= Solution
Choose $x$ upslope and $z$ normally downwards from the impermeable roof, so the buoyant layer occupies $0<z<h(x,t)$. Use the <hydrostatic approximation> and neglect the ambient return-flow <pressure> correction of relative order $h/H$. <Pressure> continuity at the interface gives the <pressure> inside the buoyant layer as
$$
p=p_0-\rho gx\sin\theta+\Delta\rho\,gh\cos\theta+(\rho-\Delta\rho)gz\cos\theta.
$$
The along-slope <Darcy velocity> is therefore
$$
v_D=-\frac{k}{\mu}\left[p_x+(\rho-\Delta\rho)g\sin\theta\right]=\frac{kg\Delta\rho}{\mu}(\sin\theta-\cos\theta\,h_x).
$$
Define $u=kg\Delta\rho\sin\theta/\mu$. Integrating over depth gives the <volume flux per unit width> $F=uh-u\cot\theta\,hh_x$. Local fluid-volume conservation, including <porosity>, is $\phi h_t+F_x=0$. Hence the <inclined porous gravity current> satisfies
$$
\boxed{\phi h_t=-\frac{kg\Delta\rho}{\mu}\sin\theta\left[h_x-\cot\theta\,(hh_x)_x\right].}
$$
The first term is upslope <advection> driven by the <mass density> difference; the second is nonlinear hydrostatic spreading. The <Darcy velocity> $u$ and the corresponding <pore velocity> $u/\phi$ must be distinguished.
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