#
Experimental and Simulation Study on Validating a Numerical Model for CO_{2} Density-Driven Dissolution in Water

^{*}

## Abstract

**:**

_{2}concentrations are typically higher. Varying concentrations of carbon dioxide were applied as boundary conditions at the top of the experimental setup, leading to the onset of convective fingering at differing times. The data were used to validate a numerical model implemented in the numerical simulator DuMu

^{x}. The model solves the Navier–Stokes equations for density-induced water flow with concentration-dependent fluid density and a transport equation, including advective and diffusive processes for the carbon dioxide dissolved in water. The model was run in 2D, 3D, and pseudo-3D on two different grids. Without any calibration or fitting of parameters, the results of the comparison between experiment and simulation show satisfactory agreement with respect to the onset time of convective fingering, and the number and the dynamics of the fingers. Grid refinement matters, in particular, in the uppermost part where fingers develop. The 2D simulations consistently overestimated the fingering dynamics. This successful validation of the model is the prerequisite for employing it in situations with background flow and for a future study of karstification mechanisms related to CO

_{2}-induced fingering in caves.

## 1. Introduction

validation is the substantiation that a computerized model within its domain of applicability possesses a satisfactory range of accuracy consistent with the intended application of the model.

## 2. Methods

#### 2.1. Experimental Setup and Procedure

#### 2.2. Numerical Simulation Model

#### 2.2.1. Governing Equations

#### 2.2.2. Numerical Solution

#### 2.2.3. Grids Used in the Simulations

#### 2.2.4. Density Variation

#### 2.2.5. Henry’s Law for Calculating Aqueous CO${}_{2}$ Concentration

#### 2.3. Characterizing Instability

## 3. Results

## 4. Discussion

## 5. Conclusions

- This study is a significant step towards a successful validation of the applied numerical model. It shows very good agreement for the onset times, for the characteristic finger velocities, and for the qualitative fingering patterns between experiments and simulations.
- We aim at applying this model concept in future studies for the evaluation of the dynamics of CO${}_{2}$ migration in the phreatic zone of caves during seasonal fluctuations of CO${}_{2}$ concentrations in cave air. In order to evaluate the relevance of nerochytic speleogenesis for karstification, various aspects need to be extended. This includes geochemical processes at the water-karst interface, very complex scenarios with varying boundary conditions with seasonal variations of CO${}_{2}$ concentration, and water background flow. It is therefore important to conclude from this study that the basic hydraulic processes and dynamics are well reproduced.
- The pseudo-3D model is computationally as cheap as the 2D model, while being capable of capturing the important effect of shear stress such that the metrics for comparison show a very good agreement with the full 3D simulations.
- The 2D model without the consideration of wall friction is not appropriate for capturing the fingering dynamics.
- Our experiments provided very nice results for lower temperatures, but at room temperature we had difficulties achieving good quality images. With the concentration of the color indicator as applied in this study, the experiments at room temperature were not successful for low CO${}_{2}$ concentrations.
- Results from the literature on linear stability analyses for convective dissolution of CO${}_{2}$ in saline aquifers cannot be transferred exactly to this setup. Here, we have Reynolds numbers in the order of 1 or slightly more, meaning that the requirements for a Darcy regime are violated, although not by much. In this transition regime, we observed that the relevant parameters for the onset time of fingers enter a functional description in a different order than in the relation found throughout the porous-media literature. A detailed analysis is, however, beyond the scope of this study.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Photo and schematic of the experimental setup: Flume filled with water and pH-sensitive color indicator; a gas pump to establish a continuous flow from a CO${}_{2}$ bag along the top of the water body. The photo gives an impression of the importance of light in this experimental setup.

**Figure 2.**Image of the extracted green light beam (

**a**) and visualized pixel matrix using a threshold value (

**b**).

**Figure 3.**The numerical grid with cell grading at the top: the left figure shows the entire grid, while the right one presents a close-up of the graded region at the top. All cells below the graded region feature equidistant spacing of 0.006 m, identical to the ungraded grid (not shown). The smallest grid cell at the top of the graded region has 0.0003 m in the vertical extent.

