A Robust Optimization Framework for Hydraulic Containment System Design Under Uncertain Hydraulic Conductivity Fields

Abstract

Effective containment of contaminant plumes in heterogeneous aquifers is critically challenged by the inherent uncertainty in hydraulic conductivity (K). Conventional, deterministic optimization approaches for pump-and-treat (P&T) system design often fail when confronted with real-world geological variability. This study proposes a novel robust simulation-optimization framework to design reliable hydraulic containment systems that explicitly account for this subsurface uncertainty. The framework integrates the Karhunen–Loève Expansion (KLE) for efficient stochastic representation of heterogeneous K-fields with a Genetic Algorithm (GA) implemented via the pymoo library, coupled with the MODFLOW groundwater flow model for physics-based performance evaluation. The core innovation lies in a multi-scenario assessment process, where candidate well configurations (locations and pumping rates) are evaluated against an ensemble of K-field realizations generated by KLE. This approach shifts the design objective from optimality under a single scenario to robustness across a spectrum of plausible subsurface conditions. A structured three-step filtering method—based on mean performance, consistency (pass rate), and stability (low variability)—is employed to identify the most reliable solutions. The framework’s effectiveness is demonstrated through a numerical case study. Results confirm that deterministic designs are highly sensitive to the specific K-field realization. In contrast, the robust framework successfully identifies well configurations that maintain a high and stable containment performance across diverse K-field scenarios, effectively mitigating the risk of failure associated with single-scenario designs. Furthermore, the analysis reveals how varying degrees of aquifer heterogeneity influence both the required operational cost and the attainable level of robustness. This systematic approach provides decision-makers with a practical and reliable strategy for designing cost-effective P&T systems that are resilient to geological uncertainty, offering significant advantages over traditional methods for contaminated site remediation.

1. Introduction

Groundwater contamination has emerged as a global environmental issue, posing significant threats to both ecosystems and human health. Major sources of aquifer pollution include industrial activities, agricultural runoff, and improper waste disposal [1,2]. To effectively address this challenge, pump-and-treat (P&T) systems have been established as a mature and widely applied remediation technology, playing a crucial role in containing and removing contaminant plumes [3,4]. This technique involves installing extraction wells within the contaminated zone to pump out polluted groundwater for ex situ treatment, thereby containing the plume and restoring aquifer quality [5].

The migration of contaminant plumes in groundwater systems is a critical issue in environmental management, particularly in heterogeneous aquifers. A primary challenge in groundwater remediation is the effective placement of extraction wells and the optimization of pumping rates under conditions of subsurface uncertainty. P&T systems are highly sensitive to subsurface heterogeneity, especially the spatial distribution of hydraulic conductivity (K), a key parameter governing groundwater flow and contaminant transport [6,7,8]. Traditional modeling approaches often assume a homogeneous or simplified heterogeneous K-field, which can lead to significant inaccuracies in predicting groundwater flow and contaminant migration.

To address these limitations, geostatistical methods have become the mainstream approach for representing heterogeneous hydraulic conductivity fields. These methods treat K as a random spatial field characterized by statistical properties such as mean, variance, and spatial correlation structures, enabling a more realistic and continuous description of subsurface heterogeneity [9,10]. Common geostatistical techniques include variogram analysis and stochastic simulation, which generate multiple realizations of the K-field consistent with observed data and spatial statistics.

Despite their advantages, geostatistical representations typically lead to high-dimensional stochastic models, posing significant computational challenges for uncertainty quantification and optimization tasks. The dimensionality arises because the K-field is discretized over many spatial locations, each treated as a random variable. To overcome this, dimensionality reduction techniques are essential. Among these, the Karhunen–Loève Expansion (KLE), a form of Principal Component Analysis (PCA) for random fields, is widely adopted in groundwater modeling to reduce the complexity of the K-field while preserving its main statistical features [11,12,13]. KLE decomposes the random field into a series of orthogonal eigenfunctions weighted by uncorrelated random variables, allowing truncation after a limited number of terms that capture most of the variance. This reduction not only facilitates efficient stochastic simulations but also enables robust optimization frameworks that evaluate system performance across multiple realizations of the reduced-dimensional K-field.

To identify the optimal design and operation of P&T systems under these stochastic conditions, simulation-optimization (S-O) frameworks are widely employed. These frameworks integrate a groundwater simulation model, which predicts the fate and transport of contaminants for a given K-field realization, with an optimization algorithm that systematically searches for the best well configurations and pumping rates. Due to the complexity and non-linearity of the objective functions (e.g., minimizing cost while meeting cleanup targets), metaheuristic algorithms have gained prominence. Popular methods include Genetic Algorithms (GAs), Particle Swarm Optimization (PSO), and Simulated Annealing (SA), which are adept at navigating vast and rugged search spaces to find near-optimal solutions without requiring gradient information [14,15,16]. While powerful, these S-O approaches can be computationally intensive, and their application is often limited to a single or a few representative K-field realizations, which may not adequately account for the full spectrum of subsurface uncertainty [17,18].

