# Application of Artificial Neural Networks for Identifying Optimal Groundwater Pumping and Piping Network Layout

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## Abstract

The simulation-optimization approach is often used to solve water resource management problem although repeated use of the simulation model enhances the computational load. In this study, Artificial Neural Network (ANN) and Bagged Decision Trees (BDT) models were developed as an approximator for Analytic Element Method (AEM) based groundwater flow model. Developed ANN and BDT models were coupled with Particle Swarm Optimization (PSO) model to solve the well-field management problem. The groundwater flow model was developed for the study area and used to generate the dataset for the training and testing of the ANN & BDT models. These coupled ANN-PSO & BDT-PSO models were employed to find the optimal design and cost of the new well-field system by optimizing discharge & co-ordinate of wells along with the cost effective layout of piping network. The Minimum Spanning Tree (MST) based model was used to find out the optimal piping network layout and checking the hydraulic constraints in the piping network. The results show that the ANN & BDT models are good approximators of AEM model and they can reduce the computational burden significantly although ANN model performs better than BDT model. The results show that the coupling of piping network model with simulation-optimization model is very significant for finding the cost effective and realistic design of the new well-field system.

## Keywords

Groundwater modeling Groundwater management Artificial neural network Bagged decision trees Particle swarm optimization## 1 Introduction

The groundwater management problems are often solved by simulation-optimization approach (Lefkoff and Gorelick 1986; Zheng and Wang 2002; Finney and Samsuhadi 1992; Emch and Yeh 1998; Gaur et al. 2011; Ayvaz 2016; Karatzas 2017). In the simulation-optimization process, simulation model is repetitively called by optimization model for generating the groundwater levels along with the velocity, concentration etc. The use of the simulation models, in the each iteration of optimization model, increases the computational load extensively. Therefore, different researchers (Rogers and Dowla 1992; Johnson and Rogers 1995; Coppola et al. 2003; ASCE 2000; Singh et al. 2004; Nikolos et al. 2008; Christelis and Mantoglou 2016) used Artificial Neural Network (ANN) models as approximators of computationally expensive numerical models. They applied ANN and optimization algorithm to solve different hydrological management problems and found this combination to be more fast and robust. Arndt et al. (2005) approximated the results of Finite Element Method (FEM) based simulation model using predictions by ANN model. Results show that the value of the objective function at the simulation based optimal solution was only 1% better than the optimal solution obtained by the ANN. Nikolos et al. (2008) coupled ANN and differential evolution algorithm for substituting the FEM for water resource management problems. The study also concluded that the ANN can be used as a good approximation model to reduce the computational burden with optimal values close to simulation model. Christelis and Mantoglou (2016) applied the radial basis functions (RBF) as an approximator model to emulate the scalar response of a multivariate function. Results show that 96% of computational time was reduced by combined RBF and evolutionary annealing-simplex algorithm for the solution of pumping optimization.

Lots of work was also carried out for the optimal design of Water Distribution System (WDS). Wu and Simpson (2002) presented the application of fast messy Genetic Algorithm (GA) for the design of WDS. Nicklow (2010) concluded that GA is flexible and powerful technique to solve complex WDS network. Bieupoude et al. (2012) worked for the optimal design of water distribution network. The geometric and multi-scale optimization was used to analytically optimize T-shaped network architectures under the constraint related to water quality. The output of the study helped in the determination of an optimal geometry of the network that minimizes the head losses. Somaida et al. (2013) developed an analytical solution for determining the optimum pumping rate in a piping network supplied from the pumping wells. The gradient technique was used and examined on a predetermined optimal water distribution system.

To the best knowledge of the authors, the development of a new well-field system by the application of machine learning technique along with piping network model was not addressed by the researchers. Moreover, very few researchers have worked on the application of coupled groundwater and piping network models (Adams and Parkin 2002; Tsai et al. 2009). Mainly, the models were applied for the aquifer modelling in a karstic region and optimal scheduling and the design of the established piping networks. Tsai et al., (2009) developed a wells management model by integrating a water distribution system model (EPANET) and a groundwater model (MODFLOW) under an optimization framework. The management model considered multiple objectives which were solved by a parallel genetic algorithm (PGA) and were found to be effective. In the study, already established piping layout model was considered for the analysis. To the best knowledge of the authors, Bagged Decision Trees (BDT) has not been used as an approximator of simulation model in the field of water resources.

