Introduction

Groundwater quality has become a serious problem in some parts of the world due to contamination by chemicals and microorganisms derived from agricultural activities, urban development, and domestic and industrial wastes. The quantity and quality of groundwater in the aquifers are also often degraded as a result of over exploitation. To track the spread or migration of a contaminated plume in space and time, groundwater flow models coupled with a contaminant transport model are used.

Analytical solutions for groundwater problems are often not suitable for complex field conditions. Numerical methods such as the finite difference method (FDM) and finite element method (FEM) are conventional techniques used to solve complex field problems with irregular boundaries. These methods (for FDM, see Zou and Parr 1995; Chakraborty and Ghosh 2013; Purnaditya et al. 2018; and for FEM, see Javadi and Al-Najjar 2007; Patil and Chore 2014), are used widely for groundwater modeling but require grid construction and preprocessing, which require much effort and computational time. Also, these methods involve model domains that require large storage space (Toutip 1999). Katsifarakis et al. (1999) used the boundary element method (BEM) with a genetic algorithm proposed by Goldberg (1989) and Michalewicz (1996) to determine the transmissivities in zoned aquifers; in remediation, this minimizes the pumping cost and facilitates control of the movement of the contaminant plume. The main drawback to this method is the need to discretize the domain into a series of internal cells to deal with the terms not taken to the boundary by the application of a fundamental solution (Toutip 1999). The dual reciprocity method (DRM) is essentially a generalized way of constructing particular solutions that can be used to solve nonlinear and time-dependent problems as well as to represent any internal source distribution. The dual reciprocity boundary element method (DRBEM) was introduced to simulate groundwater flow and transport as effectively as other numerical methods with minimal numerical dispersion (Eldho and Rao 1997). It discretizes the domain boundary only, thereby reducing the quantity of data necessary to solve a problem. The DRBEM is found to give less numerical dispersion and easier data handling (Eldho et al. 1999).

Creating connections between the mesh nodes is a major task in grid-based methods. This is overcome by the use of meshless models. The meshless method is a numerical method of solving partial differential equations (PDE) in the simulation domain without using a predefined mesh (Liu and Gu 2005). There is no need for generating the connection information between the nodes. The approximate or shape or interpolation function is a function used to approximate the state variable at any point for the nodal values (Liu and Gu 2005). The nodes are spaced at either regular or irregular intervals called nodal spacing. Based on the formulation procedure, meshless methods are divided into two types: strong form and weak form. The assumed shape function in the strong form should have a higher degree of consistency, allowing it to be differentiated up to the order of the PDE. Further, the weak form meshless methods, such as the local radial point interpolation meshless method (LRPIM), require a lower level of consistency in the shape function, which can be obtained using either the variational or weighted residual method. The weak form has an advantage over the strong form in that after discretization, the order of the governing equation is reduced, making Neumann boundary conditions easier to implement and the problem accuracy is better or comparable with FDM results. In the meshless method, it is simple to incorporate any modifications at a later time by simply adding nodes to the domain and to avoid remeshing whenever a new feature, such as a pumping/recharge well, is added to the domain.

There are different types of meshless methods as given in Liu and Gu (2005) to solve the contaminant transport problems in groundwater. The recent developments and applications of the meshless method in groundwater flow and transport work are listed in Table 1.

Table 1 Meshless methods in solving groundwater problems
Full size table

A local weak form meshless method like LRPIM has been used so far in solid and fluid mechanics problem (Wu et al. 2005; Liu and Gu 2001). LRPIM has been used to solve the groundwater flow problems in a confined aquifer (Swetha et al. 2022). In this study, LRPIM is developed further for coupled flow and transport simulation of an unconfined aquifer. LRPIM employs the radial basis function (RBF) for shape function approximation and it is a weak form approach for the discretization of governing equations. The support domain, which might be circular, square or rectangular, is the region chosen for meshless approximation with a point of interest, called the field node. This research considers a circular support domain (with just one parameter to deal with radius). Gauss quadrature points are used to carry out the integration. The area of influence of each quadrature point is the subdomain. In this study, one-dimensional (1D) and two-dimensional (2D) groundwater contaminant transport problems are solved using the developed LRPIM models. The benchmark problems are used to verify the developed model with available analytical solutions. Further, the model is applied to simulate the groundwater flow and contaminant transport for a real field aquifer problem.

Governing equations and boundary conditions

The 2D governing equation for groundwater flow in an unconfined aquifer is given as (Bear 2007):

$$\frac{\partial }{\partial x}\left[{K}_{\textrm{x}}h\frac{\partial h}{\partial x}\right]+\kern0.5em \frac{\partial }{\partial y}\left[{K}_{\textrm{y}}h\frac{\partial h}{\partial y}\right]={S}_{\textrm{y}}\frac{\partial h}{\partial t}+\sum_{w=1}^m{Q}_{\textrm{w}}\delta\ \left(r-{r}_w\right)-\kern0.5em f$$
(1)

In the case of isotropic but heterogeneous conditions, Kx and Ky are the same—i.e., Kx = Ky = K(x, y).

