1 Introduction

Groundwater is a part of the hydrologic cycle, located beneath the earth’s surface in pore spaces and fractures of geologic formations. A saturated geologic formation with a usable water supply is called an aquifer. The depth at which pore spaces of a formation become completely saturated with water indicates the water table or hydraulic head. There are hydrologic parameters such as recharge and discharge as well as aquifer parameters such as transmissivity and storativity that affect the depth and fluctuation of the water table. In reality, to simulate or predict the location of the water table, values of hydrologic and aquifer parameters should be used as input data in a groundwater model. Sometimes, field measurements of relevant parameters are lacking or there may be inaccuracies in observed data. In such cases, a model’s calibration helps to determine appropriate parameter values. The calibration process tries to minimize the difference between observed and estimated parameter values by using optimization methods.

Software packages that can simulate aquifer conditions are used to extract appropriate calibrated parameters. Although less precise than software packages, trial-and-error can be used to determine optimal/near-optimal parameters. While it is possible to calculate an optimal/near-optimal solution by trial-and-error, the probability of success is directly related to the number of times one executes trial-and-error calculations, which can be time-consuming (Fallah-Mehdipour et al. 2012). Thus, use of an optimization method together with a simulation model is recommended to calibrate groundwater models. Linear programming (LP) and nonlinear programming (NLP) are general optimization methods that can be used to calculate optimal solutions by using gradient-based techniques. However, those tools may not achieve an optimal solution in complex problems with a large number of decision variables and linear and nonlinear constraints (e.g., Bozorg Haddad et al. 2008, 2009; Afshar et al. 2011b).

Meta-heuristic algorithms have been used as optimization tools in the determination of optimal solutions in complex problems. Although the use of those algorithms does not guarantee a global optimal solution, meta-heuristic algorithms are good candidates to determine optimal/near-optimal solutions (Bozorg Haddad et al. 2010; Bozorg Haddad and Mariño 2011; Bozorg Haddad et al. 2011a, b; Ghajarnia et al. 2011; Karimi-Hosseini et al. 2011; Rasoulzadeh-Gharibdousti et al. 2011; Noory et al. 2012; Orouji et al. 2013).

Various meta-heuristic algorithms have been extensively used to solve calibration problems in the field of water resources. Samuel and Jha (2003) estimated aquifer parameters from pumping-test data by genetic algorithm (GA). The latter yielded significantly low values of the sum of square errors (SSE) as the objective function for almost all data sets, especially with an increasing generation number and population size. Schoups et al. (2005) employed multi-objective shuffled complex evolution metropolis algorithm to calibrate a surface water-groundwater flow model for an irrigated agricultural region in Mexico. They compared a four-objective calibration problem which was formulated to calculate improved estimates of the hydraulic parameters by using head data from shallow and deep aquifers, as well as data on agricultural drainage and canal seepage volumes with a single-objective approach, since the various objectives were sensitive to different parameters. Thus, there was no comparison between the performance of the shuffled complex evolution metropolis algorithm and other multi-objective procedures. Bekele and Nicklow (2007) calibrated a semi-distributed watershed model known as soil and water assessment tool (SWAT) by non-dominated sorted genetic algorithm-II (NSGA-II). They considered two calibration scenarios. The first scenario used specific objective functions to fit different portions of the time series whereas in the second scenario, the calibration was performed using data from multiple gauging stations, simultaneously. The use of multiple objectives during the calibration process resulted in improved model performance and the second scenario, in particular, provided better results partly due to the respective location of the gauging stations within the watershed. Khu et al. (2008) proposed a new multi-objective calibration approach incorporating multi-variable, multi-site, and multi-response measurements. The methodology consisted of two steps: (1) grouping of multi-site measurements into clusters using Kohonen self-organising map (SOM), and (2) multi-objective optimization with preference order genetic algorithm (POGA), which is particularly powerful for dealing with many objectives. The proposed methodology was used to calibrate a MIKE system hydrologique european (SHE) model setup of the Karup catchment in Denmark without any comparison between POGA performance and other multi-objective algorithms. Afshar et al. (2011a) proposed the PSO algorithm to calibrate a two-dimensional, large-scale, hydrodynamic, water quality model (CE-QUAl-W2). Results showed that the calibrated coefficients agreed closely with a set of field data. Dakhlaoui et al. (2012) used a modified shuffled complex evolution as the optimization tool toward a more efficient calibration scheme for a rainfall–runoff model. Sun et al. (2012) showed that patterns of groundwater storage changes inferred from a gravity recovery and climate experiment (GRACE) may be incorporated as an additional regularization mechanism for calibrating regional groundwater models. They explored the combined use of in-situ water level measurements and GRACE-derived groundwater storage changes for calibrating regional groundwater models by using NSGA-II as a multi-objective optimization tool. Because the accuracy of the calibration procedure is of paramount importance in modeling, the focus of the aforementioned investigations is of major importance. Thus, in this paper, two meta-heuristic algorithms, PSO and PS, are employed to estimate parameters in an unconfined aquifer.

