How To Calculate Hydraulic Conductivity

Hydraulic conductivity is the ease with which water moves through porous spaces and fractures in soil or rock. It is subject to a hydraulic gradient and affected by saturation level and permeability of the material. Hydraulic conductivity is generally determined either through one of two approaches. An empirical approach correlates hydraulic conductivity to soil properties. A second approach calculates hydraulic conductivity through experimentation.

1. Calculate Conductivity

Calculate hydraulic conductivity empirically by selecting a method based on grain-size distribution through the material. Each method is derived from a general equation. The general equation is:

\(K=\frac{g}{v}Cf_nd_e^2\)

Where K = hydraulic conductivity; g = acceleration due to gravity; v = kinematic viscosity; C = sorting coefficient; ƒ_n = porosity function; and d_e = effective grain diameter. The kinematic viscosity (v) is determined by the dynamic viscosity (µ) and the fluid (water) density (ρ) as:

\(v=\frac{\mu}{\rho}\)

The values of C, ƒ and d depend on the method used in the grain-size analysis. Porosity (n) is derived from the empirical relationship n=0.255 x (1+0.83^U) where the coefficient of grain uniformity (U) is given by U=d₆₀/d₁₀. In the sample, d₆₀ represents the grain diameter (mm) in which 60 percent of the sample is more fine and d₁₀ represents the grain diameter (mm) for which 10 percent of the sample is more fine.

This general equation is the basis for different empirical formulas.

2. Apply Kozeny-Carman Equation

Use the Kozeny-Carman equation for most soil textures. This is the most widely accepted and used empirical derivative based on soil grain size but is not appropriate to use for soils with an effective grain size above 3-mm or for clay textured soils:

\(K=\frac{g}{v}8.3\times 10^{-3}\frac{n^3}{(1-n)^2}d_{10}^2\)

3. Apply Hazen Equation

Use the Hazen equation for soil textures from fine sand to gravel if the soil has a uniformity coefficient less than five (U<5) and effective grain size between 0.1 mm and 3 mm. This formula is based only on the d₁₀ particle size so it is less accurate than the Kozeny-Carman formula:

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

4. Apply Breyer Equation

Use the Breyer equation for materials with a heterogeneous distribution and poorly sorted grains with a uniformity coefficient between 1 and 20 (1

\(K=\frac{g}{v}6\times 10^{-4}\log{\frac{500}{U}}d_{10}^2\)

5. Apply USBR Equation

Use the U.S. Bureau of Reclamation (USBR) equation for medium-grain sand with a uniformity coefficient less than five (U<5). This calculates using an effective grain size of d_20 and does not depend on porosity, so it is less accurate than other formulas:

\(K=\frac{g}{v}4.8\times 10^{-4}d_{20}^3d_{20}^2\)

1. Apply Darcy's Law

Use an equation based on Darcy's Law to derive hydraulic conductivity experimentally. In the lab, place a soil sample in a small cylindrical container to create a one-dimensional soil cross-section through which the liquid (usually water) flows. This method is either a constant-head test or a falling-head test depending on the flow state of the liquid. Coarse-grained soils such as clean sands and gravels typically use constant-head tests. Finer grain samples use falling-head tests. The basis for these calculations is Darcy's Law:

\(U=-K\frac{dh}{dz}\)

Where U = average velocity of fluid through a geometric cross-sectional area within the soil; h= hydraulic head; z= vertical distance in the soil; K= hydraulic conductivity. The dimension of K is length per unit of time (I/T).

2. Carry out Constant-Head Test

Use a permeameter to conduct a Constant-Head Test, the most commonly used test to determine the saturated hydraulic conductivity of coarse-grained soils in the laboratory. Subject a cylindrical soil sample of cross-sectional area A and length L is to a constant head (H2 – H1) flow. The volume (V) of the test fluid that flows through the system during time (t), determines the saturated hydraulic conductivity K of the soil:

\(K=\frac{VL}{At(H_2-H_1)}\)

For best results, test several times using different head differences.

3. Use Falling-head Test

Use the Falling-head test to determine the K of fine-grained soils in the laboratory. Connect a cylindrical soil sample column of cross-sectional area (A) and length (L) to a standpipe of cross-sectional area (a), in which the percolating fluid flows into the system. Measure the change in head in the standpipe (H1 to H2) at intervals of time (t) to determine the saturated hydraulic conductivity from Darcy's Law:

\(K=\frac{aL}{At}\ln{\frac{H_1}{H_2}}\)