^{1,2,a)}and Abbas Firoozabadi

^{1,3,b)}

### Abstract

We perform a linear stability analysis to examine the onset of buoyancy-driven convection relevant to subsurface carbon dioxide sequestration in confined, porous Cartesian and cylindrical domains. Our work amends the analysis in an earlier study on cylindrical geometries. We consider Cartesian geometries where the aspect ratio between the two horizontal dimensions is not necessarily equal to one. Two key elements of the stability analysis are: (1) the critical time and (2) the critical wavenumber. Lateral boundaries have a much greater influence on the critical wavenumber than on the critical time. The confinement due to these boundaries impedes the onset of convection to the extent that convection cannot even occur in domains that are smaller than a certain size. Large aspect ratios can significantly reduce boundary effects. Patterns of the earliest-growing perturbation mode in the horizontal plane reveal many interesting dynamics which have not been examined in previous stability analyses. We illustrate several differences between patterns in Cartesian geometries and patterns in cylindrical geometries. Based on observations from earlier papers, we hypothesize that the contrasts between the Cartesian and cylindrical patterns may lead to significantly different behavior in the two geometries after the onset of convection. Our results may guide future numerical studies that can investigate this hypothesis and may help with understanding the onset of buoyancy-driven convection in real systems where lateral boundary effects are significant.

Financial support for this work has been provided by the member companies of the Reservoir Engineering Research Institute.

I. INTRODUCTION

II. FORMULATION

A. Governing equations and initial/boundary conditions

B. Solution to the linearized perturbation equations

III. THE CRITICAL TIME AND THE CRITICAL WAVENUMBER

A. Unconfined domains

B. Cartesian versus cylindrical domains

C. Influence of the aspect ratio in Cartesian geometries

D. Laboratory-scale experiments and numerical simulations

IV. PATTERNS OF THE EARLIEST-GROWING MODE IN THE HORIZONTAL PLANE

A. Indices of the earliest-growing mode

B. Patterns in Cartesian geometries

C. Patterns in cylindrical geometries

D. Cartesian versus cylindrical geometries

V. CONCLUSIONS

### Key Topics

- Convection
- 43.0
- Carbon dioxide
- 28.0
- Boundary value problems
- 15.0
- Dissolution
- 13.0
- Porous media
- 8.0

## Figures

(a) Marginal stability (σ = 0) contour in dimensionless wavenumber–time space in an unconfined domain; (b) zoomed-in view around the critical time and critical wavenumber . The base state is unstable in the shaded region above the contour and is stable under it. The dashed line represents the large wavenumber cutoff of . Perturbation modes with wavenumbers larger than this cutoff do not become unstable.

(a) Marginal stability (σ = 0) contour in dimensionless wavenumber–time space in an unconfined domain; (b) zoomed-in view around the critical time and critical wavenumber . The base state is unstable in the shaded region above the contour and is stable under it. The dashed line represents the large wavenumber cutoff of . Perturbation modes with wavenumbers larger than this cutoff do not become unstable.

(a) and (b) vs. or ; (c) and (d) vs. or . The aspect ratio A is unity in (a) and (c) so that both and range from 0 to . The dashed line represents or . The behavior of the system is very different depending on whether it lies to the left or the right of the dashed line, as we describe in the text.

(a) and (b) vs. or ; (c) and (d) vs. or . The aspect ratio A is unity in (a) and (c) so that both and range from 0 to . The dashed line represents or . The behavior of the system is very different depending on whether it lies to the left or the right of the dashed line, as we describe in the text.

vs. for various values of the aspect ratio A: (a) A = 10; (b) A = 5; (c) A = 2; (d) A = 0.5; (e) A = 0.2; (f) A = 0.1. The dashed line represents . Large aspect ratios can significantly reduce boundary effects for both small and large values of .

vs. for various values of the aspect ratio A: (a) A = 10; (b) A = 5; (c) A = 2; (d) A = 0.5; (e) A = 0.2; (f) A = 0.1. The dashed line represents . Large aspect ratios can significantly reduce boundary effects for both small and large values of .

vs. in Cartesian geometries at two different aspect ratios: (a) A = 1; (b) A = 0.1. The dashed line represents . Finer gridding in numerical simulations may be necessary to capture the earliest-growing mode in domains where is small, especially when A < 1.

vs. in Cartesian geometries at two different aspect ratios: (a) A = 1; (b) A = 0.1. The dashed line represents . Finer gridding in numerical simulations may be necessary to capture the earliest-growing mode in domains where is small, especially when A < 1.

The indices i and j of the wavenumber of the earliest-growing mode as a function of the width and the aspect ratio A: (a) the index i in the direction; (b) the index j in the direction. The size in this direction ranges from 0 to ; (c) the index j plotted using the same colorbar axis as in (a). The indices generally increase with .

The indices i and j of the wavenumber of the earliest-growing mode as a function of the width and the aspect ratio A: (a) the index i in the direction; (b) the index j in the direction. The size in this direction ranges from 0 to ; (c) the index j plotted using the same colorbar axis as in (a). The indices generally increase with .

The indices l and m of the wavenumber of the earliest-growing mode as a function of the diameter : (a) l; (b) m. The earliest-growing mode is usually non-axisymmetric (m ≠ 0).

The indices l and m of the wavenumber of the earliest-growing mode as a function of the diameter : (a) l; (b) m. The earliest-growing mode is usually non-axisymmetric (m ≠ 0).

Patterns in the plane of the earliest-growing mode for various with A = 1: (a) ; (b) ; (c) ; (d) ; (e) ; (f) . Perturbations tend to be highly localized in very positive (red) or very negative (blue) regions. In most cases, there are channels that connect positive regions to each other and other channels that connect negative regions to each other. Pau et al. 50 have observed similar channels in their numerical simulations.

Patterns in the plane of the earliest-growing mode for various with A = 1: (a) ; (b) ; (c) ; (d) ; (e) ; (f) . Perturbations tend to be highly localized in very positive (red) or very negative (blue) regions. In most cases, there are channels that connect positive regions to each other and other channels that connect negative regions to each other. Pau et al. 50 have observed similar channels in their numerical simulations.

Patterns in the plane of the earliest-growing mode for at two different aspect ratios: (a) A = 2; (b) A = 0.5. Patterns are asymmetric with respect to an interchange of and . Two-dimensional modes tend to be preferred when one of the dimensions is small, as it is in (b).

Patterns in the plane of the earliest-growing mode for at two different aspect ratios: (a) A = 2; (b) A = 0.5. Patterns are asymmetric with respect to an interchange of and . Two-dimensional modes tend to be preferred when one of the dimensions is small, as it is in (b).

Top-down and bird's-eye views of patterns in the plane of the earliest-growing mode for smaller values of with : (a), (b) ; (c), (d) ; (e), (f) .

Top-down and bird's-eye views of patterns in the plane of the earliest-growing mode for smaller values of with : (a), (b) ; (c), (d) ; (e), (f) .

Top-down and bird's-eye views of patterns in the plane (for clarity, only one quarter of the plane is shown) of the earliest-growing mode for larger values of with : (a), (b) ; (c), (d) ; (e), (f) .

Top-down and bird's-eye views of patterns in the plane (for clarity, only one quarter of the plane is shown) of the earliest-growing mode for larger values of with : (a), (b) ; (c), (d) ; (e), (f) .

## Tables

The indices i and j of for various widths with A = 1.

The indices i and j of for various widths with A = 1.

The indices l, m of and the zeros b lm for various diameters .

The indices l, m of and the zeros b lm for various diameters .

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