The Chartership Forum

Soil Mechanics / Consolidation and settlement with time

How fast: the time factor and degree of consolidation

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

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Corrections checked in this pass

  • Replaced the discontinuous consolidation approximation with the uniform-initial-pressure series and a consistent inverse.

Checks needed for project use

  • A geotechnical reviewer must check laboratory interpretation, classification conventions and the limits of drained, undrained and one-dimensional idealisations.
  • Initial excess-pressure distribution, variation in cv, drainage and secondary compression require assessment.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

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 ueu_e:

1-D consolidation equation (Terzaghi)
∂ue∂t=cv ∂2ue∂z2\dfrac{\partial u_e}{\partial t} = c_v\,\dfrac{\partial^2 u_e}{\partial z^2}

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

ue
excess pore pressure (kPa)
cv
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
cv=kmv γwc_v = \dfrac{k}{m_v\,\gamma_w}

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

k
permeability (m/s)
mv
coefficient of volume compressibility (m²/kN)
γw\gamma_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 (Hdr = 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, ∂ue/∂t = cv·∂²ue/∂z², with cv = k/(mv·γw).
    • For the stated one-dimensional model and initial excess-pressure distribution, Tv = cv·t/Hdr² gives the corresponding U–Tv relationship.
    • Hdr is the longest drainage path: H/2 for double drainage, H for single.
    • Settlement at time t is s(t) = U(Tv)·s∞; because Hdr is squared, drainage conditions dominate the timescale.