https://orcid.org/0009-0008-1749-1164
Figures
Abstract
Groundwater sustains global agriculture but faces significant pressure from overexploitation, threatening long-term water security. Achieving a balance between agricultural productivity and sustainable groundwater use requires decision-support tools that can integrate hydrologic and economic information and be adapted to different farm and aquifer conditions. This study develops an accessible farm-level hydro-economic model that links groundwater dynamics with economic outcomes to evaluate irrigation strategies under regulatory and physical constraints. The model estimates land value over time while incorporating uncertain precipitation, irrigation practices, and regulatory limits. This research presents a novel application of Conditional Value-at-Risk to assess economic risk of groundwater irrigation by focusing on the tail of the probability curve, emphasizing potential extreme adverse outcomes rather than average performance. Applied to a representative High Plains Aquifer site, the model shows that more pumping does not guarantee greater profitability, as diminishing returns and aquifer depletion can undermine long-term benefits. Instead, irrigation strategies aligned with site-specific aquifer properties and regulatory thresholds improve both economic performance and sustainability. This scalable approach provides a useful framework to inform irrigation policy, support farmer decision-making, and promote sustainable groundwater under growing uncertainty.
Citation: Tian B, Brookfield A, Insley M (2026) Development of a coupled hydro-economic model to support groundwater irrigation decisions. PLOS Water 5(4): e0000452. https://doi.org/10.1371/journal.pwat.0000452
Editor: Lingzhong Kong, Yangzhou University, CHINA
Received: October 1, 2025; Accepted: March 4, 2026; Published: April 3, 2026
Copyright: © 2026 Tian et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All data are included in the paper.
Funding: This work was supported by a University of Waterloo Water Institute Seed Grant; University of Waterloo Trailblazer Award; and the Natural Sciences and Engineering Research Council of Canada [Discovery Grant]. All funding was awarded to Andrea Brookfield (PI) and Margaret Insley (co-PI).
Competing interests: The authors have declared that no competing interests exist.
Introduction
Groundwater supports ecosystems and human health and plays an important role in agricultural water supply. The increasing global food demand driven by population and income growth poses a significant challenge to agricultural water usage [1–3]. The use of groundwater for irrigation is increasing both in absolute terms and as a percentage of total irrigation, leading to groundwater depletion in regions where groundwater extraction is greater than groundwater recharge [4,5]. In many areas of the world, groundwater depletion has reached critical levels, forcing reductions in irrigation and subsequently crop growth [6,7]. The effects of groundwater depletion can lead to significant economic losses for the agricultural industry, reduced environmental sustainability, and increased competition between water users [8,9]. These consequences can result in severe conflicts over water resources [10,11].
The integration of hydrologic models and economic methods has been applied widely in agriculture, water management, and policy design, e.g., [12]; [13]; [14]; [15]; [16]; and [17]; and the effect of risk and uncertainty on optimal groundwater use and regulation has also been a focus in the literature: [18]; [19]; and [20]. However, none of these approaches fully account for tail risk by incorporating the less probable yet potentially most economically severe events.
This paper contributes to the literature by developing a novel coupled hydro-economic model that assesses the economic implications and tail risk of groundwater pumping decisions at the farm level. Under stochastic precipitation and regulatory constraints, the model estimates groundwater availability and expected land values while employing Conditional Value-at-Risk (CVaR) to assess potential risks.
CVaR is a widely used risk assessment tool in finance that accounts for the entire distribution of potential losses, emphasizing tail risk [21,22]. Compared to traditional risk measures like simple standard deviation or Value-at-Risk (VaR), CVaR provides a more robust evaluation of uncertainties, making it a preferred approach in decision-making under risk [21–24]. Beyond finance, CVaR has been applied in diverse fields, including inventory management [25,26], supply chain management [27,28], transportation [29,30], energy [31,32], and medicine [24,33]. However, its application in agriculture, particularly in groundwater management, remains largely unexplored. Given the increasing uncertainty in water availability due to climate change and competing demands, integrating CVaR into agricultural water management presents a valuable opportunity to enhance risk-informed decision-making. Furthermore, CVaR’s effectiveness in optimization problems makes it a promising tool for future extensions of this model, enabling more adaptive and resilient strategies for sustainable groundwater use in agriculture.