Given the inherent uncertainty in the K-field and its profound impact on the performance of hydraulic containment systems, adopting a robust optimization (RO) approach becomes essential. Robust optimization aims to find solutions that perform well across a range of possible scenarios, rather than optimizing for a single realization of the uncertain parameter [19]. Instead of seeking a solution that is optimal under a single, deterministic condition, RO identifies a solution that exhibits the best robustness across a given range of uncertainty [6,20]. This means that the remediation system is designed to effectively contain the contaminant plume with a high degree of reliability, even if the actual hydraulic conductivity field deviates from the estimated one. In the context of optimizing well placement and pumping rates, this is typically achieved through a multi-scenario analysis framework, which involves generating multiple plausible realizations of the K-field to comprehensively evaluate candidate designs, thereby ensuring that the final optimized strategy maintains its effectiveness and reliability in the face of hydrogeological uncertainty.

In this study, we propose a novel robust simulation-optimization framework that integrates the efficiency of the Karhunen–Loève Expansion (KLE) for stochastic representation of hydraulic conductivity uncertainty with an advanced Genetic Algorithm to design a cost-effective and reliable pump-and-treat (P&T) system. The primary objective is to simultaneously optimize the locations and pumping rates of extraction wells to minimize the total operational cost (represented by total pumping rate) while ensuring a high and stable probability of containing the contaminant plume across a diverse ensemble of plausible heterogeneous aquifer conditions. This is achieved by evaluating candidate well configurations against multiple K-field realizations generated via KLE, thereby identifying solutions that are not only optimal for a single deterministic scenario but also robust and reliable under significant hydrogeological uncertainty.

The remainder of this paper is structured as follows. Section 2 details the methodology, including the numerical groundwater flow model, the optimization framework, the Karhunen–Loève Expansion technique for generating heterogeneous hydraulic conductivity fields, and the robustness assessment strategy. Section 3 describes the specific experimental design, outlining the case study setup and the multi-scenario evaluation process used to validate the proposed framework. Section 4 presents and discusses the results, analyzing the performance of optimized solutions under single-scenario and multi-scenario frameworks, evaluating their robustness across varying levels of aquifer heterogeneity, and examining the characteristics of the most stable well configurations. Finally, Section 5 offers concluding remarks, summarizing the key findings, discussing the implications for groundwater management, and identifying areas for future research.

2. Methodology

This study focuses on the optimization of well placement and pumping rates for a hydraulic containment system designed to prevent contaminant plume migration in heterogeneous aquifers. Given the subsurface uncertainty, particularly in the spatial distribution of hydraulic conductivity, we adopt a robust optimization framework that evaluates candidate solutions across multiple realizations of the K-field to identify configurations that maintain effective containment under varying K-field uncertainty. The methodology integrates four core components:

• A numerical groundwater flow model based on MODFLOW for simulating hydraulic head responses under various well configurations;
• An optimization model that defines decision variables, the objective function, and constraints related to well design and contaminant containment;
• A Genetic Algorithm-based search strategy tailored for mixed-variable encoding and constrained optimization;
• A stochastic modeling approach to quantify uncertainty in the K-field and assess solution robustness.

2.1. Groundwater Flow Modeling

The groundwater flow dynamics in the study area are governed by the transient saturated flow equation based on Darcy’s law and mass conservation principles. The governing partial differential equation is expressed as

$$\nabla \cdot \left(K\nabla h\right)+Q=0$$

(1)

where $h$ is the hydraulic head [L]; $K$ is the spatially variable hydraulic conductivity tensor [LT⁻¹]; and $Q$ is the source–sink term [T⁻¹].

This equation is subject to appropriate initial and boundary conditions that define the physical behavior of the aquifer system. In this study, the model domain is discretized into a regular grid using finite difference approximations, and the resulting system of equations is solved numerically using the widely used modular finite-difference groundwater flow model MODFLOW-2005 [21].

The model was implemented via the Python package FloPy 3.9.1, which enables efficient setup, execution, and post-processing of MODFLOW simulations within a scriptable environment [22]. Each optimization iteration involves dynamically updating well locations and pumping rates, followed by re-running the MODFLOW simulation to compute the resulting hydraulic head distribution. These results are then used to evaluate the satisfaction of head-based constraints designed to contain the contaminant plume.

2.2. Optimization Model for Hydraulic Containment

The optimization problem aims to determine the optimal configuration of a set of extraction wells that minimizes total pumping effort while ensuring effective hydraulic containment of the contaminant plume.

2.2.1. Decision Variables

The decision variables in this problem consist of the spatial coordinates (row and column indices) and the corresponding pumping rates of n extraction wells, resulting in a total of 3n decision variables. Specifically, each well is represented by three variables: x_3i (row index: integer variable), x_3i+1 (column index: integer variable), and x_3i+2 (pumping rate: real number variable), where i = 0, 1, …, n corresponds to n extraction wells.

This mixed-variable encoding allows for simultaneous optimization of both discrete well locations and continuous pumping rates.

2.2.2. Objective Function

The objective function aims to minimize the total pumping rate across all wells:

$$\mathrm{Minimize} f\left(x\right)=\sum _{i=0}^{n-1}\left|x_{3i+2}\right|+P\left(x\right)$$

(2)

where $P\left(x\right)$ denotes a linear penalty term applied when any of the predefined constraints are violated. This ensures that infeasible solutions are gradually guided toward feasibility during the evolutionary process.

