[Paper Review] Rayleigh fractionation in high-Rayleigh-number solutal convection in porous media

Baole Wen, Marc A. Hesse|arXiv (Cornell University)|Jan 9, 2018

CO2 Sequestration and Geologic Interactions38 references3 citations

TL;DR

This study challenges the classical Rayleigh fractionation assumption in high-Rayleigh-number solutal convection within porous media by demonstrating that tracer fluxes depend on diffusion coefficients, not just solubility. Direct numerical simulations show that for tracer diffusivity $ D_2 \geq 10D_1 $, transverse leakage between convective plumes reduces tracer flux, invalidating Rayleigh fractionation except in the limit of extreme solubility differences.

ABSTRACT

We study the fractionation of two components between a well-mixed gas and a saturated convecting porous layer. Motivated by geological carbon dioxide (CO$_2$) storage we assume that convection is driven only by the dissolved concentration of the first component, while the second acts as a tracer with increased diffusivity. Direct numerical simulations for convection at high Rayleigh numbers reveal that the partitioning of the components, in general, does not follow a Rayleigh fractionation trend, as commonly assumed. Initially, increases in tracer diffusivity also increase its flux, because the diffusive boundary layer penetrates deeper into the flow. However, for $D_2\geq 10\, D_1$, where $D_1$ and $D_2$ are, respectively, the diffusion coefficients of CO$_2$ and the tracer in water, the transverse leakage of tracer between up- and down-welling plumes reduces the tracer flux. Rayleigh fractionation between components is only realized in the limit of two gases with very large differences in solubility and initial concentration in the gas.

Motivation & Objective

• To test the validity of Rayleigh fractionation in high-Rayleigh-number solutal convection within porous media.
• To investigate how differences in diffusion coefficients between CO2 and a tracer (e.g., He) affect mass transfer and gas composition evolution.
• To determine under what conditions the classical Rayleigh fractionation model remains applicable in geological CO2 storage scenarios.
• To provide a physically based correction to field-based estimates of CO2 dissolution rates that rely on He/CO2 ratio changes.

Proposed method

• Direct numerical simulations (DNS) of solutal convection in a porous medium at high Rayleigh numbers.
• Modeling two-component system: CO2 drives convection via solutal buoyancy, while a second component acts as a tracer with higher diffusivity.
• Deriving a dimensionless flux ratio $ F_i $ from DNS to compute the fractionation factor $ \alpha = F_1K_1 / (F_2K_2) $, where $ K_i $ is Henry’s law solubility constant.
• Using a quasi-steady-state assumption to relate gas-phase composition changes to convective fluxes via $ dn_{i,g}/dt = -F_i C_{is} q $, with $ q $ proportional to diffusivity and area.
• Applying asymptotic scaling analysis to identify regimes where Péclet number $ \text{Pe} \sim O(1) $, indicating balance between advection and diffusion.
• Deriving the gas composition evolution equation $ \mathcal{F} = 1 - (r/r^0)^{\alpha/(α-1)} $, which reduces to Rayleigh fractionation only when $ \alpha \gg 1 $.

Experimental results

Research questions

• RQ1 Does Rayleigh fractionation accurately describe component partitioning in high-Rayleigh-number porous convection when diffusion coefficients differ?
• RQ2 How does increasing tracer diffusivity affect the flux of the tracer relative to CO2 in convecting porous media?
• RQ3 Under what conditions does the classical assumption of identical fluxes ($ F_1 = F_2 $) break down in multi-component convection?
• RQ4 To what extent do transverse plume interactions reduce tracer leakage and disrupt fractionation trends?
• RQ5 Is Rayleigh fractionation valid for noble gas tracers (e.g., He, Ne, Ar) used in field estimates of CO2 dissolution in carbon sequestration?

Key findings

• For $ D_2 \geq 10D_1 $, transverse leakage of the tracer between upwelling and downwelling plumes reduces its flux, contradicting the assumption of Rayleigh fractionation.
• Rayleigh fractionation is only realized in the limit of very large solubility differences ($ K_1/K_2 \gg 1 $), not generally when diffusivities differ.
• When solubility constants are similar ($ K_1/K_2 \sim O(1) $), the flux difference $ F_1/F_2 $ has a first-order effect on fractionation, invalidating the standard model.
• The fractionation factor $ \alpha = F_1K_1/(F_2K_2) $ depends on both solubility and diffusivity, not just solubility as assumed in Rayleigh fractionation.
• For noble gases like He, Ne, and Ar with $ K_1/K_2 > 20 $, the $ O(1) $ variation in $ F_1/F_2 $ does not significantly affect the approximation $ \mathcal{F} \approx 1 - r/r^0 $, so Rayleigh fractionation holds.
• The classical assumption of negligible molecular diffusion in the 'ultimate' high-Rayleigh regime is invalid for mass transfer, as diffusion strongly influences fluxes even at high Ra.