Onset and cessation of time-dependent, dissolution-driven convection in porous media

Abstract

Motivated by convection in the context of geological carbon dioxide sequestration, we present the conditions for free, dissolution-driven convection in a horizontal, ideal porous layer from a time-dependent, pure-diffusion base state. We assume that solute as a separate phase is instantaneously placed in the pores above a given horizontal level at time zero, and gradually diffuses into the underlying liquid. As the concentration of dissolved solute in the liquid increases, its density increases and the system may eventually become gravitationally unstable and convection may begin. We define the amplitude of a perturbation as the mean square of the difference of the concentration profile and the pure-diffusion profile. To identify instability, we calculate the maximum possible instantaneous growth rate of the amplitude over all possible infinitesimal and finite perturbations. Instability exists where this growth rate is positive. We consider two scenarios. In the first scenario, the underlying liquid cannot penetrate into the upper region occupied by the separate, solute-rich phase. In this case, no instability is possible for thin porous layers corresponding to Rayleigh–Darcy numbers less than 32.50. For thicker layers, convection can occur at finite, nonzero times and wavenumbers. The earliest possible onset becomes independent of layer thickness for Rayleigh–Darcy numbers above about 75, and occurs at a time of approximately 47.9D(μϕ/[KΔρ(1−ci∗/cm∗)g])2 after solute placement, where D is the effective diffusivity, μ the liquid viscosity, ϕ the porosity and K the permeability of the medium, g the acceleration due to gravity, Δρ=ρm−ρ0 with ρm the density of saturated liquid and ρ0 the density of pure liquid, cm∗ the maximum solute concentration, and ci∗ the initial average concentration. In the second scenario, the liquid phase in the upper layer remains connected; the liquid beneath can penetrate the horizontal boundary between the two layers and immediately becomes saturated. In this case, the onset occurs sooner and increasingly so for thicker layers. We also present the mode profiles and discuss the implications for sequestration.

Impact of pressure, salt concentration, and temperature on the convective dissolution of carbon dioxide in aqueous solutions

“…These onset time and wavenumber are in good agreement with those found in several previous theoretical studies [25][26][27][28][29][30][31] . Additionally, these onset characteristics are close to those evaluated experimentally in systems analogous to the CO 2 /water system 9,11 .…”

High-Performance Modeling of Carbon Dioxide Sequestration by Coupling Reservoir Simulation and Molecular Dynamics

“…Different models have been used to describe the impact of CO 2 dissolution on the density of the solution. The following linear relation is frequently used (Slim and Ramakrishnan 2010):…”

“…CO 2 sequestration is our first example for coupling reservoir simulation and MD. Following Ennis-King et al (2005), Slim and Ramakrishnan (2010), and Allen and Sun (2012), a single-phase CO 2 -transport model is used to describe the density-driven-flow process during CO 2 sequestration. In the process, the diffusive and convective transport behavior of the dissolved CO 2 in the porous media is described by the transport equation…”

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.…”

“…Other investigations have considered approximations to the unsteady pure diffusion boundary-layer profile (cf. [7], [21]- [23]) and references therein) to take into account the development of the carbon saturated boundary layer and its instability. The approximations varied from truncated infinite series to similarity type solutions when the finite depth is asymptotically extended to infinity.…”