# Permeability

The ability of a water bearing material to transmit water. It is measured by the quantity of water passing through a unit cross section, in a unit time, under 100 percent hydraulic gradient

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

## Units

The SI unit for permeability is m2. A traditional unit for permeability is the darcy (D), or more commonly the millidarcy (mD) (1 darcy $approx$10−12m2). The unit of cm2 is also sometimes used (1 m2 = 104 cm2).

## Applications

The concept of permeability is of importance in determining the flow characteristics of hydrocarbons in oil and gas reservoirs, 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 has also many practical applications outside of geology, for example in chemical engineering (e.g., filtration).

## Description

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:

$v = frac {kappa}{mu} frac{Delta P}{Delta x}$

Therefore:

$kappa = v frac{mu Delta x}{Delta P}$

where:

$v$ 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)
$kappa$ is the permeability of a medium (m2)
$mu$ is the dynamic viscosity of the fluid (Pa·s)
$Delta P$ is the applied pressure difference (Pa)
$Delta x$ is the thickness of the bed of the porous medium (m)

In naturally occurring materials, 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, $kappa$, the permeability can be calculated as:

$kappa = K frac {mu} { ho g}$

where
• $kappa$ is the permeability, m2
• $K$ is the hydraulic conductivity, m/s
• $mu$ is the dynamic viscosity, kg/(m·s)
• $ho$ is the density of the fluid, kg/m3
• $g$ is the acceleration due to gravity, m/s2.

## 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.[1]

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 Hagen–Poiseuille equation for viscous flow in a pipe, permeability can be expressed as:

${kappa}_{I}=C cdot d^2$

where:

${kappa}_{I}$ is the intrinsic permeability [length2]
$C$ is a dimensionless constant that is related to the configuration of the flow-paths
$d$ is the average, or effective pore diameter [length].

## Intrinsic and absolute permeability

The terms intrinsic permeability and absolute permeability states that the permeability value in question is an 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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