A Simplified Approach of Pumping Rate Optimization for Production Wells to Mitigate Saltwater Intrusion: A Case Study in Vinh Hung District, Long An Province, Vietnam

Abstract

In the investigation of optimal groundwater extraction in coastal regions, conventional assumptions typically revolve around unconfined aquifers with specified boundary conditions. In such cases, intricate solutions for groundwater management have been documented. However, within extensive delta plains, the extraction wells are frequently drilled in confined aquifers with not much variable-density flow. This circumstance, characterized by paleo-saltwater intrusion, is further complicated by the placement of wells at a considerable distance from the coastal line. As a result, the design and implementation of groundwater supply systems in these areas necessitate strategic groundwater management to optimize groundwater utilization while mitigating the potential risk of saltwater intrusion. Analytical solutions and an optimization problem approach have been applied to address this challenge and solve the differential equations governing confined aquifers with salt–freshwater interfaces. These methodologies provide simplified yet dependable conditions tailored to the study area. A case study conducted in Vinh Hung district, Long An province, is focused on determining the optimal pumping rate for production wells to forestall saltwater intrusion during groundwater extraction. Here, the focus is on the migration of older saltwater towards inland pumped wells, rather than the influence of recent seawater encroachment. The findings contribute valuable insights into achieving an equilibrium between maximizing groundwater utilization and preventing saltwater intrusion in the aquifer systems by a simplified approach.

1. Introduction

Groundwater–surface water interactions are integral to the hydrological cycle and crucial for understanding their dynamic relationship. In natural environments, especially humid climates, surface water bodies like streams, lakes, seas, and oceans often serve as discharge points for groundwater. In cases of enhanced groundwater recharge from rainfall, streams and lakes function as natural drainage systems, with subsurface flow primarily directed toward them. However, water often moves from surface lakes to groundwater systems in arid and semi-arid regions or where groundwater pumping exceeds natural recharge [1]. Thus, the salinity and pollution levels of surface lakes impact groundwater quality. Coastal areas and groundwater systems, major or sole sources of freshwater, frequently experience surface water–groundwater interaction and salinization due to the high salinity of ocean water, leading to horizontal intrusion into aquifers. The rate and extent of saltwater intrusion (SWI) depend on various factors, including hydrological cycle components and water quality and quantity. Natural factors like climate change and sea level rise and human impacts such as urbanization exacerbate SWI. Hence, implementing remedial measures is crucial to minimize SWI-induced water quality deterioration [2].

In coastal plains around the world, unregulated or illegal extraction of groundwater has resulted in various negative consequences, including contamination with arsenic, intrusion of saltwater, and subsidence of land. This is particularly evident in regions such as India [3], Vietnam [4], etc. One of the most effective and economically feasible strategies for maintaining equilibrium in aquifers and addressing issues related to saltwater intrusion is controlling pumping rates [2]. Sherif et al. (2001) [5] demonstrated that redistributing pumping rates by reducing or decommissioning wells in less vulnerable areas can effectively help mitigate saline intrusion in aquifers in the Nile Delta. Similarly, Zhou et al. (2000) [6] investigated the regulation of pumping rates in existing wells within the coastal aquifer of Beihai City, China. Their findings highlighted the effectiveness of eliminating wells near the coast or across the intruded zone while also reducing pumping in wells further from the coast. Adjusting pumping schedules has been advocated as a strategy to mitigate saltwater intrusion in the Balasore coastal groundwater basin in India, particularly during periods of drought [3]. A reduction in overall groundwater withdrawal from the Akrotiri aquifer in Cyprus has been proposed during wet years when surface water adequately meets demand [7]. However, during dry years, the need for aquifer exploitation becomes more urgent. Siaka et al. (2017) [8] have recommended the periodic cessation of most pumping wells in the alluvial aquifer in Katapola, Greece, to mitigate saltwater encroachment.

The optimization of pumping strategies utilizing management models has been extensively researched in the previous literature [9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Various objective functions and constraint sets have been employed depending on the specific problem context. These include maximizing pumping rates [15,23,24], minimizing drawdown [15], minimizing pumped water [25], reducing seawater intrusion into the aquifer [22], and minimizing pumping costs [26]. Some researchers [25,27] have addressed multiple objectives simultaneously [28]. Various optimization methodologies have been applied to address coastal aquifer challenges [23,25,27,29,30].

Earlier investigations have focused on addressing issues related to variable-density flow through the use of coupled variable-density flow and solute transport models, detailed analytical solutions, or a combination of both approaches. While these studies hold significant theoretical value, their practical applicability may be limited, especially in extensive deltaic regions where groundwater extraction primarily occurs from deep confined aquifers near the freshwater–saltwater interface. In such regions, the impact of variable-density flow is minimal and can often be disregarded. Paleo-saltwater occurrences in Quaternary delta systems, resulting from Holocene transgressions, have been documented in numerous deltas and have been reported in various hydrogeological studies of Quaternary delta systems [31]. These occurrences have been observed at distances of up to several hundred kilometers from present coastlines in Quaternary delta systems, with the farthest inland observation being 300 km in Bangladesh [31]. In this context, the distribution of freshwater occurs intermittently alongside saltwater, characterized by low salinity levels. In the expansive deltas, the aquifers designated for concentrated water extraction typically comprise confined aquifers positioned at a significant distance from the coastline. This is particularly evident in two expansive deltas in Vietnam: the Vietnam’s Mekong River Delta [32] and the Red River Delta [33]. The intensive extraction of groundwater from these confined aquifers situated at a considerable distance from the coast has resulted in salinity issues within production wells, as highlighted by [4]. The emphasis here is on the inland migration of older saltwater towards pumped wells rather than the impact of recent seawater encroachment, under reasonable assumptions applicable to large delta plains. In Vietnam’s Mekong River Delta, unregulated groundwater exploitation is prevalent, particularly in households situated in remote areas and regions facing water scarcity. Therefore, the main objective of this study is to use a simplified approach to optimization problems to determine pumping rates that can mitigate salinization issues in production wells. This method was specifically applied to a water-scarce, distinct geographic region—a remote, rural area within Vinh Hung district, situated in the Long An province near the border with Cambodia.

