Introduction

Increasing demand for water to produce goods and services has led to mounting concern about water scarcity (Ahmadi and Moradkhani 2019). Globally, about 50% of people are affected by water scarcity for at least one month each year (Mekonnen and Hoekstra 2016). In 2019, about 38% of European people experienced water shortage for at least one season (Agency 2022). The Intergovernmental Panel on Climate Change (IPCC) illustrates that there is a surging trend in water scarcity and drought due to climate change, especially in arid and semi-arid regions (Wu et al. 2008).

According to the FAO report, globally, agriculture is the largest water-consuming sector with the abstraction of about 69% of fresh water. After that, the industry withdraws around 19% and the municipal sector consumes around 12%. These figures vary in different regions. In America, 47% of water is consumed in the agricultural sector, 37% in the industrial sector, and 14% in the municipal sector. These ratios for Europe are 25%, 54%, and 31%, respectively (FAO 2016). With the growth of competition between different economic sectors, the need to develop integrated economic and hydrologic approaches to estimate and allocate water as a factor input has increased.

The idea of integrating economic concepts with hydrologic models originates from the 1960s (Bierkens 2015). Hydrologic-economic models reveal the combined effects of implementing economic and/or hydrological policies on water resources and economic systems. These models can be categorized as either compartmentalized or holistic (Bauer-Gottwein et al. 2017). Compartmentalized models couple hydrologic and economic components as separate sub-models, while holistic models combine sub-models in a single framework that can endogenously consider their feedbacks on one another (Cai, Ringler and Rosegrant 2006; Dogan et al. 2018). Holistic models simultaneously integrate desired sub-models, which facilitates scenario-based analyses by avoiding separate execution of sub-models (Souza da Silva and Alcoforado de Moraes 2021). Hydrologic-economic models can be used in the form of simulation or optimization tools in water resources assessment. Optimization models have been extensively used to allocate water over a range of spatial and temporal scales. Such models are used for optimal allocation with the aim of maximizing profit or minimizing the cost of systems subject to a range of constraints including water quantity and quality as well as physical and structural requirements, such as resource capacity and environmental or ecological demands (Davidsen et al. 2015; He et al. 2019; Li et al. 2021; Macian-Sorribes and Pulido-Velazquez 2014; Sang et al. 2019; Yan et al. 2020). In addition, there are a wide range of researches that explore how simulation models assist policy evaluation in water resources management (Baccour et al. 2021; Kotir et al. 2016; Souza da Silva and Alcoforado de Moraes 2021).

Although the hydrological components in hydrologic-economic models are often built at the watershed scale, they can also be implemented at different spatial scales including a nation, a sub-watershed, or an administrative region (Harou et al. 2009). In contrast, the economic system is often represented in these coupled models as a set of economic units, starting with the smallest economic unit (for example, a farm) in the watershed and building towards the whole watershed (Amaya et al. 2021; Bauer-Gottwein et al. 2017). Achieving the desired accuracy in the coupled model requires a suitable match of spatial scale in both sub-models (Cai 2008). In addition, hydrologic models are often developed as a simulation tool focused on water balance components, such as inflows, outflows, evaporation, transpiration, groundwater inflow and outflow, and return flows (Cai and Wang 2006), whereas economic models mostly use optimization algorithms to estimate the net costs and benefits incurred under different scenarios (Candido et al. 2022).

In general, input–output models (I–O) are established based on real sectoral data representative of an economy (Matthews et al. 2015). One of the most important advantages of these models is to mirror the current state of the economy including production of economic sectors and consumed resources. These models can be used to demonstrate direct and indirect effects of an expected change in a sector, not only on itself, but also on other sectors. In addition, they are theoretically flexible enough to be established for various boundaries of the economic system ranging from crop land to a watershed. Although these models are widely applied to estimate water consumption and assess implementation of water policies in an economy (Llop 2013; Shao et al. 2017; Velazquez 2006; Wang et al. 2009), not much attention is paid to resource availability, the impact of water scarcity on sectoral production, and the linkage with hydrologic models. Brown et al. (2015) described the knowledge gap between theoretical studies offering optimal allocation policy in a watershed and water allocation in practice. Coupling an input–output model and a hydrologic model provides an improved understanding of the economic and hydrologic consequences of decision making.

Recently, Amaya et al. (2022a) utilized a hydrologic-economic modeling framework to evaluate different upstream development patterns and their impacts on watershed health in downstream locations. This coupled modeling framework was also applied to evaluate seasonal watershed impacts caused by agricultural intensification (Amaya et al. 2022b). This framework involved the loose coupling of an economic I–O model with an established watershed model that has been calibrated to represent a specific location using five years of collected data.

