# Onset of Convection in the Presence of a Precipitation Reaction in a Porous Medium: A Comparison of Linear Stability and Numerical Approaches

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, potentially leaking through any high permeability zones or artificial penetrations, such as abandoned wells [1]. Convection brings carbon dioxide-rich fluid downward and fresh fluid upward, effectively enhancing the transport of carbon dioxide into the saline aquifer. It was widely believed that these convection streams transport the dissolved and entrapped carbon dioxide in brine efficiently to depth. Recent studies, however, have questioned this very basic assumption. Preliminary results [2,3,4,5] strongly indicated that geochemical reactions between dissolved carbon dioxide and the subsurface rock matrix may have a non-negligible effect on the convective mixing in the boundary layer.

## 2. Model

#### 2.1. Governing Equations and Scaling

#### 2.2. Linear Stability Analysis

## 3. Results

#### 3.1. Growth Rate of the Perturbation

#### 3.2. Maximum Growth Rate and Corresponding Wave Number

^{2}, as shown in Figure 5. The maximum growth rate increases rapidly at early times, as the solute diffuses and accumulates in the layer, and decreases slowly over a long period of time. As Da/Ra

^{2}increases, the maximum growth rate decreases, as the density of the layer decreases owing to the removal of the solute in the form of a precipitating product from the system. The most dangerous wave number decreases from 0.075 to 0.03 as time varies between 100 and 2000. The most dangerous wave number is weakly dependent on Da/Ra

^{2}, decreasing slightly for higher Da/Ra

^{2}.

#### 3.3. Marginal Stability Curves

^{2}, convection develops over a finite period of time only. A further increase in reaction strength leads to significant shrinkage of the unstable zone. Above a critical value of 10

^{3}Da/Ra

^{2}~ 2.1, using a dominant mode analysis, the reaction stabilizes the system completely.

#### 3.4. Time for Onset of Convection

#### 3.5. Critical Wave Number at the Onset of Instability

## 4. Discussion and Conclusions

^{2}, and porosity, $\phi =0.12$, we estimate that $Da/R{a}^{2}=2\times {10}^{-3}$. In this situation, considering $\mu =4.8\times {10}^{-4}\text{}\mathrm{kg}/\left(\mathrm{m}\xb7\mathrm{s}\right),\text{}{D}_{A}=3.7\times {10}^{-9}{\mathrm{m}}^{2}/\mathrm{s},\Delta {\rho}_{0}=10.6\mathrm{kg}/{\mathrm{m}}^{3}$, we predict the time for onset of instability to be $~$1.5 month and $~$1 month using the dominant mode analysis and initial value problem analysis, respectively. The time for onset of convection based on nonlinear simulation is estimated to be$~$10 months. This means that although the reactive-diffusive layer becomes unstable ~1–1.5 months after the injection of carbon dioxide into the saline formation, substantial convective motion starts only ~10 months after injection. Only at this time does the dissolution flux of carbon dioxide into the brine increase significantly.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Solute $A$ dissolves in the underlying fluid, forming a diffusive boundary layer. Dissolution enhances the local density difference between the solute-saturated fluid at the interface and the underlying pure fluid, driving finger formation and convection. Dissolved solute A reacts with reactant species $B$ in the host rock, forming product $C$, which precipitates out from the fluid. Chemical reaction alters the spatial distribution of the solute and thereby changes the density field.

**Figure 2.**Variation of the growth rate of perturbations with wave number and time for the inert system $\left(Da/R{a}^{2}=0\right)$ using the dominant mode analysis.

**Figure 3.**Variation of the growth rate of perturbations with wave number and time for a reactive system: (

**a**) weak reaction ($Da/R{a}^{2}=0.5\times {10}^{-3}$); (

**b**) strong reaction ($Da/R{a}^{2}=2.09\times {10}^{-3}$), using the dominant mode analysis.

**Figure 4.**Growth rate of perturbations as a function of wave number for different $Da/R{a}^{2}$ at non-dimensional $t=500$.

**Figure 5.**The maximum growth rate of perturbations $({\sigma}_{1pmax})$ and the corresponding most unstable wave number $\left({k}_{max}\right)$ as a function of time, for three different $Da/R{a}^{2}$, using the dominant mode analysis.

**Figure 6.**Effect of a precipitating reaction on the marginal stability boundary using (

**a**) an initial-value problem approach [7]; (

**b**) the dominant mode analysis; and (

**c**) the quasi-steady state assumption (QSSA) method.

**Figure 7.**Comparison of neutral stability curves obtained using an initial-value problem approach (IVP), the dominant mode analysis (DM), and the quasi-steady state assumption (QSSA) method for (

**a**) $Da/R{a}^{2}=0$; (

**b**) $Da/R{a}^{2}=1.5\times {10}^{-3}$; and (

**c**) $Da/R{a}^{2}=2\times {10}^{-3}$.

**Figure 8.**Non-dimensional time for onset and cessation of convection (dimensionless) as a function of $Da/R{a}^{2}$ for a precipitating reaction, predicted using the dominant mode analysis (DM), initial-value problem approach (IVP), and nonlinear numerical simulations (NNS).

**Figure 9.**The most unstable wave number at the onset of instability (dimensionless) as a function of $Da/R{a}^{2}$ for a precipitating reaction, predicted using the dominant mode analysis (DM), the initial-value problem approach (IVP), and the quasi-steady state assumption (QSSA).

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**MDPI and ACS Style**

Ghoshal, P.; Kim, M.C.; Cardoso, S.S.S.
Onset of Convection in the Presence of a Precipitation Reaction in a Porous Medium: A Comparison of Linear Stability and Numerical Approaches. *Fluids* **2018**, *3*, 1.
https://doi.org/10.3390/fluids3010001

**AMA Style**

Ghoshal P, Kim MC, Cardoso SSS.
Onset of Convection in the Presence of a Precipitation Reaction in a Porous Medium: A Comparison of Linear Stability and Numerical Approaches. *Fluids*. 2018; 3(1):1.
https://doi.org/10.3390/fluids3010001

**Chicago/Turabian Style**

Ghoshal, Parama, Min Chan Kim, and Silvana S. S. Cardoso.
2018. "Onset of Convection in the Presence of a Precipitation Reaction in a Porous Medium: A Comparison of Linear Stability and Numerical Approaches" *Fluids* 3, no. 1: 1.
https://doi.org/10.3390/fluids3010001