Handbook of HydroInformatics: Volume III: Water Data Management Best Practices

By Saeid Eslamian

Handbook of HydroInformatics Volume III: Water Data Management Best Practices presents the latest and most updated data processing techniques that are fundamental to Water Science and Engineering disciplines.  These include a wide range of the new methods that are used in hydro-modeling such as Atmospheric Teleconnection Pattern, CONUS-Scale Hydrologic Modeling, Copula Function, Decision Support System, Downscaling Methods, Dynamic System Modeling, Economic Impacts and Models, Geostatistics and Geospatial Frameworks, Hydrologic Similarity Indices, Hydropower/Renewable Energy Models, Sediment Transport Dynamics Advanced Models, Social Data Mining, and Wavelet Transforms.

This volume is an example of true interdisciplinary work. The audience includes postgraduates and above interested in Water Science, Geotechnical Engineering, Soil Science, Civil Engineering, Chemical Engineering, Computer Engineering, Engineering, Applied Science, Earth and Geoscience, Atmospheric Science, Geography, Environment Science, Natural Resources, Mathematical Science, and Social Sciences. It is a fully comprehensive handbook which provides all the information needed  related to the best practices for managing water data. 

• Contributions from global experts in the fields of data management research, climate change and resilience, insufficient data problem, etc.  
• Thorough applied examples and case studies in each chapter, providing the reader with real world scenarios for comparison. 
• Includes a wide range of new methods that are used in hydro-modeling, with step-by-step guides on how to use them.

• Language: English
• Publisher: Elsevier Science
• Release date: Dec 6, 2022
• ISBN: 9780128219522

Book preview

Preface

Water Data Management Best Practices, Volume 3 of the Handbook of HydroInformatics, presents in 26 chapters the latest and most thoroughly updated data processing techniques that are fundamental to the water science and engineering disciplines. These include a wide range of new methods that are used in hydro-modeling, such as advantages of the grid-free analytic element method, soft-computing techniques for determining the dam outflow and breach characteristics, the hydrologic engineering center hydrologic modeling system (HEC-HMS), soft-computing methods for turbulent stormwater modeling, bed load transport assessment by conventional and fuzzy regression methods, automated flood inundation mapping, causal reasoning modeling, data assimilation and accuracy, flood routing, water resources engineering fuzzy logic applications, geographic information systems (GIS) application in flood mapping, groundwater level forecasting using hybrid soft-computing techniques, hydroinformatics methods for groundwater simulation, hydrological-hydraulic modeling of floodplain inundation, interoceanic waterways network systems, lattice Boltzmann models for hydraulic engineering problems, mathematical developments in sediment transport, wetland ecosystems simulations, multivariate linear modeling application in hydrological engineering, case-based reasoning (CBR)-supported risk response to hydrological cascading disasters, optimally pruned extreme learning machine (OP-ELM), water quality analysis based on hyperspectral remote sensor data, real-time flood hydrograph predictions, river bathymetry acquisition techniques, gene expression programming (GEP), and sediment transport by soft computing.

This volume is a true interdisciplinary work, and the intended audience includes postgraduates and early-career researchers interested in computer science, mathematical science, applied science, Earth and geoscience, geography, civil engineering, engineering, water science, atmospheric science, social science, environment science, natural resources, and chemical engineering.

The Handbook of HydroInformatics corresponds to courses that could be taught at the following levels: undergraduate, postgraduate, research students, and short course programs. Typical course names of this type include: HydroInformatics, Soft Computing, Learning Machine Algorithms, Statistical Hydrology, Artificial Intelligence, Optimization, Advanced Engineering Statistics, Time Series, Stochastic Processes, Mathematical Modeling, Data Science, Data Mining, etc.

The three-volume Handbook of HydroInformatics is recommended not only for universities and colleges, but also for research centers, governmental departments, policy makers, engineering consultants, federal emergency management agencies, and related bodies.

Key features are as follows:

•Contains contributions from global experts in the fields of data management research, climate change and resilience, insufficient data problems, etc.

