Inscience andengineering,hydraulic conductivity (K, inSI units ofmeters per second), is a property ofporous materials,soils androcks, that describes the ease with which afluid (usually water) can move through thepore space, or fracture network.^[1] It depends on theintrinsic permeability (k, unit: m²) of the material, the degree ofsaturation, and on thedensity andviscosity of the fluid. Saturated hydraulic conductivity,K_sat, describes water movement through saturated media.By definition, hydraulic conductivity is the ratio of volume flux tohydraulic gradient yielding a quantitative measure of a saturated soil's ability to transmit water when subjected to a hydraulic gradient.

There are two broad approaches for determining hydraulic conductivity:

• In theempirical approach the hydraulic conductivity is correlated to soil properties likepore-size andparticle-size (grain-size) distributions, andsoil texture.
• In theexperimental approach the hydraulic conductivity is determined from hydraulic experiments that are interpreted usingDarcy's law.

The experimental approach is broadly classified into:

• Laboratory tests using soil samples subjected to hydraulicexperiments
• Field tests (on site, in situ) that are differentiated into:
  • small-scale field tests, using observations of the water level in cavities in the soil
  • large-scale field tests, likepumping tests inwells or by observing the functioning of existing horizontaldrainage systems.

The small-scale field tests are further subdivided into:

• infiltration tests in cavitiesabove thewater table
• slug tests in cavitiesbelow thewater table

The methods of determining hydraulic conductivity and other hydraulic properties are investigated by numerous researchers and include additional empirical approaches.^[2]

Allen Hazen derived anempirical formula for approximating hydraulic conductivity from grain-size analyses:

${\displaystyle K=C(D_{10}{)}^2}$

where

${\displaystyle C}$ Hazen's empirical coefficient, which takes a value between 0.0 and 1.5 (depending on literature), with an average value of 1.0. A.F. Salarashayeri & M. Siosemarde indicate C is usually between 1.0 and 1.5, with D in mm and K in cm/s.^{[citation needed]}

${\displaystyle D_{10}}$ is thediameter of the 10percentile grain size of the material.

Apedotransfer function (PTF) is a specialized empirical estimation method, used primarily in thesoil sciences, but increasingly used in hydrogeology.^[3] There are many different PTF methods, however, they all attempt to determine soil properties, such as hydraulic conductivity, given several measured soil properties, such as soilparticle size, andbulk density.

There are relatively simple and inexpensive laboratory tests that may be run to determine the hydraulic conductivity of a soil: constant-head method and falling-head method.

Theconstant-head method is typically used on granular soil. This procedure allows water to move through the soil under a steady state head condition while the volume of water flowing through the soil specimen is measured over a period of time. By knowing the volumeΔV of water measured in a timeΔt, over a specimen of lengthL and cross-sectional areaA, as well as the headh, the hydraulic conductivity (K) can be derived by simply rearrangingDarcy's law:

${\displaystyle K=\frac{\mathrm{\Delta }V}{\mathrm{\Delta }t}\frac{L}{Ah}}$

Proof:Darcy's law states that the volumetric flow depends on thepressure differentialΔP between the two sides of the sample, thepermeabilityk and thedynamic viscosityμ as:^[4]

${\displaystyle \frac{\mathrm{\Delta }V}{\mathrm{\Delta }t}=-\frac{kA}{\mu L}\mathrm{\Delta }P}$

In a constant head experiment, the head (difference between two heights) defines an excess water mass,ρAh, whereρ is the density of water. This mass weighs down on the side it is on, creating a pressure differential ofΔP =ρgh, whereg is the gravitational acceleration.Plugging this directly into the above gives

${\displaystyle \frac{\mathrm{\Delta }V}{\mathrm{\Delta }t}=-\frac{k\rho gA}{\mu L}h}$

If the hydraulic conductivity is defined to be related to the hydraulic permeability as

${\displaystyle K=\frac{k\rho g}{\mu },}$

this gives the result.

