulation models and actual models differ; Uncertainties make groundwater optimization–simulation models more accurate.

Groundwater modeling uncertainty assessment methods include Monte Carlo-based generalized likelihood uncertainty estimation and SUFI2, Markov chain Monte Carlo, and Bayesian model averaging methods to account for all parameter uncertainty. However, both methods require many simulation model evaluations to estimate significant uncertainty. To fill this knowledge gap, some researchers have used the Differential Evolution Adaptive Metropolis algorithm to assess hydrological model uncertainty based on the differential Evolution-Markov chain method, which is improved as DREAMzs by using a small number of parallel chains for posterior sampling.

Decision-making models should evaluate optimization–simulation groundwater model results. Many surveys have resolved conflicts in decision-making models by ranking a set of possible optimal solutions in target space as the interaction curve between objectives from simulation–optimization models. Social choice rule and fallback bargaining conflict resolution models have been used in the literature to resolve decision-maker conflicts over groundwater allocation policies. Minimal computational budgets should benefit from the surrogate model. When simulation–optimization models are inaccurate, surrogate modeling can be used in groundwater modeling.

Finally, recent studies have coupled SWAT with groundwater models due to the recharge parameter’s high sensitivity. The groundwater model more accurately considers this parameter’s temporal and spatial distribution. Thus, groundwater model recharge values are more precise and distributed. Surface water models cover the constraints of semi-distributive and distributive models.

7. Prospects for Future Studies

In this paper, essential research in groundwater management is surveyed. These articles are classified and studied regarding the structure of the applied models, the method of solving optimization problems, considering uncertainties, addressing the stakeholders’ preferences, the quantitative and qualitative simulation of groundwater, and the use of surface water models.

According to the literature review, there is still room for further research in uncertainty modeling and analysis. For example, it is possible to consider the existing uncertainties in decision-making more effectively using innovative methods in uncertainty analysis, such as Markov chain Monte Carlo. Furthermore, in groundwater quality management, simulating the behavior of stakeholders, especially dischargers of the contamination load over time, and their interactions can significantly impact the actuality of the models’ output. However, a literature review shows that minimal research has been carried out in this field, which can be used for future studies.

Finally, considering that the complexity level of the original response function (which is typically unknown) plays a crucial role in determining an approximate method of proper performance, future research may be used to develop pre-analysis methods of the original response landscapes using a minimal number of samples to measure their level of complexity.