## 3 Methods
### Heat Transport Related to Hydrate Phase Transitions  
The two primary ways of heat transport relevant for the systems discussed in this work are conduction and convection. In the discussion that follows, we will consider systems where the initial amounts of water and CO\({}_{2}\) are large enough to not be consumed during the hydrate growth. In the case of hydrate nucleation and growth inside a pipeline, the new mass will continuously be supplied by the flow stream. Offshore methane gas hydrates in sediments are typically characterized by methane coming from below through the fracture systems. A continuous inflow of water though fractures connected to the seafloor above will ensure a supply of liquid water but will also cause the hydrate to dissociate due to severe lack of dissolved methane. The chemical potential of CH\({}_{4}\) in the incoming water will therefore be close to its infinite dilution chemical potential, which is typically substantially lower than the chemical potential of CH\({}_{4}\) in hydrate. A typical sediment example in the case of CO\({}_{2}\) will involve aquifer storage of CO\({}_{2}\) in reservoirs that contain regions favorable for hydrate formation. Liquid water is available in the sediments, and a continuous inflow of CO\({}_{2}\) will lead to the formation of hydrate films that will reduce vertical CO\({}_{2}\) migration. In addition to the presence of natural sealing (clay, shale), these hydrate films reduce risk of CO\({}_{2}\) leakage to the surroundings above the storage site. These two practical examples alone illustrate the importance of having a model that assumes that the original \"bulk\" phases of the water and hydrate former phase will not be totally consumed and disappear. There are numerous other relevant examples. A typical simplified heat transport model in our scenarios will involve heat conduction through the water over the growing film. The simplest approach would be a sum of symmetric heat conduction from below the hydrate growth site and heat conduction through the hydrate film towards the CO\({}_{2}\) phase. When the temperature on the hydrate surface reaches the hydrate melting point, an additional term of hydrate dissociation dynamics will enter the mass and energy balances. Additionally, mineral bedrock may also play a part in the overall energy balance, and it is entirely feasible to include the associated heat transport even within the framework of a very simple model. When the liquid water phase has been depleted in the CO\({}_{2}\) to the level of quasi-equilibrium with the CO\({}_{2}\) hydrate, a new hydrate can only form in one of the two ways: (1) CO\({}_{2}\) transport through the hydrate film and into the liquid water side of the hydrate film, or (2) water transport through the hydrate film and into the CO\({}_{2}\) side of the hydrate film. The diffusion of CO\({}_{2}\) through hydrate will be very slow and most probably limited by the existence of empty cavities; this process will trigger the temporary local destabilization of the water hydrate lattice and induce a counter diffusion of water molecules. In the absence of \"fresh\" building blocks, the first and second laws of thermodynamics will lead to a dynamic process in which the least stable hydrates (those with highest free energy) are undergoing melting to support the growth of hydrate regions with lower free energy [29]. Even by themselves, these processes can generate mass fluxes across the hydrate membrane film. Ultimately, these local free energy-governed processes can even lead to the creation of holes in the hydrate membrane, allowing for the supply of new building blocks. A common approximation in hydrate modeling, as well as in the interpretation of experimental data, is, therefore, to lump both these contributions together into an apparent conductivity:  
\[\dot{Q}=K\frac{V^{film}}{A_{\perp}}\Delta T \tag{6}\]  
where \(A_{\perp}\) is area normal to the heat transport direction, and \(V^{film}\) is the volume of hydrate film.