APPENDIX

The growth rate of a bubble, after nucleation, is predominantly controlled by the mass flux of CH₄ to the bubble. The change in mass within the bubble can be written as

(dn_CH4/dt) = 4r²Q_CH4, (A1)

where Q_CH4 is the CH₄ flux to the bubble per unit time. Q_CH4 depends on the diffusivity of CH₄ in the pore fluid, D^CH4, and its solubility, K_H^CH4, according to

Q_CH4 = Sh(D^CH4/r)K_H^CH4[P_w^CH4 – P_bub + P_H2O + (2/4)], (A2)

where

Sh = Sherwood number,

P_w^CH4 = t-dissolved CH₄ pressure of the pore fluid prior to ebullition,

P_H2O = vapor pressure of water, and

= surface tension.

The Sherwood number (Clift et al., 1978) accounts for increased mass flux to the bubble by reduction of the diffusive boundary layer around the bubble in the presence of flow; hence, when Sh = 1, the bubble growth is controlled by diffusion, and Sh > 1 can be assumed if the fluid is mixing during the ebullition process, as we expect. A characteristic timescale for the growth of a bubble of maximum radius, r_max, just before leaving the sediment into the headspace is, then,

_growth = (n/4r²Q_CH4) =

(P_bubr_max²/{3RTShD^CH4K_H^CH4[P_w^CH4– P_bub + P_H2O + (2/r_max)]}). (A3)

Because CH₄ controls the total pressure of the bubble, the bubble will scavenge trace gases (like N₂ and Ar) as it grows in a characteristic time scale:

_trace = (n/4r²Q_trace) = (r²/3RTShD^traceK_H^trace). (A4)

As the bubble grows, it will accumulate more quickly the more soluble and more diffusive trace gases from the pore fluid. The scavenging efficiency will depend on K_H^N2/K_H^CH4 (or K_H^Ar/K_H^CH4) or, and to a lesser extent, on D^N2/D^CH4 (or D^Ar/D^CH4) because for N₂, Ar, and CH₄, the greatest differences are in their solubility coefficients. As the ebullition process continues, the concentration of the trace gases in the pore fluid will decrease with time according to the ratio of _growth/_trace. Clearly, the headspace will initially accumulate the more soluble and more diffusive trace gases early on in the ebullition process.