2. Study Area

The Mekong River Delta in Vietnam is the country’s largest delta and is home to a high-density population. The distribution of salt and fresh groundwater in this region is complex and has multiple origins [34]. The demand for water supply is rapidly increasing due to domestic, agricultural, and industrial use. Besides that, climate change and upstream development cause less river discharge and, consequently, saltwater intrusion during the dry season. In certain areas, groundwater is the only source of water supply, but unsustainable abstraction practices have resulted in saltwater intrusion.

The Vietnam’s Mekong River Delta features seven main aquifers strategically interspersed with aquitards and aquicludes, as illustrated in Figure 1a,b.

Long An province is located near the border of Cambodia and is situated far from the shoreline. However, this area also faces the issue of saltwater intrusion during groundwater extraction [4,35]. Therefore, the management of water wells in this site is the objective to avoid saltwater intrusion by the optimizations of the well field’s pumping rate. Cana is a village in Vinh Hung district, Long An province, listed as one of the water scarcity areas under the national program 1553 [34] for urgent response. Geological cross-sections through Cana village’s water work and freshwater–saltwater interface in the Lower Pliocene aquifer (n₂¹) are presented in Figure 1b and Figure 1c, respectively.

At the Pilot Site, typical of aquifers in Vietnam’s Mekong River Delta, seven aquifers are present, most of which are saline at varying depths. The n₂¹ aquifer, at an average depth of 270 m, contains a mix of saltwater and freshwater distribution (Figure 1c) [36]. Presently, the district’s clean water supply station is utilizing groundwater from a pumping well PW1 situated in the aquifer n₂¹, operating at a capacity of 7–10 m³/h. However, saltwater intrusion is anticipated to eventually encroach upon the pumping well. The need for water exploitation is increasing rapidly, not only for domestic use but also for other purposes. An additional pumping well PW2 has been drilled (Figure 1d). Therefore, we focus on evaluating the hydrogeological characteristics and determining the optimal pumping rate for the production wells to ensure no saltwater intrusion and sustainable groundwater exploitation in this area.

3. Study Materials and Methods

3.1. Delineation of Freshwater–Saltwater Interface

By comparing chemical analysis data derived from groundwater samples obtained from boreholes, distinctions between water types, including fresh groundwater and saltwater, were accurately determined. Subsequently, maps were meticulously generated by extrapolating these data from the survey lines, providing precise insights into the aquifer’s composition and hydrological dynamics. Following sequential field surveys, an assessment was undertaken to evaluate the existing status of groundwater exploitation. Additionally, samples were procured to authenticate the delineation of the saltwater–fresh water interface as previously investigated. The borehole depth, field parameter measurements, and sampling protocols strictly adhered to the guidelines set forth by the Vietnam Ministry of Natural Resources and Environment (MONRE), as of the specified year. A total of fifty groundwater samples were meticulously collected from boreholes and dug wells situated within the designated study area. A correlation between Electrical Conductivity (EC) and Total Dissolved Solids (TDS) has been established through rigorous water sample analysis and on-site EC measurements. This methodology facilitated the precise delineation of the fresh and saline water interface within the experimental zone, leveraging data obtained through rigorous EC measurement campaigns consisting of 150 measurement points conducted across the study area.

3.2. Drilling an Additional Production Well and Pumping Test

In the study site, the productive aquifer for exploitation is the extensively distributed, confined aquifer denoted as n₂¹. Groundwater is currently extracted from a single borehole within this aquifer, serving the purpose of supplying water to Cana village in Vinh Hung district, Long An province. The validation and delineation of the freshwater–saltwater interface within this aquifer have been effectively confirmed through EC measurements and chemical sample analyses. These findings indicate that augmenting the groundwater extraction capacity in the existing borehole may lead to the migration of saltwater over time to the borehole. In response to the pressing water scarcity challenges faced by regions with high demand for groundwater extraction, a new borehole has been carefully drilled to provide water for around 3400 people. The borehole is designed to have a robust capacity of at least 350 m³/day, ensuring a reliable and sustainable water supply for the community. Subsequently, a pumping test with a constant pumping rate was conducted to ascertain the hydrogeological parameters of this aquifer. Furthermore, a step drawdown test incorporating four distinct pumping rate levels was undertaken to assess the operational efficacy of the groundwater extraction borehole.

The additional borehole was drilled to a depth of 296 m, strategically positioned at a specific distance from the delineated freshwater–saltwater interface as determined by EC measurements. The drilling location was chosen based on construction conditions and considering uncertainties in the saltwater interface. Given the agricultural area of the region, specifically dedicated to paddling rice fields, the drilling site must adhere to the constraints of being situated along the road and in a public area. Consequently, the village cultural house was selected as a location for the additional pumping well PW2, as depicted in Figure 1c.

Subsequent to the completion of drilling and casing procedures, the borehole underwent a development process, employing an air compressor to achieve groundwater stability, with total suspended solids measuring below 5 mg/L. The determination of the borehole pumping rate for the ensuing pumping test was predicated on the outcomes of this developmental stage. The pumping test itself entailed the use of a submersible pump with a constant pumping rate, spanning a duration of three shifts.

Throughout the pumping test, a data logger positioned at a depth of approximately 30 m continuously recorded the groundwater level, with a measurement frequency set at 1 measurement per minute. At key intervals, specifically at the commencement, midpoint, and conclusion of the pumping test, water samples were extracted for chemical composition analysis, contributing to a comprehensive assessment of the overall water quality. Subsequent to the stop of pumping, the recovery of groundwater levels was monitored until a stable state was achieved.

