Introduction

The main groundwater contamination with anthropogenic sources consists of seepage from septic tanks, absorptive wells, and sewers, and uncontrolled hazardous waste.

Declining groundwater quality and its pollution have made groundwater remediation and management an essential need. In the last few decades, groundwater contamination has become a major problem in many parts of the world reducing its availability for drinking and even agriculture. Groundwater remediation methods are very costly and time-consuming. Therefore, an efficient approach should be adopted for selecting the remediation method to eliminate the contamination efficiently. In this direction, numerical methods and optimization tools can be very beneficial for finding the best strategy for groundwater remediation (Eldho and Swathi, 2018). In the groundwater contamination literature, some researchers have studied the pollutant sources (Ghafouri and Darabi, 2007; Guneshwor et al, 2018) and some other researchers investigated the types of groundwater remediation methods such as groundwater phytoremediation (Kumar et al, 2015; Mategaonkar et al, 2018) or PAT method (Wang et al., 2018). Often, designing an efficient remediation system is a multi-objective task. For example, the PAT method components are concerned by the location of the pumping well, pumping rates of contaminated water, remediation period, and the drawdown of groundwater head during the remediation period.

To understand the response of the groundwater system to different remediation strategies, various numerical methods such as finite difference method (FDM) (He et al, 2017; Yang et al, 2018a), finite element method (FEM) (Ghafouri and Darabi, 2007; Esfahani and Datta, 2018), meshless method (Boddula and Eldho, 2017; Mategaonkar et al, 2018; Seyedpour et al, 2019) and software packages like MODFLOW (Singh and Chakrabarty, 2011; Joswig et al, 2017) have been used. Different methods are used to solve the groundwater remediation optimization problem. In the last decades, nonlinear programming methods (Gorelick et al, 1984) and Meta-Heuristic algorithms such as AMALGAM (Ouyang et al, 2017), probabilistic multi-objective genetic algorithm (PMOGA) (Singh and Minsker, 2008), niched Pareto genetic algorithm (NPGA) (Erickson et al, 2002) have been widely used. One of the goals of groundwater remediation is to reduce contaminant concentrations to the permissible level. The level of contaminant concentration can be directly or indirectly related to the carcinogenic human health risk (Yang et al, 2018b). Many researchers used Simulation–optimization (S–O) models for designing the process of an effective PAT system (Luo et al. 2014), (Sreekanth et al, 2016). Others minimized contaminant concentration and pumping costs, by optimizing the number of pumping wells and pumping rates (Alexander et al 2018) and some others focused on finding the best location for pumping wells (Wang et al., 2018; Sbai, 2019) and to minimizing groundwater remediation period (Mategaonkar et al, 2018).

The purpose of this study is to find the best location for pumping wells considering the important objective functions such as minimizing contaminant concentration, minimizing the drawdown of the groundwater head, minimizing the pumping rate, and remediation period. These objectives are used in one to four objective optimization problems. The GA-FEM and NSGA-II-FEM hybrid models are used to investigate optimization problems. We combined and used FEM method with genetic algorithm for this purpose.

Materials and methods

Case study

Hypothetical aquifer

In this study, we consider a hypothetical aquifer. This hypothetical aquifer is almost similar to that considered by Sharif et al. (2008).The domain of this aquifer is 1800 m × 1000 m in the area as shown in Fig. 1. The hydrogeological parameters for the aquifer are presented in Table 1. The entire aquifer has a storage coefficient of 0.0004, and a landfill is considered being in Zone A with a seepage rate of 0.009 m/day. It is assumed that zone A and zone C are recharged in between nodes 12–24 and 42–54 at a rate of 0.00024 and 0.00012 m/day, respectively (Fig. 1). The flow model has constant head conditions at the western and eastern boundaries with 100 and 95 m, respectively. The southern boundary is the no-flow boundary, and the aquifer is recharged only by the northern boundary at zones A and C. For the transport contaminant model, all boundaries are assumed to be impervious except the eastern boundary.

Fig. 1
figure 1

Hypothetical aquifer configuration

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Table 1 Hydrogeological aquifer data (Sharif et al. 2008)
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The source of contamination is a landfill. Heavy metals have leaked from this landfill for 30 years and have polluted the groundwater. One of the Important Heavy metals is chromium. The value of chromium concentration is 56 \({\mu g}/{\text{Lit}}\) and based on Food and Agriculture Organization (FAO) permissible limit for chromium concentration in drinking water is 50 \({\mu g}/{\text{Lit}}\). Landfill is located on 9, 10, 15, and 16 nodes. This problem is solved using Galerkin’s finite element method and the Crank-Nicolson time scheme with a structural grid in 6 × 10 and 45 squared elements shown in Fig. 2. Distance between nodes is 200 m. Five days time step is chosen for both the groundwater flow and the contaminant transport model.

