= Darcy law
{c}
{title2=$\mathbf q=-k(\nabla p-\rho\mathbf g)/\mu$}
{wiki}
= Darcy flow law
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{synonym}
= Darcy's law
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{synonym}
<Darcy law> is a linear relation between the bulk <Darcy flux> and the net driving force per unit volume. For an <isotropic> <porous medium> containing one fully saturating <Newtonian fluid>,
$$
\mathbf q=-\frac{k}{\mu}\bigl(\nabla p-\rho\mathbf g\bigr).
$$
Here $\mathbf q$ is the <Darcy velocity> relative to the solid framework, $k$ is the intrinsic <permeability of a porous medium>, $\mu>0$ is <dynamic viscosity>, $p$ is averaged <fluid pressure>, $\rho$ is <mass density>, and $\mathbf g$ is the downward <gravitational acceleration> vector. The <pressure gradient> exerts force density $-\nabla p$, while <gravity> contributes $\rho\mathbf g$; the minus sign makes the discharge follow their sum. The symbols $\mathbf u$ and $\mathbf q$ are both commonly used for <Darcy velocity>; neither denotes <pore velocity> here.
With upward coordinate $z$ and positive <gravitational acceleration> magnitude $g$, $\mathbf g=-g\mathbf e_z$. Constant $\rho$ and $g$ therefore give
$$
\nabla p-\rho\mathbf g=\nabla p+\rho g\mathbf e_z=\rho g\nabla\left(z+\frac{p}{\rho g}\right),\qquad \mathbf q=-K\nabla h,\qquad K=\frac{k\rho g}{\mu}.
$$
The <hydraulic head> $h=z+p/(\rho g)$ combines elevation and <fluid pressure>, and <hydraulic conductivity> $K$ combines the medium and <fluid> properties. <Hydrostatic pressure> satisfies $\nabla p=\rho\mathbf g$, so it produces zero <Darcy flux> even though <fluid pressure> increases downward. Conversely, equal pressures at the two ends of a vertical column do not imply zero flow: their elevations give different <hydraulic heads>.
The appeal of <Darcy law> is that a complicated connected pore geometry enters a macroscopic calculation through a measurable coefficient. It predicts filter discharge and <aquifer> flow; combined with <mass conservation>, it gives an equation for the evolving <fluid pressure>. Its simple linear form requires a small pore-scale <Reynolds number>, slow adjustment compared with pore-scale inertial relaxation, and a valid <continuum approximation>. For a characteristic pore size $d$ and mean <pore velocity> $v$, one possible convention is $\mathrm{Re}_{\rm pore}=\rho|v|d/\mu\ll1$; conventions using grain size or <Darcy velocity> have different numerical thresholds. Merely calling flow laminar does not guarantee negligible inertia. Unsaturated pores, several <fluid> phases, appreciable inertia, or a <fluid> whose stress is not Newtonian require further modeling. Woessner and Poeter discuss these limits in their authored hydrology text.https://books.gw-project.org/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow/chapter/applicability-of-darcys-law/{ref}
Henry Darcy's 1856 book \i[Les fontaines publiques de la ville de Dijon] was published by Victor Dalmont; the University Grenoble Alpes digital collection verifies this bibliographic record and supplies a digitized copy.https://bibnum-patrimoniale.univ-grenoble-alpes.fr/items/show/1643{ref} Woessner and Poeter's account connects <Darcy law> to experiments on water flowing through sand filters, relating discharge to cross-sectional area, <hydraulic head> difference and sample length.https://books.gw-project.org/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow/chapter/darcys-law/{ref} The modern intrinsic <permeability of a porous medium> notation above should not be read as a quotation of Darcy's original notation.
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