Darcy's law

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Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments[1] on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of earth sciences.

Background

Darcy's law was determined experimentally by Darcy. It has since been derived from the Navier–Stokes equations via homogenization.[2] 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.

Morris Muskat first[citation needed] refined Darcy's equation for single phase flow by including viscosity in the single (fluid) phase equation of Darcy, and this change made it suitable for the petroleum industry. Based on experimental results worked out by his colleagues Wyckoff and Botset, Muskat and Meres also generalized Darcy's law to cover 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.

Description

Diagram showing definitions and directions for Darcy's law.

Darcy's law, as refined by Morris Muskat, in the absence of gravitational forces, is a simple proportional relationship between the instantaneous flow rate through a porous medium of permeability ${\displaystyle k}$, the dynamic viscosity of the fluid and the pressure drop over a given distance in a homogeneously permeable medium.[3]

${\displaystyle Q=-{\frac {kA\left(p_{\mathrm {b} }-p_{\mathrm {a} }\right)}{\mu L}}\,.}$

This equation, for single phase (fluid) flow, is the defining equation [4] for absolute permeability (single phase permeability). 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, k (m2), the cross-sectional area to flow, A (units of area, e.g., m2), and the total pressure drop pbpa (pascals), all divided by the dynamic viscosity, μ (Pa·s) and the length over which the pressure drop is taking place L (m).

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. 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).

Dividing both sides of the equation by the area and using more general notation leads to

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

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 or Darcy velocity, 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.

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

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

${\displaystyle \mathrm {Re} ={\frac {\rho vd_{30}}{\mu }}\,,}$

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 dynamic viscosity of the fluid.

Derivation

For stationary, creeping, incompressible flow, i.e. D(ρui)/Dt ≈ 0, the Navier–Stokes equation simplifies to the Stokes equation:

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

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:

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

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

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

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

${\displaystyle q_{n}=-{\frac {k_{ni}}{\mu }}\left(\partial _{i}p-\rho g_{i}\right)\,.}$

In isotropic porous media the off-diagonal elements in the permeability tensor are zero, kij = 0 for ij and the diagonal elements are identical, kii = k, and the common form is obtained

${\displaystyle {\boldsymbol {q}}=-{\frac {k}{\mu }}\left({\boldsymbol {\nabla }}p-\rho {\boldsymbol {g}}\right)\,.}$

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

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 single fluid phase and constant fluid viscosity.

${\displaystyle Q={\frac {kA}{\mu }}\left({\frac {\partial p}{\partial x}}\right)\,,}$

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/x 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.

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.

Darcy–Forchheimer law

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.[5]

${\displaystyle {\frac {\partial p}{\partial x}}=-{\frac {\mu }{k}}q-{\frac {\rho }{k_{1}}}q^{2}\,,}$

where the additional term k1 is known as inertial permeability.

The flow in the middle of a sandstone reservoir is so slow that Forchheimer's equation is usually not needed, but the gas flow into a gas production well may be high enough to justify use of Forchheimer's equation. In this case the inflow performance calculations for the well, not the grid cell of the 3D model, is based on the Forchheimer equation. The effect of this is that an additional rate-dependent skin appears in the inflow performance formula.

Some carbonate reservoirs have lots of fractures, and Darcy's equation for multiphase flow is generalized in order to govern both flow in fractures and flow in the matrix (i.e. the traditional porous rock). The irregular surface of the fracture walls and high flow rate in the fractures, may justify use of Forchheimer's equation.

Darcy's law for gases in fine media (Knudsen diffusion or Klinkenberg effect)

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.[6][7] For a porous medium, the Knudsen equation can be given as [7]

${\displaystyle N=-\left({\frac {k}{\mu }}{\frac {p_{a}+p_{b}}{2}}+D_{\mathrm {K} }^{\mathrm {eff} }\right){\frac {1}{R_{\mathrm {g} }T}}{\frac {p_{\mathrm {b} }-p_{\mathrm {a} }}{L}}\,,}$

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).[8][9] The differential equation of transition flow in porous media based on BFM is given as[8]

${\displaystyle {\frac {\partial p}{\partial x}}=-R_{\mathrm {g} }T\left({\frac {kp}{\mu }}+D_{\mathrm {K} }\right)^{-1}N\,.}$

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

${\displaystyle {\frac {\partial p}{\partial x}}=-R_{\mathrm {g} }T\left({\frac {kp}{\mu }}+D_{\mathrm {K} }\right)^{-1}{\dfrac {p}{R_{\mathrm {g} }T}}q\,.}$