Fig. 2
figure 2

Finite Element Method Mesh for the hypothetical aquifer

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The numerical model is initialized based on the transient flow of the aquifer wherein no pumping is assumed. The entire simulation period is 30 years, with continuous contamination. The groundwater flow and the distribution of the contaminant concentrations are simulated using the finite element method. In the present case, four pumping wells are used. In the first step, the remediation period is considered 10 years. To compare the objective function value for each member of the population, we consider contaminant concentrations in eight nodes of 15, 16, 21, 22, 27, 28, 33, and 34, shown in (Fig. 2) with the square mark.

Real aquifer

In this study, the aquifer located in Ghaen basin is also investigated. Ghaen study area with an area of 929.1 km2 is located between longitudes58° 53ʹ 39ʺ to 59° 24ʹ 40ʺ east and 33° 32ʹ 07ʺ to 33° 51ʹ 20ʺ north. The aquifer is recharged from the west and south by 8.28 and 3.61 Mm3/year, respectively. 0.5 Mm3 of groundwater is discharged annually from the east side of the aquifer. There is a contaminant resource with an absorptive well. Through this absorptive well, various contaminants seepage to the ground. Two of these contaminants are chloride and nitrate-nitrogen. These pollutants have amounts higher than the standard of wastewater in Iran. Figure 3 shows the position of the contaminant source and recharge and discharge zones. Also, there are 134 pumping wells in the Ghaen aquifer. Due to the aquifer grid and distance between nodes (200 m), 17 pumping wells were placed in the nearest nodes and their pumping rate was added together. Also in Table 2 the values of contaminants and their permissible limits are given. In order to use hydraulic information for aquifer modeling, first, we create Thiessen polygon for hydraulic conductivity by using ArcGIS software. Five days’ time step is chosen for both the groundwater flow and the contaminant transport model.

Fig. 3
figure 3

Location map of the study area

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Table 2 Contaminant Values and Standard Values in Iran
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Objective functions

PAT system costs depend on the residual contaminant concentration at the end of the remediation period. Four objective functions are considered in the hypothetical and real aquifers. These objective functions are (1) Minimizing the contaminant concentration, (2) Minimizing the drawdown of the aquifer head, (3) Minimizing the mean pumping rate, (4) Minimizing the remediation period. Therefore, the purpose of this study is to determine the optimal location of pumping wells so that objective functions are minimized. Objective functions for four optimization problems are given in Table 3:

Table 3 Optimization problems and their Functions
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In the single-objective problem, we use four pumping wells at a rate of 600 m3/day (pumping at a constant rate). For the two objectives problem, in addition to the first objective function, the minimization of drawdown of aquifer head with the same pumping wells at a rate of 600 m3/day is considered. In the three objectives problem, the pumping rate is variable, and minimizing the mean of the pumping rate is also added to the set of objective functions. In the final step, considering the three previous objective functions, the remediation period is minimized. Therefore, the objective functions are defined mathematically as follows:

$$ {\text{Min}}\,\,\,\,\,F_{1} = \left( {\sum\limits_{k = 1}^{K} {C_{k} } } \right)/K $$
(1)
$$ {\text{Min}}\,\,\,\,\,F_{2} = \sum\limits_{i = 1}^{{{\text{NNODE}}}} {\left( {H_{i}^{{{\text{old}}}} - H_{i}^{{{\text{new}}}} } \right)} $$
(2)
$$ {\text{Min}}\,\,\,\,\,F_{3} = \frac{{\sum\nolimits_{j = 1}^{J} {q_{j}^{Ex} } }}{J} $$
(3)
$$ {\text{Min}}\;F_{4} = {\text{remediation}}\;\;{\text{period}} $$
(4)

where F1 to F4 are the objective functions 1 to 4, respectively.

The first objective function minimizes the mean of contaminant concentration where K is a number of nodes and Ck is the contaminant concentration in kth nodes.

In the second objective function minimizes the drawdown of the aquifer where, \(H_{i}^{old}\) and \(H_{i}^{new}\) are aquifer head before and after installing pumping wells at ith node, (m), respectively. NNODE is the total number of nodes in the aquifer domain.

The third objective function is minimizing the mean pumping rate of all wells (m3/day). J is the number of pumping wells and \(q_{j}^{Ex}\) is the pumping rate at jth pumping well (m3/day).

The fourth objective function (\(F_{4}\)) consists in minimizing the remediation period.