However, some significant points should be noted. The first point is the simulation timestep. Economic I–O models are usually implemented at an annual timestep since available I–O databases are typically aggregated to that scale and the temporal disaggregation of I–O tables has had limited attention in the literature (Avelino 2018; Sun et al. 2019). However, a sub-annual temporal scale is important to consider when estimating the environmental impacts of economic activity because temporal aggregation of annual I–O tables overlooks any variations that occur in production patterns throughout the year and cannot accurately evaluate unexpected events that generate impacts within time periods shorter than the annual timestep (Avelino 2017; Donaghy et al. 2007). Thus, we use a monthly timestep to accurately represent the interaction between the hydrologic and economic models. Another point is the level of abstraction in the coupled system, which is determined based on the purpose of the modeling and the level of the required information. When there is a need for a system-level model that provides information at the macro level, using process-level models with a large number of components and highly detailed information may not be necessary. Little et al. (2019) propose a tiered system-of-systems framework that meets such challenges by using a modular structure that couples different systems with different levels of abstraction. However, since the output from one model is usually input into the next model in a modular approach, it is an important advantage for a coupled model to have sub-systems with temporal and spatial compatibility. For example, economic and hydrological sub-models have the same timestep. In addition, water scarcity is a vital issue that is not addressed in these studies because Fauquier County does not suffer from it. This issue is especially important in areas that are facing serious water shortage as it affects the process of water allocation between economic sectors by adjusting their final demands.

Previous studies have assessed water scarcity by incorporating sectoral water use into an I–O model in varying degrees of detail (Antonelli et al. 2012; Distefano et al. 2022; Wang et al. 2020). For example, Antonelli et al. (2012) were able to distinguish between different typologies of water resources within their economic model while Wang et al. (2020) incorporated both water and energy consumption into their I-O model. However, these modeling frameworks do not account for the two-way feedback that exists between hydrologic and economic systems. Furthermore, these models are static and do not account for changes that occur over time. Indeed, Distefano et al. (2022) acknowledge that capturing temporal evolution is a necessary element for assessing policy decisions and state that a system dynamics model grounded in economics would be appropriate for future analyses. Coupling hydrologic and economic systems is a complex issue because different factors in both systems are dynamically interconnected.

Coupling hydrologic and economic systems is a complex issue because different factors in both systems are dynamically interconnected. This requires a systemic approach. System dynamics (SD) is a well-known methodology to understand the behavior of integrated complex systems as well as feedback among the key variables (Sterman 2002). This method can be applied as a conceptual and numerical modeling technique to integrate compartmentalized models (hydrologic and economic models) as a system-level framework (Little et al. 2019). SD models have been widely employed in investigating the behavior of various water management systems (Abdolabadi et al. 2016; Qin et al. 2019; Romanelli et al. 2021; Zare et al. 2019), and can be implemented for long-term policy analysis with a high level of aggregation. This approach provides an effective tool to identify the direct and indirect interactions of variables in coupled systems helping to identify the best-suited structure for the integrated model (Schoenenberger et al. 2021).

In summary, a large body of literature has explored integrating economic concepts with hydrologic models for different purposes and at different levels, but the interconnection between hydrologic and economic systems that considers the feedback of encountering water scarcity, the spatial distribution of economic activity, and the variation in watershed characteristics on the production of economic sectors and water availability has not been studied. To fill these gaps, we developed an integrated system-dynamic framework to provide insight into the management of both the supply side and the demand side by applying varied modeling assumptions (e.g. the spatial aggregation level, distribution of economic sectors, and the physical features of the watershed) and management approaches (e.g., managing the demand side). Our specific objectives are to:

  1. 1.

    Develop a system dynamics framework that couples the input–output economic and hydrologic models to account for the interaction between water use and water availability water using Vensim software (http://www.vensim.com) (Scenario 1).

  2. 2.

    Demonstrate how the level of spatial aggregation, the distribution of economic sectors, and the physical features of the watershed affect water allocation as well as water scarcity (Scenario 2).

  3. 3.

    Improve the coupled model by adjusting the final demand for the economic sectors to impose the least cost on the coupled system while partially satisfying water demand when there is water scarcity (Scenario 3).

Methodology

The coupled hydrologic-economic system dynamic model

An input–output model relates variables in an aggregated economic model via algebraic relationships that represent economic effects, including production capacity, labor, changes in demand, and resource use. Regarding water use, there are three types of water flows describing the interaction between the watershed and the economic system: (a) flow from the watershed to the economic sectors; (b) flow within the economic sectors; and (c) flow from the economic sectors into the watershed (DESA 2012). When coupled with a hydrological model, the input–output model can be used to simulate water flows between the watershed and the economic sectors. This approach should enable economists and water managers to determine how scarce water resources should be allocated to produce goods and services and provide insights into the economic consequences of diverse scenarios.

In the coupled model, the hydrologic model simulates the water cycle while the economic model endogenously provides information about water requirements and the total factor use of the economy. The components of the factor use matrix reveal the quantity of resources (including water) needed to produce the total output of each sector. Water withdrawn from the watershed by the economic sectors, which depend on their water requirements, needs to be accounted for in the hydrologic model to determine whether there is sufficient water in the watershed to allocate to the economic sectors. Figure 1 shows a schematic representation of the integrated system dynamic framework that couples two models. A more detailed structure of the framework is illustrated in next subsections and the Supplementary Information (SI).

