# Hydraulic Conductivity Calculator

Created by Rahul Dhari
Reviewed by Steven Wooding
Last updated: Nov 12, 2022

The hydraulic conductivity calculator is your go-to tool to find how much fluid would flow through soil and rock per given time. This soil property is a function of various factors like particle or grain size, uniformity coefficient, and porosity.

This article will explore several ways to estimate hydraulic conductivity, an earth sciences concept. Most of these approaches require some parameters determined using experiments or are empirical. Therefore, every formula for hydraulic conductivity has its limitation and must be used accordingly.

## What is hydraulic conductivity and its units?

Before getting into the calculations, let's understand the definition of hydraulic conductivity.

Suppose you pour some water on the ground. The water would enter the soil bed flowing through pores and fractured rocks. The parameter hydraulic conductivity is the property of ease of flow of fluid through the soil. It is measured in the units of length per time, i.e., the distance covered by the fluid per time. The suitable units for hydraulic conductivity are meters per day or feet per day. The lower the value of hydraulic conductivity, the higher the resistance for the fluid to flow through the soil.

You must be thinking, hydraulic conductivity is the same as permeability, I can just use the porosity and permeability calculator? The answer is no. Permeability is a property that depends on the soil structure and is the ability of the soil to transmit any fluid. On the other hand, the hydraulic conductivity also factors the liquid's ability to flow by considering viscosity, which is a fluid property. That said, the hydraulic conductivity is proportional to the permeability. The highly permeable soil near the coastal region has high hydraulic conductivity.

## Hydraulic conductivity formulae

So how do you calculate hydraulic conductivity? Our tool supports the following seven different types of formulations to determine hydraulic conductivity:

1. Kozeny-Carman equation
2. Darcy's Law
3. Constant head method
4. Falling head method
5. Hazen equation
6. Breyer equation
7. US Bureau of Reclamation (USBR) equation

Depending upon the parameters available to you and soil properties, you can use one or more methods to calculate hydraulic conductivity.

Kozeny-Carman equation: Based on the works of Josef Kozeny and Philip C. Carman, it is one of the most commonly used empirical equations. The equation depends on the grain size and porosity of the soil. The hydraulic conductivity, $K$, is:

$K = \frac{g}{\nu}(8.3\times10^{-3}) \frac{n^3}{(1-n)^2} d_{10}^2$

where:

• $g$ – Acceleration due to gravity;
• $\nu$ – Kinematic viscosity;
• $n$ – Porosity; and
• $d_{10}$ – Diameter of grains at which 10% of the sample is finer in size.

You can use the Kozeny-Carman equation to calculate hydraulic conductivity from grain size. However, it does not support the calculations for clay soil with particle grain size larger than 3 mm.

Darcy's Law: Formulated by Henry Darcy, we can use this hydraulic conductivity equation to determine the fluid flow through a porous medium. The law is similar to Ohm's law or heat conduction (we described the former in the Ohm's law calculator). Consider a tube containing a soil sample, having a cross-sectional area, $A$, and length, $L$. The conductivity is proportional to the pipe flow or fluid discharge, $Q$, such that:

$K = \frac{Q}{iA}$

where $i$ is the hydraulic gradient which you can estimate with the hydraulic gradient calculator. It is the ratio of difference between initial and final heads, and the distance between the two points. Mathematically,

$i = \frac{h_2 - h_1}{L}$

You can calculate the hydraulic gradient using two experimental techniques — constant head and falling head.

Constant head method: Consider a cylindrical soil sample with length $L$ and cross-sectional area, $A$. The fluid volume, $V$, across the sample per time, $t$ is:

$K = \frac{VL}{At(h_2 - h_1)}$

where $h_2$, and $h_1$ are final and initial heads.

Falling head method: For a cylindrical pipe full of soil, the water flowing through the voids and pores in the soil sample is measured over different time intervals and heads to determine the hydraulic gradient. The formula for hydraulic conductivity is:

$K = \frac{aL}{tA} \ln \frac{h_1}{h_2}$

where $a$ is the area of standpipe.

