Optimal Groundwater Exploitation and Pollution Control

Transcription

1 Otimal Groundwater Exloitation and ollution ontrol Andrea Bagnera c, Marco Massabò a, Riccardo Minciardi a,b, Luca Molini a, Michela Robba a,b,*, Roberto Sacile a,b a IMA- Interuniversity enter of Research in Environmental Monitoring b DIST- Deartment of ommunication, omuter and System Sciences c AMGA-Azienda Mediterranea Gas e Acqua * orresonding author: michela.robba@unige.it DIST - University of Genova, via Oera ia 3, 6, Genova, Italy Abstract: Aquifer management is a comlex roblem in which various asects should be taken into account. Secificall there are conflicting obectives that should be achieved. On one side, there is the necessity to satisfy the water demand, on the other the resource water should be rotected by infiltration of ollutants or substances that could reduce its availability in terms of short term and long term management. The aim of this aer is to develo a management model that is able to define the otimal uming attern for (=,, wells that withdraw water from an aquifer (characterized by ollutant contamination and hydraulically interact, with the obectives of satisfying an exressed water demand and control ollution. In order to formalize and solve the management roblem, it is necessary to consider the equations governing flow and mass transort of the biodegradable ollutants characterizing the aquifer. Such equations may be solved by using a finite-difference numerical scheme. In this work, the numerical scheme is embedded in the management model. The decision (control variables that are considered in the otimisation roblems are the water flows umed at each well at time interval t. Such flows influence the state variables of the system, that is, the hydraulic head and the ollutant concentrations in the aquifer. The obective function to be minimized in the otimisation roblem includes three terms: water demand dissatisfaction, ollutant concentrations in the extracted water, and ollutant concentrations in all cells of the discretized aquifer. Finall the otimisation roblem has been solved for a secific case study (Savona District, Italy, relevant to a confined aquifer affected by nitrate ollution deriving from agriculture activities. Keywords: Groundwater management, otimisation, ollution, decision suort system, otimal uming attern.. INTRODUTION Water is essential for human life and its rotection and sustainable exloitation are crucial tasks. Secificall it is necessary to identify the ossible water bodies that could be exloited (surface water, groundwater, reservoirs, etc. and, according to water demand needs, it is vital to define strategies that reserve the water resource from deletion and ollution and that are environmentally sustainable. The alication of otimization techniques in groundwater quantity and quality management has been deely investigated by Das and Datta (. In that work, they resent a comlete state of the art of the different otimisation aroaches that have been alied to groundwater management. Secificall the combined use of simulation and otimisation techniques is shown to be a owerful and useful method to determine lanning and management strategies for otimal design and oeration of groundwater systems. The simulation model can be combined with the management model either by using the system state equations as binding constraints in the otimisation model or by using a resonse matrix or an external simulation model. In literature, different techniques may be found to hel in finding solution to the various management roblems. Katsifarakis et al. (999 combine the boundary element method (BEM and genetic

