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  "body": "\n## Statement\n\nConsider the slow flow of a fluid of viscosity $\\mu \u003e 0$ through a porous medium of permeability $k \u003e 0$ that fills a region $\\Omega \\subseteq \\R^3$.\nLet $p \\colon \\Omega \\to \\R$ be the pressure of the fluid, and neglect gravity.\nLet $q \\colon \\Omega \\to \\R^3$ be the volumetric flux, the [[vector field|0299VCW1TH]] of the volume of fluid that crosses a unit area per unit time.\nThen\n$$q = - \\frac{k}{\\mu} \\nabla p.$$\nHere $\\nabla p$ is the [[gradient|0ABK9V9Y64]] of the pressure.\n\n## Remarks\n\nDarcy's law is an empirical law, found by Henry Darcy from experiments on water flowing through sand.\nIt can also be derived from the equations of [[Stokes flow|13CB7V2YM4]] by homogenization in a periodic porous structure.\nIt is valid when the flow is slow, that is, when the Reynolds number of the flow in the pores is small.\n\nFor an incompressible fluid without sources, the [[divergence|0MWV6ZTK6S]] of the flux is zero.\nWith constant $k$ and $\\mu$ this gives $\\Delta p = 0$, so the pressure is a harmonic function, and boundary values of $p$ determine the flow.\nWith gravity the law reads $q = -\\tfrac{k}{\\mu} (\\nabla p - \\rho g)$, where $\\rho$ is the density and $g$ the gravitational acceleration vector.\n\n## Explanations\n\n### L1 human\n\nWater seeps through sand or soil from high pressure to low pressure.\nDarcy's law says that the flow rate is proportional to the drop in pressure and to how easily the medium lets fluid through.\nIt is inversely proportional to the viscosity, that is, to how thick and sticky the fluid is.\nIt is the analogue for porous media of Ohm's law for electric current and Fourier's law for heat.\n\n### L2 human\n\nDarcy's law turns problems of groundwater, oil reservoirs and filtration into elliptic and parabolic partial differential equations for the pressure.\nCombined with conservation of mass for a compressible fluid or a gas, it leads to diffusion equations such as the porous medium equation.\nWhen the permeability varies in space, the pressure solves the divergence-form equation $\\nabla \\cdot (k \\nabla p) = 0$, a standard model of an elliptic equation with variable coefficients.\n",
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