Optimal pumping locations of skimming wells

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Hydrological Sciences–Journal–des Sciences Hydrologiques, 52(2) April 2007

Optimal pumping locations of skimming wells S. V. N. RAO1 & S. MANJU2 1 National Institute of Hydrology, Roorkee 246 667, Uttar Anchal, India 2 College of Engineering, Computer Science, Roorkee 246 667, Uttar Anchal, India [email protected]

Abstract A real-life problem involving pumping of groundwater from a series of existing wells along a river flood plain underlain with geologically saline water is examined within a conceptual framework. Unplanned pumping results in upconing of saline water. Therefore, it is necessary to determine optimal locations of fixed capacity pumping wells in space and time from a set of pre-selected candidate wells that minimize total salinity concentration in space and time. The nonlinear, non-convex, combinatorial problem involving zero–one decision variables is solved in a simulation–optimization (S/O) framework. Optimization is accomplished by using simulated annealing (SA) – a search algorithm. The computational burden is primarily managed by replacing the numerical model with a surrogate simulator – artificial neural network (ANN). The computational burden is further reduced through intuitive algorithmic guidance. The model results suggest that the skimming wells must be operated from optimal locations such that they are staggered in space and time to obtain least saline water. Key words artificial neural network; groundwater, simulated annealing; skimming wells; upconing

Localisations optimales de pompage avec des puits d’écrémage Résumé Un problème concret de pompage d’eau souterraine, avec un ensemble de puits existant le long d’une plaine alluviale qui présente des occurrences sous-jacentes d’eau saline d’origine géologique, est examiné au sein d’un cadre conceptuel. Un pompage non planifié aboutit à un soulèvement de l’interface de l’eau saline. Il est par conséquent nécessaire de déterminer les implantations optimales dans l’espace et dans le temps des puits de pompage de capacités fixées parmi un ensemble de puits candidats présélectionnés qui minimisent la concentration totale en sels dans l’espace et dans le temps. Le problème combinatoire non-linéaire non-convexe qui implique des variables de décision en zéro–un est résolu dans un cadre de simulation–optimisation. L’optimisation est réalisée en utilisant le recuit simulé – un algorithme de recherche. L’étape de calcul est gérée essentiellement en remplaçant le modèle numérique par un simulateur de substitution – réseau de neurones artificiel. L’étape de calcul est de plus réduite à travers une procédure algorithmique intuitive. Les résultats du modèle suggèrent que les puits d’écrémage doivent être utilisés en des localisations optimales de façon à les répartir dans l’espace et le temps et ainsi obtenir moins d’eau saline. Mots clefs réseau de neurones artificiel; eau souterraine; recuit simulé; puits d’écrémage; soulèvement de l’interface; variables en zéro–un

INTRODUCTION The practice of pumping fresh groundwater from flood plains along river banks is widely known. Under typical climate conditions in India the rainfall–runoff process is mostly confined to a few months during the monsoon season. The floods during this period recharge the adjacent river banks in addition to the direct rainfall recharge occurring in the alluvial flood plains in the vicinity of the river. Pumping from production wells along the banks from this naturally replenishing groundwater reservoir helps in meeting the ever-increasing demand for water during both the monsoon and non-monsoon seasons on a sustainable basis. However, pumping groundwater from a stream–aquifer system becomes complex when the aquifer is underlain with saline water due to density effects. If the location and installed capacities of pumps are fixed, then pumping patterns in space and time become crucial decision variables. By appropriate regulation, skimming wells are intended to pump freshwater (of relatively lower density) floating on saline water. The present study was motivated from a field problem involving pumping from a series of 90 existing wells (see Fig. 1) to meet drinking water needs, along the bank of Open for discussion until 1 October 2007

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Optimal pumping locations of skimming wells

77°12’30 ”

280 52’ 30”

N

Well locations

280 50’ 00”

Right marginal embankment

Left marginal embankment

280 47’ 30”

Fig. 1 The River Yamuna near Delhi showing well locations in the flood plain.

