Introduction

Artificial groundwater recharge provides an executable solution for maintaining groundwater systems in arid and semi-arid regions. However, due to the heterogeneous nature of groundwater systems, which may vary significantly from place to place, it is always necessary to perform experiments. In this regard to evaluating the applicability and feasibility of artificial recharge systems. Groundwater modeling can be defined as the process of simulating the response and behavior of groundwater systems under natural or artificial recharge/discharge events (Sherif et al. 2006). Artificial recharge plans, in addition to protecting groundwater systems in arid and semi-arid regions, can prevent flooding and damages caused by it, and also the loss of floods and excess surface water. Recharge wells, so-called injection wells, are commonly used to replenish groundwater when the aquifer is deep and separated from the surface by low permeability materials. Wells are also used to recharge unconfined aquifers where the area of available land is limited (Asano 1985).

Among the studies conducted in the field of artificial recharge using the injection well, it can be referred to the research of Ghazaw et al. (2014), which estimated the effect of artificial recharge wells on groundwater using the MODFLOW model. In this research, runoff from the rain was drained in six ponds and for each of the six pools, an injection well was designed to model the recharge to groundwater. The results showed that this research could be used because provides a good opportunity for storing and recharging the runoff that is currently wasted and will help to sustain the aquifer for a long time. Also, the life of aquifer will increase about 3 percent due to the use of proposed recharge wells, and the water table drop in the aquifer will be approximately 41.9 m for 30 years, due to the reduction in pumping rates using recharge wells which is about 36% less than the prevailing situation of the region. Händel et al. (2016a, b), evaluated the German federal state of Saxony aquifer recharge using low diameter wells in both cases Incremental injection rates and constant injection rates. The results showed that in the first case, with incremental gradual rates for a short time, the water table inside the well only increased slightly, while in the second case (at the same wells), continuously with a constant rate for 14 days shows more efficiency in water penetration. The constant injection rate of 0.75 l/s Leads to the extra volume of water to 900 m3 and shows the high-performance potential of these wells for freshwater penetration. Händel et al. (2016a, b), In Austria, evaluated the effect of shallow wells with a low diameter in the management of aquifer recharge. The results indicated that the use of these wells is a sustainable recharge method and economical in this area and its similar areas. Shi et al. (2016), investigated the effect of artificial recharge by injection wells on land subsidence control, groundwater quality, and aquifer energy storage in Shanghai, China. The results of long-term historical data show that artificial recharge is not only beneficial for rising the groundwater table and returning the land from subsidence, but also provides cheap energy sources for industrial production. Also, the regulation of confined aquifers under injection and artificial recharge wells is a major issue for groundwater recovery and land subsidence control soon. Hussain et al. (2019), Artificial recharge of groundwater in Lahore city was investigated using recharge wells through rainwater harvesting. The result showed that Lahore has a high potential in harvesting rainwater for groundwater recharge in Lahore using this recharge rate, the groundwater table rises by 3.54 ft after each monsoon period.

Also in the determination of the suitable location of these wells Suwartha et al. (2017), estimated the appropriate location for the construction of recharge wells in the Ciliwung catchment area in Indonesia using ArcGIS software. The results showed that only 2% of the total area is suitable for the construction of recharge wells and 48% of the existing wells are located in the appropriate area. Ghazavi et al. (2018), using the Fuzzy Logic technique, determined the location of recharge wells for artificial recharge in the urban area. The results showed that areas with low hydraulic conductivity as well as areas close to operation water wells are not suitable for the construction of recharge wells.

