Water Resources Management 18: 125–141, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Application of Non-Linear Simulation and Optimisation Models in Groundwater Aquifer Management NICOLAOS P. THEODOSSIOU Division of Hydraulics and Environmental Engineering, Faculty of Civil Engineering, School of Technology, Aristotle University of Thessaloniki, Thessaloniki, Greece (e-mail:
[email protected], fax: +302310995711) (Received: 5 February 2003; in final form: 19 November 2003) Abstract. The need for rational and overall water resources management has become, during the past decades, a problem of major importance due to the rising water demands. In this paper a technique is presented through which a management model that combines the use of two separate models, a flow simulation and an optimisation one, is used for groundwater management. The necessary stages for the formulation and the combined use of the two models, along with a number of problems that might arise during the development of the management model are also presented. This technique is applied to a large-scale case study problem that forms an optimisation approach with a large number of non-linear decision variables. The results of the application of the management model demonstrate the importance of the use of such models both in managing rationally available water resources and in reducing the operational cost of their exploitation. Key words: application of the response matrix technique, combined use of simulation and optimisation models, groundwater resources management, large scale case study non-linear optimisation
1. Introduction Mathematical models that simulate the function of groundwater aquifers are absolutely essential management tools, since they can be used for the evaluation of the application of different management scenarios. The basic disadvantage in using simulation models for the evaluation of management policies is that the optimal alternatives are derived through a predictor – corrector procedure. This procedure cannot exclude the existence of other equal or even better alternatives. Thus the application of simulation models is limited to the understanding of operation of the aquifer and to the estimation of its future response and not to the derivation of optimal management scenarios. In order to achieve the optimisation of management scenarios one must apply mathematical management models that combine simulation models and optimisation and operational research techniques.
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The development and application of such management models require a series of necessary stages. These stages are: • • • •
Analysis and description of the problem. Development and application of the simulation model. Formulation of the management problem. Combined use of the simulation and optimisation models.
In this paper the problems that are likely to arise during the development of such management models are presented, emphasising those corresponding to the formulation of the management problem and the combined use of simulation and optimisation models. Underlining these problems and proposing effective ways of dealing with them results to the formulation of a useful tool for groundwater resources managers. Finally at the end of the paper, the proposed techniques are presented analytically through a case study application. 2. Formulation of the Management Problem One of the most important stages in the development of management models is the formulation of the problem and the selection of the most appropriate management goals introduced in the optimisation model through the determination of the mathematical expression of the objective function and the constraints. In the work of Gorelick (1990), a comprehensive analysis on the conceptual and practical issues of the overall problem of combined groundwater simulation and nonlinear optimization was presented. 2.1. QUANTIFICATION OF NATURAL VALUES One of the most common problems that arise during the development of an optimisation model is the quantification of natural values of the variables since some of the management variables are not always quantifiable, or they are expressed in different units or with different accuracy. 2.2. SELECTION OF EQUATIONS The selection of the equations that describe the objective function and the constraints of a management problem can be expressed in many cases in several different ways with respectively different results for the same problem. Natural problems are usually multi-dimensional and non-linear. In order to describe mathematically such problems one must choose amongst a number of simplifications the most appropriate one in order to maintain a relatively acceptable simulation of the problem and at the same time getting a mathematical solution.
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The issues that have to be answered are whether to use a linear or a non-linear approach. The advantages of a linear approach are, (a) linear problems more easily and much faster result to solutions than non-linear ones, (b) linear problems are much easier to formulate while non-linear one are complex and more difficult to handle, and (c) non-linear problems, unlike linear one, often result to local optimal solutions, some times very different from the global optimum. The most important disadvantage of linear approaches is that they are formulated using too many simplifications, some times resulting to misleading solutions. As an example the problem of minimisation of cost from a system of pumping wells is presented. The pumping cost can be expressed in the following ways: (a) Linear expression Minimise qi ,
(1)
where qi the pumping rate of well i. The concept of this approach is that the less the abstracted water from an aquifer, the less the pumping cost. (b) Non-linear expression Minimise Cp hi qi ,
(2)
where Cp the pumping cost per pumping rate unit and per water table drop unit, qi the pumping rate of well i, and hi the water table drop in well i which is, in general, a non-linear function of the pumping rates of all active wells. The concept of this approach is that the pumping cost is not related only to the amount of water but also to the depth from where it is pumped. Despite the advantages of the linear function concerning the solution procedure of the problem, the results compared to those derived from the non-linear approach are not satisfactory (Latinopoulos et al., 1994). As the depth of the water table is included in the objective function (Equation (2)), the pumping rates are more normally distributed among the active wells resulting to a reduction of the function cost of the whole system. The optimal procedure is the one combining the two approaches. The results from the solution of the simplified linear problem can be used as a starting point for the solution of the more complex non-linear problem. The successful selection of a set of initial values for the non-linear variables can overcome both the problem of no solution and the problem of local optima. Another element that one has to bear in mind while formulating an optimisation model is the fact that most algorithms used for the solution of non-linear problems are designed in such way as to handle more effectively non-linearities in
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Figure 1. Procedure for the application of the response matrix method.
