# Permeability (geosciences)

The permeability ( Latin : permeare "pass through", from Latin: per "through", and Latin: meare "pass") is used in geotechnics to quantify the permeability of soils and rocks for liquids or gases (e.g. Groundwater , oil or natural gas ). The permeability coefficient explained here is very closely related to it . ${\ displaystyle K}$ ${\ displaystyle k_ {f}}$

## definition

The permeability is derived from Darcy's law and is defined as:

${\ displaystyle K = {\ frac {Q \ cdot \ eta \ cdot l} {A \ cdot \ Delta p}}}$

Here mean:

• ${\ displaystyle K}$: Permeability in m²
• ${\ displaystyle Q}$: Flow rate in m³ / s
• ${\ displaystyle \ eta}$: Dynamic viscosity of the flow medium in Ns / m²
• ${\ displaystyle l}$: length of the porous body in m
• ${\ displaystyle A}$: cross-sectional area of ​​the porous body flowed through in m²
• ${\ displaystyle \ Delta p}$: Pressure difference in Pa = N / m², which occurs after the flow.

The permeability only depends on the properties of the medium flowing through ( material parameter ), because the product of the flow rate and viscosity remains constant: ${\ displaystyle Q}$${\ displaystyle \ eta}$

${\ displaystyle Q \ cdot \ eta = {\ text {const.}}}$
I.e. the lower the viscosity, the higher the flow rate and vice versa.

Since the permeability is not influenced by the density, which in the case of gases depends on the pressure, it is well suited for gases and is therefore often used in the natural gas and oil industries. The dynamic viscosity is independent of the pressure within the range of the validity of the gas law , a temperature dependence is always given.

The SI unit for the permeability is m². Another common unit of measurement is the Darcy , named after the French scientist Henry Darcy (1803-1858) who studied the flow of water through gravel beds in 1856 :

${\ displaystyle 1 \, \ mathrm {Darcy} = 9 {,} 86923 \ cdot 10 ^ {- 13} \, \ mathrm {m} ^ {2} \ approx 10 ^ {- 12} \, \ mathrm {m } ^ {2}.}$

Since 1 Darcy is a relatively high permeability, the millidarcy (mD) or the SI unit (µm) ² are often used in geotechnical engineering and mining .

## Permeability coefficient

The permeability coefficient (or the hydraulic conductivity ) also quantifies the permeability of soil or rock, but the density and viscosity of the fluid flowing through are also included here :

${\ displaystyle k_ {f} = {\ frac {K \ cdot \ rho \ cdot g} {\ eta}} = {\ frac {Q \ cdot l \ cdot \ rho \ cdot g} {A \ cdot \ Delta p }}.}$

Here mean:

• ${\ displaystyle k_ {f}}$: Permeability coefficient in m / s
• ${\ displaystyle \ rho}$: Density of the fluid, 1000 kg / m³ for water
• ${\ displaystyle g}$: Acceleration due to gravity = 9.81 m / s²
• ${\ displaystyle \ eta}$: Dynamic viscosity of the fluid, with water 10 −3  Ns / m².

The permeability coefficient is mostly used for flowing liquids (water), i.e. in the areas of water management and hydraulic engineering . Since it can be assumed for (incompressible) liquids , the permeability coefficient can also be written in simplified form as: ${\ displaystyle \ rho s = {\ text {const.}}}$

${\ displaystyle k_ {f} = {\ frac {Q \ cdot l} {A \ cdot \ Delta h}}}$

with the height difference over which the flow occurs. ${\ displaystyle \ Delta h,}$

Unless otherwise stated, the values ​​given in the literature for usually relate to water. If the permeability coefficient for a medium through which water flows is known, then the permeability of this medium for other substances can be calculated (see below "Determination of permeability"). ${\ displaystyle k_ {f}}$

### Ranges of values

Permeability coefficients
according to DIN  18130 (water)
Permeability
> 10 −2 m / s very permeable
10 −2 to 10 −4 m / s highly permeable
10 −4 to 10 −6 m / s permeable
10 −6 to 10 −8 m / s weakly permeable
10 −8 to 10 −9 m / s very weakly permeable
<10 −9 m / s almost completely impermeable to water
Loose rock Permeability coefficient
(water)
pure gravel 10 −1 to 10 −2 m / s1
coarse-grained sand by 10 −3 m / s1
medium grain sand 10 −3 to 10 −4 m / s1
fine-grain sand 10 −4 to 10 −5 m / s1
silty sand 10 −5 to 10 −7 m / s1
clayey silt 10 −6 to 10 −9 m / s1
volume 10 −7 to 10 −12 m / s

The boundary between a permeable and an impermeable soil is around 10 −6  m / s.