2 algorithms (GAs to find otimal solution in three classes of commonly encountered groundwater flow and mass transort roblems: determination of transmissivities in zoned aquifers, minimization of uming cost from any number of wells under various constraints, hydrodynamic control of a contaminant lume by means of uming and inection wells. silovikos (999 analyses the ossibility of solving two management roblems formulated as linear rogramming and mixed integer linear rogramming through the integration of simulation and otimization ackages. The aim of this aer is to develo a management model that is able to define the otimal uming attern for (=,, wells that withdraw water from an aquifer, characterized by a oint source ollutant contamination, with the obective of satisfying the requested water demand and control ollution. Secificall three different obectives (minimization of water demand dissatisfaction, minimization of ollution in the aquifer and minimization of ollution in the extracted water have been considered. The state equations that describe the hysical behaviour of the system are embedded as constraints in the otimisation model.. THE HYSIAL-HEMIAL MODEL The overall model of the considered system may be decomosed into a hydraulic comonent and a chemical one. As regards the hydraulic comonent, The adoted model is drawn by Theim (96 and articularly focuses on the behaviour of the iezometric head at local scale, and secifically on the interaction among the various wells. The ollutant mass transort equation is solved using a finite difference scheme. The hyotheses under which our model is alied are:. confined, homogeneous and isotroic aquifer;. source terms reresented by uming wells with Q ( discharge attern for =, ; 3. wells comletely enetrating and located in (x, y, =,. The third hyothesis means that the fluid flow in the aquifer is only bi-dimensional, since the vertical comonent of the velocity field is close to zero when the wells um from all the aquifer thickness. The flow equation with the relative initial and boundary conditions are: h h h K + K = S s δ Q ( ( x x y y x y t = h ( x, t = = H ( h ( x, = H if {( x, y x + y = R } where H is the undisturbed iezometric level, R is the influence radius, K is the hydraulic conductivit h is the iezometric head in the aquifer, S s is the secific storativit and δ is the Kronecker Delta. The characteristic time scale of equation ( is: S S R Tt = K It reresents the time scale of the transition behaviour of the iezometric head within a regulation interval T R of the uming flow. When the transition time scale T t is negligible with resect to the regulation time ste T R it is ossible to consider the flow equation under steady state in each regulation time interval. In this work we consider steady state conditions for successive regulation time stes. Integrating eq. (, it is ossible to evaluate the iezometric head, in stationary condition: h( x, = H + = Q ( ln πt ( x x + ( y y R ( where T=KB is the trasmissivity of the homogeneous aquifer and B is its thickness. Deriving equation ( and using the Darcy law, it is ossible to write an analytical exression for the velocity field due to uming wells sread in the domain and having a different uming rate Q w. Let n the soil orosit and u and v the ore scale velocities of the fluid flowing in the aquifer along x and y directions, resectively. The ore scale velocities may be exressed as follows: Q ( ( x x u( x, = n π B (3 y = ( x x + ( y Q( ( y = ( x x + ( y y v( x, = (4 n π B y The knowledge of the velocity field is needed in order to solve the mass transort equation. In this work, a contaminant transort simulation model is used, which is able to redict the concentration behaviour in the aquifer for a biodegradable

3 ollutant. Since in many alication concerning the monitoring of groundwater qualit the only concentration measures that are often available are the mean value over the thickness of the samling well, the averaged mass transort equation is taken into account in this work. These equations can be obtained by vertically averaging the classical advection-disersion equation over the thickness of the aquifer system (Willis et al.,998; Bear, 97. These authors have found out the results under the following conditions: horizontal flow orosity and disersion coefficients are constant in all the aquifer the source and sink terms are reresented by uming wells recharging henomena are negligible because the aquifer is confined the bio-degradation coefficient is constant in all the aquifer. The artial differential equation for the averaged concentration is t = + D ± y ( u x = ( v + D + y x Q( δ( x x B y y k (5 where k is the bio-degradation coefficient for the ollutant concentration, considering a first order kinetics. The boundary and initial conditions needed to solve equation (5 are: o if ( x, y = ( xo, yo ( x, = otherwise x y x=, L y=, L = = (6a (6b (6c where x, y is the oint corresonding to the ollutant source. The mass transort equation (5 can be solved by using the classical central finite difference scheme in sace, and an imlicit method in time (Fletcher, 99. The stability of the methods is controlled by the disersion and advection urrant number, defined as v t D t adv = and dis =. L L The finite difference reresentation of equation (5 is for any oint, (a generic oint (x,y on the grid at any time is : ( = D = t t i+, + u i, Q B x y x ( i+, + δ ( i i x i, i, ( + D + v k i, + ( + y x + (7 where (i, is the location of the wells on the grid. 3. THE MANAGEMENT MODEL The main urose of this aer is to resent a decision model able to manage groundwater resources, satisfying the water demand and controlling the aquifer ollution. Secific control and state variables have been defined in order to formalize suitable obective functions and constraints. The control variables that characterize the system are the quantity of water that is extracted in each well in time interval t. These quantities influence both the hydraulic head and the concentration distributions in the aquifer. The state variables of the system corresond to the ollutant concentration to the hydraulic head in the aquifer. Let Q t be the control variable that reresents the quantity of water that is extracted in each well at time t. These quantities influence both the hydraulic head and the concentration distributions in the aquifer. The evolution in time and sace of ollutant and hydraulic head in the aquifer, that are the system state variables, has to be modelled as roved by ( and (7. Moreover let, t reresent the ollutant concentration in the aquifer at time t in oint (. In this work, the ollutant concentration in the extracted water from wells corresonds to the ollutant concentration i, t, in the nodes of the grid where the wells are located. Secificall reresents the ollutant concentration in well (=,, at time t (t=,,t, where =(i,. The obective function considered in this aer is comosed by three terms: minimization of water 3