3.3. Evaluation of Freshwater–Saltwater Interface Dynamics During Exploitation Using Analytical Solution

In this study, we assume a horizontal water table, which technically deviates from Dupuit’s hypothesis [37]. Nonetheless, this assumption is valid in our study area, characterized by a gentle bedrock slope and minimal piezometric gradients. Recent research on Tothian aquifers, such as the work by Huizar-Alvarez et al. (2016) [38], also supports the applicability of horizontal water table conditions in comparable environments.

Pumping wells that are located near the freshwater–saltwater interface can potentially trigger saltwater intrusion into boreholes, thereby affecting the quality of the extracted groundwater. This could lead to TDS levels exceeding the critical thresholds of 1 g/L or 1.5 g/L set by the WHO (1996) for drinking-water-quality guidelines [39] and by MONRE (2023) [40] for groundwater-quality regulations in water-scarce regions. Prior to optimizing the pumping rate for each production well to attain the maximum total pumping rate, while concurrently ensuring that the extracted water maintains a TDS below the critical thresholds, an assessment of the dynamics at the freshwater–saltwater interface during pumping will be conducted using analytical solutions. The assumptions for calculating groundwater drawdown at specific locations and predicting the migration of the initial saltwater particle from the interface between freshwater and saltwater to the production wells are outlined as follows:

-

The piezometric surface is assumed to maintain a horizontal orientation.

-

The aquifer is considered to have a seemingly infinite areal extent.

-

The aquifer is assumed to be homogenous, isotropic, and of uniform thickness.

-

Saltwater, with its low and variable TDS concentration, is treated as homogeneous, disregarding density discrepancies with freshwater.

-

Wells are assumed to fully penetrate the aquifer.

-

Water extraction from storage is assumed to occur instantaneously following a decline in hydraulic head.

-

Pumping wells are strategically positioned near the freshwater–saltwater interface, causing the development of depression cones that extend to the interface. Consequently, saltwater movement toward pumping wells is predominantly driven by the convection process.

-

Aquifer recharge is assumed to be uniform, constant, and isotropic, as a simplification to facilitate large-scale modeling approaches. This assumption has been validated in recent studies of stratified groundwater systems, particularly in regions where sediments are predominantly marine or alluvial in origin (Schiavo, 2023) [41].

-

The superposition of multiple wells is taken into account.

-

The image well method is applied to obtain the solution for aquifers with finite areal extent.

Within a confined aquifer characterized by a saltwater–freshwater interface, a series of pumping wells is identified as PW1, …, PWi, …, PWn, each associated with specific pumping rates denoted as Q₁, … Q_i, …, Q_n (see Figure 2).

The calculation of groundwater drawdown at any given position M adheres to the predefined formula [42]:

$$s_M=\sum _{i=1}^n{\displaystyle \frac{Q_i}{4\pi KD}}{\displaystyle \frac{ln2.25\frac{KD}{S}t}{r_{i–M}^2}}$$

(1)

where

sM	Groundwater drawdown at position M;
Qi	Pumping rate of the ith pumping well;
K	Hydraulic conductivity of the aquifer;
D	Thickness of the aquifer;
S	Storativity coefficient;
ri–M	Distance from the ith pumping well to the position M;
t	Groundwater extraction time;
n	The number of pumping wells for groundwater extraction.

Utilizing Formula (1) provided earlier, one can compute the groundwater drawdown at a pumping well (s_oi) as well as at the nearest freshwater–saltwater interface point to the pumping well (s_Mi).

The hydraulic gradient between the nearest saltwater interface’s point and the pumping well is as follows:

$$I_i={\displaystyle \frac{s_{oi}-s_{Mi} }{Mi}}$$

(2)

where

Mi-i	The distance from the nearest freshwater–saltwater interface point to the pumping well PWi;
Ii	The hydraulic gradient between the nearest saline interface’s point and the pumping well;
tM	The time duration during which the initial saltwater particle migrates from the freshwater–saltwater interface to the pumping well, as outlined below:

$$t_M={\displaystyle \frac{Mi-i}{v_e}}$$

(3)

where

Mi-i	The distance from the nearest freshwater–saltwater interface’s point to the pumping well PWi;
ve	Real average velocity from the nearest freshwater–saltwater interface’s point moving to the pumping well can be determined as ve = (K × Ii)/ne;
ne	The effective porosity of the aquifer.

In case the aquifer has boundary conditions of type 1 (specified head), type 2 (specified discharge), or a combination of both, the methods of image and superposition are used.

3.4. Optimization of Pumping Rates to Mitigate Saltwater Intrusion

The heterogeneity of the aquifer and the potential presence of preferential flow pathways are critical factors in understanding the dynamics of saltwater intrusion. In the current model, we assume a relatively homogeneous porous medium to simplify the simulation process, but we agree that more attention should be given to potential heterogeneity, especially near river areas or regions with higher permeability. In this study, the heterogeneity of the porous medium could be influenced by factors such as aquifer permeability and the uneven distribution of sedimentary materials. Recent studies (Schiavo, 2023) [41] show that entropy and fractality in groundwater pathways can influence saltwater intrusion. Future work will explore these dynamics in more detail using higher-resolution models to more accurately simulate the distribution and movement of groundwater.