Governing equations

Groundwater flow modeling

The governing partial differential equations describing the steady-state flow in a two-dimensional inhomogeneous, anisotropic confined and unconfined aquifers are given as (Wang and Anderson, 1982):

$$ \frac{\partial }{\partial x}\left( {T_{x} \frac{\partial H}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {T_{y} \frac{\partial H}{{\partial y}}} \right) = R $$
(5)
$$ \frac{\partial }{\partial x}\left( {k_{x} H\frac{\partial H}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {k_{y} H\frac{\partial H}{{\partial y}}} \right) = R $$
(6)

Moreover, the governing partial differential equations describing the Transient flow in a two-dimensional inhomogeneous, anisotropic confined and unconfined aquifers are given as (Wang and Anderson, 1982):

$$ \frac{\partial }{\partial x}\left( {T_{x} \frac{\partial h}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {T_{y} \frac{\partial h}{{\partial y}}} \right) = S\frac{\partial h}{{\partial t}} + Q_{w} \delta \left( {x - x_{i} } \right)\left( {y - y_{i} } \right) - q $$
(7)
$$ \frac{\partial }{\partial x}\left( {K_{x} h\frac{\partial h}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {K_{y} h\frac{\partial h}{{\partial y}}} \right) = S_{y} \frac{\partial h}{{\partial t}} + Q_{w} \delta \left( {x - x_{i} } \right)\left( {y - y_{i} } \right) - q $$
(8)

where h(x,y,t) or H(x,y,t) is the piezometric head [L], Ti(x,y) is anisotropic transmissivity [L2T−1]; Ki(x,y) is anisotropic hydraulic conductivity [LT−1]; S(x,y) is storage coefficient, Sy(x,y) is specific yield; Qw is source or sink function; (-Qw = source and Qw = sink) [LT−1]; \(\delta\) is Dirac delta function; xi and yi are the pumping or recharge well location; q(x,y,t) is vertical inflow rate [LT−1]; x and y are horizontal space variables [L] and t is the time [T].

FEM to solve governing equations for groundwater flow

Using Galerkin’s finite element method and two-dimensional element for approximation Eq. 9, the first step is to define a trial solution.

$$ \hat{h}\left( {x,y,t} \right) = \sum\limits_{L = 1}^{NP} {h_{L} \left( t \right)N_{L} \left( {x,y} \right)} $$
(9)

where hL is the unknown head, NL is the known basis function at node L, and NP is the total number of nodes in the hypothetical aquifer domain. A set of simultaneous equations is obtained when residuals weighted by each of the basis functions are forced to be zero and integrated over the entire domain \(\Omega\) (Desai et al, 2011). Thus, Eq. 9 can be written as:

$$ \iint\limits_{\Omega } {\left[ {\frac{\partial }{\partial x}\left( {T_{x} \frac{\partial h}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {T_{y} \frac{\partial h}{{\partial y}}} \right) - Q_{w} + q - S\frac{\partial h}{{\partial t}}} \right]N_{L} \left( {x,y} \right)dxdy = 0} $$
(10)
$$ \begin{gathered} \sum\limits_{e} {\iint {\left( {T_{x}^{e} \frac{{\partial \hat{h}^{e} }}{\partial x}\left\{ {\frac{{\partial N_{L}^{e} }}{\partial x}} \right\} + T_{y}^{e} \frac{{\partial \hat{h}^{e} }}{\partial y}\left\{ {\frac{{\partial N_{L}^{e} }}{\partial y}} \right\}} \right)}} dxdy\quad + \sum\limits_{e} {\iint {\left( {S\frac{{\partial \hat{h}^{e} }}{\partial t}} \right)\left\{ {N_{L}^{e} } \right\}}} dxdy \hfill \\ \quad= \sum\limits_{e} {\iint {\left( {Q_{w} } \right)\left\{ {N_{L}^{e} } \right\}}} dxdy - \sum\limits_{e} {\iint {\left( q \right)\left\{ {N_{L}^{e} } \right\}}} dxdy \hfill \\ \end{gathered} $$
(11)

where \(\left\{ {N_{L}^{e} } \right\} = \left\{ {\begin{array}{*{20}c} {N_{i} } \\ {N_{j} } \\ {N_{m} } \\ {N_{n} } \\ \end{array} } \right\}\).

For an element, Eq. 11 can be written in matrix form as:

$$ \left[ {G^{e} } \right]\left\{ {h_{I}^{e} } \right\} + \left[ {P^{e} } \right]\left\{ {\frac{{\partial h_{I}^{e} }}{\partial t}} \right\} = \left\{ {f^{e} } \right\} $$
(12)

where I = i, j, m, n are four nodes of rectangular elements and G, P and f are the element matrices known as conductance, storage matrices, and recharge vectors, respectively. Summation of elemental matrix Eq. 12 for all the elements gives the global matrix as:

$$ \left[ G \right]\left\{ {h_{I} } \right\} + \left[ P \right]\left\{ {\frac{{\partial h_{I} }}{\partial t}} \right\} = \left\{ f \right\} $$
(13)

Applying the implicit finite difference scheme for \(\frac{{\partial h_{I} }}{\partial t}\), term in time domain for Eq. 13 gives.

