# Interpreting Self-Potential Signal during Reactive Transport: Application to Calcite Dissolution and Precipitation

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## Abstract

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## 1. Introduction

## 2. Theoretical Workflow

#### 2.1. Reactive Transport in Carbonaceous System

#### 2.1.1. Carbonate Material Reactivity

#### 2.1.2. Flow and Transport

#### 2.2. The Electrical Conductivity

#### 2.2.1. The Pore Water EC

#### 2.2.2. The Sample EC

#### 2.3. The Self-Potential Method

#### 2.3.1. The Electrokinetic Contribution

#### 2.3.2. The Electro-Diffusive Contribution

#### 2.3.3. Development of a New Model for the Electro-Diffusive Potential in a Multi-Ionic System

## 3. Materials and Methods

#### 3.1. One-Dimensional Reactive Transport Simulations Using CrunchFlow

#### 3.1.1. Thermodynamic and Kinetic Data

#### 3.1.2. Initial Chemical Compositions

#### 3.1.3. Flow and Transport Properties

#### 3.1.4. Discretization

#### 3.1.5. Specific Reactive Surface Area

#### 3.2. Fully Coupled Numerical Workflow for Multi-Ionic Modeling of Electro-Diffusive Potential

#### 3.3. Experimental Setup

#### 3.4. Experimental Timeline

## 4. Results and Discussion

#### 4.1. Pore Water EC Monitoring

#### 4.2. Chemical Analysis on Outlet Water Samples

#### 4.3. Column Sample EC Measurements

#### 4.4. SP Monitoring

#### 4.4.1. Experimental Results

#### 4.4.2. CrunchFlow Simulation Results for the Reactive Transport

#### 4.4.3. Results from the Electro-Diffusive Potential Modeling

#### 4.5. Location of the Reactive Zone

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Notations

Symbol | Definition (unit) |

$Alk$ | Alkalinity ($\mathrm{mol}$ ${\mathrm{L}}^{-1}$) |

${\alpha}^{*}$ | Electro-diffusive coupling coefficient ($\mathrm{V}$) |

${\beta}_{{X}_{i}}$ | Mobility on ion ${X}_{i}$ (${\mathrm{m}}^{2}$ ${\mathrm{s}}^{-1}$${\mathrm{V}}^{-1}$) |

C | Ionic concentration of the solution ($\mathrm{mol}$ ${\mathrm{L}}^{-1}$) |

${C}^{\ominus}$ | Standard concentration ($\mathrm{mol}$ ${\mathrm{L}}^{-1}$) |

${C}_{{X}_{i}}$ | Ionic concentration of ion ${X}_{i}$ ($\mathrm{mol}$ ${\mathrm{L}}^{-1}$) |

${C}^{EK}$ | Electrokinetic coupling coefficient ($\mathrm{V}$ ${\mathrm{Pa}}^{-1}$) |

$<{d}_{g}>$ | Mean grain diameter ($\mathrm{m}$) |

${D}_{{X}_{i}}$ | Diffusion coefficient of ion ${X}_{i}$ (${\mathrm{m}}^{2}$ ${\mathrm{s}}^{-1}$) |

$\Delta t$ | Time step for the calculation of the cumulative porosity ($\mathrm{s}$) |

$\Delta V$ | Electric voltage ($\mathrm{V}$) |

E | Electric field ($\mathrm{V}$ ${\mathrm{m}}^{-1}$) |

${\eta}_{w}$ | Water dynamic viscosity ($\mathrm{Pa}$ $\mathrm{s}$) |

f | Frequency ($\mathrm{Hz}$) |

F | Formation factor (–) |

$\mathcal{F}$ | Faraday constant ($\mathrm{C}$ ${\mathrm{mol}}^{-1}$) |

${\gamma}_{{X}_{i}}$ | Activity coefficient of the ion ${X}_{i}$ (–) |

${j}_{s}$ | Source current density ($\mathrm{A}$ ${\mathrm{m}}^{-2}$) |

${j}_{s}^{diff}$ | Electrochemical coupling current density from ionic concentration gradients ($\mathrm{A}$ ${\mathrm{m}}^{-2}$) |

