  We study the gravity-driven flow of two fluid phases in a one-dimensional
homogeneous porous column when history dependence of the pressure difference
between the phases (capillary pressure) is taken into account. In the
hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett
equation with a non-monotone flux function. However, solutions for the
hysteretic case do not converge to the classical solutions in the hyperbolic
limit in a wide range of situations. In particular, with Riemann data as
initial condition, stationary shocks become possible in addition to classical
components such as shocks, rarefaction waves, and constant states. We derive an
admissibility criterion for the stationary shocks and outline all admissible
shocks. Depending on the capillary pressure functions, flux function, and the
Riemann data, two cases are identified a priori for which the solution consists
of a stationary shock. In the first case, the shock remains at the point where
the initial condition is discontinuous. In the second case, the solution is
frozen in time in at least one semi-infinite half. The predictions are verified
using numerical results.