2.2.3. Hydraulic Head Constraints

To achieve effective containment of the contaminant plume, the optimization incorporates m hydraulic head constraints designed to maintain favorable flow conditions that restrict pollutant migration. These constraints are defined between predefined pairs of monitoring points:

$$h\left({\mathrm{loc}}_j^{\mathrm{outer}}\right)-h\left({\mathrm{loc}}_j^{\mathrm{inner}}\right)\ge {\Delta h}_{min},\hspace{1em}j=1,2,\dots ,m$$

(3)

where $j$ denotes the index for the m predefined pairs of monitoring points; ${\Delta h}_{min}$ denotes a minimum head difference; and ${\mathrm{loc}}_j^{\mathrm{outer}}$ and ${\mathrm{loc}}_j^{\mathrm{inner}}$ are the locations of the outer and inner monitoring points in the j-th pair, respectively. This formulation enforces a controlled flow field that prevents downstream movement of contaminants by maintaining higher heads at outer locations relative to inner reference points.

2.2.4. Variable Bounds and Feasible Search Space

To define the feasible search space for well placement and pumping rate optimization, practical and hydrogeological considerations were incorporated. The spatial coordinates of each extraction well were constrained within a predefined subdomain of the aquifer model grid:

(1)

Row indices of well locations were bounded between $R_{\mathrm{m}\mathrm{i}\mathrm{n}}$ $\mathrm{a}\mathrm{n}\mathrm{d}$ $R_{\mathrm{m}\mathrm{a}\mathrm{x}}$,

$$x_{3\mathrm{i}}\in \left[R_{\mathrm{m}\mathrm{i}\mathrm{n}},\hspace{0.17em}{R}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right],\hspace{1em}\mathrm{i}=0,1,\dots ,m$$

(2)

Column indices were restricted within $C_{\mathrm{m}\mathrm{i}\mathrm{n}}$ $\mathrm{a}\mathrm{n}\mathrm{d}$ $C_{\mathrm{m}\mathrm{a}\mathrm{x}}$,

$$x_{3\mathrm{i}+1}\in \left[C_{\mathrm{m}\mathrm{i}\mathrm{n}},\hspace{0.17em}{C}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right],\hspace{1em}\mathrm{i}=0,1,\dots ,m$$

(3)

Pumping rates were limited to a range $\left[Q_{\mathrm{m}\mathrm{i}\mathrm{n}},Q_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$, where $Q_{\mathrm{m}\mathrm{i}\mathrm{n}}<0$ denotes the maximum allowable extraction rate (negative value indicating withdrawal), and $Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=0$ reflects the physical constraint that no injection is permitted:

$$x_{3\mathrm{i}+2}\in \left[Q_{\mathrm{m}\mathrm{i}\mathrm{n}},\hspace{0.17em}{Q}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right],\hspace{1em}\mathrm{i}=0,1,\dots ,m$$

These bounds ensure that all generated solutions are physically meaningful and operationally viable.

2.3. Optimization Framework

To solve the high-dimensional and constrained optimization problem, we employ a hybrid-variable Genetic Algorithm implemented using the pymoo optimization library [23]. Key components of the algorithm include the following:

(1)

Sampling: A mixed-variable sampling strategy that combines integer random sampling for well coordinates and uniform real-value sampling for pumping rates.

(2)

Crossover: Simulated Binary Crossover (SBX) is used for real-valued variables.

(3)

Mutation: Polynomial Mutation (PM) is applied to real variables, while random integer mutation is used for coordinate variables.

(4)

Selection: Tournament selection with a crowding distance is employed to maintain diversity in the population.

Through preliminary testing, multiple combinations of population sizes and generation limits were systematically evaluated to identify a configuration that optimally balances computational efficiency with solution quality. Although specific configurations (in

Table A1. Hyperparameters of the Genetic Algorithm (GA) used in pymoo.

Hyperparameter Category	Parameter Name	Symbol	Value	Description
Algorithm Settings	Population Size	pop_size	50	Number of individuals in each generation.
	Number of Generations	n_gen	100	Maximum number of generations for the optimization run.
Sampling	Sampling Strategy	-	Mixed-Variable Sampling	Custom strategy combining integer (for well coordinates) and real-value (for pumping rates) sampling.
Crossover	Crossover Operator	-	Simulated Binary Crossover (SBX)	Crossover operator used for real-valued variables (pumping rates).
	Crossover Probability	prob	0.9	Probability of crossover occurring between two parents.
	Distribution Index (SBX)	eta	15	Distribution index for SBX, controlling the spread of offspring around parents.
Mutation	Mutation Operator	-	Mixed-Variable Mutation	Custom mutation operator.
	Mutation Probability	prob	0.1	Probability of mutation occurring for each variable.
	Distribution Index (PM)	eta	20	Distribution index for Polynomial Mutation (PM), used for real variables.
Selection	Selection Strategy	-	Tournament Selection	Selection method (implicitly used by pymoo’s GA).
	Tournament Size	-	2	Number of individuals participating in each tournament (default for pymoo GA).
Diversity	Eliminate Duplicates	-	True	Flag to enable the removal of duplicate solutions within the population.

) demonstrated effectiveness in the case study presented in Section 3, these parameters may be modified based on the specific requirements of the optimization problem and available computational capabilities.

Each candidate solution is rigorously evaluated through full MODFLOW simulations utilizing the FloPy interface. The simulation process dynamically updates well locations and pumping rates, computes the corresponding hydraulic head distribution, and systematically evaluates constraint satisfaction. This tightly integrated coupling between the optimization engine and the groundwater flow simulator enables the identification of well configurations.

