Consolidation of Soil: Terzaghi Theory, Settlement Calculation & Worked Example

When a saturated clay layer is loaded — by a new building, embankment, or fill — the applied stress is initially carried entirely by the pore water. Over weeks, months, or years that excess pore pressure dissipates as water drains out, the soil skeleton takes over the load, and the ground surface settles. This time-dependent volume change is consolidation, and predicting how much and how fast it happens is one of the core problems in geotechnical engineering.

This guide explains Terzaghi’s one-dimensional consolidation theory, the key soil parameters, how to calculate primary settlement and time rate, and includes a fully worked numerical example.

What Is Consolidation?

Consolidation is the gradual reduction in volume of a saturated fine-grained soil due to the drainage of pore water under sustained loading. It differs from:

• Immediate (elastic) settlement — occurs instantly on loading, with no drainage; governed by undrained modulus.
• Secondary compression (creep) — time-dependent volume change at constant effective stress, after primary consolidation is complete; governed by C_α.

Consolidation is significant only in clay and silt. Sand drains almost instantly, so its “consolidation” is effectively immediate.

Terzaghi’s One-Dimensional Consolidation Theory

Karl Terzaghi (1925) derived the governing differential equation for 1-D consolidation by combining Darcy’s flow law with continuity:

∂u/∂t = c_v × ∂²u/∂z²

where u is the excess pore water pressure, t is time, z is depth within the clay layer, and c_v is the coefficient of consolidation.

Key assumptions:

• Soil is homogeneous and fully saturated.
• Compression and flow are one-dimensional (vertical only).
• Darcy’s law is valid.
• The coefficient of permeability (k) and the coefficient of compressibility (a_v) remain constant during the consolidation increment.
• Soil grains and pore water are incompressible.

Key Parameters

Parameter	Symbol	How obtained	Typical range (soft clay)
Compression index	Cc	Slope of e–log σ’ virgin curve	0.2–0.5 (can exceed 1.0 for organic clay)
Recompression index	Cr	Slope of e–log σ’ recompression curve	Cc/5 to Cc/10
Preconsolidation pressure	σ’p	Casagrande construction on e–log σ’ curve	Site-specific
Coefficient of consolidation	cv	Log-time or root-time fitting of oedometer data	0.5–5 m²/year
Void ratio	e0	From moisture content or direct measurement	0.8–2.5

Primary Consolidation Settlement Formula

For a normally consolidated clay (σ’₀ ≥ σ’_p):

S_c = [C_c / (1 + e₀)] × H × log₁₀(σ’_f / σ’₀)

For an overconsolidated clay where the final stress stays below σ’_p:

S_c = [C_r / (1 + e₀)] × H × log₁₀(σ’_f / σ’₀)

For an overconsolidated clay where loading takes it past σ’_p:

S_c = [C_r / (1 + e₀)] × H × log₁₀(σ’_p / σ’₀) + [C_c / (1 + e₀)] × H × log₁₀(σ’_f / σ’_p)

where H is the thickness of the compressible layer, σ’₀ is the initial effective stress at the mid-depth of the layer, and σ’_f = σ’₀ + Δσ is the final effective stress after loading.

Time Rate of Consolidation

The degree of consolidation U at time t is related to the time factor T_v:

T_v = c_v × t / H_dr²

where H_dr is the drainage path length — equal to H for single drainage (clay rests on an impermeable layer) or H/2 for double drainage (sand above and below).

U (%)	Tv
10	0.008
20	0.031
30	0.071
40	0.126
50	0.197
60	0.287
70	0.403
80	0.567
90	0.848
95	1.129
100	∞

Approximations: for U < 60%, T_v ≈ (π/4) × (U/100)². For U > 60%, T_v ≈ −0.9332 × log₁₀(1 − U/100) − 0.0851.

Worked Example

Given: A 4 m thick normally consolidated clay layer with double drainage (sand above and below). The groundwater table is at the surface.