•Offers thorough applied examples and case studies in each chapter, providing the reader with real-world scenarios for comparison

•Includes a wide range of new methods employed in hydro-modeling, with step-by-step guides on how to use them

Saeid Eslamian, Department of Water Engineering, College of Agriculture, Isfahan, University of Technology, Isfahan, Iran

Faezeh Eslamian, Department of Bioresource Engineering, McGill University, Montreal, QC, Canada

Chapter 1: Advantage of grid-free analytic element method for identification of locations and pumping rates of wells

Shishir Gaura; Padam Jee Omarb; Saeid Eslamianc,d    a Department of Civil Engineering, Indian Institute of Technology, Varanasi, India

b Department of Civil Engineering, Motihari College of Engineering, Motihari, India

c Department of Water Engineering, College of Agriculture, Isfahan University of Technology, Isfahan, Iran

d Center of Excellence for Risk Management and Natural Hazards, Isfahan University of Technology, Isfahan, Iran

Abstract

Solution of groundwater management problems often needs to find the best possible location and pumping rates of wells or both, which depends on the efficiency of the simulation model to define the precise location of wells and water budgeting. The present study is carried out to explore the efficiency of the analytic element method (AEM) to deal with unknown/illegal wells’ problems. The AEM-based flow model, and grid-based FDM model was coupled with the particle swarm optimization (PSO) model. Further, the developed AEM-PSO model was applied to the real field data to compute the location and pumping rates of unknown wells. The results were compared with FDM-PSO which suggests that in some conditions AEM model is more efficient to solve unknown wells’ problems. The present study gives the motivation to apply AEM based flow model for various groundwater resource management problems, where the exact location of wells and accurate water budgeting is the main decision criteria.

Keywords

Simulation-optimization; Groundwater model; Genetic algorithm; Calibration; Mathematical programming; Particle swarm optimization

1: Introduction

Satisfying the growing water demand is the most common global problem, and groundwater plays the most important role for achieving this demand. Proper management and distribution of the groundwater resources can help for fair groundwater sharing and avoid over exploitation of this resource (Tziatzios et al., 2021). Solution of groundwater management problems often needs to find the best possible location and pumping rates of wells or both, which depends on the efficiency of the simulation model to define the precise location of wells and water budgeting. Illegal extraction of this resource through pumping wells makes this problem worsen more and sets challenges in front of water agencies (Gaur et al., 2021). The unknown/illegal wells’ problem increases the unequal distribution of water share and corresponding mismanagement of the natural resources (Pu et al., 2020).

Furthermore, in order to fulfill the existing and forthcoming population’s water demand, pressure has built up to conserve and sustainable management of the groundwater to ensure the fulfillment of the future water demand (Kumar et al., 2021). Usually groundwater management problems are solved by reducing the differences between computed and observed groundwater level through inverse modeling approach (He et al., 2021). In north Bihar plains, for the assessment of the groundwater, Omar et al. (2021a) conceptualized and developed a transient multilayered groundwater flow model for the Koshi River basin. This developed model is capable of solving large groundwater problems and associated complexity with it. In north Bihar plains, the Koshi River is one of the biggest tributaries of the Ganga River system. Koshi originates from the lower part of Tibet and joins the Ganga River in Katihar district, Bihar, India. After model development, calibration of the model was also done, by considering three model parameters, to represent the actual field conditions. For validation of the model, 15 observation wells have been selected in the area. With the help of observation well data, computed and observed heads were compared. Comparison results have been found to be encouraging and the computed groundwater head matched with the observed water head to a realistic level of accuracy. Developed groundwater model is used to predict the groundwater head and flow budget in the concerned area. The study revealed that groundwater modeling is an important method for knowing the behavior of aquifer systems and to detect groundwater head under different varying hydrological stresses.

