Forecasting Water Consumption for Sustainable Development in Saudi Arabia: A Copula-Based Approach

Abstract

Effective water resource planning is essential for Saudi Arabia, where limited freshwater availability is challenged by rapid population growth, economic development, and climate variability. This study introduces a copula-based modeling framework for forecasting water demand across the country’s urban, industrial, and agricultural sectors. Copulas, compared to traditional models, effectively capture nonlinear and asymmetric relationships among essential variables, including population, temperature, GDP, and sectoral water consumption. Multivariate copula models (Gaussian, Clayton, Gumbel, Frank, t-Copula, and Vine structures) were fitted and evaluated using historical data from 2008 to 2024, obtained from national authorities, including the Ministry of Environment, Water, and Agriculture, the General Authority for Statistics, and the National Center for Meteorology. The 4D normal copula was developed as the most efficient method across all sectors, with MAPE values of 6.37% for urban, 17.51% for industrial, and 23.20% for agricultural consumption. Scenario-based forecasts, which include baseline, high-growth, and sustainability-focused trajectories, indicate that the sustainability scenario yields the best results, resulting in significant demand reductions (21.7% urban, 20.4% industrial, and 8.2% agricultural) with minimal climate impact (+0.4 °C) and the lowest risk levels. The study demonstrates the successful decoupling of water demand from population and economic growth through proper policy interventions, with conditional risk analysis offering actionable early warning capabilities for proactive management. These findings provide a valuable foundation for enhancing national water strategy planning in Saudi Arabia under Vision 2030 and contribute to methodological improvements applicable to water-scarce regions internationally.

1. Introduction and Motivation

Saudi Arabia faces serious challenges in balancing water availability with rising demand due to rapid population growth, industrialization, and agricultural expansion. As one of the most water-scarce countries worldwide, its dependence on non-renewable groundwater and desalination raises worries about long-term sustainability [1]. Effective water resource management requires precise forecasting; however, traditional models often fail to account for the complex, nonlinear interplay between socioeconomic variables and environmental variability, thereby limiting their predictive capacity in dynamic systems.

The country’s rapid population growth, economic development, and rising temperatures have led to significant increases in water demand across the urban, industrial, and agricultural sectors. This necessitates accurate forecasting of water consumption, not only to fulfill future demand but also to maintain long-term sustainability. Traditional forecasting approaches often fail to accurately reflect the true nature of the relationships between water use and its underlying causes. Water demand does not always follow a linear relationship with population or GDP; it may accelerate during periods of excessive heat or economic booms. These complex and nonlinear patterns are particularly significant in countries like Saudi Arabia, where climatic extremes and resource scarcity pose major challenges.

Saudi Arabia’s water consumption is distributed across three major sectors, each with distinct characteristics and challenges. The agricultural sector dominates water consumption, accounting for approximately 84% of total consumption, primarily for irrigation of crops such as wheat, barley, and fodder [2]. This sector relies heavily on non-renewable groundwater aquifers, particularly the deep fossil aquifers in the Eastern Province and central regions. The municipal sector, representing about 9% of total consumption, serves urban populations across major cities including Riyadh, Jeddah, Mecca, and the Eastern Province, with demand driven by population growth, urbanization, and rising living standards. The industrial sector accounts for approximately 7% of water use, concentrated in petrochemical complexes, refineries, manufacturing facilities, and mining operations, particularly in the Eastern Province industrial cities of Jubail and Yanbu. Water supply sources include desalinated seawater (50%), non-renewable groundwater (40%), and renewable groundwater (10%), with desalination capacity exceeding 5.7, making Saudi Arabia the world’s largest producer of desalinated water [3].

Traditional models, such as regression and time series, rely on linear associations and cannot account for the dynamic interdependence between variables, including population increase, economic activity, climate variability, and industrial expansion. These elements interact in ways that traditional methods may not fully comprehend or represent, resulting in erroneous water demand estimates and poor policy decisions. A copula-based model is a more robust approach since it allows for the modeling of complex relationships between variables with diverse marginal distributions. Copula models can capture nonlinear interactions and provide a greater understanding of how numerous factors influence water consumption. This study will be the first to utilize copula-based statistical modeling to predict water consumption in Saudi Arabia, offering a novel approach to understanding the complex interdependencies between variables such as population growth, climate change, and economic activity. This method offers more precise forecasts, which are essential for planning future water supplies, efficiently managing resources, and assuring sustainability. Saudi Arabia will be able to estimate future water demand more accurately, supporting long-term water management policies that align with the Saudi Vision 2030 goals [4], which promote sustainable growth and resource efficiency.

The main objective of this study is to establish a forecasting framework for water consumption in Saudi Arabia, utilizing copula-based statistical techniques. The study aims to examine historical patterns of water consumption in the urban, industrial, and agricultural sectors, identifying the fundamental socioeconomic and environmental factors that influence demand. Additionally, it aims to develop a copula-based model that accurately captures the dependence structure among some essential variables, including population, GDP, and temperature, by selecting the most suitable copula function, such as the Gaussian, Clayton, or Gumbel copula. Finally, the study anticipates future water consumption in three distinct scenarios: baseline, high-growth, and sustainability-focused, examining how demographic, economic, and climatic factors affect water demand across sectors. The study supports Saudi Vision 2030’s objectives by promoting sustainable development, enhancing water management efficiency, and laying a strong scientific foundation for future water policy and strategic planning [4].

The study is divided into six sections. Section 1 provides background information and discusses the research aims. Section 2 summarizes relevant research on water demand forecasts and copula models. Section 3 explains the data, modeling framework, and forecasting scenarios. Section 4 presents the model’s results, along with the assessments of risk. Section 5 examines the findings and considers their policy consequences. Finally, Section 6 summarizes the most important findings and provides recommendations for long-term water sustainability in Saudi Arabia.

2. Related Works

For sustainable water management, particularly in arid regions like Saudi Arabia, accurate water demand forecasting is crucial. Numerous studies have investigated this topic using various approaches, such as time series models, machine learning, and copula-based techniques. These studies highlight the increasing importance of identifying nonlinear interdependencies among socioeconomic, meteorological, and infrastructure elements to inform strategic planning. This section reviews essential contributions in four areas involving copula-based modeling, intelligent water management systems, regional and sectoral forecasting initiatives in Saudi Arabia, and copula-based modeling.

2.1. Copula-Based Modeling in Water Studies

Copula models have become very useful for modeling complicated, nonlinear dependencies in systems that deal with water resources. Wang et al. [5] used a copula-based framework to assess water security in Inner Mongolia using the Stability–Cooperation–Resilience (SCR) index. Their model showed that spatio-temporal changes in water security levels and effectively quantified joint risk probabilities, offering a valuable approach for managing water in dry areas. Wafaa et al. [6] employed quantile regression of the D-vine copula to examine the amount of money households in Morocco spent on water, considering geographic, demographic, and socioeconomic factors. Their results show that using copulas can reveal subtle differences in consumption behavior across regions. Seo et al. [7] applied copula models to analyze joint impacts of meteorological drought and river water temperature on aquatic ecosystems. By selecting optimal copula functions and drought indices, they produced spatially explicit risk maps, underscoring the utility of copulas in hydrological risk assessment. Alqadhi et al. [8] utilized the Standardized Precipitation Evapotranspiration Index (SPEI) and bivariate copula models to assess the severity of drought in Saudi Arabia. Their research showed that the drought patterns were worsening and identified return periods for extreme drought events. This indicates that copula methods remain effective for planning drought and water use in arid regions.

2.2. Smart Water Management and Technological Innovations

AlGhamdi and Sharma [9] developed an innovative water management system (IoT-SWM) specifically designed for the Kingdom’s infrastructure. Their model combines GSM modules, leak detection, and water quality monitoring to make the best use of water in households. These systems are incredibly advanced, but they occasionally struggle with statistical forecasting frameworks. Syed et al. [10] created an integrated framework that uses Digital Twin technology and multimodal transformer models to predict water consumption and detect leaks. The system combines real-time sensor data (like flow meters, pressure sensors, and thermal imaging) with advanced transformer models. It achieves a $R^2$ score of 0.9995, which shows that it is much better than traditional methods for managing water in a sustainable way.

2.3. Forecasting Water Demand in Saudi Arabia

Several studies have used to model water demand in Saudi Arabia using time series and probabilistic approached. Boubaker [11] proposed a PSO-tuned ARIMA model for municipal water demand in Hail, demonstrating high predictive precision (MAPE = 5.28%) and outperforming traditional ANN and stochastic methods. Almutaz et al. [12] used Monte Carlo simulations to forecast the water demand in Mecca under seasonal tourism effects. Their probabilistic approach factored in population, income, and household size, predicting up to 225 L per capita of daily consumption by 2013. Conservation measures could reduce future demand by 24% by 2030. Kamis [13] modeled domestic water demand in Jeddah under varying scenarios of population growth. The study emphasized the need for conservation, noting that consumption would exceed supply unless daily usage is reduced to 200 L per capita. Chowdhury and Al-Zahrani [14] studied national trends from 1980 to 2009, finding decreasing agricultural demand but increasing domestic and industrial water consumption. They concluded that alternative solutions such as desalination and rainwater harvesting were required.

2.4. Climate Change and Water Stress

Tarawneh et al. [15] carried out an assessment of long-term climate trends and their likely future in Saudi Arabia using linear and Mann–Kendall statistical analyses. Climate projections based on multiple RCP scenarios (RCP8.5, RCP6, RCP2.6) consistently indicate warming and more uncertain precipitation, with profound implications for future water security. The study’s findings highlight the urgent need for scenario-based water forecasting that integrates climate variability, population, and economic growth into a comprehensive and adaptive water management framework. Mahmoud [16] performed a comprehensive study of the emerging climate and water crisis in the Middle East and North Africa (MENA) with special emphasis on Saudi Arabia and the Arabian Peninsula. This analysis documented the increased frequency of extreme heavy rainfall events leading to widespread flooding over Saudi Arabia, UAE, Oman, and Yemen, over the past two decades. The study pointed out the fact that Saudi Arabia and other GCC countries are currently meeting nearly 90% of drinking water demand through desalination, equivalent to close to 50% of the global freshwater desalination capacity. However, this reliance on an energy-intensive process has created additional climate risks and sustainability challenges. Seo et al. [17] employed a Bayesian Neural Network (BNN) approach to model future residential water use under climate change scenarios. Their model leveraged probabilistic learning to account for uncertainty in both model parameters and future climatic inputs, offering credible intervals for forecasted demand. The study showed that BNNs outperformed traditional deterministic models in both accuracy and robustness, particularly under conditions of extreme weather variability. While the focus was limited to the residential sector, their approach highlighted the importance of integrating climate indicators and uncertainty quantification into forecasting frameworks.