This equation can be rearranged into the following equation

${\displaystyle q=-{\frac {k}{\mu }}\left(1+{\frac {D_{\mathrm {K} }\mu }{k}}{\frac {1}{p}}\right){\frac {\partial p}{\partial x}}\,.}$

Comparing this equation with conventional Darcy's law, a new formulation can be given as

${\displaystyle q=-{\frac {k^{\mathrm {eff} }}{\mu }}{\frac {\partial p}{\partial x}}\,,}$

where

${\displaystyle k^{\mathrm {eff} }=k\left(1+{\frac {D_{\mathrm {K} }\mu }{k}}{\frac {1}{p}}\right)\,.}$

This is equivalent to the effective permeability formulation proposed by Klinkenberg:[10]

${\displaystyle k^{\mathrm {eff} }=k\left(1+{\frac {b}{p}}\right)\,.}$

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).

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),

${\displaystyle \tau {\frac {\partial q}{\partial t}}+q=-k\nabla h\,,}$

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[11]),

${\displaystyle -\beta \nabla ^{2}q+q=-{\frac {k}{\mu }}\nabla p\,,}$

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 example, if a porous extracellular matrix degrades to form large pores throughout the matrix, the viscous term applies in the large pores, while Darcy's law applies in the remaining intact region. This scenario was considered in a theoretical and modelling study.[12] In the proposed model, the Brinkman equation is connected to a set of reaction-diffusion-convection equations.

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.[13] 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.[14]

References

1. ^ Darcy, H. (1856). Les fontaines publiques de la ville de Dijon. Paris: Dalmont.
2. ^ Whitaker, S. (1986). "Flow in porous media I: A theoretical derivation of Darcy's law". Transport in Porous Media. 1: 3–25. doi:10.1007/BF01036523.
3. ^ Liu, Mingchao; Wu, Jian; Gan, Yixiang; Hanaor, Dorian A. H; Chen, C. Q (2016). "Evaporation limited radial capillary penetration in porous media" (PDF). Langmuir. 32 (38): 9899–9904. doi:10.1021/acs.langmuir.6b02404. PMID 27583455.
4. ^ Zarandi, M. Amin F.; Pillai, Krishna M.; Kimmel, Adam S. (2018). "Spontaneous imbibition of liquids in glass-fiber wicks. Part I: Usefulness of a sharp-front approach". AIChE Journal. 64: 294–305. doi:10.1002/aic.15965.
5. ^ Bejan, A. (1984). Convection Heat Transfer. John Wiley & Sons.
6. ^ Cunningham, R. E.; Williams, R. J. J. (1980). Diffusion in Gases and Porous Media. New York: Plenum Press.
7. ^ a b Carrigy, N.; Pant, L. M.; Mitra, S. K.; Secanell, M. (2013). "Knudsen diffusivity and permeability of pemfc microporous coated gas diffusion layers for different polytetrafluoroethylene loadings". Journal of the Electrochemical Society. 160 (2): F81–89. doi:10.1149/2.036302jes.
8. ^ a b Pant, L. M.; Mitra, S. K.; Secanell, M. (2012). "Absolute permeability and Knudsen diffusivity measurements in PEMFC gas diffusion layers and micro porous layers". Journal of Power Sources. 206: 153–160. doi:10.1016/j.jpowsour.2012.01.099.
9. ^ Kerkhof, P. (1996). "A modified Maxwell–Stefan model for transport through inert membranes: The binary friction model". Chemical Engineering Journal and the Biochemical Engineering Journal. 64 (3): 319–343. doi:10.1016/S0923-0467(96)03134-X.
10. ^ Klinkenberg, L. J. (1941). "The permeability of porous media to liquids and gases". Drilling and Production Practice. American Petroleum Institute. pp. 200–213.
11. ^ Brinkman, H. C. (1949). "A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles". Applied Scientific Research. 1: 27–34. CiteSeerX 10.1.1.454.3769. doi:10.1007/BF02120313.
12. ^ Wertheim, Kenneth Y.; Roose, Tiina (April 2017). "A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo". Bulletin of Mathematical Biology. 79 (4): 693–737. doi:10.1007/s11538-017-0248-7. ISSN 1522-9602. PMC 5501200. PMID 28233173.
13. ^ Jin, Y.; Uth, M.-F.; Kuznetsov, A. V.; Herwig, H. (2 February 2015). "Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study". Journal of Fluid Mechanics. 766: 76–103. Bibcode:2015JFM...766...76J. doi:10.1017/jfm.2015.9.
14. ^ Arora, K. R. (1989). Soil Mechanics and Foundation Engineering. Standard Publishers.