= Darcy velocity
{c}
{title2=$\mathbf u=\epsilon(\mathbf v_l-\mathbf v_s)$}
= Darcy flux
{c}
{synonym}
= Specific discharge
{synonym}
<Darcy velocity>, also called <Darcy flux> or <specific discharge>, is <fluid> <volume flux> relative to the solid framework per unit total cross-sectional area. For a uniform axial discharge $Q$ through bulk area $A$,
$$
q=\frac QA.
$$
The area includes both solid and voids. Hence <Darcy velocity> is a superficial flux with units $\mathrm{m\,s^{-1}}$, even though no <fluid> travels through the solid part. For a nonuniform discharge, $Q=\int_A\mathbf q\cdot\mathbf n\,dA$, with $\mathbf n$ the unit normal; $Q/A$ is the area average of the normal <Darcy flux>.
If <porosity> is $\epsilon$, the pore-volume-averaged liquid <velocity> is $\mathbf v_l$, and the local solid framework <velocity> is $\mathbf v_s$, the convention used here is $\mathbf u=\epsilon(\mathbf v_l-\mathbf v_s)$. In a stationary, fully saturated framework with all pore volume participating, $\mathbf v_l=\mathbf q/\epsilon$. The notation $\phi=\epsilon$ for <porosity> is equally common. <Darcy flow through a homogeneous slab> gives $q=2.0\times10^{-5}\,\mathrm{m\,s^{-1}}$ but <pore velocity> $v=8.0\times10^{-5}\,\mathrm{m\,s^{-1}}$ at $\phi=0.25$. Keeping these quantities separate lets <Darcy law> predict filter discharge while <pore velocity> predicts ideal advective travel time.https://books.gw-project.org/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow/chapter/darcys-law/{ref}
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