Darcy's law is a constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of earth sciences.
- 1 Background
- 2 Description
- 3 Derivation
- 4 Additional forms of Darcy's law
- 5 Validity of Darcy's law
- 6 See also
- 7 References
Although Darcy's law (an expression of Newton's second law) was determined experimentally by Darcy, 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 analyze 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, as refined by Morris Muskat, at constant elevation 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., m3/s) is equal to the product of the intrinsic permeability of the medium, κ (m2), the cross-sectional area to flow, A (units of area, e.g., m2), and the total pressure drop pb − pa (pascals), all divided by the viscosity, μ (Pa·s) and the length over which the pressure drop is taking place (L). The negative sign is needed because fluid flows from high pressure to low pressure. Note that the elevation head must be taken into account if the inlet and outlet are at different elevations. If the change in pressure is negative (where pa > pb), then the flow will be in the positive x direction. Dividing both sides of the equation by the area and using more general notation leads
where q is the flux (discharge per unit area, with units of length per time, m/s) and ∇p is the pressure gradient vector (Pa/m). This value of flux, often referred to as the Darcy flux, is not the velocity which the fluid traveling through the pores is experiencing. The fluid 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 fluid velocity would be the velocity a conservative tracer would experience if carried by the fluid through the formation.
- if there is no pressure gradient over a distance, no flow occurs (these are hydrostatic conditions),
- if there is a pressure gradient, flow will occur from high pressure towards low pressure (opposite the direction of increasing gradient — hence the negative sign in Darcy's law),
- the greater the pressure gradient (through the same formation material), the greater the discharge rate, and
- the discharge rate of fluid will often be different — through different formation materials (or even through the same material, in a different direction) — even if the same pressure gradient exists in both cases.
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 less than one is clearly laminar, and it 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 ρ is the density of water (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 media (often taken as the 30% passing size from a grain size analysis using sieves — with units of length), and μ is the viscosity of the fluid.
For stationary, creeping, incompressible flow, i.e. D(ρui)/ ≈ 0, the Navier–Stokes equation simplifies 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, and κij is the second order permeability tensor. This gives the velocity in the n direction,
which gives Darcy's law for the volumetric flux density in the n direction,
Additional forms of Darcy's law
Darcy's law for short time scales
For very short time scales, 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.
Brinkman form of Darcy's law
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 1949),
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.
Darcy's law 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 fluid of constant viscosity.
where Q is the flowrate of the formation (in units of volume per unit time), k is the permeability of the formation (typically in millidarcys), A is the cross-sectional area of the formation, μ is the viscosity of the fluid (typically in units of centipoise). ∂p/ represents the pressure change per unit length of the formation. This equation can also be solved for permeability and is used to measure it, forcing a fluid of known viscosity through a core of a known length and area, and measuring the pressure drop across the length of the core.
For flows in porous media with Reynolds numbers greater than about 1 to 10, inertial effects can also become significant. Sometimes an inertial term is added to the Darcy's equation, known as Forchheimer term. This term is able to account for the non-linear behavior of the pressure difference vs flow data.
where the additional term κ1 is known as inertial permeability.
For gas flow in small characteristic dimensions (e.g., very fine sand, nanoporous structures etc.), the particle-wall interactions become more frequent, giving rise to additional wall friction (Knudsen friction). For a flow in this region, where both viscous and Knudsen friction are present, a new formulation needs to be used. Knudsen presented a semi-empirical model for flow in transition regime based on his experiments on small capillaries. For a porous medium, the Knudsen equation can be given as 
where N is the molar flux, Rg is the gas constant, T is the temperature, Deff
K is the effective Knudsen diffusivity of the porous media. The model can also be derived from the first-principle-based binary friction model (BFM). The differential equation of transition flow in porous media based on BFM is given as
This equation is valid for capillaries as well as porous media. The terminology of the Knudsen effect and Knudsen diffusivity is more common in mechanical and chemical engineering. In geological and petrochemical engineering, this effect is known as the Klinkenberg effect. Using the definition of molar flux, the above equation can be rewritten as
This equation can be rearranged into the following equation
Comparing this equation with conventional Darcy's law, a new formulation can be given as
This is equivalent to the effective permeability formulation proposed by Klinkenberg:
where b is known as the Klinkenberg parameter, which depends on the gas and the porous medium structure. This is quite evident if we compare the above formulations. The Klinkenberg parameter b is dependent on permeability, Knudsen diffusivity and viscosity (i.e., both gas and porous medium properties).
Validity of Darcy's law
Darcy's law is valid for laminar flow through sediments. In fine-grained sediments, the dimensions of interstices are small and thus flow is laminar. Coarse-grained sediments also behave similarly but in very coarse-grained sediments the flow may be turbulent. Hence Darcy's law is not always valid in such sediments. For flow through commercial pipes, the flow is laminar when Reynolds number is less than 2000, but in some sediments it has been found that flow is laminar when the value of Reynolds number is less than 1.
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