Fractional-flow tangent construction ($F^{\prime }(s_s)=[F(s_s)-F(s_0)]/(s_s-s_0)$)

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 $(s_0,F(s_0))$ to the fractional-flow curve at $s_s$. It selects the upstream shock saturation; the actual dimensional shock speed still includes total flux divided by porosity.