**Figure 4.**Comparison: Numerical results of CO${}_{2}$ mole fractions obtained from the pseudo-3D model (

**left**) and photos of the experiment at 8 °C with ${p}_{{\mathrm{CO}}_{2}}=0.25$ atm (

**right**).

**Figure 5.**Comparison: Numerical results of CO${}_{2}$ mole fractions obtained from the pseudo-3D model (

**left**) and photos of the experiment at 8 °C with ${p}_{{\mathrm{CO}}_{2}}=0.75$ atm (

**right**).

**Figure 6.**Exemplary screenshot of a 3D simulation with ${p}_{{\mathrm{CO}}_{2}}=0.75$ atm at 8 °C after 1 hour showing CO${}_{2}$ concentrations.

**Figure 7.**Onset times as a function of CO${}_{2}$ partial pressure (

**left**) and of Rayleigh number (

**right**) obtained with the graded grid.

**Figure 8.**Onset times as a function of CO${}_{2}$ partial pressure (

**left**) and of Rayleigh number (

**right**) obtained with the regular grid.

**Figure 9.**Characteristic velocities of the protruding fingers as a function of CO${}_{2}$ partial pressure (

**left**) and of Rayleigh number (

**right**) obtained with the graded grid.

**Figure 10.**Characteristic velocities of the protruding fingers as a function of CO${}_{2}$ partial pressure (

**left**) and of Rayleigh number (

**right**) obtained with the regular grid.

**Table 1.**Summarizing the relevant parameters to characterize physical instability: CO${}_{2}$ partial pressure, corresponding equlibrium mole fraction, induced density difference, Rayleigh number, characteristic convective flow velocity, Peclet number, optimal spatial discretization length, and optimal time-step size. The diffusion coefficient is $D=$ 2 × 10${}^{-9}$ m${}^{2}$/s; the dynamic viscosity is $\mu =$ 1.0 × 10${}^{-3}$ Pa s at 20 °C and 1.35 × 10${}^{-3}$ Pa s at 8 °C.

for 20 °C | ${\mathit{p}}_{\mathbf{CO}{}_{2}}$ (atm) | x^{CO2} | $\Delta \mathit{\varrho}$ (kg/m${}^{3}$) | Ra | ${\mathit{v}}_{\mathit{c}}$ (m/s) | Pe | $\Delta \mathit{l}$ (m) | $\Delta \mathit{t}$ (s) |
---|---|---|---|---|---|---|---|---|

1 | 7.04 × 10${}^{-4}$ | 3.17 × 10${}^{-1}$ | 5.18 × 10${}^{6}$ | 2.59 × 10${}^{-2}$ | 7.77 × 10${}^{4}$ | 1.54 × 10${}^{-7}$ | 5.95 × 10${}^{-6}$ | |

0.75 | 5.28 × 10${}^{-4}$ | 2.38 × 10${}^{-1}$ | 3.89 × 10${}^{6}$ | 1.94 × 10${}^{-2}$ | 5.83 × 10${}^{4}$ | 2.06 × 10${}^{-7}$ | 1.06 × 10${}^{-5}$ | |

0.5 | 3.52 × 10${}^{-4}$ | 1.59 × 10${}^{-1}$ | 2.59 × 10${}^{6}$ | 1.29 × 10${}^{-2}$ | 3.89 × 10${}^{4}$ | 3.08 × 10${}^{-7}$ | 2.38 × 10${}^{-5}$ | |

0.25 | 1.76 × 10${}^{-4}$ | 7.93 × 10${}^{-2}$ | 1.30 × 10${}^{6}$ | 6.48 × 10${}^{-3}$ | 1.94 × 10${}^{4}$ | 6.17 × 10${}^{-7}$ | 9.51 × 10${}^{-5}$ | |

0.05 | 3.52 × 10${}^{-5}$ | 1.59 × 10${}^{-2}$ | 2.59 × 10${}^{5}$ | 1.30 × 10${}^{-3}$ | 3.89 × 10${}^{3}$ | 3.08 × 10${}^{-6}$ | 2.37 × 10${}^{-3}$ | |

for 8 °C | ${\mathit{p}}_{\mathbf{CO}{}_{\mathbf{2}}}$ (atm) | x^{CO2} | $\Delta \mathbf{\varrho}$ (kg/m${}^{\mathbf{3}}$) | Ra | ${\mathit{v}}_{\mathit{c}}$ (m/s) | Pe | $\Delta \mathit{l}$ (m) | $\Delta \mathit{t}$ (s) |