Let Q_i; i = 1, 2, …, n denote the pumping rates of wells in the confined aquifer near the freshwater–saltwater interface. The primary concern lies in optimizing the total discharge from all pumping wells while ensuring that the initial saltwater particle has not yet reached the vicinity of any pumping well. Thus, the optimization problem is expressed by the following formulas:

$$\mathrm{Objective} \mathrm{function}: Q_{tot}={\sum }_{i-1}^n{Q}_i$$

(4)

$$\mathrm{Constraint} \mathrm{conditions}: { t}_{Mi }>t=\mathrm{10,000} \mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$$

(5)

$${\sum }_{i=1}^n{Q}_i\le Q_{tot max }$$

(6)

$$Q_{i }\le Q_{i, max, \mathrm{i}=1,\dots \mathrm{n} }$$

(7)

$$Q_{i }\ge 0_{, \mathrm{i}=1,\dots \mathrm{n} }$$

(8)

where

Qi	i = 1, 2, …, n: Pumping rate of pumping wells;
Qtot	Total discharge of all pumping wells.

As per Vietnamese regulations, it is imperative to note that the designated limit for a pumping well, ensuring both quantity and quality of water remain sustainable, is set at t = 10,000 days [43].

To address the aforementioned issue, the SOLVER tool, “an add-in” for Microsoft 365 Excel 2023 is employed [44]. This tool facilitates what-if analysis by determining optimal values (maximum or minimum) for a specified objective cell. It operates within defined constraints on other formula cells, known as decision variables. These variables influence the computation of formulas in both the objective and constraint cells. In essence, SOLVER adjusts the decision variable values to meet constraint limits and achieve the desired outcome for the objective cell. In practical terms, you can employ SOLVER to ascertain the maximum or minimum value of a cell by manipulating other cells, such as adjusting your projected advertising budget to observe its impact on projected profit. In this tool, the method of solution is contingent upon the nature of the data and the choices made. The Generalized Reduced Gradient (GRG) nonlinear algorithm is employed for addressing smooth nonlinear problems, while the LP simplex method is utilized for linear problems. Additionally, the Evolutionary approach is employed to tackle non-smooth problems. The Evolutionary method will be able to find a good solution to a reasonably well-scaled model. Because the Evolutionary method does not rely on derivative or gradient information, it cannot determine whether a given solution is optimal, so it never really knows when to stop. Under this heuristic stopping rule, the Evolutionary Solver will continue searching for better solutions as long as it is making a reasonable amount of progress; if it is unable to make sufficient progress in a specified time, it will stop and report the best solution found.

3.5. Validation of Pumping Rates’ Optimization Outcome Using Variable-Density Flow Model

Within the framework of this study, the follow-up is to build a numerical model for flow and saltwater transport with the above-mentioned conditions for the study area. The model was comprehensively delineated, incorporating input data encompassing hydrogeological parameters, fluid density, and boundary conditions. Scenarios were developed to predict for the transport of the freshwater–saltwater interface as the sole convection process and convection combined with dispersion. The prediction results according to the scenarios with the exploitation pumping rates as determined by the optimal problem demonstrated the transport of saltwater intrusion from the freshwater–saltwater interface to the groundwater pumping wells and evaluated the reliability of this optimization problem’s reliability.

SEAWAT was developed to simulate a three-dimensional variable-density flow and solute transport in porous aquifers [45]. This model integrates with MT3DMS and MODFLOW, adding a stepwise conversion of water density for each step period. The theoretical basis of the model is rooted in the differential equation of groundwater movement with variable density. In this case, we established a three-dimensional groundwater model at the study site utilizing a 14-layer system, integrating a variable-density flow in accordance with the SEAWAT code. The governing differential equation for variable-density flow is described as follows:

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial x}}\left(\rho K_{fx}\left({\displaystyle \frac{\partial h_f}{\partial \mathrm{x}}}+{\displaystyle \frac{\rho -{\rho }_f}{{\rho }_f}}{\displaystyle \frac{\partial z}{\partial x}}\right)\right)+{\displaystyle \frac{\partial }{\partial y}}\left(\rho K_{fy}\left({\displaystyle \frac{\partial h_f}{\partial y}}+{\displaystyle \frac{\rho -{\rho }_f}{{\rho }_f}}{\displaystyle \frac{\partial z}{\partial y}}\right)\right)\\ +{\displaystyle \frac{\partial }{\partial z}}\left(\rho K_{fz}\left({\displaystyle \frac{\partial h_f}{\partial z}}+{\displaystyle \frac{\rho -{\rho }_f}{{\rho }_f}}\right)\right)=\rho S_f{\displaystyle \frac{\partial h_f}{\partial t}}+\theta {\displaystyle \frac{\partial \rho }{\partial C}}{\displaystyle \frac{\partial C}{\partial t}}\end{array}$$

(9)

where h_f equivalent freshwater level (L); ρ_f freshwater density (ML⁻³); ρ saltwater density (ML⁻³); g gravity acceleration (LT⁻²); C solute concentration (ML⁻³).

The advection–dispersion solute transport of ions and soluble substance in aquifers from a high concentration to a low one. The one-dimensional hydrodynamic differential equation is described as follows:

$$D_L={\displaystyle \frac{{\partial }^2\mathrm{C}}{{\partial \mathrm{x}}^2}}-V_x{\displaystyle \frac{\partial C}{\partial x}}={\displaystyle \frac{\partial C}{\partial t}}$$

(10)

where

DL	Longitudinal hydrodynamic dispersion coefficient;
Vx	The average velocity of flow in the x direction;
t	Time from the start of solute transport.

In order to simulate the groundwater system, the model’s differential grid is selected to be 200 m × 200 m. Boundary conditions and hydrogeological parameters of aquifers were based on the results of the collection and investigation in this study.

4. Results

4.1. Freshwater–Saline Groundwater Distribution

The outcomes of on-site EC measurements, water sampling, and subsequent laboratory analyses were undertaken from October 2022 to January 2023. These findings are presented in Table S1, which can be referenced in Supplementary Materials S1. Utilizing the correlation between EC and TDS in groundwater, we derive the TDS of groundwater based on the EC measurement results.