$$ \left[ G \right]\left\{ {h_{I} } \right\}_{t + \Delta t} + \left[ P \right]\left\{ {\frac{{h_{t + \Delta t} - h_{t} }}{\Delta t}} \right\} = \left\{ f \right\} $$
(14)

The subscripts t and t + Δt represent the groundwater head values at earlier and present time steps. By rearranging the terms of Eq. 14, the general form of the equation can be given as (Desai et al 2011):

$$ \left[ {\left[ P \right]} \right] + \omega \Delta t\left[ G \right]\left\{ h \right\}_{t + \Delta t} = \left[ {\left[ P \right] - \left( {1 - \omega } \right)\Delta t\left[ G \right]\left\{ h \right\}_{t} + \Delta t\left( {1 - \omega } \right)\left\{ f \right\}_{t} + \omega \left\{ f \right\}_{t + \Delta t} } \right] $$
(15)

where Δt = time step size, {h}t and {h}t+Δt are groundwater head vectors at the time t and t + Δt, respectively, x is Relaxation factor which depends on the type of finite difference scheme used. For fully explicit scheme \(\omega = 0\); Crank–Nicolson scheme \(\omega = 0.5\); fully implicit scheme \(\omega = 1\).

Contaminant transport modeling

The governing differential equation is obtained based on the mass balance of any particular solute in a control volume of porous media. The final unsteady form of the equation, including adsorption and other chemical reactions, is written as follows (Ghafouri and Darabi, 2007):

$$ \frac{\partial C}{{\partial t}} = \frac{\partial }{\partial x}\left( {D_{x} \frac{\partial C}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {D_{y} \frac{\partial C}{{\partial y}}} \right) + \frac{\partial }{\partial z}\left( {D_{z} \frac{\partial C}{{\partial z}}} \right) - v_{x} \frac{\partial C}{{\partial x}} - v_{y} \frac{\partial C}{{\partial y}} - v_{z} \frac{\partial C}{{\partial z}} + {\text{CHEM}} $$
(16)

The seepage velocity required to the solute-transport model is calculated using Darcy's law (Bear 1972) and is written as: \(V_{i} = - \frac{{K_{i} }}{\theta }\frac{\partial h}{{\partial x_{i} }}\,\,,\,\,\,i = x,\,y\); where Vi(x,y) is the seepage velocity in i-direction [LT−1]; and \(\theta\) = porosity.

In which C is the concentration of contamination at any given time and space, D the dispersion coefficient and V is the seepage velocity. The first three terms at RHS show the dispersion phenomena, while the second three terms represent the advection of contaminant. Also CHEM is consisting of the followings (Ghafouri and Darabi, 2007):

$$ {\text{CHEM}} = - \frac{{\rho_{b} }}{n}\frac{\partial s}{{\partial t}} $$
(17)

Which indicates the superficial forces of adsorption and ionization processes, and

$$ {\text{CHEM}} = - \lambda \left( {C + \frac{{\rho_{b} }}{n}s} \right) $$
(18)

In the above equations ρb is the bulk soil density (ML−3), λ is the constant decay coefficient (T−1), s is the density of absorbed substance on soil grains or colloids, and n is the soil porosity. Substituting these terms into Eq. 16 and simplifying them, results in the following equation (Ghafouri and Darabi, 2007):

$$ \begin{gathered} \frac{\partial C}{{\partial t}} + \frac{{\rho_{b} }}{n}\frac{\partial s}{{\partial t}} \hfill \\ = \frac{\partial }{\partial x}\left( {D_{x} \frac{\partial C}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {D_{y} \frac{\partial C}{{\partial y}}} \right) + \frac{\partial }{\partial z}\left( {D_{z} \frac{\partial C}{{\partial z}}} \right) - v_{x} \frac{\partial C}{{\partial x}} - v_{y} \frac{\partial C}{{\partial y}} - v_{z} \frac{\partial C}{{\partial z}} - \lambda \left( {C + \frac{{\rho_{b} }}{n}s} \right) \hfill \\ \end{gathered} $$
(19)

Assuming s = kd × C, where kd is a distribution coefficient, applying the chain rule to the derivative of s and re-ordering the resulting relations, Eq. 19 may be written as follows:

$$ \begin{aligned} R\frac{\partial C}{{\partial t}} & = \frac{\partial }{\partial x}\left( {D_{x} \frac{\partial C}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {D_{y} \frac{\partial C}{{\partial y}}} \right) + \frac{\partial }{\partial z}\left( {D_{z} \frac{\partial C}{{\partial z}}} \right) \\ & \quad - v_{x} \frac{\partial C}{{\partial x}} - v_{y} \frac{\partial C}{{\partial y}} - v_{z} \frac{\partial C}{{\partial z}} - R\lambda C \\ \end{aligned} $$
(20)

Equation 16 is three-dimensional governing equation of advection–dispersion of contaminants in groundwater resources where \(R = 1 + \frac{{\rho_{b} k_{d} }}{n}\) is the retardation factor with \(\rho_{b}\) = media bulk density [ML−3], and Kd = sorption coefficient [L3M−1]; C(x,y,t) is solute concentration [ML−3]; Dx and Dy are components of dispersion coefficient tensor [L2T−1]; λ is the reaction rate constant [T−1].

For transient flow and transport analysis, the following initial conditions are used:

$$ h(x,y,0) = h_{0} (x,y);C(x,y,0) = f\,\,\,\,x,y \in \Omega $$
(21)

The flow and transport equations should be solved with appropriate boundary conditions. The boundary conditions can be written as follows(Eldho and Swathi 2018):

$$ h(x,y,t) = h_{1} (x,y,t);C(x,y,t) = g_{1} \,\,\,\,x,y \in \Gamma_{1} $$
(22)

\(T\frac{\partial h}{{\partial n}} = q_{1} (x,y,t)\) is for the confined aquifer, \(Kh\frac{\partial h}{{\partial x}} = q_{2} (x,y,t)\) is for unconfined aquifer:

$$ \left( {D_{x} \frac{\partial C}{{\partial x}}} \right)n_{x} + \left( {D_{y} \frac{\partial C}{{\partial y}}} \right)n_{y} \, = g_{2} \,\,\,\,x,y \in \Gamma_{2} $$
(23)

where Ω is flow region; \(\Gamma = \Gamma_{1} \cup \Gamma_{2}\) is region boundary; \(\frac{\partial }{\partial m}\) is normal derivative; h0(x,y) is the initial head in the flow domain [L]; h1(x,y,t) is the known head value at the boundary section \(\Gamma_{1}\)[L]; f is a given function in Ω, \(g_{1}\) and \(g_{2}\) are given functions along boundary sections \(\Gamma_{1}\) and \(\Gamma_{2}\); and nxny are the components of the unit outward normal vector to the boundary section \(\Gamma_{2}\).

FEM to solve the governing equation for contaminant transport

The analytical solution of Eq. 17 is usually unavailable for aquifers having irregular geometries and/or boundary conditions. Hence an approximate numerical solution of the equation is often sought in most cases. The method of finite elements is one of the most frequently used methods for this purpose. This method is implemented whit a variety of element types and number of nodes. The elements join together and then, the approximate solution of the sought unknown, i.e., the pollutant concentration (CL), is computed at nodal points. For any other given point, the concentration value is approximated as (Ghafouri and Darabi 2007):

$$ C\left( {x,y,z,t} \right) = \sum\limits_{L = 1}^{{{\text{NNODE}}}} {C_{L} (t).N_{L} (x,y,z)} $$
(24)

where \(N_{L} (x,y,z)\) is an arbitrary approximating function, called shape function, of node L. Applying the finite element method to Eq. 20 leads to a set of simultaneous algebraic equations as:

$$ [A]\left\{ C \right\}^{t + \Delta t} = [B]\left\{ C \right\}^{t} + \left\{ f \right\} $$
(25)

where:

$$ \left[ A \right] = \left[ G \right] + \left[ U \right] + \left[ F \right] + \frac{1}{\Delta t}\left[ P \right] $$
(26)
$$ \left[ B \right] = \frac{1}{\Delta t}\left[ P \right] $$
(27)

∆t is the length of time interval, the so-called time-step,{C}t is the vector of known concentrations at the beginning of any the time step and {C}t+∆t is the vector of sought unknown concentrations at the end of time step. [G], [U], [F] and [P] are square matrices in which the number of rows and columns are equal to the number of nodal points in the computational grid. The elements of these matrices are computed as follows (Ghafouri and Darabi 2007):

$$ G_{(L,i)}^{e} = \iint\limits_{e} {\left( {D_{x} \frac{{\partial N_{i}^{e} }}{\partial x}\frac{{\partial N_{L}^{e} }}{\partial x} + D_{y} \frac{{\partial N_{i}^{e} }}{\partial y}\frac{{\partial N_{L}^{e} }}{\partial y} + D_{y} \frac{{\partial N_{i}^{e} }}{\partial y}\frac{{\partial N_{L}^{e} }}{\partial y}} \right)dxdydz} $$
(28)
$$ P_{(L,i)}^{e} = \iint\limits_{e} {R.N_{i}^{e} N_{L}^{e} dxdydz} $$
(29)
$$ U_{(L,i)}^{e} = - \iint\limits_{e} {\left( {V_{x}^{e} \frac{{\partial N_{i}^{e} }}{\partial x}N_{L}^{e} + V_{y}^{e} \frac{{\partial N_{i}^{e} }}{\partial y}N_{L}^{e} + V_{z}^{e} \frac{{\partial N_{i}^{e} }}{\partial z}N_{L}^{e} } \right)dxdydz} $$
(30)
$$ F_{(L,i)}^{e} = - \iint\limits_{e} {R.\lambda .N_{i}^{e} N_{L}^{e} dxdydz} $$
(31)

