= Solution
Take $z$ upwards and write $\zeta=z+Wt$, so that the unperturbed interface is $\zeta=0$, with <carbon dioxide> above and <water> below. Use the sharp-interface, <incompressible flow> version of <Darcy's law>, with constant <porosity> and <permeability of a porous medium>, no <capillary pressure>, and effectively unbounded layers. The distinction between <Darcy flux> and interface speed matters: the base flux is $q_z=-\phi W$, since the printed $W$ is the speed of the interface. Thus the base <pressure gradients> satisfy
$$
p_{j,z}^0=\frac{\mu_j\phi W}{k}-\rho_jg,\qquad j=c,w.
$$
The <mass density> ordering is stabilizing, whereas injection of the less viscous fluid into the more viscous fluid produces the <Saffman–Taylor instability>.
For <linear stability analysis>, displace the interface by $\eta=\widehat\eta e^{\sigma t+i\alpha x}$. Taking the <divergence> of <Darcy's law> shows that the perturbation <pressure> satisfies the <Laplace equation>. Decay away from the interface gives, for $a_\alpha=|\alpha|>0$,
$$
p'_c=Ae^{-a_\alpha\zeta}e^{\sigma t+i\alpha x},\qquad p'_w=Be^{a_\alpha\zeta}e^{\sigma t+i\alpha x}.
$$
The <kinematic boundary condition> and continuity of normal <Darcy flux> give
$$
\phi\sigma\widehat\eta=\frac{k a_\alpha A}{\mu_c}=-\frac{k a_\alpha B}{\mu_w}.
$$
Meanwhile <pressure continuity>, evaluated on the displaced interface, gives
$$
A-B=-\widehat\eta(p_{c,z}^0-p_{w,z}^0)=\widehat\eta\left[\frac{\phi W(\mu_w-\mu_c)}k-g(\rho_w-\rho_c)\right].
$$
Eliminating $A,B$ yields the <buoyancy-modified Darcy fingering dispersion relation>:
$$
\boxed{\sigma(\alpha)=\frac{|\alpha|}{\phi(\mu_w+\mu_c)}\left[\phi W(\mu_w-\mu_c)-kg(\rho_w-\rho_c)\right].}
$$
Every nonzero <wavenumber> decays when $W<W_c$, grows when $W>W_c$, and is neutral at equality, where
$$
\boxed{W_c=\frac{kg(\rho_w-\rho_c)}{\phi(\mu_w-\mu_c)}.}
$$
The zero <wavenumber> is a uniform displacement and is neutral. Without a short-scale regularization such as <surface tension>, this model has no fastest-growing finite <wavenumber>: on the unstable side its <growth rate> increases without bound as $|\alpha|$ increases. Finite layer depths would change the <dispersion relation>; none are specified here.
Taking $g=9.81\,\mathrm{m\,s^{-2}}$ and a year of $365.25$ days, the data give
$$
\phi W_c=3.27\times10^{-6}\,\mathrm{m\,s^{-1}}\simeq103.2\,\mathrm{m\,yr^{-1}},\qquad \boxed{W_c\simeq\frac{103.2}{\phi}\,\mathrm{m\,yr^{-1}}.}
$$
The numerical paragraph supplies no <porosity>, so it cannot determine a unique numerical interface speed. If $W$ were instead intended to mean <Darcy velocity>, the numerical critical value would be $103.2\,\mathrm{m\,yr^{-1}}$, and the interface would move at $W/\phi$. These two velocity conventions must not be conflated.
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