The outcomes of establishing a linear regression between groundwater EC and TDS are depicted in Figure 3, revealing a notably strong correlation with a coefficient of determination (R²) of 0.99. The correlation equation derived from the EC and TDS measurement data is as follows:

TDS (mg/L) = 0.66EC (μS/cm) + 1.07

(11)

The correction of the freshwater–saltwater interface of the aquifer n₂¹ in the study area is depicted in Figure 1c, based on the outcomes of 150 EC measurement points.

4.2. Drilling Additional Production Well and Pumping Test

An additional pumping well PW2 was drilled and installed with casings, screens, gravel packs, and other essential components in accordance with the structural specifications detailed in Figure 4.

The preliminary results from borehole development indicate that the experimental pumping rate remains stable, reaching Q = 6 L/s. Pumping tests were conducted utilizing a submersible pump over a duration of three shifts, with the final phase demonstrating stability. To analyze hydrogeological parameters based on pumping test and step drawdown test data, AQUIFER TEST 11.0 software by Waterloo Hydrogeologic (2021) [46] was employed. Assumptions for the interpretation of the pumping test included an infinite confined aquifer, among others. Therefore, methods such as Theis, Jacov, and water level recovery were applied for interpretation, as outlined in Kruseman and de Ridder (1990) [42]. The hydrogeological parameter interpretation for pumping well PW2 is detailed in

Table 1. The outcomes of the pumping test conducted on borehole PW2 within the Lower Pliocene aquifer (n₂¹).

Aquifer Thickness, m	Pumping Rate, L/s	Static Groundwater Level, m	Groundwater Drawdown, m	Hydraulic Conductivity K, m/d	Transmissivity KD, m2/d
35	2.54	10.18	2.58	9.36	327.6

below, and with Supplementary Figure S1 in Supplementary Materials S2.

The outcomes of the pumping test conducted to assess the performance of the pumping well PW2 are presented in

Table 2. The outcomes of the step drawdown test conducted to assess the performance of the pumping well PW2.

Step Drawdown Test	Step 1	Step 2	Step 3	Step 4
Drawdown, m	0.29	1.60	2.52	4.67
Pumping rate, L/s	0.37	1.00	2.46	3.36
Efficiency, %	65.52	65.57	65.71	65.79

below:

The formula for calculating the efficiency of exploitation boreholes [42] based on the step drawdown test data is provided in Figure S2 of Supplementary Materials S2, as follows:

V = (33.9Q)/(33.9Q − 0.0469Q²)

(12)

where

V	Borehole efficiency, %;
Q	Pumping rate, L/s.

A borehole efficiency exceeding 70% is considered as satisfactory, while 65% serves as the threshold for acceptable borehole performance [47].

Hydrogeological parameters derived from the current pumping test results, along with data from previous studies, are consolidated and presented in

Table 3. The typical hydrogeological parameters of aquifer n₂¹ within the Pilot Site.

Parameters	Unit	Value	Reference Sources
Aquifer thickness, D	m	35.0	Based on the pumping well PW2’s lithology description (refer to Figure 4) in conjunction with previously investigated boreholes (outlined in Supplementary Materials S3).
Static groundwater level, Hst	m	10.18	The existing static groundwater levels were observed at two production boreholes during the cessation of pumping until complete recovery.
Hydraulic conductivity, K	m/d	9.36	The interpretation of pumping test data acquired from the additional borehole (see Supplementary Material S2) and the analysis of pumping test results obtained from previously investigated boreholes within the study area (refer to Supplementary Materials S4).
Storativity (S)	[-]	0.0026	The analysis of pumping test results obtained from the average storativity coefficient of previously investigated boreholes within the study area (refer to Table 5 and Supplementary Materials S4).
Effective porosity (ne)	[-]	0.3	According to Pham et al. (2019) [34].
Diffusivity; DK/S	m2/d	148,909	The diffusivity coefficient obtained from the above-mentioned KD and S.

.

4.3. Optimal Pumping Rate for Two Pumping Wells

The hydrogeological parameters, as outlined in Table 1, function as fundamental inputs for analytical solutions. Furthermore, the following specific data are considered for precision in calculating the optimal pumping discharge for pumping wells within the Cana village, Vinh Hung district, Long An province study area.

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Distance from the pumping well PW1 to the nearest freshwater–saltwater interface M1: 385 m;

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Distance from the pumping well PW2 to the nearest freshwater–saltwater interface M2: 570 m;

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Distance between the pumping well PW1 to the pumping well PW2: 930 m;

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Distance from the pumping well PW1 to the freshwater–saltwater interface M2: 1270 m;

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Distance from the pumping well PW2 to the freshwater–saltwater interface M1: 1130 m.

As a result of using the SOLVER tool in EXCEL to solve the optimization problem, we have determined that Q₁ of PW1 is 109.13 m³/day; Q₂ of PW2 is 245.27 m³/day, and the maximum total discharge without causing saltwater intrusion is Q_tot = 354.40 m³/day. Figure 5 illustrates the interface of the SOLVER tool, presenting both the input data for the optimization problem and its corresponding results. Further elaboration on data construction and the process of solving the optimization problem using the SOLVER tool is provided in Supplementary Materials S3.

Figure 5 also shows that changing the location of the pumping wells can also be a solution to meet a demand for the total discharge of pumping wells for water supply.

