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We study the gravity-driven flow of two fluid phases in a one-dimensionalhomogeneous porous column when history dependence of the pressure differencebetween the phases (capillary pressure) is taken into account. In thehyperbolic limit, solutions of such systems satisfy the Buckley-Leverettequation with a non-monotone flux function. However, solutions for thehysteretic case do not converge to the classical solutions in the hyperboliclimit in a wide range of situations. In particular, with Riemann data asinitial condition, stationary shocks become possible in addition to classicalcomponents such as shocks, rarefaction waves, and constant states. We derive anadmissibility criterion for the stationary shocks and outline all admissibleshocks. Depending on the capillary pressure functions, flux function, and theRiemann data, two cases are identified a priori for which the solution consistsof a stationary shock. In the first case, the shock remains at the point wherethe initial condition is discontinuous. In the second case, the solution isfrozen in time in at least one semi-infinite half. The predictions are verifiedusing numerical results.

Capillary Hysteresis and Gravity Segregation in Two Phase Flow Through Porous Media