The primary objectives of this study are (1) to build an accessible hydro-economic model focusing on groundwater availability and expected land values at a farm level; (2) to introduce CVaR analysis to quantify economic risk associated with variability in precipitation and crop price; and (3) to evaluate how different irrigation decisions affect farm value, using Monte Carlo simulations to identify optimal strategies. This is a novel application of CVaR in agricultural groundwater use, providing a new approach for assessing risk in groundwater-dependent farming. The outcomes can provide valuable insights into the sustainable use of groundwater resources in agriculture and contribute to the formulation of effective strategies for future agricultural water management.
Methods
This section presents the equations for the precipitation model, groundwater model, crop yield model, economic model, and risk assessment model, with the overall workflow illustrated in Fig 1. To simplify, we use two conceptual hydrologic models (Fig 2) representing common aquifer conceptualizations, confined and unconfined single-layer aquifers. All model components are applicable to both conditions, with differences in parameters such as specific yield or storativity explicitly stated in corresponding sections.
“Aquifer thickness” in conditional function refers to the aquifer thickness for unconfined aquifers and hydraulic head for confined aquifers.
Precipitation model
Monthly precipitation is simulated using a simple first-order Markov chain exponential model based on Richardson [34]. This method describes the occurrence of wet or dry months and then uses an exponential distribution function to predict the amount of precipitation. In this model, a month with a total rainfall of 6 mm or more is considered a wet month [34]. Pi (W/D) is the probability of a wet month in month i given a dry month in month i-1.; Pi (W/W) is the probability of a wet month in month i given a wet month in month i-1. Conversely, Pi (D/D) is the probability of a dry month in month i given a dry month in month i-1; Pi (D/W) is the probability of a dry month in month i given a wet month in month i-1. The relationships between these probabilities are:
Rainfall depth is analyzed using an exponential distribution, with the probability density function (f(R)) given by:
where is the distribution parameter [L0], which is calculated by dividing the total precipitation amount for either wet or dry months by the number of corresponding months; R is the precipitation amount [L]. A series of statistical tests and Monte Carlo simulations were conducted to assess the precipitation model’s performance, which is provided in S1 Fig.
Groundwater model
The groundwater model developed for this study simulates recharge, recovery, and resulting groundwater storage, and estimates drawdown due to pumping to assess whether aquifer thickness (or hydraulic head) is sufficient for the next growing season. Designed as an accessible tool, this analytical model provides a simplified alternative to complex numerical models, requiring less data and offering greater accessibility for policymakers and stakeholders. Since it is difficult to determine the specific area from where water is pumped within an aquifer, this model simplifies the water volume by considering changes in aquifer thickness (or hydraulic head) only, assuming an infinite areal extent. All the equations are on a yearly basis. We focus only on groundwater, excluding groundwater-surface water interactions at this time.
Groundwater recharge.
Groundwater recharge occurs as water moves from land surface through pore spaces down to the water table, which is primarily composed of infiltration of precipitation and local surface waters [35]. In addition, some water applied for irrigation also percolates downward below the root zone and recharges the aquifer [36]. Here, we use two methods to estimate recharge (Rch), suggested to be applied separately to shallow and deep aquifers:
For shallow aquifers or systems with relatively fast recharge times (i.e., where the water table is relatively close to the land surface), recharge is divided into two sources: from irrigation (RI [] and from precipitation (RP [
]). This model assumes that RI is the portion of irrigation water not used by evapotranspiration or surface runoff and is represented by:
, where Id is the irrigation water applied [
], Ie is the irrigation efficiency [L0], Irf is the percentage of irrigation runoff [L0]. Similarly, recharge from precipitation is calculated as the precipitation remaining after ET and surface runoff:
, where PY is the yearly precipitation [
], Pe is the precipitation efficiency [L0], Prf is the percentage of precipitation runoff [L0]. Appropriate values of these parameters can be selected based on site-specific irrigation practices and reported ranges in the literature. Note that Rch used in this model implicitly accounts for the storage coefficient.
Another option for recharge is to use a temporally variable rate provided by the user, which is particularly useful for deeper aquifers and those with long and complex recharge pathways. In deeper aquifers, there is often a significant delay in water recharging to the aquifer. For instance, in the High Plains Aquifer (HPA), it can take hundreds of years for recharge to reach the aquifer [37,38]. In these cases, a more accessible option is to set a prescribed recharge value (either constant or variable in time), as the aquifer depth minimizes the direct influence of weather or irrigation on recharge rates, and the influence of diffuse and focused recharge pathways are complex to discern. Regional estimates are often available in the literature, such as those for the HPA from [39–42] and other cases from [43,44].
Groundwater drawdown.
