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Darcy's law is an equation that describes the flow of a
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
through a
porous Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
medium. The law was formulated by
Henry Darcy Henry Philibert Gaspard Darcy (, 10 June 1803 – 3 January 1858) was a French engineer who made several important contributions to hydraulics, including Darcy’s law for flow in porous media. Early life Darcy was born in Dijon, France, on J ...
based on results of experiments on the flow of
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
through beds of
sand Sand is a granular material composed of finely divided mineral particles. Sand has various compositions but is defined by its grain size. Sand grains are smaller than gravel and coarser than silt. Sand can also refer to a textural class of s ...
, forming the basis of
hydrogeology Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aquif ...
, a branch of
earth science Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres ...
s. It is analogous to
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
in electrostatics, linearly relating the volume flow rate of the fluid to the
hydraulic head Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22. It is usually measured as a liquid surface elevation, expressed in units of length, ...
difference (which is often just proportional to the pressure difference) via the
hydraulic conductivity Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...
.


Background

Darcy's law was first determined experimentally by Darcy, but has since been derived from the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
via
homogenization Homogeneity is a sameness of constituent structure. Homogeneity, homogeneous, or homogenization may also refer to: In mathematics *Transcendental law of homogeneity of Leibniz * Homogeneous space for a Lie group G, or more general transformati ...
methods. It is analogous to
Fourier's law Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted . Heat spontaneously flows along a te ...
in the field of
heat conduction Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted . Heat spontaneously flows along a te ...
,
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
in the field of
electrical networks An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, ...
, and
Fick's law Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion eq ...
in
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
theory. One application of Darcy's law is in the analysis of water flow through an
aquifer An aquifer is an underground layer of water-bearing, permeable rock, rock fractures, or unconsolidated materials (gravel, sand, or silt). Groundwater from aquifers can be extracted using a water well. Aquifers vary greatly in their characterist ...
; Darcy's law along with the equation of
conservation of mass In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
simplifies to the
groundwater flow equation Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar ...
, one of the basic relationships of
hydrogeology Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aquif ...
.
Morris Muskat Morris Muskat (21 April 1906 – 20 June 1998) was an American petroleum engineer. Muskat refined Darcy's equation for single phase flow, and this change made it suitable for the petroleum industry. Based on experimental results worked out by ...
first 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 A petroleum reservoir or oil and gas reservoir is a subsurface accumulation of hydrocarbons contained in porous or fractured rock formations. Such reservoirs form when kerogen (ancient plant matter) is created in surrounding rock by the presence ...
. The generalized multiphase flow equations by Muskat and others provide the analytical foundation for
reservoir engineering Reservoir engineering is a branch of petroleum engineering that applies scientific principles to the fluid flow through porous medium during the development and production of oil and gas reservoirs so as to obtain a high economic recovery. The wo ...
that exists to this day.


