# Effects of Cemented Porous Media on Temporal Mixing Behavior of Conservative Solute Transport

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Generation of Cemented Porous Media

^{2}, as shown in Figure 1. The corresponding porosity of the generated solid grains in this two-dimensional domain was 0.45. It should be mentioned that a checking algorithm during the random distribution process of the solid grains was used to avoid the overlapping solid grains. In addition, to obtain the different degrees of the cementation in the same porous media model, we assumed that a minimum distance between any two solid grains was set to be one-fifth of the mean radius. As can be observed in Figure 1, the solid grains were totally free and non-overlapping. The solid grains’ mean radius and the coefficient of variation (COV) were set to 0.8 mm and 0.2 mm, respectively. The original porous media in Figure 1 was named as PM1. The probability distribution of the radii of the solid grains is shown in Figure 2.

#### 2.2. Flow Field and Solute Transport Models in Porous Media

#### 2.3. Quantification of Mixing

## 3. Result and Discussion

#### 3.1. Model Setup

^{3}and $\mu =1.002\times {10}^{-3}$ Pa∙s). The numerical simulation of conservative solute transport was performed. The corresponding solute molecular diffusion was set to ${D}_{m}=1\times {10}^{-9}$ m

^{2}/s. The flow field and transient solute transport models based on Equations (2)–(4) were implemented in the COMSOL Multiphysics package (COMSOL Inc., Burlington, MA, USA) using the Galerkin finite-element method. The steady-state flow field was induced by adjusting the pressure drop over the entire porous media to obtain the same Pe. The solved flow field serves as the input for the transient solute transport model. In this work, the total simulation time was 1000 s with the time step was set as 0.5 s. In order to ensure numerical stability and accuracy, the sensitivity analysis for mesh dependency was performed, and the corresponding results showed that when the porous media domain was discretized into ~196,000 triangular elements, the solutions were mesh-independent and the numerical dispersion was negligible.