Groundwater drawdown refers to the lowering of the water table due to groundwater withdrawal. Analytical solutions developed by Cooper and Jacob [45] and their corrections are used to calculate the drawdown due to pumping in a single well [46,47]:
where Saquifer is the drawdown in the aquifer over a pumping duration [L], Q is the pumping rate [], rw is the effective radius of the well [L], S is the storativity [L0], tp is the duration of pumping [T]. Taquifer is transmissivity [
], given by Taquifer = Kb, where K is the hydraulic conductivity [
], and b is the aquifer thickness [L].
Eq (5) represents drawdown in a confined aquifer. For unconfined aquifers, we use specific yield (sy [L0]) instead of storativity and account for thickness variations; other adjustments for well efficiency and neighboring well impacts are provided in S1 Text. Validation of the analytical method’s implementation is provided in S2 Fig.
Groundwater recovery.
This model assumes that groundwater pumping occurs only during the growing season. However, recovery begins simultaneously with pumping and is also influenced by when pumping ceases during the non-growing season. Theis [48] proposed the concept of recovery, or residual drawdown ( [L]):
where Qd is the discharge rate []; ts is the time in days since the start of pumping [T]; tc is the time in days since the cessation of pumping [T]. This model assumes zero natural discharge from the aquifer; therefore, it is assumed that this discharge rate equals the pumping rate (Qd = Q). Although this method is used for confined aquifers, it is applicable in unconfined aquifers for late-time recovery data [49]. Validation of the analytical method’s implementation is provided in S3 Fig.
Crop yield model
The crop yield model is based on two main components: evapotranspiration estimated using the FAO Penman-Monteith approach [50] and crop yield estimation from Martin et al. [51] and Klocke [52].
Evapotranspiration.
Evaporation is the process where liquid water is converted to water vapour and removed from the surface, while transpiration refers to the vaporization of water within plant tissues and its release to the atmosphere [50]. These processes often occur simultaneously, and they are commonly referred to as evapotranspiration (ET), which directly effects crop yield and is influenced by climate, crop characteristics, and environmental factors [50]. In this model, ET reflects crop growth status after receiving water from precipitation and irrigation and is used for further prediction of yield. To calculate potential evapotranspiration for a specific crop, reference evapotranspiration (ETr []) must first be estimated, using the FAO Penman-Monteith Equation [50]:
where is the slope of the vapor pressure curve [
]; Rn is the net radiation at the crop surface [
]; G is the soil heat flux [
];
is the psychrometric constant [
]; Tair is the average air temperature [
]; u2 is the wind speed at 2 m height [
]; es is saturation vapour pressure [
]; ea is actual vapour pressure [
].
Potential ET (ETp []) is then calculated from ETr using a daily crop coefficient (Kc [L0]). Suitable values of Kc for a variety of crops and land cover are available in published literature (e.g., [50]).
Crop yield.
In this research, a commonly used yield function, modified by Martin et al. [51] and Klocke [52], is chosen to estimate crop yield. This mathematical model aligns well with observed data and effectively captures the diminishing returns of yield gains as additional inputs are applied [52].
where Ya is the estimated yield []; Yn is the non-irrigated yield that is produced from precipitation only [
]; bslope is the slope of the yield-evapotranspiration function [
]; ETinc is the difference between the amount of water used by a fully irrigated crop for maximum yield (ETp) and the amount of water used by a non-irrigated crop, assuming equals effective precipitation during the growing season (PE) [
]; Ir is the amount of irrigation required to produce maximum yield [
]. Validation of this method is provided in S4 Fig; and the estimated annual yield is shown in S5 Fig, aligning with the exponential component of the yield function.
Maximum crop irrigation requirement.
The maximum crop irrigation requirement indicates the amount of irrigation needed to match the ETp per year. Therefore, Ir for each year can be written as:
Available water.
As mentioned at the beginning of this section, to simplify calculations and avoid the need to specify the pumping area, this model uses total available aquifer thickness (or hydraulic head), denoted as b [L], at the beginning of each growing season to represent available groundwater storage. Assuming yearly time steps , aquifer thickness is updated as:
Additionally, this model incorporates a sustainable extraction rule (), representing the maximum allowable decline in aquifer thickness (or hydraulic head), expressed as a percentage of the initial thickness. This is one possible regulatory approach to promote long-term groundwater sustainability. Detailed explanations of
are also provided in Section: Sustainable Extraction Rule.