the objective function rather than those included in the constraints (Murtagh and Saunders, 1987). Thus, wherever possible during the formulation of the problem, non-linearities must be transferred from the constraints to the objective function. For example the problem of (Murtagh and Saunders, 1987) minimising z under the constraint: F (x) − z = 0 , where F (x) a non-linear function, can be substituted by the problem of minimising F (x) without constraint (Murtagh and Saunders, 1987)
3. Combined Use of Simulation and Optimisation Models Water resources management models that combine simulation and optimisation methods can be distinguished in two categories according to the method used for the combination of the two separate models. These methods are the embedding method and the response matrix method. According to the former, the flow equations as described in the simulation model are considered as part of constraints to which the optimisation model is subjected. The response matrix method is much more complex and is presented schematically in Figure 1. An external simulation model is used for the calculation of a number of coefficients, each one of them relating unit values of the pumping rate at each active well, to water table drop at all the predefined observation nodes. All these coefficients constitute the response matrix of the aquifer that is included in the management model as a substitute of the simulation model.
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The application of this method is based on the assumption that time- and spacebased linear superposition is valid. An analytical presentation and evaluation of the applications of the two methods can be found in (Gorelick, 1983). 3.1. SELECTION OF THE COMBINATION METHOD BETWEEN SIMULATION AND OPTIMISATION MODELS
In order to select between the two methods presented above achieving the combined use of simulation and optimisation models, one must be aware of the advantages and disadvantages of each one as well as their importance in the formulated problem. Thus, the advantages of the embedding method over the response matrix method are, (a) it is more simple as a concept since the equations of the simulation model represent also the constraints of the optimisation model, (b) the optimisation model can be more easily manipulated for the same reasons, (c) it is not based on the principle of superposition and thus it has a wider range of application, and (d) it is easier to alter the management goals of the problem contrary to the latter method where the alteration of even a single decision variable requires the reconstruction of the response matrix. On the other hand the main disadvantages of the embedding method in respect to the response matrix method are, (a) the optimisation model is loaded with the entire set of equations that describe groundwater flow and as many times as the management periods investigated, regardless of the number of decision variables. On the other hand the dimensions of the response matrix depend only on the number of decision variables and observation points (not even management periods), and (b) the method cannot deal with complex implicit numerical schemes for the analysis of the flow equations but only explicit ones. For the above reasons the applications of the embedding method are limited to small-scale problems. 3.2. VALIDATION OF THE SUPERPOSITION PRINCIPLE Although the response matrix technique seems to have significant advantages compared to the embedding method, its applicability is limited by the demand that linear superposition is valid. In most aquifer related management problems the rates of pumping or injection wells are introduced as decision variables and the hydraulic head as an evaluation measure for the objective function and the constraints. In these cases the applicability of the response matrix technique depends on the validity of a linear correlation between pumping rate and hydraulic head. While in confined aquifers, this correlation is linear, in phreatic aquifers it is not as seen in Equation (3) (Bear, 1979): ∂h ∂h ∂ +Q, (3) Kij hij =S ∂xi ∂xj ∂t
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Figure 2. Variation of the hydraulic head with pumping rate in a phreatic aquifer for different values of the permeability coefficient.