4.4. Verifying Pumping Rates Optimization Outcome Using a Variable-Density Flow Model

The Localized Model utilized in the study area was derived from the comprehensive model developed for the entirety of the Southern River Delta Plain. This model was meticulously crafted by the National Center for Water Resources Planning and Investigation (NAWAPI) to facilitate the accurate forecasting of groundwater resources across the delta region [48]. The Localized Model domain spans approximately 250 km². Grid cell dimensions are systematically reduced across multiple tiers, from 2000 m to 100 m, ensuring that the finer grid step remains at least half the size of the coarser step. This meticulous meshing strategy mitigates potential calculation errors inherent in solving groundwater flow problems using the MODFLOW model. By adhering to such rigorous meshing, the model faithfully captures the study area’s boundary conditions and natural hydrogeological characteristics, thus circumventing reliance on assumed boundary conditions. The delineation of the detailed model’s area (referred to as the Localized Model) from the Regional Model, along with the grid step progression, is visually depicted in Figure 6.

The model incorporates layers derived from the Vietnam’s Mekong River Delta, South Vietnam model (the Regional Model), encompassing a total of 14 layers structured as following

Table 4. Overview of the layers incorporated within the groundwater models.

No	Layer	Lithology	Layer’s Type and Symbols
1	Layer 1	Clay silt.	Aquitard 1
2	Layer 2	Holocene sediment porous aquifer with fine sand composition.	Aquifer qh
3	Layer 3	Silty clay.	Aquitard 2
4	Layer 4	Upper Pleistocene sediment porous aquifer consisting of sand, pebbles, gravel, silt, and kaolin clay.	Aquifer qp3
5	Layer 5	Silty clay.	Aquitard 3
6	Layer 6	Middle-upper Pleistocene sediment porous aquifer composed of pebbles, gravel, sand, silt, clay silt, and clay.	Aquifer qp2–3
7	Layer 7	Silty clay.	Aquitard 4
8	Layer 8	Lower Pleistocene sediment porous aquifer containing clay, clay silt, sand, pebbles, and loose gravel.	Aquifer qp1
9	Layer 9	Silty clay.	Aquitard 5
10	Layer 10	Middle Pliocene sediment porous aquifer with fine sand mixed with grit, silty clay sand, silt, and silty clay.	Aquifer n22
11	Layer 11	Silty clay	Aquitard 6
12	Layer 12	Lower Pliocene sediment porous aquifer composed of gravelly sand, clayey silt sand, and cohesive silt.	Aquifer n21
13	Layer 13	Silty clay.	Aquitard 7
14	Layer 14	Upper Miocene sediment porous aquifer comprising gravel, mixed sand, and fine sand.	Aquifer n13

.

The delineation of primary hydrogeological parameters integrated into the Localized Model is depicted in Figures S1–S3 of Supplementary Materials S5. The variable-density flow and solute transport parameters within the Pilot Site area are determined based on findings from previous research conducted in the region or sourced from the professional literature (

Table 5. The water’s variable density and solute transport parameters applied to aquifer n₂¹ (Layer 12) within the model.

Effective Porosity	Diffusion Coefficient, D *	Longitudinal Hydrodynamic Dispersion, DL	Long and Horizontal Dispersion Ratio	Density, g/L
				Fresh Water with TDS ≤ 1 g/L	Saline Water with 5 > TDS > 1 g/L
0.3	2 × 10−9	100 (with grid size 2000 m)	DH = 0.1DL and DV = 0.01DL	1000	1001.5 *

* This value was determined through an analysis of the correlation between assumption salt concentration and densities. Specifically, densities of water TDS 1 g/L and 35 g/L correspond to 1000 g/L and 1025 g/L, respectively.

). The effective porosity values are derived from data in their study [34]. Molecular diffusion coefficients are obtained from detailed experimental findings [34], while the longitudinal and horizontal dispersion ratio is established based on empirical insights from Fetter (2017) [49]. Additionally, the density of fresh water is assumed to be 1 kg/L, while the density of brackish water with TDS ranging from 1 to 3 g/L is extrapolated from the density of seawater (1.025 kg/L). Boundary and initial conditions for the SEAWAT model are depicted in Figure S5.

The model is divided into 14 vertical layers to represent the subsurface structure, alternating between aquitard and aquifer units. Layers 1, 3, 5, 7, 9, 11, and 13 correspond to aquitard layers, which are illustrated in Dark Goldenrod. In contrast, layers 2, 4, 6, 8, 10, 12, and 14 represent aquifer layers, depicted in Byzantium. (Figure 7). The hydrogeological parameters extracted from preceding investigations [50,51,52], relevant to both the Regional Model and the Localized Model, are comprehensively outlined in

Table 6. Overview of the hydrogeological parameters characterizing the layers incorporated within the Localized Model.

Layer	Elevation, m amsl		Horizontal Hydraulic Conductivity, m/day (from–to)		Storativity [-] (from–to)
Layer 1	2.00	−5.50	1.38 × 10−1	1.91 × 100	5.00 × 10−3	1.49 × 10−1
Layer 2	−2.92	−5.60	1.92 × 100	4.23 × 100	1.00 × 10−6	2.00 × 10−1
Layer 3	−3.02	−20.20	1.00 × 10−4	2.00 × 100	2.00 × 10−8	3.10 × 10−3
Layer 4	−5.73	−61.76	2.32 × 101	3.42 × 101	5.00 × 10−3	2.50 × 10−4
Layer 5	−22.86	−69.33	1.00 × 10−4	2.00 × 100	2.00 × 10−8	3.50 × 10−5
Layer 6	−32.73	−110.40	1.78 × 10−1	2.62 × 101	5.00 × 10−3	2.80 × 10−4
Layer 7	−64.39	−122.97	1.00 × 10−4	2.00 × 100	2.00 × 10−8	3.50 × 10−5
Layer 8	−79.25	−164.22	5.47 × 100	1.07 × 101	5.00 × 10−3	2.80 × 10−4
Layer 9	−121.60	−170.66	1.00 × 10−4	2.00 × 100	2.00 × 10−8	3.50 × 10−5
Layer 10	−140.46	−232.65	5.39 × 100	2.92 × 101	5.00 × 10−3	2.80 × 10−4
Layer 11	−175.74	−242.50	1.00 × 10−4	2.00 × 100	2.00 × 10−8	3.50 × 10−5
Layer 12	−186.72	−279.51	7.18 × 100	2.56 × 101	5.00 × 10−3	2.80 × 10−4
Layer 13	−248.56	−282.36	1.00 × 10−4	2.00 × 100	2.00 × 10−8	3.50 × 10−5
Layer 14	−266.07	−331.13	2.10 × 10−2	1.66 × 100	5.00 × 10−5	2.80 × 10−4

and Supplementary Materials S4.