{f} is the load vector calculated using the following boundary integral:

$$ \left\{ f \right\} = \int\limits_{\Gamma } {\left( {D_{x} \frac{{\partial \hat{c}}}{\partial x}n_{x} + D_{y} \frac{{\partial \hat{c}}}{\partial y}n_{y} + D_{z} \frac{{\partial \hat{c}}}{\partial z}n_{z} } \right)} N_{L} d\Gamma $$
(32)

where \(\hat{c}\) denotes the given concentration values at boundary nodes, ni is directional cosines and Γ is the domain boundary.

GA and NSGA-II algorithms

In the genetic algorithm (GA) and its multi-objective version (Non-Dominated Sorting Genetic Algorithm, NSGA-II), there are crossover and mutation phases. In the single-objective version, the population is sorted by the value of the objective function, and the selection of the best individual is based on the objective function value (Akbarpour et al, 2020). In the NSGA-II algorithm, the rank of each solution in the population is based on the non-dominated sorting and crowding distance. That is, members of the population on the first front are better than those on the second front. Members on the same front are ranked by crowding distance (Felfli et al, 2002). So, in the NSGA-II algorithm, the members of the population who are in the first front are certainly better than the members of the second front and the members of the second front are better than the members of the third front, and so on. But according to Fig. 4. it can be seen that the first and second fronts have been selected for the next generation, but to select the remnant of the next generation, the selection must be made from the third front, but not all members of the third front can be present in the next generation, so they must be sorted according to the crowding distance and removed from the beginning as much as it completes the population of the next generation.

Fig. 4
figure 4

Choosing the best members of the population for the next generation

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Chain constraints (sequential selection)

In the hypothetical aquifer, we consider four pumping wells with the pumping rate of 600 m3/day. The aquifer was gridded with 60 nodes as shown in Fig. 2. In this case, it is not possible to select the boundary nodes as a pumping well location. Therefore, in the first step, these nodes are removed from the set of 60 nodes. Now, given the assumed aquifer geometry and the number of boundary nodes, the number of nodes able to select as pumping wells will be 32 nodes. After that, each time a pumping well is added to the aquifer, the node in which the pumping well is located and the nodes in the capture zone are removed from the set of selectable nodes.

Results and discussion

The results of hypothetical aquifer modeling

The hypothetical aquifer has been investigated using FEM and rectangular element. The results of groundwater flow and contaminant transport modeling for 30 years are presented in Fig. 5a and 5. As discussed in the preceding section, the hypothetical aquifer is considered with no pumping wells. The landfill is the source of contaminants.

Fig. 5
figure 5

Groundwater head and Chromium concentration distribution

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The results of one-objective optimization problem (hypothetical aquifer)

The one-objective optimization problem is solved to find the pumping wells’ location in order to obtain optimum (near-optimum) values for the first objective function. It should be noted that the first step in all remediation systems is removing pollutant sources or avoiding their dissemination and then pumping wells are installed. In this study after preventing seepage from the landfill, the contaminated groundwater is pumped by the wells, for 10 years.

Population size and maximum iteration for genetic algorithms were 10 and 20, respectively. the genetic algorithm was run five times. The results of the five runs of GA, including the best, the worst and mean values of the objective function in the five runs are presented in Table 4. As can be seen in this table, the genetic algorithm has a good performance with an average amount of first objective function equal to 41.7935 (\(\mu g/Lit\)). The best performance of GA is also shown in Fig. 6.

Table 4 Statistical characteristics of the GA performance
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Fig. 6
figure 6

The convergence trend of the GA algorithm

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The pumping well location, with 600 m3/day, (solid circle marker) and the concentration of Chromium in the aquifer domain after optimization during ten years of remediation are shown in Fig. 7. As can be seen in this figure, the best location of pumping wells is contaminant movement path. When the contamination mass is moving to the right of the aquifer, the algorithm located two more wells in front of that to remove most of the contaminants. With a constant rate of pumping (600 m3/day) and the shown location of pumping wells, the concentration of contamination throughout the aquifer is reduced to the permissible level.

Fig. 7
figure 7

Chromium concentration distribution in Aquifer \(\left( {{\mu g}/{\text{Lit}}} \right)\) and pumping well locations

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The results of two-objective optimization problem (hypothetical aquifer)

In Two-Objective optimization problem, we find the best location of wells to find the minimum of contaminant concentrations and drawdown of the aquifer head. In NSGA-II algorithm, like single-objective version, 70% of the population could be parents, and the mutation rate was set to 90%. NSGA-II is run five times. The number of solutions in the Pareto-optimal front for run 1 to run 5 are 6, 10, 5, 8 and 10, respectively. The results of all runs are shown in Fig. 8.