2.4. Uncertainty Modeling

Subsurface heterogeneity, particularly the spatial variability of hydraulic conductivity, introduces significant uncertainty in predicting groundwater flow patterns and contaminant transport dynamics. To account for this uncertainty and enhance the robustness of the optimized well configurations, a stochastic modeling framework is employed to generate multiple realizations of the K-field. These realizations represent plausible subsurface structures with varying degrees of spatial correlation and heterogeneity.

2.4.1. Stochastic Generation of Hydraulic Conductivity Fields

To account for the spatial variability and uncertainty in hydraulic conductivity, a stochastic modeling framework is employed to generate multiple realizations of the K-field using the Karhunen–Loève Expansion (KLE) method [11]. This approach allows for the representation of heterogeneous subsurface properties while maintaining statistical consistency across different realizations.

The log-conductivity field, denoted as $\mathrm{Y}\left(x,\omega \right)=\mathit{ln}K\left(x,\omega \right)$, where $x\in D$ and $\omega \in \Omega$ (a probability space), is modeled as a random process. The mean component of $\mathrm{Y}\left(x,\omega \right)$ over all possible realizations is represented by $⟨\mathrm{Y}\left(x,\omega \right)⟩$. A covariance function $C\left(x,y\right)$, which is bounded, symmetric, and positive definite, is required to construct the KLE. This covariance function can be decomposed into eigenvalues ${\tau }_i$ and eigenfunctions $f_i\left(x\right)$ according to the second kind of the homogeneous Fredholm integral equation:

$$C\left(x,y\right)={\sum }_{i=1}^{\infty }{\tau }_i{f}_i\left(x\right)f_i\left(y\right),$$

(4)

The random process $\mathrm{Y}\left(x,\omega \right)$ can be expanded as

$$\mathrm{Y}\left(x,\omega \right)=⟨\mathrm{Y}\left(x,\omega \right)⟩+{\sum }_{i=1}^{\infty }{\xi }_i\sqrt{{\tau }_i}{f}_i\left(x,\omega \right),$$

(5)

where ${\xi }_i$ are independent standard Gaussian random variables. In practice, the infinite series is truncated to a finite number N_KL of terms to obtain a computationally feasible approximation:

$$\mathrm{Y}\left(x,\omega \right)\approx ⟨\mathrm{Y}\left(x,\omega \right)⟩+{\sum }_{i=1}^{N_{KL}}{\xi }_i\sqrt{{\tau }_i}{f}_i\left(x,\omega \right),$$

(6)

This finite-dimensional approximation ensures that the generated K-fields maintain the desired statistical properties (mean, variance, and spatial correlation) while being computationally manageable. The truncation parameter N_KL determines the level of detail in representing the heterogeneity, with higher values capturing more fine-scale variability but at increased computational cost.

Each realization captures different spatial patterns of conductivity while maintaining overall statistical consistency. This approach ensures that the optimization process considers a diverse set of subsurface scenarios, thereby improving the generalizability and reliability of the resulting well designs.

2.4.2. Robustness Evaluation Across Multiple Realizations

To assess the performance stability of each candidate solution under uncertain subsurface conditions, all identified well configurations are evaluated across a total of N independently generated K-fields. For each realization, the same MODFLOW-based simulation is executed to compute the resulting hydraulic head distribution and evaluate constraint satisfaction. Key metrics used to quantify robustness include the following:

(1)

Mean Constraint Satisfaction Ratio (${\mathrm{C}\mathrm{S}\mathrm{R}}_{\mathrm{mean}}$): the average proportion of constraints satisfied across all K-fields.

(2)

Minimum CSR (${\mathrm{C}\mathrm{S}\mathrm{R}}_{\min }$): the lowest constraint satisfaction observed among all realizations, indicating the worst-case performance.

(3)

Standard Deviation of CSR (${\mathrm{C}\mathrm{S}\mathrm{R}}_{\mathrm{std}}$): a measure of performance variability across different K-fields.

(4)

Pass Rate (PR): the proportion of K-fields in which the solution meets a predefined threshold for constraint satisfaction.

These metrics enable the identification of well configurations that consistently perform well across a wide range of subsurface conditions. The integration of uncertainty quantification into the optimization workflow supports risk-informed decision-making by explicitly considering the impact of subsurface heterogeneity on system performance. This approach facilitates the selection of well designs that balance optimality and robustness, providing valuable insights for long-term planning and management of contaminated sites where subsurface information is inherently limited.

3. Case Study

3.1. Problem Description

Figure 1 illustrates the study area, which is impacted by a groundwater contaminant plume resulting from long-term leakage at a heavily polluting industrial facility located in the northwest portion of the study area. The contamination source originates from a continuous pollutant leak that has persisted for 20 years. As shown in Figure 1, the resulting contaminant plume demonstrates the spatial distribution of concentrations exceeding 5 mg/L, illustrating the extent of contamination after two decades of leakage.

To protect a designated protected zone located downstream, a hydraulic containment system must be implemented to prevent the further migration of the plume. This case study employs a two-dimensional steady-state groundwater flow model to investigate the application of a hydraulic containment system for the long-term containment of the contaminant plume. The top and bottom boundaries of the aquifer are considered impermeable. The lateral boundary conditions are defined as constant-head boundaries on the northwest and southeast sides, and as no-flow boundaries on the northeast and southwest sides.