Different researchers used this approach for identification of unknown pollution sources (Ayvaz, 2010; Singh et al., 2004; Sun et al., 2006; Omar et al., 2021b), preventing sea water intrusion through wells (Cheng et al., 2000) pump-and-treat optimization technique (Matott et al., 2006; Huang and Mayer, 1997), unknown/illegal wells problem (Saffi and Cheddadi, 2010; Ayvaz and Karahan, 2008) and parameters optimization. Hsiao and Chang (2002) solved groundwater problems by taking fixed well installation cost and pumping cost. Genetic algorithm (GA) was used to determine the number and locations of pumping wells. Constrained differential dynamic programming (CDDP) was used to evaluate the operating costs. The study concluded that well installation costs impact the optimal number and locations of wells significantly. Uddameri and Kuchanur (2007) developed simulation model for groundwater flow in the formations of the Gulf coast aquifer. The model results were analyzed with mathematical programming scheme to estimate maximum available groundwater in the county, including prevention of saltwater intrusion in the aquifer by limiting the amount of allowable drawdown in shallow aquifers. Ameli and Craig (2018) presented a new semianalytical flow and transport model for the simulation of 3D steady-state flow and particle movement between groundwater, a surface water body and a radial collector well in geometrically complex unconfined aquifers. Their presented method was grid-free based analytic element method, which handles the irregular configurations of radial wells more efficiently than grid-based methods. This method is then used to explore how pumping well location and river shape interact and together influence (1) transit time distribution (TTD) of captured water in a radial collector well and TTD of groundwater discharged into the river and (2) the percentage of well waters captured from different sources. According to Baulon et al. (2022), estimation of groundwater level development is a major issue in the context of climate change. Groundwater is a key resource and can even account in some countries for more than half of the water supply. Groundwater trend estimates are often used for describing this evolution. However, the estimated trend obviously strongly depends on available time series length, which may be caused by the existence of long-term variability of groundwater resources (Baulon et al., 2022).

Park et al. (2021) linked a groundwater flow and heat transport simulation model with a genetic algorithm (GA) as optimization technique. This coupled model can determine optimal well locations and pumping/ injection rates together or apart. Results demonstrated that simultaneous optimization of well location and flow rate can provide a better design than optimization of only well location for given flow rate. Wang and Ahlfeld (1994) performed the study by considering wells’ location as explicit decision variables for a pump-and-treat optimization problem. Barrier function technique was used along with a linear objective function. Hermite interpolation function was used to represent the well locations as a continuous function of space. Huang and Mayer (1997) developed optimization formulation for dynamic groundwater remediation management using location of wells and the corresponding pumping rates as the decision variables. They found that optimal location and pumping rate of wells obtained with the moving-well model were less expensive than solutions obtained with a comparable fixed-well model. Kayhomayoon et al. (2021) proposed a new approach for the simulation of groundwater level for an arid area. Their methodology comprises three stages as clustering, simulation, and optimization. In optimization, two advanced optimization methods, i.e., particle swarm optimization (PSO) and whale optimization algorithm (WOA) were utilized to optimize the ANN results. Mohan et al. (2007) applied Simulation-Optimization (S glyph_sbnd O) approach for opencast mine area which had dominant groundwater features and became cause of heaving and bursting of the mine floor due to excessive uplift pressure. The S glyph_sbnd O model was used to identify optimum depressurization strategy and find capable approach for solving large-scale groundwater management problems (Mohan et al., 2007).

Identification of location and discharge of unknown/illegal wells, for groundwater quantity management, has been addressed by limited researchers (Saffi and Cheddadi, 2010; Ayvaz and Karahan, 2008; Tung and Chou, 2004; Pu et al., 2021; Shekhar et al., 2021). Saffi and Cheddadi (2010) developed an algebraic expression to generate the transient influence coefficients matrix for a 1-D model. The governing equation was solved using a mixed compartment model. In the study, objective function was to minimize the errors between observed and simulated hydraulic heads to determine the illegal groundwater pumping at fixed well locations. Ayvaz and Karahan (2008) developed a simulation/optimization model for identification of unknown location and pumping rate of wells. Finite Difference Method (FDM) based flow model and GA model were used to determine the discharge rates whereas well locations were identified by iterative moving subdomain approach (Omar et al., 2020). The model was tested for both steady and transient flow conditions on two hypothetical aquifer models. Results showed that the true well locations were identified irrespective of starting point of the search process. Finally, the performance of the proposed model was compared with that of a GA solution and found that the proposed model had smaller Residual Error (RE) than the GA solution and required 14% less simulations.

The analytic element method (AEM) is based upon superposition of analytical expressions to simulate groundwater flow by considering different hydrogeological feature like streams, lakes and wells (Strack, 1989). The AEM is a grid free method and has certain advantages over grid-based methods, example wells are directly represented by their exact co-ordinates (Omar et al., 2019; Bandilla et al., 2007). The AEM flow model gives continuous solutions over the model domain and therefore gives more accurate water budget for the area. The two-dimensional implementation of the analytic element method (AEM) is commonly used to simulate steady-state saturated groundwater flow phenomena at regional and local scales. However, unlike alternative groundwater flow simulation methods, AEM results are not ordinarily used as the basis for simulation of reactive solute transport (Craig and Rabideau, 2006).

