Darcy's law

Darcy's law was first determined experimentally by Darcy, but has since been derived from the Navier–Stokes equations via homogenization methods.[2] It is analogous to Fourier's law in the field of heat conduction, Ohm's law in the field of electrical networks, and Fick's law in diffusion theory.

One application of Darcy's law is in the analysis of water flow through an aquifer; Darcy's law along with the equation of conservation of mass simplifies to the groundwater flow equation, one of the basic relationships of hydrogeology.

Morris Muskat first[3] refined Darcy's equation for a single-phase flow by including viscosity in the single (fluid) phase equation of Darcy. It can be understood that viscous fluids have more difficulty permeating through a porous medium than less viscous fluids. This change made it suitable for researchers in the petroleum industry. Based on experimental results by his colleagues Wyckoff and Botset, Muskat and Meres also generalized Darcy's law to cover a multiphase flow of water, oil and gas in the porous medium of a petroleum reservoir. The generalized multiphase flow equations by Muskat and others provide the analytical foundation for reservoir engineering that exists to this day.

Diagram showing definitions and directions for Darcy's law. A is the cross sectional area (m²) of the cylinder. Q is the flow rate (m³/s) of the fluid flowing through the area A. The flux of fluid through A is q = Q/A. L is the length of the cylinder. Δp = p_outlet - p_inlet = p_b - p_a. ${\displaystyle \nabla p}$ = Δp/L = hydraulic gradient applied between the points a and b.

Darcy's law, as refined by Morris Muskat, in the absence of gravitational forces and in a homogeneously permeable medium, is given by a simple proportionality relationship between the instantaneous flux ${\displaystyle q=Q/A}$ (units of ${\displaystyle Q}$: m³/s, units of ${\displaystyle A}$: m², units of ${\displaystyle q}$: m/s) through a porous medium, the permeability ${\displaystyle k}$ of the medium, the dynamic viscosity of the fluid ${\displaystyle \mu }$, and the pressure drop ${\displaystyle \Delta p}$ over a given distance ${\displaystyle L}$, in the form

$${\displaystyle q=-{\frac {k}{\mu L}}\Delta p}$$

This equation, for single phase (fluid) flow, is the defining equation for absolute permeability (single phase permeability).

With reference to the diagram to the right, the flux ${\displaystyle q}$, or discharge per unit area, is defined in units ${\displaystyle \mathrm {(m/s)} }$, the permeability ${\displaystyle k}$ in units ${\displaystyle \mathrm {(m^{2})} }$, the cross-sectional area ${\displaystyle A}$ in units ${\displaystyle \mathrm {(m^{2})} }$, the total pressure drop ${\displaystyle \Delta p=p_{b}-p_{a}}$ in units ${\displaystyle \mathrm {(Pa)} }$, the dynamic viscosity ${\displaystyle \mu }$ in units ${\displaystyle \mathrm {(Pa\cdot s)} }$, and ${\displaystyle L}$ is the length of the sample in units ${\displaystyle \mathrm {(m)} }$. A number of these parameters are used in alternative definitions below. A negative sign is used in the definition of the flux following the standard physics convention that fluids flow from regions of high pressure to regions of 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, then the flow will be in the positive x direction. There have been several proposals for a constitutive equation for absolute permeability, and the most famous one is probably the Kozeny equation (also called Kozeny–Carman equation).

The integral form of the Darcy law is given by:

$${\displaystyle Q={\frac {kA}{\mu L}}\,{\Delta p}}$$

where Q (units of volume per time, e.g., m³/s) is the total discharge. By considering the relation for static fluid pressure (Stevin's law):

$${\displaystyle p=\rho gh}$$

one can deduce the representation

$${\displaystyle Q={\frac {kAg}{\nu L}}\,{\Delta h}}$$

where ν is the kinematic viscosity. The corresponding hydraulic conductivity is therefore:

${\displaystyle K={\frac {k\rho g}{\mu }}={\frac {kg}{\nu }}.}$

Notice that the quantity ${\displaystyle q}$ or ${\displaystyle Q/A}$, often referred to as the Darcy flux or Darcy velocity, is not the velocity at which the fluid is travelling through the pores. The flow velocity (u) is related to the flux (q) by the porosity (φ) and takes the form

${\displaystyle u={\frac {q}{\varphi }}\,.}$

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

• 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; however, 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

${\displaystyle \mathrm {Re} ={\frac {ud}{\nu }}\,,}$

where ν is the kinematic viscosity of water, u is the specific discharge (not the pore velocity — with units of length per time), d₃₀ 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).

For stationary, creeping, incompressible flow, i.e. D(ρu_i)/Dt ≈ 0, the Navier–Stokes equation simplifies to the Stokes equation, which by neglecting the bulk term is:

${\displaystyle \mu \nabla ^{2}u_{i}-\partial _{i}p=0\,,}$

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

${\displaystyle -\left(k^{-1}\right)_{ij}\mu \varphi u_{j}-\partial _{i}p=0\,,}$

where φ is the porosity, and k_ij is the second order permeability tensor. This gives the velocity in the n direction,

${\displaystyle k_{ni}\left(k^{-1}\right)_{ij}u_{j}=\delta _{nj}u_{j}=u_{n}=-{\frac {k_{ni}}{\varphi \mu }}\partial _{i}p\,,}$

which gives Darcy's law for the volumetric flux density in the n direction,

${\displaystyle q_{n}=-{\frac {k_{ni}}{\mu }}\,\partial _{i}p\,.}$

In isotropic porous media the off-diagonal elements in the permeability tensor are zero, k_ij = 0 for i ≠ j and the diagonal elements are identical, k_ii = k, 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. [4]

${\displaystyle {\boldsymbol {q}}=-{\frac {k}{\mu }}\,{\boldsymbol {\nabla }}p\,.}$

The above equation is a governing equation for single-phase fluid flow in a porous medium.

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 et alios. Because Darcy's name is so widespread and strongly associated with flow in porous media, the multiphase equation is denoted Darcy's law for multiphase flow or generalized Darcy equation (or law) or simply Darcy's equation (or law) or simply flow equation if the context says that the text is discussing the multiphase equation of Muskat et alios. Multiphase flow in oil and gas reservoirs is a comprehensive topic, and one of many articles about this topic is Darcy's law for multiphase flow.

A number of papers have utilized Darcy's law to model the physics of brewing in a moka pot, specifically how the hot water percolates through the coffee grinds under pressure, starting with a 2001 paper by Varlamov and Balestrino,[5] and continuing with a 2007 paper by Gianino,[6] a 2008 paper by Navarini et al.[7], and a 2008 paper by W. King.[8] The papers will either take the coffee permeability to be constant as a simplification or will measure change through the brewing process.

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.[16] Hence Darcy's law is not always valid in such sediments. For flow through commercial circular pipes, the flow is laminar when Reynolds number is less than 2000 and turbulent when it is more than 4000, but in some sediments, it has been found that flow is laminar when the value of Reynolds number is less than 1.[17]

• The darcy, a unit of fluid permeability
• Hydrogeology
• Groundwater flow equation
• Mathematical model
• Black-oil equations