View source for Permeability (porous media)

{{short description|Measure of the ability of a porous material to allow fluids to pass through it}}
{{distinguish|Permeability (electromagnetism)}}
{{Multiple issues|
{{More citations needed|date=April 2008}}
{{more footnotes needed|date=September 2014}}
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[[File:In situ permeameter test.svg|thumb|100px|Symbol used to represent ''in situ'' permeability tests in geotechnical drawings]]

In [[fluid mechanics]], [[materials science]] and [[Earth science]]s, the '''permeability''' of [[porous media]] (often, a [[Rock (geology)|rock]] or [[soil]]) is a measure of the ability for fluids (gas or liquid) to flow through the media; it is commonly symbolized as ''k''.
Fluids can more easily flow through a material with high permeability than one with low permeability.<ref>{{Cite web|title=Reading: Porosity and Permeability {{!}} Geology|url=https://courses.lumenlearning.com/geo/chapter/reading-porosity-and-permeability/|access-date=2022-01-14|website=courses.lumenlearning.com}}</ref> 

The permeability of a medium is related to the ''[[porosity]]'', but also to the shapes of the pores in the medium and their level of connectedness.<ref>{{cite journal |last1=Fu |first1=Jinlong |last2=Thomas |first2=Hywel R. |last3=Li |first3=Chenfeng |title=Tortuosity of porous media: Image analysis and physical simulation |journal=Earth-Science Reviews |date=January 2021 |volume=212 |pages=103439 |doi=10.1016/j.earscirev.2020.103439|bibcode=2021ESRv..21203439F |s2cid=229386129 |url=https://cronfa.swan.ac.uk/Record/cronfa55808/Download/55808__18817__4aeefe32b0ee4ae7993bff0531362902.pdf }}</ref> 
Fluid flows can also be influenced in different [[Lithology|lithological settings]] by brittle deformation of rocks in [[Fault (geology)|fault zones]]; the mechanisms by which this occurs are the subject of [[fault zone hydrogeology]].<ref name="faultzone">{{Cite journal|last1=Bense|first1=V.F.|last2=Gleeson|first2=T.|last3=Loveless|first3=S.E.|last4=Bour|first4=O.|last5=Scibek|first5=J.|date=2013|title=Fault zone hydrogeology|url=https://linkinghub.elsevier.com/retrieve/pii/S0012825213001657|journal=Earth-Science Reviews|language=en|volume=127|pages=171–192|bibcode=2013ESRv..127..171B|doi=10.1016/j.earscirev.2013.09.008}}</ref> Permeability is also affected by the pressure inside a material.

The [[International System of Units|SI]] unit for permeability is the [[square metre]] (m<sup>2</sup>). A practical unit for permeability is the ''[[darcy (unit)|darcy]]'' (d), or more commonly the ''millidarcy'' (md) {{nowrap|(1 d ≈ 10<sup>−12</sup> m<sup>2</sup>).}} The name honors the French Engineer [[Henry Darcy]] who first described the flow of water through sand filters for potable water supply. Permeability values for most materials commonly range typically from a fraction to several thousand millidarcys. The unit of square centimetre (cm<sup>2</sup>) is also sometimes used {{nowrap|1=(1 cm<sup>2</sup> = 10<sup>−4</sup> m<sup>2</sup> ≈ 10<sup>8</sup> d).}}

== Applications ==
{{Expand section|date=February 2016}}

The concept of permeability is of importance in determining the flow characteristics of [[hydrocarbons]] in [[Petroleum|oil]] and [[gas]] reservoirs,<ref>{{cite journal  |vauthors=Guerriero V, etal |year=2012 |title= A permeability model for naturally fractured carbonate reservoirs |journal= [[Marine and Petroleum Geology]] |volume= 40 |pages= 115–134 |doi=10.1016/j.marpetgeo.2012.11.002 |bibcode=1990MarPG...7..410M }}</ref> and of [[groundwater]] in [[aquifer]]s.<ref>[http://drgan.org/wp-content/uploads/2018/10/060_Li_TiPM.pdf Multiphase fluid flow in porous media] From ''Transport in porous media''</ref>

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 in comparison with oil).  Rocks with permeabilities significantly lower than 100 md can form efficient ''seals'' (see [[petroleum geology]]).  Unconsolidated sands may have permeabilities of over 5000&nbsp;md.