Above explained before, the literature review shows different quantity or quality based groundwater management problems where optimal location, discharge of wells or both were taken as decision variable. Considering this, the present study is carried out to explore the benefits of AEM based flow model to compute the optimal location and pumping rates of wells for unknown/illegal wells’ problem. For this, AEM and grid based approach like FDM was adopted, and flow model was developed, along with Particle Swarm Optimization (PSO) model. After that, developed AEM-based flow model, and FDM model was coupled with the (PSO) model, individually. Accuracy assessment was done to know which model provides better results between AEM-PSO and FDM-PSO. Further, the developed AEM-PSO model was applied to the real field data to compute the location and pumping rates of unknown wells.

2: Limitations of the study

In spite of the fact that the present study is based on a comprehensive scientific analysis of various hydrological and hydrogeological features, it has some limitations that need to be addressed.

In the present study, to solve the partial differential equations of the groundwater flow, two different approaches have been adopted. One method is the Finite Difference Method (FDM), in which derivatives are approximated with finite differences. Another method is the Analytic Element Method (AEM), in which the boundary conditions of the flow model are discretized instead of discretization of the whole model. Another method such as the Finite Element Method (FEM), in which meshing is performed using finite elements, has not been taken into account for the flow modeling.

To optimize the pumping rate (discharge) of the well, Particle Swarm Optimization (PSO) model was also developed. After that, The PSO model was coupled with the developed AEM-based flow model, and grid-based FDM model separately. For the optimization, multiswarm optimization techniques also can be applied.

3: Methodology and formulation of the simulation-optimization model

Optimal location and discharge of pumping wells is often considered as a decision variables to solve unknown/illegal wells’ problem. Solution of these problems depends on the accuracy to forecast the position of the wells and accuracy in calculation of water budget (Eslamian and Eslamian, 2022). In the present study, The AEM and FDM flow models and the PSO model were developed. After validation of simulation models and optimization model (Gaur et al., 2011), both simulation models were coupled with PSO model to solve the unknown/illegal wells’ problem. Two cases, with and without use of moving subdomain approach (Ayvaz and Karahan, 2008), were considered to identify the optimal location and discharge of pumping wells. Comparative analysis was done for the AEM-PSO and FDM-PSO models and the efficiency of both the AEM and FDM methods to compute the location and discharge of wells for both cases were investigated. Computational efficiency of both models was also measured based on convergence of model to find optimal solution.

Elçi and Ayvaz (2014) did an intensive study to present an optimization approach to determine locations of new groundwater production wells, where groundwater is relatively less susceptible to groundwater contamination. For this, they coupled a regional-scale groundwater flow model with a hybrid optimization model that uses the Differential Evolution (DE) algorithm and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method as the global and local optimizers. In this study, several constraints such as the depth to the water table, total well length and the restriction of seawater intrusion are considered in the optimization process. The optimization problem can be formulated either as the maximization of the pumping rate or as the minimization of total costs of well installation and pumping operation from existing and new wells. After the development of simulation–optimization model, they demonstrated it on an existing groundwater flow model for the Tahtalı watershed in Izmir–Turkey. The model identifies for the demonstration study locations and pumping rates for up to four new wells and one new well in the cost minimization and maximization problem, respectively. All new well locations in the optimized solution coincide with areas of relatively low groundwater vulnerability.

3.1: AEM and FDM flow models

The analytic element method (AEM) is a numerical method used for the solution of partial differential equations. The basic principle of the analytic element method is that, for linear differential equations, elementary solutions may be superimposed to obtain more complex solutions. These analytic solutions typically correspond to a discontinuity in the dependent variable or its gradient along a geometric boundary (e.g., point, line, etc.). This discontinuity has a specific functional form and may be manipulated to satisfy Dirichlet, Neumann, or Robin (mixed) boundary conditions (Shamir et al., 1984). In AEM, groundwater flow solution is obtained by the use of potential theory where the discharge potential Ф (x, y) for an aquifer is determined by principal of superposition (Reilly et al., 1987). The linear solutions of individual elements become superimposed to find final solution and further the potential is converted into head. Each solution corresponds to particular hydraulic features (e.g., river, lakes, wells, hydraulic conductivity; Strack, 1989). AEM does not require a fixed boundary condition, which makes the development of the conceptual model less complicated (Omar et al., 2019).

