Radial Consolidation Calculator — Vertical Drains (Barron 1948)
In saturated soft clays, consolidation settlement can take years. Vertical drains — sand drains, prefabricated PVDs or geotextiles — accelerate the process by shortening the pore water flow path. The Barron 1948 theory, supplemented by Hansbo 1981, provides the degree of radial consolidation Uh as a function of time and drain spacing. This calculator combines vertical consolidation Uv (Terzaghi) with radial consolidation Uh (Barron) to obtain the combined degree U that you will use in the design of embankment pre-loading, soil improvement in port fills and site stabilisation for airport works.
What is it and when is it applied?
When a saturated clay soil is loaded, the excess pore pressure dissipates slowly by vertical flow towards permeable layers. If the vertical distance is large (thick deposit) and there is no lateral drainage, consolidation takes years. Installing vertical drains in a triangular or square grid creates radial flow paths towards each drain, reducing the effective distance to half the spacing. It is applied in road embankments on soft soils (lagoons, wetlands, consolidated landfills), ports and reclaimed areas. It is not suitable for granular soils (they already drain quickly) nor for clays with OCR > 2 (overconsolidated).
Applied formulas
Radial time factor (Barron 1948): Th = Ch · t / de², with de = equivalent diameter of the drain cell
de = 1.13·s for square grid; de = 1.05·s for triangular grid (s = drain spacing)
Degree of radial consolidation (Hansbo 1981):
Uh = 1 − exp(−8·Th / F(n)), where F(n) = ln(n) − 3/4 + (smear and well resistance effects)
n = de / dw, with dw = drain diameter (0.07 m for typical PVD, 0.30 m for sand drain)
Combined degree (Carrillo 1942):
U = 1 − (1 − Uv) · (1 − Uh)
Vertical Uv (Terzaghi): Tv = Cv · t / H²dr, Uv = 2·√(Tv/π) for Uv ≤ 60 %; Uv = 1 − (8/π²)·exp(−π²·Tv/4) for Uv > 60 %
Settlement at time t: S(t) = U · S∞, with S∞ total consolidation settlement
Calculation example
| Parameter | Value |
|---|---|
| Soft layer thickness Hdr | 8.0 m (drains top and bottom: Hdr = 4 m) |
| Cv vertical consolidation coefficient | 1.5 m²/year |
| Ch radial consolidation coefficient | 3.0 m²/year (typically 2-3 × Cv) |
| Prefabricated PVD, dw | 0.07 m |
| Spacing s (triangular grid) | 1.5 m |
| Estimated total settlement S∞ | 45 cm (from Cc, e₀, σ'v0) |
| Target consolidation period | 6 months = 0.5 years |
Equivalent cell diameter: de = 1.05 · 1.5 = 1.575 m. Ratio n = de/dw = 1.575/0.07 = 22.5. Simplified factor F(n): F(n) = ln(22.5) − 3/4 = 3.114 − 0.75 = 2.364. Radial time factor at t = 0.5 years: Th = Ch · t / de² = 3.0 · 0.5 / 1.575² = 1.5 / 2.481 = 0.605. Radial degree: Uh = 1 − exp(−8·0.605 / 2.364) = 1 − exp(−2.048) = 1 − 0.129 = 0.871 → 87.1 %. Vertical degree at the same time (Hdr = 4 m, Cv = 1.5): Tv = 1.5 · 0.5 / 4² = 0.047. Uv = 2·√(0.047/π) = 2·0.122 = 0.244 → 24.4 %. Combined: U = 1 − (1 − 0.244)·(1 − 0.871) = 1 − 0.756·0.129 = 1 − 0.0975 = 0.902 → 90.2 %. Settlement at 6 months: S(0.5) = 0.902 · 45 = 40.6 cm. Without drains (only Uv = 24.4 %), settlement in 6 months would be barely 11 cm, and reaching 90 % would take about 13 years.
Result: Uh = 87 % · Combined U = 90 % · Settlement at 6 months = 41 cm (vs 11 cm without drains).
Interpretation of results
The design with PVD at 1.5 m spacing reduces the time to reach 90 % consolidation from ~13 years to 6 months. This enables the overlying works (pavements, structures) to be built without problematic differential settlements. If Ch is uncertain, in-situ testing with piezometer monitoring during pre-loading is recommended to recalibrate. The design is validated when the measured settlement reaches 80 % of the estimated value within the predicted time.
Reference standards
- BS EN 1997-1 (Eurocode 7) — Appendix on soil improvement
- BSI standard — Subgrade improvement
- Barron, R.A. (1948). Consolidation of fine-grained soils by drain wells
- Hansbo, S. (1981). Consolidation of fine-grained soils by prefabricated drains
- Carrillo, N. (1942). Simple two- and three-dimensional cases in the theory of consolidation
- BS EN ISO 17892-11 — Standard test method for determining the flow rate of prefabricated vertical drains
- BS EN 1997-1 (Eurocode 7) — Appendix on soil improvement
Frequently asked questions
What is the difference between a sand drain and a PVD?
Sand drain: a 30-40 cm diameter borehole filled with clean well-graded sand; more expensive and slower to install but durable. PVD (Prefabricated Vertical Drain): a corrugated plastic strip wrapped in a filter geotextile, 10 cm × 4 mm, installed with a mandrel; very economical and fast (500-1,000 m per day). PVD dominates modern pre-loading projects.
Square or triangular grid?
Triangular grid is 15 % more efficient (greater influence area per drain) and is used in 90 % of cases. Square grid is only used if there are construction constraints or equipment with a mandatory square pattern. The equivalent cell diameter is 1.05·s for triangular vs 1.13·s for square.
What is the smear effect?
When installing the drain, the nearby clay is remoulded and loses permeability (Kh decreases 3-10 times). This zone of thickness 2·dw is called the smear zone and delays consolidation. In design, it is included with an additional factor in F(n): F(n) = ln(n/s) + (Kh/Ks)·ln(s) − 0.75, where s = ds/dw ≈ 1.5-3 and Ks/Kh ≈ 0.1-0.3.
How is consolidation verified on site?
Installation of vibrating wire piezometers at 3-4 depths and settlement plates at the embankment head. Excess pore pressure and vertical displacement are measured weekly during pre-loading. When excess pressure drops and settlement stabilises, consolidation is considered complete. The Asaoka 1978 method is the standard for estimating S∞ from partial data.