The groundwater recharge is allocated across the entirety of the model area, representing 1–7% of the total precipitation, while the evaporation parameter is set at 5–10% of the overall evaporation figure. These values necessitate adjustment with each model iteration. Rainfall and evaporation data are changed over a specified period, which is obtained from the Long An meteorological station. Presently, groundwater abstraction predominantly occurs through dug wells within the Holocene qh aquifer, supplemented by limited private exploitation boreholes with minor pumping rates in the qp₃ and qp_2–3 aquifers. The primary aquifer subject to intensive water extraction is the n₂¹ aquifer, where a single borehole, PW1, yields approximately 250 m³/day to serve the local people in the commune.

The calibration phase of the model spans from 2008 to 2022, while the predictive phase for groundwater fluctuations and saltwater solute transport to exploitation wells extends over 30 years, from 2022 to 2052. Throughout the predictive period, the exploitation boreholes consist of the existing PW1 borehole and the newly introduced PW2 boreholes within the n₂¹ aquifer. Pumping rates for prediction are aligned precisely with those derived from the optimization problem for exploitation boreholes. The model operates on a time step of t = 30 days.

Within the Localized Model domain, four monitoring wells situated in aquifers n₂¹ and n₁³ are integrated into the national monitoring network [53], was utilized for model calibration purposes (see in Figure S4 of Supplementary Materials S5). The calibration outcomes of both the flow models are depicted in Figure 8.

The model’s error assessment demonstrates an average error of 0.197 m, a mean absolute error of 0.861 m, and a normalized root mean square (NRMS) error of 5.9%. A significant proportion of data points are within the 95% confidence interval. Relative values consistently adhere to acceptable thresholds, guaranteeing compliance with prescribed error limits. Consequently, the evaluation confirms that all parameters meet the predetermined reliability standards. The analysis of model errors underscores that all parameters fulfill the model’s reliability criteria.

We employed the calibrated model to predict groundwater level fluctuations and saline water solute transport resulting from the optimized pumping scenario. Subsequently, we assessed the reliability of a simplified approach for optimizing production well pumping rates to mitigate saltwater intrusion. The pumping rates for the two boreholes align with the optimization problem’s outcomes: Q₁ = 109.13 m³/day for PW1 and Q₂ = 245.27 m³/day for PW2. The prediction phase spans 10,000 days, matching the optimal calculation duration precisely. Figure 9 displays the model’s predictions of groundwater level fluctuations and saline water transport at the freshwater–saltwater interface resulting from pumping.

The assessment of saltwater intrusion over 10,000 days (equivalent to 27.3 years) indicates intrusion near borehole PW1, primarily due to its closer proximity to the freshwater–saltwater interface compared to borehole PW2. However, it is worth noting that the initial saline particle element has not yet reached PW1 well.

The assessment of saltwater intrusion reveals its horizontal movement towards boreholes PW1 and PW2. However, the solute transport process progresses slowly. Prediction results over 10,000 days demonstrate that the initial saline particle from the interface disperses near PW1 but remains distant from the PW2 borehole. This finding aligns precisely with the outcomes derived from the optimization problems, where the constraints allow for the arrival of the first saline particle at the production wells after 10,000 days of exploitation.

5. Discussions

5.1. The Assumption for the Application of Optimization Problems

The assumption regarding density remains consistent with previous research problems. In the optimization problem, a uniform liquid density is considered, eliminating the necessity to incorporate this parameter into the analytical formulas for computation. This contrasts with other analytical scenarios, particularly in coastal areas, where liquid density varies. In such cases, seawater is typically assigned a density of 1.025 kg/L, corresponding to a concentration of C = 35 g/L, while freshwater is assigned a density of 1.000 kg/L, with a concentration of C = 1 g/L [54]. Our rationale for this assumption stems from the nature of groundwater exploitation in expansive plains within confined aquifers, situated far from coastal regions where salinity intrusion levels are minimal. Moreover, residual effects of paleo-saltwater intrusion have led to a substantial reduction in salt concentration (C = 1–5 g/L), consequently resulting in a significant decrease in liquid density [31].

The assumption concerning the horizontal static groundwater level within a confined aquifer is deemed acceptable under certain circumstances. Specifically, in expansive plains characterized by flat topographic surfaces, it is reasonable to consider the horizontal static groundwater level. This condition is frequently adopted in analytical problems, particularly when formulating calculation formulas for assessing groundwater interference during pumping tests [42]. While calculation errors may arise, they are typically negligible and can be disregarded.

In relation to the recharge sources within the study area, it is crucial to consider various factors. Recharge for aquifers can stem from surface water, leakage sources, or external flow into the study area. While leakage sources or external inflows are feasible for deep aquifers, the presence of small hydraulic conductivity in the aquitards between aquifers in expansive plains restricts the volume of leakage water. Consequently, leakage sources are either minimal or nonexistent. Additionally, the flow from outside the study area is limited due to the relatively horizontal groundwater level. This aspect is consistently integrated into analytical formulas for calculating pumping tests by numerous authors.

