= Solution
Let $K$ be intrinsic <permeability of a porous medium>, $\phi$ the mobile pore fraction, and let $h$ measure thickness normal to the sloping cap. Define cold and reservoir-temperature properties by
$$
\begin{aligned}
\rho_c&=\rho_w-\Delta\rho+\alpha\Delta T,&\delta_c&=\rho_w-\rho_c=\Delta\rho-\alpha\Delta T, &\mu_c&=\mu_0+\lambda\Delta T,\\
\rho_h&=\rho_w-\Delta\rho,&\delta_h&=\Delta\rho,&\mu_h&=\mu_0.
\end{aligned}
$$
The 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 velocities>
$$
\boxed{u_c=\frac{K\delta_cg\sin\theta}{\mu_c},\qquad
u_h=\frac{K\delta_hg\sin\theta}{\mu_h}.}
$$
Typically $u_h>u_c$: warming increases <buoyancy> and decreases <dynamic viscosity>.
\b[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:
$$
\boxed{h_c=\frac{Q}{\rho_cu_c}=\frac{Q\mu_c}{\rho_cK\delta_cg\sin\theta}.}
$$
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 $\phi\rho h$. Apply <mass conservation> across the moving <moving thermal front in a porous current>:
$$
\rho_h(u_h-\phi V_T)h_h=\rho_c(u_c-\phi V_T)h_c.
$$
Writing $\gamma=\phi\Gamma$ and $r=u_c/u_h$, this gives the distal warm depth
$$
\boxed{h_h=\frac{Q(1-\gamma)}{\rho_h(u_h-\gamma u_c)}
=\frac{Q}{\rho_hu_h}\frac{1-\gamma}{1-\gamma r}.}
$$
For a retarded <moving thermal front in a porous current> take $0\leq\gamma<1$; 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 $\gamma=\phi\Gamma$; <porosity> must be specified or absorbed consistently. In the stationary-front limit $\gamma=0$, the two steady depths reduce to $Q/(\rho_cu_c)$ and $Q/(\rho_hu_h)$. In the <Boussinesq approximation>, use a common reference density in the mass factors while retaining $\delta_c,\delta_h$ in the driving force.
\b[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 are
$$
\Pi_c=\frac{Q\mu_c\cot\theta}{\rho_cK},\qquad
\Pi_h=\frac{Q\mu_h\cot\theta}{\rho_hK}\frac{1-\gamma}{1-\gamma r}.
$$
For the <caprock leakage threshold> $\Delta p>0$, the candidate critical mass-injection rates are therefore
$$
\boxed{Q_c^*=\frac{\Delta p\,\rho_cK\tan\theta}{\mu_c},\qquad
Q_h^*=\frac{\Delta p\,\rho_hK\tan\theta}{\mu_h}
\frac{1-\gamma r}{1-\gamma}.}
$$
Cold-only leakage occurs for $Q_c^*\leq Q<Q_h^*$, provided $Q_c^*<Q_h^*$. Both plateaus can leak once $Q\geq\max(Q_c^*,Q_h^*)$. In the usual common-density approximation, $\mu_c>\mu_h$ and $r<1$ ensure $Q_c^*<Q_h^*$, 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 $\cos\theta\simeq1$ 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.
\b[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.
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