As shown in Figure 1, the aquifer system is assumed to be heterogeneous and isotropic, with hydraulic conductivity following a log-normal distribution. The spatial correlation structure of the log-conductivity is characterized by the following exponential covariance function.

$${\gamma (l,l^{\prime })}_{\ln K}={\sigma }_{\ln K}^2\mathrm{e}\mathrm{x}\mathrm{p}\left(-\sqrt{{\left({\displaystyle \frac{l_x-l_x^{\prime }}{{\lambda }_x}}\right)}^2+{\left({\displaystyle \frac{l_y-l_y^{\prime }}{{\lambda }_y}}\right)}^2}\right),$$

(7)

where $l=(l_x,l_y)$ and $l^{\prime }=(l_x^{\prime },l_y^{\prime })$ denote two arbitrary spatial locations, ${\sigma }_{\mathrm{l}\mathrm{n}K}^2$ is the variance, and ${\lambda }_x$ and ${\lambda }_y$ are the correlation lengths along x and y directions, respectively. Length scales of ${\lambda }_x/L_x =0.30$ and ${\lambda }_y/L_y =0.40$ are considered in this study, where $L_x$ and $L_y$ are the sizes of flow domain along the x and y directions, respectively. The mean of the $\ln K$ field is 2.30, while the variances are set to ${\sigma }_{\mathrm{l}\mathrm{n}K}^2=0.50$. In this study, the KLE method is adopted for generating multiple realizations of the K-field, and the truncation criterion for the KLE was based on capturing a target percentage (90% in this study) of the total variance of the log-conductivity field. The specific truncation parameter N_KL is automatically determined by the program based on the specified covariance model and grid. This approach was consistently applied across all scenarios, including those with varying heterogeneity levels.

To effectively contain the contaminant plume and prevent its migration toward the downstream protected zone, a series of extraction wells are proposed to be installed around the plume, forming a closed hydraulic containment system. The red dashed line in Figure 1 indicates the candidate range for well placement. The inner points (inner points) and outer points (outer points) shown in Figure 1 are used to implement the hydraulic head constraints described in Section 2.2. These points create a hydraulic gradient directed toward the center of the contaminant plume, thereby controlling its migration range and preventing further diffusion.

The primary objective of this study is to design an optimal well configuration for a hydraulic containment system that can effectively contain this plume under conditions of subsurface heterogeneity. The optimization aims to minimize the total pumping rate while ensuring that the hydraulic head constraints necessary for containment are satisfied across the domain.

3.2. Experiment Design

The experimental framework is designed to systematically evaluate the performance and robustness of optimized well configurations under significant subsurface uncertainty. The workflow is structured into three main components to enable a comprehensive assessment of solution optimality, generalizability, and resilience to geological heterogeneity.

Component 1: Single-Scenario Optimization and Cross-Validation

The first component establishes a baseline for comparison by performing deterministic optimizations under individual K-field realizations. A total of 50 independent hydraulic conductivity fields (K₁, K₂, …, K₅₀) are generated using the KLE method. The choice of 50 realizations represents a practical balance between computational feasibility and the need for a sufficiently large ensemble to capture the range of possible subsurface conditions and provide statistical robustness in the subsequent multi-scenario assessment (requiring 50 solutions × 50 K-fields = 2500 simulations). This number was also found sufficient in preliminary evaluations to observe stable trends in the distribution of solution performance metrics (e.g., mean satisfaction ratio, standard deviation) without incurring excessive computational cost, and it is consistent with the ensemble sizes used in similar stochastic groundwater modeling studies. For each K_i, a separate optimization run is conducted as described in Section 2.3. This process yields 50 candidate solutions (S₁, S₂, …, S₅₀), where S_i is the optimal well configuration identified under the K_i field.

To assess the transferability of these solutions, a cross-validation is performed. One of the 50 K-fields is selected as the true field (K_true), which represents the actual subsurface condition. Each of the 50 solutions is then evaluated on this K_true field to compute its Constraint Satisfaction Ratio (CSR). This component highlights the limitations of single-scenario optimization by demonstrating the performance gap between an estimated field (which may be obtained using geostatistics or groundwater inverse problem) and the true field.

Component 2: Multi-Scenario Robustness Assessment

To move beyond deterministic optimization and evaluate solution robustness, a comprehensive multi-scenario assessment is conducted. This comprehensive evaluation assesses the performance of each of the 50 solutions (S_i) under all 50 K-field realizations. For each (S_i, K_j), the Constraint Satisfaction Ratio is computed, resulting in a total of 2500 individual performance assessments (50 solutions × 50 K-fields). The core robustness assessment employs a three-step filtering process:

(1)

Defining the Efficacy Threshold: An efficacy threshold for CSR_mean is established based on the statistical distribution of performance across all solutions and K-fields.

(2)

Screening for Robustness: Solutions are screened based on their ability to meet or exceed this efficacy threshold in a high percentage (e.g., 75%) of the K-field realizations.

(3)

Selecting the Most Stable Solutions: Among the solutions passing the robustness screen, the final subset is selected based on the lowest ${\mathrm{C}\mathrm{S}\mathrm{R}}_{\mathrm{std}}$ across all K-fields, identifying the most stable performers.

Component 3: Application under Varying Heterogeneity Levels

This component applies the multi-scenario robustness assessment framework (Component 2) to evaluate how the level of aquifer heterogeneity influences the performance and selection of robust solutions. Three distinct levels of heterogeneity are considered, defined by the variance (σ²) of the log-conductivity field: low (σ² = 0.25), moderate (σ² = 0.50), and high (σ² = 1.00).

