Critical Hydraulic Gradient for Boiling (Quicksand) in Terms of Porosity If G is the specific gravity of sand particles (Gs) and e denotes porosity (n) of the soil, write the expression for the critical hydraulic gradient i_c for onset of boiling (quicksand) condition under upward seepage.

Introduction / Context:
The critical hydraulic gradient i_c indicates when upward seepage causes effective stress to vanish (boiling or quick condition) in cohesionless soils. Designers compare exit gradients against i_c to assess piping risk beneath hydraulic structures and at excavation bottoms under dewatering.

Given Data / Assumptions:

• Specific gravity of soil solids G (i.e., Gs).
• Porosity is denoted here by e (note: many texts use n for porosity and e for void ratio).
• Saturated, homogeneous soil; upward seepage.

Concept / Approach:

The familiar expression in terms of void ratio is i_c = (Gs − 1) / (1 + e_void), where e_void is the void ratio. If porosity n is used instead, recall the relation e_void = n / (1 − n) → 1 + e_void = 1 / (1 − n). Substituting into the standard formula gives i_c = (Gs − 1) * (1 − n). Since the problem denotes porosity as e, we write i_c = (G − 1) * (1 − e). This directly connects i_c to the fraction of solids by volume (1 − n).

Step-by-Step Solution:

Verification / Alternative check:

For typical values G ≈ 2.65 and porosity e (n) ≈ 0.40, i_c ≈ (1.65)*(0.60) ≈ 0.99, consistent with standard ranges.

Why Other Options Are Wrong:

(b) is the correct form if the symbol e meant void ratio, but the stem explicitly says e is porosity. (c), (d), and (e) are dimensionally or physically inconsistent with the boiling condition.

Common Pitfalls:

Mixing symbols: many texts use e for void ratio and n for porosity; always translate before applying formulas.

Final Answer:

i_c = (G − 1) * (1 − e)