Following Butler Jr et al. [53], the model also enforces a minimum aquifer thickness of 8 meters, below which pumping becomes technically infeasible. Therefore, groundwater extraction can proceed only if all three constraints are satisfied: (1) drawdown does not exceed the allowable sustainable extraction limit, (2) the updated aquifer thickness remains above the residual portion of the sustainable extraction limit, and (3) the minimum threshold of 8 m is maintained. These constraints ensure that pumping is not a fixed or arbitrary value, but a dynamic outcome determined by both hydrologic feedbacks and regulatory boundaries. This structure allows the model to capture realistic fluctuations in water availability and emphasize the adaptive nature of irrigation decisions under physical and policy-driven limitations.
Economic model
In this model, the state variables are precipitation PE(t), crop price P(t), available water in the aquifer B(t), and irrigation requirement for maximum yield Ir(t). The control variable is the farmer’s decision on the irrigation fraction , which represents the proportion of crop water demand met by irrigation. This economic model estimates the expected land value of a farm over a planning horizon T, including a crop price model, cost function, and irrigation strategies. Actual irrigation water applied is determined by crop water demand, groundwater availability, and operational constraints.
Price model.
This work uses a Mean Reverting (MR) process to simulate crop prices, which is a stochastic process commonly used to model commodity prices [54,55]. The MR price is given by:
where Pt is the crop price [unit of local currency] at time t; is the speed of mean reversion [L0];
is the long-run mean or equilibrium of the crop price [unit of local currency];
is the volatility [L0]; dzt is an increment in a stochastic process z that follows the standard Brownian motion. The discrete time approximation of this approach is provided by:
where is time step size, 1/12 year. Ordinary least squares regression [56] is used to estimate the coefficients, c(1) and c(2):
Where ,
,
. Estimates of these values are provided in S1 Table. Price validation is provided in S6 Fig and S1 Text.
Cost functions.
The cost includes three components: fixed costs (Cf), which remain constant regardless of yield or pumping, such as land and machinery costs; harvest costs (Cy), simplified here as fertilizer expenses, which are directly related to crop yield; and pumping costs, which depend on both energy prices (Ce) and energy usage. Pumping cost modified from [57] is given as:
where, Ce is the cost of electricity [unit of local currency]; Va is the pumping volume [L3]; is the distance for lifting water from aquifers to the ground surface [L], which is determined by each year’s groundwater level;
is the water density [
]; g is the gravity factor [
];
is the pumping efficiency [L0].
Irrigation decision.
In this model, the farmer’s irrigation decision is based on a proportional rule:
where is the control variable, while Id(t) is the resulting irrigation volume determined by the chosen
and the given crop water required to reach maximum yield Ir(t) each year (Eq 10). Although
could exceed 1 technically (pumping more than the crop water demand), such over-irrigation is generally undesirable due to potential yield losses, unnecessary pumping costs, and inefficient water use. Therefore, we restrict
, e.g.,
corresponding to full irrigation and
representing 90% of demand. The actual irrigation applied in period t, Ia(t), is the minimum of the farmer’s decision Id(t) as determined by the control variable
; the available groundwater storage B(t); and the regulatory pumping limit
:
This research evaluates alternative profiles for the regulatory limit to represent different policy scenarios. These include (i) a constant regulatory pumping limit (Qc) over the entire planning horizon (T), and (ii) a time-varying regulatory pumping limit (Qn), where the pumping limit can shift at specified breakpoints (
) (e.g., by third, or half of the horizon). This allows exploration of different regulatory rules that might be imposed including both decreasing and increasing water use over time, as well as more complex stepwise rules:
These formulations emphasize that actual irrigation depends jointly on the farmer’s decision, physical constraints, and policy constraints, resulting in a dynamic year-to-year adjustment of pumping.
Expected land values.
The land value function V depends on key state variables and the farmer’s control at each time t. Specifically, V is expressed as a function of precipitation PE(t), crop price P(t), available groundwater in the aquifer B(t), irrigation required for maximum yield Ir(t), farmer’s irrigation decision , and time t, and denoted as
. These relationships show the dynamic linkage between environmental conditions, farmer decisions, and economic outcomes.
Annual cash flow () is determined by crop yield, crop price, and associated costs. Crop yield Ya(t), determined by stochastic precipitation PE(t) and irrigation water applied Ia(t), corresponds to the actual crop yield estimated by Eq (9):
As a result, the annual cash flow is:
Then, the present value in year t = ti of annual cash flow in some future year t = t′ is calculated using a discount rate, r [L0], by:
Therefore, the expected value of the farm in year ti is the expected present value of all cash flows from time t to the end date, denoted T; where pe, p, b, denote particular realizations of the state variables PE, P, B, Ir at time ti.
