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Encyclopedia > Darcy's law

In fluid dynamics, Darcy's law is a phenomologically derived 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[1] on the flow of water through beds of sand. It also forms the scientific basis of fluid permeability used in the earth sciences. Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... The term phenomenology in modern science, especially in physics, is used to describe a body of knowledge which relates several different empirical observations of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. ... In structural analysis, constitutive relations connect applied stresses or forces to strains or deformations. ... This box:      A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of how small the applied stress. ... A porous medium or a porous material is a solid (often called frame or matrix) permeated by an interconnected network of pores (voids) filled with a fluid (liquid or gas). ... Henry Darcy. ... Impact from a water drop causes an upward rebound jet surrounded by circular capillary waves. ... For other uses, see Sand (disambiguation). ... In the earth sciences, permeability (commonly symbolized as κ, or k) is a measure of the ability of a material (typically, a rock or unconsolidated material) to transmit fluids. ... Earth science (also known as geoscience, the geosciences or the Earth Sciences), is an all-embracing term for the sciences related to the planet Earth. ...

Contents

Background

Although Darcy's law (an expression of conservation of momentum) 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. In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ... Heat flow along perfectly insulated wire Heat conduction is the transmission of heat across matter. ... Heat conduction or thermal conduction is the spontaneous transfer of thermal energy through matter, from a region of higher temperature to a region of lower temperature, and acts to equalize temperature differences. ... This article is about the law related to electricity. ... An electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... Ficks laws of diffusion describe diffusion. ... diffusion (disambiguation). ...


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. An aquifer is an underground layer of water-bearing permeable rock or unconsolidated materials (gravel, sand, silt, or clay) from which groundwater can be usefully extracted using a water well. ... The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov-Lavoisier law), states that the mass of a closed system of substances will remain constant, regardless of the processes acting inside the system. ... The groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through a porous medium (e. ... Hydrogeology (hydro- meaning water, and -geology meaning the study of the Earth) is the part of hydrology that deals with the distribution and movement of groundwater in the soil and rocks of the Earths crust, (commonly in aquifers). ...


Description

Diagram showing definitions and directions for Darcy's law.
Diagram showing definitions and directions for Darcy's law.

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

Q=frac{-kappa A}{mu}frac{(P_b-P_a)}{L}

The total discharge, Q (units of volume per time, e.g., 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 (PbPa), 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 In the earth sciences, permeability (commonly symbolized as κ, or k) is a measure of the ability of a material (typically, a rock or unconsolidated material) to transmit fluids. ... Viscosity is a measure of the resistance of a fluid to deformation under shear stress. ...

q=frac{-kappa}{mu}nabla P

where q is the flux (discharge per unit area, with units of length per time, m/s) and nabla P is the pressure gradient vector. This value of flux, often referred to as the Darcy flux, is not the velocity which the water traveling through the pores is experiencing[2]. flux in science and mathematics. ... Pressure Gradient is the change in pressure over a distance. ...


The pore 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. Porosity is a measure of the void spaces in a material, and is measured as a fraction, between 0–1, or as a percentage between 0–100%. The term porosity is used in multiple fields including manufacturing, earth sciences and construction. ...

v=frac{q}{phi}

In 3D

In three dimensions, gravity must be accounted for, as the flow is not 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, The space we live in is three-dimensional space. ... Gravity is a force of attraction that acts between bodies that have mass. ... Fluid pressure is the pressure on an object submerged in a fluid, such as water. ...

boldsymbol{q} = frac{-boldsymbol{kappa}}{mu}(nabla P - rho g hat{boldsymbol{e}}_z)

where the flux boldsymbol{q} is now a vector quantity, boldsymbol{kappa} is a tensor of permeability, nabla is the gradient operator in 3D, g is the acceleration due to gravity, hat{boldsymbol{e}}_z is the unit vector in the vertical direction, pointing downwards and ρ is the density. For other uses, see Gradient (disambiguation). ... In physics, gravitational acceleration is the acceleration of an object caused by the force of gravity from another object. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... For other uses, see Density (disambiguation). ...


Effects of anisotropy in three dimensions are addressed using a symmetric second-order tensor of permeability: Look up anisotropy in Wiktionary, the free dictionary. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In the earth sciences, permeability (commonly symbolized as κ, or k) is a measure of the ability of a material (typically, a rock or unconsolidated material) to transmit fluids. ...

boldsymbol{kappa} = begin{bmatrix} kappa_{xx} & kappa_{xy} & kappa_{xz}  kappa_{yx} & kappa_{yy} & kappa_{yz}  kappa_{zx} & kappa_{zy} & kappa_{zz} end{bmatrix}

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, kappa_{xx} = kappa_{yy} = kappa_{zz} > 0,, while all other components are 0. The permeability tensor can be interpreted through an evaluation the relative magnitudes in each component. For example, rock with highly permeable vertical fractures aligned in the x-direction will have relatively higher values for kappa_{xx}, than other component values. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... Cracks in rock resulting from stress A fracture is any local separation or discontinuity plane in a geologic formation, such as a joint or a fault that divides the rock into two or more pieces. ...


