Add solution for $S_w$ frontal advance
$$\begin{equation}
\left(\frac{dx}{dt}\right)_{S_w} = \frac{q_t}{\phi A} \left(\frac{\partial f_w}{\partial S_w}\right)_t
\end{equation}$$
where
$$\begin{equation}
f_w = \frac{1 + \frac{k_o A}{\mu_o q_t} \left(\rho_o - \rho_w\right) g \sin \alpha}{1 + \frac{k_o}{k_w}\frac{\mu_w}{\mu_o}}
\end{equation}$$
and $k_o$ and $k_w$ are dependent upon water saturation following the Brooks and Corey model.
$$\begin{equation}
k_{ro} = \left(\frac{S_o- S_{or}}{1 - S_{or} - S_{wc}- S_{gc}}\right)^{n_o}
\end{equation}$$