A phase field model for the freezing saturated porous medium

In terms of the mixture theory and phase field theory, a phase field model is developed for the saturated porous medium undergoing phase transition. In the proposed model, it is postulated that during the phase transition of the porous medium, both the solid skeleton and pore fluid will undergo phase transition. The phase states of the solid skeleton and pore fluid are characterized by respective order parameters. To simplify the proposed phase field model, the temperatures and order parameters of the solid skeleton and pore fluid are assumed to be equal. The balance laws of the porous medium are given by the conventional mixture theory. The order parameter and the porosity of the porous medium are considered as internal variables and their evolution equations are determined by the entropy inequality of the porous medium. The constitutive representations for the stresses, entropies, heat fluxes, drag force and the evolution equations for the order parameter and porosity are derived by exploitation of the entropy inequality. To illustrate the proposed model, a concrete phase field model for the freezing porous medium is established in the paper. In the model, the memory effect associated with phase transition of the porous medium is taken into account by assuming Stieltjes integral for the strain energy of the porous medium. The constitutive representations for the above variables are then derived according to the proposed free energy expression for the porous medium. Finally, the boundary condition associated with the proposed model and the determination of some parameters involved in our model are discussed in the paper briefly.