Hazen's equation: This equation depends upon the uniformity coefficient and is applicable only if the uniformity coefficient is less than 5 ($U < 5$) and grain size of 0.1 to 3 mm. The uniformity coefficient measures uniformness in the soil particle size or the ratio of the diameter of 60% finer particles to the diameter of 10% finer particles. Mathematically,

$U = \frac{d_{60}}{d_{10}}$

The Hazens equation is:

$\scriptsize K = \frac{g}{\nu} (6\times10^{-4}) ( 1 + 10 ( n - 0.26)) d_{10}^2$

Breyer equation: This equation applies to the soil with a uniformity coefficient between 1 to 20 and a grain size between 0.06 and 0.6 mm. The Breyer equation is:

$\scriptsize K = \frac{g}{\nu} (6\times10^{-4}) \log{\left(\frac{500}{U}\right)} d_{10}^2$

United States Bureau of Reclamation (USBR) equation: This equation, unlike the ones mentioned above, does not depend on porosity and uses the effective grain diameter of $d_{20}$. Therefore, the accuracy of the equation is less than the others. It applies to soils with a uniformity coefficient of less than 5. Mathematically,

$\scriptsize K = \frac{g}{\nu} (4.8\times10^{-4}) d_{20}^5$

If some of the topics we introduced above are not clear, perhaps you want to explore more with our cross sectional area calculator or pipe flow calculator. Give them a try!

## Using the hydraulic conductivity calculator – Calculating hydraulic conductivity from grain size

As you know, the calculator supports seven different equations! You can pick any equation if you have the required data. Say, you have the viscosity of the fluid, grain diameter, and porosity. You can use the Kozeny-Carman equation. Let's try an example.

Find the hydraulic conductivity of the soil having a porosity of 0.3 and grain diameter of 0.1 mm. Take the viscosity of the fluid as 1 cSt.

1. Pick the method as Kozeny-Carman.
2. Enter the kinematic viscosity of the fluid as 1 cSt.
3. Insert the grain diameter as 0.1 mm.
4. Fill in the porosity as 0.3.
5. Using the hydraulic conductivity equation:
\scriptsize \begin{align*} \qquad K &= \frac{9.81}{0.000001} (8.3\times10^{-3})\times\\\\&\qquad \frac{0.3^3}{(1-0.3)^2} 0.0001^2 \\\\ K &= 2.594 \text{ m/day} \end{align*}

Similarly, you can use other methods to calculate the hydraulic coefficient based on your sample size's grain size and uniformity coefficient. If your working fluid is water (select water as the fluid), we have prefilled the viscosity data for you at different temperatures.

You can use the advanced mode of the tool to calculate kinematic viscosity using the density and dynamic viscosity.

## FAQ

### What are some empirical methods to calculate hydraulic conductivity?

Some of the empirical methods used to calculate hydraulic conductivity are — the Kozeny-Carman equation, Hazen equation, Breyer equation, and US Bureau of Reclamation (USBR) equation. All of the above methods have a set limit of uniformity coefficient and effective grain size for which they are applicable.

### How do I calculate hydraulic conductivity using falling head method?

To calculate hydraulic conductivity using the falling head method:

1. Divide the initial head by the final head.
2. Find the natural logarithm of the head ratio.
3. Multiply the resultant by the length of the specimen and the area of the standpipe.
4. Divide the product by time interval.
5. Divide the result by cross-sectional area to obtain the hydraulic conductivity.

### How do I calculate hydraulic conductivity using constant head method?

To calculate hydraulic conductivity using the constant head method:

1. Multiply the volume of the fluid by the length of specimen.
2. Divide the product by the cross-sectional area.
3. Divide the product by the time interval.
4. Divide the result by the difference between the final and initial head to obtain the hydraulic conductivity.

### What is the porosity of the soil having uniformity coefficient of 6?

The porosity of soil with a uniformity coefficient of 6 is 0.3384. Mathematically, porosity is given by: n = 0.255 × (1 + 0.83ᵘ). Therefore, n = 0.255 × (1 + 0.83⁶) = 0.3384. You can then use Kozeny-Carman, Hazen equation, and Breyer equation to determine the hydraulic conductivity.

Rahul Dhari
Method
Kozeny-Carman
You can use this method for:
• Effective grain sizes less than 3 mm.
Fluid
Select...
Kinematic viscosity of the fluid (ν)
in²/s
Acceleration due to gravity (g)
ft/s²
Grain diameter (d₁₀)
in
Porosity (n)
Hydraulic conductivity (K)
ft/s
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