= Solution
At a fixed location behind the advancing nose, the depth decreases after passage. A fall $-dh>0$ leaves the immobile fluid volume $s\phi(-dh)$ per unit length, so total local fluid storage changes by $\phi\,dh+s\phi(-dh)=\phi(1-s)\,dh$. The thinning-region equation is therefore
$$
\phi(1-s)h_t+uh_x=0.
$$
The inlet flux remains $uh(0,t)=Qe^{-t/\tau}$. Applying the <method of characteristics> with thinning-wave speed $u/[\phi(1-s)]$ gives the <residual-trapping attenuation of a porous current>:
$$
\boxed{h(x,t)=h_0\exp\left[-\frac{t}{\tau}+\frac{\phi(1-s)x}{u\tau}\right],\qquad h_0=Q/u.}
$$
The nose advances into new pore space, which has no residual fluid yet, so its storage factor is still $\phi$ and its speed is still $u/\phi$. Consequently $X(t)=ut/\phi$ and the interior nose trace becomes $h_0e^{-st/\tau}$. 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.
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