For the first time, the entrance of computer-aided design techniques into the structural analysis field was implemented by Kagan in 1998 (Kagan et al. 1998); In that, instead of the Shape functions used in finite element, were used Spline Basis functions. In 2005, this idea evolved by Hughes et al. (2005) using NonUniform Rational B-Spline functions (NURBS) Which derive from the development of Spline functions and was called the Isogeometric Analysis method. In brief, the advantages of the Isogeometric Analysis method in comparison with other numerical methods can be pointed out such as the possibility of more precise modeling, significant precision in the convincing of boundary conditions, no need to re-networking in problems that the geometric model changes during the solving process, a significant reduction in the size of the equations system, flexibility and simplicity in network improvement problems and the applicability of this method to solving of differential equations whose their coefficients are variable functions (Kagan et al. 1998).

Shahrokhabadi et al. (2017a, b) proposed a solution to solve the Richards equation for transition flow in the Soil unsaturated area using the Isogeometric method. The results showed that the Isogeometric Analysis method can simulate the variation of pore pressure at the site of soils connection using lower degrees of freedom and higher orders of approximation compared with the conventional finite element method. Shahrokhabadi et al. (2017a, b) presented the rapid solution of convergence for the modeling of transitional flow in saturated soils using the Isogeometric Analysis method. The results showed that the higher order of NURBS can predict the wet front in nonlinear problems. The geometric analysis model is also implemented correctly across a wide range of unsaturated flow problems, while the higher order of the finite element model is divergent, especially in problems with nonlinear levels. Kalantari et al. (2017) using the Isogeometric Analysis method developed the model of groundwater flow in an unconfined aquifer. The results show the high accuracy of this model compared to the finite difference model.

The Grey wolf algorithm was first proposed by Mirjalili et al. (2014). This meta-heuristic algorithm, which is inspired by the social behavior of Grey wolves while hunting, is population based has a simple process, and can easily be generalized to large-scale problems (Mirjalili et al. 2014). One of the major advantages of the GWO algorithm compared to other optimization algorithms is that in this method very few control parameters have to be adjusted (Wang and Li 2019; Majumder and Eldho 2020).

Among the studies conducted in the field of linking the numerical method with optimization models in engineering sciences can be cited such as Mohamadian and Shojaee (2012), Shojaee et al. (2013), Roodsarabi et al. (2016a, b), Roodsarabi et al. (2016a, b), Farzampour et al. (2019), and Khatibinia and Roodsarabi (2020). Also, Majumder and Eldho (2020), for remediation of groundwater, presented the simulation–optimization model using Artificial Neural Network and Grey Wolf Optimizer. The result showed that coupling the simulation model with GWO has better stability and convergence rather than ANN-PSO and ANN-DE models. Ghordoyee Milan et al. (2021) to the prediction of optimal groundwater exploitation combined adaptive neuro-fuzzy inference system (ANFIS) with PSO, GWO, and HHO algorithms. The results showed that simulation–optimization models ANFIS-PSO, ANFIS-GWO, and ANFIS-HHO have more accuracy than the ANFIS model. The combining of the Isogeometric analysis method with optimization models has not been done so far in studies related to water engineering.

According to the positive effects of injection wells on aquifers rehabilitation and the results of past studies, based on the high accuracy of the IGA model and the GWO algorithm in groundwater modeling problems, in the present study, the linking of these two models was used to modeling of reclamation and balancing of the unconfined aquifer.

The purpose of this study is to develop a simulation–optimization model named IGA-GWO to determine the optimal location of injection wells for artificial recharge in an unconfined aquifer. In such a way that after constructing wells in optimal locations, the aim of maximum rising the water table in the injection condition is to satisfy under different values of hydraulic conductivity and specific yield. No study has been presented on the determination of injection well's location using the linking of a numerical model of Isogeometric Analysis and Grey wolf optimization algorithm in simulation–optimization problems yet.