For a transient state, the initial condition is used as follows.

$$h\left(x,y,0\right)={h}_0\left(x,y\right)\kern1.75em \textrm{where}\kern1.75em x,y\kern0.75em \in \varOmega$$
(2)

The boundary conditions used here are as follows:

$${\displaystyle \begin{array}{c}h\left(x,y,t\right)={h}_1\left(x,y,t\right)\kern1.5em \textrm{where}\kern0.75em x,y\kern1em \in \kern1em \partial {\varOmega}_1\ \left(\textrm{Dirichlet}\ \mathrm{boundary}\right),\textrm{and}\\ {} Kh\frac{\partial h}{\partial n}=q\left(x,y,t\right)\kern1.75em \textrm{where}\kern1.75em x,y\kern1em \in \kern1em \partial {\varOmega}_2\ \left(\textrm{Neumann}\ \mathrm{boundary}\right)\end{array}}$$
(3)

Here, h(x, y, t) is the piezometric or groundwater head [L]; δ(r − rw) is the Dirac delta function; r ≡ (x, y) is the point lying inside the domain; rw ≡ (xw, yw) is the location of the well; h0(x, y) is the initial head in the flow domain [L]; K is the hydraulic conductivity [LT−1]; Sy is the specific yield; Kx, Ky are the hydraulic conductivities [LT−1] along the x and y directions, respectively; Qw is the source or sink term [LT−1]; q(x, y, t) is the known inflow or outflow rate [L2T−1]; m is the number of sources; f is the recharge rate [LT−1]; \(\frac{\partial }{\partial n}\) represents the normal derivative to the boundary; h1(x, y, t) is the known head value at the boundary [L]. The flow region is represented by Ω, while the boundary is denoted by ∂Ω.

Here Darcy’s law is used to couple the groundwater flow model with the contaminant transport model. The expression to evaluate the pore velocity is given as (Bear and Cheng 2010):

$${V}_{\textrm{x}}=-\frac{K_{\textrm{x}}}{\acute{\upvarepsilon}}\frac{\partial h}{\partial x}\ \textrm{and}\kern0.5em {V}_{\textrm{y}}=-\frac{K_{\textrm{y}}}{\acute{\upvarepsilon}}\frac{\partial h}{\partial y}$$
(4)

where h is the piezometric head [L]; έ is the effective porosity; Vx, Vy are the pore velocities [LT−1] along the x and y directions, respectively. The velocities obtained from the preceding expressions are used as input to the contaminant transport model.

The governing equation for contaminant transport in groundwater is given as (Freeze and Cherry 1979; Wang and Anderson 1995; Desai et al. 2011):

$$R\frac{\partial C}{\partial t}=\frac{\partial }{\partial x}\left[{D}_{\textrm{x}\mathrm{x}}\frac{\partial C}{\partial x}\right]+\frac{\partial }{\partial y}\left[{D}_{\textrm{y}\mathrm{y}}\frac{\partial C}{\partial y}\right]-\frac{\partial }{\partial x}\left({V}_{\textrm{x}}C\right)-\frac{\partial }{\partial y}\left({V}_{\textrm{y}}C\right)-\frac{c^{\prime }w}{\varepsilon b}- RC\lambda +\frac{q_{\textrm{w}}C}{\varepsilon }$$
(5)

where \(R=1+\left[\frac{\rho_{\textrm{b}}{K}_{\textrm{d}}}{\varepsilon}\right]\)

Here, ρb is the media bulk density [ML−3]; Kd is the sorption coefficient; qw is the volumetric rate of pumping from source; Vx, Vy are the pore velocity along the x and y directions, respectively [LT−1]; Dxx, Dyy are the hydrodynamic dispersion coefficients along the longitudinal (x) and transverse (y) directions, respectively [L2T−1]; C is the concentration of the dissolved species [ML−3]; λ is the reaction rate constant [T−1]; w is the elemental recharge rate with solute concentration c′; ε is the porosity; b is the aquifer thickness [L]; R is the retardation factor.

The initial and boundary conditions considered are as follows:

$$C\left(x,y,0\right)={C}_{\textrm{O}}\left(x,y\right)$$
(6)
$$C=C_1\;\left(\text{constant}\;\text{head}\right),\;\text{for}\;\text{Dirichlet}\;\text{or}\;\text{essential}\;\text{boundary}\;\text{condition}$$
(7)
$$\begin{array}{c}D\frac{\partial C}{\partial n}=g\;\left(\text{inward}/\text{outward}\;\text{flow}\right),\;\text{for}\;\text{Newmann}\;\text{or}\;\text{natural}\;\text{boundary}\;\text{condition}\\\text{and}\;D\frac{\partial c}{\partial n}=0\left(\text{No}\;\text{flow}\right),\;\text{for}\;\text{no}\;\text{flow}\;\text{boundary}\;\text{condition}\;\end{array}$$
(8)

