# How to Calculate Hydraulic Conductivity ••• daniele2dm/iStock/GettyImages

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.

### The Empirical Approach

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; ƒn = porosity function; and de = 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+0.83U) 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.

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

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

Use the Breyer equation for materials with a heterogeneous distribution and poorly sorted grains with a uniformity coefficient between 1 and 20 (1<U<20) and an effective grain size between 0.06 mm and 0.6 mm:

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

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

### Experimental Methods - Laboratory

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).

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.

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}}

#### Tips

The small sizes of the soil samples handled in the laboratory are a point representation of the soil properties. However, if samples used in laboratory tests are truly undisturbed, the calculated value of K will represent the saturated hydraulic conductivity at that particular sampling point.

If not conducted properly, a sampling process disturbs the structure of the soil matrix and results in an incorrect assessment of actual field properties.

An inappropriate test fluid may clog the test sample with trapped air or bacteria. Use a standard solution of de-aerated 0.005 mol calcium sulfate (CaSO4) solution saturated with thymol (or formaldehyde) in the permeameter.

#### Warnings

• The auger-hole method is not always reliable when artesian conditions exist, the water table is above the soil surface, the soil structure is extensively layered or highly permeable small strata occur.

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