Material and methods

Numerical method of isogeometric analysis

The motivation for the first activities in the field of Isogeometric Analysis was due to the gap between the world of finite element analysis and computer modeling. In the early stages, one of the most important fields of research in Isogeometric Analysis is to establish a relationship between the two categories of design and analysis, and also identify the barriers and present solutions in each category (Cottrell et al. 2009). The IGA (Isogeometric Analysis) concept is based on the performance of the NURBS basis functions inaccurate modeling of geometry and solution space. NURBS basis functions are weight functions that are created by B-Spline functions interpolation (Shahrbanozadeh et al. 2014). The NURBS surface from the degree of p in knot vectors \(\Xi = \left( {\xi_{0} = 0, \ldots ,\xi_{{{\text{n}} + {\text{p}} + 1}} = 1} \right)\) direction and q in knot vectors \({\text{H}} = \left( {\eta_{0} = 0, \ldots ,\eta_{{{\text{m}} + {\text{q}} + 1}} = 1} \right)\) direction is obtained by the following equation (Peigl and Tiller 1997):

$$S\left( {\xi ,\eta } \right) = \mathop \sum \limits_{i = 0}^{{\text{n}}} \mathop \sum \limits_{j = 0}^{{\text{m}}} R_{i,j} \left( {\xi ,\eta } \right)P_{i,j} \quad 0 \le \xi ,\;\;\eta \le 1$$
(1)

In the above equations, \(S\left(\xi ,\eta \right)\) is NURBS surface, \({P}_{i,j}\)(I = 0,…, n and j = 0,…, m) are the coordinates of the control points in two directions \(\xi\) and \(\eta\) and \(R_{i,j}\) are piecewise rational basis functions. \(R_{i,j}\) is defined by the following equation:

$$R_{i,j} \left( {\xi ,\eta } \right) = \frac{{N_{{i,{\text{p}}}} (\xi )M_{{j,{\text{q}}}} (\eta )w_{i,j} }}{{\mathop \sum \nolimits_{i = 0}^{{\text{n}}} \mathop \sum \nolimits_{j = 0}^{{\text{m}}} N_{{i,{\text{p}}}} (\xi )M_{{j,{\text{q}}}} (\eta )w_{i,j} }}$$
(2)

where \({w}_{i,j}\) is the control points weight in two dimensions, \({N}_{i,\mathrm{p}}(\xi )\) and \({M}_{j,\mathrm{q}}(\eta )\) are respectively the B-Spline basis functions in the direction of the knot vectors \(\Xi\) of degree p and \({\rm H}\) of degree q. These functions are derived from the following equations (\(M_{{j,{\text{q}}}} (\eta )\) is also estimated in a similar way):

$$N_{i,0} \left( \xi \right) = \left\{ {\begin{array}{*{20}c} {1\quad \quad \xi _{i} \le \xi < \xi_{i + 1} } \\ {0\;\; \quad \quad {\text{otherwise }}} \\ \end{array} } \right.\quad {\text{for}} p = 0$$
(3)
$$N_{{i,{\text{p}}}} \left( \xi \right) = \frac{{\xi - \xi_{i} }}{{\xi_{{i + {\text{p}}}} - \xi_{i} }}N_{{i,{\text{p}} - 1}} \left( \xi \right) + \frac{{\xi_{{i + {\text{p}} + 1}} - \xi }}{{\xi_{{i + {\text{p}} + 1}} - \xi_{i + 1} }}N_{{i + 1,{\text{p}} - 1}} \left( \xi \right)\quad {\text{for}} p \ge 1$$

Discretization of the flow governing equation in unsteady state

The groundwater governing equation of the two-dimensional and unsteady state in the unconfined aquifer is defined as (4):

$$\frac{\partial }{\partial x}\left( {k_{x} H\frac{\partial H}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {k_{y} H\frac{\partial H}{{\partial y}}} \right) = \frac{{S_{y} \partial H}}{\partial t} \pm R$$
(4)

In the above equation, \(H\) is groundwater table, \({k}_{x}\) and \({k}_{y}\) are the aquifer hydraulic conductivity coefficient respectively in \(\mathrm{x}\) and \(\mathrm{y}\) directions, \({S}_{y}\) is specific yield and \(R\) is discharge or recharge.

