Capillary residual trapping ($s\in [0,1]$)

When a nonwetting phase recedes from pores, surface tension and pore geometry can retain a residual saturation $s$. If $h$ is the mobile invaded thickness and $M$ its maximum past value, stored phase volume per plan area is $\varphi [(1-s)h+sM]$. Newly invading pores have $M=h$; during recession $M$ remains fixed. With mobile Darcy velocity $u$, conservation gives advance speed $u/\varphi$ and recession speed $u/[\varphi (1-s)]$. The latter is faster for $0<s<1$.

When a thinning porous gravity current leaves fraction $s$ behind, receding storage changes by only $\varphi (1-s)dh$ after accounting for immobile fluid. Thus its advective thinning equation is $\varphi (1-s)h_t+u{h}_x=0$. Advancing into previously unwetted pores still requires the full storage $\varphi h$, so the nose moves at $u/\varphi$. With exponentially decreasing injection the mobile volume per unit well length is $V=Q\tau (e^{-st/\tau }-e^{-t/\tau })/(1-s)$. It obeys $V^{\prime }=Q{e}^{-t/\tau }-sV/\tau$; the second term becomes immobile retained volume.

For residual fraction $0<s<1$, differentiation of $V=Q\tau (e^{-st/\tau }-e^{-t/\tau })/(1-s)$ gives its unique positive maximum at the displayed time. The maximum volume is $Q\tau s^{s/(1-s)}$. When $s=0$ the mobile volume approaches $Q\tau$ monotonically, with no finite maximum. The limiting $s\to 1$ expression is $V=Qt{e}^{-t/\tau }$ with peak at $t=\tau$.

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.

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.