Permeability (fluid)
Encyclopedia
Permeability in fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

and 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 (commonly symbolized as κ, or k) is a measure of the ability of a porous material (often, a rock
Rock (geology)
In geology, rock or stone is a naturally occurring solid aggregate of minerals and/or mineraloids.The Earth's outer solid layer, the lithosphere, is made of rock. In general rocks are of three types, namely, igneous, sedimentary, and metamorphic...

or unconsolidated material) to allow fluids to pass through it.

## Units

The SI
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

unit for permeability is m2. A traditional unit for permeability is the darcy
Darcy
A 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...

(D), or more commonly the millidarcy (mD) (1 darcy 10−12m2). The unit of cm2 is also sometimes used (1 m2 = 104 cm2).

## Applications

The concept of permeability is of importance in determining the flow characteristics of hydrocarbons in oil
Petroleum
Petroleum or crude oil is a naturally occurring, flammable liquid consisting of a complex mixture of hydrocarbons of various molecular weights and other liquid organic compounds, that are found in geologic formations beneath the Earth's surface. Petroleum is recovered mostly through oil drilling...

and gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

reservoirs, and 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...

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.

For a rock to be considered as an exploitable hydrocarbon reservoir without stimulation, its permeability must be greater than approximately 100 mD (depending on the nature of the hydrocarbon - gas reservoirs with lower permeabilities are still exploitable because of the lower 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 gas with respect to oil). Rocks with permeabilities significantly lower than 100 mD can form efficient seals (see petroleum geology
Petroleum geology
Petroleum geology refers to the specific set of geological disciplines that are applied to the search for hydrocarbons .-Sedimentary basin analysis:...

). Unconsolidated sands may have permeabilities of over 5000 mD.

The concept has also many practical applications outside of geology, for example in chemical engineering
Chemical engineering
Chemical engineering is the branch of engineering that deals with physical science , and life sciences with mathematics and economics, to the process of converting raw materials or chemicals into more useful or valuable forms...

(e.g., filtration
Filtration
Filtration is commonly the mechanical or physical operation which is used for the separation of solids from fluids by interposing a medium through which only the fluid can pass...

).

## Description

Permeability is part of the proportionality constant in Darcy's law
Darcy's law
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...

which relates discharge (flow rate) and fluid physical properties (e.g. 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...

), to a pressure gradient applied to the porous media:

Therefore:

where: is the superficial fluid flow velocity
Superficial velocity
Superficial velocity , in engineering of multiphase flows and flows in porous media, is an hypothetical fluid velocity calculated as if the given phase or fluid were the only one flowing or present in a given cross sectional area. Other phases, particles, the skeleton of the porous medium, etc...

through the medium (i.e., the average velocity calculated as if the fluid were the only phase
Phase (matter)
In the physical sciences, a phase is a region of space , throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, and chemical composition...

present in the porous medium) (m/s) is the permeability of a medium (m2) is the dynamic 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 (Pa·s) is the applied pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

difference (Pa) is the thickness of the bed of the porous medium (m)

In naturally occurring materials, permeability values range over many orders of magnitude (see table below for an example of this range).

### Relation to hydraulic conductivity

The proportionality constant specifically for the flow of water through a porous media is called the hydraulic conductivity
Hydraulic conductivity
Hydraulic conductivity, symbolically represented as K, is a property of vascular plants, soil or rock, that describes the ease with which water can move through pore spaces or fractures. It depends on the intrinsic permeability of the material and on the degree of saturation...

; permeability is a portion of this, and is a property of the porous media only, not the fluid. Given the value of hydraulic conductivity for a subsurface system, k, the permeability can be calculated as:

where
• κ is the permeability, m2
• k is the hydraulic conductivity, m/s
• μ is the dynamic viscosity, kg/(m·s)
• ρ is the density of the fluid, kg/m3
• g is the acceleration due to gravity, m/s2.

## Determination

Permeability is typically determined in the lab by application of Darcy's law
Darcy's law
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...

under steady state conditions or, more generally, by application of various solutions to the diffusion equation for unsteady flow conditions.

Permeability needs to be measured, either directly (using Darcy's law
Darcy's law
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...

) or through estimation
Estimation
Estimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics,*estimation theory and estimator, for topics involving inferences about probability distributions...

using empirically
Empirical method
The empirical method is generally taken to mean the approach of using a collection of data to base a theory or derive a conclusion in science...

derived formulas. However, for some simple models of porous media, permeability can be calculated (e.g., random close packing of identical spheres
Random close pack
Random close packing is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing...

).

### Permeability model based on conduit flow

Based on Hagen–Poiseuille equation for viscous flow in a pipe, permeability can be expressed as:
where: is the intrinsic permeability [length2] is a dimensionless constant that is related to the configuration of the flow-paths is the average, or effective pore diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

[length].

## Intrinsic and absolute permeability

The terms intrinsic permeability and absolute permeability states that the permeability value in question is an intensive property
Intensive and extensive properties
In the physical sciences, an intensive property , is a physical property of a system that does not depend on the system size or the amount of material in the system: it is scale invariant.By contrast, an extensive property In the physical sciences, an intensive property (also called a bulk...

(not a spatial average of a heterogeneous block of material), that it is a function of the material structure only (and not of the fluid), and explicitly distinguishes the value from that of relative permeability
Relative 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...

.

