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Write $R=C{s}^{1/3}$, where $C^3=9DV/(2\varphi )$. At a fixed $x$ inside the current,Since $ds/dt>0$, instantaneous constancy of depth is equivalent to ${\partial }_{s}h=0$. Differentiation givesThus the lateral locations at which the depth is neither increasing nor decreasing areAt a fixed nonzero position reached by the front, the height first increases from zero, attains this maximum, and later decreases. The invasion envelope of an exponentially draining porous current is therefore the parametric curveWithin the aquifer, the previously invaded region lies between the basal boundary and this maximum-height envelope, for $|x|<R_{\max }$. The envelope tends to zero height at the limiting lateral nose. Its central singularity belongs to the ideal impulsive release, so it cannot be extended physically above the finite aquifer thickness.