Solutal-convection regimes in a two-dimensional porous medium

Abstract

We numerically characterize the temporal regimes for solutal convection from almost first contact to high dissolved solute concentration in a two-dimensional ideal porous layer for Rayleigh numbers $\mathcal{R}$ between $100$ and $5\times 10^4$. The lower boundary is impenetrable. The upper boundary is saturated with dissolved solute and either impermeable or partially permeable to fluid flow. In the impermeable case, initially there is pure diffusion of solute away from the upper boundary, followed by the birth and growth of convective fingers. Eventually fingers interact and merge, generating complex downwelling plumes. Once the inter-plume spacing is sufficient, small protoplumes reinitiate on the boundary layer and are swept into the primary plumes. The flow is now in a universal regime characterized by a constant (dimensionless) dissolution flux $F=0.017$ (the rate at which solute dissolves from the upper boundary). The horizontally averaged concentration profile stretches as a simple self-similar wedge beneath a diffusive horizontal boundary layer. Throughout, the plume width broadens proportionally to $\sqrt{t}$, where $t$ is (dimensionless) time. The above behaviour is parameter independent; the Rayleigh number only controls when transition occurs to a final $\mathcal{R}$-dependent shut-down regime. For the constant-flux and shut-down regimes, we rigourously derive upscaled equations connecting the horizontally averaged concentration, vertical advective flux and plume widths. These are partially complete; a universal expression for the plume width remains elusive. We complement these governing equations with phenomenological boundary conditions based on a marginally stable diffusive boundary layer at the top and zero advective flux at the bottom. Making appropriate approximations in each regime, we find good agreement between predictions from this model and simulated results for both solutal and thermal convection. In the partially permeable upper boundary case, fluid from the convecting layer can penetrate an overlying separate-phase-solute bearing layer where it immediately saturates. The regime diagram remains almost the same as for the impermeable case, but the dissolution flux is significantly augmented. Our work is motivated by dissolution of carbon dioxide relevant to geological storage, and we conclude with a simple flux parameterization for inclusion in gravity current models and suggest that the upscaled equations could lay the foundation for accurate inclusion of dissolution in reservoir simulators.

Plume dynamics in Hele-Shaw porous media convection

“…There is possibly a bit of negative skewness to the distribution, indicating an enhanced tendency for the plumes to slow down as in figure 5. The mean value of the plume velocity is roughly consistent with the numerical simulation value for two-dimensional porous media solutal convection at a rigid horizontal interface of 0.1 [10] although our values are somewhat less, in the range 0.03 < ṽ < 0.09.…”

“…The time scales are related: τ L = (L/H)τ c . As discussed in [10,15], the Rayleigh number Ra = g( ρ/ρ)KH/(Dν) only becomes important when the plumes reach the bottom, and during the time interval of study mass flux is predicted to be independent of Ra in the ideal two-dimensional limit. We will discuss aspects of these ideas in the context of our velocity and mass transport measurements.…”

“…Here, we describe details of our analogue approach [13] and concentrate on properties of our system rather than upon justifying the analogy or reviewing other work in this area (see, for example, [16] for an analysis of the limitations of this analogy). Of particular interest in this short report is the theoretical and experimental analysis of Hele-Shaw flows [10,15].…”

A short review on porous media convection

“…Buoyancy-driven convection in a fluid-saturated porous layer has been extensively studied due to its numerous applications in oil recovery [1], groundwater flow and geothermal energy extraction [2][3][4][5][6][7], transport in biological tissues [8], and carbon dioxide sequestration [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Moreover, porous media convection is also used as a classical example to study instabilities, bifurcations, pattern formation, and spatiotemporally chaotic dynamics [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”

A step function density profile model for the convective stability of CO $$_2$$ 2 geological sequestration

“…The approximations varied from truncated infinite series to similarity type solutions when the finite depth is asymptotically extended to infinity. Due to the time dependence of the nonlinear base profile, the instability threshold conditions are then expressed in terms of either critical times at which the boundary layer instability sets in [7] or in terms of growth rates of the critical modes corresponding to the most dangerous disturbance [23]- [24]. The critical time for instability corresponds to a boundary layer having reached its critical thickness so that it is prone to convective overturning.…”