Darcy's law
Encyclopedia
Darcy's law is a phenomenologically
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 on the flow of water
through beds of sand
. It also forms the scientific basis of fluid permeability
used in the earth science
s, particularly in hydrogeology
.
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 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 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 permeability
of the medium, k (m2), the cross-sectional area to flow, A (units of area, e.g., m2), and the pressure drop (Pa), all divided by the viscosity
, μ (Pa.s) and the length the pressure drop is taking place over. The negative sign is needed because fluids flows from high pressure to low pressure. So 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 to
where q is the flux (discharge per unit area, with units of length per time, m/s) and
is the pressure gradient
vector (Pa/m). This value of flux, often referred to as the Darcy flux, is not the velocity which the water traveling through the pores is experiencing. The pore velocity (v) is related to the Darcy flux (q) by the porosity
(n). 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.
Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that groundwater
flowing in aquifer
s exhibits, including:
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 flownet
s, 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.
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 (> nanosecond
s). 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.
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),
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.
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 relative permeability of the formation (typically in millidarcies), A is the cross-sectional area
of the formation, μ is the viscosity
of the fluid (typically in units of centipoise, and L is the length
of the porous media the fluid will flow through.
represents the pressure change per unit length of the formation. This equation can also be solved for permeability, allowing for relative permeability
to be calculated by 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.
Phenomenology (science)
The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
derived constitutive equation
Constitutive equation
In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...
that describes the flow of a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
through a porous medium. The law was formulated by Henry Darcy
Henry Darcy
Henry Philibert Gaspard Darcy was a French engineer who made several important contributions to hydraulics.-Biography:...
based on the results of experiments on the flow of water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
through beds of sand
Sand
Sand is a naturally occurring granular material composed of finely divided rock and mineral particles.The composition of sand is highly variable, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal...
. It also forms the scientific basis of fluid permeability
Permeability (fluid)
Permeability in fluid mechanics and the earth sciences is a measure of the ability of a porous material to allow fluids to pass through it.- Units :...
used in the earth science
Earth science
Earth science is an all-embracing term for the sciences related to the planet Earth. It is arguably a special case in planetary science, the Earth being the only known life-bearing planet. There are both reductionist and holistic approaches to Earth sciences...
s, particularly in hydrogeology
Hydrogeology
Hydrogeology is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust, . The term geohydrology is often used interchangeably...
.
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 equationsNavier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...
via homogenization. It is analogous to Fourier's law in the field of heat conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....
, Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
in the field of electrical networks, or Fick's law in diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
theory.
One application of Darcy's law is to water flow through an aquifer
Aquifer
An aquifer is a wet underground layer of water-bearing permeable rock or unconsolidated materials from which groundwater can be usefully extracted using a water well. The study of water flow in aquifers and the characterization of aquifers is called hydrogeology...
; Darcy's law along with the equation of conservation of mass
Conservation of mass
The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...
are equivalent to the groundwater flow equation
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 to that used in heat transfer to describe the flow...
, one of the basic relationships of hydrogeology
Hydrogeology
Hydrogeology is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust, . The term geohydrology is often used interchangeably...
. Darcy's law is also used to describe oil, water, and gas flows through petroleum reservoirs.
Description
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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 permeability
Permeability (fluid)
Permeability in fluid mechanics and the earth sciences is a measure of the ability of a porous material to allow fluids to pass through it.- Units :...
of the medium, k (m2), the cross-sectional area to flow, A (units of area, e.g., m2), and the pressure drop (Pa), all divided by the viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
, μ (Pa.s) and the length the pressure drop is taking place over. The negative sign is needed because fluids flows from high pressure to low pressure. So 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 to
where q is the flux (discharge per unit area, with units of length per time, m/s) and
is the pressure gradientPressure gradient
In atmospheric sciences , the pressure gradient is a physical quantity that describes in which direction and at what rate the pressure changes the most rapidly around a particular location. The pressure gradient is a dimensional quantity expressed in units of pressure per unit length...
vector (Pa/m). This value of flux, often referred to as the Darcy flux, is not the velocity which the water traveling through the pores is experiencing. The pore velocity (v) is related to the Darcy flux (q) by the porosity
Porosity
Porosity or void fraction is a measure of the void spaces in a material, and is a fraction of the volume of voids over the total volume, between 0–1, or as a percentage between 0–100%...
