Module 10 · Lesson 10.3

How fast: the time factor and degree of consolidation

The 1-D consolidation equation, the dimensionless time factor Tv, and settlement over time.

Why this matters

Knowing the final settlement is only half the answer. A structure cares about how much has happened by the time it is built, and how much is still to come. The rate is governed by one dimensionless group — the time factor — that ties together the soil's stiffness and permeability, the layer's thickness, and the drainage conditions.

The physics combines two things we already have. Continuity says the rate at which a slice of soil compresses equals the net rate at which water flows out of it. Darcy's law says that flow is proportional to the gradient of excess pore pressure. Putting them together, with the compressibility linking void change to effective-stress change, gives Terzaghi's one-dimensional consolidation equation — a diffusion equation for the excess pore pressure $u_{e}$ :

1-D consolidation equation (Terzaghi)

\frac{\partial u _{e}}{\partial t} = c_{v} \frac{\partial ^{2} u _{e}}{\partial z ^{2}}

What it calculates: how excess pore pressure spreads out and dies away with time and depth

u_e: excess pore pressure (kPa)
c_v: coefficient of consolidation (m²/s (or m²/yr))
z: depth (drainage direction) (m)
t: time (s)

This assumes

Saturated, homogeneous soil; 1-D flow and strain; Darcy's law holds.
cv, k and mv are constant over the stress increment; small strains.

In plain terms: It is a diffusion equation: excess pore pressure diffuses out exactly as heat diffuses through a bar, with cv playing the role of thermal diffusivity. That analogy is why the solution is universal.

Coefficient of consolidation

c_{v} = \frac{k}{m _{v} γ _{w}}

What it calculates: the single soil parameter that sets the rate of consolidation

k: permeability (m/s)
m_v: coefficient of volume compressibility (m²/kN)
$γ_{w}$: unit weight of water (kN/m³)

In plain terms: cv rises with permeability (water escapes faster) and falls with compressibility (more water must escape for a given stress change). It is the diffusivity of the consolidation problem.

From first principles

The dimensionless time factor

We want to show: Reduce the consolidation solution to a single number that governs the rate for any layer.

A diffusion equation has no intrinsic scale — only the combination of diffusivity, time and travel distance matters. Grouping cv, t and the drainage path into one dimensionless number collapses every consolidation problem onto a single curve.

Worked example

How long to reach 90% of the settlement

Given

Clay layer 6 m thick, draining both top and bottom (double drainage)
Coefficient of consolidation cv = 2.0 m²/year
Final primary settlement s∞ = 150 mm

Find

The time to reach 90% consolidation, and the settlement after 1 year.

Assumptions

Uniform clay; 1-D consolidation; cv constant over the stress range.

Practice

A 4 m clay layer drains from both faces (H_dr = 2 m) with cv = 1.0 m²/year. Using T₉₀ ≈ 0.848, how many years to reach 90% consolidation?

Check yourself

Two identical clay layers differ only in drainage: one drains both faces, the other one face only. How do their times to 90% consolidation compare?

Summary

Excess pore pressure obeys a diffusion equation, ∂u_e/∂t = cv·∂²u_e/∂z², with cv = k/(mv·γw).
The dimensionless time factor Tv = cv·t/H_dr² collapses every consolidation problem onto one U–Tv curve.
H_dr is the longest drainage path: H/2 for double drainage, H for single.
Settlement at time t is s(t) = U(Tv)·s∞; because H_dr is squared, drainage conditions dominate the timescale.

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint