The one-dimensional mass conservation equation for saturation when total Darcy flux is constant and capillary pressure and gravity are neglected. It is a scalar conservation law with dimensional characteristic speed . Shock speeds obey a Rankine-Hugoniot condition; physically admissible solutions satisfy an entropy selection. Capillary diffusion regularizes the saturation transition.
Eliminating phase pressures with gives . For decreasing capillary pressure, . This nonlinear diffusion broadens a saturation shock; it can degenerate where a phase mobility vanishes. Endpoint conservation retains the limiting Rankine-Hugoniot condition.
When a rarefaction wave attaches to a saturation shock, its terminal characteristic speed must match the Rankine-Hugoniot condition speed. This gives a tangent from the initial state to the fractional-flow curve at . It selects the upstream shock saturation; the actual dimensional shock speed still includes total flux divided by porosity.
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The Buckley–Leverett equation is a fundamental equation in petroleum engineering and reservoir engineering that describes the movement of two-phase fluids (typically oil and water) in porous media. It models the flow behavior of immiscible fluids in a reservoir when one fluid displaces another, commonly used to analyze waterflooding operations during oil recovery. The equation is derived from the conservation of mass principle and reflects the dynamics of the interfaces between the two fluids.