1 | 9.99 × 10${}^{-4}$ | 4.50 × 10${}^{-1}$ | 5.45 × 10${}^{6}$ | 2.72 × 10${}^{-2}$ | 8.17 × 10${}^{4}$ | 1.47 × 10${}^{-7}$ | 5.39 × 10${}^{-6}$ | |

0.75 | 7.49 × 10${}^{-4}$ | 3.37 × 10${}^{-1}$ | 4.09 × 10${}^{6}$ | 2.04 × 10${}^{-2}$ | 6.12 × 10${}^{4}$ | 1.96 × 10${}^{-7}$ | 9.58 × 10${}^{-6}$ | |

0.5 | 5.00 × 10${}^{-4}$ | 2.25 × 10${}^{-1}$ | 2.72 × 10${}^{6}$ | 1.36 × 10${}^{-2}$ | 4.09 × 10${}^{4}$ | 2.94 × 10${}^{-7}$ | 2.15 × 10${}^{-5}$ | |

0.25 | 2.50 × 10${}^{-4}$ | 1.13 × 10${}^{-1}$ | 1.36 × 10${}^{6}$ | 6.81 × 10${}^{-3}$ | 2.04 × 10${}^{4}$ | 5.87 × 10${}^{-7}$ | 8.62 × 10${}^{-5}$ | |

0.05 | 5.00 × 10${}^{-5}$ | 2.25 × 10${}^{-2}$ | 2.73 × 10${}^{5}$ | 1.36 × 10${}^{-3}$ | 4.09 × 10${}^{3}$ | 2.93 × 10${}^{-6}$ | 2.15 × 10${}^{-3}$ |

**Table 2.**Onset times observed in the pseudo-3D simulations. Corresponding values of ${c}_{0}$ are then fitted by using Equation (9).

${\mathit{p}}_{{\mathbf{CO}}_{2}}$ (atm) | for 8 °C: | ${\mathit{t}}_{\mathbf{onset}}$ (s) | ${\mathit{c}}_{0}$ | for 20 °C: | ${\mathit{t}}_{\mathbf{onset}}$ (s) | ${\mathit{c}}_{0}$ |
---|---|---|---|---|---|---|

1 | 290 | 1.07 × 10${}^{8}$ | 247 | 8.23 × 10${}^{7}$ | ||

0.75 | 345 | 7.14 × 10${}^{7}$ | 322 | 6.04 × 10${}^{7}$ | ||

0.50 | 387 | 3.56 × 10${}^{7}$ | 393 | 3.28 × 10${}^{7}$ | ||

0.25 | 628 | 1.45 × 10${}^{7}$ | 598 | 1.25 × 10${}^{7}$ | ||

0.05 | 1895 | 1.71 × 10${}^{6}$ | 1829 | 1.53 × 10${}^{6}$ |

${\mathit{p}}_{{\mathbf{CO}}_{2}}$ (atm) | for 8 °C: | ${\mathit{c}}_{0}/$Pe | for 20 °C: | ${\mathit{c}}_{0}/$Pe |
---|---|---|---|---|

1 | 1.31 × 10${}^{3}$ | 1.06 × 10${}^{3}$ | ||

0.75 | 1.17 × 10${}^{3}$ | 1.03 × 10${}^{3}$ | ||

0.50 | 8.72 × 10${}^{2}$ | 8.42 × 10${}^{2}$ | ||

0.25 | 7.07 × 10${}^{2}$ | 6.41 × 10${}^{2}$ | ||

0.05 | 4.19 × 10${}^{2}$ | 3.92 × 10${}^{2}$ |

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**MDPI and ACS Style**

Class, H.; Weishaupt, K.; Trötschler, O.
Experimental and Simulation Study on Validating a Numerical Model for CO_{2} Density-Driven Dissolution in Water. *Water* **2020**, *12*, 738.
https://doi.org/10.3390/w12030738

**AMA Style**

Class H, Weishaupt K, Trötschler O.
Experimental and Simulation Study on Validating a Numerical Model for CO_{2} Density-Driven Dissolution in Water. *Water*. 2020; 12(3):738.
https://doi.org/10.3390/w12030738

**Chicago/Turabian Style**

Class, Holger, Kilian Weishaupt, and Oliver Trötschler.
2020. "Experimental and Simulation Study on Validating a Numerical Model for CO_{2} Density-Driven Dissolution in Water" *Water* 12, no. 3: 738.
https://doi.org/10.3390/w12030738