Fig. 1
figure 1

Schematic of the system dynamic framework

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Economic input–output model

In holistic hydrologic-economic models, the hydrologic and economic components of a region are incorporated into a single software package, which requires simple representations of each system, but allows information to be easily transferred between the two systems (Brouwer and Hofkes 2008). This approach has been applied in many studies that focus on a comprehensive hydrologic system with some extension to include some economic variables (e.g. (Escriva-Bou et al. 2017; Kahil et al. 2016)). To address the lack of a complex representation of an economic system, a computable general equilibrium (CGE) modeling approach has also been utilized to capture interactions between hydrologic variables and a whole economy (Bohringer and Loschel 2006; Brouwer and Hofkes 2008). The CGE modeling approach is effective at capturing economy-wide impacts on hydrologic processes, but the hydrologic variables must conform to the logic of the CGE modeling framework (Scrieciu 2007; Zhang 2013). Thus, when hydrologic and economic systems are coupled in an integrated framework, typically only one system is represented in detail and extended to include variables of the other system. However, as an alternative modeling tradition to CGE models, input–output models can be used to represent the economic system within a hydrologic-economic modeling framework.

The basic I–O model is composed of two model components: the primal quantity model and the dual price model (Leontief 1961). The primal model calculates the economic output and factor use necessary to meet final demand from \(n\) economic sectors utilizing \(k\) factors of production. Factors of production are defined as required inputs that are themselves not produced, including labor and land among other resources. Specifically, water has also been incorporated as a factor of production in previous I–O studies (e.g., Lopez-Morales and Duchin 2015; Lopez-Morales and Rodriguez-Tapia 2019)). Equations 1a, 1b and 2 comprise the primal model, which is usually solved using matrix algebra. However, since a coupled hydrologic-economic model is being implemented within an integrated framework in this study, model results are obtained using system dynamics.

$$ \left( {I{-}A} \right)x = y $$
(1a)
$$ x = \left( {I{-}A} \right)^{ - 1} y = By $$
(1b)
$$ \varphi = Fx $$
(2)

In Eqs. 1a, 1b and 2, the main parameters are \(x\): the \(n \times 1\) economic output vector, \(y\): the \(n \times 1\) final demand vector, \(A\): the \(n \times n\) technical coefficient matrix, \(I\): the \(n \times n\) identity matrix, \(\varphi \): the \(k \times 1\) factor use vector, and \(F\): the \(k \times n\) factor requirement per unit of output matrix. The technical coefficients, \({a}_{ij}\), represent the output from sector \(i\) that sector \(j\) consumes to produce one unit of output. Factor requirement coefficients, (\({f}_{ij}\)), are defined as the quantities of resources \(i\) that are consumed by sector \(j\) to produce one unit of output. All these parameters have physical units.

The dual price model utilized by the economic I–O model calculates the unit cost associated with each economic sector (\(p\) vector) using a vector of prices (\(\pi \)) associated with each factor of production, or

$$ p = \left( {I{-}A^{^{\prime}} } \right)^{ - 1} F^{\prime}\pi $$
(3)

The sum of compensation for employees (labor) and capital (factors of production, e.g., land and capital goods) is referred to as total value-added (Bernstein et al. 2016). However, Duchin (2009) has also described value-added as the combination of the quantity and price components in the physical I–O model. Interpreting value-added as payment for factors of production allows land, water, and mineral resources to be included as factor inputs in value-added in addition to capital and labor. Value-added per unit of output, \(v\), is estimated by multiplying factor inputs required to produce one unit of output and the price of one unit of factor input, or

$$ v = F^{\prime}\pi $$
(4)

where \(v\) is the \(n\)-vector of value-added, \(F\) is the factor requirement per unit of output matrix, and \(\pi \) is the \(k\)-vector of factor prices. The total value-added as the total amount of money paid to factor inputs, \(Z\), is equal to the sum of value-added for each sector, \({z}_{i}\), which is calculated by multiplying value-added and output vectors, or

$$ Z = \sum\limits_{i = 1}^{n} {z_{i} } = \pi_{i}^{T} Fx \,\,\, \forall \, i = 1:n \, $$
(5)

To calculate total economic output using system dynamics, the Leontief inverse matrix, \(B= {(I-A)}^{-1}\), is used. This matrix solves for the total output from each sector that satisfies the final demand of the economy (Velazquez 2006). Each element of this matrix (\({b}_{ij}\)) shows the amount of change in the production level of sector \(i\) when the final demand of sector \(j\) changes by one unit. The total factor use vector (\(\varphi \)) is calculated by multiplying the total output vector and the factor requirement matrix, which represents the total factor inputs consumed to produce the outputs from each sector. Using system dynamics, the total output of sector \(i\) is defined as a stock variable (\({x}_{i}\)), which is calculated at a specific time and equates to the corresponding net flow from the previous timestep until the current timestep (Eq. 6). Each \({x}_{i}\) has physical units and the flow variables (\({{x}_{\mathrm{in}}}_{i}(t)\) as the inflow and \({{x}_{\mathrm{out}}}_{i}\left(t\right)\) as the outflow) are expressed in the same unit as the stock divided by the timestep.