For each heterogeneity level, a new set of K-field realizations is generated using the KLE method with the corresponding variance. The entire optimization and multi-scenario evaluation process is then repeated. This allows for a comparative analysis of solution robustness and the characteristics of the identified stable solutions under conditions of increasing subsurface uncertainty.

4. Results and Discussion

4.1. Single-Scenario Optimization and Cross-Validation

In the first component, the pymoo algorithm was independently applied to optimize well configurations under 50 randomly generated K-field realizations. Each optimization run yielded a candidate solution, representing a specific combination of well locations and pumping rates. These solutions were then evaluated on a selected true K-field—one of the 50 realizations, specifically the same K-field used as the background in Figure 1—to assess their performance in satisfying the hydraulic head constraints designed to contain the contaminant plume.

As illustrated in Figure 2, there is a strong positive correlation (correlation coefficient = 0.8541) between pumping rate and satisfaction. This quantitative relationship confirms that higher pumping rates generally lead to better constraint satisfaction, but at the potential cost of increased operational expenses. Statistical analysis reveals key characteristics: the first quartile (Q1), median (Q2), and third quartile (Q3) for pumping rates are 59.2, 84.5, and 117.8 m³/d, respectively, while for satisfaction levels, they are 0.2941, 0.3824, and 0.7500, respectively. These metrics indicate that while most solutions have moderate pumping rates, satisfaction levels show greater variability, with the majority falling below 0.75. The 95% confidence interval around the trend line further illustrates the uncertainty in the relationship, suggesting that individual solution performance can deviate significantly from the general trend due to subsurface heterogeneity.

To further evaluate the robustness of the optimized solutions, we conducted backward tracking using MODPATH to visualize the capture zones of the pumping wells under different K-field realizations. The blue pathlines represent the capture zone of the optimized pumping wells, providing insights into how effectively the contaminant plume can be contained.

Figure 3 illustrates the capture zone morphologies, highlighting the variability in performance due to subsurface heterogeneity. Figure 3a shows the optimal solution specifically for the true K-field. It achieved a pumping rate of 118 m³/d and a satisfaction level of 0.76, demonstrating effective containment of the contaminant plume. In contrast, Figure 3b presents a randomly selected solution with a pumping rate of 90 m³/d and a satisfaction level of 0.35. Its capture zone is inadequate, leading to failure in containing the contaminant plume. Another randomly selected solution (Figure 3c) exhibited a pumping rate of 98 m³/d and a satisfaction level of 0.72, showing a capture zone similar to the optimal solution, indicating robust performance. Finally, a third randomly selected solution (Figure 3d) demonstrated a high pumping rate of 160 m³/d and a satisfaction level of 0.94, achieving excellent contaminant containment but at the cost of substantial engineering expenses.

The results indicate that while some solutions perform well under their respective K-fields, their effectiveness varies significantly when applied to the true K-field. This highlights the importance of evaluating candidate solutions under different subsurface conditions rather than relying solely on deterministic optimization. The significant performance variability observed across different K-field realizations demonstrates a critical limitation of single-scenario approaches and underscores the need for a more comprehensive, multi-scenario evaluation to identify well designs that are genuinely robust and reliable under uncertain subsurface conditions.

4.2. Multi-Scenario Robustness Assessment

The single-scenario optimization results presented in Section 4.1 highlighted the significant vulnerability of solutions to geological uncertainty, despite the observed correlation between pumping rate and satisfaction. To identify well designs that are resilient to subsurface variability, the comprehensive multi-scenario evaluation described in Section 3.2 (Component 2) was conducted. This involved assessing the performance of each of the 50 optimized solutions (Si) under all 50 K-field realizations. For each solution-K-field pair (S_i, K_j), the satisfaction ratio (CSR) was computed, resulting in a total of 2500 individual performance assessments. This extensive evaluation captures the constraint satisfaction behavior of each solution under different realizations of the same geostatistical model.

The analysis is structured around the three-step filtering process outlined in Section 3.2 (Component 2) to identify solutions that are both effective and stable across diverse subsurface conditions.

Step 1: Defining the Efficacy Threshold

An efficacy threshold for the mean satisfaction ratio (CSR_mean) is established based on the statistical distribution of performance across all solutions and K-fields (as described in Section 2.4 for metric definitions). This threshold balances two criteria: (1) a sufficiently high level of constraint satisfaction, and (2) placement within the top quartile of all solutions in terms of performance.

A solution is considered effective only if its CSR_mean across all K-field realizations is at least 75%—a level deemed necessary for reliable containment. Meanwhile, this 75% satisfaction requirement aligns closely with the third quartile (Q3 = 72%) of the CSR_mean distribution shown in Figure 4, indicating that solutions meeting this threshold outperform 75% of all other solutions. Therefore, an efficacy threshold of CSR_mean = 0.75 is adopted, ensuring that only solutions with both high performance and a strong statistical standing are considered for further evaluation.

Step 2: Screening for Robustness

Building upon the efficacy threshold defined in Step 1, the second step focuses on identifying robust solutions—those that maintain their efficacy across a wide range of K-field realizations. A solution is deemed robust if its satisfaction ratio is greater than or equal to the efficacy threshold (0.75) in at least 75% of the 50 K-fields (i.e., in at least 38 out of 50 realizations). This majority rule criterion ensures that the selected solutions are not only effective on average but also consistently reliable, as their performance is not contingent on a few favorable K-field realizations.