the River Yamuna, north of Delhi (India). The river reach is recharged by floodwaters as well as rainfall recharge during the monsoon season. The freshwater in the aquifer system is underlain with aquifers of geologically saline water. In the present study, a simplified hypothetical but near-real aquifer system, that is representative of the Indian monsoon rainfall conditions, the study area, skimming wells, and aquifer parameters, is modelled in a conceptual framework. Since the existing wells have pumps of fixed capacity, it is necessary to determine which specific wells to operate from a given set of candidate wells in space and time. The decision variable is either zero or one (on or off). The nonlinear, non-convex, combinatorial problem involving discrete zero–one decision variables (pumping locations) is solved within a simulation–optimization (S/O) framework. Gradientbased methods are not suitable for discrete variables and, therefore, simulated annealing – a stochastic search technique – is used. Since all S/O problems involve high computational burden (Das & Datta, 1999; Zheng & Wang, 2002; Rao et al., 2004a,b), an artificial neural network (ANN) is used as a surrogate simulator of a variable density-driven numerical flow model. Problem specific algorithmic guidance is used to further reduce the computational burden. MODEL FORMULATION The basic objective of this study is to develop an operational model through optimal location of pumping wells from a group of existing candidate wells, in order to control Copyright © 2007 IAHS Press

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S. V. N. Rao & S. Manju

the upconing phenomenon. Mathematically, the single objective optimization problem with a view to minimizing the total salinity concentration at grid cells of well screen pumping locations in space and time may be formulated in general, within the S/O framework as follows: N

K

J

I

min J 1 = ∑∑∑∑ C in, j ,k

(1)

n =1 k =1 j =1 i =1

where I, J, K and N represent the number of rows, columns, layers and time periods, respectively, of the finite difference grid of the aquifer system; and C in, j ,k is the salinity concentration (state variable) in the production well screen grid location at the node i, j, k at the end of the nth time period. The objective is to determine optimal locations of pumping wells in space and time that minimise total salinity concentration at all grid cells at the end of each time period. However only representative screen grid cell locations where the pump is switched on would be accounted during each time period with the surrogate ANN simulator to control computational burden. The decision variable is either zero or one (on or off) at the pumping location. It is important to note that the objective function seeks to determine such pumping locations where the pump must be switched “on” from the group of candidate wells, which result in minimum total salinity. Typically, the planning horizon is taken as one year, i.e. for two time periods or seasons in a water year, when the system is restored to its original quasi-steady state condition. SOLUTION METHODOLOGY The methodology adopted in this study uses a combined S/O approach (Rao et al., 2004a, 2004b, 2006). This methodology interfaces a simulator and an optimizer. The simulator consists of a three-dimensional (3D) density-dependent flow and transport model called SEAWAT-2000 (Langevin et al. 2004). The optimiser consists of a search algorithm implementing simulated annealing (SA). An ANN (ASCE 2000) is used as a surrogate model to replace the numerical simulator at points of interest and to reduce the computational burden. As well as the ANN, problem-specific algorithmic guidance, as discussed later, further reduces the computational burden. ILLUSTRATIVE APPLICATION OF THE S/O MODEL The proposed model primarily seeks to control the pumping of groundwater from a group of wells prone to upconing of saline water. To illustrate this concept and methodology, a simplified homogeneous aquifer system representative (parsimonious) of the study area and aquifer parameters is considered. An eight-layer, 32-row, 13column finite difference grid was constructed using a pre-processor (see Fig. 2). Typically, a river boundary with constant head on one side and a groundwater divide contour on the other is considered. No flow boundaries are assumed to the north, south and bottom of the aquifer. The lower-most layer is assumed to have a constant salinity concentration of 5 kg m-3. The input variables and aquifer parameters are listed in Table 1. The SEAWAT-2000 model was implemented using a false transient approach Copyright © 2007 IAHS Press

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Production well

Fig. 2 Plan and cross-section of the simplified aquifer system.