Risk assessment model
Risk in agricultural decisions can be characterized in multiple ways. Standard deviation captures overall variability, highlighting strategies that yield high returns in good years but large variations in outcomes. However, it does not distinguish between upside and downside variability, making it a limited measure of economic risk. By contrast, Conditional Value at Risk (CVaR) focuses on the lower tail of the distribution, quantifying average outcomes in the worst-case scenarios. So this model uses CVaR as the risk assessment tool [21]:
where V refers to land values in this work, ;
is confidence level; and
for
[58].
represents the mean of the worst
fraction of outcomes, and it is continuous with respect to
and convex in V [21]. A detailed comparison between CVaR and VaR (another common tool for assessing downside risk) is provided in S1 Text and S7 Fig.
Study site
The model is designed for broad applicability across different aquifer systems, with flexibility to incorporate site-specific conditions. In this paper, it is demonstrated using a representative site informed by real-world data from the High Plains Aquifer (HPA) in Kansas. The following subsections describe the corresponding model setup.
High plains aquifer.
The HPA is the largest freshwater aquifer system in the US, covering approximately 450,700 square kilometres, and underlying parts of eight states in the Great Plains from South Dakota to Texas [59]. Approximately 30% of the groundwater used for irrigation in the U.S. comes from the HPA and approximately 20% of the irrigated land in the U.S. is located in the High Plains region [39]. Since the start of extensive irrigation development in the 1940s, groundwater levels have experienced a significant decline, with certain regions retaining less than 40% of their original saturated thickness [60,61]. As a result, crop yields in the HPA may continue to level off or decline [38]. Since groundwater is generally considered to be a a non-renewable resource without an available substitute, once it is depleted, there will be huge impacts on the quality of human life.
This paper focuses only on groundwater irrigation, as the area overlying the HPA in Kansas is a semi-arid region with limited surface-water supplies, where groundwater is used for 96% of the state’s irrigated land [62,63]. Representative data from the HPA is used to demonstrate the model (S1 Table), making it adaptable and applicable to specific farms in this region and beyond, offering valuable guidance to irrigators in various locations.
Model setup.
To demonstrate the model, we initialized predefined state variables and applied Monte Carlo simulations to account for the stochastic components: precipitation and crop price. For each scenario, we calculated the expected present value of land at time zero under different variables. This study does not identify a universally optimal strategy, as only a limited set of irrigation strategies were tested due to computational constraints of the Monte Carlo approach. In principle, a theoretically optimal policy could be obtained by solving the Hamilton-Jacobi-Bellman (HJB) equation with finite difference methods [56,64,65], but such approaches are impractical in high-dimensional problems and limit the accessibility of the model. Instead, performance is assessed using CVaR, derived as the average of the worst 5% of outcomes, to capture lower-tail risks rather than universal optimality. In addition, we evaluate relative CVaR, expressed as the ratio of CVaR to the corresponding average land value, to highlight relative changes in downside risk.
Two planning horizons (T) were considered. The first is relatively short (T = 20 years), representing decisions by a farmer or regulator planning over the next two decades. The second extends until aquifer depletion (T = TD), effectively indicating the maximum duration farming can be sustained. This TD horizon approximates an infinite horizon, as further increases in T have negligible effects on outcomes.
Price simulations were run on a monthly time step. Because the model assumes the growing season in the HPA ends in September, corn prices from October are used to calculate annual cash flows. A 150-year time series of corn prices is simulated; if the operational period exceeds this duration, the price from the 150th year is applied, as the present value of distant future cash flows becomes negligible due to discounting.
Results
This model demonstration examines how land values respond to variations in irrigation strategies, aquifer properties, and sustainable extraction rules. To capture these aspects, we selected and tested five variables representing decisions, regulatory, management, and physical conditions (Table 1).
Together, these variables illustrate how the model can assess trade-offs among farmers’ decisions, policy limits, conservation strategies, and aquifer characteristics. For each scenario, 10,000 Monte Carlo simulations were performed to ensure statistical robustness and capture the range of possible outcomes under uncertainty. Each analysis isolates and tests a single objective to evaluate sensitivity. Unless stated otherwise, results assume a 2% discount rate.
