Darcy’s law

September 16, 2016

Darcy’s law

The flow of free water through soil is governed by Darcy’s law. In 1856, Darcy demonstrated experimentally that for laminar flow in a homogeneous soil, the velocity of flow () is given by,

The velocity of flow is also known as the discharge velocity or the superficial velocity .

Eq. 8.2 is known as Darcy’s law, which is one of the corner stones of soil engineering.

The discharge is obtained by multiplying the velocity of flow () by the total cross-sectional area of soil (A) normal to the direction of flow.

The area A includes both the solids and the voids.

The coefficient of permeability can be defined using Eq. 8.2 .

If the hydraulic gradient is unity, the coefficient of permeability is equal to the velocity of flow. In other words, the coefficient of permeability is defined as the velocity of flow which would occur under unit hydraulic gradient.

The coefficient of permeability has the dimensions of velocity [L/T]. it is measured in mm/sec, cm/sec, m/sec, m/day or other velocity units. The coefficient of permeability depends upon the particle size and upon many other factors, as explained later.

Table 8.1 gives the typical values of the coefficient of permeability of different soils.

Table 8.1

Typical values of the coefficient of permeability

8.4 Validity of Darcy’s law

Darcy’s law is valid if the flow through soils is laminar.

The flow of water through soils depends upon the dimension of interstices, which, in turn, depends upon the particle size.

In fine-grained soils, the dimensions of the interstices are very small and the flow is necessarily laminar.

In coarse-grained soils, the flow is also generally laminar. However, in very coarse-grained soils, such as coarse gravels, the flow may be turbulent.

For flow of water through pipes, the flow is laminar when the Reynolds number is less than 2000.

For flow through soils, it has been found that the flow is laminar if the Reynolds number is less than unity.

For flow through soils, the characteristic length in the Reynolds number is taken as the average particle diameter (D)

Using Aleen Hazen’s equation (Eq. 8.26) for velocity, it can be shown that the maximum diameter of the particle for the flow to be laminar is about 0.50mm.

The value of the critical Reynolds number of unity is, however, conservative. It has been demonstrated that the flow remains laminar even upto the Reynolds number of 75. It has been observed that Darcy’s law is valid for flow in clays, silts and fine sands.

In coarse sands, gravels and boulders, the flow may be turbulent and Darcy’s law may not be applicable.

It is difficult to predict the exact range of the validity of Darcy’s law.

The best method to ascertain the range is to conduct experiments and determine the actual relationship between the velocity and the hydraulic gradient.

For Darcy’s law to be valid, this relationship should be approximately linear.

For flow through coarse sands, gravels and boulders, the actual relationship between the velocity and the hydraulic gradient is non-linear. Hough gave the following equation for the velocity when the flow is turbulent.

In extremely fine-grained soils, such as a colloidal clay, the interstices are very small. The velocity is very small. In such soils also, the Darcy law is not valid.

DARCY’S LAW:

This states that the discharge velocity, of water is

Proportional to the hydraulic gradient,

Where :

K = Darcy coefficient of permeability, m/s

The hydraulic gradient is the ratio of the head loss h over a distance L.

The discharge velocity v is defined as the quantity of water, q percolating through a cross-sectional area A in unit time. This is not the same as the velocity of the water percolating through the voids of the soil which is known as the seepage velocity.

The flow of water through most soils is laminar and Darcy’s law applies to most soils. however , with open, large-void gravels, flow can become turbulent when Darcy’s Law may not be valid.

The coefficient of permeability, k, is dependent on the nature of the voids(see above) and the properties of the fluid, particularly its viscosity. It is given by: