In this thesis,we aim to study the stability analysis of a moving front in porous media. Firstly, we study the theoretical stability analysis of Richards’ equation based on perturbation analysis and stability analysis of non-equilibrium Richards equation based on spectrum method. In addition, we show the numerical simulation of Richards’ equation and of non-equilibrium Richards equation to verify the theoretical results. Moreover,we show another model the modified Buckley-Leverett equation to describe preferential flow, and base on Paul.A.Zegeling’s τ -model analysis to explain how stationary points in the phase plane are connected by a special curve.