Darcy velocity field

A Darcy velocity field is a mathematical field corresponding to fluid seepage through a porous medium, specifically if the seepage is following Darcy's law. These are used geophysics, e.g., to describe water sinking through sand. Darcy's law is a general model describing such seepage, originally derived from experiments. It applies to non-compressible fluids passing through a porous medium if the seepage is sufficiently slow that inertia and turbulence are insignificant. A Darcy flow (aka Darcy flux) is a flow according to the model, specifically, an overall throughput rate per unit cross-section, slower than the mean velocity of the fluid because of the porous material through which it is passing. The model relates this to the pressure difference and length of the porous medium through which it is passing. One form of the law:

    κΔp
q = ———
     μL

• q - Darcy flow, average fluid velocity through the medium.
• κ - permeability of the medium: a fraction between 0 and 1.
• Δp - difference in pressure.
• μ - viscosity of the fluid.
• L - length of the flow through the medium.

(A minus sign is sometimes included in the equation, giving the flow a negative number to indicate a downward direction.) The total flow is:

Q = q × A

• Q - total flow, volume per unit time.
• A - cross-sectional area of the medium.

Note that the term Darcy velocity is often used for the Darcy flow, but sometimes is used for a different quantity: the Darcy flow divided by the permeability, which is an average velocity of the fluid through the pores. Darcy's law can be derived from models of such porous material and Navier-Stokes equations or the equivalent.