 One-sided constant-volume porous gravity current

A localized release of actual fluid volume $V$ per unit width on a reflecting half-line $x\ge 0$ has the Barenblatt solution $h=[L^2-x^2{]}_{+}/(6\lambda t)$, where $L^3=9\lambda Vt/\varphi$ and $\lambda =k\Delta \rho g/(\varphi \mu )$. The mass conservation normalization is $\varphi {\int }_0^{L}hdx=V$. Zero slope at the reflecting boundary gives zero Darcy flux. For symmetric spreading on the full line, the same formula has $L^3=9\lambda Vt/(2\varphi )$ when $V$ denotes the total released volume. A finite initial patch is not determined by volume alone; this exact source profile has a concentrated initial release.