Stiffness matrix, unknown vector, and force body vector after discretization of the Eq. (4) in the two-dimensional and unsteady state in the unconfined aquifer are summarized by the finite element weighted residuals method and part by part integration as relationships (5), (6), and (7), respectively (Kalantari et al. 2017):

$$\left[ K \right] = k\left[ {\iint\limits_{\Omega } {\frac{{\partial \varphi_{i} }}{\partial x}H^{{\text{n}}} \frac{\partial \varphi }{{\partial x}}{\text{d}}\Omega } + \iint\limits_{\Omega } {\frac{{\partial \varphi_{i} }}{\partial y}H^{{\text{n}}} \frac{\partial \varphi }{{\partial y}}{\text{d}}\Omega }} \right] + \iint\limits_{\Omega } {\varphi_{i} S_{y} \left( {\frac{1}{\Delta t}} \right){\text{d}}\Omega }$$
(5)
$$\left[ U \right] = H^{{{\text{n}} + 1}}$$
(6)
$$\left[ F \right] = \iint\limits_{\Omega } {\varphi_{i} S_{y} \left( {\frac{{H^{{\text{n}}} }}{\Delta t}} \right){\text{d}}\Omega } - Q_{{\text{K}}} - \iint\limits_{\Omega } {\varphi_{i} q\;{\text{d}}\Omega }$$
(7)

In the above relations \(\left[K\right]\) is the stiffness matrix, \(\left[U\right]\) is the unknown vector, and \(\left[F\right]\) is the force body vector. \({\varphi }_{i}\), \({Q}_{K}\), and q are the basis functions, concentrated discharge of wells, and distributed discharge such as rainfall or evaporation, respectively.

Isogeometric analysis method in an unconfined aquifer in the hypothetical aquifer

In this hypothetical aquifer, an unconfined aquifer with 3200 m long and 2800 m wide and a thickness of b = 100 m was considered. The effective porosity and hydrodynamic properties of the aquifer, including T (transmissivity coefficient) and Sy (specific yield) were entered into the model n = 0.3, T = 885.7 \({\mathrm{m}}^{2}/\mathrm{d}\), and Sy = 0.15. Then, the model boundary was selected in the upstream and downstream areas of the aquifer Dirichlet type at a constant elevation of 100 m, and on the left and right sides of the aquifer, the Neumann type was selected. Also, the initial values of the water table were considered equal to 100 m (Illangasekare and Doll 1989).

In the definition of the IGA model in the hypothetical aquifer, first, the order of basis functions was determined. In the two-dimensional flow problem, the degree of the basis functions was considered equal to 2 in both directions \(\xi\) and \(\eta\). Since the order of the basis functions is equal to the degree of the basis functions plus one (Peigl and Tiller 1997) the order of the basis functions of the problem was defined as 3 (p = q = 3). Also, the number of knot vector members which is obtained based on the sum of the number of control points and the order of the basis functions was estimated as follows in the direction of \(\xi\) 20 and in the direction of \(\eta\) 18:

$$k\left( \xi \right) = \left\{ {0,0,0,0,\frac{1}{13},\frac{2}{13},\frac{3}{13},\frac{4}{13},\frac{5}{13},\frac{6}{13},\frac{7}{13},\frac{8}{13},\frac{9}{13},\frac{10}{{13}},\frac{11}{{13}},\frac{12}{{13}},1,1,1,1} \right\}$$
(8)
$$k\left( \eta \right) = \left\{ {0,0,0,0,\frac{1}{11},\frac{2}{11},\frac{3}{11},\frac{4}{11},\frac{5}{11},\frac{6}{11},\frac{7}{11},\frac{8}{11},\frac{9}{11},\frac{10}{{11}},1,1,1,1} \right\}$$
(9)

Also, the number of control points in the direction \(\xi\) and \(\eta\) were defined respectively equivalent of n = 17 and m = 15 points at a distance of 200 m from each other which resulted in a total of 255 control points. Due to the two-dimensional flow, four points Gaussian integrals were used (Table 1).

