Maximum-invasion envelope of a retained current ($M_{\infty }(x)={\max }_{t\ge 0}h(x,t)$)

For the triangular current with capillary retention, the initial trailing face is already the maximum for $0<x<L$. Further upslope, the maximum occurs when the moving crest passes. HenceThe final trapped phase has saturation $s$ below this envelope. Its volume is $\varphi s\int M_{\infty }dx=\varphi a{L}^2$, exactly the initial mobile volume. Maximum invaded thickness is distinct from residual saturation: $s$ multiplies occupied pore volume rather than shrinking the geometrical envelope.