## 3 Methods
### Heat Transport Related to Hydrate Phase Transitions
Heat transport through liquid water and hydrate will be very fast, and two to three orders of magnitude faster than mass transport [21; 22]. Work is in progress on a more detailed review of available theoretical estimates of diffusion through hydrate. The available values range from \(10^{-15}\) m\({}^{2}\)/s to \(10^{-17}\) m\({}^{2}\)/s for diffusivity of CH\({}_{4}\) through hydrate, withthe corresponding CO\({}_{2}\) values likely to be slightly lower. However, the huge variation illustrates the existing uncertainty in the correct way to handle this transport, with the bulk of the studies based on some sort of cavity-jumping Monte Carlo calculations. This is basically a mathematical-statistical method, and other types of approaches might be required. One hypothesis suggests that the more vigorous process of water librations in empty cavities facilitates a temporary destabilization of the water lattice due to the motion of guest molecules in the neighboring cavities. This may eventually allow the guest molecules to enter into the previously empty cavity. This hypothesis is based on the observation of [PERSON] and [PERSON] [7] and how various guest molecules interfere with water lattice librations. Regardless of the mechanism, these phenomena will result in increased diffusion, with the mass transport through a hydrate film accordingly expected to be proportional to the square root of time with a constant rate. The first part of a film growth model is simply the time a guest molecule will need to travel across the hydrate film. The resulting heat of the hydrate formation will be distributed locally as heat transport through liquid (Kw) and through hydrate (K\({}_{\text{H}}\)) up to the hydrate dissociation temperature. The remaining fraction of the released heat will lead to the local partial dissociation of the hydrate film. The curves characterizing heterogeneous (water and a separate hydrate former phase) hydrate dissociation or hydrate formation will be given by the temperature-pressure projection of hydrate stability conditions, see Figure 1 for examples in the case of CH\({}_{4}\) and CO\({}_{2}\). See [PERSON] [24] and [PERSON] [25] for verification of the model behind Figure 1 through comparisons with experimental data. As mentioned before, the plotted curves are the solutions of Equation (2) when the free energy change is set to zero. The chemical potential for water in hydrate is calculated from Equation (3) using guest partition functions from Equation (4). The estimation of the liquid water chemical potential utilized the pure liquid water chemical potential from [PERSON] and [PERSON] [7], with activity correction added in the case of additives such as alcohols or ions present. Setting the chemical potentials of the hydrate former in the hydrate equal to its value in the separate hydrate former phase will allow one to solve Equation (2) by means of iterations. Additionally, note a very sharp increase in the hydrate stability curve in the case of CO\({}_{2}\) due to its phase transition from gas to liquid with a significantly higher density and a corresponding shift in thermodynamic properties. This changeover is frequently ignored in the published hydrate equilibrium data, with artificial smoothening of the CO\({}_{2}\) hydrate equilibrium curve creating a bias in quite a number of published datasets. A simple model describing the dynamic progress of the system following the hydrate former reaching the liquid water side will be given by  
\[ \begin{gathered}\left(dJ_{formation}A_{H_{formation}}\Delta H^{ formation}(T,P)\right)=K_{w}\Delta T_{w}dR_{w}+K_{H}\Delta T_{H}dR_{H}+\\ K_{m}\Delta T_{m}dR_{m}+dJ_{dissociation}A_{H_{a}}\Delta H_{a}^{ dissociation}(T_{a},P)\end{gathered}  \tag{7}\]  
The _\"formation\"_ superscript over the delta \(H\) denotes the enthalpy change of hydrate formation associated with the guest molecules entering the liquid water side. The simplest scenario will be represented by an evenly distributed growth flux across a planar surface of initial hydrate film.