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| '''Permeability''' in [[fluid mechanics]] and the [[earth science]]s (commonly symbolized as ''κ'', or ''k'') is a measure of the ability of a [[porous media|porous material]] (often, a [[Rock (geology)|rock]] or unconsolidated material) to allow fluids to pass through it.
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| ==Permeability==
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| Permeability is the property of rocks that is an indication of the ability for fluids (gas or liquid) to flow through rocks.
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| High permeability will allow fluids and gases to move rapidly through rocks
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| Permeability is affected by the pressure in a rock.
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| The unit of measure is called the [[darcy]], named after [[Henry Darcy]] (1803-1858).
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| Sandstones may vary in permeability from less than one to over 50,000
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| millidarcys (mD).
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| Permeabilities are more commonly in the range of tens to hundreds of
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| millidarcys.
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| A rock with 25% porosity and a permeability of 1 mD will not yield a
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| significant flow of fluids or gases. Such “tight” rocks are usually artificially
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| stimulated (fractured or acidized) to create permeability and yield a flow.
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| == Units == | |
| The [[International System of Units|SI]] unit for permeability is m<sup>2</sup>. A practical unit for permeability is the ''[[darcy]]'' (D), or more commonly the ''millidarcy'' (mD) (1 darcy <math>\approx</math>10<sup>−12</sup>m<sup>2</sup>). The name is in honor to the French Engineer [[Henry Darcy]] who first described the flow of water through sand filters for potable water supply. Permeability values for sandstones range typically from a fraction of a ''darcy'' to several ''darcys''. The unit of cm<sup>2</sup> is also sometimes used (1 cm<sup>2</sup> = 10<sup>−4</sup> m<sup>2</sup> <math>\approx</math> 10<sup>8</sup> D).
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| == Applications ==
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| The concept of permeability is of importance in determining the flow characteristics of [[hydrocarbons]] in [[Petroleum|oil]] and [[gas]] reservoirs, and of [[groundwater]] in [[aquifer]]s.
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| 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]] of gas with respect to oil). Rocks with permeabilities significantly lower than 100 mD can form efficient ''seals'' (see [[petroleum geology]]). Unconsolidated sands may have permeabilities of over 5000 mD.
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| The concept has also many practical applications outside of geology, for example in [[chemical engineering]] (e.g., [[filtration]]).
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| == Description ==
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| Permeability is part of the proportionality constant in [[Darcy's law]] which relates discharge (flow rate) and fluid physical properties (e.g. [[viscosity]]), to a pressure gradient applied to the porous media:
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| :<math>v = \frac {\kappa}{\mu} \frac{\Delta P}{\Delta x}</math>
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| Therefore:
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| : <math>\kappa = v \frac{\mu \Delta x}{\Delta P}</math>
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| where:
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| :<math>v</math> is the [[superficial velocity|superficial fluid flow velocity]] through the medium (i.e., the average velocity calculated as if the fluid were the only [[phase (matter)|phase]] present in the porous medium) (m/s)
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| :<math>\kappa</math> is the permeability of a medium (m<sup>2</sup>) | |
| :<math>\mu</math> is the dynamic [[viscosity]] of the fluid (Pa·s)
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| :<math>\Delta P</math> is the applied [[pressure]] difference (Pa)
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| :<math>\Delta x</math> is the thickness of the bed of the porous medium (m)
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| In naturally occurring materials, permeability values range over many orders of magnitude (see table below for an example of this range).
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| === Relation to hydraulic conductivity ===
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| The proportionality constant specifically for the flow of water through a porous media is called the [[hydraulic conductivity]]; 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, the permeability can be calculated as follows:
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| <math> \kappa = K \frac {\mu} {\rho g}</math>
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| :where
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| * <math>\kappa</math> is the permeability, m<sup>2</sup>
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| * <math>K</math> is the hydraulic conductivity, m/s
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| * <math>\mu</math> is the dynamic viscosity of the fluid, kg/(m·s)
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| * <math>\rho</math> is the density of the fluid, kg/m<sup>3</sup>
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| * <math>g</math> is the acceleration due to gravity, m/s<sup>2</sup>.
