# Axial Groundwater Contaminant Dispersion Modeling for a Finite Heterogeneous Porous Medium

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Contaminant Transport Model (CTM)

#### 2.1. FDM-Based Solution of the Proposed CTM

#### 2.2. CN method-Based Approximate Solution

^{th}and $\left(n+1\right)$

^{th}levels. To ensure that the resultant finite difference approximations are linearized, the retardation factor $R$ given in Equation (11) is calculated using the previous time level ($n$

^{th}time level) [40]. This yields the following approximation of Equation (11):

#### 2.3. ADI Method-Based Approximate Solution of CTM

Algorithm 1: Axial Input Source Contaminant Transport System |

Input${D}_{x0},{D}_{y0}$: Dispersion coefficients $u,v$: Advection coefficients ${c}_{0}$: initial background source ${c}_{1},{c}_{2}$: constant input sources $\phi $: porosity of soil medium $L,H$: space domains length $t$: time interval $M,N$: Number of spatial nodes in $x$ and $y$ directions $P$: Number of time steps ${m}_{1},{m}_{2}$: Decay parameters Output${C}_{h}\left(x,y,t\right)$: Concentration distribution matrix Step 1: Spatial transformation:Transform old transformation $x,y$ of space variable into new transformation $X,Y$ Step 2: Mesh formation:Construct mesh size of the space domain and time interval $\Delta X,\Delta Y,\Delta t$ Step 3: Initialize the initial and axial boundary condition concentration matrix
$c$ as follows:a) for $j\leftarrow 1:N+1$for $i\leftarrow 1:M+1$$c\left(i,j,1\right)\leftarrow {c}_{0}\times \mathrm{sin}\left(-m\times \left(\left({a}_{1}\times \left(i-1\right)\times dX\right)+\left({a}_{2}\times \left(j-1\right)\times dY\right)\right)\right)$ end for end forb) for $k\leftarrow 2:P+1$for $i\leftarrow 1:M+1$$c\left(i,1,k\right)\leftarrow {c}_{1}\times \mathrm{sin}\left(-{m}_{1}\times (k-1)\times dt\right)$ end forfor $j\leftarrow 1:N+1$$c\left(1,j,k\right)\leftarrow {c}_{2}\times \mathrm{sin}\left(-{m}_{2}\times (k-1)\times dt\right)$ end forend forStep 4: Repeat this step for P time steps to compute the following:
- A)
- Retardation matrix R (Freundlich and Langmuir)
- B)
- The penta-diagonal coefficient matrices A and B for calculating concentration values for each time step
- C)
- Compute the unknowns and solve the system of equations C = B1_inv × B × Z as follows:
- i) compute B1_inv = inverse(A)
- ii) Repeat this step for the P-1 time step to compute the final concentration values at each time step as follows:
- a)
- Compute Z
- b)
- For each column of C from index 2….M, update the values as follows: C = B×Z
End |

## 3. Results and Discussions

## 4. Conclusions

- Results of iterating over all the combinations of pollutant transport dynamics for the two-dimensional reactive system suggest that the peak concentration strengths are alike for homogeneous and heterogeneous media.
- A substantial influence of various soil media is observed for the different types of axial input sources, such as sinusoidally, asymptotically, and exponentially varying non-point sources. It is observed that with an increase in the porosity values, contaminant distribution also increases for sinusoidal and exponential cases. However, in the asymptotic case, concentration distribution decreases with increasing porosity values.
- The apparent effect of different values of $\alpha $ parameters of Langmuir sorption for the three types of axial input sources (sinusoidal, exponential, and asymptotical) is observed. It is found that with an increase in the $\alpha $ parameter, the concentration distribution also increases for sinusoidally varying input sources. In contrast, the contaminant concentration decreases when $\alpha $ parameter is increased for the exponentially varying input sources.
- A comparison of the derived solution of the proposed model with the preceding approximate model solution of the same model from several authors is successfully carried out. Further, the validation under the special case is carried out for the special case (${D}_{xx}={D}_{yy}=0.16km/yea{r}^{-2}$; $m={m}_{1}={m}_{2}=0$) of the proposed model by PDEtool and found good agreement between them. In the present study, the CN method is found to have an advantage over the ADI method.
- The influence of constant and varying velocity parameters on groundwater contaminant transport was examined. Although the effect was found to be a function of initial or boundary conditions, the order of the impact is marginal.
- Finally, the model’s stability and contaminant transport dynamics in different media are tested using the Peclet and Courant numbers. The model is found to be stable, as indicated by the observed Peclet number.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Dimension | Description |

${C}_{h}$ | $[{\mathrm{ML}}^{-3}]$ | Liquid phase contaminant concentration in heterogeneous medium |

$R$ | $[-]$ | Retardation factor |

${S}_{h}$ | $[{\mathrm{MM}}^{-1}]$ | Solid phase contaminant concentration in heterogeneous medium |

$\rho \hspace{0.17em}$ | $\hspace{0.17em}[{\mathrm{ML}}^{-3}]$ | Density |

$\phi $ | $\hspace{0.17em}[-]$ | Porosity |

$\eta \hspace{0.17em}$ | $\hspace{0.17em}[{\mathrm{T}}^{-1}]$ | First-order decay rate coefficient |

$\delta \hspace{0.17em}$ | $\hspace{0.17em}[{\mathrm{ML}}^{-3}{\mathrm{T}}^{-1}]$ | Zero-order production rate |

${u}_{x}\hspace{0.17em},{v}_{y}$ | $[{\mathrm{LT}}^{-1}]$ | Advection coefficients in $x,y$ directions |

${D}_{xx}\hspace{0.17em},{D}_{yy}$ | $[{\mathrm{L}}^{2}{\mathrm{T}}^{-1}]$ | Dispersion coefficient in $x,y$ directions |

$x,y$ | $\hspace{0.17em}[\mathrm{L}]$ | Distance variables in Longitudinal Transversal directions |

$t\hspace{0.17em}$ | $\hspace{0.17em}[\mathrm{T}]$ | Time variable |

${k}_{d}\hspace{0.17em}$ | $\hspace{0.17em}[{\mathrm{M}}^{-1}{\mathrm{L}}^{-3}]$ | Sorption coefficient |

${k}_{l}\hspace{0.17em}$ | $\hspace{0.17em}[{\mathrm{MM}}^{-1}]$ | Maximum sorption capacity |

$u,v$ | $\hspace{0.17em}[{\mathrm{LT}}^{-1}]$ | Initial groundwater velocities in $x,y$ directions |

${D}_{x0},{D}_{y0}\hspace{0.17em}$ | $[{\mathrm{L}}^{2}{\mathrm{T}}^{-1}]$ | Constant dispersion coefficients in $x,y$ directions |

${m}_{1},{m}_{2}\hspace{0.17em}$ | $\hspace{0.17em}[{\mathrm{T}}^{-1}]$ | Flow resistant coefficients |

${a}_{1},{a}_{2}$ | $\hspace{0.17em}[{\mathrm{L}}^{-1}]$ | Heterogeneity parameters of the porous medium |

${c}_{0}\hspace{0.17em}$ | $\hspace{0.17em}[{\mathrm{ML}}^{-3}]$ | Initial constant solute concentration |

${c}_{1}\hspace{0.17em},{c}_{2}$ | $\hspace{0.17em}[{\mathrm{ML}}^{-3}]$ | Sources of input concentration |

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**Figure 2.**CN method-based approximation of the contaminant concentration distribution pattern in a 2D heterogeneous domain.

**Figure 3.**ADI method-based approximation of the contaminant concentration distribution pattern in a 2D heterogeneous domain.

**Figure 4.**Contaminant concentration distribution patterns obtained by (

**a**) CN method and (

**b**) ADI method in a 2D heterogeneous domain under the assumption of linear sorption.

**Figure 5.**Contour plot-based illustration of the distribution patterns of contaminant concentration generated by the CN method (shown by dotted lines) compared to the PDE toolbox (shown by solid lines).

**Figure 7.**Contaminant concentration distribution patterns for different soil mediums at time $t=2$ years for (

**a**) periodic, (

**b**) exponential, and (

**c**) asymptotic axial input sources.

**Figure 9.**Distribution pattern of the contaminant concentration for various values of the non-linearity power $q$.

**Figure 12.**Contaminant concentration distribution patterns for non-linearity power $\alpha $ in Langmuir sorption for (

**a**) sinusoidal, (

**b**) exponential, and (

**c**) asymptotic forms of input axial source.

**Figure 14.**Contaminant concentration distribution patterns for varying and constant velocity parameters for (

**a**) exponential, (

**b**) sinusoidal, and (

**c**) asymptotical axial sources.

**Figure 15.**Contaminant concentration distribution for different values of Peclect number in different soil media.

**Figure 16.**Contaminant concentration distribution for different values of Courant number in different soil media.

**Figure 17.**Relation between the Peclet number (Pe) and Courant number (Cr) for various transport mediums.

**Table 1.**Different forms of the axial input sources ${f}_{1}\left({m}_{1}t\right)$, ${f}_{2}\left({m}_{2}t\right)$.

Sl. No. | Forms of Axial Sources | ${\mathit{c}}_{1}{\mathit{f}}_{1}\left({\mathit{m}}_{1}\mathit{t}\right)$ | ${\mathit{c}}_{2}{\mathit{f}}_{2}\left({\mathit{m}}_{2}\mathit{t}\right)$ |
---|---|---|---|

1. | Exponential | ${c}_{1}\mathrm{exp}\left(-{m}_{1}t\right)$ | ${c}_{2}\mathrm{exp}\left(-{m}_{2}t\right)$ |

2. | Sinusoidal | ${c}_{1}\mathrm{sin}\left(-{m}_{1}t\right)$ | ${c}_{2}\mathrm{sin}\left(-{m}_{2}t\right)$ |

3. | Asymptotic | ${c}_{1}\frac{{m}_{1}t}{1+{m}_{1}t}$ | ${c}_{2}\frac{{m}_{2}t}{1+{m}_{2}t}$ |

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**MDPI and ACS Style**

Radha, R.; Singh, M.K.
Axial Groundwater Contaminant Dispersion Modeling for a Finite Heterogeneous Porous Medium. *Water* **2023**, *15*, 2676.
https://doi.org/10.3390/w15142676

**AMA Style**

Radha R, Singh MK.
Axial Groundwater Contaminant Dispersion Modeling for a Finite Heterogeneous Porous Medium. *Water*. 2023; 15(14):2676.
https://doi.org/10.3390/w15142676

**Chicago/Turabian Style**

Radha, Rashmi, and Mritunjay Kumar Singh.
2023. "Axial Groundwater Contaminant Dispersion Modeling for a Finite Heterogeneous Porous Medium" *Water* 15, no. 14: 2676.
https://doi.org/10.3390/w15142676