LRPIM formulation

Radial point interpolation (RPIM) and shape function evaluation

To overcome the singularity concerns, several RBFs such as multiquadratics (MQ), exponential or Gaussian, thin plate spline, and logarithmic radial basis functions were used (Liu and Gu 2005). It is found that both MQ-RBF and exponential RBF gave equal accuracy (Singh et al. 2016). For shape function evaluation, MQ-RBF is used in this study. MQ-RBF has the following expression:

$${R}_i\left(x,y\right)={\left[{r}_i^2+\kern0.5em {C}_{\textrm{s}}^2\right]}^q={\left[{\left(x-{x}_i\right)}^2+{\left(y-{y}_i\right)}^2+{C}_{\textrm{s}}^2\right]}^q$$
(9)

where q and Cs are the shape parameters of the radial basis function and i is the node considered. In this study, the parameter q has been kept at 1.03 as given in Liu and Gu (2005). The shape parameter of the MQ-RBF, Cs, is usually defined in terms of a characteristic length (dc) (Liu and Gu 2005) i.e.

$${C}_{\textrm{s}}=\kern0.5em {\alpha}_{\textrm{c}}{d}_{\textrm{c}}$$
(10)

The characteristic length (dc) in the local support domain is determined by the nodal spacing. The value of Cs is set as 4 × dc for optimal results based on a trial-and-error method (Swetha et al. 2022). The derivatives of C(x, y) at any point (x, y) in a support domain with N neighboring nodes are derived as given in the following: (Swetha et al. 2022)

$$\frac{\partial C}{\partial x}=\frac{\partial {\varPhi}^T}{\partial x}{C}_i=\sum_{i=1}^N\frac{\partial {\varPhi}_i}{\partial x}{C}_i\kern1.5em ;\kern1.5em \frac{\partial^2C}{\partial {x}^2}=\frac{\partial^2{\varPhi}^T}{\partial {x}^2}{C}_i=\sum_{i=1}^N\frac{\partial^2{\varPhi}_i}{\partial {x}^2}{C}_i$$
(11)
$$\frac{\partial C}{\partial y}=\frac{\partial {\varPhi}^T}{\partial y}{C}_i=\sum_{i=1}^N\frac{\partial {\varPhi}_i}{\partial y}{C}_i\kern1.5em ;\kern1.5em \frac{\partial^2C}{\partial {y}^2}=\frac{\partial^2{\varPhi}^T}{\partial {y}^2}{C}_i=\sum_{i=1}^N\frac{\partial^2{\varPhi}_i}{\partial {y}^2}{C}_i$$
(12)

LRPIM formulation for contaminant transport in porous media

The governing equation for the unconfined aquifer is nonlinear and is linearized by assuming the initial value for h and convergence criteria as stated in Meenal and Eldho (2011). Here the weighted residual method is used for the LRPIM formulation of Eq. (1) as proposed by Liu and Gu (2005)

$${\int}_{\varOmega_{\textrm{s}}}\nabla \bullet \left(K\cdot h\cdot \nabla h\right)v. d\varOmega =\kern0.5em {\int}_{\varOmega_{\textrm{s}}}\sum_{w=1}^N{Q}_{\textrm{w}}\delta \left(r-{r}_{\textrm{w}}\right)v. d\varOmega -{\int}_{\varOmega_{\textrm{s}}} fv. d\varOmega +{\int}_{\varOmega_{\textrm{s}}}{S}_{\textrm{y}}\frac{\partial h}{\partial t}v. d\varOmega$$
(13)

where, \(\nabla =i\frac{\partial }{\partial x}+j\frac{\partial }{\partial y}\) is the vector differential operator such that \(\nabla h=i\frac{\partial h}{\partial x}+j\frac{\partial h}{\partial y}\) which represents the gradient in piezometric head and v is known as the Heaviside step function or weight function (Liu and Gu 2005).

The divergence theorem, subdomain method (Salih 2016) and time domain discretization (Richtmyer and Morton 1967) are applied to Eq. (13) for further simplification. An implicit forward difference formula with θ as 0.5 (Crank-Nicholson scheme) is used for the time discretization. The detailed formulation procedure as given in Swetha et al. (2022) was followed to get the final matrix form of the equation. The matrix form of Eq. (13) after simplification is given as,

$$\left[\bar {\textrm{A}}\right]\left\{{h}^{t+\Delta t}\right\}=\left[F\right]$$
(14)

