# On the Validity of the Null Current Assumption for Modeling Sorptive Reactive Transport and Electro-Diffusion in Porous Media

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical and Numerical Models

#### 2.1. General NPP Model

^{2}·s] is the total mass flux, ${D}_{i}$ [m

^{2}/s] is the species-dependent diffusion coefficient, ${C}_{D,i}$ [mol/m

^{3}] is the dissolved concentration of ith species, $F=\mathrm{96,485}$ [C/mol] is the Faraday constant, $R=8.341$ [J/K·mol] is the gas constant, $T\left[\mathrm{K}\right]$ is the absolute temperature, ${z}_{i}[-]$ is the charge number, $\psi \left[\mathrm{V}\mathrm{or}\frac{\mathrm{J}}{\mathrm{C}}\right]$ is the electric potential and $q$ [m/s] is the Darcy’s velocity.

^{3}] and ${C}_{B,i}$ [mol/kg] are the dissolved and sorbed concentration of the ith species, respectively. ${\rho}_{b}=2200$ [kg/m

^{3}] is the bulk density of the medium.

#### 2.2. Electro-Neutrality and NC Assumption

#### 2.3. Equivalency of NPP Systems and Null Current Assumption Models

#### 2.4. Numerical Models

## 3. Verification

## 4. Effect of Adsorption Reactions

^{−5}and 10

^{−4}[m

^{3}/kg] are considered, respectively. The domain is initially set to be at zero charges. All the other parameters, the boundary, and initial conditions are kept the same as in the previous section. The same COMSOL models used in the previous section are modified to include adsorption reactions and then used in the simulation of the different test cases in this section. The results are discussed in the next sections, in terms of total concentration, total charge, total current, and electrical currents, respectively.

#### 4.1. Total Concentration

#### 4.2. Total Electric Current

#### 4.3. Total Charge

#### 4.4. Electric Field

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Benchmarks Description (Boundary and Initial Conditions and Chemical System)

- Parameters, boundary, and initial conditions for Benchmark 1

Species | Left Boundary Condition (mM) | Initial Condition (mM) |
---|---|---|

${\mathrm{H}}^{+}$ | 0.001 | 0.1 |

${\mathrm{Na}}^{+}$ | 0.1 | 0.1 |

${\mathrm{Cl}}^{-}$ | 0.1 | 0.1 |

${\mathrm{NO}}_{3}^{-}$ | 0.001 | 0.1 |

Parameters | ||

Permittivity [F/m] | $7.08\times {10}^{-10}$ | |

Molecular diffusion coefficients [m^{2}/s] | ${\mathrm{Na}}^{+}$: 1.33 × 10^{−9} | |

${\mathrm{Cl}}^{-}$: 2.03 × 10^{−9} | ||

${\mathrm{NO}}_{3}^{-}$: 1.9 × 10^{−9} | ||

${\mathrm{H}}^{+}$: 9.31 × 10^{−9} |

- 2.
- Parameters, boundary, and initial conditions for Benchmark 2

Species | Left Boundary Condition (mM) (Initial Condition in the Left Half of the Domain) | Right Boundary Condition (mM) (Initial Condition in the Right Half of the Domain) |
---|---|---|

${\mathrm{H}}^{+}$ | 1 × 10^{−4} | 1 × 10^{−4} |

${\mathrm{Na}}^{+}$ | 0.5 | 0.1 |

${}_{}{}^{22}\mathrm{Na}^{+}$ | 1 × 10^{−6} | 1 × 10^{−6} |

${\mathrm{Cl}}^{-}$ | 0.5 | 0.1 |

${\mathrm{OH}}_{}^{-}$ | 1 × 10^{−4} | 1 × 10^{−4} |

Parameters | ||

Permittivity [F/m] | 5.85 × 10^{−10} | |

Molecular diffusion coefficients [m^{2}/s] | ${\mathrm{H}}^{+}$: 9.31 × 10^{−9} | |

${\mathrm{Na}}^{+},{}_{}{}^{22}\mathrm{Na}^{+}$: 1.33 × 10^{−9} | ||

${\mathrm{Cl}}^{-}$: 2.03 × 10^{−9} | ||

${\mathrm{OH}}_{}^{-}$: 5.27 × 10^{−9} |

- 3.
- Parameters and Boundary and initial conditions for Benchmark 3

Species | Initial Source (1D) and the Tracer Injection Ports (mM) | Initial Condition (1D) and Remaining Injection Ports (2D) (mM) |
---|---|---|