Step 3: Selecting the Most Stable Solutions

Among the solutions that pass the robustness screening in Step 2, the final step is to select those with the highest stability and the least sensitivity to geological variations. This is achieved by calculating the standard deviation of CSR (CSR_std) for each robust solution across all 50 K-fields. For this study, the five solutions with the smallest CSR_std are selected as the most stable solutions. This number (5) is chosen arbitrarily for demonstration purposes and can be adjusted based on the specific requirements of the analysis or the total number of solutions passing the previous screening steps. This selection minimizes the risk of encountering a scenario where the solution’s performance drops significantly below its average, providing a high degree of confidence in its long-term reliability.

The results of this three-step filtering process are presented in

Table 1. Key statistics of the most stable well configurations under geological uncertainty.

Solution_Id	Total Pumping (m3/d)	CSRmean	CSRstd	PR (CSR < 50%)
S29	174	86.82%	19.40%	10%
S19	166	86.00%	20.30%	10%
S18	156	84.47%	21.87%	12%
S28	157	85.29%	22.53%	12%
S42	142	79.88%	24.36%	20%

, which summarizes the key performance metrics for the five most stable solutions. The analysis reveals that these solutions achieve a high level of average performance, with CSR_mean ranging from 79.88% to 86.82%. The top solution (S₂₉) achieves an average CSR of 86.82%, demonstrating exceptional effectiveness across the ensemble of K-field realizations. The CSR_std ranges from 19.40% to 24.36% for these solutions. S₂₉ exhibits the highest stability with the lowest standard deviation of 19.40%, indicating that its performance is the least sensitive to geological variations. This low variability is visually confirmed in Figure 5, which presents a kernel density estimate (KDE) comparison of their satisfaction ratio distributions. The tightly clustered and right-skewed KDE curves for all five solutions provide strong quantitative evidence of their robustness. The high peak and narrow width of the curves confirm that these solutions spend the majority of their time in high-performance states, with minimal excursions into low-performance regions.

Figure 6 illustrates the capture zone morphologies of the most robust solution S₂₉ under four different randomly selected K-fields, illustrating performance variability due to subsurface heterogeneity. Figure 6a shows the capture effectiveness of S₂₉ in its corresponding hydraulic conductivity field K₂₉, where it achieved a satisfaction level of 0.75, demonstrating effective containment of the contaminant plume. In contrast, Figure 6b,d presents S29’s capture performance in randomly selected conductivity fields K₁₅ and K₄₂, respectively, with improved results where the CSR reached 0.86 and 1.00, respectively. However, Figure 6c demonstrates a failure scenario, where S29’s capture zone in K₂₁ is inadequate, with a CSR of only 0.53. These visualizations clearly demonstrate that even the most robust solution identified through the multi-scenario assessment (S₂₉) exhibits significant performance variability across different K-field realizations. While it achieves effective containment (CSR ≥ 0.75) in most scenarios (e.g., K₁₅, K₂₉, K₄₂), its performance can drop below the desired threshold in others (e.g., K₂₁, CSR = 0.53), highlighting that no single solution can guarantee universal effectiveness across all possible subsurface conditions.

However, as shown in Table 1, even the most robust solutions still exhibit a non-negligible percentage of K-field realizations (ranging from 10% to 20%) where performance drops below the desired threshold (CSR < 50%), indicating that perfect universality cannot be guaranteed.

To better understand these failure scenarios, Figure 7 presents the satisfaction ratios of the top five most stable solutions across all K-fields where at least one solution’s performance drops below the CSR of 50%. This comprehensive view reveals that within this set of challenging K-fields, there are five specific K-fields (K₂₂, K₂₃, K₃₀, K₃₅, K₄₃) where the performance degradation is particularly severe. In these five K-fields, all five of the top solutions (S29, S28, S18, S28, S42) simultaneously exhibit satisfaction ratios below 50%, indicating a systemic failure of the hydraulic containment strategy.

4.3. Evaluation of Solution Robustness Under Varying Heterogeneity Levels

The multi-scenario robustness assessment framework presented in Section 4.2 (Component 2) demonstrated its effectiveness in identifying well configurations that are resilient to subsurface uncertainty under a specific level of heterogeneity (σ² = 0.50). To further evaluate the generalizability and adaptability of this approach, Component 3 of the experimental framework (Section 3.2) applies the same multi-scenario evaluation process to scenarios with varying degrees of aquifer heterogeneity.

This component aims to investigate how the level of subsurface heterogeneity influences the performance characteristics of the identified robust solutions and the overall effectiveness of the optimization framework. Three distinct levels of heterogeneity were considered, defined by the variance (σ²) of the log-conductivity field: low (σ² = 0.25), moderate (σ² = 0.50), and high (σ² = 1.00). For each heterogeneity level, a new set of 50 independent K-field realizations was generated using the KLE method with the corresponding variance. This systematic approach allows for a direct comparison of solution performance, robustness metrics, and the characteristics of the identified stable solutions across the spectrum of subsurface uncertainty, as summarized in

Table 2. Performance summary of the most stable solution identified under each heterogeneity level.

Heterogeneity Level	Best Solution	Total Pumping (m3/d)	CSRmean	CSRstd	PR(CSR < 50%)
Low (σ2 = 0.25)	S29	202	94.82%	11.86%	2%
Medium (σ2 = 0.50)	S29	174	86.82%	19.40%	10%
High (σ2 = 1.00)	S40	207	77.06%	21.95%	12%

.

