A hydro-economic model has been developed for the entire river basin, differentiating the four basin sectors described in Fig. 1. These four sub-basins are further subdivided into smaller sections named demand units. This creates nine homogenous areas. The model combines two components: the hydrological and the economic component. Figure 2 shows the outline of the hydro-economic modelling process.
2.2.1 Hydrological component
The hydrological component is a network of nodes and links, based on the model developed by Kahil et al. (2016b) and Kahil et al. (2018). In this component, nodes stand for the basin water supply and demand units while links show the flow relationships between these nodes. The model is a reduced-form hydrological model of the Guadalquivir basin, calibrated with the observed flows by using the hydrological principles of mass balance and river flow continuity. Balances are defined for each flow, \(i\). The main flow variables, \({X}_{i}\), assigned to the model are headwater inflows, abstractions, return flows, losses, and flows. The information data obtained was measured at selected gauging points in the basin by CHG (2021) and CHG (2022). The supplied data enabled the modelling of the flow rates at each node and the spatial distribution of the available water between the different units (Fig. 3). For the purpose of the model, the basin has been further split into 18 agricultural demand zones in order to calculate the irrigation water used. Urban and industrial demand has also been included as nodes withdrawing water from the system.
The headwater inflows are defined as the total annual flows at the different headwater gauges. The inflows, \({X}_{h}\), at each headwater gauge \(h\) (a subset of \(i\)) add up to the total source supply.
The streamflow, \({X}_{v}\), at each river gauge \(v\) (a subset of \(i\)) is equal to the sum of the flows at any upstream node \(i\), whose activities affect that streamflow. These nodes include headwater inflow, river gauges, diversion, and surface return flow. The flow at each river gauge, which must be non-negative, is defined as follows:
\({X}_{v}= \sum _{i}{b}_{i,v}* {X}_{i}, \forall v\) [1]
where \({b}_{i,v}\) is a vector of coefficients that links flow nodes \(i\) to river gauge nodes \(v\). The coefficients take on values of 0 for non-contributing nodes, + 1 for nodes that add flow, and − 1 for nodes that reduce flow.
Our model considers water diversions. During drought events, a diversion limit of surface water is required to ensure that the total of the available flow at each diversion node \(d\) (a subset of \(i\)) is greater than the diverted flow \({X}_{d}\):
\({X}_{d} \le \sum _{i}{b}_{i,d}* {X}_{i}, \forall d\) [2]
where \({b}_{i,d}\) is a vector of coefficients that links flow nodes, \(i\), to diversion nodes, \(d\). The right-hand-side term represents the sum of all contributions to the flow at diversion nodes from upstream sources (headwater inflow, river gauge, diversion, and return flow). The \(b\) coefficients take on the following values 0 for non-contributing nodes, + 1 for nodes that add flow, and − 1 for nodes that reduce flow.
The applied water at each application node \(a\) (a subset of \(i\)) is defined as follows:
\({X}_{a} \le \sum _{d}{b}_{d,a}* {X}_{d}, \forall a\) [3]
where \({b}_{d,a}\) is a vector of coefficients that links application nodes to diversions. The coefficients take on values of + 1 for application nodes withdrawing water from available sources, and 0 for those not withdrawing water.
The total irrigation water applied at each agricultural node \({X}_{a}^{ag}\) is defined in equation [4]:
\({X}_{a}^{ag}= \sum _{j,k}{b}_{a,j,k}\left(\sum _{d}{b}_{d,a}* {L}_{d,j,k}\right), \forall a\) [4]
where:
\(j\) : crops.
\(k\) : irrigation technologies.
\({b}_{a,j,k}\) : water application per ha.
\({L}_{d,j,k}\) : irrigated area.
