# Universal Relationship between Mass Flux and Properties of Layered Heterogeneity on the Contaminant-Flushing Process

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{L}), transverse dispersivities (α

_{T}), porosities (θ), and retardation factors (R). In this system, a constant flux of solute-free clean water is introduced at the inlet located at the left boundary (x = 0) to flush out a previous contaminated domain of length L where the exit at x = L is an open boundary. For the simplicity of demonstration, the initial concentrations in both layers are assumed to be identical, denoted as C

_{0}, and it is flushed by solute-free clean water at the same time. The upper and lower boundaries of the two-layer system are impermeable to flow and transport. The solute transport in the two layers is coupled with continued concentrations and mass fluxes at the interface of the two layers.

^{3}], $v=q/\theta $ is the advection velocity [L/T], q is Darcy velocity which is along the x-axis [L/T], θ is the aquifer porosity [dimensionless], t is time [T], x is the distance along the flow path [L], D

_{x}and D

_{y}are the principal hydrodynamic dispersion coefficients along the x and y directions, respectively [L

^{2}/T], and R is the retardation factor [dimensionless]. Notably, ${D}_{x}={\alpha}_{x}v+{D}^{*}$ and ${D}_{y}={\alpha}_{y}v+{D}^{*}$, where ${\alpha}_{x}$ and ${\alpha}_{y}$ are the longitudinal and transverse dispersivities [L], respectively, and ${D}^{*}$ is the effective molecular diffusion coefficient which depends on the free-water diffusion and a geometric factor called tortuosity [L

^{2}/T]. For most cases of advective velocities concerned here, the contribution of molecular diffusion to the overall hydrodynamic dispersion is secondary and negligible. However, for some special cases, the contribution of molecular diffusion should be considered or may even play a major role. For instance, when one of the two layers in Figure 1 is bedrock in which groundwater is almost motionless, then molecular diffusion will be the primary driving force for mass exchange between the bedrock and the adjacent aquifer. In another example, if one of the two layers in Figure 1 is an aquitard consisting of clay or silt in which groundwater pore velocity is very small but not zero, then the contribution of molecular diffusion may be somewhat similar to the contribution of the mechanical dispersion term.

## 3. Results and Discussion

#### 3.1. Numerical Setup

^{−9}m

^{2}/s, while the default values of other parameters are shown in the following sections.

#### 3.2. Flushing Processing

_{0}). Secondly, the concentration contour lines were slanted near the interface of two layers, but they were vertical when approaching the upper boundary of the layer-1 and lower boundary of the layer-2.

#### 3.3. Effects of Porosity

#### 3.3.1. Effect of Porosity for Case 1

^{2}/d at x = 14 m for t = 200 d, while it was 0.112 mmol/m

^{2}/d at x = 37 m for t = 600 d. Secondly, mass flux appeared as an asymmetric bell-shape distribution with distance of x at any given time, with a steeper rising limb, a relatively flat top, and a less steeper falling limb. For instance, the range of the flat top was not apparent for t = 200 d, but for t = 600 d, it can be seen from x = 37 m to 58 m with a range of 21 m. Thirdly, as time passed, the range of the relatively flat tops increased. For example, when t = 600 d, the range of the flat top was 21 m (from x = 37 m to 58 m). For t = 100 d, the range increased to 38 m (from x = 60 m to 98 m).

#### 3.3.2. Effects of Porosity for Cases 1 and 2

^{2}/d at x = 37 m (for Case 1); in Figure 3b, it increased to 0.152 mmol/m

^{2}/d at x = 20 m (for Case 2). Secondly, the rising limb of the mass flux-distance curve shifted towards the left boundary (at x = 0) but the falling limb of the mass flux-distance curve remained the same, leading to a broader mass flux spatial distribution (i.e., a greater distance between points A and D in Figure 3d). For the purposes of illustration, the cutoff mass flux for the starting and ending points were set at 0.001 mmol/m

^{2}/d in the following discussion. For instance, when t = 600 d, Figure 3a shows that the mass flux started from x = 23 m and ended at x = 64 m with a range of 41 m (for Case 1); Figure 3b shows that the range of mass flux was from 10 m to 64 m with a range of 54 m (for Case 2).

