= Solution
When the hydrostatic spreading term is small, the depth equation reduces to $h_t+c h_x=0$, where $c=u/\phi$. Its <characteristic curves> have $x=c(t-t_0)$, and the depth is constant along each characteristic. At the inlet the imposed flux gives $uh(0,t_0)=Qe^{-t_0/\tau}$. Therefore $h_0=h(0,0)=Q/u$ in this advective approximation and
$$
\boxed{u=\frac{kg\Delta\rho}{\mu}\sin\theta,\qquad h(x,t)=h_0\exp\left[-\frac{t-\phi x/u}{\tau}\right].}
$$
This 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 $0<x<ut/\phi$.
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.
Back to article page
