Hydraulic conductivity is the ease with which water can move through porous spaces and fractures in soil or rock, subject to a hydraulic gradient and conditioned upon the level of saturation and permeability of the material. Hydraulic conductivity is generally determined either through an empirical approach by which the hydraulic conductivity is correlated to soil properties or through an experimental approach whereby the hydraulic conductivity is calculated by experimentation. Methods in each approach are presented here.

## 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=(g/v)_C_ƒ(n)*(d_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=µ/ρ. The values of C, ƒ(n) and d depend on the method used in the grain-size analysis. Porosity (n) is derived from the empirical relationship n=0.255(1+0.83^U) where the coefficient of grain uniformity (U) is given by U=d_60/d_10. In the sample, d_60 represents the grain diameter (mm) in which 60% if the sample is more fine and d_10 represent the grain diameter (mm) for which 10% of the sample is more fine.

The different empirical formulas are based on this general 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 clayey textured soils:

K=(g//v)_8.3_10^-3[n^3/(1-n)^2]*(d_10)^2

Use the Hazen equation for soil textures from fine sand to gravel, providing the soil has a uniformity coefficient less than five (U<5) and effective grain size between 0.1 mm and 3 mm. As this formula is based only on the d_10 particle size, it is therefore less accurate than the Kozeny-Carman formula:

K=(g/v)*(6_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=(g/v)*(6_10^-4)_log(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). Because it calculates using an effective grain size of d_20 and does not depend on porosity, it is less accurate than other formulas:

K=(g/v)*(4.8_10^-4)*(d_20)^3_(d_20)^2

## Experimental Methods - Laboratory

Use an equation based on Darcy's Law to derive hydraulic conductivity experimentally. In the lab, a soil sample is placed in a small cylindrical container creating a one-dimensional soil cross-section through which the liquid, usually water, flows. This method is classified as either a constant-head test or a falling-head test depending on the flow state of the liquid. Constant-head tests are usually used on coarse-grained soils such as clean sands and gravels. Falling-head tests are used on finer grain samples. The basis for these calculations is Darcy's Law:

U= -K(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. It is the most commonly used test for determining the saturated hydraulic conductivity of coarse-grained soils in the laboratory. A cylindrical soil sample of cross-sectional area A and length L is subjected 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=VL/[At(H2-H1)]

For best results, test several times using different head differences, H2 - H1.

Use the Falling-head test for determining the K of fine-grained soils in the laboratory. In the falling-head method, a cylindrical soil sample column of cross-sectional area A and length L is connected to a standpipe of cross-sectional area a, in which the percolating fluid flows into the system. By measuring the change in head in the standpipe, H1 to H2, at intervals of time (t), the saturated hydraulic conductivity can be determined from Darcy's Law:

K=(aL/At)ln(H1/H2)

## Experimental Methods - Field

Use the Auger-Hole field procedure on an unconfined aquifer with homogeneous soil properties and a shallow water table. It is most commonly used for determining saturated hydraulic conductivity of soils. It requires preparing a hole that partially penetrates the aquifer, with a minimal disturbance of the soil. When the water in the equalizes with the water table level, all the water is removed from the hole and the rate of rise of the water level within the cavity is measured until the level again equalizes with the water table. There is no simple equation for accurately determining the hydraulic conductivity. One calculation used is:

Kh = F(Ho-Ht) / t

Where Kh = horizontal saturated hydraulic conductivity (m/day); H = depth of the water level in the hole relative to the water table in the soil (cm); Ht = H at time t, Ho = H at time t = 0, t = time (in seconds) since the first measurement of H as Ho, and F is a factor derived from the geometry of the hole:

F = 4000r / h'(20+D/r)(2−h'/D)

Where r = radius of the cylindrical hole (cm), h' is the average depth of the water level in the hole relative to the water table in the soil (cm) , found as:

h'=(Ho+Ht)/2

And D is the depth of the bottom of the hole relative to the water table in the soil (cm).

Use a Piezometer Method for soils in an unconfined aquifer with a shallow water table level. Designed for application in layered soil aquifers and for determining either horizontal or vertical components of the saturated hydraulic conductivity. This method consists of installing a piezometer tube or pipe, long enough to partially penetrate the unconfined aquifer, into an auger hole drilled through the subsurface system without disturbing the soil. The walls of the tube are totally closed except at its lower end, where the open tube is screened to form a cylindrical cavity of radius r and height hc within the aquifer. The water in the piezometer tube is first removed to clean the system and is then allowed to equilibrate with the groundwater level before removing the water from the pipe and then measuring the rate of the rise of the water within the pipe. The saturated hydraulic conductivity is a function of the dimensions in the piezometer tube, the dimensions of the aquifer, and the measured rate of rise of the water table in the tube. The value for the hydraulic conductivity is calculated with the help of a nomograph and tables. The piezometer method is particularly useful in calculating the hydraulic conductivity of the individual layers in a stratified subsurface systems.

Use the Well Slug-test method for calculating saturated hydraulic conductivity in the soil of unconfined and confined aquifers. This test requires removing a predetermined measure, or slug, of water all at once from a well and then measuring the rate of water recovery back into the well. This test provides a representation of the soil hydraulic conductivity averaged over a larger volume of the soil than either the piezometer or auger-hole methods. The results primarily reflect the K value in the horizontal direction. Using the Bouwer & Rice equation to calculate K:

K=[(r_c^2_ln(R_e/r_w))/2L_e]_(1/t)*ln(h_0/h)

Where K=hydraulic conductivity; r_c=radius of the well casing; r_w=radius of the well including gravel envelope; R_e=radial distance over which the head is dissipated; L_e=length of the screen; t=time elapsed since h=h_0; h_0=draw down at time t=0; h=draw down at time t=t.

#### Tip

Choosing a specific method will most often depend on the objectives desired.

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, then the calculated value of K will represent the saturated hydraulic conductivity at that particular sampling point.

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

The test sample can get clogged with entrapped air, bacteria, and fines when an inappropriate test fluid is selected. A standard solution of de-aerated 0.005 mol calcium sulfate (CaSO4) solution saturated with thymol (or formaldehyde) can be used in the permeameter.

#### Warning

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