• γ_sat = 18 kN/m³
• e₀ = 1.1
• C_c = 0.35
• c_v = 2.5 m²/year
• Applied surface load Δσ = 80 kPa (uniform, wide — so no stress dissipation with depth)

Step 1 — Initial effective stress at mid-layer (z = 2 m):

σ’₀ = (γ_sat − γ_w) × z = (18 − 9.81) × 2 = 16.4 kPa

Step 2 — Final effective stress:

σ’_f = 16.4 + 80 = 96.4 kPa

Step 3 — Primary consolidation settlement:

S_c = [0.35 / (1 + 1.1)] × 4 × log₁₀(96.4 / 16.4)

S_c = 0.1667 × 4 × log₁₀(5.878)

S_c = 0.6667 × 0.7693 = 0.513 m = 513 mm

Step 4 — Time for 90% consolidation:

H_dr = 4/2 = 2 m (double drainage)

T_v at 90% = 0.848

t = T_v × H_dr² / c_v = 0.848 × 4 / 2.5 = 1.36 years

Result: The clay will settle about 513 mm total, reaching 90% of that (462 mm) in approximately 1.4 years.

Overconsolidation Ratio (OCR)

OCR = σ’_p / σ’₀

• OCR = 1: normally consolidated (NC) — the soil has never been under higher stress than now.
• OCR > 1: overconsolidated (OC) — previously loaded and then unloaded (e.g. by erosion, glacier retreat, or excavation). Settlement is much smaller in the recompression range.
• OCR < 1: underconsolidated — excess pore pressure still exists from a prior load that hasn’t fully dissipated. Rare but dangerous.

The Oedometer Test

All the parameters above come from the one-dimensional consolidation test (oedometer or consolidometer). A thin soil specimen (~20 mm tall, 50–75 mm diameter) is confined in a rigid ring, loaded vertically in increments (usually doubling: 25, 50, 100, 200, 400, 800 kPa), and settlement is recorded against time at each increment.

• e–log σ’ curve gives C_c, C_r, and σ’_p (via the Casagrande graphical method).
• Settlement–log time curve at each load step gives c_v (via the Casagrande log-time method or Taylor root-time method).
• End-of-primary void ratio at each step gives the e–σ’ relationship.

Methods to Accelerate Consolidation

Method	How it works	Typical application
Prefabricated vertical drains (PVDs / wick drains)	Shorten the drainage path from H to the drain spacing (1–2 m)	Highway embankments, reclamation fills
Surcharge preloading	Apply temporary extra load to force faster pore-pressure dissipation	Building sites on soft ground
Vacuum consolidation	Apply suction via PVDs to create an effective pressure increase without surcharge	Very soft clays near stability limits
Electro-osmosis	Apply DC voltage to drive pore water toward cathode drains	Research/specialty — rarely cost-effective
Stone columns	Provide both drainage and load-sharing through the soft layer	Moderate-depth soft clay, structural loads

Frequently Asked Questions

What is the difference between consolidation and compaction?

Compaction is the densification of unsaturated soil by mechanical effort (rollers, rammers) — it expels air. Consolidation is the volume reduction of saturated soil by expelling pore water under sustained load — it takes much longer and cannot be forced mechanically.

Why does sand not consolidate?

Sand has high permeability, so excess pore pressure dissipates almost instantly when loaded. Its “consolidation” is complete before you can measure a time lag. The settlement is effectively immediate (elastic).

How long does consolidation take in practice?

It depends on the drainage path and the clay’s permeability. A 3 m thick double-drained soft clay might reach 90% consolidation in 1–3 years. A 15 m thick single-drained layer could take decades — which is why vertical drains are used to shorten the drainage path.

What is secondary compression?

After primary consolidation is complete (all excess pore pressure dissipated), the soil continues to creep slowly under constant effective stress. This is secondary compression, quantified by the coefficient of secondary compression C_α. It is typically 1–5% of C_c and becomes significant for organic soils and peat.