Table 1 Gaussian points and related weight coefficients
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Two extraction wells in position (1400,1400), (1800,1400) and one observation well in position (1000,1000) on the control points with pumping rates \(Q_{{{\text{p}}1}} = 1142.85 \;{\text{m}}^{3} /{\text{d}}\) at the first well and \(Q_{{{\text{p}}2}} = 1428.57 \;{\text{m}}^{3} /{\text{d}}\) at the second well during the 210 days were defined (Fig. 1).

Fig. 1
figure 1

Location of extraction wells and observation well on the control points. In an unconfined aquifer

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After running the model to validate the model, the simulated groundwater table was compared by analytical solution method using the following evaluation criteria:

Evaluation criteria in unstable conditions (Sadeghi-Tabas et al. 2016):

$${\text{ME}} = \frac{{\mathop \sum \nolimits_{j = 1}^{{\text{m}}} \mathop \sum \nolimits_{i = 1}^{{\text{n}}} h_{0} - h_{{\text{s}}} }}{m \times n}$$
(10)
$${\text{MAE}} = \frac{{\mathop \sum \nolimits_{j = 1}^{{\text{m}}} \mathop \sum \nolimits_{i = 1}^{{\text{n}}} \left| {h_{0} - h_{{\text{s}}} } \right|}}{m \times n}$$
(11)
$${\text{RMSE}} = \sqrt {\frac{{\mathop \sum \nolimits_{j = 1}^{{\text{m}}} \mathop \sum \nolimits_{i = 1}^{{\text{n}}} \left( {h_{0} - h_{{\text{s}}} } \right)^{2} }}{m \times n}}$$
(12)

In the above relations, \(h_{0}\), \(h_{{\text{s}}}\), \(n\), and m, respectively, show the values of the observed groundwater table and simulated groundwater table by the model, n is the number of observation wells, and m is the number of periods.

Then to the recovery of the hypothetical aquifer, after simulation of groundwater table of the aquifer under extraction conditions, the Grey wolf optimization model was used to find the optimized site of injection wells in the aquifer without the extraction wells. In the artificial recharge condition, two scenarios were defined in which the aquifer was divided into three zones with different hydraulic conductivity k1 = 14.8, k2 = 29.5 in scenario 1 and k3 = 36.9 m2/day (Fig. 2) and three zones with different specific yield Sy1 = 0.09, Sy2 = 0.1 and Sy3 = 0.3 (Fig. 3) in scenario 2. Table 2 shows the running conditions of the IGA simulation model under injection test.

Fig. 2
figure 2

Schematic view of the hypothetical aquifer under injection test with different hydraulic conductivity (scenario 1)

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Fig. 3
figure 3

Schematic view of the hypothetical aquifer under injection test with different specific yield (scenario 2)

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Table 2 Running conditions of IGA simulation model under injection test
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Also, the amount of injection rate for each of the injection wells was estimated based on the following equation:

$$Q = kiA$$
(13)

where Q is injection rate (m3/day), K is hydraulic conductivity (m/day), i is hydraulic gradient and A is a cross section (m2).

Grey wolf optimization algorithm

In the mathematical modeling of the wolf social hierarchy at GWO algorithm, the best solutions are named alpha (α), beta (β), and delta (δ) wolves, respectively. The optimization is led by them, and the rest of the candidate’s solutions are omega (ω) wolves that follow these three wolves in search of the global optimum. In addition to social leadership, the following equations are presented to simulate the behavior of grey wolves during hunt Mirjalili et al. (2014):

$$\vec{D} = \left| {\vec{C} \cdot \vec{X}_{{\text{p}}} \left( t \right) - \vec{X}\left( t \right)} \right|$$
(14)
$$\vec{X}\left( {t + 1} \right) = \vec{X}_{{\text{p}}} \left( t \right) - \vec{A} \cdot \vec{D}$$
(15)