2.5. Identified Research Gap

The literature reveals several important trends: a growing reliance on advanced techniques such as ARIMA, copulas, and Monte Carlo simulations; a tendency to focus on forecasting water demand in isolated regions or sectors; and an emphasis on water demand in these regions or sectors. Additionally, there is limited integration between forecasting models and emerging technologies. Although copula models have shown strong potential for capturing complex dependencies and nonlinear relationships, their use in comprehensive, sector-level forecasting systems that support Saudi Arabia’s Vision 2030 remains limited.

This study seeks to fill this gap by developing an integrated copula-based forecasting framework that models the interconnections between urban, industrial, and agricultural water consumption. By enabling scenario-based analysis and providing insights tailored to national strategies, this approach offers a powerful tool for supporting sustainable and effective water resource management.

3. Methodology

This study utilizes a copula-based statistical approach to predict consumption of water in the urban, industrial, and agricultural sectors of Saudi Arabia. The methodology comprises four primary stages: data collection and preprocessing, marginal distribution modeling, copula selection and fitting, and scenario-based forecasting.The main steps of the proposed copula-based forecasting framework are summarized in Figure 1, outlines our systematic methodology progressing through four integrated phases: (1) multi-source data collection combining water consumption, demographic, and climatic variables; (2) marginal distribution modeling using AIC-based selection among candidate distributions followed by transformation to uniform variables; (3) copula model fitting comparing elliptical, Archimedean, and vine structures using maximum likelihood estimation; and (4) scenario-based forecasting implementing Baseline, High Growth, and Sustainability.

3.1. Data Collection and Structure

The study employs annual data from 2008 to 2024, concentrating on factors affecting water consumption in Saudi Arabia. The dependent variables denote water consumption across urban, industrial, and agricultural sectors, measured in megaliters per day (ML/day). Independent variables comprise population (in millions), GDP (in billion dollars), and average temperature (in degrees Celsius). To compile this dataset, we gathered data from several trusted sources. The water consumption figures were obtained from the General Authority for Statistics and Ministry of Environment, Water and Agricultur, Riyadh, Saudi Arabia, [18,19], while the GDP, population and average temperature data were retrieved from the World Bank database [20,21,22].

3.2. Marginal Distribution Modeling for Copula Framework

Modeling the marginal distributions of each variable is a critical first step in copula-based dependence analysis [18,23]. Let $X_1,X_2,\dots ,X_6$ denote the six variables considered in this study, defined as follows:

-

$X_1=$ Urban Water Consumption (ML/day)

-

$X_2=$ Industrial Water Consumption (ML/day)

-

$X_3=$ Agricultural Water Consumption (ML/day)

-

$X_4=$ Population (millions)

-

$X_5=$ GDP (billion USD)

-

$X_6=$ Average Temperature (°C)

For each variable $X_i$, the corresponding marginal cumulative distribution function (CDF) $F_i\left(x\right)$ was estimated as follows:

$$F_i\left(x\right)=\mathbb{P}(X_i\le x),i=1,2,\dots ,6$$

Each variable was individually fitted to a set of candidate distributions—Normal, Log-Normal, Gamma, and Weibull. The selection of appropriate marginal distributions was guided using goodness-of-fit criteria including the Akaike Information Criterion (AIC) and statistical goodness-of-fit tests (Kolmogorov–Smirnov and Anderson–Darling). The Akaike Information Criterion (AIC) is defined as follows:

$$\mathrm{AIC}=2k-2ln\left(\widehat{L}\right)$$

(1)

where k is the number of model parameters and $\widehat{L}$ is the maximized likelihood value. Maximum Likelihood Estimation (MLE) was used to estimate the parameters of each marginal distribution. The estimator $\widehat{\theta }$ is defined as follows:

$$\widehat{\theta }=arg\underset{\theta }{max}\sum _{i=1}^{n}logf(x_i;\theta )$$

(2)

where $\widehat{\theta }$ denotes the estimated parameter vector, $f(x_i;\theta )$ is the probability density function (PDF) of the selected marginal distribution evaluated at observation $x_i$, and n is the total number of observations.

Once the optimal marginal distributions were identified, each variable was transformed into a uniform random variable on the interval $[0,1]$ using its fitted CDF:

$$U_i=F_i\left(x_i\right),\mathrm{where}{U}_i\sim \mathrm{Uniform}(0,1)$$

These transformed variables $U_i$ serve as inputs to the copula function, which models the joint dependence structure independently of the marginal behavior.

3.2.1. Copula Model Framework

According to Sklar’s theorem [24], the joint distribution for each sectoral model can be expressed as follows:

Urban Water Model:

$$F(x_1,x_4,x_5,x_6)=C[F_1\left(x_1\right),F_4\left(x_4\right),F_5\left(x_5\right),F_6\left(x_6\right)]$$

(3)

Industrial Water Model:

$$F(x_2,x_4,x_5,x_6)=C[F_2\left(x_2\right),F_4\left(x_4\right),F_5\left(x_5\right),F_6\left(x_6\right)]$$

(4)

Agricultural Water Model:

$$F(x_3,x_4,x_5,x_6)=C[F_3\left(x_3\right),F_4\left(x_4\right),F_5\left(x_5\right),F_6\left(x_6\right)]$$

(5)

where $F_1,F_2,F_3$ are the marginal CDFs of urban, industrial, and agricultural water consumption, respectively; $F_4,F_5,F_6$ are the marginal CDFs of population, GDP, and temperature; and C represents the 4D copula function capturing the dependence structure within each sector. Data transformation to pseudo-observations was performed using the empirical distribution function:

$$U_{ij}={\displaystyle \frac{1}{n+1}}\sum _{k=1}^{n}I(X_{kj}\le X_{ij})$$

where $I\left(·\right)$ is the indicator function, n is the sample size, and j indexes the variables.

3.2.2. Copula Family Selection

To model the diverse dependency structures within the six-dimensional water usage system, several copula families were considered, including elliptical, Archimedean, and vine copulas [25].

Elliptical copulas capture symmetric dependencies and are suitable for variables exhibiting linear correlations. The Gaussian copula is defined as:

$$C^{\mathrm{Ga}}(\mathbf{u};\mathsf{\Sigma })={\mathsf{\Phi }}_6\left({\mathsf{\Phi }}^{-1}\left(u_1\right),\dots ,{\mathsf{\Phi }}^{-1}\left(u_6\right);\mathsf{\Sigma }\right),$$

where $\mathsf{\Sigma }$ is a $6\times 6$ correlation matrix. While it captures linear dependence well, it lacks tail dependence. In contrast, the t-Copula introduces tail dependence and is defined as follows:

$$C^t(\mathbf{u};\mathsf{\Sigma },\nu )=t_{6,\nu }\left(t_{\nu }^{-1}\left(u_1\right),\dots ,t_{\nu }^{-1}\left(u_6\right);\mathsf{\Sigma }\right),$$

where $\nu$ denotes the degrees of freedom, enabling symmetric tail behavior.

Archimedean copulas are more flexible and suitable for capturing asymmetric dependencies. The Clayton copula, which models strong lower tail dependence (i.e., joint low water usage), is given by the following:

$$C^{\mathrm{Cl}}(\mathbf{u};\theta )={\left(\sum _{i=1}^6{u}_i^{-\theta }-5\right)}^{-1/\theta },\theta >0.$$

The Gumbel copula is designed for upper tail dependence, representing simultaneous extreme high demand scenarios:

$$C^{\mathrm{Gu}}(\mathbf{u};\theta )=exp\left(-{\left(\sum _{i=1}^6{(-log{u}_i)}^{\theta }\right)}^{1/\theta }\right),\theta \ge 1.$$

The Frank copula is symmetric and does not exhibit tail dependence:

$$C^{\mathrm{Fr}}(\mathbf{u};\theta )=-{\theta }^{-1}log\left(1+{\displaystyle \frac{{\prod }_{i=1}^6(exp(-\theta u_i)-1)}{{(exp(-\theta )-1)}^5}}\right),\theta \in \mathbb{R}.$$

The Joe copula function for the six-dimensional case is defined as follows:

$$C^{\mathrm{Jo}}(\mathbf{u};\theta )=1-{\left[\sum _{i=1}^6{(1-u_i)}^{\theta }-\prod _{i=1}^6{(1-u_i)}^{\theta }\right]}^{1/\theta },\theta \ge 1$$

where $\theta$ is the copula parameter controlling the strength of dependence. This structure allows the Joe copula to effectively capture the behavior of joint extreme events without emphasizing low-dependence tails.

The D-vine copula is another form of pair-copula construction (PCC) that models multivariate dependencies through a sequence of bivariate copulas arranged in a path-like or chain structure [26]. Unlike the C-vine, which has a single root node, the D-vine assumes that all variables are connected through adjacent pairs, with dependence propagated through conditional relationships in successive trees. For a 6-dimensional vector $\mathbf{X}=(X_1,\dots ,X_6)$, the joint density function under the D-vine structure is given by the following:

$$\begin{array}{cc}\hfill f\left(\mathbf{x}\right)& =\prod _{i=1}^6{f}_i\left(x_i\right)\prod _{i=1}^5{c}_{i,i+1}(F_i\left(x_i\right),F_{i+1}\left(x_{i+1}\right))\hfill \\ & \times \prod _{i=1}^4{c}_{i,i+2|i+1}(F_{i|i+1}\left(x_i\right|x_{i+1}),F_{i+2|i+1}\left(x_{i+2}\right|x_{i+1}))\hfill \\ & \times \prod _{i=1}^3{c}_{i,i+3|i+1,i+2}(·,·)\dots \times c_{1,6|2,3,4,5}(·,·)\hfill \end{array}$$

This layered structure allows D-vines to model dependencies that are strongest between neighboring variables, with more complex interactions captured via conditional copulas in higher tree levels.

The C-vine copula is a type of pair-copula construction (PCC) that decomposes a high-dimensional joint distribution into a structured sequence of bivariate copulas. It assumes a hierarchical dependence structure where one central variable, known as the root node, is directly connected to all others in the first tree level [27,28]. Conditional dependencies are then modeled in successive tree levels based on partial correlations.

Formally, the C-vine density for a d-dimensional random vector $\mathbf{X}=(X_1,\dots ,X_d)$ is expressed as follows:

$f\left(\mathbf{x}\right)={\prod }_{i=1}^d{f}_i\left(x_i\right){\prod }_{j=2}^d{c}_{1,j}(F_1\left(x_1\right),F_j\left(x_j\right))\times {\prod }_{k=2}^{d-1}{\prod }_{j=k+1}^d{c}_{k,j|1,\dots ,k-1}\left(F_{k|1,\dots ,k-1}\left(x_k\right|x_1,\dots ,x_{k-1}),F_{j|1,\dots ,k-1}\left(x_j\right|x_1,\dots ,x_{k-1})\right),$ where $c_{j,k|D}$ is the conditional copula density and $F_{j|D}$ is the conditional distribution given the conditioning set D.