Kennedy and Eberhart (1995) developed the PSO algorithm as a meta-heuristic algorithm based on the social behavior of birds or fishes to reach a destination. In the water resources field, Chuanwen and Bompard (2005) applied a self-adaptive chaotic PSO algorithm for short-term, hydroelectric-system scheduling hydroelectric plant dispatch model based on the rule of maximizing the benefit in a deregulated environment. The proposed approach introduces chaos mapping and the self-adaptive chaotic PSO algorithm increased both mapping convergence rate and resulting precision. Suribabu and Neelakantan (2006) used the East Palo Alto network (EPANET) and the PSO algorithm as simulation and optimization models, respectively, to design a water distribution pipeline network. Results indicated that the PSO algorithm is more efficient than other optimization algorithms with a better (lower) value of the objective function. Matott et al. (2006) identified the PSO algorithm as an effective technique for solving pump-and-treat optimization problems by using analytic element method flow models. Izquierdo et al. (2008) applied the PSO algorithm to the optimization of a wastewater collection network. Results showed the algorithm’s performance and the obtained results were consistent with those calculated by using dynamic programming to solve the same problem under the same conditions. Fallah-Mehdipour et al. (2011) proposed a multi-objective PSO (MOPSO) approach in multipurpose multireservoir operation. Results showed that the proposed algorithm was successful in computing near-optimal Pareto fronts. Ostadrahimi et al. (2012) employed a multi-swarm PSO to extract multireservoir operation rules. Results indicated that this algorithm may significantly outperform the common implicit stochastic optimization approach in the real-time operation of a reservoir system. Fallah-Mehdipour et al. (2013) extracted multi-crop planning rules in a reservoir system by using the PSO algorithm, GA, and shuffled frog leaping algorithm (SFLA). They maximized the total net benefit of the water resource system by supplying irrigation water for a proposed multi-cropping pattern over the planning horizon. Gaur et al. (2012) coupled ANN and PSO for the management of the Dore river basin in France. An analytic element method (AEM) based flow model was used to generate the dataset for the training and testing of the ANN. Then, to minimize the pumping cost of the wells, the developed ANN-PSO was used and compared to AEM-PSO. Results showed that the ANN-PSO reduced the computational burden significantly as it is able to analyze various management scenarios. Saadatpour and Afshar (2013) used MOPSO in a pollution spill response management model in reservoirs. They coupled CE-QUAL-W2 with the MOPSO algorithm to obtain a desirable near-optimal reservoir operation strategy and/or emergency planning in a selective withdrawal framework. Kumar and Reddy (2012) employed elitist-mutated PSO (EMPSO) as an efficient and reliable swarm intelligence-based approach in multipurpose reservoir operation. Results demonstrated that EMPSO consistently performed better than the PSO algorithm.

Hooke and Jeeves (1961) called PS a family of numerical optimization methods that do not require the gradient of the problem to be optimized. Although PS is a capable tool to determine an appropriate solution in optimization problems, there are few investigations that use PS in the water resources area. Tung (1984) employed the PS algorithm without any comparisons by using other meta-heuristic algorithms to calibrate the Muskingum model. Neelakantan and Pundarikanthan (1999) extracted an optimal hedging rule for water supply reservoir systems. They used a neural network model to speed up the optimization process without considering the number of function evaluations for the simulation of the reservoir system operation instead of a conventional simulation model.

This paper addresses the accuracy of calculation of water levels in an unconfined aquifer by using calibrated parameters with two objective functions, SSD and SAD, as well as comparison criteria. Moreover, the PSO and PS algorithms, which are widely used in water management, are tested in the groundwater model calibration.