The present research work is performed for the development of a new well-field system without considering any pre-defined layout of piping network, discharge and location of the wells which makes this work quite distinctive. Also the comparison of ANN and BDT models in the groundwater management problems is not addressed so far. In this regard, applicability of Particle Swarm Optimization (PSO) based ANN-PSO & BDT-PSO models were examined to minimize the pumping cost of the wells, where the ANN & BDT models were developed to approximate the Analytic Element Method (AEM) based groundwater simulation model. Both ANN & BDT models were trained and tested through dataset generated by AEM model. Piping layout network model was incorporated to design the optimal piping network. These coupled models were applied to find the optimal piping network by minimizing the pumping cost for the given water demand & maximum discharge limit of the wells. The results of ANN-PSO & BDT-PSO models were compared with AEM-PSO models. The study shows that combination of ANN & BDT models with piping layout design model is the efficient way for the design of new well-field system and can be applied on other areas.

## 2 Methodology

The present study incorporates the strength of both machine learning techniques and optimization algorithm to solve the well-field management problem. AEM based groundwater flow model was developed in the first stage and was used to generate the groundwater head for the training and testing of ANN & BDT models. The co-ordinate and discharge of the wells were taken as inputs and the groundwater head at the periphery of the well was taken as output for the development of ANN & BDT models. The random discharge & co-ordinate of the wells were generated by random number generator. Further, developed ANN & BDT models are integrated with PSO and used as a proxy simulator to AEM-PSO models. Kruskal’s algorithm based model was developed and coupled with ANN-PSO & BDT-PSO model to apply Minimum Spanning Tree (MST) method for generating the optimal piping layout network. The coupled models were applied for identifying the optimal layout of piping network, with minimum total length, in each iteration of the optimization model. Whereas, the hydraulic design of piping network was also checked in the each iteration by computing head losses at each segment of piping network and checking the continuity of flow in the network. A penalty function method was used to handle the constraints in the optimization function. The major steps involved in the above procedure are explained in the following sections.

## 3 Objective Function and Constraints

The objective function was to minimize the pumping and the piping network cost. The pumping cost consists of the cost of well installation, pump cost and electricity cost for the pumping. In this study, all the wells were considered to be of the same depth and diameter. Therefore the cost of each well was taken constant and the total cost was calculated by multiplying the installation cost of one well by number of wells. Detail description about the function is given below.

*Piping Cost*: In this study, the piping length was computed by MST algorithm which creates the network by joining all the nodal points and also gives the minimum length of that network. The piping cost is defined as

*L*

_{i}= pipe length for

*i*

^{th}segment,

*A*

_{d}= total cost for the development of per meter pipe network of specific diameter

*d*and

*N*

_{p}= total number of pipe segments.

*A*

_{d}consist of the cost of earthwork, pipes, connectors and platforms.

*Pumping Cost*is a major factor that influences the cost of WFS. The pumping cost is influenced by quantity of the water to be pumped

*Q*, pumping head

*H*along with specific weight of the water

*γ*, efficiency

*η*of pump and energy

*R*

_{E}(Sharma and Swamee 2006; Moradi et al. 2003). Total cost of pumping consists of the cost of pump units and the electricity cost (Swamee and Sharma 1990; Swamee 1996) which can be defined as,

*γ* = 9810 N/m^{3}; *η* = 85%; *R*_{E} = 0.085 euros/kwh; *r* = the rate of interest in “euros per euros per year” (euros/euros/year) and taken as 6%, *T* = 25 project life, *N*_{w} = total number of wells, *H*^{n} = total required head at each well location/node to lift the water at storage tank and k_{p} = pump cost on the basis of required pump power. As the same pump unit was adopted for each well, therefore a constant value was adopted in the function. The well construction & installation cost was taken as 4000 Euros per well. The location of storage tank was taken as X = 687,000 and Y = 218,000.

*P*

_{n}= penalty term which varies linearly with the magnitude of constraint violation and.

*β*

_{n}= weighting factor which is selected on the basis of constraints within the range of 10

^{3}to 10

^{7}.