where Kij is the permeability coefficient, hij is the hydraulic head, S is the storativity coefficient, Q is the term related to the quantity of water pumped from or injected to the aquifer per area unit, xi , xj are the coordinates, and t is time. This observation does not exclude necessarily the response matrix technique to be applied to phreatic aquifer problems. Under certain circumstances the assumption of linear superposition can be applied even in phreatic aquifers. In the following figures the variation of hydraulic head according to the pumping rate is presented. In Figure 2 this variation is presented for several values of the permeability coefficient K (from 1 × 10−5 to K = 1 × 10−3 ) as derived from the solution of the well known flow governing equation for phreatic aquifers (Bear, 1979): R Q 2 2 ln . (4) H −h = πK r From this figure one can conclude that for high values of the permeability coefficient it can be assumed that there is a linear correlation between pumping rate and hydraulic head. One can assume linear correlation even for smaller values of the permeability coefficient but only for small pumping rate variations. Similar conclusions can be derived, as seen in Figure 3, by the variation of the hydraulic head according to the pumping rate (solution of Equation (4)) for different initial hydraulic heads in a phreatic aquifer (from H = 50 m to H = 150 m). As stated by Bear (1979) for thick aquifers and small drawdowns Equation (4) can be linearised since the difference between the initial (H ) and the resulting (h) hydraulic head is not significant. This is represented in Figure 3 by the higher curve (H = 150 m). It can also be observed from Figure 3 that the higher the initial hydraulic head value (representing thick aquifers) and the lesser the pumping rate (resulting to small drawdowns) the effect of the non-linearity is minimised. And even for smaller values of the initial head and larger values of the pumping rate there are linear intervals.
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Figure 3. Variation of the hydraulic head with pumping rate in a phreatic aquifer for different initial values of the hydraulic head.
From the above observations one can conclude that if a detailed analysis is performed, an area of applicability of the linear superposition and thus the response matrix method can be safely defined. It is also noted that one can take advantage of the small linear intervals of non-linear variations. By selecting the ‘unit value’ of the pumping rate to be close to the expected optimised one, the error introduced can be relatively small. This can be achieved either by experience or by several iterations. From the equation of groundwater flow in a phreatic aquifer (Equation (3)) it is shown that the pumping rate is not in linear correlation with the hydraulic head but with the square of the hydraulic head. So, if the management model is formulated to use the square of the hydraulic head, then the response matrix technique can be applied without any problem. Such cases concern problems that do not require the actual values of the hydraulic head but the difference between two values of the hydraulic head. For example for the stabilisation of pollution plume the difference between the hydraulic head outside and inside the plume boundaries is controlled (Latinopoulos et al., 1994; Mylopoulos, Y. et al., 1999). Another typical application is the dewatering of an excavation where the differences between hydraulic head and the bottom of the excavation can be used as control variables for the management model (Latinopoulos et al., 1985). In these cases one must ensure that no negative values are implied. This can be easily achieved by altering the reference level. 4. Case Study Application The mathematical management model is applied to the Kokkinohoria area in southeastern Cyprus (Theodossiou, 1994b). The underlying phreatic aquifer is the only resource in the area, and water demands are met with groundwater abstractions. It
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is mainly an agricultural region which has been functioning under no water policy management. Farmers used to drill their own pumping wells which reached the number of 1400 in a 60 km2 area. This situation led to a dramatic lowering of the water table in the aquifer, forcing the farmers to drill even more wells to meet their needs. Due to the large number of wells and the relatively low permeability of the aquifer, the water level distribution is highly non-smooth. The Water Development Department of the Ministry of Agriculture and Natural Resources of the Republic of Cyprus has designed a project called ‘Southern Conveyor Project’ through which water from the mountainous western part of the island is to be conveyed and distributed along the south coast to regions with high demands including the Kokkinohoria area (Water Development Department, 1982). Under the new development of the conveyance of excess water to meet the demands of the region and the expected reduction of the discharge from the aquifer, the management aim is to minimise the pumping cost in the aquifer. 