Recently, the observa�on of nonmonotonicity of traveling wave solu�ons for satura�on profiles during constant-flux infiltra�on experiments has highlighted the shortcomings of the tradi�onal, seventy year old mathema�cal model for immiscible displacement in porous media. Several recent modifica�ons have been proposed to explain these observa-�ons. The present paper suggests that nonmonotone satura�on profiles might occur even at zero flux. Specifically, nonmonotonicity of satura�on profiles is predicted for hydrosta�c equilibrium, when both fluids are at rest. It is argued that in tradi�onal theories with the widely used single-valued monotone cons�tu�ve func�ons, nonmonotone profiles should not exist in hydrosta�c equilibrium. The same applies to some modifica�ons of the tradi-�onal theory. Nonmonotone satura�on profiles in hydrosta�c equilibrium arise within a generalized theory that contains the tradi�onal theory as a special case. The physical origin of the phenomenon is simultaneous occurrence of imbibi�on and drainage. It is argued that indica�ons for nonmonotone satura�on profiles in hydrosta�c equilibrium might have been observed in past experiments and could become clearly observable in a closed column experiment. Nonmonotonic saturation profiles (saturation overshoot) occur as travelling waves in gravity driven fingering. They seem important for preferential flow mechanisms and have found much attention recently. Here, we predict them even for hydrostatic equilibrium when all velocities vanish. We suggest that hysteresis suffices to explain the effect
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