Abstract. 1 Introduction

Transcription

1 Pumping cost analysis in groundwater management, using the MODMAN (modflow management) model Aris Psilovikos and Christos Tzimopoulos Department of Rural & Surveying Engineering, Aristotle University ofthessaloniki. 546 Thessaloniki, Greece arts ccf.auth. gr tzimop@edessa. topo. eng. auth.gr Abstract The aim of the present study is the minimization of a linear objective function of the form 2, Q Q in afieldof wells, where Q are the cost coefficients and i Qi the pumping recharge. A number of linear constraints, concerning the piezometric level and water balance must be satisfied. Two different cases are discussed below. The first one is the quantitative management. In this case all the cost coefficients in the objective function are equal to «1» and the goal is the maximization of water extraction. The second is the economic management, in which all the coefficients are different to each other and they depend on the piezometric level of each different pumping well, the goal being the pumping cost minimization. The comparison of the above two management cases as well as the results obtained by the optimization, makes the applications of MODMAN model very interesting. 1 Introduction Water is a valuable resource for life, therefore its exploitation has been the object for several studies, projects and works from the ancient times up to present. Recently the geometric increase in global population, associated with the unequal distribution of water in space and time, makes the quantitative management of water a real necessity. The use of numerical simulation groundwater models such as GROUND - multiple cells '*, MODFLOW - three dimensional finite difference method*'^

2 28 Computer Methods in Water Resources XII Transactions on Ecology and the Environment vol 17, 1998 WIT Press, ISSN and FEM -finiteelement method^ in conjunction with several different scenarios in pumping recharge distribution, consists a groundwater management tool. This approach does not ensure optimum management, and the questions that has to do with optimum water management in industrial, agricultural and urban use still remain. 2 Problem Description The complicated issues in groundwater management, and the possibility of using combined simulation - optimization models has forced the researchers to the creation of water management models. Such a model, is MODMAlf (MODflow MANagement, RM Greenwald 1994, Geotrans), which is an extension of AQMAN (Lefkoff, and Gorelick, 1987, USGS) offering a large variety of management options and features not available in AQMAN code such as mixed integer - linear programming. It works as a linkage between the three dimensional simulation model MODFLOW^ of groundwater movement, and the special modified version of LINDO (Schrage L., 1991) that is an optimization software based on Mathematical Programming. MODMAN is based on response matrix method^ that has prevailed as a linkage method between simulation and optimization models. An external numerical simulation model is used to calculate coefficients each one of them associated with unit rates in each pumping well in the aquifer with unit drawdowns to the wells that are used as control points. The organization of these coefficients is called response matrix and is included in the management model as a substitute of the simulation model based on the acceptance of space superposition (for steady - state problems) and moreover on space and time superposition (for transient problems). According to the optimization procedure (Fig. 1) MODMAN^ calls (N+l) times subroutine MM FLOW, which is a slightly modified version of MODFLOW, where N is the number of managed wells. During the first simulation the «unmanaged heads» are computed which are the hypothetical heads in the wells with the assumption that no pumping at all takes place in the aquifer. During the next N simulations, the response matrix is computed that contains the unit head drawdowns referred in specific control points in the aquifer which are due to the separate pumping of each managed well. The control points are some of the pumping well points that are in the most disadvantaged position which means that head drawdowns in the end of the irrigation period are very low. The unit pumping rates are equal to «1» nrvd and «influence surfaces)) (Psilovikos, 1996 *) are produced in the well

3 Computer Methods in Water Resources XII 29 Transactions on Ecology and the Environment vol 17, 1998 WIT Press, ISSN field according to the theory of elasticity (Fig. 2). The reason why the unit rates are chosen equal to 1 mvd, is to avoid probable scaling problems that would have occurred in comparison with the real pumping rates that have a range of 1* to 5*1*. 3 Optimization Process The next step in the procedure is the solution of the optimization problem based on Linear Programming. The ASCII output file produced from MODMAN (Modmps) constitutes the inputfilefor the optimization code LINDO (fig. 1). This file iswritten in IMPS'* (Mathematical Programming System) format which is an international code that can be read from all the commercial optimization packages such as LINDO, MINOS, GAMS, WHAT'S BEST etc. So the scope in this stage of the research is the optimization of an objective function of the form below : k=\ j=l Constraints in maximum permissible drawdown, based on response matrix method in control points in the aquifer have to be satisfied (Psilovikos, 1996*): * j* k _ ~<*~*> * * - ^ k=l j=l Balance Constraints referred to the entire quantity of extracting water for each managing period, have to be equal to the water demand for irrigation. Both the physical and the mathematical problems are satisfied T N k=l y=l y -"' (3.) Finally, constraints in minimum and maximum pumping rates are formulated : <&*<G^ajc (4) where i = 1,..,3 control point in piezometric level. j = l,...ll pumping well. k=l,...3 managing period. Ui* unmanaged head at control point i during the k managing period. H ^ managed head at control point i during the k managing period.