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| == Determination ==
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| Permeability is typically determined in the lab by application of [[Darcy's law]] under steady state conditions or, more generally, by application of various solutions to the [[diffusion equation]] for unsteady flow conditions.<ref>{{cite web
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| |url=http://www.calctool.org/CALC/eng/fluid/darcy
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| |title=CalcTool: Porosity and permeability calculator
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| |publisher=www.calctool.org
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| |accessdate=2008-05-30}}</ref>
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| Permeability needs to be measured, either directly (using [[Darcy's law]]), or through [[Estimation theory|estimation]] using [[Empirical method|empirically]] derived formulas. However, for some simple models of porous media, permeability can be calculated (e.g., [[Random close pack|random close packing of identical spheres]]).
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| ===Permeability model based on conduit flow===
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| Based on [[Hagen–Poiseuille equation|Alen Hazen]], permeability can be expressed as:
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| :<math>{\kappa}_{I}=C \cdot d^2</math>
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| where:
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| :<math>{\kappa}_{I}</math> is the intrinsic permeability [length<sup>2</sup>]
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| :<math>C</math> is a dimensionless constant that is related to the configuration of the flow-paths
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| :<math>d</math> is the average, or effective pore [[diameter]] [length].
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| ==Intrinsic and absolute permeability==
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| The terms ''intrinsic permeability'' and ''absolute permeability'' states that the permeability value in question is an [[Intensive and extensive properties|intensive property]] (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]].
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| ==Permeability to gases==
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| 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<ref>L. J. Klinkenberg, "The Permeability Of Porous Media To Liquids And Gases", Drilling and Production Practice, 41-200, 1941 [http://www.onepetro.org/mslib/servlet/onepetropreview?id=API-41-200&soc=API&speAppNameCookie=ONEPETRO (abstract)].</ref> when the gas [[mean free path]] is comparable to the pore size (about 0.01 to 0.1 μm at standard temperature and pressure). See also [[Knudsen diffusion]] and [[constrictivity]]. For example, measurement of permeability through sandstones and shales yielded values from 9.0x10<sup>−19</sup> m<sup>2</sup> to 2.4x10<sup>−12</sup> m<sup>2</sup> for water and between 1.7x10<sup>−17</sup> m<sup>2</sup> to 2.6x10<sup>−12</sup> m<sup>2</sup> for nitrogen gas.<ref>J. P. Bloomfield and A. T. Williams, "An empirical liquid permeability-gas permeability correlation for use in aquifer properties studies". Quarterly Journal of Engineering Geology & Hydrogeology; November 1995; v. 28; no. Supplement_2; p.S143-S150. [http://qjegh.geoscienceworld.org/cgi/content/abstract/28/Supplement_2/S143 (abstract)]</ref> Gas permeability of [[reservoir rock]] and [[source rock]] is important in [[petroleum engineering]], when considering the optimal extraction of [[shale gas]], [[tight gas]], or [[coalbed methane]].
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| == Tensor permeability ==
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| <!-- Note: This section is linked to from Darcy's law, also fix there if you change the name if this section -->
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| To model permeability in [[anisotropic]] media, a permeability [[tensor]] 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 realised using a 3 by 3 [[Matrix (mathematics)|matrix]] being both [[Symmetric matrix|symmetric]] and [[Positive-definite matrix|positive definite]] (SPD matrix):
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| * The tensor is symmetric by the [[Onsager reciprocal relations]].
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| * The tensor is positive definite as the component of the flow [[Parallel (geometry)|parallel]] to the pressure drop is always in the same direction as the pressure drop.
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| 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.
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| ==Ranges of common intrinsic permeabilities==
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| These values do not depend on the fluid properties; see the table derived from the same source for values of [[hydraulic conductivity]], which are specific to the material through which the fluid is flowing.
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| As far as soils concerned, typical ranges of permeability coefficient for different soils can be found on [http://www.geotechdata.info/parameter/permeability.html Geotechdata.info database].