$$ \frac{{d(x_{i} )}}{dt} = x_{{{\text{in}}_{i} }} (t) - x_{{{\text{out}}_{i} }} (t) $$
(6)
$$ X_{{{\text{in}}_{i} }} (t) = \sum\limits_{j = 1}^{n} {b_{ij} y_{j} } (t) \, $$
(7a)
$$ x_{{{\text{out}}_{i} }} (t) = x_{i} (t) $$
(7b)

In Eq. 7a, \({b}_{ij}\) is the \(ij\) th element of the Leontief inverse matrix, and \({y}_{j}\) is the final demand from sector \(j\). Both \(i\) and \(j\) denote the economic sectors and \(n\) refers to the number of sectors. To calculate the total factor use for each economic sector based on Eq. 2, the \(ij\) th element of the factor requirement matrix (\({f}_{ij}\)) is multiplied by the output from each sector:

$$ \varphi_{ji} \, = f_{ij} \times x_{j} \, $$
(8)

where \(j\) is the sector number and \(i\) represents each factor input.

Finding an appropriate level of aggregation for the economic sectors is important from a modeling point of view as well as for gathering the required data. There are various classification systems, such as the Statistical Classification of Economic Activities in the European Community (NACE) and the North American Industry Classification System (NAICS), which serve different purposes (Tukker et al. 2006). The International Standard Industrial Classification of all economic activities (ISIC), which is a well-recognized standardized classification system, can also be used to develop the input–output data tables. However, in this paper, because we are assessing the capability of the coupled model to track water through a watershed, we provide and solve a numerical description of a hypothetical economy comprised of only three sectors, Crops, Machine Production, as well as Power Generation (referred to as Sector 1, 2, and 3, respectively), and using three factor inputs (i.e. water, land, and labor) to produce their outputs. In addition, we include return water in the factor requirement matrix to account for the effect of return flow on the water budget. Because the model is implemented at a monthly timestep, it is possible to vary the exogenous variables and parameters, such as the final demand of the economy, factor prices, and factor requirements throughout the seasons. Nonetheless, the equations remain unchanged during each monthly timestep. Appendix A in the SI provides more information about model, including figures, tables, and results.

Conceptual hydrologic model

In this study, we use System Dynamics to develop the hydrologic model with two sub-systems (i.e., a rainfall-runoff and a simple snowfall-snowpack model) to estimate runoff volume and available water for a watershed for a monthly timestep. We utilize a simple conceptual lumped hydrologic model called the Tank model, which was originally developed by Sugawara (1995). We briefly describe the underlying structure of the hydrologic model, its governing equations, and the assumptions we made to estimate available water and water scarcity.

The conceptual model involves a simplified representation of a series of vertically-arranged tanks. The number of tanks and model structure depend on watershed characteristics such as geology, soil type, and land use. Basri (2013) illustrates how the model structure changes for different land uses, proposing five versions of the Tank model for Settlement, Paddy, Garden, Vacant, and Forest land uses. For instance, the 4-tank model fits best for the humid and forest regions. However, for settlement land use, the model contains one tank as it is impermeable to account only for the surface flow. In the 4-tank model, the first tank (top) has two side outlets and one bottom outlet. Precipitation is the main input into the top tank, and evapotranspiration is the sink subtracted from the tank. The flow through side outlets accounts for surface runoff and the bottom flow represents infiltration. The second and the third tanks have one side and one bottom outlet. Flow from side outlets forms the interflow. Percolated water into the last tank discharges gradually by the only side outlet as the base flow (Fig. 2b).

Fig. 2
figure 2

Schematic representation of the hydrologic system dynamics model. a The conceptual hydrologic system dynamics model. b The Tank model and the Available Water stock. Available Water represents the corresponding volume of water in the river channel at each timestep, which discharges from the watershed as the outflow

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There are two types of the parameter in the tank model that represent different hydrological processes, including side and bottom outlet coefficients (a, b, c, and d) as well as storage heights (height of side outlet, HA, HB, and HC). These coefficients need to be calibrated by comparing observed and simulated data. (SugaWara 1979) proposed an automatic calibration program based on a trial-and-error procedure to adjust parameters based on the shape and volume of the hydrograph. However, many researchers used varied optimization algorithms to calibrate the tank model coefficients (Jaiswal et al. 2020; Lee et al. 2020; Ou et al. 2017).

In this study, we assume that the 3-tank model is suitable for modeling a single entity watershed, and we use the 4-tank and 2-tank models when the watershed is disaggregated into two sub-watersheds (Scenario 2). As Vensim allows us to optimize model parameters (constants) by utilizing optimization algorithms, we can calibrate all the coefficients within the model. The merit is that the model is much faster and can be simply applied to watersheds with different characteristics. This feature becomes more important when there is a need to develop an integrated framework or coupled hydrologic system that operates at different spatial and temporal scales.