2 Mathematical Model

Groundwater flow in an unconfined aquifer can be approximated by the equation of flow in a compressible, isotropic, and heterogeneous confined aquifer, as follows:

$$ \frac{\partial }{{\partial x}}\left( {T\frac{{\partial h}}{{\partial x}}} \right)+\frac{\partial }{{\partial y}}\left( {T\frac{{\partial h}}{{\partial y}}} \right)\pm W=S\frac{{\partial h}}{{\partial t}} $$
(1)

in which T = aquifer transmissivity; h = hydraulic head; S = aquifer storativity; W = total recharge and discharge for each unit of aquifer model, e.g., a cell in the mesh/grid; and x, y, t = space and time coordinates.

Most of the models based on Eq. (1) use numerical methods such as finite differences and finite elements to solve groundwater problems. In the finite difference method (FDM), the aquifer area is transferred to a network with the same size cells. The FDM solution technique can be classified into two groups: implicit and explicit methods. Implicit methods are interpolative and require a solution of simultaneous equations for each time interval by using matrix algebra and require computing the inverse of the matrix for each time interval. Thus, computations are complicated and time-consuming. Explicit methods are extrapolative and require a simple solution of an algebraic equation for one unknown for each point and for each time interval. The CPU time is usually much smaller than in the implicit technique (Karahan and Ayvaz 2005). The alternating direction implicit method (ADIM) is one of the main techniques employed in FDM. Extended equations of ADIM for each cell based on Fig. 1 can be expressed as:

$$ {q_2}-{q_1}+{q_4}-{q_3}+W\times \varDelta x\times \varDelta y=S\frac{{\partial h}}{{\partial t}}\times \varDelta x\times \varDelta y $$
(2)
$$ \begin{array}{*{20}c} {{{{\left( {{T_{i+1,j }}+{T_{i,j }}} \right)}} \left/ {2} \right.}\times \left( {\frac{{h_{i+1,j}^t-h_{i,j}^t}}{{\varDelta {x^2}}}} \right)-{{{\left( {{T_{i-1,j }}+{T_{i,j }}} \right)}} \left/ {2} \right.}\times \left( {\frac{{h_{i,j}^t-h_{i-1,j}^t}}{{\varDelta {x^2}}}} \right)+\ldots } \hfill \\ {{{{\left( {{T_{i,j+1 }}+{T_{i,j }}} \right)}} \left/ {2} \right.}\times \left( {\frac{{h_{i,j+1}^t-h_{i,j}^t}}{{\varDelta {y^2}}}} \right)-{{{\left( {{T_{i,j-1 }}+{T_{i,j }}} \right)}} \left/ {2} \right.}\times \left( {\frac{{h_{i,j}^t-h_{i,j-1}^t}}{{\varDelta {y^2}}}} \right)+\ldots } \hfill \\ {\frac{{{W_{i,j }}}}{{\varDelta x\times \varDelta y}}=S\times \left( {\frac{{h_{i,j}^t-h_{i,j}^{t-1 }}}{{\varDelta t}}} \right)} \hfill \\ \end{array} $$
(3)
Fig. 1
figure 1

Scheme of ADIM for each cell

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As it is shown in Fig. 1, q 1, q 2, q 3 and q 4 = inflow and outflow for each cell; Δx and Δy = horizontal and vertical differences in the ADIM network; and Δt = time interval.

Equation (3) has been summarized by Todd and Mays (2005) as follows:

$$ \begin{array}{*{20}c} {{A_{i,j }}=\frac{{\left( {{T_{i+1,j }}+{T_{i,j }}} \right)\times B{C_{i+1,j }}}}{{2{{{\left( {\varDelta x} \right)}}^2}}},} \hfill & {{B_{i,j }}=\frac{{\left( {{T_{i-1,j }}+{T_{i,j }}} \right)\times B{C_{i-1,j }}}}{{2{{{\left( {\varDelta x} \right)}}^2}}},} \hfill & {{C_{i,j }}=\frac{{\left( {{T_{i,j+1 }}+{T_{i,j }}} \right)\times B{C_{i,j+1 }}}}{{2{{{\left( {\varDelta y} \right)}}^2}}}} \hfill \\ {{D_{i,j }}=\frac{{\left( {{T_{i,j-1 }}+{T_{i,j }}} \right)\times B{C_{i,j-1 }}}}{{2{{{\left( {\varDelta y} \right)}}^2}}},} \hfill & {{E_{i,j }}=\frac{{{W_{i,j }}}}{{\varDelta x\times \varDelta y}}+\frac{{{S_{i,j }}}}{{\varDelta t}}\times h_{i,j}^{t-1 }, } \hfill & {{F_{i,j }}=\frac{{{S_{i,j }}}}{{\varDelta t}}} \hfill \\ \end{array} $$
(4)
$$ h_{i,j}^t=\frac{{{A_{i,j }}h_{i+1,j}^t+{B_{i,j }}h_{i-1,j}^t+{C_{i,j }}h_{i,j+1}^t+{D_{i,j }}h_{i,j-1}^t+{E_{i,j }}}}{{{A_{i,j }}+{B_{i,j }}+{C_{i,j }}+{D_{i,j }}+{F_{i,j }}}} $$
(5)