## Permeability to gases

Sometimes permeability to gases can be somewhat different that those for liquids in the same media. One difference is attributable to "slippage" of gas at the interface with the solid when the gas mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

is comparable to the pore size (about 0.01 to 0.1 μm at standard temperature and pressure). See also Knudsen diffusion
Knudsen diffusion
Knudsen diffusion is a means of diffusion that occurs in a long pore with a narrow diameter because molecules frequently collide with the pore wall.Consider the diffusion of gas molecules through very small capillary pores...

and constrictivity
Constrictivity
Constrictivity is a dimensionless parameter used to describe transport processes in porous media.Constrictivity is viewed to depend on the ratio of the diameter of the diffusing particle to the pore diameter. The value of constrictivity is always less than 1...

. For example, measurement of permeability through sandstones and shales yielded values from 9.0x10−19 m2 to 2.4x10−12 m2 for water and between 1.7x10−17 m2 to 2.6x10−12 m2 for nitrogen gas.

## Tensor permeability

To model permeability in anisotropic media, a permeability tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

is needed. Pressure can be applied in three directions, and for each direction, permeability can be measured (via Darcy's law in 3D) in three directions, thus leading to a 3 by 3 tensor. The tensor is realized using a 3 by 3 matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

being both symmetric and positive definite
Positive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....

(SPD matrix):
• The tensor is symmetric by the Onsager reciprocal relations
Onsager reciprocal relations
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists....

.
• The tensor is positive definite as the component of the flow parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...

to the pressure drop is always in the same direction as the pressure drop.

The permeability tensor is always diagonalizable (being both symmetric and positive definite). The eigenvectors will yield the principal directions of flow, meaning the directions where flow is parallel to the pressure drop, and the eigenvalues representing the principal permeabilities.

## Ranges of common intrinsic permeabilities

These values do not depend on the fluid properties; see the table derived from the same source for values of hydraulic conductivity
Hydraulic conductivity
Hydraulic conductivity, symbolically represented as K, is a property of vascular plants, soil or rock, that describes the ease with which water can move through pore spaces or fractures. It depends on the intrinsic permeability of the material and on the degree of saturation...

, which are specific to the material through which the fluid is flowing.
 Permeability Pervious Semi-Pervious Impervious Unconsolidated SandSandSand 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... & GravelGravelGravel is composed of unconsolidated rock fragments that have a general particle size range and include size classes from granule- to boulder-sized fragments. Gravel can be sub-categorized into granule and cobble... Well Sorted Gravel Well Sorted Sand or Sand & Gravel Very Fine Sand, Silt, LoessLoessLoess is an aeolian sediment formed by the accumulation of wind-blown silt, typically in the 20–50 micrometre size range, twenty percent or less clay and the balance equal parts sand and silt that are loosely cemented by calcium carbonate..., LoamLoamLoam is soil composed of sand, silt, and clay in relatively even concentration . Loam soils generally contain more nutrients and humus than sandy soils, have better infiltration and drainage than silty soils, and are easier to till than clay soils... Unconsolidated Clay & Organic PeatPeatPeat is an accumulation of partially decayed vegetation matter or histosol. Peat forms in wetland bogs, moors, muskegs, pocosins, mires, and peat swamp forests. Peat is harvested as an important source of fuel in certain parts of the world... Layered ClayClayClay is a general term including many combinations of one or more clay minerals with traces of metal oxides and organic matter. Geologic clay deposits are mostly composed of phyllosilicate minerals containing variable amounts of water trapped in the mineral structure.- Formation :Clay minerals... Unweathered Clay Consolidated Rocks Highly Fractured Rocks Oil ReservoirPetroleum geologyPetroleum geology refers to the specific set of geological disciplines that are applied to the search for hydrocarbons .-Sedimentary basin analysis:... Rocks Fresh SandstoneSandstoneSandstone is a sedimentary rock composed mainly of sand-sized minerals or rock grains.Most sandstone is composed of quartz and/or feldspar because these are the most common minerals in the Earth's crust. Like sand, sandstone may be any colour, but the most common colours are tan, brown, yellow,... Fresh LimestoneLimestoneLimestone is a sedimentary rock composed largely of the minerals calcite and aragonite, which are different crystal forms of calcium carbonate . Many limestones are composed from skeletal fragments of marine organisms such as coral or foraminifera...., DolomiteDolomiteDolomite is a carbonate mineral composed of calcium magnesium carbonate CaMg2. The term is also used to describe the sedimentary carbonate rock dolostone.... Fresh GraniteGraniteGranite is a common and widely occurring type of intrusive, felsic, igneous rock. Granite usually has a medium- to coarse-grained texture. Occasionally some individual crystals are larger than the groundmass, in which case the texture is known as porphyritic. A granitic rock with a porphyritic... κ (cm2) 0.001 0.0001 10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12 10−13 10−14 10−15 κ (millidarcy) 10+8 10+7 10+6 10+5 10,000 1,000 100 10 1 0.1 0.01 0.001 0.0001

Source: modified from Bear, 1972

• Hydraulic conductivity
Hydraulic conductivity
Hydraulic conductivity, symbolically represented as K, is a property of vascular plants, soil or rock, that describes the ease with which water can move through pore spaces or fractures. It depends on the intrinsic permeability of the material and on the degree of saturation...

• 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...

• Permeation
Permeation
Permeation, in physics and engineering, is the penetration of a permeate through a solid, and is related to a material's intrinsic permeability...

• Petroleum geology
Petroleum geology
Petroleum geology refers to the specific set of geological disciplines that are applied to the search for hydrocarbons .-Sedimentary basin analysis:...

• Relative permeability
Relative 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...