where

$$\bar {\textrm{A}}={K}_j\left({\int}_{{{\partial \Omega}}_s}\left(\frac{\partial {\varPhi}_i(r)}{\partial x}{h}_i^{t+\Delta t}{n}_{\textrm{x}}+\frac{\partial {\varPhi}_i(r)}{\partial y}{h}_i^{t+\Delta t}{n}_{\textrm{y}}\right){d\Omega}_j\right)\uptheta -{\int}_{\varOmega_{\textrm{s}}}\left[\frac{S_{\textrm{y}}}{\Delta t}\right] d\varOmega$$
(15)
$$ \begin{aligned}F=&{\int}_{\varOmega_{\textrm{s}}}\sum_{w=1}^N{Q}_{\textrm{w}}\delta \left(r-{r}_{\textrm{w}}\right) d\varOmega -{f}_j{A}_{\Omega}-\kern1.25em {K}_j\left({h}_i^t{\int}_{{{\partial \Omega}}_{\textrm{s}}}\left(\frac{\partial {\varPhi}_i(r)}{\partial x}{h}_i^t{n}_{\textrm{x}}+\frac{\partial {\varPhi}_i(r)}{\partial y}{h}_i^t{n}_{\textrm{y}}\right){d\Omega}_j\right)\\&\left(1-\theta \right)-{\int}_{\varOmega_{\textrm{s}}}{h}_i^t\left[\frac{S_{\textrm{y}}}{\Delta t}\right] d\varOmega\end{aligned} $$
(16)

where j represents the index of the node considered; i is the neighbouring node number; nx and ny are the normal derivatives in x and y directions, respectively; θ is the relaxation parameter (ranging between 0 and 1, 0.5 for Crank-Nicholson scheme) and AΩ is the subdomain area. Here, the Cartesian coordinate is used in the study.

The head value obtained by solving Eq. (14) is used in Eq. (4) to calculate the pore velocity and couple the flow model with the contaminant transport model.

In a similar way, the transport model is developed by discretizing Eq. (5). The detailed formulation procedure given in Swetha et al. (2022) is followed for Eq. (5) in a similar procedure to get the matrix form as follows.

$$\left[M\right]\left\{{C}^{t+\Delta t}\right\}=\left[\hat{F}\right]$$
(17)

where

$$ M=\left(\left({D}_j\left(\ {\int}_{{{\partial \Omega}}_s}\left(\frac{\partial {\varPhi}_i(r)}{\partial x}{n}_{{x}}+\frac{\partial {\varPhi}_i(r)}{\partial y}\ {n}_{{y}}\right)d\Omega \right) -V\left(\ {\int}_{\Omega_s}\left(\frac{\partial {\varPhi}_i(r)}{\partial x}+\frac{\partial {\varPhi}_i(r)}{\partial y}\ \right)d\Omega \right)\right)\uptheta -{\int}_{\varOmega_{\textrm{s}}}\frac{R}{\Delta t} d\varOmega \right)$$
(18)
$$\hat{F}=\left(\left(\left(-{D}_j\right)\left({\int}_{{{\partial \Omega}}_s}\left(\frac{\partial {\varPhi}_i(r)}{\partial x}{n}_{\textrm{x}}+\frac{\partial {\varPhi}_i(r)}{\partial y}\ {n}_{\textrm{y}}\right)\ d\Omega \right)+V\ \left({\int}_{\Omega_s}\left(\frac{\partial {\varPhi}_i(r)}{\partial x}+\frac{\partial {\varPhi}_i(r)}{\partial y}\ \right)d\Omega \right)\ \right)\left(1-\theta \right)-{\int}_{\varOmega_{\textrm{s}}}\frac{R}{\Delta t} d\varOmega +{\int}_{\varOmega_{\textrm{s}}} R\lambda d\varOmega -{\int}_{\varOmega_{\textrm{s}}}\frac{q_w}{\varepsilon } d\varOmega \right){C}_i^t+{\int}_{\varOmega_{\textrm{s}}}\frac{c^{\prime }w}{\varepsilon b} d\varOmega$$
(19)

Here also, an implicit forward difference scheme is used for the discretization.

Model development and verification

A flow chart for the developed groundwater flow and contaminant transport model is shown in Fig. 1. The following step-by-step approach is used for developing the LRPIM coupled flow and transport model, called LRPIM-CFT.

  1. Step 1.

    Aquifer parameters such as longitudinal and transverse dispersion coefficient, transmissivity, contaminant concentration, porosity, aquifer thickness, and discharges and recharges are collected from field observations.

  2. Step 2.

    Model parameters such as shape of the support domain, size of the subdomain and the support domain, time-step size, and total simulation period are provided as input.

  3. Step 3.

    Nodes are generated within the problem domain and at the boundary. The initial value of groundwater head and contaminant concentration are applied throughout the problem domain.

  4. Step 4.

    The shape function is calculated for every node and variable in the support domain using the MQ-RBF shape function. Swetha et al. (2022) suggested a circular subdomain and support domain with an optimized radius of 0.6 and 4 times the nodal spacing, respectively. The same is used in all the cases considered in this study.

  5. Step 5.

    The boundary conditions (Dirichlet and Neuman) are used to solve the model for groundwater (or piezometric) head throughout the domain.

  6. Step 6.

    The piezometric head values obtained from the flow model are used to calculate the pore velocity using Darcy’s law. The calculated velocity is used as an input parameter to the contaminant transport model.

  7. Step 7.

    The M and F matrices are formed using Eqs. (18) and (19). The resulting global system of equations (Eq. (17)) is then solved using the direct approach of matrix inversion to get the concentration values throughout the domain.