The concept also has many practical applications outside of geology, for example in [[chemical engineering]] (e.g., [[filtration]]), as well as in Civil Engineering when determining whether the ground conditions of a site are suitable for construction.

== Description ==
{{also|Fick's laws of diffusion}}

Permeability is part of the proportionality constant in [[Darcy's law]] which relates discharge (flow rate) and fluid physical properties (e.g. [[dynamic viscosity]]), to a pressure gradient applied to the porous media:<ref>[https://imechanica.org/files/JCIS-2019-Tailoring%20porous%20media%20for%20controllable%20capillary%20flow_0.pdf Controlling Capillary Flow], an application of Darcy's law, at iMechanica</ref>

:<math>v  = \frac {k}{\eta} \frac{\Delta P}{\Delta x}</math> (for linear flow)

Therefore:
: <math>k = v \frac{\eta \, \Delta x}{\Delta P}</math>

where:
:<math>v</math> is the [[superficial velocity|fluid velocity]] through the porous medium (i.e., the average flow velocity calculated as if the fluid was the only [[phase (matter)|phase]] present in the porous medium) (m/s)
:<math>k</math> is the permeability of a medium (m<sup>2</sup>)
:<math>\eta</math> is the [[dynamic viscosity]] of the fluid (Pa·s)
:<math>\Delta P</math> is the applied [[pressure]] difference (Pa)
:<math>\Delta x</math> is the thickness of the bed of the porous medium (m)

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

=== Relation to hydraulic conductivity ===
The global proportionality constant for the flow of water through a [[porous medium]] is called the [[hydraulic conductivity]] ({{mvar|K}}, unit: m/s). Permeability, or intrinsic permeability, ({{mvar|k}}, unit: m<sup>2</sup>) is a part of this, and is a specific property characteristic of the solid skeleton and the microstructure of the porous medium itself, independently of the nature and properties of the fluid flowing through the pores of the medium. This allows to take into account the effect of temperature on the viscosity of the fluid flowing though the porous medium and to address other fluids than pure water, ''e.g.'', concentrated [[brine]]s, [[petroleum]], or [[organic solvent]]s. Given the value of hydraulic conductivity for a studied system, the permeability can be calculated as follows:

:<math> k = K \frac {\eta} {\rho g}</math>

:where
* <math>k</math> is the permeability, m<sup>2</sup>
* <math>K</math> is the hydraulic conductivity, m/s
* <math>\eta</math> is the dynamic viscosity of the fluid, Pa·s
* <math>\rho</math> is the density of the fluid, kg/m<sup>3</sup>
* <math>g</math> is the acceleration due to gravity, m/s<sup>2</sup>.

=== Anisotropic permeability ===
Tissue such as brain, liver, muscle, etc can be treated as a heterogeneous porous medium. Describing the flow of biofluids (blood, cerebrospinal fluid, etc.) within such a medium requires a full 3-dimensional [[Anisotropy|anisotropic]] treatment of the tissue.  In this case the [[Scalar (physics)|scalar]] hydraulic permeability is replaced with the hydraulic permeability [[tensor]] so that Darcy's Law reads<ref>{{Cite journal|last=Sowinski|first=Damian|title=Poroelasticity as a Model of Soft Tissue Structure: Hydraulic Permeability Reconstruction for Magnetic Resonance Elastography in Silico|journal=Frontiers in Physics|year=2021|volume=8|page=637|arxiv=2012.03993|doi=10.3389/fphy.2020.617582|pmid=36340954 |pmc=9635531 |bibcode=2021FrP.....8..637S|doi-access=free}}</ref>