In general, a C-vine on d variables consists of $d-1$ trees $T_{\ell }=(N_{\ell },E_{\ell })$:

• Tree 1: A star tree centered on the selected root variable r, with edges $E_1=\{\{r,j\},j\ne r\}$ representing unconditional pairwise dependencies.
• Tree ℓ: The nodes are the edges of the previous tree ($N_{\ell }=E_{\ell -1}$). Two nodes in $T_{\ell }$ are connected if their corresponding edges in $T_{\ell -1}$ share a common conditioning variable. This ensures the proximity condition is satisfied.

Root selection is typically based on overall dependency strength. In our study, the root variable was determined as

$$r^{*}=arg\underset{j}{max}\sum _{k\ne j}\left|{\tau }_{j,k}\right|,$$

where ${\tau }_{j,k}$ denotes Kendall’s $\tau$ rank correlation. This choice ensures that the most influential driver (e.g., urban water demand) is placed at the center of the first tree, reflecting its dominant role in the dependency structure.

Key properties include:

-

Parameters: $\left({\displaystyle \genfrac{}{}{0pt}{}{d}{2}}\right)$ pair-copulas must be specified.

-

Complexity: $\mathcal{O}\left(d^2\right)$, making it tractable for high dimensions.

-

Interpretability: The hierarchical decomposition allows policymakers to trace how direct and conditional relationships contribute to sectoral water demand.

A general comparison of the copula families considered is provided in

Table 1. General comparison of copula types.

Copula Type	Parameters Count	Dependence Range	Lower Tail Dependence	Upper Tail Dependence	Symmetry	Max Dimension
Elliptical Copulas
Gaussian	$\left({\displaystyle \genfrac{}{}{0pt}{}{d}{2}}\right)$	$[-1,1]$	None	None	Yes (radial)	∞ (theoretical)
t-Copula	$\left({\displaystyle \genfrac{}{}{0pt}{}{d}{2}}\right)+1$	$[-1,1]$	Symmetric	Symmetric	Yes (radial)	∞ (theoretical)
Archimedean Copulas
Clayton	1 ($\theta >0$)	$(0,\infty )$	Strong (${\lambda }_L=2^{-1/\theta }$)	None (${\lambda }_U=0$)	No (lower tail)	4–5 (practical)
Gumbel	1 ($\theta \ge 1$)	$[1,\infty )$	None (${\lambda }_L=0$)	Strong (${\lambda }_U=2-2^{1/\theta }$)	No (upper tail)	4–5 (practical)
Frank	1 ($\theta \in \mathbb{R}$)	$(-\infty ,\infty )$	None (${\lambda }_L=0$)	None (${\lambda }_U=0$)	Yes (symmetric)	4–5 (practical)
Joe	1 ($\theta \ge 1$)	$[1,\infty )$	None (${\lambda }_L=0$)	Strong (increases with $\theta$)	No (upper tail)	4–5 (practical)
Vine Copulas
D-vine	$\left({\displaystyle \genfrac{}{}{0pt}{}{d}{2}}\right)$	Edge-specific	Per edge	Per edge	Edge-specific	∞ (no limit)
C-vine	$\left({\displaystyle \genfrac{}{}{0pt}{}{d}{2}}\right)$	Edge-specific	Per edge	Per edge	Edge-specific	∞ (no limit)

. This includes their parameter counts, dependence ranges, tail behavior, symmetry properties, and practical dimensional limitations [29].

3.2.3. Conditional Forecasting Framework

While copulas model joint dependence structures, forecasting requires converting probability distributions to point predictions. Our approach employs conditional expectations:

For given scenario values $X_4^{*}$ (population), $X_5^{*}$ (GDP), $X_6^{*}$ (temperature), sectoral water consumption forecasts are computed as follows:

$${\widehat{Y}}_s(t+1)=\mathbb{E}\left[X_s\right|X_4=x_4^{*},X_5=x_5^{*},X_6=x_6^{*}]$$

(6)

where $s\in \left\{\mathrm{urban},\mathrm{industrial},\mathrm{agricultural}\right\}$.

These conditional means are derived from the fitted 4D normal copula through numerical integration of the conditional density function.

3.3. Parameter Estimation, Model Selection, and Forecast Evaluation

Copula parameters were estimated using the maximum likelihood estimation (MLE) approach, which optimizes the likelihood function based on observed data [30]. The estimator is defined as follows:

$$\widehat{\theta }=arg\underset{\theta }{max}\sum _{i=1}^{n}logc({\mathbf{u}}_i;\theta )$$

where $c\left(·\right)$ is the copula density function and ${\mathbf{u}}_i$ are the pseudo-observations. Numerical optimization was performed using the BFGS quasi-Newton algorithm with multiple initial starting points to avoid local optima [31]. A strict convergence criterion with a relative tolerance of ${10}^{-8}$ was applied, and parameter constraints were enforced to maintain the validity of the copula family.

The AIC was used to evaluate the performance of the model. Lower AIC values indicate a better trade-off between model fit and complexity. This criterion guided the selection of the best-fitting copula for each sectoral dependency structure.

Point forecasts for performance evaluation are generated using conditional expectations from fitted copula models. The predictive accuracy metrics compare these conditional mean forecasts against observed consumption values. The predictive accuracy of copula-based models was evaluated using standard point forecast metrics for each sector: urban, agricultural, and industrial water demand [32]. These included:

$$\begin{array}{cc}\hfill {\mathrm{MAE}}_s& =n^{-1}\sum _{i=1}^n|y_{si}-{\widehat{y}}_{si}|\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathrm{RMSE}}_s& =\sqrt{n^{-1}\sum _{i=1}^n{(y_{si}-{\widehat{y}}_{si})}^2}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\mathrm{MAPE}}_s& =n^{-1}\sum _{i=1}^n{\displaystyle \frac{|y_{si}-{\widehat{y}}_{si}|}{y_{si}}}\times 100\%\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathrm{NRMSE}}_s& ={\displaystyle \frac{\sqrt{n^{-1}{\sum }_{i=1}^n{(y_{si}-{\widehat{y}}_{si})}^2}}{y_{s,\max }-y_{s,\min }}}\hfill \end{array}$$

(10)

where $s\in \{\mathrm{urban},\mathrm{agricultural},\mathrm{industrial}\}$, $y_{si}$ is the observed value, and ${\widehat{y}}_{si}$ is the predicted value. These metrics collectively assess the magnitude, variance, and relative scale of forecasting errors.

A strict convergence criterion with a relative tolerance of ${10}^{-8}$ was applied, and parameter constraints were enforced to maintain the validity of the copula family.

Statistical analysis was performed using R (Version 4.3.1, R Foundation for Statistical Computing, Vienna, Austria). Copula modeling was implemented using the VineCopula package (Version 2.4.4) and the copula package (Version 1.1-0). Maximum likelihood estimation was conducted using the optim function with BFGS algorithm convergence tolerance of ${10}^{-8}$. Bootstrap resampling ($N=1000$) was performed using base R functions. Data visualization and plotting were created using ggplot2 (Version 3.4.2).

The AIC was used to evaluate the performance of the model.

3.4. Scenario-Based Forecasting Framework

Three scenarios were developed in alignment with Saudi Vision 2030 objectives [4], each considering differential impacts across water consumption sectors. The baseline scenario assumes moderate development with a growth in urban population of 2.4% annually, a growth in GDP of 3.5% annually, and a temperature increase of 0.06 °C annually, resulting in moderate growth in all water sectors with balanced development. The High Growth Scenario represents an accelerated development pathway with urban population growth of 3.2% annually, GDP growth of 5.5% annually, and temperature increases of 0.09 °C annually, leading to accelerated urban and industrial water demand with potential agricultural sector intensification. In contrast, the Sustainability Scenario emphasizes controlled development with urban population growth of 1.8% annually, GDP growth of 2.5% annually, and temperature increases of 0.04 °C annually, promoting controlled urban expansion, enhanced water efficiency across all sectors, and sustainable agricultural practices.

3.5. Water Demand Efficiency Analysis

To evaluate the sustainability performance of different development scenarios, this study introduces a Water Demand Efficiency metric that quantifies the relationship between demographic growth and water consumption patterns. This metric enables assessment of how effectively water resources are utilized under varying development pathways and provides insights into the potential for decoupling population growth from water demand increases.

The Water Efficiency Ratio (WER) is defined as the proportional relationship between population growth and corresponding changes in water demand:

$$\mathrm{WER}={\displaystyle \frac{\mathrm{Population}\mathrm{Growth}(\%)}{\mathrm{Water}\mathrm{Demand}\mathrm{Change}(\%)}}$$

(11)

where Population Growth (%) represents the percentage change in population from baseline year to forecast horizon and Water Demand Change (%) represents the percentage change in total water consumption over the same period. The Water Efficiency Ratio provides several interpretative insights: WER > 1 Water demand is growing slower than population growth, indicating improved water use efficiency per capita, WER = 1 Proportional growth between water demand and population (neutral efficiency), WER < 1 Water demand is growing faster than population growth, indicating declining water use efficiency, and Negative WER indicates absolute water demand reduction despite population growth (optimal efficiency) [1]. While, in this study, a Negative WER is interpreted as “optimal efficiency,” reflecting successful decoupling of water demand from population growth, it is important to note that in other contexts (e.g., parts of Africa), similar patterns may also indicate economic water scarcity resulting from supply constraints or affordability barriers. Thus, interpretation should always be context-specific.

3.6. Uncertainty Quantification

Model uncertainty evolution is captured through an exponential decay framework [33]:

$$\sigma \left(t\right)={\sigma }_0·e^{-\lambda t}$$

(12)

where $\sigma \left(t\right)$ represents forecast uncertainty at time t, ${\sigma }_0$ is initial uncertainty, and $\lambda$ is the decay Equation (12) provides the theoretical framework for uncertainty quantification. In practice, forecast uncertainty bounds were calculated empirically using bootstrap resampling ($N=1000$). The observed pattern of decreasing uncertainty (e.g., from $\pm 22.9$ to $\pm 3.7$ ML/day) reflects the conditional nature of scenario-based forecasting, where predetermined development pathways reduce forecast variance over time.

3.7. Assessing Water Demand Risks

A comprehensive risk assessment approach was designed to detect and evaluate any possible weaknesses in water demand under bad conditions [34]. This methodology incorporates three interrelated components: conditional risk analysis, Value-at-Risk (VaR) estimation, and stress testing [35].

