= Solution
For a stationary solid skeleton, the <Darcy velocity> $\mathbf u$ is liquid volume transport per unit total cross-sectional area, while the <pore velocity> $\mathbf v$ is the intrinsic mean fluid <velocity> within the connected pore space. Thus
$$
\boxed{\mathbf u=\phi\mathbf v.}
$$
For an incompressible liquid in a saturated region, <conservation of mass> is $\phi_t+\nabla\cdot\mathbf u=0$. With time-independent but spatially varying <porosity>, it becomes
$$
\boxed{\nabla\cdot\mathbf u=\nabla\cdot(\phi\mathbf v)=0,}
$$
not generally $\nabla\cdot\mathbf v=0$. Neglecting dispersion, a fluid interface is advected by the pore <velocity>; its mean normal speed is $V_n=\mathbf v\cdot\mathbf n=\mathbf u\cdot\mathbf n/\phi$.
For the thin layer on the inclined plane, the <gravitational acceleration> components are $g\sin\theta$ downslope and $-g\cos\theta$ normal to the plane. The <hydrostatic approximation> and constant <pressure> $p_a$ above the current give
$$
p=p_a+\rho g\cos\theta(h-z).
$$
With <permeability of a porous medium> $k(z)$, <Darcy's law> therefore gives
$$
u_x=\frac{\rho gk(z)}\mu(\sin\theta-\cos\theta\,h_x),
\qquad u_y=-\frac{\rho gk(z)}\mu\cos\theta\,h_y.
$$
Here $u_x,u_y$ denote the components of the Darcy <velocity>. Integrate over depth, using the prescribed power laws:
$$
\mathcal V=\int_0^h\phi(z)\,dz=\frac P\alpha h^\alpha,
$$
$$
\mathbf Q=\int_0^h(u_x,u_y)\,dz
=\frac{\rho gK}{\beta\mu}h^\beta
(\sin\theta-\cos\theta\,h_x,-\cos\theta\,h_y).
$$
Both integrals are finite for the positive exponents in the model. Conservation of the stored fluid gives $\mathcal V_t+\nabla_{x,y}\cdot\mathbf Q=0$, or
$$
\frac P\alpha\partial_t h^\alpha+
\frac{\rho gK\sin\theta}{\beta\mu}\partial_xh^\beta
=\frac{\rho gK\cos\theta}{\beta\mu}
\nabla_{x,y}\cdot(h^\beta\nabla_{x,y}h).
$$
For $0<\theta<\pi/2$, set $X=x\tan\theta$, $Y=y\tan\theta$ and
$$
T=\frac{\alpha\rho gK\sin\theta\tan\theta}{\beta\mu P}t.
$$
Since $\cos\theta\tan^2\theta=\sin\theta\tan\theta$, the <depth-dependent inclined porous-current equation> is
$$
\boxed{\partial_T h^\alpha+\partial_Xh^\beta
=\partial_X(h^\beta h_X)+\partial_Y(h^\beta h_Y).}
$$
The longitudinal flux carries fluid downslope, while the thickness-gradient terms spread it. The degenerate horizontal or vertical slope limits need their own coordinate scaling.
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