Description

Darcy's law, as refined by
Morris Muskat Morris Muskat (21 April 1906 – 20 June 1998) was an American petroleum engineer. Muskat refined Darcy's equation for single phase flow, and this change made it suitable for the petroleum industry. Based on experimental results worked out by ...
, in the absence of gravitational forces and in a homogeneously permeable medium, is given by a simple proportionality relationship between the instantaneous
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
q = Q/A (units of Q: m3/s, units of A: m2, units of q: m/s) through a
porous medium A porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid ( liquid or gas). The skeletal material is us ...
, the permeability k of the medium, the dynamic
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
of the fluid \mu, and the pressure drop \Delta p over a given distance L, in the form q =- \frac \Delta p This equation, for single phase (fluid) flow, is the
defining equation Defining equation may refer to: *Defining equation (physics) *Defining equation (physical chemistry) In physical chemistry, there are numerous quantities associated with chemical compounds and reactions; notably in terms of ''amounts'' of subst ...
for absolute permeability (single phase permeability). With reference to the diagram to the right, the flux q, or discharge per unit area, is defined in units \mathrm, the permeability k in units \mathrm, the cross-sectional area A in units \mathrm, the total pressure drop \Delta p = p_b - p_a in units \mathrm, the
dynamic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
\mu in units \mathrm, and L is the length of the sample in units \mathrm. 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 direction. There have been several proposals for a
constitutive equation In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
for absolute permeability, and the most famous one is probably the Kozeny equation (also called
Kozeny–Carman equation The Kozeny–Carman equation (or Carman–Kozeny equation or Kozeny equation) is a relation used in the field of fluid dynamics to calculate the pressure drop of a fluid flowing through a packed bed of solids. It is named after Josef Kozeny and ...
). The integral form of the Darcy law is given by: Q = \frac \, where (units of volume per time, e.g., m3/s) is the total discharge. By considering the relation for static fluid pressure ( Stevin's law): p = \rho g h one can deduce the representation Q = \frac \, where ν is the
kinematic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
. The corresponding
hydraulic conductivity Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...
is therefore: : K = \frac=\frac. Notice that the quantity q or 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 In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
() is related to the flux () by the
porosity Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
() and takes the form : u=\frac q \varphi \,. Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that
groundwater Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidate ...
flowing in
aquifer An aquifer is an underground layer of water-bearing, permeable rock, rock fractures, or unconsolidated materials (gravel, sand, or silt). Groundwater from aquifers can be extracted using a water well. Aquifers vary greatly in their characterist ...
s exhibits, including: * if there is no pressure gradient over a distance, no flow occurs (these are
hydrostatic Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
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 Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar ...
(based on Darcy's law and the conservation of mass) is in the construction of
flownet A flow net is a graphical representation of two-dimensional steady-state groundwater flow through aquifers. Construction of a flow net is often used for solving groundwater flow problems where the geometry makes analytical solutions impractical. T ...
s, to quantify the amount of
groundwater Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidate ...
flowing under a
dam A dam is a barrier that stops or restricts the flow of surface water or underground streams. Reservoirs created by dams not only suppress floods but also provide water for activities such as irrigation, human consumption, industrial use ...
. Darcy's law is only valid for slow,
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
flow; however, most groundwater flow cases fall in this category. Typically any flow with a
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
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 : \mathrm = \frac\,, where is the
kinematic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
of
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
, is the specific discharge (not the pore velocity — with units of length per time), 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).


Derivation

For stationary, creeping, incompressible flow, i.e. , the Navier–Stokes equation simplifies to the Stokes equation, which by neglecting the bulk term is: : \mu\nabla^2 u_i -\partial_i p=0\,, where is the viscosity, is the velocity in the direction, is the gravity component in the direction and is the pressure. Assuming the viscous resisting force is linear with the velocity we may write: : -\left(k^\right)_ \mu\varphi u_j-\partial_i p=0\,, where is the
porosity Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
, and is the second order permeability tensor. This gives the velocity in the direction, : k_\left(k^\right)_ u_j= \delta_ u_j = u_n = -\frac \partial_i p\,, which gives Darcy's law for the volumetric flux density in the direction, : q_n=-\frac \, \partial_i p\,. In
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
porous media A porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid (liquid or gas). The skeletal material is usu ...
the off-diagonal elements in the permeability tensor are zero, for and the diagonal elements are identical, , 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. : \boldsymbol=-\frac \, \boldsymbol p \,. The above equation is a
governing equation The governing equations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables) change when one or more of the known (i.e. independent) variables change. Mass balance A mass balance, also called a ...
for single-phase fluid flow in a porous medium.


Use 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 The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
. 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 Muskat (Muskat the spice merchant, compare Kanehl, Zimt, Safran and so on) is a surname. Notable people with the surname include: * Joyce Muskat, American television writer * Lisa Muskat, American film producer *Morris Muskat (1906–1998), Amer ...
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 Morris Muskat et al.Muskat M. and Meres M.W. 1936. The Flow of Heterogeneous Fluids Through Porous Media. Paper published in J. Appl. Phys. 1936, 7, pp 346–363. https://dx.doi.org/10.1063/1.1745403Muskat M. and Wyckoff R.D. and Botset H.G. and M ...
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 Muskat (Muskat the spice merchant, compare Kanehl, Zimt, Safran and so on) is a surname. Notable people with the surname include: * Joyce Muskat, American television writer * Lisa Muskat, American film producer *Morris Muskat (1906–1998), Amer ...
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 Morris Muskat et al.Muskat M. and Meres M.W. 1936. The Flow of Heterogeneous Fluids Through Porous Media. Paper published in J. Appl. Phys. 1936, 7, pp 346–363. https://dx.doi.org/10.1063/1.1745403Muskat M. and Wyckoff R.D. and Botset H.G. and M ...
.