Fig. 8
figure 8

Comparison of the Pareto fronts in 5 runs of NSGA-II algorithm

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As it can be seen in this figure, NSGA-II algorithm finds the solution with a minimum of the first objective function in run 3 and 5. Also, NSGA-II algorithm finds the solution with a minimum of the second objective function in run 4. But, we can consider run 2 as the best performance of NSGA-II algorithm. Because this run of NSGA-II algorithm has good coverage on solution space. Also, run 2 could find the minimum for two objective functions.

The results of three-objective optimization problem (hypothetical aquifer)

In Three-Objective optimization problem, we find the best location of wells to find the minimum of contaminant concentrations, the drawdown of the aquifer head and mean pumping rate. In this problem, NSGA-II algorithm is run five times and the solutions in the Pareto front for run 1 to run 5 are 10, 9, 10, 10 and 10, respectively. The results of NSGA-II algorithm indicate that the number of solutions in the Pareto-fronts for NSGA-II algorithm is higher than in the case of the two-objective optimization problem. In this case, the algorithm is able to get any value as a pumping rate in a range of [500,600]. In Fig. 9, the Pareto front of NSGA-II algorithm in the last run is shown.

Fig. 9
figure 9

Pareto front of NSGA-II algorithm in three objective function

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For the chosen solution that is shown in red color in Fig. 9, the pumping rates of four pumping wells are 553.4, 579.4, 558.6 and 558.2 (m3/day), respectively. The location of pumping wells in the groundwater head and contaminant concentration is shown in Fig. 10. As shown in this figure, two wells with higher pumping rates are located close to in the contaminant source, but one of them is located close to the left boundary (constant head boundary), thus, in addition to minimizing contaminant concentration, the drawdown of groundwater head will also be minimized.

Fig. 10
figure 10

a Groundwater head, b contaminant concentration

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The results of four-objective optimization problem (hypothetical aquifer)

In four-objective optimization problem, we find the best location of wells to find the minimum of contaminant concentrations, the drawdown of the aquifer head, mean pumping rate and remediation period.

In the last step, the optimization problem with four-objective functions is also investigated. In single-objective to three-objective problems, the remediation period was equal to 10 years. The fourth objective function is the remediation period minimization.

Population size and maximum iteration for NSGA-II algorithm were 10 and 20, respectively. NSGA-II algorithm was run five times. The results of the second run of NSGA-II algorithm are shown in Fig. 11. Each solution is shown as a solid curve in two-dimensional space, and the 3rd dimension (Contaminant Concentration) is shown by size of curve and the 4th dimension (remediation period) is shown by the color bar graph. Each of these solutions can be a groundwater remediation alternative for the PAT system. As shown in this figure, changing the remediation policy from one option to another can reduce the remediation period by almost 200 days without exceeding the permissible limit of contaminant concentrations at the end of the remediation period.

Fig. 11
figure 11

Pareto front of NSGA-II algorithm in four objective functions

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In Table 5, the solutions of the NSGA-II algorithm are presented with the minimum of each objective function. As shown in this table, the first solution is the best solution. Because in this solution not only the first objective function but also four objective function are minimized. In rows 2 and 3 of Table 5, it can be seen, find a better location for pumping wells can reduce the mean pumping rate and remediate contaminants at the same time and the same amount.

Table 5 Comparison of solutions with the minimum of each objective function
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The results of real aquifer modeling

The real aquifer has been investigated using FEM and rectangular element. The results of groundwater flow and Chloride transport modeling for 5 years are presented in Fig. 12a and b. The real aquifer is considered with pumping wells.

Fig. 12
figure 12

Groundwater head and contaminant distribution after 5 years

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The results of one-objective optimization problem (real aquifer)

One-Objective optimization problem is solved to find the pumping wells’ location in order to obtain optimum (near-optimum) values for the first objective function. In this case, the contamination concentration affects only a small part of the aquifer. Therefore, it is not necessary to explore the entire aquifer domain. As heuristic information, we limited the search domain to determine the best location of the pumping wells. The new search domain is around contaminant distribution. Therefore, members of the population in the genetic algorithm search only a domain 1600 × 2200 m. The new study domain is a structural grid in 12 × 9 and 88 squared elements. The distance between nodes is 200 m. Also, we amused 20 nodes around the contaminant source for calculating the first objective function. The remediation period is considered 3 years after preventing contaminant seepage. Monitoring nodes were also considered around the contaminant source. Population size and maximum iteration for genetic algorithms were 10 and 20, respectively. The results of GA are shown in Fig. 13.

