= Invasion envelope of an exponentially draining porous current
{title2=$x_*^2=R^2(1+2\Omega s)/3$}
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 $x_*(s)=C s^{1/3}\sqrt{(1+2\Omega s)/3}$, $h_*(s)=C^2(1-\Omega s)^2/(9D s^{1/3})$. The source solution is singular at the initial line release; finite source size and aquifer depth limit its near-source validity.
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