Darcy's law

Remarks

Darcy's law is an empirical law, found by Henry Darcy from experiments on water flowing through sand. It can also be derived from the equations of Stokes flow by homogenization in a periodic porous structure. It is valid when the flow is slow, that is, when the Reynolds number of the flow in the pores is small.

For an incompressible fluid without sources, the divergence of the flux is zero. With constant $k$ and $\mu$ this gives $\Delta p = 0$, so the pressure is a harmonic function, and boundary values of $p$ determine the flow. With gravity the law reads $q = -\tfrac{k}{\mu} (\nabla p - \rho g)$, where $\rho$ is the density and $g$ the gravitational acceleration vector.

Going deeper

For a student

Darcy's law turns problems of groundwater, oil reservoirs and filtration into elliptic and parabolic partial differential equations for the pressure. Combined with conservation of mass for a compressible fluid or a gas, it leads to diffusion equations such as the porous medium equation. When 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.