## Properties and influencing factors

The permeability of soils depends primarily on their porosity , that of rock on its porosity and / or its fissures . The porosity of soils, in turn, depends on the grain sizes , their distribution and thus on the pore volume of the soil.

Permeability and permeability coefficient quantify the flow rate through a permeable medium in a similar way as a function of the pressure difference, only their units are different: ${\ displaystyle Q}$${\ displaystyle \ Delta p}$

• for permeability it is an area (m²)
• for the permeability coefficient it is a speed (m / s).

Both quantities can be direction-dependent and are then represented as tensors .

In addition, both quantities are constant over the flow rate , provided the following conditions are met: ${\ displaystyle Q}$

1. Laminar flow
2. No interaction between rock surface and flowing medium
3. Only one phase in the pore space at one hundred percent saturation with this phase.

## determination

### Metrological

Determination of liquid permeabilities

In rock physics laboratories, the permeability is routinely determined on cylindrical samples with a diameter of 30 mm and a length of 40 to 80 mm; in the United States, 1 "  ×  1 12  " specimens are common for routine measurements  . For investigations where a large pore volume is important (for example relative permeability), sample diameters of 40 mm are also common. The orientation of the samples is as standard parallel to the layering .

A special case is the determination of the permeability anisotropy on cubes with an edge length of 30 or 40 mm. These are to be worked out from the core material in such a way that two surfaces are oriented parallel to the layering and thus data can be determined parallel and perpendicular to the layering on a sample.

The outlined measuring arrangement determines for water: ${\ displaystyle k_ {f}}$

${\ displaystyle k _ {\ mathrm {f}} = {\ frac {Q \ cdot l} {A \ cdot (h_ {1} -h_ {2})}}}$     (Symbols from sketch and as in the above formulas)

${\ displaystyle K}$can then be calculated using the dynamic viscosity of the water and its density : ${\ displaystyle \ eta}$${\ displaystyle \ rho}$

${\ displaystyle K = {\ frac {k _ {\ mathrm {f}} \ cdot \ eta} {\ rho \ cdot g}}}$

If the permeability coefficient for the medium through which water flows is determined experimentally, the above-mentioned relationship can be used to determine the permeability coefficient of this medium for other fluids, e.g. B. for crude oil, by inserting its density and dynamic viscosity, calculate: ${\ displaystyle k_ {f}}$

{\ displaystyle {\ begin {aligned} K \ cdot g & = {\ frac {k _ {\ mathrm {f}} \ cdot \ eta} {\ rho}} = {\ text {const.}} \\\ Rightarrow k_ {\ mathrm {f2}} & = k _ {\ mathrm {f1}} \ cdot {\ frac {\ rho _ {2} \ cdot \ eta _ {1}} {\ rho _ {1} \ cdot \ eta _ {2}}} \ end {aligned}}}

or using the kinematic viscosity : ${\ displaystyle \ nu}$

${\ displaystyle \ nu = {\ frac {\ eta} {\ rho}}}$
${\ displaystyle \ Rightarrow k _ {\ mathrm {f2}} = k _ {\ mathrm {f1}} \ cdot {\ frac {\ nu _ {1}} {\ nu _ {2}}}}$

### Mathematically from the particle size distribution curve

For soils, it is possible to estimate the permeability coefficient for water in m / s from the grain distribution curve (Hazen, 1893) : ${\ displaystyle k_ {f}}$

${\ displaystyle k _ {\ mathrm {f}} \ approx 0 {,} 0116 \ cdot {d _ {\ mathrm {w}}} ^ {2} \ approx 0 {,} 0116 \ cdot {d_ {10}} ^ {2}}$

Here mean:

• ${\ displaystyle d_ {w}}$: Effective grain diameter in mm
• ${\ displaystyle d_ {10}}$: Grain diameter for the weight fraction m = 10% of the grain distribution curve.