4 demand dissatisfaction, minimization of ollutant concentration in extracted water, minimization of ollutant concentration in all nodes of the discretized aquifer. Every obective is weighed with secific coefficients in an overall obective function. The otimisation roblem turns out to be non linear. 3. Minimization of water demand dissatisfaction The water demand dissatisfaction corresonds to the difference between the requested water and the extracted water from the wells, when such a difference is ositive or zero. Thus this obective function (to be minimized among this difference and zero, can be exressed as max Q ( REQ Q t, (8 = t = where: Q REQ reresents the overall requested water flow, exressed in l/s, over the whole decision horizon; N is the number of available wells; T is the lanning horizon. 3. Minimization of ollutant resence in extracted water Another obective of the otimization roblem is to minimize the imact of the ollutant in the water extracted from wells. Let ( be the ollutant concentration, exressed in mg/l, of the water extracted from well in the t-th time interval. This obective function can be formalized as follows: Q( F = t = where [ ( ] [ ( ] (9 F is a function of ollutant concentration and has been considered to be [ ( ] ( F = ( 3.3 Minimization of ollutant concentration in the aquifer The aquifer ollution should be limited for two imortant reasons: the reservation of the water resource and the ossibility to satisfy water demand for a longer time in the future. Indicating with (, T the ollutant concentration [mg/l] at node ( at the end of the otimisation eriod, the obective function to be minimized is I J (, T ( i = = where i and are the coordinates of the nodes of the grid reresenting the aquifer. 3.4 The overall obective function The overall obective function to be minimized is given by the weighted sum of functions (8, (9, and (, each one multilied by a secific weighting factor. Then, the overall obective function is the minimization of by min imize β Q( F = t = = t = { α max Q Q( REQ I [ ( ] + γ (, T } J i = =, + ( where α, β, and γ are suitable weighting coefficients. 3.5 The constraints There are different kinds of constraints that should be considered in the model. The first class of constraints reresents the state equations that reresent the dynamics of the ollutant concentrations and of the hydraulic head, as driven by the control variables. The other constraints are: the hydraulic head limitations due to hydraulic conditions that must be resected, the wells caacit and the constraints that avoid to extract water from wells when the ollutant concentration exceeds the one imosed by regulations. Besides, one can make that the equation on which the hysical model is based hold only under secific hyothesis. One of them is that the aquifer is in ressure, that is to say: 4

5 h(, > B (3 where B is the aquifer thickness. Besides the water flow extracted from a well must be less or equal to its caacit namely Q W =,, t =,,T (4 t Finall the water extracted must have a concentration of ollutant not exceeding a secific bound defined by regulations. In other words, this means that: * t > Q t = =,, t=,,t (5 where * is the maximum ollutant concentration allowed by regulation. 4. THE ASE STUDY The model has been alied to a study area of 5mx5m in which three wells um water from a confined aquifer that is affected by nitrate ollution. The satial location of the uming wells resect to the source of ollution sees well as the nearest to the ollutant source, while well 3 is the most far. The case study is located within the eriale Municiality (Savona, Italy, and the confined aquifer is affected by nitrate ollution due to agricultural ractices. The well field is used to extract water for drinking use, but it is eriodically closed because of the ollution due to nitrates infiltration. The alication of the otimisation model allows finding the otimal uming attern in order to satisfy the water demand needs and to control the advancing of the ollutants in the aquifer. The otimisation roblem has been solved over a three months eriod. The aquifer has been discretized in sace ( m, and in time ( hours. The total water demand is 6 l/min, while the ollutant concentration is 5 mg/l. The initial value of hydraulic head is m, while the aquifer thickness is equal to 5 m. The roblem has been solved for two different cases: ase (each well is able to um the total amount of the water demand l/s, and ase (the three wells can um at maximum the same quantity of water (3.33 l/s. The management roblem formalized in the revious section has been solved, in both cases, over a time horizon of three months. In ase, only well (the nearest to the ollution source overcomes the law limit (5 mg/l reaching a concentration value of about 6 mg/l. Well and well 3 (the farthest from the ollution source reach a maximum concentration of 4 and mg/l, resectivel far below the threshold for the whole length of time horizon. Figure shows the attern of the concentration over time, for the three wells. Figure. ollutant concentration in the extracted water (first case [m g /l] 5 T [ h ] 6 In ase, see Figure, no management olicy can be alied, as, in order to satisfy the water demand, every well has to um the maximum flow for the whole management horizon. Note that water umed bywell overcomes the limit after 7 days, whereas, well reaches such a limit after 6 days. Only well 3 can work over 3 months. Figure. The ollutant concentration in the extracted water (ase 8 [m g /l] 5 T [h ] 6 W e ll lim = 5 W e ll W e ll 3 W ell W ell W ell 3 Finall a sensitivity analysis has been erformed on the weight coefficients of the obective function. Secificall the value of the weighting factors is changed in order to make one obective more significant resect to the others. Figure 3 reorts the results, assuming the weighting factor α, relevant to water demand satisfaction obective (equation (8, as revailing. This assumtion forces the system to maximize the extracted water quantit setting each well uming rate to the maximum and stoing extraction when ollutant concentration is over the limit. This strategy causes a very quick overcoming of the fixed otability threshold in the extracted water: well is over the limit in 7 days, well in about 46 days, well 3 in 59. 5