Assumptions

Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that groundwater flowing in aquifers exhibits, including: Groundwater is water located beneath the ground surface in soil pore spaces and in the fractures of lithologic formations. ... An aquifer is an underground layer of water-bearing permeable rock or unconsolidated materials (gravel, sand, silt, or clay) from which groundwater can be usefully extracted using a water well. ...

  • if there is no pressure gradient over a distance, no flow occurs (this is 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. Hydrostatics, also known as fluid statics, is the study of fluids at rest. ... The groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through a porous medium (e. ... The construction of a Flownet is a graphical method used to solve two-dimensional steady-state groundwater flow problems through aquifers. ... Groundwater is water located beneath the ground surface in soil pore spaces and in the fractures of lithologic formations. ... This article is about structures for water impoundment. ...


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 for 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 Viscosity is a measure of the resistance of a fluid to deformation under shear stress. ... In fluid mechanics, the Reynolds number may be described as the ratio of inertial forces (vsρ) to viscous forces (μ/L) and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions. ...

Re = frac{rho v d_{30}}{mu}

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. For other uses, see Density (disambiguation). ... This box:      A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of how small the applied stress. ... Particle size, also called grain size, refers to the diameter of individual grains of sediment, or the lithified particles in clastic rocks. ...


Additional forms of Darcy's law

Time derivative of flux

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), Heat flow along perfectly insulated wire Heat conduction is the transmission of heat across matter. ...

taufrac{partial q}{partial t}+q=-Knabla 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. To help compare orders of magnitude of different times this page lists times between 10−9 seconds and 10−8 seconds (1 nanosecond and 10 nanoseconds) See also times of other orders of magnitude. ... The groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through a porous medium (e. ... The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... For hyperbole, the figure of speech, see hyperbole. ...


Brinkman term

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),

betanabla^{2}q+q=-Knabla h,

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.


Multiphase flow

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. In fluid mechanics, multiphase flow is a generalisation of the modelling used in two-phase flow to cases where the two phases are not chemically related (e. ... In multiphase flow in porous media, relative permeability is a dimensionless measure of the effective permeability of each phase. ...


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

q_i = -frac{kappa_i}{mu_i} nabla P_i

where κi is the phase permeability for phase i. From this we also define relative permeability κri for phase i as In multiphase flow in porous media, relative permeability is a dimensionless measure of the effective permeability of each phase. ...

κri = κi / κ

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


Forchheimer equation for non-Darcy flow

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

-nabla P = frac{mu}{kappa} V + beta rho V^2

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


In membrane operations

In pressure-driven membrane operations, Darcy's law is often used in the form, Membrane operation or membrane process is considered like an unit operation in chemical engineering. ...

J = frac{Delta P - Delta Pi} {mu (R_f + R_m)}

where,

  • J is the volumetric flux (m.s − 1),
  • ΔP is the hydraulic pressure difference between the feed and permeate sides of the membrane (Pa),
  • ΔΠ is the osmotic pressure difference between the feed and permeate sides of the membrane (Pa),
  • μ is the dynamic viscosity (Pa.s),
  • Rf is the fouling resistance (m − 1), and
  • Rm is the membrane resistance (m − 1)

References

  1. ^ H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris (1856).
  2. ^ See Stauffer, Philip H. (2006). "Flux Flummoxed: A Proposal for Consistent Usage". Ground Water 44 (2): 125–128. doi:10.1111/j.1745-6584.2006.00197.x.  for a discussion of the many, sometimes confusing names given to (q) in the ground water literature.

A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...

See also

For the 19th-century French scientist, see Henry Darcy. ... In the earth sciences, permeability (commonly symbolized as κ, or k) is a measure of the ability of a material (typically, a rock or unconsolidated material) to transmit fluids. ... Hydrogeology (hydro- meaning water, and -geology meaning the study of the Earth) is the part of hydrology that deals with the distribution and movement of groundwater in the soil and rocks of the Earths crust, (commonly in aquifers). ... Groundwater discharge is the volumetric flow rate of groundwater through an aquifer. ... The groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through a porous medium (e. ... The Richards equation represents the movement of water in unsaturated soils, and was formulated by Lorenzo A. Richards in 1931. ... The Darcy friction factor is a dimensionless number used in internal flow calculations. ...

External links

Browser-based numerical calculator of permeability using Darcy's law.


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