With aircraft icing occurring in increasingly complex environments, its modelling and simulation remains a challenging task. This is particularly the case for ice crystal icing on a heated substrate where effects such as liquid-solid phase change and liquid imbibition can take place. In this communication, a model based on the enthalpy method, that takes into account the melting and accretion of the ice layer in several dimensions, is proposed. This type of approach is able to efficiently capture the melting front and accounts for temperature gradients. In order to improve the predictions, a diffusion flux model is proposed in this work to account for the potential liquid water transport in the ice layer due to imbibition. The model is formulated in terms of a partial differential equation expressed on the time-evolving domain. A moving finite element method is used to solve the equations in weak form. After describing the model and numerical resolution scheme, a verification case based on the Stefan problem is presented and numerical convergence properties are discussed. The method is then used to simulate a heated flat plate configuration in ice crystal icing conditions, for which experimental data is available. The simulations are first performed in one dimension and several properties of the methodology are assessed. In particular, the effect of taking the imbibition process into account is investigated. The capabilities of the proposed methodology are then demonstrated with a two-dimensional simulation of the heated plate.