#### 3.2. Flow Fields

#### 3.3. Characteristic of Solute Transport

#### 3.3.1. Plume Dilution

#### 3.3.2. Global SDR

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Sale, T.C.; Zimbron, J.A.; Dandy, D.S. Effects of reduced contaminant loading on downgradient water quality in an idealized two-layer granular porous media. J. Contam. Hydrol.
**2008**, 102, 72–85. [Google Scholar] [CrossRef] [PubMed] - Bagalkot, N.; Kumar, G.S. Numerical modeling of two species radionuclide transport in a single fracturematrix system with variable fracture aperture. Geosci. J.
**2016**, 20, 627–638. [Google Scholar] [CrossRef] - Song, X.; Hong, E.; Seagren, E.A. Laboratory-scale in situ bioremediation in heterogeneous porous media: Biokinetics-limited scenario. J. Contam. Hydrol.
**2014**, 158, 78–92. [Google Scholar] [CrossRef] [PubMed] - Anna, P.; Jimenez-Martinez, J.; Tabuteau, H.; Turuban, R.; Le Borgne, T.; Derrien, M.; Meheust, Y. Mixing and reaction kinetics in porous media: An experimental pore scale quantification. Environ. Sci. Technol.
**2014**, 48, 508–516. [Google Scholar] [CrossRef] [PubMed] - Soltanian, M.R.; Ritzi, R.W.; Dai, Z.; Huang, C.C. Reactive solute transport in physically and chemically heterogeneous porous media with multimodal reactive mineral facies: The lagrangian approach. Chemosphere
**2015**, 122, 235–244. [Google Scholar] [CrossRef] [PubMed] - Dou, Z.; Chen, Z.; Zhou, Z.; Wang, J.; Huang, Y. Influence of eddies on conservative solute transport through a 2d single self-affine fracture. Int. J. Heat Mass Transf.
**2018**, 121, 597–606. [Google Scholar] [CrossRef] - Dou, Z.; Zhou, Z.; Wang, J.; Huang, Y. Roughness scale dependence of the relationship between tracer longitudinal dispersion and peclet number in variable-aperture fractures. Hydrol. Process.
**2018**, 32, 1461–1475. [Google Scholar] [CrossRef] - Shapiro, M.; Brenner, H. Dispersion of a chemically reactive solute in a spatially periodic model of a porous medium. Chem. Eng. Sci.
**1988**, 43, 551–571. [Google Scholar] [CrossRef] - Pini, R.; Vandehey, N.T.; Druhan, J.; O’Neil, J.P.; Benson, S.M. Quantifying solute spreading and mixing in reservoir rocks using 3-d pet imaging. J. Fluid Mech.
**2016**, 796, 558–587. [Google Scholar] [CrossRef] - Bear, J. Dynamics of Fluids in Porous Media; Courier Corporation: New York, NY, USA, 2013. [Google Scholar]
- Rolle, M.; Chiogna, G.; Hochstetler, D.L.; Kitanidis, P.K. On the importance of diffusion and compound-specific mixing for groundwater transport: An investigation from pore to field scale. J. Contam. Hydrol.
**2013**, 153, 51–68. [Google Scholar] [CrossRef] - Swanson, R.D.; Binley, A.; Keating, K.; France, S.; Osterman, G.; Day-Lewis, F.D.; Singha, K. Anomalous solute transport in saturated porous media: Relating transport model parameters to electrical and nuclear magnetic resonance properties. Water Resour. Res.
**2015**, 51, 1264–1283. [Google Scholar] [CrossRef][Green Version] - Bijeljic, B.; Mostaghimi, P.; Blunt, M.J. Insights into non-fickian solute transport in carbonates. Water Resour. Res.
**2013**, 49, 2714–2728. [Google Scholar] [CrossRef] - Fiori, A.; Janković, I.; Dagan, G. Modeling flow and transport in highly heterogeneous three-dimensional aquifers: Ergodicity, gaussianity, and anomalous behavior—2. Approximate semianalytical solution. Water Resour. Res.
**2006**, 42. [Google Scholar] [CrossRef] - Neuman, S.P.; Tartakovsky, D.M. Perspective on theories of non-fickian transport in heterogeneous media. Adv. Water Resour.
**2009**, 32, 670–680. [Google Scholar] [CrossRef] - Heidari, P.; Li, L. Solute transport in low-heterogeneity sandboxes: The role of correlation length and permeability variance. Water Resour. Res.
**2014**, 50, 8240–8264. [Google Scholar] [CrossRef] - Hou, Y.; Jiang, J.; Wu, J. Anomalous solute transport in cemented porous media: Pore-scale simulations. Soil Sci. Soc. Am. J.
**2018**, 82, 10. [Google Scholar] [CrossRef] - Edery, Y.; Guadagnini, A.; Scher, H.; Berkowitz, B. Origins of anomalous transport in heterogeneous media: Structural and dynamic controls. Water Resour. Res.
**2014**, 50, 1490–1505. [Google Scholar] [CrossRef] - Voller, V.R. A direct simulation demonstrating the role of spacial heterogeneity in determining anomalous diffusive transport. Water Resour. Res.
**2015**, 51, 2119–2127. [Google Scholar] [CrossRef] - Bijeljic, B.; Raeini, A.; Mostaghimi, P.; Blunt, M.J. Predictions of non-fickian solute transport in different classes of porous media using direct simulation on pore-scale images. Phys. Rev. E
**2013**, 87, 013011. [Google Scholar] [CrossRef] - Gramling, C.M.; Harvey, C.F.; Meigs, L.C. Reactive transport in porous media: A comparison of model prediction with laboratory visualization. Environ. Sci. Technol.
**2002**, 36, 2508–2514. [Google Scholar] [CrossRef] - Chiogna, G.; Cirpka, O.A.; Herrera, P.A. Helical flow and transient solute dilution in porous media. Transp. Porous Media
**2015**, 111, 591–603. [Google Scholar] [CrossRef] - Rolle, M.; Hochstetler, D.; Chiogna, G.; Kitanidis, P.K.; Grathwohl, P. Experimental investigation and pore-scale modeling interpretation of compound-specific transverse dispersion in porous media. Transp. Porous Media
**2012**, 93, 347–362. [Google Scholar] [CrossRef] - Barros, F.P.; Dentz, M.; Koch, J.; Nowak, W. Flow topology and scalar mixing in spatially heterogeneous flow fields. Geophys. Res. Lett.
**2012**, 39. [Google Scholar] [CrossRef] - Le Borgne, T.; Dentz, M.; Villermaux, E. Stretching, coalescence, and mixing in porous media. Phys. Rev. Lett.
**2013**, 110, 204501. [Google Scholar] [CrossRef] [PubMed] - Dou, Z.; Zhou, Z.-F.; Wang, J.-G. Three-dimensional analysis of spreading and mixing of miscible compound in heterogeneous variable-aperture fracture. Water Sci. Eng.
**2016**, 9, 293–299. [Google Scholar] [CrossRef] - Rolle, M.; Eberhardt, C.; Chiogna, G.; Cirpka, O.A.; Grathwohl, P. Enhancement of dilution and transverse reactive mixing in porous media: Experiments and model-based interpretation. J. Contam. Hydrol.
**2009**, 110, 130–142. [Google Scholar] [CrossRef] - Kitanidis, P.K. The concept of the dilution index. Water Resour. Res.
**1994**, 30, 2011–2026. [Google Scholar] [CrossRef] - Kapoor, V.; Kitanidis, P.K. Concentration fluctuations and dilution in two-dimensionally periodic heterogeneous porous media. Transp. Porous Media
**1996**, 22, 91–119. [Google Scholar] [CrossRef] - Cirpka, O.A. Choice of dispersion coefficients in reactive transport calculations on smoothed fields. J. Contam. Hydrol.
**2002**, 58, 261–282. [Google Scholar] [CrossRef] - Molins, S.; Trebotich, D.; Steefel, C.I.; Shen, C. An investigation of the effect of pore scale flow on average geochemical reaction rates using direct numerical simulation. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Yang, A.; Miller, C.; Turcoliver, L. Simulation of correlated and uncorrelated packing of random size spheres. Phys. Rev. E.
**1996**, 53, 1516. [Google Scholar] [CrossRef] - Pope, S.B.; Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Rolle, M.; Kitanidis, P.K. Effects of compound-specific dilution on transient transport and solute breakthrough: A pore-scale analysis. Adv. Water Resour.
**2014**, 71, 186–199. [Google Scholar] [CrossRef] - Dreuzy, J.R.; Carrera, J.; Dentz, M.; Le Borgne, T. Time evolution of mixing in heterogeneous porous media. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef][Green Version] - Dou, Z.; Sleep, B.; Mondal, P.; Guo, Q.; Wang, J.; Zhou, Z. Temporal mixing behavior of conservative solute transport through 2d self-affine fractures. Processes
**2018**, 6, 158. [Google Scholar] [CrossRef] - Le Borgne, T.; Dentz, M.; Bolster, D.; Carrera, J.; de Dreuzy, J.-R.; Davy, P. Non-fickian mixing: Temporal evolution of the scalar dissipation rate in heterogeneous porous media. Adv. Water Resour.
**2010**, 33, 1468–1475. [Google Scholar] [CrossRef]

