An adaptive moving mesh finite difference method is presented to solve a modified Buckley Leverett equation with a dynamic capillary pressure term from porous media. The effects of the dynamic capillary coefficient, the infiltrating flux rate and the initial and boundary values are systematically studied using a traveling wave ansatz and efficient numerical methods. Special attention is paid to the non-monotonic profiles. The governing equation is discretized with an adaptive moving mesh finite difference method in the space direction and an implicit-explicit method in the time direction. In order to obtain high quality meshes, an adaptive time dependent monitor function with directional control is applied to redistribute the mesh grid in every time step, and a diffusive mechanism is used to smooth the monitor function. The behavior of the central difference flux, the standard local Lax-Friedrich flux and the local Lax-Friedrich flux with reconstruction is investigated by solving a 1D modified Buckley-Leverett equation. With the moving mesh technique, a good mesh quality and a high numerical accuracy are obtained. A collection of one-dimensional and two-dimensional numerical experiments is presented to demonstrate the accuracy and effectiveness of the numerical method.