*n*is the number of constraints in the problem. Following constraints were taken in the model,

Hydraulic design constraints to maintain the flow in piping network are defined below.

*i*

^{th}inlet pipe at junction node – head loss in

*i*

^{th}inlet pipe = head at junction node.

*Q*

_{i}is the potential discharge from

*i*

^{th}well.

*i = 1, 2.......Nw*.

*Q*

_{i,min}and

*Q*

_{i,max}are the minimum and maximum discharge limit for

*i*

^{th}well,

*h*

_{i,min}= minimum allowable head of groundwater at

*i*

^{th}well,

*S*

_{w,min}= minimum distance between any pair of wells;

*x*

_{j}and

*y*

_{j}are co-ordinates of remaining well i.e.

*i*≠

*j*. where

*h*

_{i}is the water head at the

*i*

^{th}well and

*i =*1, 2….,

*N*

_{w.}

*hd*

_{i}is the head at particular well node and

*hf*

_{i}is the head loss in the pipe connecting well node

*i*to next well node.

The study area consists of two rivers Allier and Dore, where Dore river is an important tributary of Allier River. Dore river catchment is situated in the eastern part of the Massif-Central in France. The study area lies between 45054’N to 460 N latitude and 3025′E to 3029′10″E longitude. The developed model was applied on the part of the basin to identify the optimal location and discharge of the wells by satisfying the above said constraints. Depending on the rate of clay deposits, the hydraulic conductivity in the study area varies from 1 × 10^{−3} to 3 × 10^{−3} m/s. For the development of groundwater model, the rivers of the study area were represented by 39 head line-sink elements. Discharge wells were represented by well elements. A total of 12 piezometric measurements were available in the study area which were used for calibration of the model. The water level in the rivers was under observation at 11 different locations and were used to develop the AEM model (Gaur et al. 2011). Whereas, the constraint 4a was defined to insure the maximum and minimum discharge limit of the single well which was taken as 120 m^{3}/h to 280 m^{3}/h. The constraint 4b was taken to insure the minimum water discharge from all the wells and was fixed on the basis of minimum water demand of the area and taken as 980 m^{3}/h. The constraint 4c was taken to limit the drawdown of the groundwater i.e. 253 m. The constraint 4d was taken to keep the wells in specified zone for the development of well-field. The constraint 4e was taken to insure the minimum distance between the wells. To provide the protective zone around the wells a minimum distance of 300 m between any two wells was defined for protective zones. The constraint 4f & 4 g were taken as hydraulic constraints to maintain the continuity of flow and conservation of energy in the piping network (Tsai et al. 2009).

### 3.1 Model Development

AEM is a computational method based upon the superposition of analytical expression to represent the two dimensional vector fields. Ground water flow model was developed in MATLAB 7.0 (Strack 1989; Gaur et al. 2011; Math Works Inc 2001). The ANN and BDT models were developed and trained for a defined area which had the highest probability for occurrence of wells. This specific area was selected as potential alluvium aquifer between the two rivers and considering the output of AEM-PSO model (Gaur et al. 2011). 2000 sets of random discharge and location of wells were generated for the training and testing of both the models and AEM model was used to compute the groundwater head on the basis of this data. Total 21 input variables were used for the training and testing of ANN and BDT models which consists of co-ordinates (*x, y*) and the discharge (*Q*) of all the seven wells and corresponding groundwater head. Total seven ANN models were developed where each model gave the value of one selected well out of the seven wells. Therefore, all seven models were capable to give the head on the periphery of all the seven wells. Finally, to minimize the cost of all seven wells, PSO was combined with ANN and BDT models.