  8. Step 8.

    The concentration values obtained at the current time step are used as the initial concentration for the next time step. Step 7 is then repeated throughout the simulation period.

Fig. 1
figure 1

Flow chart for the groundwater flow and transport model

Full size image

Based on the aforementioned procedure, a computer model, LRPIM-CFT, was developed in the MATLAB platform to solve 1D and 2D problems using the flowchart as shown in Fig. 1. The data handling is carried out by MATLAB’s built-in functions and formats.

Model verification: case 1—pure diffusion in 1D

In this case, a 1D diffusion-dominated contaminant transport problem with an analytical solution is solved using LRPIM-CFT. A domain length of 100 m is considered, with 101 nodes at a nodal spacing of 1 m. A continuous contaminant source with a concentration of 1 mg/L is applied at one end (left) and no flow boundary condition is applied at the other end, with an initial concentration of zero at all other nodes as shown in Fig. 2a. The parameters used for this case are Dxx= 0.0005 m2/day and V = 0 m/day. For this problem an analytical solution was given by Crank (1979) as follows:

$$C={C}_0\operatorname{erfc}\left(\frac{x}{\sqrt{4{D}_{\textrm{L}}t}}\right)$$
(20)
Fig. 2
figure 2

a Aquifer and nodal configurations for pure diffusion; b Concentration variation along the length after 10,000 days

Full size image

The solute transport model is developed with a time step of 1 day over a simulation period of 10,000 days. The support domain is four times the nodal spacing in size (Fig. 2a). The result obtained from the meshless LRPIM model is compared with the analytical solution and is shown in Fig. 2b. It demonstrates that the meshless LRPIM and analytical solution results match very well. When the concentration simulated after 10,000 days was compared to the analytical solution at various distances from the source, it was found to be in good agreement.

Further, a sensitivity study is carried out by varying the value of the dispersion coefficient (Dxx) by ±50%, around 0.0005 m2/day, and the results are plotted in Fig. 3. When the value of dispersion is increased (Dxx = 0.00075 m2/day), the concentration value also increases throughout the length of the domain. When the dispersion coefficient is decreased (Dxx = 0.00025 m2/day), the concentration along the length of the domain also decreases as compared to the concentration when Dxx = 0.0005 m2/day. The same trend is observed in both the meshless LRPIM and analytical solution as shown in Fig. 3a,b. The simulation is repeated for 10,000 days with different time steps of 2 and 5 days over the simulation period to check the numerical stability. To quantify the agreement between meshless LRPIM and the analytical solution, the following root mean square error (RMSE) is used:

$$\mathrm{RMSE}=\sqrt{\frac1p{\textstyle\sum_{i=1}^p}\left(C_i^{\mathrm{analytical}}-C_i^{\mathrm{predicted}}\right)^2}\mathrm{where},\;i=\mathrm{nodes}\;1,\;2,\;3\;\dots\;p$$
(21)
Fig. 3
figure 3

Comparison of (C/C0) for various dispersion values along the domain length after 10,000 days using: a LRPIM; b Analytical solutions

Full size image

The RMSE values were calculated between the analytical and simulated values with different time steps. It is found that the RMSE obtained between the analytical and predicted values for the aforementioned cases are very small suggesting good agreement between the meshless LRPIM model and the analytical solution. In this case, the changes in the dispersion curve are noticed in the first 15 nodes. Hence, RMSE is calculated using the first 15 nodes and is found to be 0.01. Further, a study was also carried out by changing the nodal spacing in the domain to 5 and 10 m. In this case, the RMSE value increased to 0.02 and 0.05 when the nodal spacing was increased to 5 and 10 m, respectively.

Model verification: case 2—instantaneous source (1D)

In this case, an instantaneous source (injected mass, \(\overset{\check{} }{M}\)) is injected at x0 with zero initial concentration in the rest of the domain. The analytical solution, as given by Marino (1974), is:

$$C\ \left(x,t\right)=\frac{\overset{\check{} }{M}}{\sqrt{4{\pi D}_{\textrm{x}}t}}\exp \left[\frac{-{\left(x-{x}_0-{v}_{\textrm{x}}t\right)}^2}{\sqrt{4{D}_{\textrm{x}}t}}\right]$$
(22)

where vx is the flow velocity in the x direction, Dx is the dispersion coefficient in the x direction, and t is time.

For a contaminant injection of 500 mg/L at x0 = 50 m, as in Fig. 4, concentration variation at the end of the first and second year is evaluated. The other parameters used for this case are vx = 0.0259 m/day, Dx = 0.036 m2/day and a time-step size of 1 day. Over a domain length of 100 m, numerical simulation is performed utilising 101 nodes with a nodal spacing of 1 m. Figure 5a,b depict the concentration fluctuation after 1 and 2 years, respectively. The peak concentration in the aquifer has shifted nearer to 60 m when the first year ends and close to 70 m when the second year ends, as shown in Fig. 5a,b. The figures show a decline in peak value of roughly 12 mg/L from the end of the first to the end of the second year. From Fig. 5, it can be seen that the results compared to the analytical solution are in good agreement.