#### 3.3.3. Effects of Porosity for Cases 2 and 3

^{2}/d at x = 8 m when t = 200 d, while in Figure 3c, the peak value of mass flux was 0.197 mmol/m

^{2}/d at x = 7 m at the same time. Secondly, at a given time, the range of mass flux (i.e., section AD in Figure 3d) would become shorter (with the rising limb remaining fixed and the falling limb shifting towards the left boundary) when the porosity of layer-1 increased. For example, when t = 600 d, Figure 3b shows the mass flux starting approximately from x = 10 m and ending at x = 64 m with a range of 54 m (for Case 2), while Figure 3c shows the range decreased to 27 m (from x = 10 m to 37 m) (for Case 3).

#### 3.3.4. Effects of Porosity on Temporal Distribution of the Maximum Vertical Mass Flux

^{2}/d at t = 90 d, whereas it was 0.289 mmol/m

^{2}/d at t = 50 d in Case 2). Conversely, when the porosity of layer-2 was unchanged, increasing the porosity of layer-1 led to a lower maximum mass flux (i.e., the maximum mass flux in Case 2 was 0.289 mmol/m

^{2}/d at t = 50 d, whereas it was 0.210 mmol/m

^{2}/d at t = 150 d in Case 3). The changes caused by the increased porosity of layer-2 were greater than the changes caused by the increased porosity of layer-1. Secondly, we observed that with a constant porosity of layer-1, increasing the porosity of layer-2 prolonged the time needed for mass transfer. For example, Case 1 took 2200 days to complete the solute flushing (when the maximum mass flux dropped below 0.001 mmol/m

^{2}/d, the flushing was regarded as completed), whereas it took 4600 days for Cases 2 and 3 to complete the flushing. After about 400 days, the maximum mass flux-time curves in Cases 2 and 3 coincided with each other.

#### 3.4. Effect of Transverse Dispersivity

#### 3.4.1. Effect of Transverse Dispersivity for Cases 4–6

^{2}/d in Case 4, while it was 0.185 mmol/m

^{2}/d in Case 5. On the other hand, increased transverse dispersivity of layer-1 also increased the peak value of the mass flux at a given time when inspecting Figure 5b,c. For instance, the peak value of mass flux was 0.185 mmol/m

^{2}/d at t = 200 d in Case 5, which was 0.240 mmol/m

^{2}/d in Case 6.