Conditional risk analysis assesses how the probability of high water demand changes when a related variable, such as population or temperature, exceeds a critical threshold. This is formally expressed as follows [36]:

$$P(W_s>w_{extreme}\mid X_j>x_{stress})={\displaystyle \frac{C(F_s\left(w_{extreme}\right),F_j\left(x_{stress}\right))-F_j\left(x_{stress}\right)}{1-F_j\left(x_{stress}\right)}}$$

(13)

Here, $W_s$ denotes the water demand in sector s, and $X_j$ represents an influencing factor under stress (e.g., population or temperature). The function C is the fitted copula capturing the joint behavior of the variables, while $F_s$ and $F_j$ denote their respective marginal cumulative distribution functions. This equation quantifies the increased risk of extreme demand under specific stress conditions. This enables the model to estimate how the likelihood of extreme demand intensifies under scenarios such as rapid urban population growth or abnormal climatic events. By combining this equation with Value-at- Risk (VaR) and stress testing, we provide a comprehensive framework for identifying potential vulnerabilities in Saudi Arabia’s water system and generating early-warning indicators for policymakers. Value-at-Risk (VaR) was used to set essential water demand thresholds by estimating the highest expected demand under normal conditions at specific confidence levels (e.g., 90%, 95%, and 99%). These thresholds act as early warning indicators, emphasizing demand levels that are unlikely to be exceeded under normal conditions. Formally, the VaR at confidence level $\alpha$ is defined as [37]:

$${\mathrm{VaR}}_{\alpha }=F_s^{-1}\left(\alpha \right)$$

(14)

In this expression, $F_s^{-1}$ is the inverse distribution function for water demand in sector s, and $\alpha$ is the chosen confidence level.

Stress testing was applied to assess the resilience of the water demand system under hypothetical extreme conditions, following established hydrological risk assessment methods [38]. The approach included three main components: (i) univariate stress scenarios, which examined the impact of shocks to single variable (e.g., a only population increases); (ii) multivariate stress scenarios, which analyzed the effects of simultaneous changes in multiple factors (e.g., concurrent increases in population, GDP, and temperature); and (iii) tail dependence analysis, which measured how relationships between variables intensify during extreme events [25].

The tail dependence analysis calculated risk amplification factors (RAF) to quantify these effects:

$$RAF={\displaystyle \frac{{\lambda }_{\mathrm{extreme}}}{{\rho }_{\mathrm{baseline}}}}$$

where ${\lambda }_{\mathrm{extreme}}$ represents the tail dependence coefficient under crisis conditions, and ${\rho }_{\mathrm{baseline}}$ denotes the correlation under normal operating conditions [39].

4. Results and Analysis

This section presents the comprehensive analysis of water consumption patterns in Saudi Arabia using copula-based modeling. The results are organized into descriptive statistics, marginal distribution analysis, copula model selection and validation, forecasting performance, and scenario-based projections aligned with Saudi Vision 2030 objectives.

4.1. Descriptive Statistics and Data Characteristics

The dataset encompasses 17 years of annual observations (2008–2024) covering six variables representing Saudi Arabia’s water usage system.

Table 2. Summary statistics for Saudi Arabia water usage data (2008–2024).

Variable	Mean	Std Dev	Min	Max	Skewness	CV (%)
Urban Water Usage (ML/day)	3038.4	585.2	2007.0	3850.0	−0.294	19.26
Industrial Water Usage (ML/day)	954.4	279.9	628.0	1680.0	1.227	29.34
Agricultural Water Usage (ML/day)	15,447.3	3931.2	8500.0	21,200.0	−0.122	25.45
GDP	2.9	0.7	1.6	4.2	0.277	24.14
Population	29.0	3.3	23.3	34.3	−0.337	11.38
Temperature	26.6	0.4	25.8	27.3	−0.438	1.50

presents the summary statistics for all variables.

As shown in Table 2, the descriptive analysis highlights agriculture as the most water-intensive sector in Saudi Arabia, with an average consumption of 15,447.3 (ML/day), accounting for approximately 79% of the total water demand. Urban water usage exhibits a consistent upward trend, supported by a moderate coefficient of variation (CV = 19.3%), indicating stable growth in line with the Vision 2030 development goals. The industrial sector exhibits the highest positive skewness (1.227), suggesting occasional surges in water demand driven by periods of intensified industrial activity. The temperature remains relatively stable over the observed period ($\sigma =0.4$ °C), with a slight negative skewness, indicating infrequent cooler intervals within the predominantly arid climate.

4.2. Correlation and Dependence Structure Analysis

The correlation analysis (

Table 3. Correlation matrix for Saudi Arabia water consumption variables.

	Urban	Industrial	Agricultural	Population	GDP	Temperature
Urban Water	1.000			0.882 ***	0.676 ***	0.537 ***
Industrial Water		1.000		0.354	0.042	−0.014
Agricultural Water			1.000	−0.378	−0.390	−0.377
Population	0.882 ***	0.354	−0.378	1.000	0.816 ***	0.389 **
GDP	0.676 ***	0.042	−0.390	0.816 ***	1.000	0.283 *
Temperature	0.537 ***	−0.014	−0.377	0.389 **	0.283 *	1.000

Notes: Significance: *** p < 0.001, ** p< 0.01, * p< 0.05.

) reveals important insights regarding the dynamics of water consumption in Saudi Arabia. The strongest correlation is observed between urban water usage and population ($r=0.882$, $p<0.001$), indicating that urbanization is the primary driver of municipal water demand. Economic development also plays a crucial role, as evidenced by strong correlations between GDP and both urban water consumption ($r=0.676$, $p<0.001$) and population ($r=0.816$, $p<0.001$).

Climate effects are evident through moderate positive correlations between temperature and urban water demand ($r=0.537,p<0.001$), as well as population ($r=0.389,p<0.01$). These findings highlight the influence of arid conditions on consumption behaviors, particularly in terms of cooling needs. Agricultural water consumption shows consistent negative correlations with all socioeconomic variables (population: $r=-0.378$; GDP: $r=-0.390$; temperature: $r=-0.377$), suggesting a structural shift toward more efficient agricultural practices and potential competition for water resources among sectors. Industrial water exhibits weak correlations with all influencing factors, indicating unique consumption patterns that are less dependent on traditional demographic and economic variables.

4.3. Marginal Distribution Analysis

The marginal distribution of each variable was evaluated using multiple candidate distributions.

Table 4. Marginal distribution selection results.

Variable	Best Distribution	AIC	Parameters	K-S Test p-Value	Goodness-of-Fit
Urban Water	Weibull	268.0	shape = 6.27, scale = 3277.8	0.912	Excellent
Industrial Water	Log-Normal	239.1	$\mu$ = 6.82, $\sigma$ = 0.27	0.654	Good
Agricultural Water	Weibull	333.1	shape = 4.58, scale = 16,966.7	0.859	Excellent
GDP	Gamma	509.4	$\alpha$ = 16.6, $\beta$ = 0	0.830	Excellent
Population	Weibull	561.4	shape = 10.73, scale = 30,487,864.4	0.885	Excellent
Temperature	Weibull	23.3	shape = 73.45, scale = 26.8	0.971	Excellent

summarizes the best-fitting distributions based on AIC criteria.

The analysis of marginal distributions shows that five out of six study variables (83.3%) follow Weibull distributions, indicating underlying growth-related processes with predictable patterns. All variables exhibit strong statistical fits, as evidenced by K-S test p-values ranging from 0.654 to 0.971, supporting confidence in the chosen distributional assumptions. The temperature shows the best fit ($p=0.971$, AIC = 23.3) with a notably high shape parameter (73.45), reflecting the climatic stability in the study area. In contrast, Industrial Water follows a Log-Normal distribution ($\mu =6.82$, $\sigma =0.27$), suggesting multiplicative growth dynamics and higher variability. This distinct behavior highlights the need for specialized modeling approaches to account for its inherent uncertainty.

The dominance of Weibull distributions in urban water, agricultural water, GDP, population, and average temperature supports the application of reliability-based forecasting methods for robust water resource planning. However, the Log-Normal nature of Industrial Water requires alternative analytical techniques to address its greater uncertainty.

4.4. Multivariate Copula Analysis Results for Urban Water

This section analyzes urban water consumption through multivariate copula modeling, focusing on the interdependencies between urban demand and three variables: population, GDP, and temperature. The analysis evaluates various copula families to determine the most effective forecasting framework and offers scenario-based estimates for Saudi Vision 2030.

4.4.1. Model Selection and Performance

Table 5. Comprehensive 4D Copula Model Performance Comparison for Urban Water Consumption.

Rank	Model	Type	AIC	BIC	Log-Lik	Parameters
1	4D Normal	Elliptical	−74.83	−69.83	43.41	6
2	4D Student’s t	Elliptical	−73.71	−67.87	43.85	7
3	4D Gumbel	Archimedean	−49.08	−48.24	25.54	1
4	4D Clayton	Archimedean	−48.41	−47.57	25.20	1
5	4D Frank	Archimedean	−47.62	−46.79	24.81	1
6	4D Joe	Archimedean	−40.86	−40.02	21.43	1

presents a comprehensive comparison of 4D copula models for forecasting urban water consumption. The 4D normal copula demonstrated superior performance, with the lowest AIC value (−74.83) and the highest log-likelihood (43.41), making it the optimal choice for capturing the joint behavior between urban water consumption and its predictors (population, GDP, and temperature). The elliptical copulas (Normal and Student’s t) significantly outperformed Archimedean copulas, indicating that urban water dependencies are best characterized by symmetric, linear relationships rather than asymmetric tail dependencies.

Pairwise copula models were also evaluated to understand the individual relationships between urban water consumption and each predictor variable.

Table 6. Pairwise copula model performance for urban water consumption forecasting.

Rank	Model	Type	AIC	BIC	Log-Lik	Variables
1	2D Student’s t	Pairwise	−47.80	−46.96	24.90	Urban|Population
2	2D Normal	Pairwise	−19.75	−18.91	10.87	Urban|GDP
3	2D Gumbel 180°	Pairwise	−12.13	−11.29	7.06	Urban|Temperature

presents these results. The strongest dependency between urban water and population was observed, best captured by the 2D Student’s t copula (AIC = 47.80). GDP showed a moderate association, with the Normal copula performing best, while temperature exhibited the weakest dependency, modeled effectively using the Gumbel 180° copula.

4.4.2. Parameter Estimates and Statistical Significance

The 4D normal copula model was estimated using the maximum likelihood method to capture the dependency structure between urban water consumption and its predictors (see

Table 7. 4D Normal copula parameter estimates and statistical significance.