In AEM, the model conceptualization was done in GIS environment (Geographic Information System) using the base map files, which was created in the DXF format. Boundary conditions are defined in the hydrological element itself as the head. The model domain was defined before inputting any model parameters. The term domain is referring to the regions within which the aquifer properties are constant.

In FDM, the governing groundwater flow equation can be defined as:

si1_e

   (1)

where Kx, Ky, and Kz are the hydraulic conductivity (HC) of the aquifer in all three directions (x, y, and z), W is the volumetric flux (flow) per unit volume, SS is the specific storage of aquifer’s porous material, h is the potentiometric head, and t is the time.

Numerical solution, i.e., Finite Difference Method (FDM) of the equation gives the variability of groundwater head (h) in an aquifer. In the FDM, the mathematical approximation is used in solving the groundwater flow equation while in the AEM harmonic function is used to solve the groundwater flow equation (Laplace equation) which produces more accurate results. As the AEM provides a continuous groundwater level surface while the FDM provides solutions at discrete points in the grid, the AEM approach is suitable to follow the sharp changes of the groundwater level while the FDM approach provides the groundwater surface only at discrete points/grids.

3.2: Particle swarm optimization

Optimization techniques can be classified into two types. The first is deterministic optimization techniques, which include linear programming (LP), nonlinear programming (NLP), and dynamic programming (DP). The second type is stochastic optimization including Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Shuffled Complex Evolution, Simulating Annealing (SA), etc. Groundwater management problems are usually nonlinear and nonconvex mathematical programming problems (McKinney and Lin, 1994). For such problems, using deterministic optimization techniques can lead to some unforeseen situations. These techniques usually require good initial solutions to produce an optimal solution. Furthermore, they rely on the local gradient of the objective function to determine the search direction, and thus, can converge to local optimal solutions (Ayvaz, 2009). Therefore, the use of stochastic optimization techniques is generally preferred because of their ability to find solutions without the need for gradients and initial solutions (El-Ghandour and Elsaid, 2013).

Particle swarm optimization (PSO) is a stochastic population-based optimization algorithm inspired by the interactions of individuals in a social world. This algorithm is widely applied in various areas of water resource problems. Particle Swarm Optimization (PSO), which is also an evolutionary computation technique, which is an efficient method for solving large and complex optimization problems (Kennedy and Eberhart, 1995). PSO is a member of wide category of swarm intelligence based methods and efficient in global optimization problems. PSO considers two factors for achieving the goal: the particle’s own best previous experience (i.e., pbest) and the best experience of all other members (i.e., gbest). The model was developed on the MATLAB (Gaur et al., 2011).

4: Model application and results

Both AEM-PSO and FDM-PSO models were employed on real field data. Both cases were analyzed and comparative study was performed. Objective function of the problem was defined to minimization the Residual Error (RE) between observed and computed values.

si2_e

   (2)

where, hcomuptedi are the values computed by AEM-PSO model, hactuali are observed values in the presence of pumping wells and Ne are the total number of control points.

The study area is located in the central part of Indo-Gangetic plain of the Indian subcontinent. Varanasi is the oldest and religious district of the Utter Pradesh, India. Varanasi has been the cultural center of North India for several thousand years and is closely linked to the Ganges. It is situated at the bank of Ganga River. The study area lies in Eastern Uttar Pradesh and bounded between two major rivers the Ganga River and Gomati River. The River Ganga lies in the southern and eastern part of the study area and Gomati River lies in northern part. Two small tributaries, namely Basuhi and Morwa lie in the western side of the area. The Varuna River divides the study area into almost two equal parts. The study area covers three districts; Varanasi, and some areas of Sant Ravi Das Nagar and Jaunpur. Sant Ravi Das Nagar is also known as Bhadohi, situated in the plains of the River Ganga. Jaunpur is situated on the bank of the Gomati River.

The study area covers about 2785 km² area in and around Varanasi district, of which 1535 km² located in Varanasi district. The study area lies between the latitude 25°05′16″ N—25°40′01″ N and longitude 82°22′05″ E—83°11′35″ E as shown in Fig. 1. The entire area is divided into 10 administrative blocks in which 8 blocks Pindara, Cholapur, Baragaon, Harhua, Chiraigaon, Sevapuri, Arajiline, and Kashi Vidya Peeth (KVP) exist in Varanasi District.

