 Residual-trapping attenuation of a porous current

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.