The motion of two-phase flow in a porous medium under the condition of vertical equilibrium can be described by a viscous conservation law that involves a non-convex flux function with two inflection points. In [5], a first order Godunov scheme was used to numerically approximate solutions of the model. In this paper we show that using instead the high resolution Weighted Essentially Non Oscillatory (WENO) technology, and an IMEX strategy to handle the capillary term by an implicit

more »

... n, leads to a noticeable increase in resolution power and efficiency. We carefully discuss the implementation of WENO schemes for the model equation, paying special attention to the choice of the definition of the numerical viscosity. We also present numerical simulations when the capillary number is negligible (i.e., the model is a homogeneous conservation law) and non-negligible (i.e. the model equation becomes a 'viscous' conservation law). The numerical results are compared with those obtained with the method proposed in [5] in terms of accuracy, resolution power and global efficiency.