:<math>\boldsymbol q = -\frac{1}{\eta}\boldsymbol \kappa \cdot\nabla P </math>

* <math>\boldsymbol q</math> is the Darcy flux, or filtration velocity, which describes the bulk (not microscopic) velocity field of the fluid, <math>[\text{Length}][\text{Time}]^{-1}</math> 
* <math>\eta</math> is the dynamic [[viscosity]] of the fluid, <math>[\text{Mass}][\text{L}]^{-1}[T]^{-1}</math>
* <math>\boldsymbol \kappa </math> is the hydraulic permeability [[tensor]], <math>[\text{L}]^2</math>
* <math>\nabla </math> is the [[Del|gradient operator]], <math>[\text{L}]^{-1}</math>
* <math>P</math> is the [[pressure]] field in the fluid, <math>[\text{M}][\text{L}]^{-1}[\text{T}]^{-2}</math>

Connecting this expression to the isotropic case, <math>\boldsymbol \kappa = k\mathbb  1</math>, where k is the scalar hydraulic permeability, and 1 is the [[Identity matrix|identity tensor]].

== Determination ==
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
|url=http://www.calctool.org/CALC/eng/fluid/darcy
|title=CalcTool: Porosity and permeability calculator
|publisher=www.calctool.org
|access-date=2008-05-30}}</ref>

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]]).

===Permeability model based on conduit flow===
Based on the [[Hagen–Poiseuille equation]] for viscous flow in a pipe, permeability can be expressed as:
:<math>k_{I}=C \cdot d^2</math>
where:
:<math>k_{I}</math> is the intrinsic permeability [length<sup>2</sup>]
:<math>C</math> is a dimensionless constant that is related to the configuration of the flow-paths
:<math>d</math> is the average, or effective pore [[diameter]] [length].

==Absolute permeability (aka intrinsic or specific permeability<ref>{{Cite web |date=2016-09-08 |title=Chapter 2: Physical Properties and Principles {{!}} Freeze and Cherry Groundwater Book |url=https://fc79.gw-project.org/english/chapter-2/ |access-date=2023-05-02 |language=en-US}}</ref>)==
''Absolute permeability'' denotes the permeability in a porous medium that is 100% saturated with a single-phase fluid.  This may also be called the ''intrinsic permeability'' or ''specific permeability.''  These terms refer to the quality that the permeability value in question is an [[Intensive and extensive properties|intensive property]] of the medium, not a spatial average of a heterogeneous block of material {{clarify|date=June 2018|reason=https://fc79.gw-project.org/english/chapter-2/|text=[https://fc79.gw-project.org/english/chapter-2/ equation 2.28]}}{{explain|date=June 2018}}; and that it is a function of the material structure only (and not of the fluid). They explicitly distinguish the value from that of [[relative permeability]].

==Permeability to gases==
Sometimes, permeability to gases can be somewhat different than that for liquids in the same media. One difference is attributable to the "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 9.0×10<sup>−19</sup> m<sup>2</sup> to 2.4×10<sup>−12</sup>&nbsp;m<sup>2</sup> for water and between 1.7×10<sup>−17</sup>&nbsp;m<sup>2</sup> to 2.6×10<sup>−12</sup>&nbsp;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; pp. 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 gas from [[Unconventional (oil & gas) reservoir|unconventional]] sources such as [[shale gas]], [[tight gas]], or [[coalbed methane]].

== Permeability tensor {{anchor|Tensor permeability}} ==
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):
* The tensor is symmetric by the [[Onsager reciprocal relations]]
* The tensor is positive definite because the energy being expended (the [[inner product]] of fluid flow and negative pressure gradient) is always positive

The permeability tensor is always [[diagonalizable]] (being both symmetric and positive definite). The [[eigenvectors]] will yield the principal directions of flow where flow is parallel to the pressure gradient, and the [[eigenvalues]] represent 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]], which are specific to the material through which the fluid is flowing.<ref>Bear, Jacob, 1972. ''Dynamics of Fluids in Porous Media,'' Dover.  {{ISBN|0-486-65675-6}}</ref>