4.1. THE SIMULATION MODEL In order to face the particularities of the Kokkinohoria aquifer a completely new mathematical model was developed. It is a two-dimensional numerical scheme solving the governing flow equations (Equation (3)) using the finite differences method. According to this discretisation 403 cells contain active wells. The implicit set of equations derived is solved using the alternating directions iteration method (A.D.I.). Apart from the hydraulic related particularities of the problem (non-smooth distribution of the hydraulic head, large number of pumping wells, spatial heterogeneity etc.) the new developed model was designed in order to be able to be connected to other commercial models, an option that was considered essential for its inclusion in the management model. 4.2. THE RESPONSE MATRIX The simulation model was used to develop the elements of the response matrix. These elements represent the response of the aquifer to the applied unit stress at any location. In every possible well point that is to be included in the response matrix, a unit pumping rate is applied. The resulting water level changes show their influence on the aquifer. In the Kokkinohoria aquifer, after the finite differences discretisation, there are 403 cells that contain active wells. Through an automatic computer procedure each one of these cells, one at a time, was considered to have a unit pumping rate which was actually the maximum value of the original pumping rates of the aquifer. The simulation model was solved under these stress conditions, and the results were compared with those derived from the simulation model under no stress conditions. Their differences were recorded as the response of the aquifer to the application of unit stress to the cell under consideration. This procedure was repeated 403 times,
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and the assemblage of these records created the response matrix. From a total 804 cells (defined as control cells) derived from the discretisation of the aquifer, 403 cells (defined as active cells) contained wells. As noted above, the concept of the response matrix method is based on the assumption of the validity of linear superposition over space in the aquifer. This requires linearity in the groundwater flow equations. In spite of the fact that the Kokkinohoria aquifer is phreatic and the flow equations are non-linear, the assumption of linear superposition between water abstraction and drawdown was considered valid for the following reasons: (a) this assumption is valid only when drawdown is small, and this is the case in the Kokkinohoria aquifer because of the relatively small pumping rates; (b) the finite differences approximation simulates pumping as distributed discharge over the area of each cell. This means that the assumption of linearity has already been applied over drawdown; (c) linearity was tested and found valid, using several scenarios of pumping wells that functioned separately or as a group; and (d) the resulting pumping rates derived from the optimisation model were verified through their application back to the simulation model.
4.3. THE OPTIMISATION MODEL The study area has been divided into ten irrigation zones according to the yield of the wells (Figure 4). The optimisation procedure has two targets. First, to respecify the pumping rates of all wells inside each irrigation zone in order to pump the same quantities of water in lesser cost, and second to optimise the distribution among the irrigation zones of the excess water according to its total amount in order to accomplish maximum reduction of operational pumping cost (Theodossiou, 1994a). 4.3.1. Objective Function of the Optimisation Model The management policy demands the minimisation of the water pumping cost from the aquifer. The non-linear quadratic approach of the objective function as shown in Equation (5) does not fully represent the actual pumping cost but a respective measure.
minimize
n i=1
where
n 1 Qi Di + Qj S1i,j , Q 1 j =1
(5)
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Figure 4. Aquifer discretisation in irrigation zones.
Qi Di Q1
= = =
S1i,j
=
n
=
the pumping rate from cell i [L/T]; the initial water level depth from the ground [L]; the constant values of the initial ‘unit pumping rate’ used for the estimation of the response matrix [L/T]; the water table drawdown in cell i, due to the implementation of ‘unit pumping rate’ in active cell j [L]; the number of active cells (equal to 403).
The 403 partial derivative of the non-linear objective function necessary for the solution of the optimisation model are expressed in Equation (6). n, j =i n 1 1 1 ∂F = Di + Qi S1i,j + Qj S1i,j + Qj S1j,i , (6) ∂Qi Q1 Q1 Q1 j =1 j =1 where F = the objective function.
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4.3.2. Constraints of the Optimisation Model There are two groups of constraints. The first group ensures the hydraulic function of the 804 ‘control cells’ demanding the water table drawdown to be less than the thickness of the saturated zone (Equation (7)). n 1 for i = 1, 804 , (7) Qj S1i,j ≤ (Hi − Bi ) Q 1 j =1 where Hi Bi
= =
the initial value of the hydraulic head at control cell i [L]; the depth of the impermeable base of the aquifer at control cell i [L].
The second group of constraints ensures the total amount of groundwater pumped from each one of the 10 irrigation zones. QTi ≥ Ai
for i = 1, 10 ,
(8)
where QTi
=
Ai
=
the total amount of groundwater pumped from irrigation zone i (as the sum of the pumping rates Qi of the active cells in zone i) [L/T]; the predefined total amount of demanded groundwater from zone i [L/T].
