Question

If the \(D_{10}\) is the effective grain size, the coefficient of permeability of a soil is proportional to

• A. \(D_{10}\)
• B. \((D_{10})^2\)
• C. \((D_{10})^3\)
• D. \((D_{10})^{\frac{1}{2}}\)

Hint:

The coefficient of permeability \(k\) measures how easily water flows through a soil. It depends strongly on the size of the soil particles and the pore spaces between them.

Answer 1

The Correct Option is B

Step 1: What permeability depends on.
The coefficient of permeability \(k\) tells how easily water moves through soil. It depends mainly on the size of the soil grains, because grain size controls the size of the pore channels.

Step 2: Meaning of \(D_{10}\).
The effective size \(D_{10}\) is the grain diameter such that 10 percent of the soil by weight is finer than it. It represents the small grains that mostly control flow.

Step 3: Bigger grains, easier flow.
Larger grains leave wider pores, and wider pores let water pass far more easily, so permeability rises sharply as grain size grows.

Step 4: Hazen's relation.
Allen Hazen found from experiments that \[ k = C\,(D_{10})^{2}, \] where \(C\) is a constant. The permeability is set by the square of the effective size.

Step 5: Read off the power.
So \(k\) is proportional to \((D_{10})^{2}\), not to the first power or the cube.

Step 6: Answer.
Hence the coefficient of permeability is proportional to \((D_{10})^{2}\).

\[ \boxed{k \propto (D_{10})^{2}} \]