${j}_{s}^{EK}$ | Electrokinetic coupling current density ($\mathrm{A}$ ${\mathrm{m}}^{-2}$) |

${J}_{tot}$ | Total electric current density ($\mathrm{A}$ ${\mathrm{m}}^{-2}$) |

${k}_{B}$ | Boltzmann constant ($\mathrm{J}$ ${\mathrm{K}}^{-1}$) |

${K}_{{A}_{1}}$ | Acidity constant of bicarbonate ion (–) |

${K}_{{A}_{2}}$ | Acidity constant of carbonate ion (–) |

${K}_{h}$ | Hydration constant (–) |

${K}_{sp}$ | Solubility product of calcite (–) |

${\Lambda}_{{X}_{i}}$ | Molar conductivity of ion ${X}_{i}$ ($\mathrm{S}$ ${\mathrm{m}}^{-2}$ ${\mathrm{mol}}^{-1}$) |

m | Cementation exponent (–) |

${m}_{Ca\phantom{\rule{3.33333pt}{0ex}}cumul}$ | Cumulative mass of calcium (g) |

$\omega $ | Angular frequency ($\mathrm{rad}$ ${\mathrm{s}}^{-1}$) |

$\Omega $ | Saturation index (–) |

${p}_{e}$ | Empirical parameter for temperature compensation (–) |

$Pe$ | Péclet number (–) |

${\mathrm{P}}_{j}$ | One of the measurement electrodes |

${\mathrm{P}}_{ref}$ | Reference electrode |

$\varphi $ | Porosity (–) |

${\varphi}_{end}$ | Final porosity (–) |

${\varphi}_{init}$ | Initial porosity (–) |

Q | Flow rate (${\mathrm{m}}^{3}$ ${\mathrm{s}}^{-1}$) |

${\widehat{Q}}_{v}$ | Volumetric excess charge ($\mathrm{C}$ ${\mathrm{m}}^{-3}$) |

$\mathcal{R}$ | Molar gas constant ($\mathrm{J}$ ${\mathrm{mol}}^{-1}$ ${\mathrm{K}}^{-1}$) |

REV | Representative elementary volume |

${\rho}_{{\mathrm{CaCO}}_{3}}$ | Calcite volumetric mass ($\mathrm{k}$$\mathrm{g}$ ${\mathrm{m}}^{-3}$) |

$\sigma $ | Sample EC ($\mathrm{S}$ ${\mathrm{m}}^{-1}$) |

${\sigma}_{bulk}$ | Bulk conductivity ($\mathsf{\mu}$$\mathrm{S}$ $\mathrm{c}$${\mathrm{m}}^{-1}$) |

${\sigma}_{in}$ | Inlet pore water EC ($\mathsf{\mu}$$\mathrm{S}$ $\mathrm{c}$${\mathrm{m}}^{-1}$) |

${\sigma}_{out}$ | Outlet pore water EC ($\mathsf{\mu}$$\mathrm{S}$ $\mathrm{c}$${\mathrm{m}}^{-1}$) |

${\sigma}_{surf}$ | Surface conductivity ($\mathrm{S}$ ${\mathrm{m}}^{-1}$) |

${\sigma}_{w}$ | Pore water electrical conductivity ($\mathrm{S}$ ${\mathrm{m}}^{-1}$) |