${\mathrm{K}}^{+}$ | 0.29 | 1 × 10^{−6} |

${\mathrm{Mg}}^{2+}$ | 0.29 | 1 × 10^{−6} |

${\mathrm{Cl}}^{-}$ | 0.87 | 3 × 10^{−6} |

Parameters | ||

Permittivity [F/m] | 5.85 × 10^{−10} | |

Molecular diffusion coefficients [m^{2}/s] | ${\mathrm{K}}^{+}$: 1.77 × 10^{−9} | |

${\mathrm{Mg}}^{2+}$: 6.26 × 10^{−10} | ||

${\mathrm{Cl}}^{-}$: 2.03 × 10^{−9} | ||

Transverse dispersion coefficients [m^{2}/s] | ${\mathrm{K}}^{+}$: 2.405 × 10^{−9} | |

${\mathrm{Mg}}^{2+}$: 1.745 × 10^{−9} | ||

${\mathrm{Cl}}^{-}$: 2.425 × 10^{−9} |

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**Figure 1.**Comparison of the concentration results of the NPP, NC, and ‘Crunch’ models for the three 1D benchmarks. Dashed and continuous lines are indistinguishable showing perfect agreement between models.

**Figure 2.**Comparison of the concentration results of the NPP (Colors), NC (solid black lines), and ‘Crunch’ (dashed red lines) models for 2D simulations of benchmark 3.

**Figure 3.**Comparison of total concentration results of the NPP and NC-models for 1D simulations of the three benchmarks in the case of adsorption reactions. Test case 1 deals with low sorptivity while test case 2 deals with high sorptivity.

**Figure 4.**Comparison of total concentration results of the NPP (Colors) and NC (Lines) models for simulation of benchmark 3-2D with sorption: low sorptivity (test case 1—

**left**), high sorptivity (test case 2—

**right**).

**Figure 5.**Comparison of total concentration results of the NPP and NC models for Benchmark 2 with linear initial condition Figure 1.—

**left**), high sorptivity (test case 2—

**right**).

**Figure 7.**The total electrical current for the 2D benchmark in the horizontal (

**left**) and vertical directions (

**right**). The NPP model is represented with colors and the NC model with lines: low sorptivity (test case 1), high sorptivity (test case).

**Figure 9.**The total charge for the 2D benchmark with the NC-model: low sorptivity (test case 1), high sorptivity (test case 2).

**Figure 11.**Benchmark3-2D. (

**a**) The electric field with the NPP-model for the case of no sorption (colors), low sorption (black lines), and high sorption (red dashed lines). (

**b**) The electric field with the NC-model for the case of no sorption (colors), low sorption (black lines), and high sorption (red dashed lines).

Benchmark | Primary Components | Process | Dimension |
---|---|---|---|

1 | ${\mathrm{H}}^{+},{\mathrm{NO}}_{3}^{-},{\mathrm{Na}}^{+},{\mathrm{Cl}}^{-}$ | (Molecular/electro) Diffusion | 1D |

2 | ${\mathrm{Na}}^{+},{\mathrm{Cl}}^{-},{}_{}{}^{22}\mathrm{Na}^{+}{}^{1},{\mathrm{H}}^{+},{\mathrm{OH}}^{-}$ | (Molecular/electro) Diffusion | 1D |

3 | ${\mathrm{K}}^{+},{\mathrm{Cl}}^{-},{\mathrm{Mg}}^{++}$ | (Molecular/electro) Diffusion/Advection | 1D/2D |

^{1}${}_{}{}^{22}\mathrm{Na}^{+}$ is another isotope of ${\mathrm{Na}}^{+}$ which is treated as a distinct component.

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**MDPI and ACS Style**

Tabrizinejadas, S.; Carrayrou, J.; Saaltink, M.W.; Baalousha, H.M.; Fahs, M. On the Validity of the Null Current Assumption for Modeling Sorptive Reactive Transport and Electro-Diffusion in Porous Media. *Water* **2021**, *13*, 2221.
https://doi.org/10.3390/w13162221

**AMA Style**

Tabrizinejadas S, Carrayrou J, Saaltink MW, Baalousha HM, Fahs M. On the Validity of the Null Current Assumption for Modeling Sorptive Reactive Transport and Electro-Diffusion in Porous Media. *Water*. 2021; 13(16):2221.
https://doi.org/10.3390/w13162221

**Chicago/Turabian Style**

Tabrizinejadas, Sara, Jerome Carrayrou, Maarten W. Saaltink, Husam Musa Baalousha, and Marwan Fahs. 2021. "On the Validity of the Null Current Assumption for Modeling Sorptive Reactive Transport and Electro-Diffusion in Porous Media" *Water* 13, no. 16: 2221.
https://doi.org/10.3390/w13162221