Parameter	Description	Estimate	Std. Error	t-Statistic	p-Value
${\rho }_{12}$	Urban—Population	0.7834	0.0892	8.781	<0.001 ***
${\rho }_{13}$	Urban—Temperature	0.4267	0.1245	3.428	$0.003$ **
${\rho }_{14}$	Urban—GDP	0.6521	0.1038	6.284	<0.001 ***
${\rho }_{23}$	Population—Temperature	0.3892	0.1167	3.337	$0.004$ **
${\rho }_{24}$	Population—GDP	0.8156	0.0756	10.789	<0.001 ***
${\rho }_{34}$	Temperature—GDP	0.2834	0.1334	2.125	$0.045$ *

Notes: Significance codes: *** p < 0.001, ** p < 0.01, * p < 0.05.

). The results indicate strong and highly significant correlations between urban water demand and both population (${\rho }_{12}=0.7834,p<0.001$) and GDP (${\rho }_{14}=0.6521,p<0.001$), confirming these as the primary determinants of municipal water consumption. Temperature exhibits a moderate but statistically significant relationship with urban water consumption (${\rho }_{13}=0.4267,p=0.003$), highlighting the impact of climate factors, particularly in terms of cooling needs.

Among the predictors, there is a significant interdependency between population and GDP (${\rho }_{24}=0.8156,p<0.001$), underscoring the strong relationship between demographic factors and economic growth. The population shows a moderate correlation with temperature (${\rho }_{23}=0.3892,p=0.004$), while the relationship between temperature and GDP is weaker but still statistically significant (${\rho }_{34}=0.2834,p=0.045$). These findings reveal a clear order of importance for factors influencing urban water demand: population is the strongest influence, followed by GDP, which is the second most significant factor, and temperature has a smaller but still meaningful impact. The conditional forecasting approach was validated using leave-one-out cross-validation given the limited 17-year dataset. Bootstrap confidence intervals (N = 1000) were computed for all accuracy metrics to assess forecast reliability.

4.4.3. Forecasting Performance

Table 8. Performance metrics for 4D normal copula model.

Metric	Value	Performance Level
RMSE	231.74 units	Low prediction error variance
MAE	188.03 units	Consistent reliability
MAPE	6.37%	Excellent accuracy
NRMSE	12.57%	Acceptable precision

summarizes the accuracy results of the 4D normal copula model based on the evaluation metrics.

The model achieved an MAPE of 6.37%, indicating that predictions typically fall within 6.4% of observed values, which represents excellent accuracy for urban water consumption forecasting. The RMSE of 231.74 units and MAE of 188.03 units demonstrate low prediction variance and consistent reliability, respectively. The NRMSE of 12.57% indicates acceptable precision relative to the data scale.

Model validation through residual analysis confirmed the suitability of the chosen copula structure. The Shapiro–Wilk normality test yielded a W statistic of 0.9787 with a p-value of 0.9443, indicating that the residuals follow a normal distribution (p > 0.05), thereby validating the underlying model assumptions. This statistical confirmation, combined with the low prediction errors, establishes the robustness of the model to capture complex multivariate dependencies between urban water demand and its predictors: population, GDP, and temperature. The comprehensive performance metrics demonstrate the suitability of the model for strategic planning of water resources and operational forecasting in urban environments, providing a reliable foundation for policy making and infrastructure development in support of Saudi Vision 2030.

Table 9. Urban water consumption forecasts (ML/day) under different scenarios (2025–2030).

Year	Baseline		High Growth		Sustainability
	Urban (ML/Day)	±Uncertainty	Urban (ML/Day)	±Uncertainty	Urban (ML/Day)	±Uncertainty
2025	3722.5	±22.9	3780.4	±21.5	3609.1	±20.8
2026	3657.1	±18.4	3770.6	±17.3	3438.0	±16.4
2027	3591.4	±14.2	3760.1	±14.7	3273.2	±12.3
2028	3528.6	±10.5	3749.3	±10.8	3115.9	±8.7
2029	3465.4	±7.7	3739.6	±7.3	2966.7	±6.3
2030	3403.4	±3.7	3728.1	±4.2	2824.3	±3.1

summarizes the forecasted urban water consumption for 2025–2030 under three scenarios: Baseline, High Growth, and Sustainability. In the Baseline scenario, urban demand decreases gradually from 3722.5 ML/day in 2025 to 3403.4 ML/day in 2030, with forecast uncertainty narrowing significantly from ±22.9 to ±3.7 ML/day. The High Growth scenario, characterized by accelerated population and economic expansion, maintains high consumption levels, declining only slightly from 3780.4 to 3728.1 ML/day, with persistently elevated uncertainty (±4.2 ML/day by 2030). In contrast, the Sustainability scenario exhibits the steepest decline in water use, dropping from 3609.1 to 2824.3 ML/day, accompanied by the lowest uncertainty range (±3.1 ML/day in 2030). These results indicate that while the High Growth pathway sustains high demand and risk, the Sustainability trajectory achieves substantial demand reduction and enhanced forecasting confidence.

Figure 2 depicts the historical trend in urban water demand from 2008 to 2024, together with forecasts for the years 2025–2030 under three scenarios: Baseline, High Growth, and Sustainability. Historical data indicate a consistent rise in urban water demand, which is expected to reach its peak around 2024. The Baseline scenario anticipates a steady decrease in demand, whereas the High Growth scenario sustains generally stable and heightened levels owing to swift demographic and economic growth. Conversely, the Sustainability scenario predicts the most significant decrease in demand, indicative of the effects of efficiency enhancements and conservation strategies. These findings are particularly relevant to the goals of Vision 2030 [4], which emphasizes improved municipal service delivery, reduced non-revenue water, and promotion of water conservation technologies in cities. The copula-based model captures how urban water demand responds to population growth and climate variables, supporting smart urban planning and infrastructure investments aligned with these goals.

Table 10. Comparative analysis of urban water scenarios by 2030.

Parameter	Baseline	High Growth	Sustainability
Population Growth (%)	13.1	30.7	7.7
Temperature Increase (°C)	+1.0	+2.3	+0.4
GDP Growth (%)	18.8	37.1	13.1
Urban Water Demand Change (%)	$-8.6$	$-1.4$	$-21.7$
Final Uncertainty (ML/day)	$\pm 3.7$	$\pm 4.2$	$\pm 3.1$
2030 Consumption (ML/day)	3403.4	3728.1	2824.3
Water Efficiency Ratio	1.52	21.93	0.35
Risk Level	Good	Excellent	Low

provides a comparative summary of several indicators under three future scenarios for 2030. The High Growth scenario reflects the most intensive pathway, characterized by a 30.7% increase in population, a 2.3 °C rise in temperature, and a 37.1% expansion in GDP. Despite these substantial growth rates, it achieves only a marginal reduction in water demand (−1.4%) and exhibits the highest forecast uncertainty (±4.2 ML/day), indicating elevated risk levels. The Baseline scenario assumes moderate demographic and economic growth, leading to an 8.6% decline in urban water demand with a forecast uncertainty of ±3.7 ML/day. In contrast, the Sustainability scenario delivers the most favorable outcomes, achieving a 21.7% reduction in demand despite a 7.7% population increase, reaching the lowest consumption level (2824.3 ML/day) and showing the smallest uncertainty (±3.1 ML/day). In the Baseline scenario (WER = 1.52), results show good efficiency with demand growing slower than population. The High Growth scenario (WER = 21.93) reflects excellent efficiency, as demand growth is far below rapid population growth. The Sustainability scenario (WER = 0.35) appears as lower efficiency, but actually represents a positive decoupling, since water demand declines (−21.7%) despite population growth (+7.7%). These results highlight the important role of integrated policy and planning strategies in shaping future water security and ensuring resilience under varying growth pathways.

4.4.4. Multivariate Risk Assessment of Urban Water Demand

The multivariate copula-based risk assessment identified significant weaknesses in urban water demand during stress conditions. When population and temperature stresses occur simultaneously, the probability of extreme water demand rises sharply from baseline levels of 10–15% to over 60%, representing a four-fold increase in risk.

The Value-at-Risk (VaR) analysis identified extreme demand thresholds of 3720.94 ML/day (90% confidence), 3792.93 ML/day (95% confidence) and 3838.77 ML/day (99% confidence), with tail risk premiums decreasing from 69.56 to 5.54 ML/day as confidence levels increased. The relatively light-tailed distribution indicates a concentrated risk in the 90 to 95% range rather than in extreme outliers.

Stress testing confirmed the scenario of ‘Combined Stress’ as the most severe, producing demand multipliers of 1.8 to 2.2 times the baseline level. Isolated temperature stress alone increased demand by 1.4–1.6 times baseline, underscoring temperature as the primary risk factor. In particular, tail correlation analysis showed an increase in 71% in the temperature–demand relationship during extreme events, with the correlation increasing from 0.34 to 0.58. This suggests that standard risk measures can underestimate actual exposure to the crisis.

Conditional risk dependencies offer early warning indicators: during temperature stress, the likelihood of extreme urban water demand is 45%. This quantified relationship facilitates proactive management of water systems and enhances emergency preparedness by allowing for timely interventions to mitigate potential crises.

4.5. Multivariate Copula Analysis Results for Industrial Water Demand

This section examines industrial water consumption through multivariate copula modeling, focusing on the relationships between industrial demand and three critical variables: population, GDP, and average temperature.

4.5.1. Model Selection and Performance

Table 11. Comparison of 4D Copula Models for Industrial Water Demand.

No.	Model	Type	AIC	BIC	LogLik	Parameters
1	4D NORMAL	Elliptical	$-18.84$	$-13.84$	$15.42$	6
2	4D T	Elliptical	$-17.23$	$-11.40$	$15.62$	7
3	4D CLAYTON	Archimedean	$-8.42$	$-7.58$	$5.21$	1
4	4D GUMBEL	Archimedean	$-5.86$	$-5.02$	$3.93$	1
5	4D FRANK	Archimedean	$-5.78$	$-4.95$	$3.89$	1
6	4D JOE	Archimedean	$-3.25$	$-2.41$	$2.62$	1

presents a comparison of 4D copula models fitted to industrial water demand data. Among the candidate models, the 4D normal copula provided the best overall fit, achieving the lowest AIC ($-18.84$) value. This indicates that the dependency structure of industrial water demand is best represented through symmetric, linear relationships.

The pairwise analysis of the copula, detailed in

Table 12. Pairwise copula models for industrial water demand.

No.	Model	Type	AIC	BIC	LogLik	Variables
1	2D Gumbel_180	Pairwise	$-0.44$	$0.40$	$1.22$	Industrial|Population
2	2D Frank_270	Pairwise	$0.28$	$1.12$	$0.86$	Industrial|Temperature
3	2D t	Pairwise	$0.94$	$1.77$	$0.53$	Industrial|GDP

, indicates that industrial water consumption demonstrates the most significant individual dependency on the population, effectively represented by the Gumbel_180 copula. The relationships with average temperature and GDP are considerably weaker, indicating different consumption patterns less directly related to conventional socioeconomic predictors.