t | Time ($\mathrm{s}$) |

T | Temperature ($\mathrm{K}$) |

${t}_{{X}_{i}}$ | Transference number (–) |

$\theta $ | Temperature ($\xb0\mathrm{C}$) |

U | Darcy velocity ($\mathrm{m}$ ${\mathrm{s}}^{-1}$) |

v | Particle velocity ($\mathrm{m}$ ${\mathrm{s}}^{-1}$) |

V | Electric potential ($\mathrm{V}$) |

${V}_{tot}$ | Volume of the column (${\mathrm{m}}^{3}$) |

$\left({X}_{i}\right)$ | Ionic activity of the ion ${X}_{i}$ (–) |

${z}_{{X}_{i}}$ | Valence of the ion ${X}_{i}$ (–) |

## Appendix A. Temperature Monitoring

**Figure A1.**Temperature monitoring of the room (gray curve), the inlet pore water (black triangles), and the outlet pore water (blue triangle).

## Appendix B. Electro-Diffusive Coupling Computation from Pore Water EC Computation

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**Figure 1.**Numerical workflow that couples geochemical simulation (blue dashed framework) with geophysical monitoring (red dashed framework) for the computation of the electro-diffusive potential.

**Figure 2.**(

**a**) Schematic drawing of the experimental setup. C${}_{1}$ and C${}_{2}$ are the current electrodes made of stainless steel, while P${}_{1}$ to P${}_{4}$ are the silver–silver chloride (Ag-AgCl) unpolarizable potential electrodes. (

**b**) Picture of the setup during geo-electrical measurements. (

**c**) Picture of the column, which is a Plexiglas cylinder filled with calcite grains. The tightening structure clamps the column and the current electrodes with four threaded rods between two rigid plates.

**Figure 3.**Timeline of the experiment successive stages and events. Blue, pink, and green rectangles indicate the different stages: Stage I consists of the initialization to reach a stationary state close to equilibrium with calcite; Stage II corresponds to calcite dissolution with the constant injection of hydrochloric acid (solution S1); Stage III refers to calcite precipitation with the constant injection of a calcite over-saturated solution (S2). The four stars of the timeline are related to specific events. Event 0 refers to the turning on of the peristaltic pump. Event 1 refers to the connection between the inlet and outlet for recirculation of the solution in the sample. Event 2 refers to the disconnection between the inlet and outlet compartments and to the start of injecting solution S1. Event 3 refers to the start of injection of solution S2.

**Figure 4.**Measurements were performed during the laboratory experiment. The dashed boxes are the delimitation of the zoomed-in views presented in Figure 5. (

**a**) Sample EC. (

**b**) Inlet and outlet pore water EC. The data gap of the outlet water EC curve comes from an acquisition interruption due to the conductivity meter flat battery. (

**c**) SP curves were measured on three channels of acquisition, using electrode P${}_{4}$ as the reference. The white squares represent the interruption of SP acquisition to measure sample EC instead. (

**d**–

**f**) pH, calcium concentration ${C}_{{\mathrm{Ca}}^{2+}}$, and alkalinity of the water sampled at the outlet of the column, respectively. (

**g**) Saturation index for calcite computed from outlet pore water chemical analysis.

**Figure 5.**Zoomed views of Figure 4 centered on Events 2 and 3. (

**a**,

**b**) Sample EC measured on three channels of acquisition corresponding to different dipoles of potential electrodes. (

**c**,

**d**) Inlet and outlet pore water EC. (

**e**,

**f**) SP curves measured on three channels of acquisition corresponding to different dipoles of potential electrodes, using electrode P${}_{4}$ as the reference. The white squares represent the interruption of SP acquisition to measure sample EC instead. (

**g**,

**h**) Calcium concentration, alkalinity, and pH variations.

**Figure 7.**Measured and modeled pH, calcium concentration, and bicarbonate concentration. The data points come from the chemical analyses conducted on the outlet pore water samples, and the curves are obtained from CrunchFlow simulations. (

**a**,

**c**,

**e**) Comparison between the measured data and the corresponding simulations for dissolution. (

**b**,

**d**,

**f**) Comparison between the measured data and the corresponding simulations for precipitation.

**Figure 8.**(

**a**,

**b**) SP data were corrected from the linear decrease generated by the intrinsic potential drift of the reference electrode P${}_{4}$. Time zero corresponds to the timing of Events 2 (injection of solution S1) and 3 (injection of solution S2), respectively. (