\({b}_{d,a}\) : binary matrix to set nodes
Water consumption, \({X}_{c}\), at each consume node, \(c\) (a subset of \(i\)), is an empirically determined proportion of applied water, \({X}_{a}\). In irrigation, water consumption is the amount of water used through crop evapotranspiration (ET), whereas, in urban scenarios, it is defined as the proportion of the urban water supply that is not returned through the sewer system. Water consumption always has a positive value, and it is defined in equation [5]:
\({X}_{c}= \sum _{a}{b}_{a,c}* {X}_{a}, \forall c\) [5]
where \({b}_{a,c}\), are coefficients to denote the portion of applied water that is used in each consume node. Water consumed at agricultural nodes is measured as follows:
\({X}_{c}^{ag}= \sum _{j,k}{b}_{c,j,k}* {L}_{c,j,k}, \forall c\) [6]
Equation [6] states that irrigation water consumed, \({X}_{c}^{ag}\), is equal to the sum over crops (\(j\)) and irrigation technologies (\(k\)) of empirically estimated ET per ha, \({b}_{c,j,k}\), multiplied by irrigated area, \({L}_{c.j.k}\), for each crop and irrigation technology.
Return flows, \({X}_{r}\), at each return flow node, \(r\) (a subset of \(i\)), is a proportion of water applied, \({X}_{a}\), that returns to the river system. Return flows are defined as follows:
\({X}_{r}= \sum _{a}{b}_{a,r}* {X}_{a}, \forall r\) [7]
where \({b}_{a,r}\) are coefficients which indicate the proportion of total water applied that is returned to the river system. Return flows at agricultural nodes are defined as follows:
\({X}_{r}^{ag}= \sum _{j,k}{b}_{r,j,k}\left(\sum _{u}{b}_{d,r}* {L}_{d,j,k}\right), \forall r\) [8]
Equation [8] estimates the irrigation return flows, \({X}_{r}^{ag}\). Water applied must equal water consumed plus water returned.
\({X}_{r}^{ag}\) : total irrigation return flows.
\(j\) : crops.
\(k\) : irrigation technologies.
\({b}_{r,j,k}\) : empirically estimated return flows per ha.
\({L}_{d,j,k}\) : irrigated area.
\({b}_{d,r}\) : binary matrix
The calibration of the hydro-economic model has been achieved using a set of slack variables for each river section, allowing the model to replicate the real observed flows. The slack variables represent unobserved inflows and outflows (e.g., groundwater flows, evaporation, and returns). They are calculated as the difference between the initially estimated flows and those measured at the gauge points. Their inclusion in the model is required to achieve a mass-balance model.
2.2.2 Socio - economic component
The socio - economic component consists of an optimisation model for agricultural activities. To this end, the agricultural activity of the basin has been divided into 10 Irrigation Demand Areas (IDAs) based on the Irrigation Zones defined by the hydrological network according to current supply infrastructure as reported by CHG (2021). The farmers' private gross margin on crop production, subject to technical and resource constraints, has been customised for each IDA. In the model, it is assumed that costs and product prices are constant and that the yield functions are linear and decreasing with crop expansion. A perennial land-fallowing penalty has been included in the objective function, to quantify the possible future yield losses if farmers decide to fallow perennial land. The formulation of the optimisation problem is as follows:
\(Max {TGM}^{ag}= \sum _{d,j,k}{GM}_{d,j,k}^{ag}* {L}_{d,j,k}- Pty\) [9]
where the variable to maximise, \({TGM}^{ag}\), is the total gross margin obtained by the agricultural sector on all irrigated areas in the basin. \({GM}_{d,j,k}^{ag}\) is the gross margin achieved per ha of crop \(j\) using irrigation technology \(k\) at the diversion node \(d\).
\({L}_{d,j,k}\) is the irrigation area of crop \(j\) using irrigation technology \(k\) at the diversion node \(d\).
Finally, \(Pty\)represents a perennial land fallowing penalty, indicating possible future yield losses, if farmers decide to fallow perennial crop lands.
\(Pty= \sum _{d,per,k} \left[PT * {AF}_{d,per,k}\right]\) [10]
where \(per\) is the subset of \(j\) of perennial crops, PT is the penalty coefficient, and \({AF}_{d,per,k}\) is the area of perennial crops fallowed in the diversion node \(d\),
subject to:
\(\sum _{j,k}{L}_{d,j,k}^{ag} \le {Tland}_{d} , \forall d\) [11]
\({X}_{d }\le {Twater}_{d}\) [12]
\({[L}_{d,per,k}+ {L}_{d,per,k}^{IS}] \le {l}_{d,per,k}, \forall d,per,k\) [13]
Equation [11] states the land constraint. Here the irrigated area variable at each node \(d\) cannot be greater than the observed area at that node, \({Tland}_{d}\). Equation [12] links the farm activity optimisation model to the hydrological component by setting an available water constraint, \({Twater}_{d}\). Finally, Equation [13] prevents the area allocated to normal and survival irrigation from exceeding the observed area of each perennial crop in the baseline scenario (\({l}_{d,per,k}\)).
