Critical exit gradient (boiling) in soils: dependence on parameters The critical exit gradient icr for upward seepage in soils is functionally related to specific gravity Gs and void ratio e. Which statement best reflects this dependence?

Introduction / Context:
The critical exit gradient indicates the onset of quicksand or boiling conditions when upward seepage reduces effective stress to zero. Understanding parameter dependence helps in the safe design of filters, cutoffs, and dewatering systems.

Given Data / Assumptions:

• Soil is cohesionless for boiling criterion.
• Upward seepage through a saturated soil mass.
• Standard derivation assumes equilibrium of submerged unit weight and seepage force.

Concept / Approach:

The classical expression is icr = (Gs − 1) / (1 + e). Hence, icr increases with increasing Gs (through Gs − 1) and decreases with increasing e. The most precise qualitative statement among the options is that icr varies inversely with void ratio e.

Step-by-Step Solution:

Verification / Alternative check:

Setting e → 0 (very dense) gives larger icr; lower-density soils (higher e) have smaller icr, consistent with field observations of piping susceptibility.

Why Other Options Are Wrong:

(a) Direct with e contradicts formula; (b) inverse with Gs is incorrect; (c) “direct with Gs” is imprecise because the exact proportionality is to (Gs − 1), not strictly to Gs; hence (d) is the best correct statement.

Common Pitfalls:

Confusing proportionality to Gs with (Gs − 1); overlooking the (1 + e) term in the denominator; applying the criterion to cohesive soils without caution.

Final Answer:

icr is inversely proportional to the void ratio e