Porous-media flow

Porous-media flow is fluid motion through an interconnected pore space. At small pore-scale Reynolds number, its volume-averaged velocity is commonly governed by Darcy's law.

Darcy's law gives the relative seepage velocity through a porous packing as ${\mathbf{u}}_f-{\mathbf{u}}_s=-(k/\eta )\nabla p$, where $k$ is permeability.

A porous gravity current spreads because a density difference creates a lateral hydrostatic-pressure gradient. For a two-dimensional current of thickness $h$, density deficit $\Delta \rho$, permeability $k$, viscosity $\mu$, and porosity $\varphi$,

A leaky porous gravity current loses flux through a localized fracture or a distributed permeable boundary. At a point fracture, mass conservation makes the horizontal Darcy flux discontinuous by exactly the leakage rate.