{|  class="wikitable"
! style="text-align:right" | Permeability
! colspan="4" | Pervious
! colspan="4" | Semi-pervious
! colspan="5" | Impervious
|-
! style="text-align:right" | Unconsolidated sand and gravel
| colspan="2" align="center" | Well sorted [[gravel]]
| colspan="3" align="center" | Well sorted [[sand]] or sand and gravel
| colspan="4" align="center" | Very fine sand, silt, [[loess]], [[loam]]
| colspan="4" |
|-
! style="text-align:right" | Unconsolidated clay and organic
|  colspan="4" |
|  colspan="2" align="center" | [[Peat]]
|  colspan="3" align="center" | Layered [[clay]]
|  colspan="4" align="center" | Unweathered clay
|-
! style="text-align:right" | Consolidated rocks
|  colspan="4" align="center" | Highly fractured rocks
|  colspan="3" align="center" | [[Petroleum geology|Oil reservoir]] rocks
|  colspan="2" align="center" | Fresh [[sandstone]]
|  colspan="2" align="center" | Fresh [[limestone]], [[Dolomite (rock)|dolomite]]
|  colspan="2" align="center" | Fresh [[granite]]
|-
! style="text-align:right" | ''k'' (cm<sup>2</sup>)
| 0.001
| 0.0001
| 10<sup>−5</sup>
| 10<sup>−6</sup>
| 10<sup>−7</sup>
| 10<sup>−8</sup>
| 10<sup>−9</sup>
| 10<sup>−10</sup>
| 10<sup>−11</sup>
| 10<sup>−12</sup>
| 10<sup>−13</sup>
| 10<sup>−14</sup>
| 10<sup>−15</sup>
|-
! style="text-align:right" | ''k'' (m<sup>2</sup>)
| 10<sup>−7</sup>
| 10<sup>−8</sup>
| 10<sup>−9</sup>
| 10<sup>−10</sup>
| 10<sup>−11</sup>
| 10<sup>−12</sup>
| 10<sup>−13</sup>
| 10<sup>−14</sup>
| 10<sup>−15</sup>
| 10<sup>−16</sup>
| 10<sup>−17</sup>
| 10<sup>−18</sup>
| 10<sup>−19</sup>
|-
! style="text-align:right" | ''k'' (millidarcy)
| 10<sup>+8</sup>
| 10<sup>+7</sup>
| 10<sup>+6</sup>
| 10<sup>+5</sup>
| 10,000
| 1,000
| 100
| 10
| 1
| 0.1
| 0.01
| 0.001
| 0.0001
|}

==See also==
* [[Fault zone hydrogeology]]
* [[Hydraulic conductivity]]
* [[Hydrogeology]]
* [[Permeation]]
* [[Petroleum geology]]
* [[Relative permeability]]
* [[Klinkenberg correction]]
* [[Electrical resistivity measurement of concrete]]
* [[Permeability of soils]]

==References==
{{reflist}}

==Further reading==
* Wang, H. F., 2000. Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press. {{ISBN|0-691-03746-9}}

==External links==
* [https://careers.slb.com/inside-schlumberger/transformative-technology/defining-permeability Defining Permeability ]
* [https://imechanica.org/files/JCIS-2019-Tailoring%20porous%20media%20for%20controllable%20capillary%20flow_0.pdf Tailoring porous media to control permeability]
* [http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/chapter-2-the-porous-medium/permeability/ Permeability of Porous Media]
* [http://techalive.mtu.edu/meec/module06/Permeability.htm Graphical depiction of different flow rates through materials of differing permeability]
* [http://www.calctool.org/CALC/eng/fluid/darcy Web-based porosity and permeability calculator given flow characteristics]
* [http://drgan.org/wp-content/uploads/2018/10/060_Li_TiPM.pdf Multiphase fluid flow in porous media]
* [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] {{Webarchive|url=https://web.archive.org/web/20110616123355/http://www.dot.state.fl.us/statematerialsoffice/administration/resources/library/publications/fstm/methods/fm5-578.pdf |date=2011-06-16 }}

{{Geotechnical engineering|state=collapsed}}

{{DEFAULTSORT:Permeability (Earth Sciences)}}
[[Category:Aquifers]]
[[Category:Hydrology]]
[[Category:Soil mechanics]]
[[Category:Soil physics]]
[[Category:Porous media]]
[[Category:In situ geotechnical investigations]]
[[Category:Physical quantities]]