in which, BC = boundary condition; A, B, C, D, E, F = coefficients.

To solve Eq. (5) for each cell, Karahan and Ayvaz (2005) proposed an iterative alternating direction implicit method (IADIM). Accordingly, the aquifer is divided into rectangular grid intervals both in x and y directions in order to carry out the iterative calculation. Input data such as Δx, Δy, Δt, aquifer parameters, and initial head are introduced in the model. The IADIM process consists of two inner and outer loops. The inner loop computes hydraulic heads for a given time interval until a convergence criterion is met. Then, the outer loop governs the time dimensions for further use in the inner loop. More information about IADIM is contained in Karahan and Ayvaz (2005).

In this paper, T, S, recharge and discharge to different cells are decision variables. The hydraulic head is estimated by considering the preceding parameters and the IADIM. The main objective of the optimization algorithm is considered as follows:

$$ Min.\quad SAD=\sum\limits_{t=1}^{12 } {\sum\limits_{i=1}^{row } {\sum\limits_{j=1}^{col } {\left| {h_{i,j}^t-\widehat{h}_{i,j}^t} \right|} } } $$
(6)
$$ Min.\quad SSD=\sum\limits_{t=1}^T {\sum\limits_{i=1}^{row } {\sum\limits_{j=1}^{col } {{{{\left( {h_{i,j}^t-\widehat{h}_{i,j}^t} \right)}}^2}} } } $$
(7)

where \( h_{i,j}^t \) and \( \widehat{h}_{i,j}^t \) = estimated and observed hydraulic head for the i th and j th cell at the t th time step, and T, row, and col = number of time step, row, and column of the groundwater network, respectively.

3 Meta-heuristic Algorithms

Meta-heuristic algorithms are computational methods that can calculate an optimal solution by generating a random solution and an iterative search for optimality. These algorithms make few or no assumptions about the problem being optimized, even in a large decision space. In this paper, two meta-heuristic algorithms, PSO and PS, are employed to determine optimal values of aquifer parameters.

3.1 Particle Swarm Optimization (PSO) Algorithm

The PSO algorithm is a swarm intelligence algorithm that mimics the movement of a bird organism in a swarm. Kennedy and Eberhart (1995) identified each bird as a particle and a collection of birds as a swarm in this algorithm. The movements of each particle is guided by its own best known position in the search-space (Pbest) as well as the entire swarm’s best known position (Gbest).

The PSO starts by generating random solutions. Each particle as the individual solution (X k  = [x k,1,x k,2,…,x k,N ]) is moved around in the search-space according to its velocity, which has N-dimensional vectors (V k  = [v k,1,v k,2,…,v k,N ]), the same as the number of decision variables.

The corresponding fitness value of each particle is calculated and the velocity of the k th particle is presented, as follows:

$$ \begin{array}{*{20}c} {v_{k,d}^{iter+1 }=w.v_{k,d}^{iter }+{c_1}.r_1^{iter}\left( {Pbest_{k,d}^{iter }-x_{k,d}^{iter }} \right)+{c_2}.r_2^{iter}\left( {Gbest_d^{iter }-x_{k,d}^{iter }} \right)} \hfill \\ {for\,k=1,2,\ldots,N\,and\,iter=1,2,\ldots,n} \hfill \\ \end{array} $$
(8)

in which \( v_{k,d}^{iter+1 } \) = velocity of the d th dimension of the k th particle in the iter + 1th iteration; w = inertia weight that regulates the effects of the current velocity; \( v_{k,d}^{iter } \) = velocity of the d th dimension of the k th particle in the iter th iteration; c 1 and c2 = cognitive and social parameters, respectively, that regulate movement toward Pbest and Gbest; \( r_1^{iter } \) and \( r_2^{iter } \) = random numbers within [0,1]; \( Pbest_{k,d}^{iter } \) = best position of the d th dimension for the k th particle up to the iter th iteration; \( Gbest_d^t \) = d th dimension of the best position of the swarm up to the iter th iteration.