Fig. 4
figure 4

Aquifer configuration for an instantaneous source

Full size image
Fig. 5
figure 5

Contaminant concentration variation: a after 1 year; b after 2 years

Full size image

Model verification: case 3—2D contaminant transport

In this case, a continuous source of 5 ppm is injected throughout the left boundary of the aquifer with a domain size 50 m × 50 m (Fig. 6a). The top and bottom sides of the aquifer are considered to have a zero concentration boundary. A Neumann boundary condition with zero concentration is considered on the right side of the aquifer as shown in Fig. 6a. Figure 6b shows the nodes generated with a nodal spacing (∆x or ∆y) of 1.25 m in the problem domain. The simulation parameters are Dxx = 0.1 and Dyy = 0.1 m2/day, with a resultant velocity (v) of 0.1 m/day. The contaminant source is a continuous release that lasts for 300 days. The simulation is run for 300 days with 1-day time steps.

Fig. 6
figure 6

a Aquifer configurations for 2D transport; b Nodal representation of the problem domain

Full size image

Since the source is released continuously for 300 days, the contaminant plume starts moving along the length of the domain as in Fig. 7a. Figure 7b shows the comparison of concentration curves obtained from LRPIM with the analytical solution at selected nodes in the domain after 300 days. The plume reaches the right boundary after 300 days from the beginning of the simulation. The results were compared with the analytical solution given by Goltz and Huang (2017) as shown in Fig. 7a,b.

Fig. 7
figure 7

a Concentration variation after 300 days; b Comparison of concentration curves obtained from LRPIM with the analytical solution at selected nodes

Full size image

Figure 7b depicts the concentration build-up with time at three nodes: node 361 (10 m, 10 m), node 689 (20 m, 10 m), and node 1017 (30 m, 10 m). These nodes were chosen for the purposes of comparison with the analytical solutions. The LRPIM solutions are found to be in good agreement with the analytical solutions having an RMSE value of 0.17 (calculated for all the nodes in the model domain).

Further, a sensitivity study was carried out by varying the values of dispersion and velocity. It is seen that after increasing (2 × D) the dispersion value, the changes observed in the result is very small as shown in Fig. 8a. However, when the velocity is increased to tgroundwater flow and MT3DMS wice the earlier values (2 × v, Fig. 8b) the plume reaches the earlier maximum concentration in almost half the time it took earlier. This indicates that the problem is advection dominant with a Peclet number \(\left(\frac{v}{{~}^{D}\!\left/ \!{~}_{\Delta x}\right.}\right)\) of 1.25. Here, v is the velocity, D is the dispersion coefficient and ∆x is the consecutive nodal distance.

Fig. 8
figure 8

Concentration variation at different points in the domain: a Sensitivity study on dispersion; b Sensitivity study on velocity

Full size image

Application to a real field case study

An unconfined aquifer in the Kodaganar River basin in the Dindigul district of Tamil Nadu, India, was studied using the data provided in Mondal and Singh (2012). The study area lies between 10̊13′44′′−10̊26′47′′N latitudes and 77̊ 53′ 08′′−78̊ 01′ 24′′E longitudes. The location map of the study area is given in Fig. 9. During the previous four decades, unregulated discharge of effluents by 80 operating tanneries has contaminated both surface water and groundwater in this area (Peace Trust 2000). The simulation domain has a total area of approximately 75.56 km2. The aquifer with its boundary condition is shown in Fig. 10a.

Fig. 9
figure 9

Location of study area. Note: The hydrogeological location map is taken from the UN map of the world (UN 2022)

Full size image
Fig. 10
figure 10

a Aquifer configuration for the real field case; b Hydraulic conductivity (K) and specific yield (Sy) zones in the model domain; c Recharge zones in the model domain

Full size image

The average annual rainfall during the period of 1971−2007 was about 905.3 mm. Archaean granites and gneisses dominate the study area, which is intruded by dykes (Chakrapani and Manickyan 1988). In this region, the most common groups of rocks are granite, granodiorites, gneissic granite, and gneisses (Krishnan 1982). These formations cause rock weathering by forming a network of joints and fractures, which allows groundwater to move and be stored more easily. The weathered zone’s estimated thickness ranges from 5 to 27 m. The aquifer receives both direct and indirect recharge from rainfall and seepage from irrigated areas and irrigation tanks. At the southern and eastern boundaries, constant head boundaries are applied. The Kodaganar River forms the western and northwestern boundary. Throughout the simulation, the values used in the boundary are kept constant. As shown in Fig. 10b, permeability and specific yield of the study area are classified into three zones, whereby the hydraulic conductivity of the southern part is 4.5 m/day, the central part is 6 m/day, and the northern part is 9 m/day. Likewise, the southern portion has a specific yield (Sy) of 0.015, the central portion has a yield of 0.02 and the northern portion has a yield of 0.03. The effective and total porosity are given as 0.15 and 0.20, respectively, throughout the region (GWREC 1997).