In the falling-head method, the soil sample is first saturated under a specific head condition. The water is then allowed to flow through the soil without adding any water, so the pressure head declines as water passes through the specimen. The advantage to the falling-head method is that it can be used for both fine-grained and coarse-grained soils..^[5] If the head drops fromh_i toh_f in a timeΔt, then the hydraulic conductivity is equal to

${\displaystyle K=\frac{L}{\mathrm{\Delta }t}\ln \frac{h_f}{h_i}}$

Proof: As above, Darcy's law reads

${\displaystyle \frac{\mathrm{\Delta }V}{\mathrm{\Delta }t}=-K\frac{A}{L}h}$

The decrease in volume is related to the falling head byΔV = ΔhA.Plugging this relationship into the above, and taking the limit asΔt → 0, the differential equation

${\displaystyle \frac{dh}{dt}=-\frac{K}{L}h}$

has the solution

${\displaystyle h(t)=h_i{e}^{-\frac{K}{L}(t-t_i)}.}$

Plugging in${\displaystyle h(t_f)=h_f}$ and rearranging gives the result.

In compare to laboratory method, field methods gives the most reliable information about the permeability of soil with minimum disturbances. In laboratory methods, the degree of disturbances affect the reliability of value of permeability of the soil.

Pumping test is the most reliable method to calculate the coefficient of permeability of a soil. This test is further classified into Pumping in test and pumping out test.

There are also in-situ methods for measuring the hydraulic conductivity in the field.
When the water table is shallow, the augerhole method, aslug test, can be used for determining the hydraulic conductivity below the water table.
The method was developed by Hooghoudt (1934)^[6] in The Netherlands and introduced in the US by Van Bavel en Kirkham (1948).^[7]
The method uses the following steps:

1. an augerhole is perforated into the soil to below the water table
2. water is bailed out from the augerhole
3. the rate of rise of the water level in the hole is recorded
4. theK-value is calculated from the data as:^[8]

1. ${\displaystyle K=F\frac{H_o-H_t}{t}}$

where:

• K is the horizontal saturated hydraulic conductivity (m/day)
• H is the depth of the water level in the hole relative to the water table in the soil (cm):
  • H_t =H at timet
  • H_o =H at timet = 0
• t is the time (in seconds) since the first measurement ofH asH_o
• F is a factor depending on the geometry of the hole:

  ${\displaystyle F=\frac{4000r}{h^{\prime }}\left(20+\frac{D}{r}\right)\left(2-\frac{h^{\prime }}{D}\right)}$

where:

• r is the radius of the cylindrical hole (cm)
• h' is the average depth of the water level in the hole relative to the water table in the soil (cm), found as${\displaystyle h^{\prime }=\frac{H_o+H_t}{2}}$
• D is the depth of the bottom of the hole relative to the water table in the soil (cm).

The picture shows a large variation ofK-values measured with the augerhole method in an area of 100 ha.^[9] The ratio between the highest and lowest values is 25. The cumulative frequency distribution islognormal

The transmissivity is a measure of how much water can be transmitted horizontally, such as to a pumping well.

Transmissivity should not be confused with the similar wordtransmittance used inoptics, meaningthe fraction of incident light that passes through a sample.

Anaquifer may consist ofn soil layers. The transmissivityT_i of a horizontal flow for theith soil layer with asaturated thicknessd_i and horizontal hydraulic conductivityK_i is:

${\displaystyle T_i=K_i{d}_i}$

Transmissivity is directly proportional to horizontal hydraulic conductivityK_i and thicknessd_i. ExpressingK_i in m/day andd_i in m, the transmissivityT_i is found in units m²/day.
The total transmissivityT_t of the aquifer is the sum of every layer's transmissivity:^[8]

${\displaystyle T_t=\sum T_i}$

Theapparent horizontal hydraulic conductivityK_A of the aquifer is:

${\displaystyle K_A=\frac{T_t}{D_t}}$

whereD_t, the total thickness of the aquifer, is the sum of each layer's individual thickness:$D_t=\sum d_i.$

The transmissivity of an aquifer can be determined frompumping tests.^[10]

Influence of the water table
When a soil layer is above thewater table, it is not saturated and does not contribute to the transmissivity. When the soil layer is entirely below the water table, its saturated thickness corresponds to the thickness of the soil layer itself. When the water table is inside a soil layer, the saturated thickness corresponds to the distance of the water table to the bottom of the layer. As the water table may behave dynamically, this thickness may change from place to place or from time to time, so that the transmissivity may vary accordingly.
In a semi-confined aquifer, the water table is found within a soil layer with a negligibly small transmissivity, so that changes of the total transmissivity (D_t) resulting from changes in the level of the water table are negligibly small.
When pumping water from an unconfined aquifer, where the water table is inside a soil layer with a significant transmissivity, the water table may be drawn down whereby the transmissivity reduces and the flow of water to the well diminishes.

