would be valid to apply Darcy's law. Experimental tests have shown that flow regimes with Reynolds numbers up to 10 may still be Darcian, as in the case of groundwater flow. The Reynolds number (a dimensionless parameter) for porous media flow is typically expressed as where is the kinematic viscosity of water, is the specific discharge (not the pore velocity — with units of length per time), is a representative grain diameter for the porous media (the standard choice is d30, which is the 30% passing size from a grain size analysis using sieves — with units of length). Derivation For stationary, creeping, incompressible flow, i.e. , the Navier–Stokes equation simplifies to the Stokes equation, which by neglecting the bulk term is: where is the viscosity, is the velocity in the direction, is the gravity component in the direction and is the pressure. Assuming the viscous resisting force is linear with the velocity we may write: where is the porosity, and is the second order permeability tensor. This gives the velocity in the direction, which gives Darcy's law for the volumetric flux density in the direction, In isotropic porous media the off-diagonal elements in the permeability tensor are zero, for and the diagonal elements are identical, , and the common form is obtained as below, which enables the determination of the liquid flow velocity by solving a set of equations in a given region. The above equation is a governing equation for single-phase fluid flow in a porous medium. Use in petroleum engineering Another derivation of Darcy's law is used extensively in petroleum engineering to determine the flow through permeable media — the most simple of which is for a one-dimensional, homogeneous rock formation with a single fluid phase and constant fluid viscosity. Almost all oil reservoirs have a water zone below the oil leg, and some have also a gas cap above the oil leg. When the reservoir pressure drops due to oil production, water flows into the oil zone from below, and gas flows into the oil zone from above (if the gas cap exists), and we get a simultaneous flow and immiscible mixing of all fluid phases in the oil zone. The operator of the oil field may also inject water (and/or gas) in order to improve oil production. The petroleum industry is therefore using a generalized Darcy equation for multiphase flow that was developed by Muskat