The utilization of averaging parameters in analytical solutions may introduce errors during calculations. Parameters such as hydraulic conductivity, aquifer thickness, storativity coefficient, and those involved in solute transport are all subject to averaging when determining groundwater level drawdown and the migration of saline water from the freshwater–saltwater interface. Nevertheless, in the context of extensive plains within a localized study area, sediment formation conditions often exhibit homogeneity, thus justifying the averaging of these values. Furthermore, various authors, including Mantoglou (2003) [19] and Kruseman and de Ridder (1990) [42], acknowledge and accept this potential error when employing analytical solutions for calculations. The sensitivity analysis of the parameters input into the optimization model reveals that the hydraulic conductivity coefficient has minimal influence on the timing of the initial saline intrusion into the borehole (see Table S1 of Supplementary Materials S6). This is likely applicable in areas with a depression cone near the pumping wells and in scenarios where the saline water–freshwater boundary is close to the borehole. In contrast, the effective porosity of the aquifer has a significant impact on the calculated intrusion time (see Table S2 of Supplementary Materials S6). As this study did not include an independent experiment to determine effective porosity, relying instead on reference documents, a more comprehensive evaluation will be conducted in the near future. This parameter is essential in assessing saline intrusion and should be a key consideration in future investigations. Additionally, the duration of groundwater exploitation is another factor that requires attention (see Table S2 of Supplementary Materials S6). In practice, groundwater recharge to the exploitation boreholes increases as the radius of influence expands. While the occurrence of the first saline particle alters the TDS, it does not immediately affect the water quality standards of the extracted water.

5.2. A Comparison Is Made Between the Groundwater Model’s Results and Those Generated by the Optimization Problem

The regional groundwater model developed for the Vietnam’s Mekong River Delta integrates input data from comprehensive studies conducted by Minderhoud et al. (2017) [32] and Pham et al. (2019) [34]. Furthermore, it incorporates the findings of recent investigations conducted by NAWAPI [48]. Distinct from the Regional Model, a Localized Model is developed to simulate boundary and hydrogeological conditions accurately, minimizing man-made influences [55]. Detailed grid steps and hydrogeological parameterization enhance the model’s realism.

Upon comparing the model outcomes with those derived from the optimization problem, it was observed that, after 10,000 days, the freshwater–saltwater interface extended to production borehole PW1 when the predicted pumping rate matched the rate calculated by the optimization problem. Notably, a discrepancy arose between the optimization problem’s results and those of the groundwater model: while the optimization problem stipulated that upon reaching the 10,000-day mark, the first saline particle on the freshwater–saltwater interface would reach both production boreholes, the groundwater model predicted saline intrusion only at borehole PW1, with PW2 remaining unaffected. This variance can be attributed to the following factors:

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The variance between the detailed representation of parameters in the predicted groundwater model and their averaged values in the optimization problem likely constitutes the primary underlying cause of this disparity.

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In the optimization problem, the assessment of the first saline water particle’s migration to production boreholes disregarded the molecular diffusion process and simplified the advection process by assuming direct movement along the shortest path from the freshwater–saline water interface. Conversely, the model’s depiction of flow paths offers a more realistic representation. Consequently, the duration for the first saline water particle to reach borehole PW2 from the fresh saltwater interface exceeds 10,000 days, aligning appropriately with real-world conditions.

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It is noteworthy that the optimal pumping rate for the additional borehole PW2 was determined to be Q₂ = 245.27 m³/day or 2.84 L/s. Upon substitution into Formula (12), the efficiency of this borehole is calculated to be V = 65.74%, meeting the requisite performance standards for a production borehole. However, conventional air compressor pumping methods often prove ineffective in the deep aquifer n₂¹ (from 238–288 m). Consequently, a change in approach is warranted for subsequent well-development endeavors. Additionally, when the groundwater level in the borehole experiences a significant drop, the borehole efficiency falls below 65%. Nevertheless, it is crucial to note that the performance of the production borehole solely influences the groundwater level within the casing and does not impact the aquifer’s groundwater level, thus exerting no influence on saline water migration in the optimization problem.

The simplification applied to the optimization problem does not significantly alter the optimal calculation outcomes. Moreover, employing this simplified approach typically enhances safety in practical applications. Consequently, there is potential for broader utilization of this optimal calculation method in similar cases.

6. Conclusions

Study results on a simplified approach of pumping rate optimization for production wells to mitigate saltwater intrusion for groundwater extraction in confined aquifers located close to the freshwater–saline water interface can draw some conclusions as follows:

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Given the conditions mentioned above, it is acceptable to use a simplified approach that minimizes the consideration of the disparity in variable-density flow between saltwater and freshwater while disregarding external natural recharge in order to address the optimization problem effectively.

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Utilizing the SOLVER tool within EXCEL facilitates the resolution of optimization problems pertaining to the management of exploitation wells in proximity to the freshwater–saltwater interface. This enables the minimization of saltwater intrusion into exploitation wells in a manner that is both efficient and effective.

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A simplified approach may suffice in extensive delta plains, where groundwater extraction occurs within confined, relatively uniform aquifers characterized by intricate distributions of freshwater and saltwater away from the coastline. However, addressing challenges associated with variable-density flow and heterogeneous hydrogeological parameters of the coastal aquifer necessitates employing more sophisticated methodologies.

In the context of assessing the dynamics of the freshwater–saltwater interface during exploitation, as detailed in Section 3.3, this study utilizes an analytical solution methodology, aiming to clarify the underlying assumptions inherent in its application. Furthermore, the ensuing discussion in Section 5 underscores the recognition of the study’s limitations, primarily contingent upon the extent of the fulfillment of these assumptions. Therefore, it is necessary to adopt more sophisticated methodologies if these prerequisites are not met. Particularly in cases characterized by significant groundwater recharge, the development of analytical solutions is necessary before using the SOLVER tool.