Stability analysis of dissolution-driven convection in porous media

Abstract

We study the stability of dissolution-driven convection in the presence of a capillary transition zone and hydrodynamic dispersion in a saturated anisotropic porous medium, where the solute concentration is assumed to decay via a first-order chemical reaction. While the reaction enhances stability by consuming the solute, porous media anisotropy, hydrodynamic dispersion, and capillary transition zone destabilize the diffusive boundary layer that is unstably formed in a gravitational field. We perform linear stability analysis, based on the quasi-steady-state approximation, to assess critical times, critical wavenumbers, and neutral stability curves as a function of anisotropy ratio, dispersivity ratio, dispersion strength, material parameter, Bond number, Damköhler number, and Rayleigh number. The results show that the diffusive boundary layer becomes unstable in anisotropic porous media where both the capillary transition zone and dispersion are considered, even if the geochemical reaction is significantly large. Using direct numerical simulations, based on the finite difference method, we study the nonlinear dynamics of the system by examining dissolution flux, interaction of convective fingers, and flow topology. The results of nonlinear simulations confirm the predictions from the linear stability analysis and reveal that the fingering pattern is significantly influenced by combined effects of reaction, anisotropy, dispersion, and capillarity. Finally, we draw conclusions on implications of our results on carbon dioxide sequestration in deep saline aquifers.

Instability of a Diffusive Boundary Layer beneath a Capillary Transition Zone

“…However, the capillary transition zone allows vertical flow across the interface between the capillary transition zone and the diffusive boundary layer, affecting the onset of the natural convection. The effects of the capillary transition zone on the onset of convection have been studied by considering either a permeable upper boundary condition in a single-phase system [2][3][4][5] or a transition zone in a two-phase system [6][7][8]. These studies demonstrated that the capillary transition zone could potentially decrease the onset time of instability by threefold [5], fivefold [4], or even sixfold [6,8].…”

“…In this study, we further examined the effect of the capillary transition zone on the onset of instability and natural convection by extending the linear stability analysis (LSA) presented in [6,8] to remove three assumptions. First, we considered a three-dimensional, two-phase (gas and water), two-component (CO 2 and H 2 O) physical system with Darcy's velocities in x-, y-, and z-direction instead of using two-dimensional stream function equations.…”

“…First, we considered a three-dimensional, two-phase (gas and water), two-component (CO 2 and H 2 O) physical system with Darcy's velocities in x-, y-, and z-direction instead of using two-dimensional stream function equations. Second, we included the variation of relative permeability as a function of distance in the flow equation (i.e., Equation (12) in [6] and Equation (13) in [8]). As demonstrated in the next section, this consideration results in the addition of a saturation term to the stability equation for the perturbed velocity, i.e., Equation (24).…”

“…As demonstrated in the next section, this consideration results in the addition of a saturation term to the stability equation for the perturbed velocity, i.e., Equation (24). Third, we took account of the variation of water-phase density due to CO 2 dissolution in the saturation equation (i.e., Equation (13) in [6] and Equation (14) in [8]), which was neglected in the previous studies [6,8]. As discussed in the subsequent section, this consideration would add five terms to the stability equation for perturbed saturation, i.e., Equation (25).…”

Numerical Analyses of the Effect of the Impurity in the Gas on the Solubility Trapping in the CO2 Sequestration

“…The increase of the density is heterogeneous in the different part of the formation. A local increase of density, coupled with the heterogeneity of the aquifer such as permeability heterogeneity [18] , will result in the instability of the gravity and prompt the onset of the viscous finger [19][20][21][22] . The viscous finger will lead to the convective mix, accelerating the process of CO2 solution in the brine and reducing the possibility of CO2 leakage.…”