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For the proposed exponential profile, put $\ell =u\tau /\varphi$. Then $h_x=h/\ell$ and $(h{h}_x{)}_x=2h^2/{\ell }^2$. Thus the exact ratio of the discarded spreading term to the retained advective term isThis is a quantitative validity condition for the advection limit of an inclined porous current. For a specified small tolerance $\eta >0$, at a fixed $x$ it is enough that the inlet characteristic has arrived andIf the logarithm is negative, arrival itself already gives this tolerance. A large-distance estimate comparing the overall current length $ct$ with a height-induced pressure length $h\cot \theta$ is $t\gg \varphi h\cot \theta /u$, but the local gradient criterion above is the sharper test for this particular injection history.

At the advective nose, $x=ut/\varphi$, the exponential factor is one. Therefore large time alone is insufficient for uniform validity: the outer profile also needs $2\varphi h_0\cot \theta /(u\tau )\ll 1$, and its abrupt nose needs an inner spreading layer. This limitation follows by direct substitution, not from a missing constant in the solution.

Indeed, with $\xi =x-ut/\varphi$, the full equation becomes $h_t=D(h{h}_{\xi }{)}_{\xi }$, $D=u\cot \theta /\varphi$. Once a finite-volume injection pulse is effectively over, nonlinear diffusion continues to broaden it. Dimensional balances give width ${\ell }_d\sim (DMt{)}^{1/3}$ and height $M/{\ell }_d$, where $M=\int hd\xi$ is the conserved area. A fixed exponential translating profile cannot be the uniform infinite-time limit of that equation. The printed formula is consequently an outer advection approximation with the displayed smallness condition.

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.