4 3 Computer Methods in Water Resources XII Oij ^"^ average drawdown in each i* observation well at the end of the T* pumping period due to a unit rate of pumping at the j* managed well applied throughout the k^ pumping period^ pumping rate at well j, during the k managing period. maximum managed drawdown at control point i at the end of the last irrigation period T. Hi,miiJ minimum allowable head at control point i at the end of the last irrigation period T. The following application has to do with 11 managed wells in three successive time periods - months. The control points in piezometric level are the threefirstmanaged wells, G1284, G1318 and G1267. Two different cases are discussed below : a) Quantitative management : In this case all the coefficients in the objective function (Ac. 1) are equal to «1». The objective is to maximize the total pumping rate from all the wells while head and balance constraints must be satisfied. The coefficients are dimensionless. b) Economic Management: In this case all the coefficients are different to each other. They depend on the piezometric level of each different pumping well at the end of the last pumping period T. The objective is the minimization of the total pumping cost including the number of working employees, the cost of the groundwater supply network, and the total head. The total head consists of i) the beginning head difference between the ground surface and the unmanaged head, ii) the head difference between the unmanaged head and the latest heads after 3 periods of pumping and iii) 62 meters of manometric head that is the minimum required for the operation of the closed conduit irrigation supplies (Table 1). The head losses are included in the 62 m manometric head. The dimension of the coefficients is Monetary Units / Time [MU/T]. The maximum allowable pumping recharges of the simulation model are increased by 5-1% in the management model (Table 2), so that the new optimum values of pumping rates are allowed to have a wider range of fluctuation. 3.1 Cost coefficients optimization The power of a pump is given from the formula below*": = 1.512

5 Computer Methods in Water Resources XII 31 where : y [N/m*] Q [mvd] Hman [ttl] n =.75 (yield coefficient - dimensionless). The cost in drachmas ( 1$ «28 drachmas) per month is: 6 = jg f^ - g = #^ ' G where p is the KWh price and it is equal to 15.5 drachmas. There is an increase in (3 if the salaries of the employees and the pumping cost operation is included. So the formula for the cost in drachmas per month finally is : 6 = /? #_ g = #, ' 8 [drachmas/month], (5) where the new P is equal to 21.5 drachmas. The table of the coefficients (Table 1.) is obtained based on this analysis. 4 Conclusions From the above two optimization methods we obtain : a) For the first case (coefficients = «1»), emphasis is given in the preservation of the piezometric level in very high levels. b) For the second case (coefficients ^ «1»), the piezometric level reaches its lower permissible drawdown constraints. Emphasis is given in the minimization of the total pumping cost, so the pumping stops from the wells that have the bigest coefficients (fig. 2). c) For both cases the same piezometric (fiic. 2) and balance constraints (fiic. 3) are satisfied in the control point wells and managed wells respectively. Only the coefficients in the objective function change. d) From the cost analysis obtained from the above two optimization methods ( a and b ) as well as from the simulation model Modflow (a scenario), the most economic method is the case b, that gives total cost of drachmas. The least economic is case a, that gives drachmas, while the one derived from the simulation model gives an intermediate cost of drachmas.(fig. 3).

6 References 1. Colarullo, S. M. Heidari, T Maddock III, Identification of an optimal groundwater management strategy in a contaminated aquifer, Water Resources Bulletin, vol. 2, no. 5, pp , Gorelick, S. M., A Review of Distributed Parameter Groundwater Management Modeling Methods, Water Resources Research, vol. 19, no. 2, pp , Greenwald, R. M., MODflow MANagement: An Optimization Module for MODFLOW, Geotrans, Virginia, version 3.2, Me Donald, M. G, and AW Harbaugh,. A modular three - dimensional finite - difference groundwater flow model, U.S. Geological Survey, Book6, Chapter Al, Moutsopoulos C And C Tzimopoulos. Groundwater flow simulation : a comparison between the method of multiple cells and the FEM, Proc ofhydrosoftconf, Porto Carras, Chalkidiki, Greece, Psilovikos Aris. Optimum management in aquifer studies using the Linear Programming (LP) method. An application to Eidomeni - Evzones area, MSc Thesis, Thessaloniki,pp. 155, Psilovikos A, C Moutsopoulos, C Tzimopoulos, S. Giannopoulos, Water mass balance estimation in the aquifer of Eidomeni - Evzones using the Modflow model, Proc. of the 2"** Conf. on Water Resources Management, Larisa, Greece, pp , Tzimopoulos C, A. Psilovikos, C Tzimourtas, Simulation of the hydrogeological basin of Eidomeni - Evzones area using the Multiple Cells method, Proc of the 6* Conf of Hellenic Hydrotechnical Union, Thessaloniki, Greece, pp , (N+l) Simulations MODMAN MODFLOW MODMAN LINDO 1ODMAN DATA 1DFLOW DATA MODFLOW OUTPUT MODMPS - MODOPT Figure. 1 : Flow chart between simulation - management and optimization models.

7 Figure. 2 ((Influence surface» from unit rate pumping from well Table 1. Wells G1284 G1318 G1267 E1+E2 G2459 G246 G2461 G2538 GN1 GN2 GN3 Ground level Cost Coefficients Beg. AM Piezom Final piezom Ah total Ah Hman Cost Coefficients

8 Table 2. Pumping Rates [m^/d] Pumping rates applied in Maximum permissible simulation model MODFLOW. pumping rates given in MODMAN input data Periods / Wells G1284 G1318 G1267 E1+E2 G2459 G246 G2461 G2538 GN1 GN2 GN3 jst "i rd * "" rd * optimum solution. Coefficients = «1». r 25 4 nnd rd * optimum solution. Coefficients ^ «1». jst 25 2"" 25 3* 25 G1284 G1318 G1267 E1+E2 G2459 G246 G2461 G2538 GNl GN2 GN3 Figure 3. Pumping cost in drachmas for each pumping well and total cost for the sum of wells, for the Modflow simulation, the T' and the 2** optimization methods respectively.