6 8 [mg/l] 5 3 Figure 3. The ollutant concentration in the extracted water (maximum extraction case In order to see how solution varies when the minimization of the ollutant concentration in the extracted water is taken as the rimary obective, the weighting factor β relevant to equation (9 is increased in order to be redominant resect to the other coefficients. Figure 4 reorts the results for the otimisation roblem. Secificall this strategy can satisfy the quality of the umed water (see Figure 4 but it turns out that in the most art of the time interval the overall umed water is far below the request. [m g /l] 9 5 Figure 4. ollutant concentrations in umed water 5. ONLUSIONS T [h] 6 W ell W ell W ell 3 T [h ] 6 W e ll W e ll W e ll 3 The management of an aquifer is a very comlex task since it is necessary to link together otimisation and simulation models in order to find strategies that are able to take into account several asects (hysical, economic, environmental, etc.. In this work, a mathematical formulation of the management roblem has been resented, with reference to the extraction of water form wells, in order to satisfy three conflicting obectives (namel the minimization of water demand dissatisfaction, the minimization of ollutant concentration in extracted water, and the minimization of ollutant concentration in all nodes of the discretized aquifer. Every obective is weighted by a factor whose value is set by the decision maker. Otimal solutions, which may suort the decision makers in the evaluation of an extraction strateg can be obtained solving the related mathematical rogramming roblem that embeds a simlified simulation model of the aquifer dynamics. A reliminary sensitivity analysis on the arameters reresenting the weighting factors is reorted. Future develoments may regard the definition of other decision variables that can give the ossibility of considering the ossibility of the installation of a treatment lant or of the introduction of wells for the inection of water in the aquifer (in order to control the direction of the contaminant lume. Moreover, the hysical model comlexity may be increased: the most restrictive assumtion is the homogeneity of the hydraulic conductivity of the aquifer (the management model can be adated to this case using an aroriate solution for the velocity field. Finall a different aroach for groundwater management might be the identification of emirical models (both from simulation runs and real data collection able to describe the resonse of the aquifer in every grid oint (in terms of hydraulic head and ollutant concentration to the uming from the different wells. 6. REFERENES Das, A., Datta, B, Alication of otimisation techniques in groundwater quantity and quality management, Sadhana, 6( , K.L. Katsifarakis, D.K. Karouzos, N. Theossiou, ombined use of BEM and genetic algorithms in groundwater flow and mass transort roblems, Engineering analysis with boundary elements, 3, ,999 Fletcher,. A. J, omutational Techniques for Fluid Dynamics. Sringer Series in omutational hysics. Sringer-Verlag, st edition,. 53, 99 silovikos, A.A. Otimization models in groundwater Management, Based on Linear and Mixed Integer rogramming. An alication to a Greek Hydrogeological Basin. hys.hem. Earth (B 4( , 999 Theim, G. Hydrologische Methoden, Gebhardt, Leizig,. 56, 96 (in German 6