Theresistance to vertical flow (R_i) of theith soil layer with asaturated thicknessd_i and vertical hydraulic conductivityK_vi is:

${\displaystyle R_i=\frac{d_i}{K_{v_i}}}$

ExpressingK_vi in m/day andd_i in m, the resistance (R_i) is expressed in days.
The total resistance (R_t) of the aquifer is the sum of each layer's resistance:^[8]

${\displaystyle R_t=\sum R_i=\sum \frac{d_i}{K_{v_i}}}$

Theapparent vertical hydraulic conductivity (K_vA) of the aquifer is:

${\displaystyle K_{v_A}=\frac{D_t}{R_t}}$

whereD_t is the total thickness of the aquifer:$D_t=\sum d_i.$

The resistance plays a role inaquifers where a sequence of layers occurs with varying horizontal permeability so that horizontal flow is found mainly in the layers with high horizontal permeability while the layers with low horizontal permeability transmit the water mainly in a vertical sense.

When the horizontal and vertical hydraulic conductivity ($K_{h_i}$ and$K_{v_i}$) of the$i\text{-th}$ soil layer differ considerably, the layer is said to beanisotropic with respect to hydraulic conductivity.
When theapparent horizontal and vertical hydraulic conductivity ($K_{h_A}$ and$K_{v_A}$) differ considerably, theaquifer is said to beanisotropic with respect to hydraulic conductivity.
An aquifer is calledsemi-confined when a saturated layer with a relatively small horizontal hydraulic conductivity (the semi-confining layer oraquitard) overlies a layer with a relatively high horizontal hydraulic conductivity so that the flow of groundwater in the first layer is mainly vertical and in the second layer mainly horizontal.
The resistance of a semi-confining top layer of an aquifer can be determined frompumping tests.^[10]
When calculating flow todrains^[11] or to awell field^[12] in an aquifer with the aim tocontrol the water table, the anisotropy is to be taken into account, otherwise the result may be erroneous.

Because of their high porosity and permeability,sand andgravelaquifers have higher hydraulic conductivity thanclay or unfracturedgranite aquifers. Sand or gravel aquifers would thus be easier to extract water from (e.g., using a pumpingwell) because of their high transmissivity, compared to clay or unfractured bedrock aquifers.

Hydraulic conductivity has units with dimensions of length per time (e.g., m/s, ft/day and (gal/day)/ft² ); transmissivity then has units with dimensions of length squared per time. The following table gives some typical ranges (illustrating the many orders of magnitude which are likely) forK values.

Hydraulic conductivity (K) is one of the most complex and important of the properties of aquifers in hydrogeology as the values found in nature:

• range over manyorders of magnitude (the distribution is often considered to belognormal),
• vary a large amount through space (sometimes considered to berandomly spatially distributed, orstochastic in nature),
• are directional (in generalK is a symmetric second-ranktensor; e.g., verticalK values can be several orders of magnitude smaller than horizontalK values),
• are scale dependent (testing a m³ of aquifer will generally produce different results than a similar test on only a cm³ sample of the same aquifer),
• must be determined indirectly through fieldpumping tests, laboratory column flow tests or inverse computer simulation, (sometimes also fromgrain size analyses), and
• are very dependent (in anon-linear way) on the water content, which makes solving theunsaturated flow equation difficult. In fact, the variably saturatedK for a single material varies over a wider range than the saturatedK values for all types of materials (see chart below for an illustrative range of the latter).

Table of saturated hydraulic conductivity (K) values found in nature

Values are for typical freshgroundwater conditions — using standard values ofviscosity andspecific gravity for water at 20 °C and 1 atm.See the similar table derived from the same source forintrinsic permeability values.^[13]

Source: modified from Bear, 1972

Hydraulic conductivity at Liquid Limit for several Clays^[14][15]

Soil Type	Liquid Limit, LL (%)	Void Ratio at Liquid Limit,${\displaystyle e_L}$ (%)	Hydraulic conductivity,${\displaystyle {10}^{-7}}$ cm/s
Bentonite	330	9.24	1,28
Bentonite${\displaystyle +}$ sand	215	5,91	2,65
Natural marine soil	106	2,798	2,56
Air-dried marine soil	84	2,234	2,42
Open-dried marine soil	60	1,644	2,63
Brown soil	62	1,674	2,83

• Aquifer test
• Hydraulic analogy
• Pedotransfer function – for estimating hydraulic conductivities given soil properties

• Hydraulic conductivity calculator

• Hydrology
• Hydraulic engineering
• Soil mechanics
• Soil physics
• Physical quantities

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