Onset of convection in a gravitationally unstable diffusive boundary layer in porous media

Abstract

We present a linear stability analysis of density-driven miscible flow in porous media in the context of carbon dioxide sequestration in saline aquifers. Carbon dioxide dissolution into the underlying brine leads to a local density increase that results in a gravitational instability. The physical phenomenon is analogous to the thermal convective instability in a semi-infinite domain, owing to a step change in temperature at the boundary. The critical time for the onset of convection in such problems has not been determined accurately by previous studies. We present a solution, based on the dominant mode of the self-similar diffusion operator, which can accurately predict the critical time and the associated unstable wavenumber. This approach is used to explain the instability mechanisms of the critical time and the long-wave cutoff in a semi-infinite domain. The dominant mode solution, however, is valid only for a small parameter range. We extend the analysis by employing the quasi-steady-state approximation (QSSA) which provides accurate solutions in the self-similar coordinate system. For large times, both the maximum growth rate and the most dangerous mode decay as $t^{1/4}$. The long-wave and the short-wave cutoff modes scale as $t^{1/5}$ and $t^{4/5}$, respectively. The instability problem is also analysed in the nonlinear regime by high-accuracy direct numerical simulations. The nonlinear simulations at short times show good agreement with the linear stability predictions. At later times, macroscopic fingers display intense nonlinear interactions that significantly influence both the front propagation speed and the overall mixing rate. A dimensional analysis for typical aquifers shows that for a permeability variation of 1—3000 mD, the critical time can vary from 2000 yrs to about 10 days while the critical wavelength can be between 200 m and 0.3 m.

Convective Instability and Mass Transport of Diffusion Layers in a Hele-Shaw Geometry

“…However, similar to thermal layers [3], the diffusion layer must thicken enough to become unstable. Analytical and numerical calculations have been performed [4][5][6][7][8][9] to predict the instability incubation time and subsequent mass transport in CO 2 -brine systems. However, the difficulties in realizing a laboratory experiment for the purposes of testing the scenario are considerable, e.g.…”

“…Non-dimensional scalings and solution techniques for the conservation of momentum, mass, and concentration are described in detail elsewhere [4][5][6][7][8][9]. Here, we discuss the di- mensionless parameters and a few details relevant to our experiment.…”

“…Near the finger tips and the finger cores the Pelect number can approach 50, and the advection-diffusion equation may be modified by Taylor dispersion [17] with the effective diffusion coefficient 15 times larger than molecular diffusion. Neglecting the impact of P e within the finger cores, the Rayleigh number Ra = ∆ρgKH/ρνD governs the flow [4][5][6][7][8][9]. Here, g is the acceleration due to gravity and all fluid properties are evaluated at c w = 0.3.…”

Nonmonotonic Rayleigh-Taylor Instabilities Driven by Gas–Liquid CO2 Chemisorption

“…reason, RTI are also increasingly studied nowadays in the framework of carbon dioxide (CO 2 ) sequestration in deep saline aquifers, [8][9][10] a process for which mixing is expected to add up to the effect of chemical reactions in boosting the overall efficiency. , obtained for the absorption of CO 2 in an aqueous solution of sodium bicarbonate (NaHCO 3 , 790 mol/m 3 ) and sodium carbonate (Na 2 CO 3 , 625 mol/m 3 ) shows a quite unusual behavior.…”

Convective Mixing Induced by Acid–Base Reactions

“…Conversely, one can address the question: how can chemical reactions influence natural convection or even be at the very source of hydrodynamic motion? These issues are at the heart of numerous applications in combustion, 1,2 polymer processing, 3,4 extraction techniques, 5,6 microfluidic devices, 7À9 bioconvection, 10 traveling fronts, 11À13 and CO 2 sequestration, 14,15 to name a few.To answer such questions, experimental studies have for instance investigated chemically driven convective mixing and enhanced extraction from one phase to another, induced by reactions between reactants initially contained separately in immiscible solvents. 5,16À18 In that case, it has been shown that the flow around the interface and within the bulk solutions result from (i) the coupling between transfer of chemical species at the interface, (ii) changes by the reaction of the density of the solutions which can trigger buoyancy-driven convective motions, and (iii) reaction-induced Marangoni effects, that is, fluid motion generated by surface tension changes at the immiscible interface.…”