## 2 Materials and Methods  
### Modeling Thawing of Frozen Sediments  
Heat transport in the subsurface occurs through conduction or convection, where conduction is commonly believed to be the dominant process in permafrost environments except for during extensive spring thawing [18; 19]. Heat is conducted downward into the ground if surface temperature is higher than ground temperature, and opposite. Surface temperature is primarily controlled by air temperature, therefore heat flux into the ground is typically largest during summer, but other factors such as snow cover and vegetation also affect surface temperature [3]. If the ground is initially frozen, which is common in the Arctic, this heat flux may cause thawing. In the Arctic today, winter temperatures are generally cold enough so that summer thawing is completely reversed during winter, but if Arctic temperatures increase as they are expected to do [2; 20], we will likely see increased thawing of currently frozen ground in the near future. By considering conduction only, we can describe the strength of the vertical heat flux by the heat transport equation:  
\[C\frac{\partial T}{\partial t}+\
abla\cdot(-\kappa VT)=Q. \tag{1}\]  
Here, \(C\) (Jkg\({}^{-1}\)K\({}^{-1}\)) is the volumetric specific heat capacity (in the following referred to as heat capacity), \(T\) (\({}^{\circ}\)C) is temperature at depth \(z\) (m), \(t\) is time, \(\kappa\) (Wm\({}^{-1}\)K\({}^{-1}\)) is thermal conductivity, and \(Q\) (Wm\({}^{-2}\)) is external heat flux. The effective thermal properties of a composite depend on the thermal properties of the individual constituents and their volume fractions. For permafrost that is typically a matrix, water, ice, and air (in the following defined by subscripts m, w, i, and a, respectively). We also need to include latent heat \(L\) (kJ/kg) when one or more of the materials involved changes phase (here: water/ice, \(L\) = 333.6 kJ/kg), and following [21], we do this by replacing \(C\) with an equivalent heat capacity \(C_{eq}\) that includes latent heat:  
\[C_{eq}=\theta_{m}\rho_{m}c_{m}+\theta_{a}\rho_{a}c_{a}+\theta_{w}\rho_{w}\left( c_{w}+\frac{\partial\Phi}{\partial T}L\right)+\theta_{i}\rho_{i}\left(c_{i}+ \frac{\partial\Phi}{\partial T}L\right). \tag{2}\]  
Density, heat capacity, and volume fraction of each constituent are defined by \(\rho_{n}\), \(c_{n}\), and \(\theta_{n}\), respectively, with subscripts as defined above. Total porosity is \(\theta=\theta_{a}+\theta_{w}+\theta_{i}\), and the volume fractions of water and ice are given by \(\theta_{w}=(\theta-\theta_{a})\Phi\) and \(\theta_{i}=(\theta-\theta_{a})-\theta_{w}\), where \(\Phi\) is a variable changing from 0 to 1 for fully frozen to fully unfrozen material over the phase change temperature interval. Note that saturation here denotes total fluid content in the pore space, i.e., \(\frac{\theta_{w}+\theta_{i}}{\theta}\), while ice saturation denotes the amount of the fluid that is frozen, i.e., \(\frac{\theta_{i}}{\theta_{w}+\theta_{i}}\). Furthermore, we parameterize thermal conductivity using the approximation of [PERSON] [22]:  
\[\kappa_{eq}=\frac{f_{m}\theta_{m}\kappa_{m}\ +\ f_{a}\theta_{a}\kappa_{a}\ +\ f_{w}\theta_{w}\kappa_{w}\ +\ f_{i}\theta_{i}\kappa_{i}}{\theta_{m}f_{m}\ +\ \theta_{a}f_{a}\ +\ \theta_{w}f_{w}\ +\ \theta_{i}f_{i}}. \tag{3}\]  
The shape factors \(f_{n}\) are given by [19] as:  
\[f_{n}=\left[1+\frac{1}{3}\left(\frac{\kappa_{n}}{\kappa_{0}}-1\right)\right]^ {-1}, \tag{4}\]  
where \(\kappa_{0}\) denotes the thermal conductivity of the background medium (here: \(\kappa_{0}=\kappa_{m}\)). By assuming that the only heat source is the surface temperature (\(Q=0\)), we can model how temperature distribution evolves with time and depth by using Equation (1) with thermal properties as given by Equations (2)-(4) in a finite-element scheme as provided by COMSOL [23]. Porosity, saturation, and salinity all affect thawing depth in frozen ground because the effective thermal properties of a composite are sensitive to the volume fractions of each constituent.