## Non-modal growth of perturbations in density-driven convection in porous media.(English)Zbl 1147.76030

Summary: In the context of geologic sequestration of carbon dioxide in saline aquifers, much interest has been focused on the process of density-driven convection resulting from dissolution of CO$$_{2}$$ in brine in the underlying medium. Recent investigations have studied the time and length scales characteristic of the onset of convection based on the linear stability theory. It is well known that the non-autonomous nature of the resulting matrix does not allow a normal mode analysis, and previous researchers have either used a quasi-static approximation or solved the initial value problem with arbitrary initial conditions. In this manuscript, we describe and use the recently developed non-modal stability theory to compute maximum amplifications possible, optimized over all possible initial perturbations. Non-modal stability theory also provides us with the structure of the most-amplified (or optimal) perturbations. We also present the details of three-dimensional spectral calculations of the governing equations. The results of the amplifications predicted by non-modal theory compare well to those obtained from spectral calculations.

### MSC:

 76E20 Stability and instability of geophysical and astrophysical flows 76E06 Convection in hydrodynamic stability 76S05 Flows in porous media; filtration; seepage 86A05 Hydrology, hydrography, oceanography

Eigtool
Full Text:

### References:

 [1] DOI: 10.1016/j.advwatres.2005.05.008 [2] DOI: 10.1016/S0735-1933(03)00201-X [3] DOI: 10.1063/1.865832 · Zbl 0608.76087 [4] Straughan, The Energy Method, Stability and Nonlinear Convection. (2004) · Zbl 1032.76001 [5] Schmid, Stability and Transition in Shear Flow. (2001) · Zbl 0966.76003 [6] DOI: 10.1146/annurev.fluid.38.050304.092139 [7] DOI: 10.1063/1.1699561 [8] DOI: 10.1063/1.1707601 · Zbl 0063.02071 [9] DOI: 10.1017/S002211207300008X · Zbl 0262.76027 [10] DOI: 10.1007/s11242-005-6088-1 [11] Golub, Matrix Computations. (1996) [12] Epherre, Intl Chem. Engng 17 pp 615– (1977) [13] DOI: 10.1063/1.2033911 · Zbl 1187.76141 [14] Garcia, Tech. Rep. (2001) [15] Ennis-King, SPE J. 10 pp 349– (2005) [16] DOI: 10.1063/1.1761393 [17] Chandrasekhar, Hydrodynamic and Hydromagnetic Stability. (1961) [18] DOI: 10.1175/1520-0469(1996)0532.0.CO;2 [19] Castinell, Intl Chem. Engng 17 pp 605– (1977) [20] DOI: 10.1175/1520-0469(1996)0532.0.CO;2 [21] Canuto, Spectral Methods in Fluid Dynamics. (1988) [22] DOI: 10.1093/qjmam/33.1.47 · Zbl 0423.73087 [23] DOI: 10.1063/1.1446885 · Zbl 1185.76534 [24] Bauer, The Qualitative Theory of Ordinary Differential Equations: An Introduction. (1969) [25] DOI: 10.1017/S0022112005007494 [26] DOI: 10.1029/2002WR001290 [27] Peyret, Spectral Methods for Incompressible Viscous Flow. (2002) · Zbl 1005.76001 [28] DOI: 10.1002/aic.10896 [29] Nield, Convection in Porous Media. (1999) · Zbl 0924.76001 [30] DOI: 10.1016/B978-008044276-1/50078-7 [31] Lapwood, Proc. Camb. Phil. Soc. 44 pp 508– (1948) [32] Trefethen, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.