= Solution
After transverse equilibration, a localized release on a sufficiently long aquifer has the <advection-diffusion heat-kernel solution>
$$
\boxed{C(x,t)=\frac{\mathcal M}{\sqrt{4\pi D_{\rm eff}t}}
\exp\left[-\frac{(x-x_0-Ut)^2}{4D_{\rm eff}t}\right],}
$$
where $\mathcal M=\int C\,dx$ is conserved. The actual tracer amount per unit source length is $\phi h\mathcal M$ in this normalization. Its center moves at the mean <interstitial velocity> $U$, its variance is $2D_{\rm eff}t$, and its peak decreases as $t^{-1/2}$. A finite initial packet is convolved with this kernel.
Before times of order $h^2/D$, this effective Gaussian description need not hold. The parabolic velocity profile initially shears a packet, and particles initially at different depths acquire different along-flow displacements. Transverse <diffusion> subsequently erases depth memory and produces the long-time <Taylor dispersion> law. A short pulse can therefore have an initially asymmetric or strongly distorted longitudinal shape before entering the averaged regime.
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