#### 3.4.2. Effect of Transverse Dispersivity on Temporal Distribution of the Maximum Vertical Mass Flux

#### 3.4.3. Effect of Transverse Dispersivity on Total Vertical Mass Flux

#### 3.5. Effect of Retardation Factor

_{c}, with a retardation factor R and a porosity of θ is

## 4. Applications and Limitations

#### 4.1. Applications

#### 4.2. Limitations and Future Works

## 5. Conclusions

- (1)
- With all the other parameters remaining the same, increasing the porosity of layer-2 (which has a slower flushing velocity) would (a) lead to increased mass flux across the interface of two layers, (b) shift the rising limb of the mass flux-distance curve towards the left boundary where solute-free water is introduced for flushing, resulting in a larger mass flux range at a given time. Thus, the total amount of mass flux at a given time would be greater. However, if keeping all parameter unchanged but increasing the porosity of layer-1 (which has a faster flushing velocity) would (a) lead to decreased mass flux, (b) shift the falling limb of the mass flux-distance curve towards the left boundary, causing less total mass flux. Furthermore, increasing the porosity of layer-2 would also prolong the time required for completely flushing out the solute from the system.
- (2)
- When increasing the transverse dispersivity in either layer-1 or layer-2, the mass flux would increase. Changing the transverse dispersivity has little effect on the longitudinal transport, and so, the time needed for completing the flushing process will not be affected.
- (3)
- Retardation factor plays a similar role with porosity. When all the other parameters remain unchanged, the increased retardation factor of layer-2 would increase the mass flux and expand the spatial range (along the layering or bedding direction) of the vertical mass flux. In contrast, an increased retardation factor in layer-1 would decrease the mass flux and lead to a reduced range of the vertical mass flux. Furthermore, increasing the retardation factor of layer-2 would also prolong the time needed for completely flushing out the solute from the system.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Chakraborti, D.; Rahman, M.M.; Mukherjee, A.; Alauddin, M.; Hassan, M.; Dutta, R.N.; Pati, S.; Mukherjee, S.C.; Roy, S.; Quamruzzman, Q. Groundwater arsenic contamination in Bangladesh—21 Years of research. J. Trace Elem. Med. Biol.
**2015**, 31, 237–248. [Google Scholar] [CrossRef] - Wang, D.; Wu, J.; Wang, Y.; Ji, Y. Finding high-quality groundwater resources to reduce the hydatidosis incidence in the Shiqu County of Sichuan Province, China: Analysis, assessment, and management. Expo. Health
**2020**, 12, 307–322. [Google Scholar] [CrossRef] - Su, Z.; Wu, J.; He, X.; Elumalai, V. Temporal changes of groundwater quality within the groundwater depression cone and prediction of confined groundwater salinity using Grey Markov model in Yinchuan area of northwest China. Expo. Health
**2020**, 12, 447–468. [Google Scholar] [CrossRef] - Li, P.; Karunanidhi, D.; Subramani, T.; Srinivasamoorthy, K. Sources and consequences of groundwater contamination. Arch. Environ. Contam. Toxicol.
**2021**, 80, 1–10. [Google Scholar] [CrossRef] - Tatti, F.; Papini, M.P.; Torretta, V.; Mancini, G.; Boni, M.R.; Viotti, P. Experimental and numerical evaluation of Groundwater Circulation Wells as a remediation technology for persistent, low permeability contaminant source zones. J. Contam. Hydrol.
**2019**, 222, 89–100. [Google Scholar] [CrossRef] [PubMed] - Padhye, L.P.; Srivastava, P.; Jasemizad, T.; Bolan, S.; Hou, D.; Sabry, S.; Rinklebe, J.; O’Connor, D.; Lamb, D.; Wang, H. Contaminant containment for sustainable remediation of persistent contaminants in soil and groundwater. J. Hazard. Mater.
**2023**, 455, 131575. [Google Scholar] [CrossRef] [PubMed] - Ibrahim, M.; Nawaz, M.H.; Rout, P.R.; Lim, J.