 Solution

When the hydrostatic spreading term is small, the depth equation reduces to $h_t+c{h}_x=0$, where $c=u/\varphi$. Its characteristic curves have $x=c(t-t_0)$, and the depth is constant along each characteristic. At the inlet the imposed flux gives $uh(0,t_0)=Q{e}^{-t_0/\tau }$. Therefore $h_0=h(0,0)=Q/u$ in this advective approximation andThis is the advection limit of an inclined porous current, behind its leading nose and after the relevant characteristic has left the inlet. For injection starting into an empty aquifer, the advective support is $0<x<ut/\varphi$.

The qualification “sufficiently large time” must refer to an advection-dominated outer region, not an exact solution of the full nonlinear equation. Substituting this exponential into the full equation leaves a nonzero spreading term. The next part quantifies when it can be neglected and explains the nonuniform leading-edge limit.