Irrigation Strategies
In this section, we evaluate six groups of irrigation strategies (Table 2), designed to reflect local conditions and common irrigation practices, and representing a range of decision-making, regulatory, and management approaches: (A) no regulatory pumping limit () with varying irrigation fractions (
); (B) constant regulatory pumping limits with varying irrigation fractions; (C) time-varying pumping limits (two-stage) with a fixed irrigation fraction, following a large-to-small pattern; (D) time-varying pumping limits with a fixed irrigation fraction, following a small-to-large pattern; (E) time-varying irrigation fractions (two-stage) with a fixed pumping limit, following a large-to-small pattern; and (F) time-varying irrigation fractions with a fixed pumping limit, following a small-to-large pattern. The tested strategies draw from real-world irrigation practices, focusing on options that are both practically relevant and economically meaningful.
Fig 3 presents the expected land value and 95% CVaR results for Group B under T = 20 years and time to depletion (TD). In Fig 3d, the expected land value peaks at ~$2.5 million when (full irrigation) and the pumping limit is 3800 m3/day. This represents the fair market value of the land if sold. Beyond this point, expected land value is largely insensitive to further increases in the pumping limit. At the same pumping limit, the 95% CVaR is ~$1.9 million (In Fig 3e), indicating that in the worst 5% of cases, land value could fall to roughly ~$1.9 million. When the irrigation fraction is in the range
, land value initially increases with the regulatory pumping limit and then stabilizes, as the regulatory pumping rate exceeds the farmer’s decision and no longer constrains pumping. However, when
, the regulatory pumping limit does not constrain water use because actual irrigation remains fixed at 50% of crop water demand, resulting in relatively little variation in land value. The variations observed in Fig 3c and 3f arise because, when the pumping limit is ~2400 m3/day, annual profits shift from negative to positive values, resulting in a small total land value and a large difference from the average, leading to a sudden change in the CVaR percentage. In addition to the main cases, we also evaluate a scenario with a 5% discount rate. Relative to the baseline 2% case, this generates a narrower range of land value and CVaR, but a greater relative CVaR. The higher discount rate places greater weight on near-term cash flows, reducing the present value of expected future income. A demonstration of the Monte Carlo results is provided in S8 Fig and S1 Text.
(a)-(c) T = 20; (d)-(f) T = TD.
Fig 4a and 4c illustrate the relationship between expected land value and CVaR, with strategies (Table 2) labelled for comparison. Overall, higher expected land values are associated with higher CVaR results, indicating that increases in average land value also correspond to better outcomes in the worst case scenario (bottom 5% of the distribution). Group C (front-loaded pumping with a higher limit in the early period) produces the largest values, particularly under T = TD, as discounting places greater weight on near-term income. For the scenarios with no regulatory limits, strategy A2, which applies 90% of crop water demand, yields substantially lower values than full-demand irrigation (A1). For the remaining strategies, differences in land value are relatively modest, generally less than 10%.
Fig 4b and 4d further illustrate these dynamics. Expected land values and standard deviations follow a characteristic concave (“C-shaped”) pattern, with higher land values generally associated with greater risk.
For farmer’s irrigation decision, the decision variable, we examined the effects of varying irrigation fractions () under
(Fig 5a–5c). Expected land value and CVaR both increase with higher irrigation fractions, as greater water application enables more crop demand to be met. However, this gain comes at a cost: operational years decline sharply from 375 to 171 as
increases, illustrating the trade-off between short-term profitability and long-term resource availability. While
achieves ~90% of the full-irrigation land value and ~80% of its CVaR, reducing
to 0.8 cuts land value nearly in half and reduces CVaR to ~30%. Below
, land values fall to ~15% and CVaR becomes negative, highlighting increased downside risk and potential financial losses.
To explore dynamic decision-making, we tested eight irrigation strategies about (Groups E and F in Table 2, Fig 5d–5g). Strategies that start with higher
and gradually taper (Group E) consistently outperform others. In the T = TD case, most strategies fall within a high-return/high-variability regime (top-right of Fig 5g), indicating that producers often face a trade-off between maximizing profitability and reducing interannual volatility. However, CVaR analysis shows an almost linear relationship with expected land value, indicating that lower mean returns are associated with worse downside outcomes. High pumping strategies result in increased variability: they generate high returns in favorable years but become more sensitive to unfavorable conditions (e.g., droughts, low prices). Notably, strategies with median land values also exhibit median CVaR, reflecting moderate downside risk, yet their standard deviations remain high. This highlights that variability captured by standard deviation does not necessarily reflect greater downside risk, results may fluctuate widely in the middle of the distribution without significantly worsening the worst-case outcomes.