4.5.2. Parameter Estimates and Dependency Structure

The analysis of the parameter estimates in (

Table 13. Parameter estimates with statistical significance.

Parameter	Description	Estimate	Std Error	t-Value	p-Value
${\rho }_{12}$	Ind-Pop	0.3538	0.2368	1.4938	0.1352
${\rho }_{13}$	Ind-Temp	$-0.0136$	0.2916	$-0.0468$	0.9627
${\rho }_{14}$	Ind-GDP	0.0420	0.2827	0.1487	0.8818
${\rho }_{23}$	Pop-Temp	0.7089	0.1077	6.5833	<0.001 ***
${\rho }_{24}$	Pop-GDP	0.8024	0.0721	11.1354	<0.001 ***
${\rho }_{34}$	Temp-GDP	0.6510	0.1254	5.1912	<0.001 ***

Notes: Significance levels: *** p < 0.001.

) shows weak and statistically insignificant correlations between industrial water demand and its covariates: population (${\rho }_{12}$ = 0.3538, $p=0.1352$), temperature (${\rho }_{13}$ = $-0.0136$, $p=0.9627$), and GDP (${\rho }_{14}$ = 0.0420, $p=0.8818$). This can be explained by the fact that the majority of industrial water demand is concentrated in planned areas such as Yanbu and Jubail, where water is distributed according to set government agreements and provided by desalination or treated wastewater, making it less sensitive to population or GDP changes. Furthermore, average temperature changes have minimal impact on companies that frequently use closed-loop water recycling systems. Unless there are significant expansions or new projects, industrial water use remains constant, in contrast to municipal demand.

4.5.3. Forecasting Performance

As shown in

Table 14. Performance metrics of the 4D normal copula model for industrial water consumption.

Metric	Value	Performance Level
RMSE	272.07	Low prediction error variance
MAE	182.22	Consistent reliability
MAPE	17.51%	Reasonable accuracy
NRMSE	25.86%	Moderate precision

, the 4D normal copula model demonstrates moderate predictive accuracy for industrial water consumption. The model’s MAPE of 17.51% reflects reasonable but not high forecasting precision. The RMSE (272.07 ML/day) and MAE (182.22 ML/day) indicate occasional large prediction errors, while the NRMSE of 25.86% suggests moderate precision relative to the scale of the data.

As shown in

Table 15. Industrial water consumption forecasts (ML/day) under different scenarios (2025–2030).

Year	Baseline		High Growth		Sustainability
	Demand (ML/Day)	±Uncertainty	Demand (ML/Day)	±Uncertainty	Demand (ML/Day)	±Uncertainty
2025	980.2	±230.6	1013.2	±249.6	960.3	±237.7
2026	984.3	±239.7	1026.6	±254.3	915.0	±223.6
2027	959.9	±228.7	1024.1	±255.5	879.0	±218.7
2028	939.3	±224.3	1004.8	±250.5	841.3	±204.0
2029	923.0	±222.1	1001.2	±247.4	797.4	±194.0
2030	921.3	±234.3	1000.5	±247.8	764.9	±192.8

, the forecasts of industrial water consumption reveal distinct performance in three scenarios from 2025 to 2030. The sustainability scenario achieves the best results, with a 20.4% reduction (from 960.3 to 764.9 ML/day) and a decrease in uncertainty (±237.7 to ±192.8 ML/day), demonstrating both water conservation and improved forecast reliability. The High Growth scenario maintains high consumption levels with minimal decline (1.3%), peaking at 1026.6 ML/day in 2026 before stabilizing around 1000 ML/day, but carries consistently high uncertainty (±247–255 ML/day). The baseline scenario shows moderate performance with a reduction of 6. 0% to 921.3 ML/day by 2030. Importantly, all scenarios demonstrate successful decoupling of industrial water demand from economic growth, with uncertainty levels generally decreasing over time, indicating that the forecasting model becomes more reliable for longer-term strategic planning aligned with Saudi Arabia’s Vision 2030 objectives.

Figure 3 displays the historical trend in industrial water demand from 2008 to 2024, alongside future predictions for 2025 to 2030 across three development scenarios. Historical data show a steady increase in industrial demand until 2020, followed by a sharp decline likely due to operational or policy changes. Forecasts indicate divergent pathways beyond 2025: the High Growth scenario maintains nearly constant demand, reflecting intensive industrial activity; the Baseline scenario shows a moderate decline, suggesting gradual efficiency improvements; and the Sustainability scenario realizes the most pronounced reduction, underscoring the effects of conservation strategies and technological progress. These trends emphasize the important role of policy interventions in influencing future industrial consumption patterns and ensuring compliance with long-term resource sustainability objectives. In the context of Vision 2030, the projected industrial expansion must be balanced with optimized water use [4]. Vision 2030 aims to diversify the economy through industrial growth, particularly in mining, logistics, and manufacturing. The model enables forecasting of industrial water demand under various growth pathways, allowing for the design of resource-efficient industrial policies and water allocation strategies that ensure sustainable industrialization.

The analysis presented in

Table 16. Comparative analysis of industrial water scenarios by 2030.

Parameter	Baseline	High Growth	Sustainability
Population Growth (%)	13.1	30.7	7.7
Temperature Increase (°C)	+1.0	+2.3	+0.4
GDP Growth (%)	18.8	37.1	13.1
Industrial Water Demand Change (%)	$-6.0$	$-1.3$	$-20.4$
Final Uncertainty (ML/day)	$\pm 234.3$	$\pm 247.8$	$\pm 192.8$
2030 Consumption (ML/day)	921.3	1000.5	764.9
Water Efficiency Ratio	2.18	23.62	0.38
Risk Level	Good	Excellent	Low

shows all three 2030 scenarios achieve water demand reductions despite economic growth, demonstrating successful decoupling through efficiency improvements. The Sustainability scenario performs best with a 20.4% reduction to 764.9 ML/day and lowest risk. The Baseline scenario achieves moderate results with 6.0% reduction to 921.3 ML/day. The High Growth scenario, despite having the highest efficiency gains, shows only 1.3% reduction due to substantial industrial expansion, resulting in the highest consumption (1000.5 ML/day) but maintains excellent risk management. All scenarios confirm that industrial growth does not necessitate proportional water consumption increases when coupled with efficiency measures.

4.5.4. Multivariate Risk Assessment of Industrial Water Demand

The 4D normal copula risk assessment revealed substantial weaknesses in industrial water demand when subjected to multivariate stress conditions. Conditional risk analysis revealed an important finding: the probability of extreme industrial water demand increases to 42.8% under population stress, in contrast to baseline probabilities ranging from 16% to 26%. This signifies an almost threefold increase in risk, positioning population growth as the primary factor over temperature changes.

The Value-at-Risk (VaR) analysis determined extreme demand thresholds of 1400.00 ML/day (90% confidence), 1470.75 ML/day (95%), and 1634.31 ML/day (99%), with tail risk premiums declining from 71.97 ML/day to 23.92 ML/day at higher confidence levels. This suggests that extreme events are well-bounded, with limited catastrophic tail risks.

Stress testing confirmed the High Population scenario as the most critical, showing demand multipliers of 1.14 times baseline levels. Population stress alone increased demand by 14%, while temperature stress contributed only a 1% These conditional risk dependencies offer actionable early warning insights: when population stress is observed, there is a 42.8% probability of extreme industrial water demand. This provides a strong basis for demographic-focused management strategies and enhances the operational value of planning and capacity management in industrial water systems.

4.6. Multivariate Copula Analysis Results for Agriculture Water

This section investigates agricultural water consumption using multivariate copula modeling, emphasizing the interrelationships between agricultural demand and three predictors: population, GDP, and average temperature. The analysis assesses various copula families to determine the most effective forecasting framework and provides scenario-based forecasts for Saudi Vision 2030.

4.6.1. Model Selection and Performance for Agricultural Water Consumption

The multivariate copula analysis of agricultural water consumption evaluated six distinct copula models to determine the most effective approach to forecasting.

Table 17. Copula model comparison for agricultural water consumption.

Rank	Model	Type	AIC	BIC	Log-Likelihood	Parameters
1	4D Normal	Elliptical	$-14.44$	$-9.44$	$13.22$	6
2	4D t	Elliptical	$-12.39$	$-6.56$	$13.20$	7
3	4D Clayton	Archimedean	$-0.97$	$-0.14$	$1.48$	1
4	4D Gumbel	Archimedean	$1.13$	$1.96$	$0.44$	1
5	4D Frank	Archimedean	$1.17$	$2.00$	$0.42$	1
6	4D Joe	Archimedean	$1.77$	$2.60$	$0.12$	1

shows the results of the entire model comparison. The 4D normal copula was identified as the superior model, with the lowest AIC value of −14.44, and was much more than two AIC points superior to all other models. This suggests that relationships between agricultural water consumption, population, average temperature, and GDP are more accurately represented by symmetric linear dependencies typical of multivariate Gaussian models.

These results confirm that agricultural water demand is best modeled by symmetric linear dependence structures, with population emerging as a primary influencing factor.

An important observation during the analysis was the presence of structural breaks in the agricultural water consumption time series. These breaks, occurring around 2020, coincide with major policy interventions and shifts in agricultural subsidy programs under Vision 2030. The identified structural change indicates a departure from historical consumption patterns, suggesting that traditional forecasting models relying on stationarity assumptions may be inadequate. The copula-based approach proved effective in adapting to these regime changes, capturing altered dependency structures between agricultural demand and its drivers, such as GDP and temperature. Recognizing structural breaks is crucial for accurate forecasting and for ensuring that future policy planning accounts for both historical trends and recent transformations in the agricultural sector.

4.6.2. Pairwise Dependency Analysis

To understand individual relationships between agricultural water consumption and each predictor variable, pairwise copula models were evaluated (see

Table 18. Two-dimensional copula models (pairwise dependencies for agriculture).

Model	Type	AIC	BIC	LogLik	Variables
2D frank	Pairwise	0.33697	1.17018	0.83152	Agricultural|GDP
2D clayton	Pairwise	0.67019	1.50341	0.66490	Agricultural|Temperature
2D normal	Pairwise	0.73128	1.56449	0.63436	Agricultural|Population

). The results outline a distinct structure of an indicator impact on agricultural water consumption. The population exhibits the most substantial individual reliance, optimally represented by the normal copula, underscoring its central influence on agricultural demand. The GDP shows a moderate correlation, accurately reflected by the Frank copula, but the temperature has the poorest relationship, characterized by the Clayton copula.

4.6.3. Parameter Estimates and Statistical Significance

Table 19. Four-dimensional normal copula parameter estimates and statistical significance.

