Residual-trapping attenuation of a porous current ($h=h_0\exp [-t/\tau +\varphi (1-s)x/(u\tau )]$)

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 $t=\tau$.