In the above equations t, \(\overrightarrow {{X_{{\text{p}}} }}\) and X are the current iteration, the position vector of the prey, and the position vector of a grey wolf, respectively. Also, A and C are coefficient vectors that are calculated as follows:

$$\vec{A} = 2\vec{a} \cdot \vec{r}_{1} - \vec{a}$$
(16)
$$\vec{C} = 2 \cdot \vec{r}_{2}$$
(17)

where elements of a decrease from 2 to 0 in each iteration, linearly and r1, r2 are random vectors in [0,1].

The GWO algorithm determines the optimal solution of optimization problems using the simulated social leadership and encircling mechanism. This algorithm stores the first three best solutions obtained and obliges other search agents (including the Omega) to update their positions in connection with them. To simulate hunting and find promising areas of the search space, the following formulas are constantly used for each search agent Mirjalili et al. (2016):

$$\vec{D}_{\alpha } = \left| {\vec{c}_{1} \cdot \vec{X}_{\alpha } - \vec{X}} \right|$$
(18)
$$\vec{D}_{\beta } = \left| {\vec{c}_{2} \cdot \vec{X}_{\beta } - \vec{X}} \right|$$
(19)
$$\vec{D}_{\delta } = \left| {\vec{c}_{3} \cdot \vec{X}_{\delta } - \vec{X}} \right|$$
(20)
$$\vec{X}_{1} = \vec{X}_{\alpha } - \vec{A}_{1} \cdot \left( {\vec{D}_{\alpha } } \right)$$
(21)
$$\vec{X}_{2} = \vec{X}_{\beta } - \vec{A}_{2} \cdot \left( {\vec{D}_{\beta } } \right)$$
(22)
$$\vec{X}_{3} = \vec{X}_{\delta } - \vec{A}_{3} \cdot \left( {\vec{D}_{\delta } } \right)$$
(23)
$$\vec{X}\left( {t + 1} \right) = \frac{{\vec{X}_{1} + \vec{X}_{2} + \vec{X}_{3} }}{3}$$
(24)

This algorithm creates the initial population as a set of random solutions to start optimization. During optimization, alpha, beta, and delta solutions are stored and for each omega wolf (search agent except α, β, and δ), the position is updated based on (18) to (24) formulas. Finally, after satisfying the stop condition, the alpha wolf position is stored as the best solution obtained during optimization.

The optimization problem

The purpose of running the GWO optimization model is to determine the optimized location of injection wells that have the greatest effect on rising the water table. Therefore, to define the objective function for this problem, the difference of groundwater table after injection and before that was calculated as follows:

$${\text{Max}}\;\;f\left( h \right) = \mathop \sum \limits_{i = 1}^{{{\text{n}}_{{\text{w}}} }} \left( {h_{i} - h_{0} } \right) - \beta_{1} p\left( h \right) - \beta_{2} p\left( Q \right)$$
(25)

In the above relation \(f\left( h \right)\) is the objective function and \(n_{{\text{w}}}\) is the number of injection wells. Also, two parameters \(h_{i}\) and \(h_{0}\), respectively, are groundwater table after injection through well in i location (m) and groundwater table before injection (m). Also, \(\beta_{1 } \;{\text{and}}\; \beta_{2 }\) weight factors that are determined based on the problem (\(\beta_{1 } = \beta_{2 } = 0.1\)) and \(p\left( h \right)\) and \(p\left( Q \right)\) are penalty parameters that change linearly with the amount of violation of the problem constraints. \(p\left( h \right)\) and \(p\left( Q \right)\) are equal to:

