 Triangular current with capillary retention

For capillary residual trapping of an initial triangular current, the method of characteristics translates its leading and trailing faces at $v_A=u/\varphi$ and $v_R=u/[\varphi (1-s)]$. Their intersection has height $a[L-(v_R-v_A)t/2]$. Extinction occurs at $t_{\ast }=2L/(v_R-v_A)$ and position $x_{\ast }=2L/s$. These formulas require $0<s<1$; at zero residual saturation the triangle translates indefinitely.