under average rainfall–recharge conditions for a long time period (5000 days) until steady-state conditions in terms of heads and concentrations were achieved. Optimal location of wells The illustrative example (Fig. 2) was conceived and motivated by the real-world problem shown in Fig. 1. The planning horizon of one year is assumed to be divided into two stress periods (seasons) of 180 days each. The two stress periods correspond to the monsoon and non-monsoon seasons, typical of Indian rainfall conditions. Uniform recharge is assumed to occur only during the monsoon season. Some additional recharge in the flood plain (two grid cells along the river boundary) was assumed. Copyright © 2007 IAHS Press

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Table 1 Aquifer and other parameters used for the SEAWAT-2000 model. Particulars Hydraulic conductivity in X, Y and Z directions Specific yield, specific storage Longitudinal and transverse dispersivity (αl , αt) Uniform rainfall recharge Grid in X and Y directions (Δx, Δy) Grid in Z direction (Δz) No. of rows, columns and layers Stress period No. of stress periods, time steps in each period Concentration of freshwater Max. conc. of saline water (bottom-most layer) Maximum density of saline water Density of freshwater Aquifer top and bottom elevation Constant head in river Courant number, coupling parameter DNSCRIT

Values 40, 40 and 4 m d-1 0.15, 0.001 (m-1) 30 and 10 m 0.12 m per monsoon season 50 m 10 m 32, 13 and 8 6 months (180 days) 2, 18 0 5 kg m-3 1003.5 kg m-3 1000 kg m-3 80 m, 0 m 75 m ≤1, 0.01 kg m-3

A series of eight candidate wells is considered (Fig. 2). It is assumed that only part of the wells (four wells) operate at a fixed rate (500 m3 d-1) during any given stress period or season. It is required to determine their optimal location in space and time. All the wells were assumed to pump from the uppermost, i.e. third layer (barring the first and second layers for possible variation in drawdown due to pumping). The idea of pumping from the third layer is obvious, as the salinity concentration is expected to be least towards the topmost layer in a vertical direction in a density-driven flow phenomenon. The illustrative problem was designed such that the optimal solution is known intuitively, as a proof of concept. The use of eight candidate wells considered in this study implies 16 decision variables for the two stress periods (see Table 1). A typical data set of seven data patterns with zero–one as input variables and corresponding concentration at eight locations at the end of two stress periods is presented in Table 2. If each decision variable takes two values, i.e. zero or one, this results in 216 possible configurations. For any given set of pumpages, the SEAWAT-2000 model takes, on an average, 60 seconds to execute two stress periods (360 days) involving the iterative solution of flow and transport on a desktop PC. Clearly, a brute force technique is impractical, i.e. evaluating every configuration; besides, the optimization (SA) process involves several thousands of function calls to the simulator. Therefore a surrogate ANN simulator was developed to reduce the computational burden. To generate training sets (patterns) for the ANN, the SEAWAT-2000 model was repeatedly executed to pumping at random pumping locations. The model was implemented in transient mode for two stress periods beginning with the monsoon season. The initial groundwater levels corresponded to the steady-state conditions discussed earlier. During each run, and each stress period, random locations were generated at any four locations out of eight possible candidate wells and were assigned a fixed pumping rate of 500 m3 d-1. The remaining four locations were assigned zero pumping. After the model execution at the end of each realization, the input pumping (0 or 500 m3 d-1) and its corresponding aquifer responses (output) at each well in terms Copyright © 2007 IAHS Press

Input: Output: Zero–one variables at eight locations for two time Concentration (kg m-3) at screen grid locations at the periods (16 decision variables) end of first time period (180 days)

Concentration (kg m-3)at screen grid locations at the end of second time period (360 days)

