= Solution
Use a constant <porosity> $\phi$ and let $v$ denote pore <velocity>. <Darcy law> gives the <Darcy velocity> $q=\phi v$. If the <velocity> convention has already absorbed <porosity>, set $\phi=1$ in the following formulas. Assume positive <permeability of a porous medium> throughout the layer, so $k_0>0$ and $k_0+k_1>0$.
$$
v(y)=\frac{\Delta P}{\mu\phi L}\left(k_0+\frac{k_1y}{H}\right),\qquad U=\frac1H\int_0^H v(y)\,dy=\frac{\Delta P}{\mu\phi L}\left(k_0+\frac{k_1}{2}\right).
$$
Here $\mu$ is <dynamic viscosity>. The <flux-weighted residence time in a porous layer> is the pore volume divided by throughput. Equivalently, weighting each streamline's transit time $L/v(y)$ by its inlet flux $v(y)\,dy$ gives
$$
\boxed{\overline\tau_Q=\frac{\mu\phi L^2}{\Delta P(k_0+k_1/2)}=\frac LU}.
$$
This is also the advective travel scale after cross-layer mixing has homogenized the <concentration>. For <resident-weighted transit time in a layered flow>, sampling the initial fluid uniformly by resident volume, while suppressing cross-stream exchange, is a different experiment: its mean is
$$
\overline\tau_{\rm resident}=\frac{\mu\phi L^2}{\Delta P k_1}\log\left(\frac{k_0+k_1}{k_0}\right),
$$
with the continuous $k_1=0$ limit $\mu\phi L^2/(\Delta P k_0)$. Specifying the sampling convention avoids confusing the reciprocal of the mean speed with the mean reciprocal speed.
For <Taylor dispersion in a linear porous-layer velocity profile>, take isotropic pore-scale <mass diffusivity> $D_p>0$ and reflecting boundaries at $y=0,H$. Write $v-U=\gamma(y-H/2)$, where $\gamma=\Delta P k_1/(\mu\phi LH)$. Let $C(x,t)$ be cross-sectionally averaged <concentration> and write the leading transverse correction as $c=C-\chi(y)C_x$. Substitution into <scalar transport> $c_t+v c_x=D_p(c_{xx}+c_{yy})$ yields the cell problem
$$
-D_p\chi''=v-U,\quad\chi'(0)=\chi'(H)=0,\quad\langle\chi\rangle=0.
$$
Thus $\chi'=\gamma y(H-y)/(2D_p)$, and averaging the axial solute flux gives $UC-(D_p+D_m)C_x$, with
$$
\boxed{D_m=\langle(v-U)\chi\rangle=D_p\langle(\chi')^2\rangle
=\frac{\gamma^2H^4}{120D_p}
=\frac{(\Delta P k_1)^2H^2}{120\mu^2\phi^2L^2D_p}}.
$$
The <integration by parts> has no boundary term. It also proves nonnegativity, even when $k_1<0$. The total effective <mass diffusivity> is $\mathcal D=D_p+D_m$. For anisotropic pore dispersion, the denominator uses the transverse coefficient and the added molecular term uses the axial coefficient.
There are two separate tests for significance. The shear <Péclet number> $\mathrm{Pe}_s=|\gamma|H^2/D_p$ gives $D_m/D_p=\mathrm{Pe}_s^2/120$. A large value implies substantial enhancement of axial spreading. But the <Taylor dispersion> limit additionally needs
$$
\boxed{\frac LU\gg\frac{H^2}{D_p}}.
$$
Otherwise particles can cross the rock before transverse mixing occurs, and a constant long-time $D_m$ is not an adequate breakthrough model. The relative front width is of order $\sqrt{\mathcal D/(UL)}$ once the long-time model applies.
For a constant tracer input flux, normalize $j_0$ as solute flux per pore cross-sectional area, and use the initially tracer-free half-line model
$$
C_t+UC_x=\mathcal D C_{xx},\quad C(x,0)=0,\quad
UC(0,t)-\mathcal D C_x(0,t)=j_0,\quad C(\infty,t)=0\ \text{for finite }t.
$$
This is a <constant-flux tracer inlet solution>; prescribing a flux is distinct from prescribing the inlet <concentration>. A <Laplace transform> in time has decaying spatial root $r=(U-\sqrt{U^2+4\mathcal D p})/(2\mathcal D)$ and gives
$$
\widetilde C(x,p)=\frac{2j_0}{p\left(U+\sqrt{U^2+4\mathcal D p}\right)}e^{rx}.
$$
Define $\xi=UL/\mathcal D$, $\vartheta=U^2t/\mathcal D$, and $z_\pm=(\xi\pm\vartheta)/(2\sqrt\vartheta)$. Inverting, or differentiating the following expression to check the equation and flux boundary condition, gives the requested arrival history:
$$
\boxed{\frac{C(L,t)}{j_0/U}=\frac12\operatorname{erfc}(z_-)
+\sqrt{\frac{\vartheta}{\pi}}e^{-z_-^2}
-\frac12(1+\xi+\vartheta)e^\xi\operatorname{erfc}(z_+)}.
$$
It tends from zero to $j_0/U$. For $UL/\mathcal D\gg1$ near breakthrough, its leading front is
$$
\boxed{C(L,t)\simeq\frac{j_0}{2U}\operatorname{erfc}\left(\frac{L-Ut}{2\sqrt{\mathcal D t}}\right)}.
$$
If a well-stirred inlet instead holds $C(0,t)=C_0=j_0/U$, the exact <constant-concentration inlet solution> is $C(L,t)=\tfrac{C_0}{2}\{\operatorname{erfc}(z_-)+e^\xi\operatorname{erfc}(z_+)\}$. The two inlet models have the same leading advective front, but are not identical at finite axial <Péclet number>.
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