Fig. 1

Fig. 1 Location of the study area with observation wells and pumping wells.

4.1: Physiography and topography of the area

The River Ganga is present as a trans-boundary between India and Bangladesh, the point of origin of the 2525 km (1569 mile) long River is Western Himalaya in the Indian state of Uttarakhand, and it flows to the south and east Gangetic plains of North India into Bangladesh where it drains in the Bay of Bengal (Gupta and Deshpande, 2004; Jain et al., 2007). Ganga is the longest River of India and is ranked second as the greatest River in terms of water discharge. The average annual discharge of the River Ganga is about 16,650 m³/s (Jain et al., 2007). Ganga basin is of magnificent variation in altitude, usage of the land, the pattern of crops, and climate. One of the tributaries of River Ganga is Gomati River which originates from the Gomat taal, Pilibhit, India. The length of the river Gomati is around 900 km extending from the state of Uttar Pradesh and meets the Ganga near Saidpur, Kaithi in Ghazipur. One of the minor tributaries of the Ganga is River Varuna. Throughout the entire course, both the rivers receive a large amount of agricultural run-off consisting of pesticides and fertilizers from the catchment area.

The topography of study area is characterized by significant variation in the elevation. It varies from 33 to 101 m mean sea level (MSL). The highest elevation in the study area has been observed near around Bhadohi. The average elevation of the area is 80.71 m from mean sea level (MSL). The Ganga River has an elevation of 66.27 m MSL at the point where it enters in to the study area and 60.78, where it leaves the area in the downstream. The Gomati River elevation from top to bottom ranges from 68.32 to 60.78 m MSL, respectively.

The groundwater flow model was conceptualized on the basis of geological, climatic, and hydro-geological characteristics of the study area. Parameters have been used to develop the FDM model as well. Grid dimension has been taken 100 × 100 m for the development of FDM model. Further, the AEM-PSO and FDM-PSO models were applied on case 1 and case 2 to compute the location and discharge of unknown wells.

Initially 5 wells were placed at different locations and corresponding groundwater heads were computed by AEM and head values were recorded at predefined 12 observation points. These recorded values were considered as observed values. Further, the AEM-PSO and FDM-PSO models were employed to identify the location and discharge of those five wells by considering different sets of 1–5 wells. Corresponding optimal values of objective functions, i.e., RE values for different set of wells have been recorded and the number of wells was finalized based on minimum value of RE. The results of AEM-PSO and FDM-PSO models were applied on two cases (1) optimal location and discharge of wells, (2) Computational efficiency of both models.

Case 1:

In this case, optimal discharge and location of pumping wells were identified by straight application of AEM-PSO and FDM-PSO model. Efficiency of model was calculated by identifying the difference in discharge values and location displacement between computed and observed values. Location displacement was computed by accumulating the difference between observed and computed location of all wells, which is defined as,

si3_e    (3)

where, di is the distance between two adjacent wells and N = total number of wells.

Results indicate that AEM-PSO has error of 1.9%–4.1% and FDM has error of 3.4%–7.3% in discharge rate for 5 wells. Whereas, AEM-PSO has error of 128 m and FDM has error of 437.7 m in location displacement of 5 wells. Table 1 shows the location, discharge rate of wells and groundwater head values by both AEM-PSO and FDM-PSO models. Results describe that the optimal locations differ in the AEM and FDM model, where AEM gives more accurate values in comparison of FDM. Although present model was developed by considering small size of cells, but it shows that increasing the size of cells can lead to more error in the final results.

Table 1

Maximum number of iterations, i.e., 1000 was considered for the convergence of the optimization model. The model convergence showed that the AEM-PSO model converged after 905 iterations for the set of 5 wells, whereas the FDM-PSO model was found to have converged after 825 iterations for the set of 5 wells, respectively. The results show that the FDM-PSO model converged with less iteration than the AEM-PSO model. The parameters of the PSO model were considered as linearly varying inertia weight from 2.0 to 1.8, acceleration constant 2.0–2.0.