**Figure 1.**The distribution of the generated solid grain in two-dimensional domain. The boundary conditions were applied for the solute transport simulation in all porous media.

**Figure 3.**Flow fields in four porous media with different characteristic of pore space. The local velocity plots at different cross-sections (x = 20 mm and x = 50 mm). The color code represents the computed pore-scale velocities at the case Peclet number (Pe) = 100. The velocity distribution of the cross-sections at a-a’, b-b’, c-c’, d-d’, e-e’, f-f’, g-g’, and h-h’ were shown in Figure 3

**a**–

**h**, respectively.

**Figure 4.**The calculated dilution index for solute in the different degrees of heterogeneous pore-scale domains at the case of Pe = 100 (

**a**–

**c**), Pe = 200 (

**d**–

**f**) and Pe = 400 (

**g**–

**i**) respectively.

**Figure 5.**The zoomed snapshots (Pv = 0.04, 0.055, 0.07 and 0.10) of local solute transport in the different heterogeneous porous media (PM1, PM4) for the Pe = 100 (

**a**,

**b**) and Pe = 400 (

**c**,

**d**).

**Figure 6.**Scalar dissipation rate estimated in different degree of heterogeneous pore-scale domains at the case Pe = 100, Pe = 200, and, Pe = 400 respectively.

Porous Media | Porosity | ${\mathit{r}}_{\mathit{a}\mathit{v}\mathit{e}}$ (mm) | ${\mathit{N}}_{\mathit{c}}$ | ${\mathit{N}}_{\mathit{a}}$ | ${\mathit{d}}_{\mathit{a}\mathit{v}\mathit{e}}$ (mm) | ${\mathit{N}}_{\mathit{g}}$ |
---|---|---|---|---|---|---|

PM1 | 0.45 | 0.785 | 0 | 0 | 0.36355 | 1679 |

PM2 | 0.40 | 0.818 | 0 | 0 | 0.29635 | 1679 |

PM3 | 0.35 | 0.854 | 0 | 0 | 0.22507 | 1679 |

PM4 | 0.30 | 0.888 | 154 | 4.62 | 0.23562 | 1112 |

Dimensionless Pe and Re | Porous Media | τ_{a} (s) | ${\mathit{\sigma}}_{\mathit{u}}$ (-) | ${\mathit{u}}_{\mathit{a}\mathit{v}}$ (m/s) | $\mathit{C}{\mathit{V}}_{\mathit{U}}$ (-) | $\overline{\mathit{u}}$ (m/s) | ${\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}$ (m/s) |
---|---|---|---|---|---|---|---|

