Permeability (Earth sciences)

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 ${\displaystyle \approx }$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 ${\displaystyle \approx }$ 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 [2].

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), as well as in Civil Engineering when determining whether the ground conditions of a site are suitable for construction.

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 [3]:

${\displaystyle v={\frac {k}{\mu }}{\frac {\Delta P}{\Delta x}}}$ (for linear flow)

Therefore:

${\displaystyle k=v{\frac {\mu \Delta x}{\Delta P}}}$

where:

${\displaystyle v}$ 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)

${\displaystyle k}$ is the permeability of a medium (m²)

${\displaystyle \mu }$ is the dynamic viscosity of the fluid (Pa·s)

${\displaystyle \Delta P}$ is the applied pressure difference (Pa)

${\displaystyle \Delta x}$ 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).

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.[4]

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).

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.

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[5] 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.[6] 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.

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.

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.[7]

• Hydraulic conductivity
• Hydrogeology
• Permeation
• Petroleum geology
• Relative permeability
• Klinkenberg correction
• Electrical resistivity measurement of concrete

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

• Defining Permeability
• Tailoring porous media to control permeability
• Permeability of Porous Media
• Graphical depiction of different flow rates through materials of differing permeability
• Web-based porosity and permeability calculator given flow characteristics
• Multiphase fluid flow in porous media
• Florida Method of Test For Concrete Resistivity as an Electrical Indicator of its Permeability