The main objective of this research is to provide a thorough understanding of the non- monotonic behaviors of solutions to the two-phase ow equations in porous media. To achieve this objective, numerical modeling and simulation have been performed. Speci c objectives are as follows.
Simulate the saturation overshoot phenomenon observed in laboratory experiments and study the e ects of di erent models with and without hysteretic e ects.
Numerically investigate the relationship between the overshoot saturation and the dynamic capillary coe cient, the in ltrating ux rate, the initial and boundary values, and resolve the steep wave fronts by using an adaptive moving mesh method.
Study the non-monotonic solutions of the non-equilibrium Richards equation and the mod- i ed Buckley-Leverett equation by applying a moving mesh method and choosing di erent ux reconstruction schemes.
Design an accurate discontinuous Galerkin method for the modi ed Buckley-Leverett equa- tion and study the in uences of order and limiting strategy.
Because of the inspiration of the thin lm ow, Cueto-Felgueroso and Juanes  proposed a phase eld model for the two-phase porous media ow. Hence, we will investigate the thin lm ow equation and resolve the non-monotonic structures using an adaptive moving mesh nite element method.