#
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

- Garcia, J. Density of Aqueous Solutions of CO
_{2}; Technical Report, LBNL Report 49023; Lawrence Berkeley National Laboratory: Berkeley, CA, USA, 2001. - Weir, G.J.; White, S.P.; Kissling, W.M. Vertical convection in an aquifer column under a gas cap of CO
_{2}. Energy Convers. Manag.**1996**, 23, 37–60. [Google Scholar] [CrossRef] - Lindeberg, E.; Wessel-Berg, D. Reservoir storage and containment of greenhouse gases. Transp. Porous Media
**1997**, 38, S229–S234. [Google Scholar] [CrossRef] - IPCC. Special Report on Carbon Dioxide Capture and Storage; Technical Report; Metz, B., Davidson, O., de Conink, H.C., Loos, M., Meyer, L.A., Eds.; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2005. [Google Scholar]
- Ennis-King, J.; Paterson, L. Rate of dissolution due to convective mixing in the underground storage of carbon dioxide. In Greenhouse Gas Control Technologies; Gale, J., Kaya, Y., Eds.; Elsevier: Amsterdam, The Netherlands, 2003; Volume 1, pp. 507–510. [Google Scholar]
- Ennis-King, J.; Paterson, L. Role of convective mixing in the long-term storage of carbon dioxide in deep saline formations. In Proceedings of the Annual Fall Technical Conference and Exhibition, Denver, CO, USA, 5–8 October 2003; Society of Petroleum Engineers: London, UK, 2003. Number SPE-84344. [Google Scholar]
- Riaz, A.; Hesse, M.; Tchelepi, H.; Orr, F. Onset of convection in a gravitationally unstable diffusive boundary layer in porous media. J. Fluid Mech.
**2006**, 548, 87–111. [Google Scholar] [CrossRef] - Hassanzadeh, H.; Pooladi-Darvish, M.; Keith, D. Modelling of convective mixing in CO
_{2}storage. J. Can. Pet. Technol.**2005**, 44. [Google Scholar] [CrossRef][Green Version] - Hassanzadeh, H.; Pooladi-Darvish, M.; Keith, D. Stability of a fluid in a horizontal saturated porous layer: Effect of non-linear concentration profile, initial, and boundary conditions. Transp. Porous Media
**2006**, 65, 193–211. [Google Scholar] [CrossRef] - Hassanzadeh, H.; Pooladi-Darvish, M.; Keith, D. Scaling behavior of convective mixing, with application to geological storage of CO
_{2}. Am. Inst. Chem. Eng. J.**2007**, 53, 1121–1131. [Google Scholar] [CrossRef] - Pau, G.; Bell, J.; Pruess, K.; Almgren, A.; Lijewski, M.; Zhang, K. High resolution simulation and characterization of density-driven flow in CO
_{2}storage in saline aquifers. Adv. Water Resour.**2010**, 33, 443–455. [Google Scholar] [CrossRef] - Green, C.; Ennis-King, J. Steady flux regime during convective mixing in three-dimensional heterogeneous porous media. Fluids
**2018**, 3, 58. [Google Scholar] [CrossRef][Green Version] - Scherzer, H.; Class, H.; Weishaupt, K.; Sauerborn, T.; Trötschler, O. Nerochytische Speläogenese: Konvektiver Vertikaltransport von gelöstem CO
_{2}—Ein Antrieb für Verkarstung in der phreatischen Zone im Bedeckten Karst. Laichinger Höhlenfreund**2017**, 52, 29–35. [Google Scholar] - Oberkampf, W.; Trucano, T. Verification and validation in computational fluid dynamics. Prog. Aerosp. Sci.
**2002**, 38, 209–272. [Google Scholar] [CrossRef][Green Version] - Thomas, C.; Lemaigre, L.; Zalts, A.; D’Onofrio, A.; Wit, A.D. Experimental study of CO
_{2}convective dissolution: The effect of color indicators. Int. J. Greenh. Gas Control**2015**, 42, 525–533. [Google Scholar] [CrossRef] - Flekkøy, E.G.; Oxaal, U.; Feder, J.; Jøssang, T. Hydrodynamic dispersion at stagnation points: Simulations and experiments. Phys. Rev. E
**1995**, 52, 4952–4962. [Google Scholar] [CrossRef] [PubMed] - Venturoli, M.; Boek, E.S. Two-dimensional lattice-Boltzmann simulations of single phase flow in a pseudo two-dimensional micromodel. Phys. A Stat. Mech. Appl.
**2006**, 362, 23–29. [Google Scholar] [CrossRef] - Laleian, A.; Valocchi, A.; Werth, C. An Incompressible, Depth-Averaged Lattice Boltzmann Method for Liquid Flow in Microfluidic Devices with Variable Aperture. Computation
**2015**, 3, 600–615. [Google Scholar] [CrossRef] - Kunz, P.; Zarikos, I.M.; Karadimitriou, N.K.; Huber, M.; Nieken, U.; Hassanizadeh, S.M. Study of Multi-phase Flow in Porous Media: Comparison of SPH Simulations with Micro-model Experiments. Transp. Porous Media
**2015**, 114, 581–600. [Google Scholar] [CrossRef][Green Version] - Koch, T.; Gläser, D.; Weishaupt, K.; Ackermann, S.; Beck, M.; Becker, B.; Burbulla, S.; Class, H.; Coltman, E.; Emmert, S.; et al. DuMu
^{x}3—An open-source simulator for solving flow and transport problems in porous media with a focus on model coupling. arXiv**2019**, arXiv:1909.05052. [Google Scholar] [CrossRef] - Koch, T.; Gläser, D.; Weishaupt, K.; Ackermann, S.; Beck, M.; Becker, B.; Burbulla, S.; Class, H.; Coltman, E.; Fetzer, T.; et al. DuMuX 3.0; Zenodo: Genève, Switzerland, 2018. [Google Scholar] [CrossRef]
- Poling, B.; Prausnitz, J.; O’Connel, J. The Properties of Gases and Liquids; McGraw-Hill, Inc.: New York, NY, USA, 2001. [Google Scholar]
- Unver, A.; Himmelblau, D. Diffusion coefficients of CO
_{2}, C_{2}H_{4}, C_{3}H_{6}and C_{4}H_{8}in water from 6 to 65 C. J. Chem. Eng. Data**1964**, 9, 428–431. [Google Scholar] [CrossRef] - The International Association for the Properties of Water and Steam (IAPWS). Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam. 1997. Available online: http://www.iapws.org/relguide/IF97-Rev.pdf (accessed on 8 March 2020).
- Sander, R. Compilation of Henry’s law constants (version 4.0) for water as solvent. Atmos. Chem. Phys.
**2015**, 15, 4399–4981. [Google Scholar] [CrossRef][Green Version] - Bénard, H. Les tourbillons cellulaires dans une nappe liquide.—Méthodes optiques d’observation et d’enregistrement. J. Phys. Theor. Appl.
**1901**, 10, 254–266. [Google Scholar] [CrossRef] - Lord Rayleigh, O.F. LIX. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1916**, 32, 529–546. [Google Scholar] [CrossRef][Green Version] - Emami-Meybodi, H.; Hassanzadeh, H.; Green, C.; Ennis-King, J. Convective dissolution of CO
_{2}in saline aquifers: Progress in modeling and experiments. Int. J. Greenh. Gas Control**2015**, 40, 238–266. [Google Scholar] [CrossRef] - Gresho, P.; Lee, R. Don’t suppress the wiggles—They’re telling you something! Comput. Fluids
**1981**, 9, 223–253. [Google Scholar] [CrossRef][Green Version]

**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