(n). 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.
Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that groundwater
Groundwater
Groundwater is water located beneath the ground surface in soil pore spaces and in the fractures of rock formations. A unit of rock or an unconsolidated deposit is called an aquifer when it can yield a usable quantity of water. The depth at which soil pore spaces or fractures and voids in rock...
flowing in aquifer
Aquifer
An aquifer is a wet underground layer of water-bearing permeable rock or unconsolidated materials from which groundwater can be usefully extracted using a water well. The study of water flow in aquifers and the characterization of aquifers is called hydrogeology...
s 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
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 to that used in heat transfer to describe the flow...
(based on Darcy's law and the conservation of mass) is in the construction of flownet
Flownet
A flownet is a graphical representation of two-dimensional steady-state groundwater flow through aquifers. Construction of a flownet is often used for solving groundwater flow problems where the geometry makes analytical solutions impractical...
s, to quantify the amount of groundwater
Groundwater
Groundwater is water located beneath the ground surface in soil pore spaces and in the fractures of rock formations. A unit of rock or an unconsolidated deposit is called an aquifer when it can yield a usable quantity of water. The depth at which soil pore spaces or fractures and voids in rock...
flowing under a dam
Dam
A dam is a barrier that impounds water or underground streams. Dams generally serve the primary purpose of retaining water, while other structures such as floodgates or levees are used to manage or prevent water flow into specific land regions. Hydropower and pumped-storage hydroelectricity are...
.
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
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
(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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
of the fluid.
Additional forms of Darcy's law
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 (> nanosecond
Nanosecond
A nanosecond is one billionth of a second . One nanosecond is to one second as one second is to 31.7 years.The word nanosecond is formed by the prefix nano and the unit second. Its symbol is ns....
s). The main reason for doing this is that the regular groundwater flow equation
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 to that used in heat transfer to describe the flow...
(diffusion equation) leads to singularities
Mathematical singularity
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...
at constant head boundaries at very small times. This form is more mathematically rigorous, but leads to a hyperbolic
Hyperbolic
Hyperbolic refers to something related to or in shape of hyperbola , or to something employing the literary device of hyperbole .The following topics are based on the hyperbola etymology:...
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.
Another extension to the traditional form of Darcy's law is the Brinkman
Brinkman
Brinkman may refer to:* Hero Brinkman , Dutch politician* Jacques Brinkman , Dutch retired field hockey player* Bert Brinkman , Dutch retired water polo player* Charles Brinkman , American retired figure skater...
term, which is used to account for transitional flow between boundaries (introduced by Brinkman in 1947),
-
,
where β is an effective viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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.
Another derivation of Darcy's law is used extensively in petroleum engineering
Petroleum engineering
Petroleum engineering is an engineering discipline concerned with the activities related to the production of hydrocarbons, which can be either crude oil or natural gas. Subsurface activities are deemed to fall within the upstream sector of the oil and gas industry, which are the activities of...
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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
.
-
,
where Q is the flowrate of the formation (in units of volume per unit time), k is the relative permeability of the formation (typically in millidarcies), A is the cross-sectional area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
of the formation, μ is the viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
of the fluid (typically in units of centipoise, and L is the length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
of the porous media the fluid will flow through.
represents the pressure change per unit length of the formation. This equation can also be solved for permeability, allowing for relative permeabilityRelative permeability
In multiphase flow in porous media, the relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability...
to be calculated by 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.
See also
- The darcyDarcyA darcy and millidarcy are units of permeability, named after Henry Darcy. They are not SI units, but they are widely used in petroleum engineering and geology. Like other measures of permeability, a darcy has the same units as area.-Definition:Permeability measures the ability of fluids to...
unit of fluid permeability - HydrogeologyHydrogeologyHydrogeology is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust, . The term geohydrology is often used interchangeably...
- Groundwater flow equationGroundwater flow equationUsed 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 to that used in heat transfer to describe the flow...