To limit the variation of a particle’s velocity, allowable lower (v Min.) and upper (v Max.) bounds for velocity are controlled and the particle’s position is calculated by using Eq. (10):

$$ {v_{Min. }}\leq v_{k,d}^{iter+1}\leq {v_{Max. }} $$
(9)
$$ x_{k,d}^{iter+1 }=x_{k,d}^{iter }+v_{k,d}^{iter+1 } $$
(10)

The new swarm is used as the new members to calculate the corresponding objective function value for the next iteration. The PSO searching process is commonly terminated when a maximum number of iterations is reached.

3.2 Pattern Search (PS) Algorithm

First named by Hooke and Jeeves (1961), PS is an optimization method that does not require gradient search to achieve an optimal solution. The method starts the searching process by selecting a random solution from a decision space. This solution has a vector with N dimensions, the same as the number of decision variables (X 1 = [x 1,1,x 1,2,…,x 1,N ]), which is commonly selected from feasible ranges of constraints.

The PS method generates a sequence of solutions that produces a mesh around the first solution and approaches an optimal solution. There are two patterns to generate those solutions: (1) generalized pattern search (GPS) and (2) mesh adaptive direct search (MADS). In the GPS, the number of generated solutions around the first solution is 2N, which are produced as follows:

$$ X_1^1=m\times {{\left[ {\begin{array}{*{20}c} 1 & 0 & \ldots & 0 \\ \end{array}} \right]}_{{1\times N}}}+{X_1} $$
(11)
$$ \begin{array}{*{20}c} {X_1^2=m\times {{{\left[ {\begin{array}{*{20}c} 0 & 1 & \ldots & 0 \\ \end{array}} \right]}}_{{1\times N}}}+{X_1}} \hfill \\ \vdots \hfill \\ \end{array} $$
(12)
$$ X_1^N=m\times {{\left[ {\begin{array}{*{20}c} 0 & 0 & \ldots & 1 \\ \end{array}} \right]}_{{1\times N}}}+{X_1} $$
(13)
$$ X_1^{N+1 }=m\times {{\left[ {\begin{array}{*{20}c} {-1} & 0 & \ldots & 0 \\ \end{array}} \right]}_{{1\times N}}}+{X_1} $$
(14)
$$ \begin{array}{*{20}c} {X_1^{N+2 }=m\times {{{\left[ {\begin{array}{*{20}c} 0 & {-1} & \ldots & 0 \\ \end{array}} \right]}}_{{1\times N}}}+{X_1}} \hfill \\ \vdots \hfill \\ \end{array} $$
(15)
$$ X_1^{2N }=m\times {{\left[ {\begin{array}{*{20}c} 0 & 0 & \ldots & {-1} \\ \end{array}} \right]}_{{1\times N}}}+{X_1} $$
(16)

in which m = mesh size.

In the MADS, N + 1 new solutions are generated around the X 1, as follows:

$$ X_1^1=m\times {{\left[ {\begin{array}{*{20}c} 1 & 0 & \ldots & 0 \\ \end{array}} \right]}_{{1\times N}}}+{X_1} $$
(17)
$$ \begin{array}{*{20}c} {X_1^2=m\times {{{\left[ {\begin{array}{*{20}c} 0 & 1 & \ldots & 0 \\ \end{array}} \right]}}_{{1\times N}}}+{X_1}} \hfill \\ \vdots \hfill \\ \end{array} $$
(18)
$$ X_1^N=m\times {{\left[ {\begin{array}{*{20}c} 0 & 0 & \ldots & 1 \\ \end{array}} \right]}_{{1\times N}}}+{X_1} $$
(19)
$$ X_1^{N+1 }=m\times {{\left[ {\begin{array}{*{20}c} {-1} & {-1} & \ldots & {-1} \\ \end{array}} \right]}_{{1\times N}}}+{X_1} $$
(20)