Parameter	Description	Estimate	Std. Error	t-Statistic	p-Value
${\rho }_{12}$	Agricultural—Population	−0.3780	0.2344	−1.6124	0.1069
${\rho }_{13}$	Agricultural—Temperature	−0.3766	0.2358	−1.5966	0.1104
${\rho }_{14}$	Agricultural—GDP	−0.3899	0.2313	−1.6855	0.0919
${\rho }_{23}$	Population—Temperature	0.7285	0.0960	7.5851	<0.001 ***
${\rho }_{24}$	Population—GDP	0.8161	0.0642	12.7127	<0.001 ***
${\rho }_{34}$	Temperature—GDP	0.6571	0.1204	5.4557	<0.001 ***

Notes: Significance codes: *** p < 0.001.

demonstrates the parameter estimates for the 4D normal copula. Agricultural water demand exhibits negative correlations with all covariates. Nonetheless, the correlation with GDP (${\rho }_{14}=-0.3899$, $p=0.0919$) is marginally significant at the 10% level. This pattern may indicate ongoing structural changes in the agricultural sector, including improved water-use efficiency and dependence on non-renewable groundwater resources.

4.6.4. Model Performance Assessment

As shown in

Table 20. Analytical method performance metrics for agricultural water consumption.

Metric	Value	Result Interpretation
RMSE	3970.350	High prediction error variance
MAE	3364.693	Moderate reliability
MAPE	23.20%	Acceptable accuracy
NRMSE	31.26%	Low-to-moderate precision

, the model demonstrates moderate forecasting performance. The MAPE of 23.20% indicates acceptable long-term prediction accuracy, but relatively high RMSE (3970.35 ML/day) and NRMSE (31.26%) values suggest limited precision and potential for refinement in the modeling approach.

Table 21. Agricultural water consumption forecasts (ML/day) under different scenarios (2025–2030).

Year	Baseline		High Growth		Sustainability
	Agricultural (ML/Day)	±Uncertainty	Agricultural (ML/Day)	±Uncertainty	Agricultural (ML/Day)	±Uncertainty
2025	12,841.4	±3192.6	13,309.7	±3336.5	12,521.0	±3181.8
2026	12,747.8	±3238.5	13,145.7	±3285.8	11,998.0	±3064.7
2027	12,185.5	±3109.8	12,927.5	±3213.0	10,989.1	±2730.6
2028	11,701.6	±2961.5	12,564.3	±3136.5	10,434.2	±2684.0
2029	11,408.2	±2879.9	12,140.5	±3125.7	9687.9	±2503.8
2030	11,031.4	±2839.2	11,597.4	±2808.0	9105.8	±2295.8

indicates a steady decrease across all scenarios from 2025 to 2030. The sustainability scenario realizes the greatest significant reduction, decreasing demand by 27.3% (from 12521 ML/day to 9105.8 ML/day) while demonstrating the least uncertainty. The high growth scenario sustains the highest consumption levels, showing a slight decrease of 12.9%, whereas the baseline scenario indicates a reduction of 14.1%. These trends underscore the potential for substantial conservation advancements through sustainability-oriented policy. These reductions align with Vision 2030 agricultural reforms that aim to transition to low-water-use crops, increase treated wastewater reuse, and modernize irrigation. The forecasting framework assesses the impact of such efficiency policies under the sustainability scenario, providing data-driven insight to support agricultural transformation while reducing environmental stress.

4.6.5. Scenario-Based Forecasts for Agricultural Water Demand (2025–2030)

Figure 4 shows the historical evolution of agricultural water demand in Saudi Arabia, which rose steadily until peaking before 2016, followed by a sharp decline and subsequent stabilization around 2024. The projections for 2025–2030 illustrate three alternative development pathways: the Green Agricultural Scenario, which produces the steepest reduction in demand; the Baseline Scenario, which sustains moderate consumption levels with a gradual decline; and the High Agricultural Growth Scenario, which maintains comparatively higher demand with only a slight reduction.

4.6.6. Comparative Scenario Analysis of Agricultural Water by (2030)

Table 22. Comparative analysis of agricultural water scenarios by 2030.

Parameter	Baseline	High Growth	Sustainability
Population Growth (%)	13.1	30.7	7.7
Temperature Increase (°C)	+1.0	+2.3	+1.0
GDP Growth (%)	18.8	37.1	13.1
Agricultural Water Demand Change (%)	+12.5	+28.9	$-8.2$
2030 Consumption (ML/day)	2156.7	2847.3	1764.2
Water Efficiency Ratio	0.68	0.61	0.85
Risk Level	Lower	Lower	Lower

presents a comprehensive comparison of the three scenarios and their implications for 2030. The high growth scenario results in a significant 28.9% increase in water demand, accompanied by the lowest water efficiency ratio (0.61), indicating considerable risk. In contrast, the sustainability scenario achieves an 8.2% decrease in water demand and attains the lower efficiency ratio (0.85), indicating successful conservation and negligible risk. The baseline scenario yields moderate performance accompanied by balanced growth.

4.6.7. Multivariate Risk Assessment

The 4D normal copula risk assessment revealed moderate weaknesses in agricultural water demand in multivariate stress situations. Conditional risk analysis indicated a comparable probability of exceedance for the stress scenarios of population (14.5%), temperature (14.3%), and GDP (13.8%), implying the absence of a predominant single risk component. However, GDP surfaced as the main weakness, with low GDP scenarios yielding the largest demand multiplier (1.12 times baseline levels).

The Value-at-Risk analysis revealed severe demand limits of 20,234.76 ML/day (90% confidence), 20,902.99 ML/day (95% confidence), and 21,136.28 ML/day (99% confidence). Tail risk premiums decreased from 601.38 ML/day to 32.77 ML/day as confidence levels increased, indicating well-defined catastrophic risks.

Stress testing identified the “Low GDP” scenario as the most severe. In contrast, tail correlation analysis revealed positive risk amplification across all variables (e.g., GDP: +0.244), suggesting that dependencies intensify during extreme events. These findings highlight the importance of economically oriented management options in addressing weaknesses in agricultural water systems.

5. Discussion

This study highlights the effectiveness of copula-based models in forecasting water demand across urban, industrial, and agricultural sectors in Saudi Arabia. These models capture complex, nonlinear, and asymmetric dependencies between water consumption and its drivers, namely population, GDP, and average temperature, offering advantages over traditional linear approaches.

5.1. Comparison and Discussion of Sectoral Model Performance

This section integrates the results of the urban, industrial, and agricultural sectors to evaluate the performance of the forecast, the results of the scenarios, and the characteristics of the risk.

The urban water model demonstrated the highest predictive accuracy, achieving a MAPE of 6.37% and showing strong dependency on population growth and GDP. This confirms that urbanization and economic development are the primary drivers of municipal water consumption. In contrast, the industrial model exhibited weaker correlations with its predictors and a MAPE of 17.51%, suggesting more stable demand patterns influenced by centralized industrial planning and limited sensitivity to climatic factors. The agricultural sector model, with a MAPE of 23.20%, showed the greatest variability and sensitivity to population and climate conditions, reflecting structural shifts in irrigation practices and resource allocation.

The presence of structural breaks in the agricultural water demand data particularly around 2020, reflects the impact of major policy interventions and agricultural subsidy reforms introduced under Vision 2030. These shifts mark a clear deviation from historical consumption trends and reinforce the necessity of using forecasting frameworks that do not rely on strict stationarity assumptions.

The weak and statistically insignificant correlations in the industrial water model (${\rho }_{12}=0.3538$, $p=0.1352$ for population; ${\rho }_{14}=0.0420$, $p=0.8818$ for GDP) indicate that conventional socioeconomic drivers do not adequately explain industrial demand in Saudi Arabia. This results in higher forecasting uncertainty (MAPE = 17.51% vs. 6.37% for urban) and reflects the sector’s reliance on centralized allocation, government contracts, and recycling systems that decouple demand from market forces. Policy efforts should therefore emphasize capacity planning, efficiency standards, and supply reliability rather than demand forecasting. While this independence suggests resilience during downturns, alternative monitoring indicators are needed for effective early warning in the industrial sector.

The copula based model demonstrated strong adaptability to such structural changes, accurately capturing the evolving dependency relationships between agricultural water use and its key drivers. From a policy standpoint, recognizing and incorporating these structural breaks is essential for designing resilient and forward-looking strategies that align with Saudi Arabia’s transition toward sustainable agriculture. Failing to account for regime shifts could lead to under- or overestimation of future demand, ultimately affecting water allocation, pricing, and investment decisions.

5.1.1. Sectoral Forecasting Performance Comparison

The 4D normal copula exhibited optimal performance in all three sectors, albeit with differing levels of precision and accuracy.

Table 23. Comparison of forecasting accuracy and risk by sector in 2030.

Sector	Best Model	MAPE (%)	Dominant Risk Driver	Risk Level (2030)
Urban	4D Normal Copula	6.37	Population & Temperature	Very Low (Sustainability)
Industrial	4D Normal Copula	17.51	Population Stress	High (High Growth)
Agricultural	4D Normal Copula	23.20	GDP Stress	Critical (High Growth)

presents the main performance indicators and risk insights for each sector. The analysis indicates that the urban sector achieved the highest predictive accuracy (MAPE = 6.37%), with water demand significantly affected by population growth and variations in temperature. The Sustainability scenario in this sector presents optimal results, achieving substantial reductions in water consumption while minimizing associated risks. The industrial sector demonstrates moderate prediction accuracy (MAPE = 17.51%), showing a reduced direct reliance on predictors. However, it is highly sensitive to stress related to population growth, underscoring the need for proactive management strategies. The agricultural sector exhibits the lowest predictive precision (MAPE = 23.20%) and the highest level of uncertainty. In this context, GDP is identified as a major risk factor, with the high growth scenario leading to serious water stress. However, the sustainability scenario continues to provide considerable conservation advantages, highlighting the efficacy of focused policy measures.

5.1.2. Scenario-Based Comparative Analysis

Table 24. Scenario-based forecast comparison across sectors in 2030.

Sector	Scenario	Demand Change (%)	2030 Demand (ML/Day)	WER	Risk Level
Urban	Baseline	−8.6	3403.4	1.52	Good
	High Growth	−1.4	3728.1	21.93	Excellent
	Sustainability	−21.7	2824.3	0.35	Low
Industrial	Baseline	−6.0	921.3	2.18	Good
	High Growth	−1.3	1000.5	23.62	Excellent
	Sustainability	−20.4	764.9	0.38	Low
Agricultural	Baseline	+12.5	11,031.4	0.68	Low
	High Growth	+28.9	13,309.7	0.61	Low
	Sustainability	−8.2	9105.8	0.85	Low

provides a comparative analysis of various sectors for the year 2030, focusing on projected demand, WER, and related risk levels. The analysis reveals that the sustainability scenario consistently delivers optimal outcomes across all sectors, achieving substantial reductions in water demand (21.7% in urban, 20.4% in industrial, and 8.2% in agricultural) while maintaining low risk levels throughout.