In this area, 80% of the water extracted is used for irrigation, while the remaining 20% is used for domestic and industrial purposes. Water abstraction for irrigation is dispersed across the landscape, whereas domestic and industrial abstractions are concentrated in and around towns. The total abstraction from the study area is approximately 23,588 m3/day (PWD 2000). The entire aquifer region is divided into five zones with recharge values of 80, 120, 160, 180, and 250 mm/year (Mondal and Singh 2010), as shown in Fig. 10c.

In the LRPIM modeling approach, an aquifer model with a regular nodal spacing of 250 m in both the x and y directions is created. A total of 1,310 nodes were employed as indicated in Fig. 11a. To get the groundwater heads, the LRPIM groundwater flow model (Swetha et al. 2022) is used. The groundwater head values obtained are given in Fig. 12. The groundwater flow velocity is calculated from the resultant heads using Darcy’s law.

Fig. 11
figure 11

a Nodal representation for the LRPIM method; b Grid representation for MODFLOW/MT3DMS (red mark represents constant head boundary; yellow mark represents source location)

Full size image
Fig. 12
figure 12

Comparison of groundwater head values

Full size image

The concentration of chloride (Cl−) in the surface effluent is 17,230 mg/L (Basker 2000). Only 30% of this enters the groundwater system, while the other 70% is carried away by runoff or retained in the unsaturated zone. The amount of chloride entering the groundwater system from tannery effluents was assumed to be 5,169 mg/L (Peace Trust 2000). The longitudinal and transverse dispersivity is taken as 6.86 m and 0.686 m, respectively. The initial concentration throughout the study area is obtained by interpolating the chloride values from the observation well during April 2001 as shown in Fig. 13a.

Fig. 13
figure 13

a Initial chloride concentration; b Chloride concentration comparison after 10,000 days (case A); c Chloride concentration comparison after 10,000 days (case B)

Full size image

Two cases were considered for the study. In case A, continuous seepage of the chloride contaminant into the aquifer system throughout the simulation period, and in case B, the chloride contaminant is taken only as an initial condition (no further release for the rest of the simulation period).

Case A

In this case, chloride from the industrial effluent is entering as a continuous source into the groundwater system. The locations of these sources (tanneries) are shown in Fig. 10a. The velocity obtained from the flow model along with other transport parameters are given as an input to the contaminant transport model. It is assumed that a chloride concentration of 5,169 mg/L is entering continuously for 10,000 days. The amount of effluent entering into the aquifer is the same as the recharge rate. The model is run from 1 April 2001 to 30 September 2028, for 10,000 days with a time step of 10 days. For comparison purposes, an aquifer model with the same input parameter is constructed using MODFLOW (McDonald and Harbaugh 1988) for groundwater flow and MT3DMS (Zheng and Wang 1999) for contaminant transport (Aquaveo - GMS10.2 2017). In MODFLOW, the entire study area is divided by constructing a grid of size 250 m × 250 m, as shown in Fig. 11b. The total number of grids in the region is 1209.

The groundwater head contours obtained from the MODFLOW and meshless LRPIM method are plotted in Fig. 12, for comparison purposes. The groundwater head follows the direction of the flow vector in the study area (i.e., higher to lower elevation). To quantify the agreement between the meshless LRPIM and the FDM solution, RMSE is used.

$$\mathrm{RMSE}=\sqrt{\frac1n\sum_i\left(h_i^{\mathrm{analytical}}-h_i^{\mathrm{predicted}}\right)^2},\;i=\mathrm{nodes}\;1,\;2,\;\dots,\;n$$
(23)

A good agreement, with an RMSE value of 1.14, is observed for the LRPIM model with MODFLOW. The value was found by including all the nodes in the model domain. On comparison with MODFLOW, the maximum and minimum ranges of difference obtained are 3.976 and 0.0004 m, respectively. Further, the LRPIM contaminant model is simulated for the case considered for 10,000 days. The concentrations at the end of 10,000 days are plotted in Fig. 13.

For comparison purposes, the chloride concentrations (contaminant plume) obtained at the end of 10,000 days from the MT3DMS and meshless LRPIM method are plotted in Fig. 13b and they are found to be comparable with a normalized RMSE value of 0.136. Here the normalized RMSE is defined as:

$$\textrm{Normalized}\ \textrm{RMSE}=\frac{\textrm{RMSE}}{C_{\textrm{max}}-{C}_{\textrm{min}}}$$
(24)

where Cmax and Cmin are the maximum and minimum concentrations obtained from the MT3DMS model.

Concentration variation at different points in the domain after 10,000 days of continuous source seepage is shown in Fig. 14. These nodes were chosen arbitrarily to represent the behaviour of the concentration throughout the region. The results from both models were comparable and the higher and lower peaks were captured by both models.