Pe = 100 Re = 0.0996 | PM1 | 24.622 | $4.953\times {10}^{-5}$ | $5.250\times {10}^{-5}$ | 0.943 | $6.373\times {10}^{-5}$ | $4.375\times {10}^{-4}$ |

PM2 | 26.776 | $5.322\times {10}^{-5}$ | $4.847\times {10}^{-5}$ | 1.098 | $6.111\times {10}^{-5}$ | $5.766\times {10}^{-4}$ | |

PM3 | 29.155 | $6.405\times {10}^{-5}$ | $4.253\times {10}^{-5}$ | 1.506 | $5.857\times {10}^{-5}$ | $9.244\times {10}^{-3}$ | |

PM4 | 31.530 | $8.296\times {10}^{-5}$ | $3.577\times {10}^{-5}$ | 2.319 | $5.632\times {10}^{-5}$ | $2.724\times {10}^{-3}$ | |

Pe = 200 Re = 0.1992 | PM1 | 12.311 | $9.906\times {10}^{-5}$ | $1.050\times {10}^{-4}$ | 0.943 | $1.275\times {10}^{-4}$ | $8.749\times {10}^{-4}$ |

PM2 | 13.388 | $1.064\times {10}^{-4}$ | $9.694\times {10}^{-5}$ | 1.098 | $1.222\times {10}^{-4}$ | $1.153\times {10}^{-3}$ | |

PM3 | 14.578 | $1.281\times {10}^{-4}$ | $8.507\times {10}^{-5}$ | 1.506 | $1.171\times {10}^{-4}$ | $1.855\times {10}^{-3}$ | |

PM4 | 15.765 | $1.659\times {10}^{-4}$ | $7.155\times {10}^{-5}$ | 2.319 | $1.126\times {10}^{-4}$ | $5.449\times {10}^{-3}$ | |

Pe = 400 Re = 0.3985 | PM1 | 6.1555 | $1.981\times {10}^{-4}$ | $2.100\times {10}^{-4}$ | 0.943 | $2.549\times {10}^{-4}$ | $1.749\times {10}^{-3}$ |

PM2 | 6.6940 | $2.128\times {10}^{-4}$ | $1.939\times {10}^{-4}$ | 1.098 | $2.445\times {10}^{-4}$ | $2.306\times {10}^{-3}$ | |

PM3 | 7.2887 | $2.562\times {10}^{-4}$ | $1.701\times {10}^{-4}$ | 1.506 | $2.343\times {10}^{-4}$ | $3.696\times {10}^{-3}$ | |

PM4 | 7.8825 | $3.316\times {10}^{-4}$ | $1.431\times {10}^{-4}$ | 2.319 | $2.253\times {10}^{-4}$ | $1.090\times {10}^{-2}$ |

Dimensionless Pe | The Peak Value of Dilution Index | |||
---|---|---|---|---|

PM1 | PM2 | PM3 | PM4 | |

Pe = 100 | $5.7914\times {10}^{-4}$ | $5.0943\times {10}^{-4}$ | $6.1390\times {10}^{-4}$ | $5.4365\times {10}^{-4}$ |

Pe = 200 | $5.8992\times {10}^{-4}$ | $5.9883\times {10}^{-4}$ | $6.1525\times {10}^{-4}$ | $5.2547\times {10}^{-4}$ |

Pe = 400 | $6.0294\times {10}^{-4}$ | $6.0676\times {10}^{-4}$ | $6.1612\times {10}^{-4}$ | $5.1070\times {10}^{-4}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dou, Z.; Zhang, X.; Chen, Z.; Yang, Y.; Zhuang, C.; Wang, C. Effects of Cemented Porous Media on Temporal Mixing Behavior of Conservative Solute Transport. *Water* **2019**, *11*, 1204.
https://doi.org/10.3390/w11061204

**AMA Style**

Dou Z, Zhang X, Chen Z, Yang Y, Zhuang C, Wang C. Effects of Cemented Porous Media on Temporal Mixing Behavior of Conservative Solute Transport. *Water*. 2019; 11(6):1204.
https://doi.org/10.3390/w11061204

**Chicago/Turabian Style**

Dou, Zhi, Xueyi Zhang, Zhou Chen, Yun Yang, Chao Zhuang, and Chenxi Wang. 2019. "Effects of Cemented Porous Media on Temporal Mixing Behavior of Conservative Solute Transport" *Water* 11, no. 6: 1204.
https://doi.org/10.3390/w11061204