Darcys Law

Although Darcy's law (an expression of conservation of momentum) was originally determined experimentally by Henry Darcy (during 1855–1856), it has since been derived from the Navier-Stokes equations via homogenization. It is analogous to Fourier's law in the field of heat conduction, Ohm's law in the field of electrical networks, or Fick's law in diffusion theory.

One application of Darcy's law is to water flow through an aquifer. Darcy's law along with the equation of conservation of mass are equivalent to the groundwater flow equation, one of the basic relationships of hydrogeology. Darcy's law is also used to describe oil, water, and gas flows through petroleum reservoirs.

Darcy's law is a simple proportional relationship between the instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance.

The total discharge, Q (units of volume per time, e.g., ft³/s or m³/s) is equal to the product of the permeability (κ units of area, e.g. m²) of the medium, the cross-sectional area (A) to flow, and the pressure drop (Pb − Pa), all divided by the dynamic viscosity μ (in SI units e.g. kg/(m·s) or Pa·s), and the length L the pressure drop is taking place over. The negative sign is needed because fluids flow from high pressure to low pressure. So if the change in pressure is negative (in the x-direction) then the flow will be positive (in the x-direction). Dividing both sides of the equation by the area and using more general notation leads to

where q is the filtration velocity or Darcy flux (discharge per unit area, with units of length per time, m/s) and is the pressure gradient vector. This value of the filtration velocity (Darcy flux), is not the velocity which the water traveling through the pores is experiencing.

The pore (interstitial) velocity (v) is related to the Darcy flux (q) by the porosity (φ). The flux is divided by porosity to account for the fact that only a fraction of the total formation volume is available for flow. The pore velocity would be the velocity a conservative tracer would experience if carried by the fluid through the formation.

In three dimensions, gravity must be accounted for, as the flow is now affected by the vertical pressure drop caused by gravity when assuming hydrostatic conditions. The solution is to subtract the gravitational pressure drop from the existing pressure drop in order to express the resulting flow,

where the flux is now a vector quantity, is a tensor of permeability, is the gradient operator in 3D, g is the acceleration due to gravity, is the unit vector in the vertical direction, pointing downwards and ρ is the density of the fluid.

Effects of anisotropy in three dimensions are addressed using a symmetric second-order tensor of permeability:

where the magnitudes of permeability in the x, y, and z component directions are specified. Since this a symmetric matrix, there are at most six unique values. If the permeability is isotropic (equal magnitude in all directions), then the diagonal values are equal, , while all other components are 0. The permeability tensor can be interpreted through an evaluation of the relative magnitudes of each component. For example, rock with highly permeable vertical fractures aligned in the x-direction will have higher values for than other component values.

Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that groundwater flowing in aquifers exhibits, including:

A graphical illustration of the use of the steady-state groundwater flow equation (based on Darcy's law and the conservation of mass) is in the construction of flownets, to quantify the amount of groundwater flowing under a dam.

Darcy's law is only valid for slow, viscous flow; fortunately, most groundwater flow cases fall in this category. Typically any flow with a Reynolds number (based on a pore size length scale) less than one is clearly laminar, and it would be valid to apply Darcy's law. Experimental tests have shown that flow regimes with values of Reynolds number up to 10 may still be Darcian. Reynolds number (a dimensionless parameter) for porous media flow is typically expressed as

where ρ is the density of the fluid (units of mass per volume), v is the specific discharge (not the pore velocity — with units of length per time), d30 is a representative grain diameter for the porous medium (often taken as the 30% passing size from a grain size analysis using sieves), and μ is the dynamic viscosity of the fluid.

When the acceleration terms are neglected - that is when ρDui / Dt = 0, the Navier-Stokes equation is simplified to the Stokes equation:

where μ is the viscosity, ui is the velocity in the i direction, gi is the gravity component in the i direction and p is the pressure. Assuming the viscous resisting force is linear with the velocity we may write:

where φ is the porosity; − (kij) − 1 is a proportionality factor (a second order tensor). This gives the velocity:

which gives Darcy's law:

For very short time scales or high frequency oscillations, a time derivative of flux may be added to Darcy's law, which results in valid solutions at very small times (in heat transfer, this is called the modified form of Fourier's law),

where τ is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at "normal" times (> nanoseconds). The main reason for doing this is that the regular groundwater flow equation (diffusion equation) leads to singularities at constant head boundaries at very small times. This form is more mathematically rigorous, but leads to a hyperbolic groundwater flow equation, which is more difficult to solve and is only useful at very small times, typically out of the realm of practical use.

Another extension to the traditional form of Darcy's law is the Brinkman term, which is used to account for transitional flow between boundaries (introduced by Brinkman in 1947),

where β is an effective viscosity term. This correction term accounts for flow through medium where the grains of the media are porous themselves, but is difficult to use, and is typically neglected.

For multiphase flow, an approximation is to use Darcy's law for each phase, with permeability replaced by phase permeability, which is the permeability of the rock multiplied with relative permeability. This approximation is valid if the interfaces between the fluids remain static, which is not true in general, but it is still a reasonable model under steady-state conditions.

Assuming that the flow of a phase in the presence of another phase can be viewed as single phase flow through a reduced pore network, we can add the subscript i for each phase to Darcy's law above written for Darcy flux, and obtain for each phase in multiphase flow

where κi is the phase permeability for phase i. From this we also define relative permeability κri for phase i as

where κ is the permeability for the porous medium, as in Darcy's law.

For a sufficiently high flow velocity, the flow is nonlinear, and Dupuit and Forchheimer have proposed to generalize the flow equation to

where V is the flow velocity and β is a factor to be experimentally deduced.

In pressure-driven membrane operations, Darcy's law is often used in the form,

where,