Architecture of ANN model was finalized on the basis of trial-and-error procedure (Atiya and Ji 1997; Morshed and Kaluarachchi 1998). The ANN model was trained using the back-propagation with Levenberg-Marquardt (L-M) technique and the sigmoid transfer functions ‘tansig’, log-sigmoid ‘logsig’ and linear ‘purelin’ were chosen for transfer functions of both the hidden layer and the output layer. The decision trees are also popularly used for various machine learning applications. The individual decision trees are highly sensitive to noise in its training dataset. The trees that grow very deep may learn a highly irregular pattern lead to low bias, but very high variance. The output from multiple trees can be taken instead of output from just a single tree. Although training multiple trees from same dataset could give strongly correlated trees or may even give the same tree. Bagging Technique was used to avoid this problem. Bagging or Bootstrap aggregating (Breiman 1994) is a machine learning technique used to improve accuracy and stability of machine learning algorithms. In order to improve the accuracy of decision trees, it combines results from different trees trained using randomly generated dataset. Thus, the trees get de-correlated. Given a training dataset of samples *N*, bagging generates \( \mathcal{M} \) new datasets of size *S*, by taking samples from dataset uniformly with replacement. Further, \( \mathcal{M} \) decision trees are trained for these new \( \mathcal{M} \) datasets. Output is decided on the basis of average of all the trees. This concept of sample bagging can be extended on the features rather than selecting all the features at the same time only a fraction of features can be selected to further avoid correlation.

The efficiency & accuracy of the models were increased by two modifications. In this regard, the velocity term of PSO models were modified to deal with the wells coordinate. The decimal number values generated by the velocity term generally increase the iteration of PSO models exceptionally. To handle this, restrictions were applied on velocity term for integer number and ‘units position’ rounded up to 5 or 10 accordingly. This implies that from each previous location, new search location will be at least 5 m away. The number of iterations got reduced by 11% due to this modification. It was observed that the modified model converged in less than 1000 iterations whereas more than 1000 iterations were required to converge the model without this modification. In the second modification, additional input features were introduced in the ANN and BDT models which consisted of the distances of the concerning well, for which model was developed, from the remaining wells. These inputs helped to account the impact on the groundwater head due to distances between the wells. Significant improvements were observed as the NRMSE value was found to be lower than the previous case. It was also observed that with the increase in the number of input features from 21 to 27, the number of neurons in the hidden layer required for the ANN model reduced notably which was found 36 for 27 input features. The similar trends were observed in case of BDT model. The number of trees required for the model decreased with the increase in the number of input features. For BDT model with 27 parameters, optimum number of trees was found out to be 39.

## 4 Model Application

Three scenarios were considered to find out the optimal cost of WFS. In the first two scenarios, the model was run by considering single diameters of all pipes in the piping network and an optimal cost was obtained for those diameters of pipes. The piping diameters were taken as 200 mm and 250 mm in the first two scenarios respectively. In the third scenario, diameter of the pipes, in the network system, was also considered as one of the decision variables. As the diameter with a continuous value is not practically feasible, discrete values were taken in the optimization process i.e. 200 and 250 mm.

A spanning tree of a network connects all the vertices of that network together (wells in this case), and MST is a spanning tree with minimum possible total weight, where the weights are measured as the distances between the edges/wells. In every iteration of ANN-PSO & BDT-PSO models, the piping network model first developed the five optimal piping network layouts through minimum spanning tree method. Once the piping network layouts were finalized on the basis of total minimum lengths, total discharge at each junction node from all input pipe segments and discharge of well at that node was considered as total discharge of that node. Once the network layout was finalized, the hydraulic design model was applied to calculate the total discharge through connected pipes, at each junction node. A single well was considered at each node of piping network layout. The discharge value at node, which has only one output pipe segment and no input pipe segment, was assigned to the discharge value corresponding to the discharge of that well only. Whereas, discharge at junction nodes, which had one or more input pipes, was computed by hydraulic design model. Once all the discharge values were known at each single & multi pipe nodes, the hydraulic design model computed the head losses in each pipe segment on the basis of the discharge value and the length of the corresponding pipe segment. The model started adjusting the head losses by adding it in the head ‘H^{n}’ at each well. At the same time, the hydraulic design model also adjusted the final value of ‘H^{n}’ at wells in such a way that all connected pipes started having same head at nodes i.e. junctions. This helped to satisfy the constraints 4 f. The model started the adjustment from the storage tank and moved along the network till it reached the starting point(s) of the network. The corresponding, pumping and pump cost was computed on the basis of final “H^{n}” computed by the piping network model. Total 21 decision variables were taken for S-1 & S-2 which accounted the discharge and co-ordinates of the wells. Whereas, 28 decision variables were taken for S-3 by accounting the discharge, co-ordinates of the wells and diameter of the pipes. In the optimization process, a population of PSO particles was initialized with random values of decision variables i.e. X & Y co-ordinates, discharge of wells and diameter of pipes, in the problem space. In each iteration of the PSO optimisation, the pumping cost was calculated by using groundwater head values. The coupled ANN & BDT model computed the head values at given locations and discharges of wells for each PSO particle at each iteration. Meanwhile, the layout of the piping network model also identified the optimal layout in each iteration of PSO model & the hydraulic design model was used for computing head values at junctions of piping network. Finally, the PSO model converged to search the optimal location and discharge of wells along with optimal layout and diameters of piping network.

