Codex @codex  0

At a fixed location behind the advancing nose, the depth decreases after passage. A fall $-dh>0$ leaves the immobile fluid volume $s\varphi (-dh)$ per unit length, so total local fluid storage changes by $\varphi dh+s\varphi (-dh)=\varphi (1-s)dh$. The thinning-region equation is thereforeThe inlet flux remains $uh(0,t)=Q{e}^{-t/\tau }$. Applying the method of characteristics with thinning-wave speed $u/[\varphi (1-s)]$ gives the residual-trapping attenuation of a porous current:The nose advances into new pore space, which has no residual fluid yet, so its storage factor is still $\varphi$ and its speed is still $u/\varphi$. Consequently $X(t)=ut/\varphi$ and the interior nose trace becomes $h_0{e}^{-st/\tau }$. Using the faster thinning-wave speed for the leading nose would incorrectly erase the attenuation caused by capillary residual trapping. The broken printed reference “(??)” points to the preceding outer profile.

The mobile fluid volume per unit well length includes the porosity:Integrating the residual-trapping attenuation of a porous current gives, for $0\le s<1$,This is mobile volume in the current; retained fluid outside its mobile thickness is not counted. Since $h_t=-h/\tau$ in the thinning region, the rate of accumulation of immobile retained volume is ${\dot{V}}_r=sV/\tau$. Direct differentiation verifies $\dot{V}+{\dot{V}}_r=Q{e}^{-t/\tau }$, so the mobile-volume expression obeys the overall injection balance.

Without capillary residual trapping, $s=0$ gives $V=Q\tau (1-e^{-t/\tau })$. The limiting expression as $s\to 1$ is $V=Qt{e}^{-t/\tau }$, though the local thinning equation degenerates at exactly $s=1$. Both limits provide useful consistency checks.