-W.; Mainali, B.; Shahid, M.K. Advances in Produced Water Treatment Technologies: An In-Depth Exploration with an Emphasis on Membrane-Based Systems and Future Perspectives. Water
**2023**, 15, 2980. [Google Scholar] [CrossRef] - Alazaiza, M.Y.; Albahnasawi, A.; Ali, G.A.; Bashir, M.J.; Copty, N.K.; Amr, S.S.A.; Abushammala, M.F.; Al Maskari, T. Recent advances of nanoremediation technologies for soil and groundwater remediation: A review. Water
**2021**, 13, 2186. [Google Scholar] [CrossRef] - Hadley, P.W.; Newell, C.J. Groundwater remediation: The next 30 years. Groundwater
**2012**, 50, 669–678. [Google Scholar] [CrossRef] - Reddy, K.R. Physical and chemical groundwater remediation technologies. In Overexploitation and Contamination of Shared Groundwater Resources; Springer: Berlin/Heidelberg, Germany, 2008; pp. 257–274. [Google Scholar] [CrossRef]
- Sharma, P.K.; Mayank, M.; Ojha, C.; Shukla, S. A review on groundwater contaminant transport and remediation. ISH J. Hydraul. Eng.
**2020**, 26, 112–121. [Google Scholar] [CrossRef] - Mackay, D.M.; Cherry, J.A. Groundwater contamination: Pump-and-treat remediation. Environ. Sci. Technol.
**1989**, 23, 630–636. [Google Scholar] [CrossRef] - Palmer, C.D.; Fish, W. Chemical Enhancements to Pump-and-Treat Remediation. In Epa Environmental Engineering Sourcebook; Superfund Technology Support Center for Ground Water, Robert S. Kerr Environmental Research Laboratory: Ada, OK, USA, 1992; pp. 59–86. [Google Scholar]
- Voudrias, E. Pump and treat remediation of groundwater contaminated by hazardous waste: Can it really be achieved. Glob. Netw. Environ. Sci. Technol.
**2001**, 3, 1–10. Available online: https://journal.gnest.org/sites/default/files/Journal%20Papers/voudrias.pdf (accessed on 11 September 2023). - Rittmann, B.E.; Seagren, E.; Wrenn, B.A. In Situ Bioremediation, 2nd ed.; Elsevier Science: New York, NY, USA, 1994. [Google Scholar]
- Raymond, R.L.; Brown, R.A.; Norris, R.D.; O’neill, E.T. Stimulation of Biooxidation Processes in Subterranean Formations. U.S. Patent No. 4,588,506, 13 May 1986. [Google Scholar]
- Farhadian, M.; Vachelard, C.; Duchez, D.; Larroche, C. In situ bioremediation of monoaromatic pollutants in groundwater: A review. Bioresour. Technol.
**2008**, 99, 5296–5308. [Google Scholar] [CrossRef] [PubMed] - Liu, D.; Li, Q.; Liu, E.; Zhang, M.; Liu, J.; Chen, C. Biomineralized nanoparticles for the immobilization and degradation of crude oil-contaminated soil. Nano Res.
**2023**, 1–8. [Google Scholar] [CrossRef] - Park, M.; Wu, S.; Lopez, I.J.; Chang, J.Y.; Karanfil, T.; Snyder, S.A. Adsorption of perfluoroalkyl substances (PFAS) in groundwater by granular activated carbons: Roles of hydrophobicity of PFAS and carbon characteristics. Water Res.
**2020**, 170, 115364. [Google Scholar] [CrossRef] - Bin Jusoh, A.; Cheng, W.; Low, W.; Nora’aini, A.; Noor, M.M.M. Study on the removal of iron and manganese in groundwater by granular activated carbon. Desalination
**2005**, 182, 347–353. [Google Scholar] [CrossRef] - Liu, C.J.; Werner, D.; Bellona, C. Removal of per-and polyfluoroalkyl substances (PFASs) from contaminated groundwater using granular activated carbon: A pilot-scale study with breakthrough modeling. Environ. Sci. Water Res. Technol.
**2019**, 5, 1844–1853. [Google Scholar] [CrossRef] - National Research Council. Alternatives for Ground Water Cleanup; National Academies Press: Washington, DC, USA, 1994.
- US EPA. Guidance on Remedial Actions for Contaminated Ground Water at Superfund Sites; Environmental Protection Agency: Washington, DC, USA, 1988.
- Chen, Z.; Wang, Y.; Zhan, H. Universal Relationship Between Mass Flux and the Properties of Layered Heterogeneity on the Contaminant Flushing Process. In Proceedings of the AGU Fall Meeting Abstracts, New Orleans, LA, USA, 13–17 December 2021. [Google Scholar]
- Barry, D.; Parker, J. Approximations for solute transport through porous media with flow transverse to layering. Transp. Porous Media
