## 3 Methods
### Heat Transport Related to Hydrate Phase Transitions
R is the distance of heat transport. Subscript w denotes liquid water, subscript H stands for hydrate film, and m indicates minerals. In the case of liquid water and hydrate, the reference temperature is that of hydrate formation; it is approximated by the average of the surface area normal to the mean mass flux through the sediment in the case of minerals. For the hydrate film, the heating contribution (the second term) will continue to increase up to the hydrate dissociation point corresponding to the actual pressure (see Figure 1). After that, heat transported in this direction will be split between the dissociation of hydrate and raising the hydrate temperature. As can be seen from Table 2 below and the associated discussion, the reported experimental values of heat conductivity through hydrate vary significantly. The difference in liquid water heat conductivity is, however, limited. If the K-values for hydrate and liquid water are assumed to be identical, then the temperature difference from one time step to the next will be the same whether the conduction is through liquid water or hydrate. To simplify conductivity for both the hydrate and water phase, they are the same up to two limits. The first limit is when hydrate dissociation begins; another limit is if the heat transport through the hydrate approaches the hydrate former phase, which is almost heat insulating. If we ignore the mineral surfaces for a moment, Equation (7) can be rewritten into the following one-dimensional form:  
\[ \begin{gathered}\Big{(}dI_{formation}A_{\perp}\Delta H^{ formation}(T,P)\Big{)}=K\Delta T_{w}d\overset{\rightarrow}{R}+K\Delta T_{H}d \overset{\rightarrow}{R}+\\ d\Big{]}_{dissociation}A_{\perp}\Delta H^{dissociation}_{\alpha}(T_{ \alpha},P)\end{gathered}  \tag{8}\]  
The purpose of vectorial R is only to indicate the direction relative to a reference point. With the reference point set so that \(R=0\) at the hydrate surface, the hydrate former entering the aqueous phase will correspond to positive \(R\), while transport through the hydrate film will be indicated by negative \(R\) values. The second reasoning behind the vector notation in the context of (8) is to ensure absolute values for both the first and the second terms. Both the left-hand side and the third term on the right-hand side, are also absolute values, allowing Equation (8) to describe the distribution of heat released during hydrate formation. While Equation (8) is clearly oversimplified, there is some justification for it for the systems in consideration. Heat transport through liquids is normally two to three orders of magnitudes faster than mass transport [21; 22; 23; 24; 25; 26; 27; 28; 29], which also explains the difference in flux rather than associated heat release (or consumption). Transporting the guest molecules through a hydrate film from the gas side to the liquid side to sustain the growth may be 8 to 10 orders of magnitudes slower than diffusion of the same molecules though liquid water. As such, it will be fairly safe to assume that any guest molecules that reach the liquid water side will instantly dispose the hydrate formation heat due to the combination of the three terms in Equation (8). A number of theoretical methods for estimating thermal conductivity are available in the literature as well as substantial amounts of experimental data from different research groups worldwide. It is far outside the focus of this work to conduct a detailed review of theoretical models and experimental data for thermal conductivities of relevance to this work. As such, the literature values in Table 2 are listed with reference to the actual sources without any additional comment. Generally, gas hydrates exhibit a glass-like behavior where it comes to the temperature and pressure dependence of their thermal conductivity. This feature makes them drastically different in comparison to ice and other molecular crystals [33]. In hydrates, the water framework is much more distorted, with the hydrogen bond more strained than in ice.