Coefficient of permeability is a measure of how easily fluids can flow through a porous medium, such as soil or rock. It is an important parameter in geotechnical engineering and hydrogeology. The coefficient of permeability is denoted by the symbol k and has units of velocity.

The effective size of a particle refers to the diameter of a hypothetical sphere that has the same settling velocity as the particle in question. It is a measure of the size of the particle in terms of its influence on fluid flow.

To understand why the coefficient of permeability is directly proportional to the square of the effective size of the particle, we need to consider Darcy's law, which relates the flow rate (Q) through a porous medium to the hydraulic gradient (i), the cross-sectional area (A), and the coefficient of permeability (k) as follows:

Q = k * A * i

From this equation, we can see that the coefficient of permeability is directly proportional to the flow rate. In other words, a higher coefficient of permeability means that fluids can flow more easily through the porous medium.

Now let's consider the effect of particle size on the coefficient of permeability. When the effective size of the particles is increased, there are fewer available flow paths for the fluid to pass through the porous medium. This is because larger particles tend to have larger void spaces between them, which restrict the movement of fluids.

As a result, the flow rate decreases, and the coefficient of permeability decreases as well. This indicates an inverse relationship between the coefficient of permeability and the effective size of the particle.

However, when we square the effective size of the particle, we effectively square the decrease in flow rate. This means that the decrease in flow rate is amplified, resulting in a larger decrease in the coefficient of permeability. Therefore, the coefficient of permeability is directly proportional to the square of the effective size of the particle.

In summary, the coefficient of permeability is directly proportional to the square of the effective size of the particle because larger particles restrict fluid flow more significantly, leading to a decrease in the coefficient of permeability. Squaring the effective size amplifies this decrease, resulting in a direct proportional relationship between the coefficient of permeability and the square of the effective size of the particle.