The general water balance of a watershed is composed of the water storage variation (ds/dt (mm/month)), the precipitation (P (mm/month), the evapotranspiration (ET (mm/month)), and the outflow (Q (mm/month)) (Chow et al. 1988).

$$ \frac{ds}{{dt}} = P - E - Q $$
(9)

Figure 1 shows the conceptual schematic of the hydrologic, system dynamics model. The SD model includes two main components of the hydrologic processes in the watershed: rainfall-runoff and snow accumulation. The water balance for the tanks (A, B, C, and D) is formulated by Equations B.2 through B.14 (see Appendix B. Hydrologic model in the SI). The total runoff (QT) is computed as:

$$ Q_{T} = Q_{1A} + Q_{2A} + Q_{1B} + Q_{1C} + Q_{1D} $$
(10)

where, Q1A = surface flow; Q2A = sub surface flow; Q1B = inter flow; Q1C = sub base flow; Q1D = base flow. Runoff Volume (QV) is computed by multiplying the total runoff and the watershed area (WA) for the time interval (\({Q}_{V}={Q}_{T}*WA\)).

The available water volume, a stock variable termed Available Water (m3), shows the volume of water in the river channel, which finally discharges from the watershed as the outflow at each timestep (Eq. 11). We defined this variable to allow the hydrologic and economic model to be linked. Figure 2b shows schematically the Available Water at a specific point in time as well as inflows and outflows controlling the quantity of this stock. The dashed arrows indicate that the amount of the flow depends on the amount of the stock variable. All the flow variables have units of cubic meters per month.

$$ \frac{d}{dt}({\text{Available}}\,{\text{Water}}) = Q_{V} - {\text{ET}} - Q_{{{\text{out}}}} + {\text{Sources}} - {\text{Sinks}} $$
(11)

QV is the main flow influencing the Available Water stock. Qout is the outlet flow from the watershed and ET represents Evapotranspiration from the river and its riparian zone (Fig. 2a). Sources are the flows directly discharging into the river channel. The return flows from the economic sectors are the most significant sources in the watershed. Sinks include water withdrawal by the economic sectors from the Available Water stock. These variables are adjusted by checking if the available water satisfies the environmental water demand. The required water for Crops, Machines, and Power are directly added as flow variables from the economic model (Sinks). The return flow from the economic sectors (Sources) is calculated as

$$ Q_{R} \, = \sum\limits_{i = 1}^{n} {C_{rwi} } *{\text{ SWR}}_{i} $$
(12)

where Crwi is the return water coefficient (representing the fraction of water that is returned to the watershed after use) for each sector and SWRi (m3/month) is the water requirement for each sector. Finally, the Outflow, which indicates the quantity of the available water that flows out of the watershed via the river, can be computed as:

$$ Q_{{{\text{out}}}} = {\text{Available}}\begin{array}{*{20}c} {} \\ \end{array} {\text{Water}}/t_{g} $$
(13)

where tg is the average residence time to discharge the water. As a simplifying assumption, we assume that tg is equal to the modeling timestep (1 month). However, tg can be refined during the calibration process. To estimate the effect of seasonal temperature variation on the hydrologic model, we used a simple single-layered, snowpack, lumped model to estimate Snow Water Equivalent Depth (SWED). This represents the amount of water contained within the snowpack and is equal to the depth of melted water of the entire snowpack (Ghashghaei et al. 2013). This stock will melt when the temperature rises above the threshold 0 °C; in that case, the melted snow is added to the top tank. More information about the hydrologic model is provided in Appendix B. Hydrologic model (SI). In addition, all the variables, parameters, and equations used in the system dynamic model are described in Tables B.1 and B.2 (Appendix B).

After developing the model, we applied it to the Upper Broad Run Sub-watershed within the Occoquan watershed (Appendix B1) to test the accuracy of the model and quantified the model’s error based on the Nash-Sutcliff efficiency coefficient, the Pierson correlation coefficient, and the standard error. The validation procedure, model parameters and the results of the tests are presented in Appendix B2. Calibration and Validation in the SI.

Environmental flows are the required flows that guarantee the health of the ecosystem as well as its provided goods and services over time (Merritt et al. 2010). To calculate the environmental flow components, including extreme low flows, low (base) flows, high flow pulses, and floods, we used the Indicators of Hydrologic Alteration (IHA) software (Mathews and Richter 2007). We selected the low flow indicating the amount of water that is available for most of the year as the environmental water demand (EF, m3/month). This flow guarantees the required quantity and quality of water needed to maintain ecosystem health. Like other flow components, low flow can vary in time and space, but for simplicity, we used a constant value describing the minimum flow in the summer.

Simulation scenarios

Because we emphasize the dynamic behavior of the system generated by the model structure, we use three simple scenarios to illustrate the relationship between the hydrologic and economic models. They represent different ways to calculate available water, total water withdrawal, allocation percentage, and water scarcity in a watershed. It is important to note that satisfying the EF is the priority of water allocation in all scenarios.

  • Scenario 1—the baseline coupled model

In the baseline scenario, we calculate the water requirement for all sectors using the economic model and link the water requirement for each sector to the flow variables withdrawing water from the Available Water stock and computing return flow as well as allocation percentage in the hydrologic model for each timestep. This approach assumes that the watershed is a single entity and calculates all the desired variables for the watershed regardless of the location of the economic sectors. Figure 3a shows the coupled model algorithm indicating how the models are linked. Figure 3b describes the first scenario in which all the economic sectors are randomly located within the watershed. Each sector withdraws water from the river and discharges its return flow back to the river. The return flow coefficients are 80% for Power, 20% for Crops, and 0 for Machines, which means, for example, that 20% of the water extracted by Power is lost and 80% returns to the river. Detailed information is elaborated in Appendix C1.

  • Scenario 2—the spatially-resolved coupled model

Fig. 3
figure 3

The coupled model algorithm (a) and the location of economic sectors in the watershed (b). Note that the water balance is made at the outlet and that all flows are treated as if they are either added to or withdrawn from this point

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In this scenario, the spatial distribution of the economic sectors is considered by dividing the watershed into upstream and downstream sub-watersheds (Fig. 4). This disaggregation reveals spatial differences in water scarcity. The hypsographic curve is used to determine the relationship between altitude and the cumulative surface area of a watershed. After separating the two sub-watersheds, the coupled model is applied to estimate available water, water scarcity, and outflow from each sub-watershed. We assume that the two sub-watersheds have different physical features, including soil type and land cover, which have either more or less available water. For example, one sub-watershed has soil and ground cover that allows a larger percent of rainfall to become runoff when compared to the other sub-watershed. The process of estimating the available water of the hydrologic model is like the main process described in the previous sections except for the sinks, which are determined based on the location of each economic sector and land use. Considering less permeable land use for upstream (Settlement-Vacant area), the runoff volume is calculated using a 2-Tank model, having larger runoff potential than downstream (4-Tank model). The downstream has an additional inflow variable to its Available Water stock accounting for the outflow of the upstream sub-watershed into the downstream sub-watershed. Figure C2.1, shows the coupled model for the upstream and downstream sub-watersheds (see Appendix C2 in the SI).

  • Scenario 3—the temporal coupled model with adjusted demand

Fig. 4
figure 4

Dividing the watershed into upstream and downstream sub-watersheds

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In this scenario, a temporally resolved coupled model illustrates how the approach can account for and eliminate water scarcity in the watershed by automatically adjusting the final demand of the less efficient economic sectors. This scenario focuses on demand management. To achieve this goal, we introduce a value-added per unit water ratio based on the objective function for minimizing the total factor use established by Duchin and López-Morales (2012). This ratio describes the contribution of each sector to the total value-added for consuming one unit and helps prioritize alternatives to adjust the final demands. Figure 5 illustrates the algorithm of the coupled model considering the feedback of water scarcity on adjusting final demands in the watershed. Detailed information and equations are provided in Appendix C3 (SI).

Fig. 5
figure 5

The temporal coupled model algorithm

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Results and discussion

Before implementing the coupled model, we solved the input–output model (Eqs. 1a, 1b and 2) to estimate the total output, factor requirements, and the price vectors. Detailed results are available in Appendix A. Input–Output Model in the SI. However, results reveal that Power needs to extract more water (about 3700 and 3400 (m3) in high and low demand seasons, respectively) than the other sectors (about 1100 and 600 (m3) for Sector 1 and 2, respectively). In addition, the economy produces about 19 (ton) agricultural products, 110 (unit) machines, and 49 and 44 (MWh) electricity (see Table A.2).

Scenario 1—the baseline coupled model

In this scenario, we assume that the EF is 500 (m3/month) and the lumped land use is a vacant area. The coupled model is solved at a monthly timestep, with results shown in Table 1.

Table 1 The results of the coupled model based on the baseline scenario (dam3)
Full size table

As shown in Table 1, the watershed and economy suffer from water scarcity in the summer months. Water scarcity begins in June with a water shortage of about 100 m3. In July and September, about 75% of the total water requirement can be withdrawn by economic sectors. The extra water needed to satisfy the water requirement is about 1400 and 1300 m3 in these two months. The highest water scarcity (about 1700 m3) is recorded in August when precipitation is low and evapotranspiration is high. Similar logic can be used to interpret the amount of water scarcity and allocation percentage in July and September. Return flow is about 3200 m3 in summer and winter, and 2900 m3 for other months when there is no water scarcity. The amount of return flow will change based on variation in the total water withdrawal. Eventually, the outflow of the watershed is computed by adding all the inflow variables to and subtracting all losses from the Available Water stock. As expected, the outflow reaches a low of 2900 m3 in August. In addition, the low outflow in January and February compared to other months is due to low precipitation in these months (see Table B1.1).

Scenario 2—the spatially-resolved coupled model

The holistic approach in Scenario 1 can omit important details when estimating water scarcity. Comparing the total water requirement of economic sectors with the available water and the EF, regardless of their location in the watershed, the allocation percentage and the amount of water scarcity are calculated at each time step. This approach may be subject to the lumping of some spatial characteristics of the watershed. For instance, scarcity in some areas is balanced by abundance in other areas. Another point is the possibility of downstream sectors using the return flow from the economic sectors located upstream. In Scenario 1, we did not account for water available to the lower sectors because of return flow from the upstream sectors. Accounting for the spatial distribution of the economic sectors can avoid over- or under-estimating water scarcity. Now we take four situations into account. First, the watershed is separated into the two sub-watersheds (upstream and downstream). In Case 1 (see Fig. 4 for illustrations of Case 1 through 4), the most water consuming sector (Power) is located upstream with Crops and Machines downstream. Then, in Case 2, we relocate the economic sectors to determine how the water balance will change and whether there is water scarcity. Therefore, Crops and Machines are located upstream and Power is located downstream. In addition, different land uses are assigned to the upstream and downstream. The upstream has a higher potential runoff. As a result, Case 3 and 4 have the same distribution of sectors as Cases 1 and 2, but land uses have been changed. The return flow coefficients are the same as Scenario 1 for all cases. Table 2 shows the estimated water scarcity in the upstream and downstream for all cases. Detailed results are provided in the SI (Appendix C2).

Table 2 The results of the spatially-resolved coupled model for water scarcity (dam3)
Full size table

Dividing the watershed into upstream and downstream, there is no water scarcity in the upstream in all cases. With Power upstream, the amount of available water, including the return flow, will be enough to satisfy the EF for the ecosystem as well as the water requirement for Power. Moreover, because the total water requirement of Crops and Machines is about 1700 m3, which is far less than the water requirement of Power (about 3700 and 3400 m3), and there is always sufficient water to satisfy the total water requirement of these sectors as well as the EF in the upstream, the allocation percentage is 100% for all months for all Cases.

In the downstream sub-watershed, the Available Water stock is fed by the outflow from upstream and the runoff volume obtained from precipitation in the downstream sub-watershed. Assuming all hydrological features of the model, except the location of sectors, are the same in Case 1 and 2, there is no water scarcity in the downstream in Case 1. Therefore, dividing the watershed into two sub-watersheds and distributing sectors between them in this way allows water scarcity to be eliminated. However, the economy suffers from water scarcity during the summer months (i.e. July, August, and September) in Case 2. Since Crops consumes 80% of its water requirement and there is no return water from Machines, the total amount of water consumed by these two sectors is about 1500 m3. On the other hand, Power uses 20% of its water requirement. These results indicate that since the return rates of Crops and Machines are small (20% and 0% respectively), the amount of return flow contributing to the available water for the downstream will decrease and, as a result, there is insufficient water to satisfy the water requirement of Power. Comparing the results of Case 2 and Scenario 1, there is a considerable difference in intensity of water scarcity in July and August. The amount of water scarcity in July and August are about 500 and 1200 m3, which are decreased by about 64% and 30%, respectively.

Although the results of Case 3 are similar to Case 1, they are nevertheless interesting. As can be seen, there is no water scarcity in the upstream and downstream even when the potential of runoff downstream in Case 3 is less than its corresponding amount downstream in Case 1 (comparing land uses). Furthermore, the physical features of the sub-watersheds in Case 3 are the same as Case 4. Nevertheless, the watershed encounters water scarcity in the latter case in January as well as the summer months like Case 2. Comparing Cases 4 and 2, it is evident that the different physical features of the watershed can directly affect the water availability so that the probability of confronting water scarcity will increase if the runoff potential reduces (e.g. encountering water scarcity in January). Thus, in Cases 2 and 4, the reversed spatial location of the economic sectors leads to confronting water scarcity.

Overall, the results reveal that if Power operates upstream and Crops and Machines are placed downstream (the upstream Power return flow is available for withdrawal by Crops and Machines downstream), there will be no water scarcity. Consequently, the spatial distribution of economic sectors, as well as the physical features of the watershed have a significant impact on estimating available water and water scarcity.

Scenario 3—the temporal coupled model with adjusted demand

Like the previous scenarios, we assume that the final demand remains unchanged to calculate outputs for each sector until the available water is less than the total water requirement. If there is water scarcity, the model will automatically correct itself to eliminate water scarcity. To decrease the total water requirement, we need to decrease the total output of each sector. According to the input–output model, the output of each sector includes both intermediate and final demand, which can adjust the final output. We assume that the model will adjust the final demand when there is water scarcity. Table 3 shows the value-added per water ratio of each sector (R), their adjusted final demands (y), and their outputs.

