Permeability of soil varies

Question

Examoogle · Accepted Answer

Option B: as square of grain size

Explanation & Concepts:
According to Allen Hazen's empirical relationship and the Kozeny-Carman equation, the coefficient of permeability ($k$) of soil is directly proportional to the square of the grain size ($D^2$). This relationship assumes that permeability depends on the size of the pores, which in turn scales with the particle diameter.

Working Principle: In laminar flow through porous media (Darcy's Law), the resistance to flow is dictated by the specific surface area of the soil particles. Since the surface area per unit volume is inversely proportional to the grain size, and flow velocity is dictated by pore geometry, the hydraulic conductivity follows a square relationship with the characteristic particle diameter.

Key Formulas: $k = C \cdot D_{10}^2$ — Allen Hazen's formula where $C$ is a constant and $D_{10}$ is the effective size; $k = \frac{\gamma_w}{\mu} \cdot \frac{e^3}{1+e} \cdot C_s \cdot D_s^2$ — Kozeny-Carman generalized equation

Key Points: Permeability is highly sensitive to the effective grain size ($D_{10}$).; For coarse-grained soils, permeability increases significantly with larger particle diameters.; The constant of proportionality depends on the shape of particles and the fluid properties.; This relationship is strictly valid for laminar flow regimes.

Advantages: Allows quick estimation of $k$ from grain size distribution curves.; Useful for filter design and drainage calculations.

Disadvantages: Does not account for stratification or anisotropic conditions.; Empirical constants ($C$) vary significantly with packing density.

Applications: Dewatering system design.; Seepage analysis for dams and embankments.; Assessment of aquifer recharge potential.

Additional Info: The constant $C$ in Hazen's formula generally ranges from 100 to 150 when $D_{10}$ is in cm and $k$ is in cm/s.. Option A is incorrect as permeability increases with grain size, not decreases.

Answer: B is correct — The coefficient of permeability is directly proportional to the square of the particle diameter ($k \propto D^2$).

Exam Tip: Always verify if the question assumes Allen Hazen's simplified approach ($D^2$) or the broader Kozeny-Carman theory, though both agree on the square-law dependence for grain size.

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Option A: inversely as square of grain size

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Option C: as grain size

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Option D: inversely as void ratio

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