One-dimensional consolidation: If C_v is the coefficient of consolidation, t is the elapsed time, and d is the drainage path length, the dimensionless time factor T_v is given by:

Introduction / Context:
Terzaghi’s one-dimensional consolidation theory uses two dimensionless groups: the time factor T_v and the average degree of consolidation U. Correctly forming T_v is the key to relating lab or field drainage times to soil layer thickness and drainage conditions (single or double drainage).

Given Data / Assumptions:

• C_v has units of length^2/time.
• t is time since loading.
• d is the maximum drainage path length (half thickness for double drainage, full thickness for single drainage).

Concept / Approach:

Dimensional analysis and the consolidation solution define T_v = C_v * t / d^2. This non-dimensional time collapses different layer thicknesses and drainage conditions into a common framework to read U–T_v relationships.

Step-by-Step Solution:

Verification / Alternative check:

Standard charts (e.g., U vs. T_v) rely on this definition; laboratory oedometer test interpretations use the same form.

Why Other Options Are Wrong:

(b) is the reciprocal; (c) and (d) include an unjustified factor 4; (e) has incorrect dimensions.

Common Pitfalls:

Using layer thickness instead of drainage path; forgetting that for double drainage d = H/2.

Final Answer:

T_v = (C_v * t) / d^2