Permeability (earth sciences)

Permeability in fluid mechanics and the earth sciences (commonly symbolized as k) is a measure of the ability of a porous material (often, a rock or an unconsolidated material) to allow fluids to pass through it.

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.


Permeability is the property of rocks that is an indication of the ability for fluids (gas or liquid) to flow through rocks. High permeability will allow fluids to move rapidly through rocks. Permeability is affected by the pressure in a rock. The unit of measure is called the darcy, named after Henry Darcy (1803–1858). Sandstones may vary in permeability from less than one to over 50,000 millidarcys (md). Permeabilities are more commonly in the range of tens to hundreds of millidarcies. A rock with 25% porosity and a permeability of 1 md will not yield a significant flow of water. Such “tight” rocks are usually artificially stimulated (fractured or acidized) to create permeability and yield a flow.


The SI unit for permeability is m2. A practical unit for permeability is the darcy (d), or more commonly the millidarcy (md) (1 darcy 10−12m2). 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 sandstones range typically from a fraction of a darcy to several darcys. The unit of cm2 is also sometimes used (1 cm2 = 10−4 m2 108 d).


The concept of permeability is of importance in determining the flow characteristics of hydrocarbons in oil and gas reservoirs[1], and of groundwater in aquifers.

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.

The concept also has many practical applications outside of geology, for example in chemical engineering (e.g., filtration).


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:

(for linear flow)



is the superficial fluid flow velocity through the medium (i.e., the average velocity calculated as if the fluid were the only phase present in the porous medium) (m/s)
is the permeability of a medium (m2)
is the dynamic viscosity of the fluid (Pa·s)
is the applied pressure difference (Pa)
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 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:

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


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.[2]

Permeability needs to be measured, either directly (using Darcy's law), or through estimation using empirically derived formulas. However, for some simple models of porous media, permeability can be calculated (e.g., 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:


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 [length].

Absolute permeability (aka intrinsic or specific permeability)

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 property of the medium, not a spatial average of a heterogeneous block of material; 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 those for liquids in the same media. One difference is attributable to "slippage" of gas at the interface with the solid[3] 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 9.0×10−19 m2 to 2.4×10−12 m2 for water and between 1.7×10−17 m2 to 2.6×10−12 m2 for nitrogen gas.[4] 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.

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 being both symmetric and positive definite (SPD matrix):

  • The tensor is symmetric by the Onsager reciprocal relations.
  • The tensor is positive definite as the component of the flow parallel 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, which are specific to the material through which the fluid is flowing.[5]

Permeability Pervious Semi-pervious Impervious
Unconsolidated sand and gravel Well sorted gravel Well sorted sand or sand and gravel Very fine sand, silt, loess, loam
Unconsolidated clay and organic Peat Layered clay Unweathered clay
Consolidated rocks Highly fractured rocks Oil reservoir rocks Fresh sandstone Fresh limestone, dolomite Fresh granite
k (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
k (millidarcy) 10+8 10+7 10+6 10+5 10,000 1,000 100 10 1 0.1 0.01 0.001 0.0001

See also


  1. ^ Guerriero V, et al. (2012). "A permeability model for naturally fractured carbonate reservoirs". Marine and Petroleum Geology. 40: 115–134. Bibcode:1990MarPG...7..410M. doi:10.1016/j.marpetgeo.2012.11.002.
  2. ^ "CalcTool: Porosity and permeability calculator". Retrieved 2008-05-30.
  3. ^ L. J. Klinkenberg, "The Permeability Of Porous Media To Liquids And Gases", Drilling and Production Practice, 41-200, 1941 (abstract).
  4. ^ 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. (abstract)
  5. ^ Bear, Jacob, 1972. Dynamics of Fluids in Porous Media, Dover. ISBN 0-486-65675-6


  • Wang, H. F., 2000. Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press. ISBN 0-691-03746-9

External links

Center for Biofilm Engineering

The Center for Biofilm Engineering (CBE) is an interdisciplinary research, education, and technology transfer institution located on the central campus of Montana State University in Bozeman, Montana. The center was founded in April 1990 as the Center for Interfacial Microbial Process Engineering with a grant from the Engineering Research Centers (ERC) program of the National Science Foundation (NSF). The CBE integrates faculty from multiple university departments to lead multidisciplinary research teams—including graduate and undergraduate students—to advance fundamental biofilm knowledge, develop beneficial uses for microbial biofilms, and find solutions to industrially relevant biofilm problems. The center tackles biofilm issues including chronic wounds, bioremediation, and microbial corrosion through cross-disciplinary research and education among engineers, microbiologists and industry.