Use in coffee brewing

A number of papers have utilized Darcy's law to model the physics of brewing in a
moka pot The moka pot is a stove-top or electric coffee maker that brews coffee by passing boiling water pressurized by steam through ground coffee. Named after the Yemeni city of Mocha, it was invented by Italian engineer Alfonso Bialetti in 1933 and ...
, specifically how the hot water percolates through the coffee grinds under pressure, starting with a 2001 paper by Varlamov and Balestrino, and continuing with a 2007 paper by Gianino, a 2008 paper by Navarini et al., and a 2008 paper by W. King. The papers will either take the coffee permeability to be constant as a simplification or will measure change through the brewing process.


Additional forms


Differential expression

Darcy's law can be expressed very generally as: :\mathbf=-K\nabla h where q is the volume flux vector of the fluid at a particular point in the medium, ''h'' is the total
hydraulic head Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22. It is usually measured as a liquid surface elevation, expressed in units of length, ...
, and ''K'' is the
hydraulic conductivity Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, at that point. The hydraulic conductivity can often be approximated as a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
. (Note the analogy to
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
in electrostatics. The flux vector is analogous to the current density, head is analogous to voltage, and hydraulic conductivity is analogous to electrical conductivity.)


Quadratic law

For flows in porous media with
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
s greater than about 1 to 10,
inertial In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
effects can also become significant. Sometimes an
inertial In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
term is added to the Darcy's equation, known as Forchheimer term. This term is able to account for the
non-linear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
behavior of the pressure difference vs flow data. : \frac=-\fracq-\fracq^2\,, where the additional term 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 many 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 the use of Forchheimer's equation.


Correction 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 The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
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 :N=-\left(\frac\frac+D_\mathrm^\mathrm\right)\frac\frac\,, where is the molar flux, is the gas constant, is the temperature, 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 : \frac=-R_\mathrmT\left(\frac+D_\mathrm\right)^N\,. This equation is valid for
capillaries A capillary is a small blood vessel from 5 to 10 micrometres (μm) in diameter. Capillaries are composed of only the tunica intima, consisting of a thin wall of simple squamous endothelial cells. They are the smallest blood vessels in the body: ...
as well as porous media. The terminology of the Knudsen effect and Knudsen diffusivity is more common in
mechanical Mechanical may refer to: Machine * Machine (mechanical), a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement * Mechanical calculator, a device used to perform the basic operations of ...
and
chemical engineering Chemical engineering is an engineering field which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials int ...
. 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 : \frac=-R_\mathrmT\left(\frac+D_\mathrm\right)^\dfracq\,. This equation can be rearranged into the following equation : q=-\frac\left(1+\frac\frac\right)\frac\,. Comparing this equation with conventional Darcy's law, a new formulation can be given as : q=-\frac\frac\,, where :k^\mathrm=k\left(1+\frac\frac\right)\,. This is equivalent to the effective permeability formulation proposed by Klinkenberg: : k^\mathrm=k\left(1+\frac\right)\,. where 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 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 Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted . Heat spontaneously flows along a te ...
), : \tau \frac+q=-k \nabla h\,, where is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at "normal" times (>
nanosecond A nanosecond (ns) is a unit of time in the International System of Units (SI) equal to one billionth of a second, that is, of a second, or 10 seconds. The term combines the SI prefix ''nano-'' indicating a 1 billionth submultiple of an SI unit ( ...
s). The main reason for doing this is that the regular
groundwater flow equation Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar ...
( diffusion equation) leads to singularities at constant head boundaries at very small times. This form is more mathematically rigorous but leads to a
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
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), : -\beta \nabla^2 q +q =-\frac \nabla p\,, where is an effective
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
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.


Validity of Darcy's law

Darcy's law is valid for laminar flow through
sediment Sediment is a naturally occurring material that is broken down by processes of weathering and erosion, and is subsequently transported by the action of wind, water, or ice or by the force of gravity acting on the particles. For example, sand an ...
s. In fine-grained sediments, the dimensions of
interstices An interstitial space or interstice is a space between structures or objects. In particular, interstitial may refer to: Biology * Interstitial cell tumor * Interstitial cell, any cell that lies between other cells * Interstitial collagenase, ...
are small and thus flow is laminar. Coarse-grained sediments also behave similarly but in very coarse-grained sediments the flow may be
turbulent In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
. 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.


See also

* The darcy, a unit of fluid permeability *
Hydrogeology Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aquif ...
*
Groundwater flow equation Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar ...
*
Mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
* Black-oil equations


References

{{Hydrogeology Water Civil engineering Soil mechanics Soil physics Hydrology Transport phenomena