After generating new points from either GPS or MADS, the PS algorithm computes the objective function at the mesh points and selects one whose value is better than the first solution’s obtained objective function value. If there is a point/solution with a better obtained objective in the generated solutions, the searching point will be transferred to this new point. In this state, the expansion coefficient is used to generate new solutions. Thus, the new mesh size is larger than the previous one. In contrast, if there is no better solution in the generated solutions, the contraction coefficient will be applied and the mesh size is limited to the smaller value. Commonly, the algorithm terminates when a maximum number of iterations is reached. More information about the PS is contained in Hooke and Jeeves (1961) and Lewis and Torczo (2000).

4 Methodology

In groundwater modeling, an aquifer is mapped to cells in a mesh or grid, in which their hydrological and hydrogeological parameters are assumed by a value for each cell. To simplify the problem, those values are defined for an area of considerable extent, which may be different in a cell area. In this paper, to extract the best recharge and discharge in each active cell, two simulation (ADIM) and optimization (PSO or PS) techniques are coupled and an iterative searching process performed to achieve an optimal or near-optimal solution. Figure 2 shows a flowchart of employed research methodology to calibrate a problem. As shown, a random solution(s) is generated, in which each individual is a set of recharge and discharge. Then, the corresponding SSD or SAD of each individual is calculated. The random solution(s) is improved by use of an optimization algorithm and the new solution(s) is input to the simulation part in the next iteration. This iterative process continues until a stopping criterion is satisfied and the final solution as the best values of recharge and discharge is reported.

Fig. 2
figure 2

Flowchart of research methodology

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5 Case Study

The case study focuses on the Ghaen plain, located in eastern Iran. The study area is 306 × 106 m2 and most of the discharge wells are located in the center of this area. Accordingly, 60, 30, and 10 percent of these discharges have been allocated to agricultural, municipal, and industrial sectors, respectively. On the other hand, no permanent loss from a river exists in the Ghaen plain. The aquifer recharges from precipitation, temporary rivers, and recharging wells. In this study, to simulate aquifer behavior in the Ghaen aquifer, seven observation wells and their water level variation in a 12-month period were selected as the groundwater indicator. The aquifer was represented by a network with 17 rows and 18 columns. Changes in water table elevation are very small compared to the saturated thickness of the unconfined aquifer. Aquifer transmissivity, storativity, and total recharge and discharge for each active unit/cell of the aquifer model were considered as the decision variables. Thus, the total number of decision variables depends on number of active cells. In the Ghaen plain model, the number of active cells is 137, in which transmissivity and storativity are assumed constant during different periods; therefore, there are 3,562 decision variables.

The preceding parameters are input data to the mathematical models while values of hydraulic head in 12 months are calculated by using IADIM. These estimated heads are then compared with observed heads of piezometer wells for computing SAD and SSD. The aquifer parameters are then improved until the values of estimated and observed hydraulic heads are nearly the same.

6 Results and Discussion

The PSO and PS algorithms were coded in the software package Matlab7.1 and run on a PC/WindowsXP/256MB RAM/2GZ computer. Fifty particles and 200 iterations were considered for the PSO algorithm and 200 iterations for the PS algorithm. In meta-heuristic algorithms, the number of computational efforts needed to achieve an optimal/near-optimal solution is equal to number of function evaluations. This value in each iteration of PSO and PS algorithms is equal to the number of particles and solutions in a mesh, respectively. Thus, the total number of function evaluations in the search process for PSO and PS algorithms is respectively equal to the number of particles and solutions in a mesh, which is multiplied to the number of iterations. In this paper, to conduct different runs with the same condition, the number of function evaluations is equal to 1,000. In the PSO algorithm, c 1 and c 2 are cognitive and social parameters. Moreover, v Min. and v Max., which control a particle’s velocity are equal to -1 and 1, respectively. In the PS algorithm, contraction and expansion coefficients are equal to 0.5 and 2, respectively. According to the applied models, recharge and discharge amounts for each cell are decision variables and SAD and SSD as the objective functions are improved by an iterative process with 200 iterations as the stopping criterion.

