• *Advection-di*ff*usion equation for [H2]*

It will take some time for gas moving into the subsurface to reach the sensors at 1 m depth, and on the way, there will be mixing with the gas in the subsurface. The advection–diffusion equation describes these phenomena. The 1D flux, *jH2*, of *H*2 (expressed as the volume fraction of *H*2 in air) by diffusion and advection is:

$$j\_{H2} = -D\_E \frac{\partial [H2]}{\partial z} + V\_{a\dot{r}} [H\_2] \tag{14}$$

where [*H*2] is the volume fraction of *H*2 in air. By mass conservation:

$$\frac{\partial \rho[H\_2]}{\partial t} = -\nabla \bullet j\_{H2} = \frac{\partial}{\partial z} D\_E \frac{\partial [H\_2]}{\partial z} - V\_{air} \frac{\partial [H\_2]}{\partial z} \tag{15}$$

where *DE* is the effective diffusion constant of *H*2 in the vent, and *Vair* is the vertical flux of air (m<sup>3</sup> of air passing through a plan area or 1 m<sup>2</sup> per second) in the portion of the vent considered (in this case the uppermost part). The effective diffusion constant *DE* = *D*φτ , where *D* is the diffusion constant of *H*2 in air, ϕ is the porosity of the vent sediments, and τ is the tortuosity of the pores in the sediments, which we take to equal 2. If we make *t, z* and [*H*2] non-dimensional by defining:

$$\begin{cases} t = \overline{t} \frac{b^2 \rho}{D\_E} = \overline{t} \tau\_D\\ z = \overline{z} b\\ [H\_2] = [\overline{H}\_2][H\_2]\_{\text{dep}}\\ N\_{\text{pr}} = \frac{V b}{D\_E} \end{cases} \tag{16}$$

Equation (15) becomes:

$$\frac{\partial \left[ \overline{H}\_2 \right]}{\partial \overline{t}} = \frac{\partial^2 \left[ \overline{H}\_2 \right]}{\partial \overline{z}^2} - N\_{p\epsilon} \frac{\partial \left[ \overline{H}\_2 \right]}{\partial \overline{z}} \tag{17}$$

where *Npe* is the Peclet number (ratio of advection to diffusion).