Table 3 The results of the temporal coupled model
Full size table

As can be seen, Power has the smallest value-added for consuming a unit of water (R3). Thus, the model starts to decrease the final demand until it reaches zero. Then, it calculates how much water is needed to eliminate water scarcity. This amount of water will be compensated by adjusting (reducing) the final demand of the next sector having smaller R. Machines has the second smallest value-added per water ratio after Power. As Power’s final demand decreased by about 4.8% and 82% in June and July, respectively, the output of this sector declined from 49 to 47 and 28 MWh, and its water requirement reduced by 100 and 1500 m3, respectively. In August and September, the final demand of Power is diminished so that it reaches its lowest amount of zero. In this situation, the output of this sector reaches the minimum amount of 24 MWh, which only satisfies the intermediate demand of the economy. In other words, economic sectors need to consume this amount of electricity to produce their outputs. Although the total reduction of Power’s final demand leads to about 48% reduction in its water requirement (about 3700 to 1800 (24*75) m3), there is still need for curtailment of about 150 m3 water to eliminate water scarcity. Since Crops has the second smallest R, the model begins curtailing its final demand so that there is 16% decline in August. As a result, like Power, its output diminishes by 2.6 ton in this month. There is also a small change in the final demand of Machines in the summer. The water requirement, final demand, and the sectors’ output are approximately the same as their corresponding values in other months. The minimum amount of total water requirement (about 3300 m3) recorded in August decreased by about 37%. Eventually, although there is no water scarcity in the watershed, the total value-added is reduced by about 9%, 13% and 11% in July, August, and September, respectively.

Comparing the results of Scenario 1 (Table 1) and the temporally resolved coupled model, the outflow of the watershed does not change except in summer when there is a slight increase due to the adjusting capability of the model that decreases the final demands and water consumption. In the baseline coupled model, although using the Water Withdrawal Condition Coefficient (WWCC) forces economic sectors to reduce their water withdrawal so that the available water satisfies the environmental flow requirement, there is still water scarcity because the water requirements for sectors do not change. On the other hand, the temporally resolved coupled model allows economic sectors to adjust their final demands to eliminate water scarcity.

Conclusions

In this paper, we show how the intimate coupling of hydrologic and economic models helps to understand the interaction and inter-sectoral relationships more completely between the economy and the watershed. We simulated the dynamic behavior of the coupled system under three scenarios to demonstrate how these changes can affect available water, total and sectoral water requirements, and water scarcity. The coupled model provides two key benefits. First, it directly links the supply side (using the hydrologic model to estimate water availability) to the demand side (using the economic model to estimate water use by sector) using a common framework that accounts for the interdependence of the two models. This linkage allows us to analyze water allocation and calculate the intensity of water scarcity. Second, it enables us to consider the effect of both spatial distribution of economic activities and different physical characteristics of the sub-watersheds on the hydrologic model and prevents either under or over estimating water scarcity. Without this spatial disaggregation, a shortfall in one sub-watershed may be averaged with abundance in another sub-watershed.

Considering a watershed as a single entity, we used the coupled model to represent a baseline situation (Scenario 1). We restricted economic sectors to withdraw the required water only if the EF is satisfied. Otherwise, only a proportion of the water requirement will be met. Here, water scarcity is calculated as an endogenous variable by estimating the difference between required water and withdrawn water. Results show that the watershed encounters water scarcity in summer months when the lowest allocation percentage (or largest water scarcity) is recorded. Comparing the total water requirement for economic sectors with the available water and the EF regardless of their location in the watershed and lumping some physical characteristics of the watershed may omit important details in computing water scarcity. For instance, the results of the model for Cases 1 and 3 demonstrate that water scarcity can be eliminated by disaggregating the watershed into upstream and downstream sub-watersheds and using the return flows from the economic sectors located upstream for sectors downstream.

Scenario 2 reveals that, although having different physical features in sub-watersheds can cause changes in water availability, the proper spatial distribution of economic sectors based on the potential return flows can constructively balance the frequency and intensity of water scarcity. Results indicated that when a sector with potentially low return flow is upstream and a sector having high return flow operates downstream, the latter is at a disadvantage because the return flow from upstream cannot be used. This situation is less optimistic in comparison with the situation assuming the downstream sector withdraws its water requirement from upstream. Therefore, unless the economic sectors are logically distributed within the watershed, there would be water scarcity.

Under Scenario 3, the improved coupled model endogenously allows for controlling the final demand of less efficient sectors. The result showed that in some situations, decreasing the final demand of the less efficient sector cannot provide an adequate amount of water to compensate for water scarcity. In this case, the model starts diminishing the final demand of the second less efficient sector and this process continues until the required water and available water are equal. Although this approach finally eliminates water scarcity, in the real world, we cannot arbitrarily reduce the final demand of a sector regardless of other vital factors including social, political and security criteria. For instance, the electricity demand of final users such as households must be satisfied.

Although the coupled model is sufficiently flexible to assess more complex situations in its current state, including varied spatial disaggregation and feedbacks, future analyses will implement a precise distributed hydrologic model as well as a real survey-based, input–output model to prevent under or overestimating the results. With this coupled modeling framework, the long-term impacts on watershed health resulting from changes in economic activity can be evaluated along with how these impacts drive technological innovation within the economic system over time, such as the implementation of desalination plants in arid environments. Additionally, the long-term effectiveness of policies imposed on the economic system, such as restrictions on water withdrawals, can be evaluated to determine if such policies substantially alleviate impacts on watershed health or merely inhibit economic development. Thus, linking the hydrologic and the input–output economic models provides insight into the emergent behavior of the coupled system, which facilitates the design and implementation of management policies such as considering technological choices for economic sectors, or best management practices in the watershed.