Under the sustainability scenario, the low WER values (0.35 urban, 0.38 industrial, 0.85 agricultural) indicate that demand reductions significantly outpace population growth, reflecting successful efficiency interventions. Conversely, the high growth scenario exhibits exceptionally high WER values (21.93 urban, 23.62 industrial), suggesting that despite rapid population expansion, water demand changes remain relatively modest due to efficiency measures, though absolute consumption levels remain elevated.

Sector-specific analysis reveals distinct vulnerability patterns: the urban sector demonstrates exceptional demand management potential under sustainability conditions; the industrial sector shows capacity for maintaining stable consumption while supporting economic growth; and the agricultural sector presents unique challenges, being the only sector showing demand increases under baseline (+12.5%) and high growth (+28.9%) scenarios, yet maintaining consistently low risk levels across all scenarios.

Risk assessments incorporate multiple factors beyond WER calculations, including absolute consumption levels, climate impacts (+0.4 °C for sustainability vs. +2.3 °C for high growth), and system resilience. The agricultural sector’s consistent low risk ratings across scenarios (despite demand increases in two scenarios) suggest inherent sector resilience, while urban and industrial sectors show risk level improvements correlating with demand reduction magnitude.

5.1.3. Risk Assessment Comparison

The conditional risk behavior and Value-at-Risk (VaR) metrics exhibit substantial variation among the three sectors.

Table 25. Risk Assessment Summary Across Water Sectors (2030).

Sector	Extreme Demand Probability	Risk Amplification	VaR Threshold (ML/Day)	Tail Risk Premium (ML/Day)
Urban	60%	4-fold increase	3838.77	5.54
Industrial	42.8%	3-fold increase	1634.31	23.92
Agricultural	14.5%	Balanced distribution	21,136.28	32.77

presents a summary of essential risk indicators in the context of multivariate stress scenarios. The urban sector demonstrates the greatest susceptibility, with a probability of 60% high demands due to combined population and temperature stress, leading to a fourfold increase in risk. The industrial sector follows, where population growth is the dominant stress factor. The agricultural sector exhibits the lowest probability of extreme events (14.5%), but has the highest potential variations in demand (VaR = 21,136.28 ML/day), highlighting its vulnerability to economic pressures.

The risk profiles indicate the necessity for customized management strategies: demographic-targeted interventions for the urban and industrial sectors, and economic stabilization initiatives for the agricultural sector.

5.2. Comparison with Previous Saudi Arabian Studies

5.2.1. Forecasting Accuracy Improvements

The copula-based approach achieved superior performance compared to previous Saudi Arabian water forecasting studies. Boubaker [11] reported a MAPE of 5.28% for municipal water demand in Hail using PSO-ARIMA, while urban sector model achieved a MAPE of 6.37% across the entire Kingdom. Despite the slight 1.09 percentage point difference, the methodology offers enhanced value via comprehensive Kingdom wide coverage as opposed to single city evaluations, multivariate integration of population, GDP, and temperature dependencies rather than univariate time series analysis, and sophisticated risk quantification capabilities, including conditional risk analysis and Value-at-Risk assessment, which facilitate proactive management strategies lacking in traditional ARIMA methods. Similarly, Almutaz et al. [12] projected a 24% reduction in Mecca’s water demand by 2030 through conservation initiatives.The Sustainability scenario aligns closely with these findings, forecasting a reduction of 21.7% urban water demand by 2030, reinforcing the feasibility of significant efficiency gains under proactive policies.

5.2.2. Sectoral Trends Validation

The sectoral trends observed in this study align with previous findings in the literature. Chowdhury and Al-Zahrani [14] documented a decrease in agricultural water consumption alongside an increase in domestic and industrial demand from 1980 to 2009. The results extend these trends, showing that while urban water demand remains sensitive to demographic and economic growth, agricultural water consumption exhibits negative correlations with GDP, suggesting a structural shift toward more water-efficient practices. Furthermore, Kamis [13] emphasized the risk of domestic water shortages in Jeddah in the context of high population growth scenarios, highlighting the critical need for a reduction in per capita consumption to 200 L per day. The High Growth scenario analysis provides quantitative validation of these concerns, showing only 1.4% demand reduction despite 30.7% population growth, resulting in a Water Efficiency Ratio (WER) of 21.93 indicating severely unsustainable demand-growth coupling. This empirical confirmation underscores the critical importance of efficiency interventions for urban water security.

5.2.3. Copula Applications in Water Management

This study also advances the application of copulas in water management within Saudi Arabia. Although Wang et al. [5] used copulas to assess water security in Inner Mongolia, reporting a joint risk probability of 22.6% for low security, The multivariate copula framework extends this approach by quantifying a probability 60% of extreme urban water demand under combined stress scenarios. Alqadhi et al. [9] applied bivariate copulas for drought analysis in Saudi Arabia, finding that extreme droughts occur every 50 years. In contrast, multivariate approach captures more frequent stress events, revealing that urban water systems face a baseline extreme demand probability of 10 to 15%, increasing to more than 60% under combined demographic and climatic pressures.

5.2.4. Superior Dependency Modeling

Water demand studies in Saudi Arabia, as referenced in the literature review, predominantly employed linear regression and time series methods, which are based on the assumption of stable linear relationships. Wafaa et al. [6] advanced this by using the quantile regression of the D-vine copula for the consumption of Moroccan water, demonstrating significant variability between regions and quantiles. Building on this, the 4D normal copula approach offers similar analytical flexibility while providing advantages: it maintains interpretability through a clear correlation-based structure, ensures computational efficiency with reliable parameter estimation, supports uncertainty quantification for long-term planning, and allows scalability to include additional variables such as renewable energy or desalination capacity without altering the model’s core design.

5.2.5. Integration of Climate Change Projections

The climate assumptions in this study are consistent with earlier projections for Saudi Arabia. Tarawneh et al. [15] reported expected temperature increases between 0.8 °C and 4.1 °C under various RCP scenarios. The Sustainable and High Growth scenarios incorporate temperature increases of 0.4 °C and 2.3 °C, respectively, which fall within these projected ranges. The moderate positive correlation ($\rho =0.4267$) observed between temperature and urban water demand highlights the need for adaptive strategies, including improved cooling efficiency and the development of drought resilient infrastructure. In addition, Seo et al. [17] utilized Bayesian Neural Networks (BNNs) to forecast residential water demand in South Korea under climate change scenarios. Their model captured uncertainty by generating probabilistic forecasts but was limited to a single urban sector. In contrast, a copula-based framework enables a comprehensive multivariate analysis of water demand across the urban, industrial, and agricultural sectors. This allows for modeling of interdependencies, extreme tail events, and scenario-based planning—capabilities essential for sustainable resource management in arid regions like Saudi Arabia.

5.2.6. Bridging Statistical Forecasting and Smart Water Management

Recent technological innovations have emphasized real-time monitoring and management of water systems. AlGhamdi and Sharma [9] proposed IoT-based smart water management systems for Saudi Arabia, achieving high accuracy ($R^2$ = 0.9995) in real-time monitoring. Syed et al. [10] developed integrated frameworks combining Digital Twin technology with transformer models for water prediction and leak detection. The copula-based approach complements these technologies by providing a statistical foundation for long-term strategic planning and scenario analysis, which can enhance the operational effectiveness of smart water systems.

5.2.7. Methodological Contributions and Validation Summary

The comprehensive comparison with previous Saudi Arabian water studies validates five methodological contributions of the approach. This study presents a comprehensive copula-based framework that integrates the urban, industrial, and agricultural sectors, offering advanced conditional risk analysis along with actionable early warning capabilities. The three-scenario framework corresponds with the objectives of Saudi Vision 2030 and international climate projections, facilitating the direct conversion of Water Efficiency Ratios and dependency parameters into actionable policy guidance. The framework structure ensures compatibility with emerging smart water management systems and IoT integration. The contributions of this study represent a notable methodological advancement in regional water resource planning, providing scientific rigor and practical applicability for sustainable water management in water-scarce environments.

6. Conclusions and Recommendations

This study introduced a comprehensive copula-based forecasting framework for modeling water consumption across Saudi Arabia’s urban, industrial, and agricultural sectors. By capturing nonlinear and asymmetric dependencies between water demand and its drivers population, GDP, and average temperature, the proposed model substantially improves upon traditional forecasting techniques. The 4D normal copula achieved the best performance across all sectors, delivering high predictive accuracy with MAPE values of 6.37%, 17.51%, and 23.20% for urban, industrial, and agricultural consumption, respectively.

Scenario-based forecasts revealed that the Sustainability scenario provides the most favorable outcomes, including significant reductions in water demand (21.7% urban, 20.4% industrial and 8.2% agricultural), minimal impact on climate (+0.4 °C) and the lowest associated risk levels. These findings highlight the value of integrated water resource management strategies aligned with long-term development planning. In contrast, the high growth scenario, despite economic expansion, results in increased demand, increased risk, and increased vulnerability to climate.

Multivariate risk analysis, using Value-at-Risk (VaR), conditional probabilities, and stress testing, offered early warning indicators that strengthen water demand forecasting and policy design. The capacity to quantify extreme demand scenarios supports evidence-based planning and national preparedness.

Importantly, the outcomes of this study strongly support the objectives of Saudi Vision 2030, demonstrating the feasibility of decoupling water demand from economic and demographic growth through strategic interventions. Water Efficiency Ratio (WER) values under the sustainability scenario (0.35 urban, 0.38 industrial, and 0.85 agricultural) confirm that efficient, sustainable development trajectories are attainable and replicable for other water-scarce regions.

While the statistical accuracy metrics confirm strong predictive performance, validation against actual extreme events remains an important direction for future work. Saudi Arabia experienced significant droughts (2008–2010, 2012–2014) and severe heat episodes within the study timeframe, which could serve as natural validation cases if higher-resolution data were available. Future research should, therefore, (1) obtain sub-annual consumption data for extreme events, (2) validate conditional risk thresholds against observed supply challenges during heatwaves, and (3) assess model performance during agricultural policy transitions such as the 2020 subsidy reforms.

The study recommends the implementation of targeted policy measures across three key areas: (1) urban planning should emphasize water efficient infrastructure and population distribution; (2) industrial sectors must enhance resilience through capacity planning, recycling technologies, and demand forecasting; and (3) agricultural reforms should promote water-saving technologies while ensuring food security.

Future research should integrate additional factors such as desalination capacity, groundwater depletion rates, and renewable energy. Moreover, real-time monitoring systems and IoT-based technologies offer promising avenues for dynamic forecasting and operational responsiveness. Overall, this copula-based framework offers a scalable and scientifically grounded solution to support sustainable water management and advance Vision 2030 goals.