Fig. 13
figure 13

The convergence trend of GA algorithms

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The pumping wells’ location, with 800 m3/day, (red circle marker) and the concentration of Nitrate-Nitrogen in the aquifer domain after optimization during 3 years of remediation are shown in Fig. 14. As can be seen in this figure, the best location of pumping wells where the concentration of pollutant is maximum at the beginning of the remediation period and a round of that point. When the contamination mass is moving to the right of the aquifer, the algorithm located one well in front of that to remove most of the contaminants. But there is pumping well for agricultural use in this location. Therefore, in multi-objective problems, nodes in which there are wells for agricultural use, similar to nodes located at the aquifer boundary, are removed from the selectable nodes.

Fig. 14
figure 14

Contaminant distribution after 3 years remediation

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The results of two-objective optimization problem (real aquifer)

In Two-Objective optimization problem, we find the best location of wells to find the minimum of contaminant concentrations and drawdown of the aquifer head. In this case, the nodes with pumping wells for agriculture use is not in selectable nodes set. The parameter of NSGA-II in the real case studies is the same hypothetical case studies. The number of solutions in the Pareto-optimal front was 9. The results of these runs are shown in Fig. 15.

Fig. 15
figure 15

Pareto fronts in NSGA-II algorithm (Real Aquifer)

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After that, we selected one of the solutions in optimal pareto-front. Groundwater flow and Nitrate-Nitrogen concentration for this solution are shown in Fig. 16a and b, respectively.

Fig. 16
figure 16

Selected solution a Groundwater head, b contaminant concentration

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The results of three-objective optimization problem (real aquifer)

In Three-Objective optimization problem, we find the best location of wells to find the minimum of contaminant concentrations, the drawdown of the aquifer head and mean pumping rate. In this case, NSGA-II algorithm is able to get any value as a pumping rate in a range of [500,800]. In Fig. 17, the Pareto front of NSGA-II algorithm is shown.

Fig. 17
figure 17

Pareto front of NSGA-II algorithm (Real Case Study)

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The results of four-objective optimization problem (real aquifer)

In four-objective optimization problem, we find the best location of wells to find the minimum of contaminant concentrations, drawdown of the aquifer head, mean pumping rate and remediation period.

Population size and maximum iteration for NSGA-II algorithm was 10 and 20, respectively. The results of NSGA-II algorithm is shown in Fig. 18. Each solution is shown as a solid curve in two-dimensional space, and the 3rd dimension (drawdown of the aquifer head) is shown by the size of the curve and the 4th dimension (remediation period) is shown by the color bar graph. Each of these solutions can be a groundwater remediation alternative for the PAT system. As shown in this figure and due to \(\Delta T = 5\,\,{\text{days}}\), changing the remediation policy from one option to another can reduce the remediation period by almost 50 days without exceeding the permissible limit of contaminant concentrations at the end of the remediation period.

Fig. 18
figure 18

Pareto front of NSGA-II algorithm (Real Aquifer)

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Conclusion

In this study, hybrid optimization-simulation models, GA-FEM and NSGA-II-FEM are used to solve a groundwater remediation problem by PAT method. The hypothetical aquifer was almost similar to the aquifer considered by Sharif et al. 2008. The optimization problem with these models was solved in single-objective and multi-objective cases. In solving the single-objective optimization problem, the objective was to determine the optimal location of four pumping wells with a rate of 600 m3/day to minimize the mean of contaminant concentration. The results indicated that the GA-FEM model has a good efficiency with 41.79 \(\left( {{\mu g}/{\text{Lit}}} \right)\) and 126.04 (mg/lit) for a hypothetical and real case study, respectively. In the two-objectives problem, the drawdown of groundwater head was also considered. The optimization problem was investigated with four pumping wells at a constant pumping rate of 600 m3/day. The results indicated that the number of solutions in the Pareto front was not high, due to the constant pumping rate and a low number of selectable nodes in aquifer the domain. However, among the solutions provided by each model, the most efficient solution was able to reduce the contaminant concentration in the aquifer to the standard pollutant concentrations and, conversely, minimize the drawdown of groundwater head. In general, among the Pareto-optimal solutions, the solution selected should establish a balance between the two objective functions.

When considering the three-objectives optimization problem, and given of the possibility of selecting any value for the pumping rate of wells, solutions in the Pareto front were more than the two-objective problem. In some cases, even the entire population of the NSGA-II-FEM model appeared in the Pareto front. In the last step, four-objectives problem was also examined, in which the fourth objective function was minimizing the remediation period. The results of this section indicated that by changing the PAT policy, the remediation period can be reduced to 7 and 2 months for the hypothetical and real case study, respectively, without exceeding the permissible amount of pollutants at the end of the remediation period.