0 1 0 0 0 0 1

0.873 0.848 0.847 0.847 0.873 0.873 0.855

0 0 1 0 0 1 1

0 0 1 1 1 0 0

1 1 0 0 1 0 0

1 0 0 0 1 1 1

1 0 0 1 1 1 0

1 1 1 1 0 1 1

0 1 1 1 0 0 1

1 0 0 0 0 1 1

1 0 0 0 1 0 0

0 1 1 0 1 0 0

1 1 1 1 0 1 0

0 1 0 0 1 1 1

0 0 0 1 0 1 0

1 0 1 1 1 0 1

0.877 0.857 0.857 0.857 0.877 0.877 0.877

0.784 0.765 0.784 0.784 0.784 0.784 0.767

0.429 0.431 0.408 0.430 0.430 0.408 0.403

0.260 0.260 0.257 0.264 0.264 0.267 0.267

0.628 0.628 0.646 0.646 0.628 0.646 0.646

0.549 0.576 0.570 0.571 0.549 0.560 0.560

0.639 0.669 0.669 0.648 0.639 0.640 0.673

0.787 0.787 0.787 0.787 0.807 0.787 0.787

0.766 0.763 0.782 0.782 0.782 0.766 0.757

0.404 0.428 0.406 0.428 0.410 0.409 0.405

0.267 0.264 0.270 0.261 0.267 0.263 0.263

0.618 0.618 0.626 0.627 0.626 0.628 0.644

0.553 0.553 0.573 0.572 0.554 0.549 0.558

0.637 0.670 0.665 0.636 0.646 0.625 0.671

0.782 0.788 0.782 0.782 0.786 0.787 0.782


Table 2 Typical data set used for input–output ANN training concentration (kg m-3) at eight pumping locations (fixed pumping at 500 m3 d-1) at the end of two time periods.

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of salinity concentration at the screen-located grid cells were recorded in an output file. Repeated execution of the SEAWAT-2000 model involved some 30 h of computer run to generate more than 1800 realizations of input–output (data sets). Here, 1800 realisations is a conservative estimate and the number of data patterns are generally justified in terms of goodness of fit (R2), as discussed below. Initially the input variables (0 or 500 m3 d-1) pertaining to pumping are converted into zero–one variables. A typical data set of seven patterns with zero–one as input variables and corresponding concentration at eight locations during two time periods is presented in Table 2. The input–output patterns are then standardized before ANN training. For this purpose, the input–output data series (patterns) are scaled between zero (0.0) and one (1.0). A three-layer feed-forward network with an input, and sigmoid and linear output layers, was trained using the ANN toolbox of MATLAB (2000) to obtain optimal weights and biases for each network. The supervised training was accomplished with the help of a back-propagation algorithm, as implemented in MATLAB. Typically, to train an 8-6-1 ANN architecture for concentration of solute at one of the locations for the first stress period would mean eight input neurons, six hidden neurons and one output neuron. Similarly there will be 16 neurons as input and six hidden neurons and one output neuron for training of concentration at any one location at the end of the second stress period. This training procedure was repeated for each output variable, i.e. salinity concentration at each well screen location (grid cell in the third layer) and at the end of each stress period. Training only one output at a time generally takes only a few seconds. Training more than one output at a time takes much longer. The network with optimal weights and biases in the form of a small subroutine involves only simple matrix operations to convert the SEAWAT-2000 model into a surrogate simulator (only at points of interest). The behaviour of the ANN surrogate model in general showed very high goodness of fit (R2 = 0.97–0.98). Other similar details have been discussed in Rao et al. (2004a,b) and are therefore not presented here. The surrogate model was subsequently interfaced within the S/O model to replace the SEAWAT-2000 simulator. The annealing parameters for SA were arrived at through trial and error (Dougherty et al., 1991; Cunha, 1999; Rao et al., 2004a,b). The initial SA temperature (set at 0.2) was arrived at such that more than 80% of the feasible configurations are accepted in the beginning. The chain length (equilibrium criterion) was set in the range of 80–90 times the number of decision variables and the cooling factor (rate of reducing the SA temperature) was varied in the range of 0.7–0.9. The SA procedure was terminated when four successive temperature reductions did not yield an improvement in the solution. The optimal solution is presented in Table 3. The evolution of the model solution using the SA procedure is depicted in Figs 3 and 4. The optimal solution was found to be along expected lines and consistent with intuition. In the first stress period, the model allocates the fixed pumpages (500 m3 d-1) in locations 1, 2, 5 and 7, while in the second stress period it chooses locations 2, 4, 6 and 8. Intuitively, the only other alternative optimal solution, which the model could find with same value of objective function, would be to interchange the locations between the first and second stress periods. This is because of the density-driven flow phenomenon. The salinity concentration in the wells, which were pumping during the first stress period, was relatively higher than neighbouring wells (see Table 2). Copyright © 2007 IAHS Press