Agricultural production has been represented by 55 crops, and economic indicators are based on the Study of Costs and Incomes of Agricultural Farms (Ministry of Agriculture, Fisheries and Food, MAPA). Crop area is based on the Statistical Yearbook (MAPA). We have converted the provincial information to hydrological IDAs with the support of the official Common Agricultural Policy crop declarations (municipality scale). Likewise, crop yield from the Statistical Yearbook of the Ministry of Agriculture has been adapted using the Common Agricultural Policy Regionalisation Plan (county scale). Table 3 outlines the sources of the main parameters used.
Table 3
Summary of main indicators and their sources of information
Indicator/ variable | Description | Source |
Water resource supply | Annual resources | Hydrological plan (CHG 2021) |
Observed river flow | Water flow at selected locations (current, historical) | (CHG 2022) |
Water supply network | Network with abstraction points and allocated resources | Hydrological Plan (CHG 2021) |
Crop cost (NUTS2) | Study of Costs and Incomes of Agricultural Farms | MAPA (mean 2011–2018) |
Crop area (NUTS3) | Statistical Yearbook & 1T Sheet | MAPA (2018) |
Crop Yield (NUTS3) | Statistical Yearbook & CAP regionalisation plan | MAPA (mean 2011–2018) |
Crop prices | Statistical Yearbook | MAPA (mean 2011–2018) |
MAPA = Ministry of Agriculture Fisheries and Forestry |
The gross margin per hectare \({GM}_{d,j,k}^{ag}\) is given by Eq. 14
\({GM}_{d,j,k}^{ag}= {P}_{j}* {Y}_{d,j,k}- {C}_{d,j,k}\) [14]
Where \({P}_{j}\) is the price of the crop \(j\), \({Y}_{d,j,k}\) is the yield of crop \(j\) under irrigation technology \(k,\) in node \(d,\) and \({C}_{d,j,k}\) are the variable costs of crop \(j\) under technology \(k,\) in node \(d\).
Positive mathematical programming (PMP) was used in the calibration of the crop model to obtain the observed water and land use solution in the baseline scenario (Howitt, 1995). The main advantage of adopting PMP is its ability to produce smooth changes as a result of implementing new management policies to face droughts, while ensuring optimised results that match observed outcomes (Gohar and Cashman 2016). The variant of PMP by Dagnino and Ward (2012) was used in the calibration. This variant enables the estimation of the parameters for a linear yield function from first-order gross margin maximisation conditions. The yield function used [15] is a linear function with diminishing returns, which follows the Ricardian rent principle in which the rents that are used first are those with the highest yields and, therefore, the yield of a crop decreases as its scale of production increases.
\({Y}_{ijk}= {B}_{{0}_{i,j,k}}+ {B}_{{1}_{i,j,k}}* {L}_{d,j,k}\) [15]
Urban and industrial water use has been calculated using the CHG (2015) database. The methodology to calculate the spill over effect is based on the Input/Output (I/O) accounts. The I/O accounts follow the Leontief model, which applies the use of two indicators called backward linkage and forward linkage. Based on Muñoz-Repiso et al. (2013) the spill-over effect estimator used in this study is 1.8252 (i.e., an increase of 1 EUR in the primary sector generates an increase of 0.8252 EUR in the rest of the economy due to the spill-over effect). This value is in line with those mentioned in the introduction of Howitt et al. (2015), Gómez-Ramos and Pérez (2012) or Rodríguez-Chaparro (2013) (values ranged from 1.49 to 3.43). The agricultural production multipliers cited have a range between 1.40 and 3.43. Hence,, 1.825 will be used as an estimate of the spill-over effect proposed by Muñoz-Repiso et al. (2013) as a mean value between both extremes and supported by the I/O table methodology.