This estimate is only valid provided that the degree of non-uniformity is (uniform soil). ${\ displaystyle C_ {u} = d_ {60} / d_ {10} <5}$

According to Beyer is

${\ displaystyle k _ {\ mathrm {f}} \ approx C \ cdot d_ {10} ^ {2}}$

${\ displaystyle C}$is a coefficient that depends on the degree of non-uniformity: For certain is , so that the formula agrees with Hazen's. ${\ displaystyle C = f (C_ {u}).}$${\ displaystyle C_ {u}}$${\ displaystyle C = 0 {,} 0116}$

## application

These material parameters are applied when soil or rock are traversed by liquids or gases: groundwater flows, drinking water , extraction of oil or natural gas, calculations of water huge intake of buildings and tunnels , determining the tightness of dams and levees , even in contaminated soils and the Injection of carbon dioxide .

### Extraction of oil and natural gas

The production rate is an important factor influencing the profitability of the production of crude oil and natural gas. She hangs u. a. on the permeability of the geological formations from which these raw materials are extracted. However, since the world market price also decides on profitability, no permanent limit values ​​can be specified here.

### Planning of sealing and compensation injections

Knowledge of the hydraulic properties is important when planning injections for sealing and / or improving the mechanical properties of loose rock.

## Transmissibility and transmissivity

The transmissibility is defined as the product of permeability and the thickness of the water-bearing soil or rock layer ( aquifer ): ${\ displaystyle T _ {\ mathrm {K}}}$${\ displaystyle K}$ ${\ displaystyle M}$

${\ displaystyle T _ {\ mathrm {K}} = K \ cdot M}$

Similarly, the transmissivity is defined as the product of the permeability coefficient and thickness: ${\ displaystyle T _ {\ mathrm {k}}}$${\ displaystyle k_ {f}}$

${\ displaystyle T _ {\ mathrm {k}} = k _ {\ mathrm {f}} \ cdot M}$

If the aquifer consists of i layers with different permeability and thickness, then the respective products are added:

${\ displaystyle T_ {k} = k _ {\ mathrm {f1}} \ cdot M_ {1} + k _ {\ mathrm {f2}} \ cdot M_ {2} + k _ {\ mathrm {f3}} \ cdot M_ { 3} + \ ldots + k _ {\ mathrm {fi}} \ cdot M _ {\ mathrm {i}}}$

The purpose of the two quantities becomes clear from the last formula: they represent the integrals of the respective permeability values ( or ) over the aquifer thickness. This takes into account that the permeability is usually not the same over the entire height of the aquifer: the aquifer is inhomogeneous in terms of it its permeability. ${\ displaystyle K}$${\ displaystyle k_ {f}}$

Because of the common factor, there is the same relationship between the two quantities as between and : ${\ displaystyle M}$${\ displaystyle k_ {f}}$${\ displaystyle K}$

${\ displaystyle {\ frac {T _ {\ mathrm {k}}} {T _ {\ mathrm {K}}}} = {\ frac {k _ {\ mathrm {f}}} {K}} = {\ frac { \ rho \ cdot g} {\ eta}}}$

The more common of the two variables is, based on DIN, transmissivity, as it plays an important role in the extraction of groundwater as drinking water .

The transmissivity is usually determined via a pumping test. However, you only get the overall transmissivity and no information about the above. Inhomogeneities with regard to the permeability of the aquifer.

Remarks:

• The transmissibility is also used in ophthalmic optics as permeability measurement for contact lenses used. Here it indicates how much oxygen (in cm³ / s) flows through an area of ​​1 cm² of a membrane at 1 mmHg pressure on both sides. See: permeability (solids) , Barrer .
• The degree of transmission used in physics is often (scientifically and casually) referred to as transmissivity .