Definitions

# Permeability (earth sciences)

Permeability in the earth sciences (commonly symbolized as κ, or k) is a measure of the ability of a material (typically, a rock or unconsolidated material) to transmit fluids. It is of great importance in determining the flow characteristics of hydrocarbons in oil and gas reservoirs, and of groundwater in aquifers. It is typically measured 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.

## Formula

The intrinsic permeability of any porous material is:
$\left\{kappa\right\}_\left\{I\right\}=C cdot d^2$
where
$\left\{kappa\right\}_\left\{I\right\}$ is the intrinsic permeability [L2]
$C$ is a dimensionless constant that is related to the configuration of the flow-paths
$d$ is the average, or effective pore diameter [L]

Permeability needs to be measured, either directly (using Darcy's law) or through estimation using empirically derived formulas.

A common unit for permeability is the darcy (D), or more commonly the millidarcy (mD) (1 darcy $approx$10−12m2). Other units are cm2 and the SI m2.

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. The proportionality constant specifically for the flow of water through a porous media is the hydraulic conductivity; permeability is a portion of this, and is a property of the porous media only, not the fluid. In naturally occurring materials, it ranges over many orders of magnitude (see table below for an example of this range).

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.

## 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 realized 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.
 Permeability Pervious Semi-Pervious Impervious Unconsolidated Sand & Gravel Well Sorted Gravel Well Sorted Sand or Sand & Gravel Very Fine Sand, Silt, Loess, Loam Unconsolidated Clay & Organic Peat Layered Clay Fat / Unweathered Clay Consolidated Rocks Highly Fractured Rocks Oil Reservoir Rocks Fresh Sandstone Fresh Limestone, Dolomite Fresh Granite 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

## References

• 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 0691037469