In addition, we examined water depth dynamics under a constant regulatory pumping limit and full-demand irrigation (Group B1) over a 20-year horizon, with results summarized in Table 3. Differences across Monte Carlo simulations and under alternative pumping limits are not significant. This outcome is because in these simulations the 180-day non-pumping period in each year allows for substantial groundwater recovery. Moreover, given that the crop water demand (Ir) is around 3600 m3/day, the tested pumping limits () are relatively close to Ir. When repeated across 10,000 Monte Carlo simulations, the average actual pumping converges to similar values, leading to negligible differences in simulated water depth. It is anticipated that shorter recovery periods, or lower conductive materials that would require additional time to recover, leading to greater differences between these simulations.
Hydraulic conductivity
In this section, we examined the effects of two key aquifer properties: hydraulic conductivity and specific yield. Hydraulic conductivity influences both groundwater drawdown and recovery, while specific yield primarily affects drawdown. For this analysis, no regulatory pumping limits are imposed, and irrigation decisions are assumed to fully meet crop water demand for maximum yield (Group A1).
Fig 6(a1)–6(c1) presents land value and CVaR results across different hydraulic conductivities. Over a 20-year horizon, both metrics increase slightly with higher conductivity. Greater hydraulic conductivity enhances aquifer recovery (Eq 6), improves water replenishment, and reduces drawdown (Eq 5), thereby lowering pumping costs and raising land values. When evaluated over the depletion horizon (T = TD), these effects become more noticeable: a conductivity of K = 10 m/day supports 52 years of operation, while K = 40 m/day extends this to 164 years. As shown in Fig 6(c1), the relative percentage differences decrease. These results confirm the role of aquifer properties in shaping economic outcomes, linking higher conductivity to greater returns and reduced risk.
However, the influence of specific yield on land value is less noticeable in our preliminary analysis. Variations in specific yield do not affect land value or CVaR across either planning horizon, as other physical constraints dominate the simulations. Supporting simulation results are provided in S9 Fig and S1 Text.
Sustainable extraction rule
The sustainable extraction rule discussed in this section refers to the allowable proportion of groundwater use (). For example, if annual withdrawals are limited to 30% of the initial aquifer thickness (or hydraulic head) at the start of year 1, pumping continues through the growing season until the annual threshold is reached (i.e., conditional function in Fig 1), ensuring seasonal rather than mid-season cessation.
With a 20-year time frame, stored groundwater is sufficient to support irrigation, resulting in land value and CVaR being nearly identical as increases in Fig 6(a2)–6(c2). However, when T = TD, land values and CVaR increase with relaxed extraction rules before stabilizing, and the relative CVaR decreases. At stricter sustainable extraction rules, relaxing the constraint (
) leads to higher land values, as increased water availability significantly impacts both economic returns and associated risks. Despite this, even with an extension of operational years from 65 to 172 due to greater water availability, rising pumping costs and the effect of discounting future revenues contribute to a relatively stable expected land value over the long term.
Discussion
This study applies a newly developed hydro-economic model that uses Conditional Value-at-Risk (CVaR) and land values to examine how groundwater management strategies influence economic outcomes. By simulating a range of scenarios, the analysis identifies key relationships among groundwater use, land value, and financial risk.
For the demonstration case, a pumping limit of 3800 m3/day generates the highest expected land values for most scenarios, since annual crop water demand ranges from 3600 to 3900 m3/day depending on precipitation. Thus, setting the pumping limit at 3800 m3/day effectively balances water demand with sustainable extraction, providing both regulatory control and economic benefit. Across all irrigation strategies, allowing a larger pumping limit initially and then reducing it over time generates the highest expected land values in both T = 20 and T = TD cases. This outcome arises because discounting places greater weight on near-term revenues.
For farmers’ direct decisions (), higher irrigation fractions generally raise land values by better meeting crop water demand. Irrigation of less than 80% of water demand is not optimal for the farmer, but the marginal benefits diminish sharply between 0.9 and 1.0. Although shorter operational periods can generate higher near-term value, this approach compromises long-term sustainability by leaving less water for future generations as indicated by the time to aquifer depletion and reducing flexibility to address emerging challenges.
CVaR analysis shows that strategies with higher expected land values are also associated with lower risk (higher CVaR), indicating that improvements in average returns are accompanied by stronger performance under unfavorable conditions. Unlike standard deviation, which captures both upward and downward variability, CVaR focuses on the lower tail of the distribution and therefore provides a clearer measure of downside risk. This makes CVaR a more informative tool for identifying management options that safeguard against unfavorable scenarios, rather than penalizing strategies simply because they exhibit greater overall variability.