Factors affecting permeability of soils

A number of factors affect the permeability of soils, from particle size, impurities in the water, void ratio, the degree of saturation, and adsorbed water, to entrapped air and organic material.

Greaseproof paper

Greaseproof paper is paper that is impermeable to oil or grease and is normally used in cooking or food packaging. Normally greaseproof paper is produced by refining the paper stock and thus create a sheet with very low porosity. This sheet is supercalendered to further improve the density, creating a paper called glassine. The glassine is treated with starches, alginates or carboxymethyl cellulose (CMC) in a size press to fill pores or treat the paper chemically to make it fat repellent. Basis weights are usually 30–50 g/m².In Australia, greaseproof paper is also referred to as SKAN Greaseproof.

Index of soil-related articles

This is an index of articles relating to soil.

Lower oceanic crust

The lower oceanic crust is the lower part of the oceanic crust and represents the major part of it (volumetrically biggest part). It is generally located 4–8 km below the ocean floor and the major lithologies are mafic (ultramafic and gabbroic rocks) which derive from melts rising from the earth's mantle. This part of the oceanic crust is an important zone for processes such as melt accumulation and melt modification (fractional crystallisation and crustal assimilation). And the recycling of this part of the oceanic crust, together with the upper mantle has been suggested as a significant source component for tholeiitic magmas in Hawaiian volcanoes. Although the lower oceanic crust builds the link between the mantle and the MORB, and can't be neglected for the understanding of MORB evolution, the complex processes operating in this zone remain unclear and there is an ongoing debate in Earth Sciences about this.

Paper-based microfluidics

Paper-based microfluidics are microfluidic devices that consist of a series of hydrophilic cellulose or nitrocellulose fibers that guide liquid from an inlet to a desired outlet by imbibition. The technology builds on the conventional lateral flow test which is capable of detecting many infections agents and chemical contaminants. The main advantage of this is that it is largely a passively controlled device unlike more complex microfluidic devices. Development of paper-based microfluidic devices began in the early 21st century to meet a need for inexpensive and portable medical diagnostic systems.


In physics and engineering, permeation (also called imbuing) is the penetration of a permeate (such as a liquid, gas, or vapor) through a solid. It is directly related to the concentration gradient of the permeate, a material's intrinsic permeability, and the materials' mass diffusivity. Permeation is modeled by equations such as Fick's laws of diffusion, and can be measured using tools such as a minipermeameter.

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. It can be viewed as an adaptation of Darcy's law to multiphase flow.

For two-phase flow in porous media given steady-state conditions, we can write

where is the flux, is the pressure drop, is the viscosity. The subscript indicates that the parameters are for phase .

is here the phase permeability (i.e., the effective permeability of phase ), as observed through the equation above.

Relative permeability, , for phase is then defined from , as

where is the permeability of the porous medium in single-phase flow, i.e., the absolute permeability. Relative permeability must be between zero and one.

In applications, relative permeability is often represented as a function of water saturation; however, owing to capillary hysteresis one often resorts to a function or curve measured under drainage and another measured under imbibition.

Under this approach, the flow of each phase is inhibited by the presence of the other phases. Thus the sum of relative permeabilities over all phases is less than 1. However, apparent relative permeabilities larger than 1 have been obtained since the Darcean approach disregards the viscous coupling effects derived from momentum transfer between the phases (see assumptions below). This coupling could enhance the flow instead of inhibit it. This has been observed in heavy oil petroleum reservoirs when the gas phase flows as bubbles or patches (disconnected).