Optimal discharge and co-ordinates of wells in the Scenarios 1, 2 & 3

Well discharges | Co-ordinate (X, Y) | |||||
---|---|---|---|---|---|---|

S1 | S2 | S3 | S1 | S2 | S3 | |

1 | 137.7 | 135.1 | 162.3 | (687,292, 2,108,072) | (686,768, 2,108,190) | (686,423, 2,107,839) |

2 | 128.4 | 173.7 | 120.2 | (686,700, 2,107,971) | (687,300, 2,108,002) | (687,006, 2,107,384) |

3 | 123.6 | 131.7 | 139.6 | (686,704, 2,108,290) | (687,119, 2,107,725) | (686,855, 2,108,270) |

4 | 132.6 | 142.1 | 131.8 | (687,007, 2,108,301) | (687,204, 2,107,436) | (686,727, 2,107,874) |

5 | 167.0 | 132.0 | 129.3 | (687,178, 2,107,758) | (686,439, 2,107,781) | (687,011, 2,107,690) |

6 | 147.2 | 139.6 | 123.8 | (686,429, 2,108,120) | (686,725, 2,107,873) | (686,575, 2,108,408) |

7 | 143.6 | 125.8 | 173.1 | (686,877, 2,107,721) | (686,483, 2,108,287) | (687,292, 2,108,068) |

To understand the importance of piping layout model, model was run and piping cost was computed without optimizing piping layout. In this run, each well was directly connected to the storage tank. The results show that optimal cost of WFS system is high in this condition. The total cost in this run is increased by 4.21% in comparison to the cost in S-3, which is 58,045 Euros higher. On the other side, this price is lower in comparison to S-1which shows that piping layout model is effective when optimal selection of diameter is also incorporated to compute the total cost. It is also found that decreasing the diameter of pipe becomes cause of increased head losses and corresponding higher pumping cost, which highly dominates the piping cost of the system. The results also show that application of piping network model is effective and capable to reduce the cost of system & found the realistic network. The major advantage of the ANN-PSO model that was found is that it reduces the computational time. Where the AEM-PSO model approximately takes 6–7 h for 1000 iterations of model convergence, ANN-PSO model takes only 4–5 min and BDT-PSO takes 10–12 min (when run in intel core i7 processor) for the same number of iterations.

## 5 Summary and Conclusions

In this study, the AEM based simulation model was approximated to ANN & BDT models. Further these models were coupled with PSO algorithm to minimize the pumping cost of the wells. The optimal cost by ANN-PSO models was found more close to AEM-PSO models within the limit of 1.48% to 0.9% in S-1 & S-2. The cost by BDT-PSO model was found within the range of 7.9% to 8.7% of the cost obtained from AEM-PSO models. The result obtained from ANN-PSO models was found close to the result of AEM-PSO model with the error of less than 2%. Whereas, BDT-PSO model was found to be more inferior in comparison of AEM-PSO model with error of less than10%.

Even though the cost in the case of the BDT-PSO models is quite low but it is not in accordance with the AEM-PSO models as it fails to account interference. Study concluded that number of iterations and computation time for the convergence of BDT-PSO model is more in comparison of ANN-PSO model. Developed ANN model was successful in accounting well interference and also required lesser data for training. The study concludes that use of piping network model is very important to identify realistic and cost effective piping network. The diameter of the pipe is found to be a very influencing parameter and can vary the location and arrangement of the wells significantly. The use of coupled models increases the computational time significantly, further which can be reduced by parallel processing.

## Notes

### Acknowledgements

A previous shorter version of the paper was presented in the 10th world congress of EWRA “Panta Rei” Athens, Greece, 5-9 July 2017.

### Compliance with Ethical Standards

### Conflict of Interest

The authors declare that they have no conflict of interest.

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