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Table 3 Optimal pumping (m3 d-1; upper half-row) location and salinity (kg m-3; lower half-row) in screenlocated grid cells in the third layer at the end of each stress period. Objective function value

Stress period (d)

Pumping locations: 1 2

3

4

5

6

7

8

First

500.0 0.30205 0.0 0.32155

500.0 0.27857 0.0 0.29555

0.0 0.15508 500.0 0.34725

500.0 0.27522 0.0 0.29017

0.0 0.15203 500.0 0.34413

500.0 0.28448 0.0 0.30293

2.5322 0.0 0.14449 500.0 0.37013

1000

10000

(1–180)

Second

Obj. function (concentration kg/m 3)

(180–360)

0.0 0.15380 500.0 0.36425

2.8 2.7 2.6 2.5 1

10

100

No of function calls

Fig. 3 Evolution of model solution using the SA algorithm

80 60 40 20 1

0.1

0.01 Temperature

% of acceptance

100

0 0.001

Fig. 4 Graphic representation of temperature reduction with % of acceptance using the SA algorithm

Therefore, during the second stress period, the model prefers not to choose the same wells. The net effect is to stagger the pumpages in space and time. The model staggers in space and time in order to minimize the effect of interference from neighbouring wells, which enhances the advective velocities leading to increase in concentration in the grid cells from which it has been decided to carry out pumping, as indicated in Table 2. The study therefore leads to the inference that skimming wells must be operated such that they are staggered in space and time to obtain groundwater of minimum salinity. Computational burden and algorithmic guidance The CPU time in general depends on a number of factors. This includes the time consumed by the simulator, the number of decision variables, the tightness of conCopyright © 2007 IAHS Press

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straints, the speed of the processor, and annealing parameters (initial temperature, cooling factor, chain length or equilibrium and termination criteria). The SA procedure in the present methodology introduces a computational time burden that has two distinct components. The first component is due to the time consumed by the function calls to the simulator and is associated with every feasible trial configuration. This virtually reduces to near zero with ANN as the surrogate simulator. The second component is the average time consumed in generating feasible solutions until equilibrium and termination criteria are met. The second component can be kept to a minimum through efficient coding and algorithmic guidance, so that the number of infeasible trial configurations is kept to a minimum. The total CPU time is determined by the sum of the two components multiplied by the total number of iterations or chains. At initial temperature, the number of iterations is large mainly due to infeasible solutions. At final temperature, the uphill moves are too many in general. The total number of iterations is problem specific and, therefore, can be determined only after actual model execution. For the unconstrained problem discussed so far, the optimal solution was attained after 1224 calls to the simulator with CPU time of less than 60 s. However, the number of calls depends on the beginning search point, which actually depends on the random seed. In any event, the computational burden is largely controlled with the ANN as the simulator. If the problem is constrained for salinity (say 0.4 kg m-3) at each pumping location (which is on) the computational burden increases to 1075 s. This is due to the increase in the number of infeasible calls that are rejected by the constraint. Here, the computational burden arises from the second component, also as discussed earlier. This can only be controlled through efficient algorithmic guidance. A simple problem-specific, algorithmic guidance was designed to introduce suitable bias for early convergence towards the optimal solution presented in Table 3. A small subroutine was coded to ensure that the trial random allocations of fixed pumpages were staggered in space and time. Computationally, this was achieved along space and time as follows: the staggered allocations in space were made by ensuring that the centre of gravity of allocated pumpages lies somewhere in the middle band (near mid-point) along the arm of the series of pumping locations, beginning at Location 1 (see Fig. 2). Along time, this is achieved by avoiding allocation at the same location in the next time period. With this simple approach, the computational burden could be reduced to 120 s. However, it is important to note that this approach has been applied to a simplified aquifer system. Real systems involve many other aspects, such as influence of external wells, aquifer properties/geometry, varying depth of the saline–freshwater interface, as well as boundary and confining conditions. Nevertheless, the concept of staggering in space and time can still be extended in general to real systems to determine the optimal locations of skimming wells and to reduce the computational burden via algorithmic guidance. SUMMARY AND CONCLUSIONS A simplified aquifer system that is representative of the field problem in terms of study area, existing wells, input variables and aquifer parameters is solved in a conceptual Copyright © 2007 IAHS Press