Determining the values of Ai , the minimum total amount of water pumped to meet the water demands of each irrigation zone is defined starting with values that represent the current conditions. The conveyance of excess surface water from other sources (Southern Conveyor Project) can be introduced by the respective reduction of Ai . In this way the second management goal to optimise the distribution among the irrigation zones of the excess water, according to its total amount is embedded. Thus the optimisation model is formulated using 403 non-linear decision variables within the objective function while subjected to 814 linear constraints. It must be noted that the response matrix is introduced in the management model through as a means for the determination of the water table drawdown. As mentioned earlier, the coefficients of the response matrix express the water level drawdown at each control cell from the application of a unit pumping rate at another active cell. The principle of linear superposition defines that: (a) the water level drawdown at a control cell due to the application of pumping rate at an active cell is given by: Si,j =
1 Qj S1i,j , Q1
(9)
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where Q1
=
S1i,j
=
Qj
=
Si,j
=
the constant values of the initial ‘unit pumping rate’ used for the estimation of the response matrix [L/T]; the water table drawdown in cell i due to the implementation of ‘unit pumping rate’ in active cell j [L]; the actual value of the pumping rate at active cell j (decision variable of the optimisation problem); the resulting water table drawdown at control cell i due to the application of pumping rate Qj at the active cell j .
and (b) the final water level drawdown at a certain control cell defined by the sum of all partial drawdowns corresponding to the function of the active cells that affect it is given by: si
403
Si,j
for i = 1, 804 ,
(10)
j =1
where si = the final water level drawdown at control cell i. Summarising the water level drawdown at each control cell of the aquifer is given by the equation: S1,1 S1,2 S1,3 · · · S1,403 Q1 S1 S2 S2,1 S2,2 S2,3 · · · S2,403 Q2 (11) = .. .. . .. .. .. .. . . . . . . S804
S804,1 S804,2 S804,3 · · · S804,403
Q403
4.4. SOLUTION PROCEDURE Since the combined simulation and optimisation models had to be applied a number of times in order to investigate all possible management scenarios, the solution procedure was designed so that all necessary steps be executed automatically through a computer program without any user interference. This is a concept that has been used in several large scale, multi-level optimisation applications (Esposito and Floudas, 2000; Finley et al., 1998; Mylopoulos, N. et al., 1999; Mylopoulos, Y. et al., 1999). These steps are as follows: Step 1: Selection of initial values. Two sets of initial values were used, the actual current values of the pumping rates, and those derived from the solution of
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Step 3:
Step 4:
Step 5:
Step 6: Step 7:
Step 8:
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the linear problem (Equation (1)). Both applications resulted at the same optimum. Production of the input files necessary for the application of the flow simulation model. Execution of the simulation model for the construction of the response matrix. The model is executed 403 times, as many times as the decision variables. From the output files produced from the application of the flow simulation model, the computer program recognized the elements that expressed the correlation of the pumping rates and the hydraulic head drop at each control cell. Construction of the response matrix. Preparation of the input files required by the optimisation software (MINOS) based on the objective function (Equation (5)) and the set of constraints (Equations (7) and (8)). Preparation of a subroutine describing the non-linear objective function and its 403 partial derivatives. Compilation and link with all the other routines of MINOS. Execution of the optimisation software. Identification from the output files of MINOS of the values of the decision variables as well as the status of the final solution (if it was an optimal solution, a nearly optimal solution, or an infeasible problem). Execution of the simulation model using the derived optimal values of the decision variables. Evaluation of the results.
MINOS (Murtagh and Saunders, 1987) was used not only because it is widely recognised as an efficient method for solving non-linear optimisation problems but also because it permits interference to the programming code, an option that was considered necessary in order to jointly using of the simulation and optimisation models as described above. 4.5. RESULTS The solution of the optimisation model showed that all ten constraints described by Equation (8), demanding at least a certain amount of water pumped from each irrigation area, were always satisfied to their limit. This was expected since both the objective function and the rest of the constraints tend to minimise the pumping rates (to reduce the total operational cost, in the objective function and to ensure that the total drawdown does not exceed the thickness of the saturated layer, in the constraints). The only constraints that tend to increase the values of the pumping rates are constraints 8, and thus they are satisfied to their limit. In order to present the results of the previously described management procedure one must first record the current situation. Figure 5a presents the pumping rates distribution over the ten irrigation zones. The significant differences are justified by the unequal size of the irrigated areas and the variable productivity of the aquifer.