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| {| border="1" width="600"
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| | bgcolor="#FAEBD7" | Permeability
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| | bgcolor="#FAEBD7" colspan="4" align="center" | Pervious
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| | bgcolor="#FAEBD7" colspan="4" align="center" | Semi-Pervious
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| | bgcolor="#FAEBD7" colspan="5" align="center" | Impervious
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| |-
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| | bgcolor="#FAEBD7" | Unconsolidated [[Sand]] & [[Gravel]]
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| | colspan="2" align="center" | Well Sorted Gravel
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| | colspan="3" align="center" | Well Sorted Sand or Sand & Gravel
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| | colspan="4" align="center" | Very Fine Sand, Silt, [[Loess]], [[Loam]]
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| | colspan="4" |
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| |-
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| | bgcolor="#FAEBD7" | Unconsolidated Clay & Organic
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| | colspan="4" |
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| | colspan="2" align="center" | [[Peat]]
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| | colspan="3" align="center" | Layered [[Clay]]
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| | colspan="4" align="center" | Unweathered Clay
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| |-
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| | bgcolor="#FAEBD7" | Consolidated Rocks
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| | colspan="4" align="center" | Highly Fractured Rocks
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| | colspan="3" align="center" | [[Petroleum geology|Oil Reservoir]] Rocks
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| | colspan="2" align="center" | Fresh [[Sandstone]]
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| | colspan="2" align="center" | Fresh [[Limestone]], [[Dolomite]]
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| | colspan="2" align="center" | Fresh [[Granite]]
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| |-
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| | bgcolor="#FAEBD7" | ''κ'' (cm<sup>2</sup>)
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| | 0.001
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| | 0.0001
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| | 10<sup>−5</sup>
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| | 10<sup>−6</sup>
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| | 10<sup>−7</sup>
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| | 10<sup>−8</sup>
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| | 10<sup>−9</sup>
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| | 10<sup>−10</sup>
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| | 10<sup>−11</sup>
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| | 10<sup>−12</sup>
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| | 10<sup>−13</sup>
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| | 10<sup>−14</sup>
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| | 10<sup>−15</sup>
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| |-
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| | bgcolor="#FAEBD7" | ''κ'' (millidarcy)
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| | 10<sup>+8</sup>
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| | 10<sup>+7</sup>
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| | 10<sup>+6</sup>
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| | 10<sup>+5</sup>
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| | 10,000
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| | 1,000
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| | 100
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| | 10
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| | 1
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| | 0.1
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| | 0.01
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| | 0.001
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| | 0.0001
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| |}
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| Source: modified from Bear, 1972
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| ==See also==
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| * [[Hydraulic conductivity]]
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| * [[Hydrogeology]]
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| * [[Permeation]]
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| * [[Petroleum geology]]
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| * [[Relative permeability]]
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| * [[Klinkenberg correction]]
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| * [[Electrical resistivity measurement of concrete]]
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| ==Footnotes==
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| {{reflist}}
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| ==References==
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| * Bear, Jacob, 1972. Dynamics of Fluids in Porous Media, Dover. — ISBN 0-486-65675-6
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| * Wang, H. F., 2000. Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press. ISBN 0-691-03746-9
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| ==External links==
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| * [http://techalive.mtu.edu/meec/module06/Permeability.htm Graphical depiction of different flow rates through materials of differing permeability]
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| * [http://www.calctool.org/CALC/eng/fluid/darcy Web-based porosity and permeability calculator given flow characteristics]
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| * [http://www.dot.state.fl.us/statematerialsoffice/administration/resources/library/publications/fstm/methods/fm5-578.pdf Florida Method of Test For Concrete Resistivity as an Electrical Indicator of its Permeability]
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| {{Geotechnical engineering|state=collapsed}}
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| {{DEFAULTSORT:Permeability (Earth Sciences)}}
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| [[Category:Aquifers]]
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| [[Category:Hydrology]]
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| [[Category:Soil mechanics]]
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| [[Category:Porous media]]
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