*3.4. Thermal Effects*

The heat transfer is modeled through the following advection diffusion equation

$$
\rho\_b c\_b \frac{\partial T}{\partial t} + \rho\_f c\_f \mathbf{v}\_D \cdot \nabla T - \nabla \cdot (b \nabla T) = q\_{r\text{\textquotedblleft}} \tag{7}
$$

where *T* is the temperature field, *ρf* and *c f* are the fluid density and specific heat, respectively; *ρb* and *cb* are the bulk density and the bulk specific heat defined as *ρbcb* = *φρf c f* + (1 − *φ*)*ρscs* with *φ*, *ρs* and *cs* being the porosity, the solid density and specific heat capacity; **v***D* is the Darcy velocity defined as **v***D* = − *K* ∇*p*, *b* is the bulk thermal conductivity which is an average of the conductivity of the solid and the fluid phase and finally *qr* is the heat source. Concerning the initial condition, we consider an homogeneous temperature field, namely *T* = 0 ∀**x** ∈ Ω(*t* = *<sup>t</sup>*0). According to the nomenclature shown in Figure A1, we consider the following boundary conditions,*T* = *Tice*(*t*) on <sup>Γ</sup>(*t*), ∇*T* · **n** = 0 on *∂*Ω(*t*) \ <sup>Γ</sup>(*t*), where **n** is the unit outward normal to the boundary and *Tice*(*t*) is the basin top temperature. Following a widely used approach in literature, see for example [16,25], the presence of the ice on top of the basin is taken into account by means of a variation of the top temperature of the basin (*Tice*(*t*)) during the glaciation cycle.
