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**Onset of convection over a transient base-state in anisotropic and layered porous media.**
*(English)*
Zbl 1183.76716

Summary: The topic of density-driven convection in porous media has been the focus of many recent studies due to its relevance as a long-term trapping mechanism during geological sequestration of carbon dioxide. Most of these studies have addressed the problem in homogeneous and anisotropic permeability fields using linear-stability analysis, and relatively little attention has been paid to the analysis for heterogeneous systems. Previous investigators have reduced the governing equations to an initial-value problem and have analysed it either with a quasi-steady-state approximation model or using numerical integration with arbitrary initial perturbations. Recently, Rapaka et al. (J. Fluid Mech., vol. 609, 2008, pp. 285-303) used the idea of non-modal stability analysis to compute the maximum amplification of perturbations in this system, optimized over the entire space of initial perturbations. This technique is a mathematically rigorous extension of the traditional normal-mode analysis to non-normal and time-dependent problems. In this work, we extend this analysis to the important cases of anisotropic and layered porous media with a permeability variation in the vertical direction. The governing equations are linearized and reduced to a set of coupled ordinary differential equations of the initial-value type using the Galerkin technique. Non-modal stability analysis is used to compute the maximum growth of perturbations along with the optimal wavenumber leading to this growth. We show that unlike the solution of the initial-value problem, results obtained using non-modal analysis are insensitive to the choice of bottom boundary condition. For the anisotropic problem, the dependence of critical time and wavenumber on the anisotropy ratio was found to be in good agreement with theoretical scalings proposed by Ennis-King et al. (Phys. Fluids, vol. 17, 2005, paper no. 084107). For heterogeneous systems, we show that uncertainty in the permeability field at low wavenumbers can influence the growth of perturbations. We use a Monte Carlo approach to compute the mean and standard deviation of the critical time for a sample permeability field. The results from theory are also compared with finite-volume simulations of the governing equations using fully heterogeneous porous media with strong layering. We show that the results from non-modal stability analysis match extremely well with those obtained from the simulations as long as the assumption of strong layering remains valid.

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### References:

[1] | DOI: 10.1017/S030500410002452X |

[2] | DOI: 10.1017/S0022112065000332 · Zbl 0224.76048 |

[3] | DOI: 10.1063/1.2759978 · Zbl 1182.76376 |

[4] | Joseph, Stability of Fluid Motions I (1976) |

[5] | DOI: 10.1017/S0022112082000135 · Zbl 0493.76087 |

[6] | IPCC Special Report on Carbon Dioxide Capture and Storage. Prepared by Working Group III of the Intergovernmental Panel on Climate Change (B. Metz, O. Davidson, H. C. de Coninck, M. Loos and L. A. Meyer) (2005) |

[7] | DOI: 10.1063/1.1707601 · Zbl 0063.02071 |

[8] | Hitchon, Aquifer Disposal of Carbon Dioxide: Hydrodynamic and Mineral Trapping (1996) |

[9] | DOI: 10.1007/s11242-005-6088-1 |

[10] | DOI: 10.1016/j.advwatres.2005.05.008 |

[11] | Golub, Matrix Computations (1996) |

[12] | Wooding, NZ J. Sci. 27 pp 219– (1978) |

[13] | DOI: 10.1017/S0022112091002422 · Zbl 0718.76100 |

[14] | DOI: 10.1063/1.1761393 |

[15] | Trefethen, Spectra and Pseudospectra: The Behaviour of Nonnormal Matrices and Operators (2005) |

[16] | Ferziger, Computational Methods for Fluid Dynamics (2001) |

[17] | DOI: 10.1063/1.865832 · Zbl 0608.76087 |

[18] | DOI: 10.1175/1520-0469(1996)053<2041:GSTPIN>2.0.CO;2 |

[19] | Straughan, The Energy Method, Stability and Nonlinear Convection (2004) · Zbl 1032.76001 |

[20] | DOI: 10.1111/j.1745-6584.2006.00197.x |

[21] | DOI: 10.1115/1.1351164 |

[22] | DOI: 10.1016/S0169-7722(01)00160-7 |

[23] | DOI: 10.1146/annurev.fluid.38.050304.092139 |

[24] | DOI: 10.1175/1520-0469(1996)053<2025:GSTPIA>2.0.CO;2 |

[25] | DOI: 10.1017/S0022112005007494 |

[26] | Epherre, Intl Chem. Engng 17 pp 615– (1977) |

[27] | DOI: 10.1063/1.2033911 · Zbl 1187.76141 |

[28] | DOI: 10.1017/S0022112008002607 · Zbl 1147.76030 |

[29] | DOI: 10.1017/S0022112067000576 |

[30] | DOI: 10.1029/2002WR001290 |

[31] | DOI: 10.1017/S0022112067000850 |

[32] | DOI: 10.1115/1.2910390 |

[33] | Cheng, Adv. Heat Transfer 14 pp 1– (1978) |

[34] | DOI: 10.1007/s11242-006-9045-8 |

[35] | Castinel, Intl Chem. Engng 17 pp 605– (1977) |

[36] | DOI: 10.1007/s11242-008-9297-6 |

[37] | DOI: 10.1017/S0022112093002162 · Zbl 0780.76072 |

[38] | Nield, Convection in Porous Media (2006) · Zbl 1256.76004 |

[39] | DOI: 10.1063/1.1694096 |

[40] | DOI: 10.1023/A:1006549930540 |

[41] | DOI: 10.1016/0142-727X(94)90019-1 |

[42] | DOI: 10.1017/S0022112082001104 · Zbl 0483.76094 |

[43] | DOI: 10.1017/S0022112080002170 · Zbl 0427.76072 |

[44] | DOI: 10.1029/WR018i005p01379 |

[45] | DOI: 10.1017/S0022112079002445 · Zbl 0393.76057 |

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