= Residual-trapping attenuation of a porous current
{title2=$h=h_0\exp[-t/\tau+\phi(1-s)x/(u\tau)]$}
When a thinning <porous gravity current> leaves fraction $s$ behind, receding storage changes by only $\phi(1-s)\,dh$ after accounting for immobile fluid. Thus its advective thinning equation is $\phi(1-s)h_t+uh_x=0$. Advancing into previously unwetted pores still requires the full storage $\phi h$, so the nose moves at $u/\phi$. 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 $V'=Qe^{-t/\tau}-sV/\tau$; the second term becomes immobile retained volume.
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