The numerical simulation of the one-dimensional displacement of oil by water in a vertical porous slab is studied. The water saturation is governed by a quasilinear diffusion-convection equation. The diffusion term vanishes for the extreme values of the saturation. The transport term may be nonmonotone (for small or zero water injection rates in presence of gravity). Various boundary conditions are used: a generalized Dirichlet condition (for the pure transport case), and a condition taking into account the so-called "well effect" on the production boundary (for the case with capillary diffusion). The above problem is solved using discontinuous-finite elements together with mixed-finite elements, following the ideas of Lesaint, Raviart and Thomas. Godunov's generalization of upstream weighting is also incorporated. A computer code of this algorithm works satisfactorily under a wide range of experimental conditions, going from the pure diffusion case (ƒ = 0) to the pure transport case (α = 0). Results of various runs on idealized data as well as on laboratory data are shown.