 Invasion envelope of an exponentially draining porous current

For an impulsive leaky porous gravity current, write $s=(1-e^{-\Omega t})/\Omega$ and $h=(1-\Omega s)[R^2-x^2{]}_{+}/(6Ds)$, where $R=C{s}^{1/3}$. At a fixed lateral location, $h_t=0$ when $x^2=R^2(1+2\Omega s)/3$. The maximum invaded thickness is the envelope $x_{\ast }(s)=C{s}^{1/3}\sqrt{(1+2\Omega s)/3}$, $h_{\ast }(s)=C^2(1-\Omega s{)}^2/(9D{s}^{1/3})$. The source solution is singular at the initial line release; finite source size and aquifer depth limit its near-source validity.