= Dupuit flow in a channel with cubic wetted area
{c}
{title2=$Q=\alpha K(H^4-\eta^4)/(12L)$}
For saturated <Darcy flow> through a long dam occupying a channel of width $\alpha z^2$, let hydraulic conductivity be $K=k\rho g/\mu$ and free-surface height $h(x)$. The <Dupuit approximation> gives $Q=-K\alpha h^3h_x/3$. Fixed heads $h(0)=H$ and $h(L)=\eta$ give $h^4=H^4-(H^4-\eta^4)x/L$ and the displayed steady discharge. <Porosity> relates pore velocity to superficial <Darcy velocity> but does not independently multiply this discharge when $k$ is intrinsic permeability. For depth-dependent permeability, integrate the horizontal conductivity over the saturated cross-section first.
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