A key finding is that a consistent trend emerges regarding solution robustness and reliability. As heterogeneity increases from low (σ² = 0.25) to high (σ² = 1.00), the CSR_mean of the most stable solution decreases (from 94.82% to 77.06%). Concurrently, the performance stability, measured by CSR_std, deteriorates from 11.86% to 21.95%, and the risk of encountering a failure scenario (CSR < 50%) increases from 2% to 12%. These findings underscore the complex interplay between aquifer heterogeneity, containment effectiveness, and system reliability, highlighting the critical need to explicitly account for the specific characteristics of geological uncertainty during the design phase.

However, the relationship between aquifer heterogeneity and the total pumping rate required for effective containment is not monotonic. The most stable solution under moderate heterogeneity (σ² = 0.50) requires the lowest total pumping rate (174 m³/d), while higher rates are needed under both low (202 m³/d) and high (207 m³/d) heterogeneity levels. This non-monotonic behavior suggests that the optimal operational cost is not simply dictated by the degree of geological variability.

A comparative analysis of the capture zone morphologies across different heterogeneity levels (Figure 8 and Figure 9) more clearly illustrates the impact of heterogeneity intensity. Under low heterogeneity (σ² = 0.25), the capture zones for solution S₂₉ are highly stable and effective in most cases (e.g., K₈, K₁₆, CSR = 0.94, 1.00), with only a few instances showing slightly worse performance (e.g., K₃₇, CSR = 0.47). This is consistent with the extremely low performance standard deviation (11.86%) and only 2% failure rate in Table 2 for this scenario. However, under high heterogeneity (σ² = 1.00), the capture zones for the best-performing solution (S₄₀) exhibit significant instability, showing a stark contrast from good performance in K₁₈ (CSR = 0.94) to clear failure in K₅ (CSR = 0.41). This directly corresponds to the highest performance standard deviation (21.95%) and 12% failure rate for this scenario in Table 2. Overall, as heterogeneity increases from low to high, the capture zone morphology of the robust solution transitions from highly consistent and reliable to significantly variable and prone to failure.

5. Conclusions

(1) This study presents a robust optimization framework for the design of hydraulic containment systems in heterogeneous aquifers, specifically addressing the challenge of contaminant plume migration. The framework integrates an optimization algorithm (pymoo) with a numerical groundwater flow model (MODFLOW) to simultaneously optimize well placement and pumping rates. Its key innovation lies in the explicit consideration and mitigation of subsurface uncertainty through a multi-scenario evaluation approach. Rather than relying on a single deterministic K-field, the method assesses candidate solutions across an ensemble of K-field realizations generated via the Karhunen–Loève Expansion (KLE) method. This allows for the identification of well configurations that are not only optimal under a specific scenario but also robust and reliable across a spectrum of plausible subsurface conditions.

(2) The effectiveness of the proposed framework was demonstrated through a numerical case study. The results confirm that traditional single-scenario optimization is highly sensitive to the specific realization of the K-field used for training, often leading to significant performance degradation when applied to different subsurface conditions. In contrast, the multi-scenario robustness assessment successfully identified a small subset of stable solutions (e.g., S29, S28, S18, S28, S42) that maintained a high CSR_mean (e.g., 79.88% to 86.82%) and low performance variability (e.g., CSR_std: 19.40% to 24.36%) across diverse K-field scenarios. These robust solutions effectively balanced containment performance with operational cost.

(3) Moreover, the application of the framework under varying heterogeneity levels revealed valuable insights, such as the consistent trend of decreasing robustness (lower mean satisfaction, higher variability) with increasing subsurface uncertainty, and the non-monotonic relationship between heterogeneity and required pumping rates. This underscores the framework’s capability to provide nuanced guidance for risk-informed decision-making in groundwater management.

Despite its demonstrated effectiveness, this study has several limitations.

First, the computational cost is significant. The requirement for numerous, independent MODFLOW simulations (2500 in the main assessment) represents a substantial computational burden. This brute-force approach, while effective for demonstrating the framework’s conceptual feasibility, may limit its application to larger domains, finer grids, or more complex, transient, three-dimensional models without access to significant computational resources. To address this critical limitation, future research will prioritize the integration of surrogate modeling techniques. Methods such as Gaussian Process Regression (Kriging), Polynomial Chaos Expansion, or advanced neural networks (e.g., Physics-Informed Neural Networks) could be employed to approximate the expensive MODFLOW simulations. These surrogate models, once trained on a subset of simulations, can dramatically reduce the number of required full physics-based model runs (potentially from thousands to hundreds or tens), thereby making the robust optimization framework computationally feasible for real-world, large-scale applications.

Second, the current framework focuses on a two-dimensional, steady-state flow model and a conservative contaminant. Extending the approach to three-dimensional, transient flow and transport models, including reactive transport, would be necessary for broader applicability to real-world sites with complex hydrogeology and diverse pollutants.

Third, this study assumes that the statistical properties of the K-field (mean, variance, correlation structure) are known or can be accurately estimated, which may not always be the case in practical scenarios with limited data.

Future research will focus on enhancing computational efficiency through surrogate modeling or adaptive sampling, incorporating more complex physical processes (3D, transient, reactive transport), exploring multi-objective optimization frameworks (e.g., balancing cost, time, and robustness), investigating methods for robust design under imprecise probabilistic information or incomplete probabilistic information, and applying the framework to field data or benchmark sites to validate its operational effectiveness.