Hydraulic conductivity

Hydraulic conductivity, symbolically represented as K, is a property of vascular plants, soil or rock, that describes the ease with which water can move through pore spaces or fractures. It depends on the intrinsic permeability of the material and on the degree of saturation. Saturated hydraulic conductivity, K_sat, describes water movement through saturated media. One application of it is the Starling equation, which calculates flow across walls of capillaries.

Contents 1 Estimation of hydraulic conductivity 1.1 Direct estimation 1.2 Empirical estimation 1.3 Pedotransfer function 2 Experimental approach 2.1 Constant-head method 2.2 Falling-head method 2.3 Augerhole method 3 Transmissivity 4 Resistance 5 Anisotropy 6 Relative properties 7 Ranges of values for natural materials 8 See also 9 References

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 ^[1]) .

Empirical estimation

Shepherd^[2] derived an empirical formula for approximating hydraulic conductivity from grain size analyses:

K = a(D₁₀)^b

where

a and b are empirically derived terms based on the soil type, and

D₁₀ is the diameter of the 10 percentile grain size of the material

Note: Shepherd's Figure 3 clearly shows the use of d₅₀, not d₁₀, measured in mm. Therefore the equation should be K = a(d₁₀)^b. His figure shows different lines for materials of different types, based on analysis of data from others with d₅₀ up to 10 mm.

Pedotransfer function

A pedotransfer function (PTF) is a specialized empirical estimation method, used primarily in the soil sciences, however has increasing use 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 soil particle size, and bulk density.

Experimental approach

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.

Constant-head method

The constant-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 quantity (volume) of water flowing through the soil specimen is measured over a period of time. By knowing the quantity Q of water measured, length L of specimen, cross-sectional area A of the specimen, time t required for the quantity of water Q to be discharged, and head h, the hydraulic conductivity can be calculated:

where v is the flow velocity. Using Darcy's Law:

and expressing the hydraulic gradient i as:

where h is the difference of hydraulic head over distance L, yields:

Solving for K gives:

Falling-head method

The falling-head method is very similar to the constant head methods in its initial setup; however, the advantage to the falling-head method is that can be used for both fine-grained and coarse-grained soils. The soil sample is first saturated under a specific head condition. The water is then allowed to flow through the soil without maintaining a constant pressure head^[4].

Augerhole method

There are also in-situ methods for measuring the hydraulic conductivity in the field.
When the water table is shallow, the augerhole method can be used for determining the hydraulic conductivity below the water table.
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. the K-value is calculated from the data thus obtained using an appropriate equation

The picture shows a large variation of K-values measured with the augerhole method in an area of 100 ha ^[5] . The ratio between the highest and lowest values is 25. The cumulative frequency distribution is lognormal and was made with the CumFreq program.

Transmissivity

The transmissivity, T, of an aquifer is a measure of how much water can be transmitted horizontally, such as to a pumping well:

Transmissivity is directly proportional to average horizontal permeability (K_h) and aquifer thickness (D). 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.

Resistance

The resistance R of a soil layer with thickness d to vertical flow is ^[1] :

The average vertical permeability K_v of a layered aquifer with thickness D and total resistance R_t of the layers is:

Anisotropy

When the average permability K_h for horizontal flow in an aquifer and the average vertical permeability K_v differ, the permeability is anisotropic and the aquifer may be semi-confined.
When calculating flow to drains ^[6] or to wells ^[7] in an aquifer, the anisotropy is to be taken into account.

Relative properties

Because of their high porosity and permeability, sand and gravel aquifers have higher hydraulic conductivity than clay or unfractured granite aquifers. 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)/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) for K 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 many orders of magnitude (the distribution is often considered to be lognormal),
• vary a large amount through space (sometimes considered to be randomly 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.^[8]

K (cm/s)

10²

101

100=1

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

10−9

10−10

K (ft/day)

105

10,000

1,000

100

10

1

0.1

0.01

0.001

0.0001

10−5

10−6

10−7

Relative Permeability

Pervious

Semi-Pervious

Impervious

Aquifer

Good

Poor

None

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

Source: modified from Bear, 1972

See also

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

References

1. ^ ^{a b} 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: [1] or directly at [2].
2. ^ Shepherd, Russell G. (1989). "Correlations of permeability and grain-size". Ground Water 27 (5): 633–638. doi:10.1111/j.1745-6584.1989.tb00476.x. 
3. ^ Wösten, J.H.M., Pachepsky, Y.A., and Rawls, W.J. (2001). "Pedotransfer functions: bridging the gap between available basic soil data and missing soil hydraulic characteristics". Journal of Hydrology 251: 123–150. doi:10.1016/S0022-1694(01)00464-4. 
4. ^ Liu, Cheng "Soils and Foundations." Upper Saddle River, New Jersey: Prentice Hall, 2001 ISBN 0-13-025517-3
5. ^ R.J.Oosterbaan, 2002. Drainage research in farmer's fields: analysis of data. On line: [3]
6. ^ R.J.Oosterbaan, 1997, The energy balance of groundwater flow applied to subsurface drainage in anisotropic soils by pipes or ditches with entrance resistance. On line: [4]. The corresponding free EnDrain program can be downloaded from: [5] .
7. ^ R.J.Oosterbaan, 2002, Subsurface drainage by (tube)wells, 9 pp. On line: [6]. The correspondig free WellDrain program can be downloaded from: [7] .
8. ^ Bear, J. (1972). Dynamics of Fluids in Porous Media. Dover Publications. ISBN 0-486-65675-6. 

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