Table 1 shows results of PSO and PS algorithms and statistical measures using SAD and SSD as the objective function. As shown, the minimum value of SAD yielded by the PSO algorithm was 27.35, which is 2.41 percent worse than the 26.71 obtained by the PS algorithm. Although the minimum and average of the SAD by the PS algorithm is less than the corresponding values by the PSO, the maximum value of SAD by PS was 1.21 percent higher than the maximum of the PSO algorithm. Thus, the standard deviation and coefficient of variation of the optimal SAD yielded by the PS algorithm were higher than those of the PSO algorithm.

Table 1 Results of 10 runs and statistical measures using PSO and PS algorithms
Full size table

The aforementioned results were the same for SSD. Accordingly, the best (minimum) value of SSD was yielded by the PS algorithm, 0.20 percent less than the corresponding value calculated by the PSO algorithm. Moreover, the average value of SSD by the PS algorithm was 0.21 percent less than the 17.620 calculated by the PSO algorithm. Although PS produced the best value of SSD, the coefficient of variation of this algorithm was 65.33 percent higher than in the PSO algorithm. According to these results, the PS algorithm is capable to search and yield the best solution, whereas the probability of computing this solution is less than in the PSO algorithm.

To illustrate the best solution produced by the PS algorithm, the estimated hydraulic head for seven different observation wells are presented in Fig. 3. As shown, estimated heads for all observation wells are closer to the values of observed head. Moreover, estimated hydraulic heads for the second, third, and fifth observation wells differ more from observed values. These differences can be decreased by developing a more precise mathematical simulation model or by increasing the accuracy of field measurements.

Fig. 3
figure 3

Estimated hydraulic head for a first; b second; c third; d fourth; e fifth; f sixth; and g seventh observation wells

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Based on the best SAD and SSD produced by the PS algorithm, the estimated hydraulic heads using SSD are closer to the observed hydraulic heads. The minimum difference between estimated and observed heads by the SSD was 99.86 percent less than the corresponding value by the SAD. This result was indicated by the average of differences by the SAD and SSD. Thus, the SSD was more capable to calculate optimal parameter values than the SAD.

Figures 4 and 5 show the best values of transmissivity and storativity in the Ghaen aquifer, respectively. As presented, values of T and S are between 250 to 700 m2/day and 2 to 4 percent in the Ghaen aquifer, respectively. Accordingly, values of total optimal recharge and discharge in the Ghaen aquifer are 0.15 and 1.87 × 106 m3, respectively. However, measured recharge and discharge are respectively 0.079 and 1.05 × 106 m3, which indicates inaccuracy of groundwater flow measurements in this aquifer. Based on these results, all parameters affecting aquifer performance should be calibrated at the same time.

Fig. 4
figure 4

Calibrated pattern of transmissivity (m2/day) in Ghaen aquifer

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Fig. 5
figure 5

Calibrated pattern of storativity in Ghaen aquifer

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7 Concluding Remarks

Use of meta-heuristic algorithms to calibrate optimal parameters in groundwater models resulted in performance measures much better than traditional gradient-based techniques. In the water industry and real-world, various groundwater simulation models are employed to determine different states of aquifer under several recharges and discharges. In groundwater simulation, selection of appropriate optimal aquifer parameters is an important issue in hydraulic head calculation, which is directly related to the management of groundwater quality and quantity. Thus, in this paper, to achieve appropriate values of aquifer parameters such as transmissivity, storativity, and total recharge and discharge for each unit of aquifer model, two meta-heuristic algorithms (namely, PSO and PS) were used and compared in a real aquifer (the Ghaen aquifer in Iran). The main comparison criteria considered were SAD and SSD, in which SSD emphasizes more penalties to each unit of difference between observed and calculated hydraulic heads by using total squared differences. In contrast, the formulation of SAD considered the absolute value of those differences.

Results from application of the aforementioned meta-heuristic algorithms were compared by using the same objective function for each algorithm. The best (minimum) obtained value of SAD was calculated by using the PS algorithm, resulting in 2.36 percent lower (better) than the value computed by using the PSO algorithm. Moreover, a similar result was obtained by using SSD. Accordingly, the best (minimum) obtained value of objective function was calculated by the PS algorithm, resulting in 0.20 percent lower (better) than the value computed by using the PSO algorithm. Results indicated that the accuracy and efficiency of the aquifer parameters’ calibration improved by employing the PS algorithm. Development of this algorithm for other types of water-resource optimization problems as well as application of other evolutionary algorithms are recommended for future studies.