Case 2:

In this case, moving subdomain approach was applied and its efficiency in both AEM and FDM model were analyzed. Detailed description about this technique can be found in Ayvaz and Karahan, 2008. In this study, initial location of subdomain was taken same in AEM and FDM with equal size of 300 m × 300 m in AEM and 3 × 3 cells in FDM. In AEM, domain size was allowed to reduce till 20 m × 20 m from 300 m × 300 m as AEM is not bounded by minimum size of grids. Predefined well location was assumed on the center of the domain and center of its sides. Contraction of domain was allowed when the location of well was not found improved in subdomain by flipping in all directions as shown in Fig. 2. In each contraction of subdomain, again it was allowed to move in all direction to find any other possible best location. The procedure remained continue till the subdomain size approaches to minimum limit, i.e., 20 × 20 m. Contraction was employed to the two sides, by the shifting rate of − 10 m, having wells with more RE values than wells of remaining two sides. The RE value for addition of each subdomain was computed and used to find the optimal number of wells. Table 2 shows number of wells and the different RE values. Whereas reduction of subdomain in the FDM model was restricted and the subdomain was allowed to move until, the optimal location was not found. Optimal location was finalized when location of well, in specific cell of grid, was not improved. To finalize the location of cell, subdomain was allowed to flip in all directions (Fig. 2).

Fig. 2

Fig. 2 Flip and reduction of subdomain.

Table 2

In the presence of optimal location of well, location of final cell was not found to be improved. Table 3 shows the location, discharge rate of wells and groundwater head values by both AEM-PSO and FDM-PSO models for Case 2.

Table 3

Results show that optimal number of wells is five, which shows minimum RE values. Results indicate that cumulative location displacement in FDM was found 270.7 m more in comparison of the AEM model. Whereas computed pumping rates was found with the error of 1.3%–2.1% in AEM and 2.1%–3.6% in FDM. Total number of iterations in AEM-PSO model was found 22% more in comparison of FDM model. It shows that simulation model needs more iteration to get closer values to real values.

The comparison between results of Case 1 and Case 2 shows (Table 4) that moving subdomain approach is efficient to find more accurate results. It was also observed that location displacement error was more improved than pumping rates error in Case 2. It was found that location displacement error was improved by 42.7 m in AEM model and 81.7 m in FDM model from Case 1 to Case 2. Discharge values were improved by 1.1% in AEM and 1.5% in FDM. In Case 2, only 5 decision variables were taken to optimize the pumping rate of the wells, whereas total 15 decision variables were taken in Case 1 to optimize the location and pumping rates. Both AEM-PSO and FDM-PSO models were employed for single run to find the optimal solution in the Case 1. Whereas in Case 2, both models took 10 to 15 runs to find the optimal solution. Although model run was more in Case 2, but due to presence of five variables, model converges very early in comparison of Case 1.

Table 4

5: Conclusions

The present study is carried out to explore the efficiency of the Analytic Element Method (AEM) to deal with unknown/illegal wells’ problems. The AEM-based flow model and grid based FDM model was coupled with the Particle Swarm Optimization (PSO) model. Further, the developed AEM-PSO model was applied to the real field data to compute the location and pumping rates of unknown wells. The results were compared with FDM-PSO, which suggests that in some conditions The AEM model is more efficient to solve unknown wells’ problems. The present study gives the motivation to apply the AEM based flow model for various groundwater resource management problems, where the exact location of wells and accurate water budgeting is the main decision criteria.

In the present study, the advantage of AEM was investigated for solving unknown/illegal wells’ problem by identifying location and discharge through simulation and optimization technique. In the study, PSO model was coupled with AEM and FDM method. The AEM-PSO model was compared with the FDM-PSO model and both were applied on real field data. Two cases were considered with and without use of subdomain approach. The results show that the AEM flow model is very effective to solve unknown wells’ problem with some advantage over grid based method. The AEM model computes more accurate location of wells, in both cases and provides a more accurate discharge rates as well. The results of the study show that AEM model is itself efficient to solve unknown well problems in comparison of FDM model in the both cases. It was also found that efficiency of the AEM and FDM models can be improved through subdomain approach. Although number of iterations was found more in the AEM-PSO model in comparison of the FDM-PSO model.

This study also suggested that the AEM method is very efficient to deal with groundwater management problems where location of wells and water budgeting are main decision variables. In the AEM, Computational burden depends on the number of hydrogeological features and their discretization level. Thus, the present study is more useful to perform effective study for the large geographic areas without excessive computation time. The study concluded that moving subdomain approach is effective to find the optimal well locations in less number of iterations.

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