$$p\left( h \right) = \left\{ {\begin{array}{*{20}c} {h_{0} - h_{i} \quad if\;h_{i} < h_{0} } \\ {0\quad \quad \;\;\;if\;h_{i} \ge h_{0} } \\ \end{array} } \right.$$
(26)
$$p\left( Q \right) = \left\{ {\begin{array}{*{20}c} {\mathop \sum \limits_{i = 1}^{{n_{w} }} q_{i} - Q\quad if\;\mathop \sum \limits_{i = 1}^{{{\text{n}}_{{\text{w}}} }} q_{i} > Q} \\ {0\quad \quad \quad \;\;\;if\; \mathop \sum \limits_{i = 1}^{{{\text{n}}_{{\text{w}}} }} q_{i} \le Q } \\ \end{array} } \right.$$
(27)

where \(q_{i}\) is equal to the flow rate injected into the aquifer at location i m3/day (Eq. 13) and \(Q\) is equal to the total flow rate available for injection (\(Q\) = 2500 m3/day).

Also, the constraints of the minimum allowable distance between the injection wells were defined through the following equation:

$$\sqrt {\left( {x_{i} - x_{j} } \right)^{2} + \left( {y_{i} - y_{j} } \right)^{2} } \ge S_{w \min }$$
(28)

where \(x_{i}\) and \(y_{i}\) are the injection wells location in place i and in the direction x and y and \(x_{j}\) and \(y_{j}\) are the injection wells location in place j and in the direction x and y. Also, \(S_{{\text{w min}}}\) (\(S_{{\text{w min}}}\) = 400 m) is the minimum distance between injection wells (m).

Table 3 shows the running conditions of the GWO optimization algorithm under injection test. Also, Fig. 4 presents the IGA-GWO numerical model flowchart. This optimization model was run considering 15 Grey wolves and 50 iterations.

Table 3 Running conditions of GWO optimization algorithm under injection test
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Fig. 4
figure 4

Flowchart of the IGA-GWO numerical model

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Results and discussion

IGA model validation

To evaluate the performance of the model in simulation of groundwater table, after running the IGA numerical model code, the model results were compared with the analytical solution presented by Illangasekare and Doll (1989) using ME, MAE, and RMSE evaluation criteria. The results of the analytical solution showed that after 210 days from the start of operation of the aquifer through two wells, the groundwater table has decreased by 42.5 cm at the observation well. This value reaches 39.90 cm in the same conditions in the numerical solution of the IGA model. Also, the results of evaluation criteria with values of ME = − 0.96%, MAE = 1.11%, and RMSE = 0.0146 m showed that the IGA numerical model estimates the groundwater table with acceptable accuracy (Table 4). Figure 5 indicates the groundwater table fluctuations simulated by the IGA model at intervals of 60, 90, 120, 150, 180, and 210 days, in which the decrease in the aquifer water table is evident over time.

Table 4 Calculation of mean error, absolute mean error and root-mean-square error in unsteady state
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Fig. 5
figure 5

Fluctuations of simulated water table using IGA model on internal control points

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Result of injection test in three zone with different hydraulic conductivity

After ensuring the accuracy of the model, the code of the GWO optimization algorithm was linked to it and the IGA-GWO simulation–optimization model was created. In this scenario, the purpose of creating this model is to determine the optimal locations for the construction of 10 injection wells to increase the groundwater table under different values of hydraulic conductivity. The result shows the model converges in the fifteenth iteration. Also, the optimal value of the objective function was estimated at 0.525. This value is equivalent to the maximum increase in the water table equal to 52.5 cm on average in 10 injection wells. According to the general rule of the algorithm, which is based on the minimum distance between Grey wolves and prey, was maximized the problem of the study by considering the inverse of the objective function. Also, the optimal solution of the problem was expressed based on the position of Alpha Wolf. Figure 6 shows the optimal values of the objective function based on the alpha wolf position at 50 iterations.

