  We study a one dimensional model for two-phase flows in heterogeneous media,
in which the capillary pressure functions can be discontinuous with respect to
space. We first give a model, leading to a system of degenerated non-linear
parabolic equations spatially coupled by non linear transmission conditions. We
approximate the solution of our problem thanks to a monotonous finite volume
scheme. The convergence of the underlying discrete solution to a weak solution
when the discretization step tends to 0 is then proven. We also show, under
assumptions on the initial data, a uniform estimate on the flux, which is then
used during the uniqueness proof. A density argument allows us to relax the
assumptions on the initial data, and to extend the existence-uniqueness frame
to a family of solution obtained as limit of approximations. A numerical
example is then given to illustrate the behavior of the model.