**c**,

**d**) Electro-diffusive voltage computed from the numerical workflow.

**Figure 9.**Representations along the column. Locations of measurement electrodes P${}_{1}$, P${}_{2}$, P${}_{3}$, and P${}_{4}$ are indicated on top of each graphic. (

**a**) Porosity difference along the column for CrunchFlow simulations of calcite dissolution and precipitation. (

**b**) The SP voltage is modeled at different time steps during calcite dissolution. (

**c**) The SP voltage is modeled at different time steps during calcite precipitation.

**Table 1.**Equations for the carbonate system in equilibrium with calcite in aqueous media, e.g., [30].

Equations | Thermodynamic Constants (25 $\xb0$C) |
---|---|

CO${}_{2}$ + H${}_{2}$O ↔ H${}_{2}$CO${}_{3}$ | ${K}_{h}={10}^{-1.47}$ |

H${}_{2}$CO${}_{3}$ ↔ H${}^{+}$ + HCO${}_{3}^{-}$ | ${K}_{{A}_{1}}={10}^{-6.35}$ |

HCO${}_{3}^{-}$ ↔ H${}^{+}$ + CO${}_{3}^{2-}$ | ${K}_{{A}_{2}}={10}^{-10.33}$ |

Ca${}^{2+}$ + CO${}_{3}^{2-}$ ↔ CaCO${}_{3}$ | ${K}_{sp}={10}^{-8.42}$ |

Ionic Species | Molar Conductivity ${\Lambda}_{{\mathit{X}}_{\mathit{i}}}$ | Mobility ${\mathit{\beta}}_{{\mathit{X}}_{\mathit{i}}}$ |
---|---|---|

($\mathbf{S}$ $\mathbf{c}$$\mathbf{m}$${}^{2}$ ${\mathbf{mol}}^{-1}$) | (${10}^{-3}$ $\mathbf{m}$${}^{2}$ ${\mathbf{s}}^{-1}$ ${\mathbf{V}}^{-1}$) | |

Ca${}^{2+}$ | 119.1 | 0.62 |

H${}^{+}$ | 349.6 | 3.62 |

Na${}^{+}$ | 50.0 | 0.52 |

CaCl${}^{+}$ | 50.9 | 0.53 |

CaHCO${}_{3}^{+}$ | 19.0 | 0.20 |

CaOH${}^{+}$ | 39.1 | 0.41 |

Cl${}^{-}$ | 76.2 | 0.79 |

HCO${}_{3}^{-}$ | 44.3 | 0.46 |

CO${}_{3}^{2-}$ | 143.5 | 0.74 |

NaCO${}_{3}^{-}$ | 22.0 | 0.23 |

HO${}^{-}$ | 197.9 | 2.05 |

**Table 3.**Input parameters used for the CrunchFlow simulations. Input pore water concentration values come from the outlet pore water chemical analyses, and inlet solution concentration values come from S1 and S2 compositions.

Experiment | Dissolution | Precipitation |
---|---|---|

Rock composition | Calcite | |

Reactive specific surface area (m${}_{mineral}^{2}$/m${}_{bulk}^{3}$) | 1.5 | |

Temperature ($\xb0\mathrm{C}$) | from experimental measurements | |

Initial pore water properties | ||

pH | 6.9 | 7.4 |

${C}_{{\mathrm{Ca}}^{2+}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 2.9 | 1.4 |

${C}_{{\mathrm{HCO}}_{3}^{-}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 7.1 | 1.4 |

${C}_{{\mathrm{Cl}}^{-}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 2.0 | 1.0 |

${C}_{{\mathrm{Na}}^{+}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 2.0 | 0.0 |