Resin-bound paving

Resin-bound paving is a mixture of aggregate stones and resin used to pave footpaths, driveways, etc. It is a kind of permeable paving solution.

It is a flexible surfacing material, so is resistant to cracking.

The system is mixed on site and cold applied, using a high-quality clear resin binder to coat the aggregate particles prior to laying. Unlike resin-bonded surfacing, where a thin layer of resin is applied to the surface and then the aggregate scattered on top (which can then become loose over time and is impermeable), resin and aggregates are thoroughly mixed together prior to laying, ensuring that the aggregate is completely coated and so providing a totally bound surface. As a result, a resin-bound surface is more durable and requires less maintenance – it needs to be swept or power washed at least twice a year, to avoid the buildup of detritus and prevent the growth of moss or algae.

Resin-bound paving is a fully permeable paving solution which allows water to freely drain through the surface. Meeting the requirements of Sustainable urban drainage systems (SUDS) standards, this helps to prevent standing water and largely eliminates surface water runoff.

Resin-bound surfacing does not currently conform to any British standards apart from BS 8204-6:2008+A1:2010 Appendix B for slip resistance. Therefore, great care must be taken to research any company that is offering resin and aggregate services.

The quality of the resin-bound surfacing is dependent on a variety of factors.

The polyurethane resin should be sourced from a manufacturer who can demonstrate that they have BS EN 9001 quality management standards. All companies conforming to this will be able to provide documentation on request. The resin-bound aggregate mix will generally be a mix of natural aggregates that must be kiln dried to prevent moisture coming in contact with polyurethanes and causing discolouration and poor performance.

The resin sourced should be a two-part, quality-controlled polyurethane or similar with the activator added during the manufacturing process to eradicate mistakes on site made by operatives who are not qualified chemists. Too much activator or too little can affect the performance of the product.

The ratio of resin to aggregate should be at the optimal amount depending on the application and environment type, and stone type which should be dried mixed aggregate incorporating 5-8mm sizes to ensure that the aggregate gets sufficiently coated and also to meet the standard requirements when tested to BS 8204-6:2008+A1:2010 Appendix B for slip resistance. This ration is sufficient to install at a minimum depth of 15mm at a coverage rate of approximately 4.5m2 and is suitable for pedestrian traffic. For vehicular traffic of up to 7.5 tonne a minimum of 18mm is required.

Although resin-bound paving is a slip-resistant permeable decorative paving system, it has the ability to last for many years if sourced and installed correctly.

Natural aggregate mix blends tested to BS 8204-6:2008+A1:2010 Appendix B for slip resistance will, when installed correctly, provide a slip-resistant, permeable, decorative finish suitable for pedestrian and light vehicular traffic.

Decorative coloured recycled glass and pigmented quartzes are suitable generally for visual purposes only, as they are susceptible to damage due to having low crush values, so will become damaged if walked on.

Natural aggregate and recycled rubber blends are available from BS EN 14001 manufacturers and are suitable for pathways and nature walks where environmental benefits are required from the specifier. The resins available are predominantly polyurethanes.

The usage of higher-quality polyurethane resins ensures greater flexural performance for external applications and the resin will not yellow in appearance when exposed to UV light, the presence also provides greater protection from UV degradation. Non light-stable polyurethanes, although strong in performance, are more suited to internal applications where they are not in direct sunlight. Internal resin-bound applications are generally known as stone carpet.

Resin-bound systems incorporating 10 mm dried aggregates and larger sizes are generally used as tree surrounds known as tree pits. These are a cost-effective and practical alternative to metal tree grilles that are stolen for scrap value, are costly to purchase and harbour litter thus increasing maintenance costs for local authorities and tax payers. Tree pit systems work in a similar way to resin-bound surfacing systems only using larger aggregates to allow more water to permeate through to feed the trees they surround.

Tree pit systems are generally installed at a depth of 40 mm to 50 mm, depending on the specification; they have a reduced ratio of 6% light stable polyurethane resin to 100 kg kiln dried aggregate.

Retaining walls
Numerical analysis

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