Due to faster aquifer recovery and smaller drawdown, higher hydraulic conductivity enhances both land values and CVaR for T = 20 and TD cases, reflecting reduced financial risk as groundwater availability improves. This positive correlation highlights the importance of aligning pumping rules with aquifer properties to sustain farm-level economic stability under variable conditions. The results of sustainable extraction rules suggest that greater water availability does not always lead to higher land values. When , diminishing returns occur, additional water use only provides limited economic benefit. This implies that regulators may have flexibility to reduce allowable extraction without significantly compromising farm profitability. Such adjustments could support long-term aquifer sustainability while mitigating risks of over-extraction.
A key limitation of this study is the assumption of a homogeneous aquifer, which neglects spatial and temporal variations in aquifer thickness or hydraulic head associated with the area of influence of pumping. Boundary conditions such as hydraulic connections with streams or lakes are also not explicitly represented. Crop pattern variability is not considered. A more detailed 2D or 3D hydrologic numerical model could better capture groundwater dynamics; however, this study prioritizes an easily accessible approach that minimizes data, technical, and financial requirements.
Future research could build on this framework by addressing the current limitations, including spatial heterogeneity in aquifer properties, explicit boundary conditions, and crop pattern variability. Incorporating more complex hydrologic models could better capture groundwater dynamics and groundwater-surface water interactions. Expanding the analysis to include climate change scenarios, market volatilities, and crop management practices would enhance the robustness and realism of the model. Additionally, exploring the science-policy interface could help transfer hydro-economic insights into actionable strategies for sustainable water management, supporting decisions by farmers, regulators, and other stakeholders.
Conclusion
This study develops a farm-level hydro-economic model that links groundwater dynamics with economic outcomes to evaluate irrigation strategies under physical and regulatory constraints. By incorporating Conditional Value-at-Risk (CVaR), the model captures not only average outcomes but also downside risks, offering a more robust decision-making framework. Results show that higher pumping does not necessarily increase revenues or reduce risks, as diminishing returns and aquifer depletion can offset short-term gains. Differences in aquifer properties are important in shaping both economic outcomes and risk profiles. The analysis indicates that current extraction decisions have long-term implications for farm viability, as reflected in projected aquifer depletion timelines. Designed to be adaptable to site-specific conditions, this model provides a flexible tool for farmers, policymakers, and regulators seeking to align irrigation decisions with long-term groundwater sustainability.
Supporting information
S1 Fig. Simulated and observed monthly precipitation.
https://doi.org/10.1371/journal.pwat.0000452.s001
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S2 Fig. Comparison of Chapuis drawdown data and calculated results, demonstrating a good fit with late-time data.
https://doi.org/10.1371/journal.pwat.0000452.s002
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S3 Fig. Comparison of Case et al. recovery data and calculated results.
https://doi.org/10.1371/journal.pwat.0000452.s003
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S4 Fig. Comparison of Klocke crop yield data and calculated results.
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S5 Fig. Yield model: (a) Annual yield at different pumping rates; (b) Zoomed-in view for pumping rates between 3000 and 5000 m3/day.
https://doi.org/10.1371/journal.pwat.0000452.s005
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S6 Fig. MLE simulated data and historical prices, HPA.
Note: The one-year shift occurs because the simulation requires an initial year to establish the starting price.
https://doi.org/10.1371/journal.pwat.0000452.s006
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S7 Fig. Demonstration of CVaR: (a) Normal distribution of land value for a hypothetical site; (b) illustration of VaR and CVaR.
https://doi.org/10.1371/journal.pwat.0000452.s007
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S8 Fig. Monte Carlo simulation results of expected land value for the demonstration site (n = 10,000) under irrigation strategy B1.
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S9 Fig. Demonstration of specific yield: (a) Land values, (b) 95% CVaR, and (c) relative CVaR for different specific yields.
https://doi.org/10.1371/journal.pwat.0000452.s009
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S1 Table. Model parameter values used in the hydro-economic simulations.
https://doi.org/10.1371/journal.pwat.0000452.s010
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S1 Text. Additional description of aquifer drawdown, hydrologic model validation, price model, CVaR and VaR, Monte Carlo results, and specific yield.
https://doi.org/10.1371/journal.pwat.0000452.s011
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