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framework. The nonlinear, non-convex, combinatorial model is solved as a zero–one problem using the S/O approach. The illustrative example seeks to address the issue of upconing and provides an insight into the solution through optimal pumping locations from a series of candidate wells. The model clearly suggests that skimming wells must be operated such that they are staggered in space and time so as to obtain the least saline water. The computational burden is primarily reduced by replacing the variable density simulator with an ANN. The burden is further reduced through efficient algorithmic guidance based on intuitive understanding of the problem. For application to real systems this approach can work with moderate computational burden involving 25–30 decision variables. For larger systems, parallel processors may be required. Acknowledgements The author is grateful to Dr K. D. Sharma (Director National Institute of Hydrology, Roorkee, India) for permission and encouragement to publish this paper. The author is very grateful to the anonymous reviewers for help in improving the quality of the paper. REFERENCES ASCE (Task Committee on Application of Artificial Neural Networks in Hydrology) (2000) Artificial neural networks in hydrology. I: Preliminary concepts. J. Hydrol. Engng ASCE 5(2), 115–123. Cunha, M. D. C. (1999) On solving aquifer management problems with simulated annealing algorithms. Water Resour. Manage. 13, 153–169. Das, A. & Datta, B. (1999) Development of multi objective management models for coastal aquifers. J. Water Resour. Plan. Manage. ASCE 125(2), 76–87. Dougherty, D. E. & Marryott, R A. (1991) Optimal groundwater management. 1: Simulated annealing. Water Resour. Res. 27(10), 2493–2508. Langevin, C. D., Shoemaker, W. B., Guo, W. & Missimer, C. D. M. (2004) MODFLOW-2000: the USGS modular groundwater model. Documentation of the SEAWAT 2000 Version with Variable-Density Flow process (VDF) and the integrated MT3DMS Transport Process (IMT). USGS Open File Report 03-426. MATLAB (2000) Neural Network Tool Box for Use with Matlab User Guide Version 4. The Mathwork, Inc. 3, Apple Hill Drive, Massachusetts, USA. Rao, S. V. N., Murty Bhallamudi, S., Thandaveswara, B. S. & Mishra, G. C. (2004a) Conjunctive use of surface and groundwater for coastal and Deltaic systems. J. Water Resour. Plann. Manage. ASCE 130(3), 255–267. Rao, S. V. N., Srinivasulu, V., Murty Bhallamudi, S., Thandaveswara, B. S. & Sudheer, K. P. (2004b) Planning groundwater development in coastal aquifers. Hydrol. Sci. J. 49(1), 155–170. Rao, S. V. N., Kumar, S., Shekher, S. & Chakravorty, D. (2006) Optimal pumping from skimming wells. J. Hydrol. Engng ASCE 11(5), 464–471. Zheng, C. & Wang, P. P. (2002) A field demonstration of the simulation optimisation approach for remediation system design. Groundwater 40(3), 258–265.

Received 18 July 2005; accepted 2 January 2007
