Capillary diffusion ($D_c=-K{\lambda }_w{\lambda }_n{p}_c^{\prime }/({\lambda }_w+{\lambda }_n)$)

Eliminating phase pressures with $p_c=p_n-p_w$ gives $\varphi s_t+QF(s{)}_x=(D_c{s}_x{)}_x$. For decreasing capillary pressure, $D_c\ge 0$. This nonlinear diffusion broadens a saturation shock; it can degenerate where a phase mobility vanishes. Endpoint conservation retains the limiting Rankine-Hugoniot condition.