**1987**, 2, 65–82. [Google Scholar] [CrossRef] - Anderson, J.L.; Bouma, J. Water movement through pedal soils: I. Saturated flow. Soil Sci. Soc. Am. J.
**1977**, 41, 413–418. [Google Scholar] [CrossRef] - Vogel, H.-J.; Roth, K. Moving through scales of flow and transport in soil. J. Hydrol.
**2003**, 272, 95–106. [Google Scholar] [CrossRef] - Tang, D.H.; Frind, E.O.; Sudicky, E.A. Contaminant transport in fractured porous media: Analytical solution for a single fracture. Water Resour. Res.
**1981**, 17, 555–564. [Google Scholar] [CrossRef] - Chen, K.; Zhan, H. A Green’s function method for two-dimensional reactive solute transport in a parallel fracture-matrix system. J. Contam. Hydrol.
**2018**, 213, 15–21. [Google Scholar] [CrossRef] - Tan, J.; Cheng, L.; Rong, G.; Zhan, H.; Quan, J. Multiscale roughness influence on hydrodynamic heat transfer in a single fracture. Comput. Geotech.
**2021**, 139, 104414. [Google Scholar] [CrossRef] - Zhan, H.; Wen, Z.; Gao, G. An analytical solution of two-dimensional reactive solute transport in an aquifer-aquitard system. Water Resour. Res.
**2009**, 45, W10501. [Google Scholar] [CrossRef] - Zeng, C.-F.; Xue, X.-L.; Zheng, G.; Xue, T.-Y.; Mei, G.-X. Responses of retaining wall and surrounding ground to pre-excavation dewatering in an alternated multi-aquifer-aquitard system. J. Hydrol.
**2018**, 559, 609–626. [Google Scholar] [CrossRef] - Filippini, M.; Parker, B.L.; Dinelli, E.; Wanner, P.; Chapman, S.W.; Gargini, A. Assessing aquitard integrity in a complex aquifer–aquitard system contaminated by chlorinated hydrocarbons. Water Res.
**2020**, 171, 115388. [Google Scholar] [CrossRef] - Hendrickx, J.M.; Flury, M. Uniform and preferential flow mechanisms in the vadose zone. In Conceptual Models of Flow Transport in the Fractured Vadose Zone; National Academies Press: Washington, DC, USA, 2001; pp. 149–187. [Google Scholar]
- Gerke, H.H. Preferential flow descriptions for structured soils. J. Plant Nutr. Soil Sci.
**2006**, 169, 382–400. [Google Scholar] [CrossRef] - Coats, K.H.; Smith, B.D. Dead-end pore volume and dispersion in porous media. Soc. Pet. Eng. J.
**1964**, 4, 73–84. [Google Scholar] [CrossRef] - Van Genuchten, M.T.; Wierenga, P.J. Mass transfer studies in sorbing porous media I. Analytical solutions. Soil Sci. Soc. Am. J.
**1976**, 40, 473–480. [Google Scholar] [CrossRef] - Gao, G.; Zhan, H.; Feng, S.; Fu, B.; Ma, Y.; Huang, G. A new mobile-immobile model for reactive solute transport with scale-dependent dispersion. Water Resour. Res.
**2010**, 46, W08533. [Google Scholar] [CrossRef] - Dou, Z.; Tang, S.; Zhang, X.; Liu, R.; Zhuang, C.; Wang, J.; Zhou, Z.; Xiong, H. Influence of shear displacement on fluid flow and solute transport in a 3D rough fracture. Lithosphere
**2021**, 2021, 1569736. [Google Scholar] [CrossRef] - Blackmore, S.; Pedretti, D.; Mayer, K.; Smith, L.; Beckie, R. Evaluation of single-and dual-porosity models for reproducing the release of external and internal tracers from heterogeneous waste-rock piles. J. Contam. Hydrol.
**2018**, 214, 65–74. [Google Scholar] [CrossRef] [PubMed] - Wood, B.D.; Dawson, C.N.; Szecsody, J.E.; Streile, G.P. Modeling contaminant transport and biodegradation in a layered porous media system. Water Resour. Res.
**1994**, 30, 1833–1845. [Google Scholar] [CrossRef] - Skopp, J.; Gardner, W.R.; Tyler, E.J. Solute movement in structured soils: Two-region model with small interaction. Soil Sci. Soc. Am. J.
**1981**, 45, 837–842. [Google Scholar] [CrossRef] - Gerke, H.H.; Van Genuchten, M.T. A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour. Res.
**1993**, 29, 305–319. [Google Scholar] [CrossRef] - Liang, X.; Zhang, Y.K.; Liu, J.; Ma, E.; Zheng, C. Solute transport with linear reactions in porous media with layered structure: A semianalytical model. Water Resour. Res.
**2019**, 55, 5102–5118. [Google Scholar] [CrossRef] - Wu, M.-C.; Hsieh, P.-C. Analytical modeling of solute transport in a two-zoned porous medium flow. Water
**2022**, 14, 323. [Google Scholar] [CrossRef] - Kurasawa, T.; Takahashi, Y.; Suzuki, M.; Inoue, K. Laboratory Flushing Tests of Dissolved Contaminants in Heterogeneous Porous Media with Low-Conductivity Zones. Water Air Soil Pollut.
**2023**, 234, 240. [Google Scholar] [CrossRef] - Dorchester, L.; Day-Lewis, F.D.; Singha, K. Evaluation of Dual Domain Mass Transfer in Porous Media at the Pore Scale. Groundwater
**2023**. [Google Scholar] [CrossRef] - Chen, Z. Flushing of Contaminated Homogenous and Heterogenous Aquifers. Master’s Thesis, Texas A&M University, College Station, TX, USA, 26 April 2021. [Google Scholar]