Fig. 6
figure 6

The convergence process of simulation–optimization model IGA-GWO under scenario 1

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Also, the locations determined in the 50th iteration were considered as the optimal locations of 10 injection wells defined for the aquifer. Figure 7 shows the optimal location of injection wells at the aquifer surface. Examination of the location of the well shows that the majority of injection wells (7 wells) due to different values of hydraulic conductivity and defined boundary conditions are located in zone 3 (with maximum hydraulic conductivity). These results are consistent with the results of Ghazavi et al. (2018) based on the fact that regions with high hydraulic conductivity are suitable for the construction of injection wells. This is proved by Eq. (13) and the direct effect of hydraulic conductivity on increasing the injection rate and consequently increasing the water table. Also, 3 wells were located in Zone 2 due to the drop caused by the extraction wells and the created hydraulic gradient. Table 5 shows the average groundwater table in each injection well.

Fig. 7
figure 7

Optimal location of injection wells under scenario 1

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Table 5 Average of water table rising in each injection well under scenario 1
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Result of injection test in three zone with different specific yield

In this scenario, with the definition of 3 regions with different values of specific yield, the location of injection wells was estimated. After running the IGA-GWO model, the results showed that the model converges in the third iteration (Fig. 8). Also, Fig. 8 shows that the best solution of the model is 0.532, which implies that the groundwater table on average, in 10 injection wells, increases equal to 53.2 cm.

Fig. 8
figure 8

The convergence process of simulation–optimization model IGA-GWO under scenario 2

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Figure 9 shows the location of injection wells under scenario 2. Due to the direct effect of the specific yield parameter on increasing the water table, 4 injection wells were located in the region with the highest specific yield rate. Also, the majority of wells are located in the vicinity of extraction wells. This is due to the higher hydraulic gradient and the drop created based on the water extraction in this area. Table 6 shows the location of injection wells and the rising of the water table in each injection well.

Fig. 9
figure 9

Optimal location of injection wells under scenario 2

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Table 6 Average of water table rising in each injection well under scenario 2
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Conclusions

Nowadays, the use of various artificial recharge methods to regenerate and balance aquifers with declining groundwater tables, especially in arid and semi-arid regions, is widely used. In the present study, the IGA-GWO simulation–optimization model was designed to determine the optimal location of injection wells to increase the groundwater table under different values of hydraulic conductivity and specific yield. In this regard, first, the water table simulation code was written based on the IGA numerical model. Then, by examining the conformity of the model results with the analytical solution of the problem based on the evaluation criteria ME = − 0.96%, MAE = 1.11%, and RMSE = 0.0146 m, the accuracy of the model was checked. After ensuring the accuracy of the model, the GWO optimization algorithm was linked to the simulation model. After running the IGA-GWO simulation–optimization model code in 50 iterations was specified that the optimal solution of the model is equal to 52.5 cm and 53.2 cm rise in the groundwater table in scenario 1 and scenario 2, respectively. Based on the results, it can be seen that the use of the injection well method in artificial recharge of the aquifer plays a positive role in increasing the water table. Also, according to the parameter of aquifer hydraulic conductivity and boundary conditions defined for the problem, the majority of injection wells in scenario 1 were located in zone 3 (k3 = 36.9 m3/day). Therefore, due to the direct effect of hydraulic conductivity in increasing the water table, regions with high hydraulic conductivity are estimated to be suitable regions for artificial recharge by injection well method. Also, the results of Scenario 2 show that the water table in regions with high specific yield and also regions with a drop (due to higher hydraulic gradient) are among the optimal locations for the construction of injection wells. Therefore, from the results of the present study, can be seen that the parameters such as hydraulic conductivity, specific yield, hydraulic gradient, and aquifer boundary conditions are the important parameters in determining the appropriate location for the construction of injection wells. Based on the above results, it can be seen that proper recharge of aquifers by determining suitable injection sites based on simulation–optimization methods, has a significant effect on increasing the groundwater table. Also, the continuation of this process helps to regenerate and balance the aquifers and prevent damage due to their decline.