HydraulicConductivity

Hydraulic conductivity is linked to permeability by:

$$ K=\frac{\kappa}{\mu}\cdot \rho g $$

Hydraulic conductivity \(K\), can be estimated from empirical formulas ( 7 , 12) as function of porosity and effective grain diameter. In a general form:

$$ K=\frac{g}{\nu} \cdot \beta \cdot \theta(n) \cdot d_{e}^{2} $$

where:

• \(n\) is the porosity
• \(d_{e}\) effective grain diameter usually defined for each formula
• \(\nu\) is the water kinematic viscosity at \(10 C\)

\(\nu=1.307\cdot10^{-6}\frac{m^2}{s} \qquad \nu=\frac{\mu}{\rho}\)

It is possible to find the value at the following link Kaye and Laby Physical properties of sea water the value at \(20 C\) is: \( 1.05 \cdot 10^{-6} m^2 s^{-1}\)

• \(U=d_{60}/d_{10}\) Uniformity coefficent
• \(g\) gravity acceleration

Summary Table

Author	\(d_e\)	Applicability
Beyer	\(d_{10}\)	\(0.06mm < d_{e} < 0.6mm\)
Hazen	\(d_{10}\)	\(0.1mm < d_{e} < 3mm\)
Kozeny	\(d_{10} \)	\(0.06 < d_{e} < 3mm\)
Sauerbrei	\(d_{17}\)	\( d_{e} < 0.5mm \)
Terzaghi	\(d_{10}\)	large grain sand

Hazen

Hazen formula was originally developed for determination of hydraulic conductivity of uniformly graded sand but is also useful for fine sand to gravel range, provided the sediment has a uniformity coefficient less than 5 and effective grain size between 0.1 and 3mm.

$$ K=\frac{g}{\nu}\cdot 6 \cdot 10^{-4}\cdot(1+10\cdot(n-0.26))\cdot d_{10}^{2} $$

Kozeny-Carman:

The Kozeny-Carman equation is one of the most widely accepted and used derivations of permeability as a function of the characteristics of the soil medium. This equation was originally proposed by Kozeny (1927) and was then modified by to become the Kozeny-Carman equation .

$$ K=\frac{g}{\nu}\cdot 8.3 \cdot 10^{-3}\cdot \left( \frac{n^3}{ (1-n)^2 } \right) \cdot d_{10}^{2} $$

It is not appropriate for either soil with effective size \( d_{10} > 3mm \) and for clayey soils. In the above formula all the dimension are in the SI and the result is in m/s

Beyer

This method does not consider porosity and therefore, porosity function takes on value 1. Breyer formula is often considered most useful for materials with heterogeneous distributions and poorly sorted grains with uniformity coefficient between 1 and 20. For consistent units

$$ K=\frac{g}{\nu}\cdot 6 \cdot 10^{-4}\cdot \log{ \frac{500}{U} }\cdot d_{10}^{2} $$

where

$$ \beta= 6 \cdot 10^{-4} \cdot \log{ \frac{500}{U} } $$

validity for:

• effective grain size between \(0.06mm < d_{10} < 0.6mm\)
• uniformity coefficient \( 1< U < 20\)

Sauerbrei

For fine sand and sandy clay Sauerbrei introduced the following formula:

$$ K=\frac{g}{\nu}\cdot \beta_z \cdot \tau \left( \frac{n^3}{ (1-n)^2 } \right) \cdot d_{17}^{2} $$

where:

• \(\beta_z = 3.75 \cdot 10^{-3}\)
• \(\tau\) is a temperature correction factor that can be linearly interpolated

Temp.	\(\tau\)
\( 0 C\)	\( 0.588 \)
\( 60 C\)	\( 2.231 \)

Terzaghi

Terzaghi formula is most applicable for large-grain sand

$$ K=\beta_0 \frac{\mu_{10°C}}{\mu_{t}} \cdot \left( \frac{n-0.13}{\sqrt[3]{1-n}} \right)^2 \cdot d_{10}^{2} $$

where the

• \( \mu_{t} \) the dynamic viscosity of the fluid
• \(d_{10} \) effective grain diameter expressed in "cm"
• \(\beta_0\) is a function of the grain size and shape for consistent unit it can range

Sediment type	values of \(\beta_0\)
Sea Sand	\( 750\div 663 \)
Dune Sand	\( 800 \)
Pure RIver Sand	\( 696\div 460 \)
Muddy river sand	\( 203 \)

For consistent units the formula can be expressed as:

$$ K=\frac{g}{\nu}\cdot \beta_{T} \left( \frac{n-0.13}{\sqrt[3]{1-n}} \right)^2 \cdot d_{10}^{2} $$

where the parameter \(\beta_{T}\) can range from:

	values of \(\beta_{T}\)
smooth grains	\(10.7\cdot 10^{-3}\)
coarse grain	\( 6.1 \cdot 10^{-3}\)

Reference

[7] Justine Odong, Evaluation of Empirical Formulae for Determination of Hydraulic Conductivity based on Grain-Size Analysis, Journal of American Science, 3(3), 2007,

[12] Michael Kasenow, Determination of hydraulic conductivity from grain size analysis, water resource pubblication

-- RobertoBernetti - 21 Feb 2010