**Figure 1.**Schematic diagram of a flushing model in the layered heterogeneous aquifer. The thickness of the layer is denoted as B, L is the length of domain, and the subscripts 1 and 2 represent the parameters in layer-1 and layer-2, respectively.

**Figure 3.**The vertical mass flux of Cases 1–3 between the layers varies with distance at different times. (

**a**) Case 1; (

**b**) Case 2; (

**c**) Case 3; (

**d**) An example curve (Case 1 at t = 600 d) with turning points highlighted.

**Figure 5.**The vertical mass flux of Cases 4–6 across the interface of the layers varies with distance at different times. (

**a**) Case 4; (

**b**) Case 5; (

**c**) Case 6.

**Table 1.**Summary of the latest relevant studies including their approaches, main focuses, and differences from this investigation.

Literatures | Methods | Main Points | Differences from This Study |
---|---|---|---|

[44] | Semi-analytical model | The model considers transverse dispersion and linear reactions in a layered medium, and the mass exchange between the zones is determined by the transverse dispersion across the interface. | This paper focused only on the transverse dispersion but did not consider other influence factors. |

[45] | Analytical model | The modeling results show that the pollutant concentration is more sensitive to the Peclet number than the retardation factor and the first-order decaying coefficient in uniform groundwater flow. | The model was based on 1-D ADE, and the flow direction was perpendicular to the interface of two layers. |

[46] | Laboratory model | The effects of the geometry of low-conductivity zones, conductivity contrast, and flow regime on solute flushing. | This paper focused only on the conductivity contrast but did not consider other influence factors. |

[47] | Synthetic pore-scale millifluidics simulation | They compared the length scales associated with mass transfer rate and the calculation of the Peclet number and found that the Peclet number is commonly larger than the characteristic length scale associated with mass transfer rate. | The simulations were using a millifluidics device, which might not fully represent the complex and heterogeneous nature of real-world porous media. |

Case No. | θ_{1} | θ_{2} |
---|---|---|

1 | 0.1 | 0.2 |

2 | 0.1 | 0.4 |

3 | 0.2 | 0.4 |

Case No. | α_{T}_{1} | α_{T}_{2} |
---|---|---|

4 | 0.01 | 0.02 |

5 | 0.01 | 0.04 |

6 | 0.02 | 0.04 |

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**MDPI and ACS Style**

Chen, Z.; Zhan, H.
Universal Relationship between Mass Flux and Properties of Layered Heterogeneity on the Contaminant-Flushing Process. *Water* **2023**, *15*, 3292.
https://doi.org/10.3390/w15183292

**AMA Style**

Chen Z, Zhan H.
Universal Relationship between Mass Flux and Properties of Layered Heterogeneity on the Contaminant-Flushing Process. *Water*. 2023; 15(18):3292.
https://doi.org/10.3390/w15183292

**Chicago/Turabian Style**

Chen, Zehao, and Hongbin Zhan.
2023. "Universal Relationship between Mass Flux and Properties of Layered Heterogeneity on the Contaminant-Flushing Process" *Water* 15, no. 18: 3292.
https://doi.org/10.3390/w15183292