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The imposed pressure gradient and Darcy's law give a linear velocity profile. Use $U$ as the mean pore velocity in the contaminant advection-diffusion equation; thenThe last conditions express zero transverse solute flux under the standard closed-layer interpretation. Transverse exchange through the layer boundaries would change this cell problem and requires additional boundary data. If the prescribed mean speed is a Darcy velocity, its corresponding pore velocity is that speed divided by porosity.

The given parameter is the ratio of transverse mixing time to travel time:Thus transverse dispersion mixes the contaminant across the layer many times during its passage along the long layer. Different fluid speeds still matter: repeated exchange between slow and fast portions produces Taylor dispersion, rather than transport solely at each parcel's original speed. The averaged description applies after $t\gg H^2/D_T$ and on longitudinal scales large compared with the distance travelled during that mixing time.

Write $C=\overline{c}$ and, to leading correction order, $c=C+\chi (y)C_x+\cdots$, with $\overline{\chi }=0$. Taking $C_t\simeq -U{C}_x$ in the local equation gives the cell problemWith $\eta =y/H$, its solution isAveraging the local advection-diffusion equation now yields $C_t+U{C}_x=[D_L-\overline{(u-U)\chi }]C_{xx}$. An integration by parts in the cell problem gives $-\overline{(u-U)\chi }=D_T\overline{({\chi }^{\prime }{)}^2}$. Since ${\int }_0^1({\eta }^2-\eta {)}^{2}d\eta =1/30$, the effective equation isThis is Taylor dispersion in a linear porous-layer velocity profile. For a localized pulse away from the inlet and outlet, its mean position advances at $U$ and its longitudinal variance grows as $2D_{\mathrm{eff}}t$.

For the inlet step, use the semi-infinite inlet approximation $x>0$, $C(0,t)=1$, $C(x,0)=0$ and $C\to 0$ as $x\to \infty$ at fixed $t$. The resulting constant-concentration inlet solution isOne derivation is to Laplace transform in time: $\tilde{C}(x,p)=p^{-1}\exp [(U-\sqrt{U^2+4D_{\mathrm{eff}}p})x/(2D_{\mathrm{eff}})]$. Inverting gives the two complementary error functions. Their sum is one at $x=0$; for fixed $x>0$ both terms vanish as $t\downarrow 0$, and substitution verifies the averaged equation. Retaining only the first term gives the familiar moving error-function front far from the inlet when longitudinal advection dominates diffusion, but does not satisfy the inlet condition exactly.

For a literal finite layer $0<x<L$, a downstream boundary condition is also needed once the outlet influences the solution. The PDF's step-inlet request specifies only the initially clean region $x>0$, so the formula above is the semi-infinite/long-layer solution, not a claim of a unique finite-interval solution with unspecified outlet data.

Let $K$ be intrinsic permeability of a porous medium, $\varphi$ the mobile pore fraction, and let $h$ measure thickness normal to the sloping cap. Define cold and reservoir-temperature properties byThe imposed inequality makes ${\delta }_c>0$, so the cold fluid is still buoyant. Assume $K>0$, $\sin \theta >0$, positive densities and viscosities, and the usual positive thermal-expansion and viscosity coefficients. Away from localized fronts, neglect the along-current thickness gradient. Hydrostatic water pressure and Darcy's law then give the upslope Darcy velocitiesTypically $u_h>u_c$: warming increases buoyancy and decreases dynamic viscosity.

Current depths and the moving moving thermal front in a porous current. For this slope-driven porous gravity current, the injected mass flux per unit well length fixes the cold plateau thickness:The moving thermal front in a porous current moves at $V_T=\Gamma u_c$. Since $u_c$ is explicitly a Darcy velocity, the fluid mass stored per unit horizontal area is $\varphi \rho h$. Apply mass conservation across the moving moving thermal front in a porous current:Writing $\gamma =\varphi \Gamma$ and $r=u_c/u_h$, this gives the distal warm depthFor a retarded moving thermal front in a porous current take $0\le \gamma <1$; otherwise this two-plateau geometry needs reconsideration. The factor $(1-\gamma )/(1-\gamma r)$ is important: the distal mass flux is not generally $Q$, because mass is being stored as the cold region replaces the warm region. Setting the two fluxes equal would silently assume a stationary moving thermal front in a porous current.

If the model absorbs porosity into storage and uses pore velocity as its transport speed, the same formulas use $\gamma =\Gamma$. The explicit Darcy convention in the PDF instead gives $\gamma =\varphi \Gamma$; porosity must be specified or absorbed consistently. In the stationary-front limit $\gamma =0$, the two steady depths reduce to $Q/({\rho }_c{u}_c)$ and $Q/({\rho }_h{u}_h)$. In the Boussinesq approximation, use a common reference density in the mass factors while retaining ${\delta }_c,{\delta }_h$ in the driving force.

Leakage thresholds. A normal thickness $h_i$ generates cap overpressure ${\Pi }_i={\delta }_{i}g{h}_i\cos \theta$. Before leakage modifies the current, the two plateau values areFor the caprock leakage threshold $\Delta p>0$, the candidate critical mass-injection rates are thereforeCold-only leakage occurs for $Q_c^{\ast }\le Q<Q_h^{\ast }$, provided $Q_c^{\ast }<Q_h^{\ast }$. Both plateaus can leak once $Q\ge \max (Q_c^{\ast },Q_h^{\ast })$. In the usual common-density approximation, ${\mu }_c>{\mu }_h$ and $r<1$ ensure $Q_c^{\ast }<Q_h^{\ast }$, giving the expected sequence: no leakage, cold-only leakage, then leakage from both cold and warm regions. Without that approximation the ordering must be checked; the given inequality $\Delta \rho >\alpha \Delta T$ alone does not establish it. These are onset criteria computed on the nonleaking current, not a post-leakage mass budget.

For small slopes $\cos \theta \simeq 1$ and $\tan \theta \simeq \sin \theta \simeq \theta$. If another thickness convention is used, its hydrostatic column and projected flux must be changed consistently; one should not mix a normal thickness with a vertical-pressure formula lacking the cosine.

Cross-current heat conduction. Heat transfer from the warm formation makes temperature vary across the carbon dioxide depth and introduces a warming time controlled by thickness, thermal diffusivity and the surrounding rock's heat capacity. It smooths the sharp thermal adjustment and causes gradual changes in buoyancy, viscosity and the velocity profile. The cold fluid generally warms, becomes more mobile and requires less depth to carry a prescribed mass flux; the enhanced cold-region overpressure and its distinct leakage zone tend to shrink. Warming may occur before a parcel reaches the idealized advective moving thermal front in a porous current, especially for a thin current. A single supplied $\Gamma$ no longer describes all heat transport. Quantitative depths and thresholds then require a coupled temperature equation and thermal boundary data; their exact changes cannot be inferred from the linear property laws alone.