Use the <Dupuit approximation>: the saturated region is shallow enough that <hydrostatic pressure> is $p=\rho g(h-z)$ and flow is predominantly horizontal. <Darcy's law> then gives the horizontal <Darcy velocity> $u_x=-u_b(1+\beta z)h_x$, where $u_b=k_0\rho g/\mu$. Integrating through the saturated depth gives the <volume flux per unit width>
$$
q=\int_0^h u_x\,dz=-u_b\left(h+\frac\beta2h^2\right)h_x.
$$
The stored water volume per unit area is $\phi h$, so <mass conservation> yields the <unconfined aquifer with depth-dependent permeability> equation
$$
\boxed{\phi h_t=u_b\partial_x\left[\left(h+\frac\beta2h^2\right)h_x\right]+R,\qquad h(0,t)=0,\qquad q(L,t)=0.}
$$
Define the positive river discharge by $Q(t)=-q(0,t)$. A useful check on all subsequent results is the integrated <mass conservation> law
$$
\phi\frac d{dt}\int_0^Lh\,dx=RL-Q.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 332 / 1 / Solution

With <permeability of a porous medium> $k(z)=k_0(1+\beta z)$, the <Dupuit approximation> gives
$$
q=-u_b\left(h+\frac\beta2h^2\right)h_x,\qquad \phi h_t+q_x=R,\qquad u_b=\frac{k_0\rho g}{\mu}.
$$
Here $\phi$ is <porosity> and $R$ is recharge. The low-height limit has mobility proportional to $h$, whereas large heights have mobility proportional to $h^2$. This changes both the <forced filling similarity for power-law diffusion> and the <separable draining profile for power-law diffusion>.

Unconfined aquifer with depth-dependent permeability

{c}

The Dupuit approximation treats flow in a shallow <unconfined aquifer> as predominantly horizontal, with <hydrostatic pressure> $p=\rho g(h-z)$ and horizontal pressure gradient independent of depth. Integrating <Darcy's law> over the saturated depth then gives a <volume flux per unit width> depending only on the groundwater height and its slope.

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 Incoming links: Dupuit approximation