Inlet solutions properties | ||

pH | 3.0 | 8.5 |

${C}_{{\mathrm{Ca}}^{2+}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | $1.3\times {10}^{-25}$ | 1.2 |

${C}_{{\mathrm{HCO}}_{3}^{-}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 1.4 | 4.8 |

${C}_{{\mathrm{Cl}}^{-}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 1.0 | 2.4 |

${C}_{{\mathrm{Na}}^{+}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | $1.0\times {10}^{-27}$ | 4.9 |

Flow and transport properties | ||

Effective diffusion coefficient ($\mathrm{m}$${}^{2}$ ${\mathrm{s}}^{-1}$) | 3.0$\times {10}^{-9}$ | |

Dispersivity ($\mathrm{m}$) | 0.9$\times {10}^{-2}$ | |

Darcy velocity ($\mathrm{m}$ ${\mathrm{s}}^{-1}$) | 2.7$\times {10}^{-6}$ |

**Table 4.**Experimental conditions, chemical composition from high-performance liquid chromatography (HPLC) and inductively coupled plasma atomic emission spectroscopy (ICP-OES) analysis, experimental pH and alkalinity of the injected solutions (concentrations are given in $\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$), and computed saturation index for calcite and activity coefficients. The bicarbonate concentration is assumed to present identical values with alkalinity measurements in the pH range. ${\gamma}_{1}$ corresponds to the activity coefficient of all ions with a valence of 1 (e.g., ${\mathrm{H}}^{+}$ and ${\mathrm{HCO}}_{3}^{-}$), and ${\gamma}_{2}$ corresponds to the activity coefficient of all ions with a valence of 2 (e.g., ${\mathrm{Ca}}^{2+}$).

Experiment | Dissolution | Precipitation |
---|---|---|

Experimental conditions | ||

Temperature ($\xb0\mathrm{C}$) | 21.8 ± 1.14 | |

Pressure ($\mathrm{bar}$) | 1 | |

pCO${}_{2}$ ($\mathrm{bar}$) | 10${}^{-3.5}$ | |

Sample | Crushed and sifted calcite with diameter | |

ranging from 125 to 250 $\mathsf{\mu}$$\mathrm{m}$ | ||

Inlet solution | S1 | S2 |

Diluted hydrochloric acid | Over-saturated brine | |

Inlet solutions’ properties | ||

pH | 3.0 | 8.5 |

${C}_{{\mathrm{Ca}}^{2+}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 0 | 1.2 |

$Alk$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | - | 4.8 |

${C}_{{\mathrm{Cl}}^{-}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 1.0 | 2.4 |

${C}_{{\mathrm{Na}}^{+}}$ ($\mathrm{m}$$\mathrm{mol}$ ${\mathrm{L}}^{-1}$) | 0 | 4.9 |

Saturation index and activity coefficients | ||

$\Omega $ | 0 | 14 |

${\gamma}_{1}$ | 0.96 | 0.90 |

${\gamma}_{2}$ | - | 0.67 |

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**MDPI and ACS Style**

Rembert, F.; Jougnot, D.; Luquot, L.; Guérin, R.
Interpreting Self-Potential Signal during Reactive Transport: Application to Calcite Dissolution and Precipitation. *Water* **2022**, *14*, 1632.
https://doi.org/10.3390/w14101632

**AMA Style**

Rembert F, Jougnot D, Luquot L, Guérin R.
Interpreting Self-Potential Signal during Reactive Transport: Application to Calcite Dissolution and Precipitation. *Water*. 2022; 14(10):1632.
https://doi.org/10.3390/w14101632

**Chicago/Turabian Style**

Rembert, Flore, Damien Jougnot, Linda Luquot, and Roger Guérin.
2022. "Interpreting Self-Potential Signal during Reactive Transport: Application to Calcite Dissolution and Precipitation" *Water* 14, no. 10: 1632.
https://doi.org/10.3390/w14101632