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DERIVATION THROUGH DARCY'S LAW

Hydraulic conductivity is the proportionality constant in Darcy's Law , which relates the amount of water which will flow through a unit cross-sectional area of Aquifer under a unit gradient of Hydraulic Head . It is analogous to the thermal conductivity of materials in heat conduction, or 1/resistivity in electrical circuits. The hydraulic conductivity (''K'' — the English Letter "kay") is specific to the flow of a certain fluid (typically water, sometimes oil or air); intrinsic permeability (''κ'' — the Greek Letter "kappa") is a parameter of a porous media which is independent of the fluid. This means that, for example, ''K'' will go up if the water in a porous medium is heated (reducing the viscosity of the water), but ''κ'' will remain constant. The two are related through the following equation

:K = rac{\kappa \gamma}{\mu}
where
:K is the hydraulic conductivity or m s-1 ;
:\kappa is the Intrinsic Permeability of the material or m2 ;
:\gamma is the Specific Weight of water or N m-3 , and;
:\mu is the Dynamic Viscosity of water or kg m-1 s-1 .


ESTIMATION OF HYDRAULIC CONDUCTIVITY


Direct estimation

Hydraulic conductivity can be measured by applying Darcy's Law on the material. Such experiments can be conducted by creating a hydraulic gradient between two points, and measuring the flow rate (Oosterbaan and Nijland R.J.Oosterbaan and H.J.Nijland, 1994, Determination of the Saturated Hydraulic Conductivity. In: H.P.Ritzema (ed.) Drainage Principles and Applications, ILRI Publication 16, p.435-476. International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands. ISBN 90 70754 3 39.
Free download from the Articles page of waterlog.info .).


Empirical estimation

Shepherd1 derived an Empirical formula for approximating hydraulic conductivity from grain size analyses:
:K = a (D_{10})^b
where
:a and b are empirically derived terms based on the soil type, and
:D_{10} is the Diameter of the 10 Percentile grain size of the material
Note: Shepherd's Figure 3 clearly shows the use of d_{50}, not d_{10}, measured in mm. Therefore the equation should be K = a (d_{10})^b. His figure shows different lines for materials of different types, based on analysis of data from others with d_{50} up to 10 mm.


Pedotransfer function

A Pedotransfer Function (PTF) is a specialized empirical estimation method, used primarily in the Soil Science s, however has increasing use in hydrogeology2. 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 soil Particle Size , and Bulk Density .


TRANSMISSIVITY

The transmissivity, T, of an Aquifer is a measure of how much water can be transmitted horizontally, such as to a pumping well:
: T = K_s \, b
Transmissivity is directly proportional to the aquifer thickness. For a confined aquifer, this remains constant, as the saturated thickness remains constant. The aquifer thickness of an unconfined aquifer is from the base of the aquifer (or the top of the Aquitard ) to the Water Table . The water table can fluctuate, which changes the transmissivity of the unconfined aquifer. This may provide Positive Feedback of a pumping well that is pumping more than can be provided by the aquifer, where the transmissivity drops as the well pumps, thus eventually reducing the aquifer to the height of the pumping well screen.

Transmissivity should not be confused with similar word Transmittance (used in Optics ),
which means fraction of incident light that passes through a sample.


RELATIVE PROPERTIES

Because of their high porosity and permeability, Sand and Gravel Aquifer s have higher hydraulic conductivity than Clay or unfractured Granite aquifer. Sand or gravel aquifers would thus be easier to extract water from (e.g., using a pumping Well ) 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/ft&2) ); 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) for ''K'' values.

Hydraulic conductivity (''K'') is the most complex and important of the hydrogeologic aquifer properties; values found in nature:
  • range over many Orders Of Magnitude (the distribution is often considered to be Lognormal ),

  • vary a large amount through space (sometimes considered to be Random ly spatially distributed, or Stochastic in nature),

  • are directional (in general ''K'' is a symmetric second-rank Tensor ; e.g., vertical ''K'' values can be several orders of magnitude smaller than horizontal ''K'' 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 field pumping tests, laboratory column flow tests or inverse computer simulation, (sometimes also from Grain Size analyses), and

  • are very dependent (in a Non-linear way) on the water content, which makes solving the Unsaturated Flow equation difficult. In fact, the variably saturated ''K'' for a single material varies over a wider range than the saturated ''K'' values for all types of materials (see chart below for an illustrative range of the latter).



RANGES OF VALUES FOR NATURAL MATERIALS

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

Values are for typical fresh Groundwater conditions — using standard values of Viscosity and Specific Gravity for water at 20°C and 1 atm.
See the similar table derived from the same source for Intrinsic Permeability values.3

Source: modified from Bear, 1972


SEE ALSO



REFERENCES