Fig. 14
figure 14

Concentration variation at different points in the domain (case A)

Full size image

Case B

In this case, the starting chloride concentration is derived from April 2001 in the model domain. It is presumed that no more effluent is added from any of the sources. The model is run over a simulation period of 10,000 days, with a 10-day time step. The same flow and transport model is constructed with MODFLOW and MT3DMS for comparison purposes. Figure 13c is obtained by plotting the results from the meshless LRPIM and MT3DMS. It is found to have a good agreement with a normalized RMSE value of 0.104. Concentration variation at different points in the domain after 10,000 days for case B is shown in Fig. 15.

Fig. 15
figure 15

Concentration variation at different points in the domain (case B)

Full size image

Discussion

As discussed in earlier sections, the meshless LRPIM-CFT model has been developed for simulating the groundwater flow and contaminant transport in the aquifer considered. The developed model is verified with benchmark problems having analytical solutions. Both 1D and 2D transport problems were used for the LRPIM model verification and found to be satisfactory. Further, a sensitivity analysis was carried out by varying the time step size involved in the simulation period to check the model stability. It is found that the model gives almost the same concentration values, with an RMSE value less than 1. In addition, the impact of advection and dispersion parameters on concentration distribution was studied. The changes in concentration depend more on velocity than dispersion. This suggests that the problem considered here for the study is advection dominant. This study demonstrates that the meshless LRPIM model can be effectively used in groundwater flow and contaminant transport simulations.

The developed LRPIM model is applied to a field case study in a region with tannery industries. For the purpose of comparison, the same simulation was carried out using MODFLOW and MT3DMS. It was observed that the results of meshless LRPIM agree well with MODFLOW and MT3DMS. This demonstrates the effectiveness of the meshless LRPIM approach in simulating the advection-dispersion problems.

In the southern half of the study region, groundwater flows northwest, while in the northern section, groundwater flows in the north and northeast directions. The flow of the contaminant plume takes place from the central region towards the northwestern region. The contaminant plume obtained from case A is found to have a higher concentration nearer to the source, whereas in case B, the plume is found to move along the flow direction. The high recharge rate found in the south and southwestern region moves/pushes the plume towards the northeastern region. The plume is found to touch the boundary of the region by the end of the simulation period (10,000 days).

The chloride concentration, due to the effluent release from the source, is predicted with and without the source for up to 10,000 days. Based on the direction of movement of the contaminant, suitable remediation techniques can be chosen to bring down the contaminant concentration in the aquifer in the study area.

The simulation of a real field problem predicting the concentration after 10,000 days (with 1,000 time steps) using the LRPIM method takes around 279 s to run on a device with an Intel(R) Core 335 (TM) i5 (8265U)–1.6 GHz processor and 16 GB RAM. Meshless LRPIM requires much less preprocessing time than grid-based methods like FEM and FDM. It needs the nodal information only to run the simulation. This eliminates the high computational costs associated with meshing or remeshing. In the strong forms of meshless methods, a higher degree of consistency is needed for the approximation function, such that it has to be differentiable up to the order of the PDE. The LRPIM requires weaker consistency on the approximation function owing to its weak form discretization; however, meshless approaches have a few flaws that need to be resolved. The size and shape of the support domain, as well as the optimal values of the shape parameters, are all subjective. The LRPIM approach cannot capture variation in flow contours when the changes in hydraulic parameters are of the order 103 or larger (Singh 2017) from one zone to another in the problem domain. It is one of the model’s shortcomings. This occurs due to the smoothness of the RBF; however, the RBFs were used to avoid the issue of singularity associated with the polynomials (Liu and Gu 2005). The computational efficiency of LRPIM is comparatively lower because the system matrices (stiffness) are asymmetric in nature, as reported in Liu and Gu (2005).

Conclusions

The LRPIM is a meshless method that uses a local weak form formulation and it has been used to model contaminant movement in an unconfined aquifer in this study. In LRPIM models, the Dirichlet boundary conditions are easily applied because the method satisfies the Kronecker delta property and the state variables are approximated with a multiquadratic radial basis function. Further, the Neumann boundary conditions are effectively managed, as LRPIM is a weak form formulation. The solution obtained using the meshless LRPIM for the coupled groundwater flow and solute transport model was verified with the analytical solutions of benchmark problems. It is found that the meshless LRPIM solution is not significantly affected by time-step size for the 1D problem considered in the study. The purely diffusion transport and advection-dispersion transport problems were also simulated with LRPIM. In addition, the coupled model is used to simulate flow and transport in an unconfined aquifer contaminated by chloride in a large heterogeneous field problem. The model is used to predict the migration of contaminant plumes. The meshless LRPIM results were compared to those of MT3DMS and found to be in good agreement. The results show that meshless LRPIM technique can be used to solve large-scale groundwater flow and solute transport problems with irregular and complex domains. In contrast to FDM and FEM, where complete remeshing is required with changes in domain features such as the introduction of any wells, changes in boundary or various features can be readily incorporated by adding or removing field nodes in meshless LRPIM models. Meshless methods also avoid and save time involved in the remeshing process of adaptive analysis. The developed model can be effectively used in field-level aquifer management and remediation problems.