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Figure 5a-d. (a) Initial distribution of pumping rates over the ten irrigation zones. (b) Initial value of objective function over the ten irrigation zones. (c) Pumping cost per unit of abstracted water for the irrigation zones in descending order. (d) Cumulative water demands of the ten irrigation zones.
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Figure 5e. (e) Variation of the value of the objective function due to the allocation of water.
The variation within the ten irrigation zones of the value of the objective function (Equation (5)) under the current pumping conditions is demonstrated in Figure 5b. The accomplishment of the second optimization target, the distribution of conveyed surface water within the irrigation areas in order to reduce the total operational pumping cost is as mentioned earlier, indirect. The problem is to determine in successive order the irrigation zones whose water demands must be first met according to the amount of allocated water for the objective to be achieved and the total cost reduced. The total amount of water allocated to the area is not known a priori, and is not even constant over time but subjected to the availability of water at its origin and to the competitive water demands that must be also met. This concept results to the formulation of a management problem that is dynamic over time. Considering the total amount of allocated water as a discrete (not continuous) variable whose minimum value could be zero and maximum value equal to the total water demands of the area, the problem is solved successively for values ranging from 0 to 100% using a 10% step. While the marginal scenarios are obvious (0% means that all water needs will be met by water abstracted from the aquifer and 100% that no water will be abstracted) the problem focuses on the allocation of water equal to 10 to 90% of the water demands. Although indicative, Figure 5b cannot be used for the determination of the order according to which water must be allocated within the irrigation zones, since the pumping cost presented there is a function of the total abstracted amount of water. The criterion that must be used in order to achieve maximum reduction of the operational cost is the pumping cost per unit of abstracted water. This is expressed in Figure 5c as the value of the total pumping cost for each irrigation area (Figure 5b) divided by the respective amount of abstracted water (Figure 5a). In Figure 5c the irrigation zones are presented in descending order according to the pumping cost per unit of abstracted water.
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Following the successive order of the irrigation zones (Figure 5c), Figure 5d was prepared presenting the cumulative water demands. Using this figure one can determine the irrigation zones whose water demands must be met with the allocated water according to its total amount, in order to achieve maximum reduction in the total operational cost. Thus, for example, if 10 000 m3 d−1 were to be allocated in the area, the water demands of irrigation zones 10, 8 and 1 should be first met and the rest should be allocated in zone 4. It must be noted that these calculations are always performed following the application of the previously presented management model, which determines which pumping wells should be active in order to minimise the objective function (Equation (5)) under the constraints (Equations (7) and (8)). The final results from the application of the management model are presented in Figure 5e. The amount of water allocated is distinguished in eleven different levels from 0 to 100% with a 10% step. According to the amount of allocated water that each level represents and using the order presented in Figure 5d, the irrigation zones whose water demands will be first met are determined. This means that the rest of the water demands for each irrigation zone will be met using abstracted groundwater. The amount of the necessary groundwater needed to meet these demands is expressed through the coefficients Ai in the constraints 8 of the optimisation model. Thus the re-distribution of the active wells within each irrigation zone and their respective pumping rates are determined through the application of the optimisation model in order to cover the water demands that are not met using the allocated surface water. The final value of the objective function for each level of discretisation of the allocated surface water is presented in Figure 5e. The observation derived from the comparison of the two first columns of Figure 5e is quite remarkable. The first column presents the value of the objective function (which is a measure of the operational cost) without the application of the management model. The second column presents the respective value, for the same amount of abstracted groundwater, after the application of the management model and the re-distribution of pumping rates. The reduction of the total operational cost, without even allocating surface water, is of the order of 33%. This emphasises the importance of the application of the management model.
5. Conclusions Since water scarcity tends to become one of the most important global problems, rational water resources management becomes more essential than ever demanding detailed and analytical approaches. From the application presented in this paper one can easily conclude that the use of extended mathematical management models that include all necessary components such as simulation and optimisation techniques is of major importance in water resources management. Such models can serve as significant operational tools for water managers providing the means to optimise the use of the available water resources at a minimum operational cost.
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