Stability of a moving drainage front, say of the interface between two immiscible fluids, of contrasting viscosities and densities, where the one with non-wetting properties displaces the wetting one, is of paramount importance in several fields of engineering. In the case of an interface between dry air and water progressing through initially saturated soils, modeling of the drainage process allows to characterize the evolution of soils' hydro-mechanical properties in drought conditions, which are becoming increasingly widespread as a consequence of drastic climate change. The purpose of the research presented in this communication is to compare two different approaches to modeling two-phase fluid front evolution based on the one hand on a phase-field model, describing the two immiscible fluids as a heterogeneous mixture whose energy accounts for possible coexistence between the two \lq\lq phases\rq\rq, say the gaseous and the liquid one, endowed with a suitable second gradient regularization term, in the spirit of Cahn-Hilliard model, see \cite{CahnHilliard} and \cite{Ommi1,Ommi2}, and on the other hand adopting a formulation based on the notion of dynamic capillary pressure, introduced by Hassanizadeh \& Gray \cite{Hassa}, which provides a regularization of sharp fronts via space-time mixed derivatives. A comparison is provided between the two formulations in particular in what concerns the construction of a traveling wave solution simulating the propagation of a drought front through an initially water saturated porous soil and possible loss of stability of the front with respect to transversal perturbation (fingering instabilities) according with the chosen boundary conditions.