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Average degree of consolidation U against time factor Tv: finite differences and the exact series
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How to use
Pick a preset or move the sliders; everything is live. The left drawing is a vertical section: a clay layer between drains, a load on top, and seven glass standpipes (piezometers) whose water level shows u/γw at seven depths. The blue tint of the clay is its excess pore pressure; the clay strips squeeze unevenly (more near the drains). The chart on the right shows the pore-pressure isochrones u(z, t): the orange curve is the present state, the coloured ones are earlier moments, the dashed green one the exact series. Move the mouse over the chart to read u and σ' at any depth. The graphs below show U against Tv, settlement against time, and the two oedometer constructions (Taylor and Casagrande), plus the e–log σ' line. Things to try: (1) press Jump to t90 and note the time, then switch to Single drainage and see it become four times larger; (2) set a 20 mm specimen and compare it with the 6 m layer; (3) choose the ramp preset and see how the peak pressure never reaches the full load; (4) raise k by two orders of magnitude (silty clay) and watch weeks replace years; (5) compare the lake-bed clay (Cc = 1.4): a small stress increase makes a large settlement. Drag to orbit, wheel to zoom. Space pauses, R resets.
The laboratory bench, the oedometer (cell with brass ring, porous stones, loading cap, lever hanger with slotted weights and a dial gauge), the spring-and-piston analogue, the glass soil tank with clay and sand layers and the standpipes were modelled in Blender 5.2 with baked PBR textures (clay, sand, rock, wood, brushed steel) and exported as one Draco GLB. Three.js r160 compresses the clay in 24 slices, moves the dial needle, raises the water in the standpipes and draws the isochrone on the glass from the same finite-difference model as the 2D page.
The science
When a building, embankment or specimen load is placed on a saturated clay, the water in the pores cannot escape instantly because clay is almost impermeable. At the first instant the water carries the whole extra load as excess pore pressure u, the soil skeleton feels nothing, and nothing settles. As water slowly seeps out through the drained faces, the load moves onto the grains (effective stress) and the layer compresses. That delay is consolidation.
1. Effective stress. Terzaghi's principle: σ' = σ − u. Compression and strength depend on σ', not on the total stress.
2. The equation. Darcy's law for vertical flow, water continuity and a linear soil skeleton (∂ε/∂σ' = mv) give the diffusion equation
∂u/∂t = cv ∂²u/∂z², cv = k / (mv γw)
With the time factor Tv = cv t / Hdr² the problem has no free parameters left. Hdr is the longest drainage path: H/2 for double drainage (sand above and below), H for single drainage. Drained faces: u = 0. Sealed faces: ∂u/∂z = 0.
3. Solution for a uniform initial pressure u0.
u(Z,Tv) = Σ (2u0/M) sin(MZ) exp(−M²Tv), M = π(2m+1)/2
U(Tv) = 1 − Σ (2/M²) exp(−M²Tv)
U ≈ √(4Tv/π) for U < 60 %, Tv(50 %) = 0.197, Tv(90 %) = 0.848
The isochrones in the chart are exactly these curves (the dashed green line is the series). Because t = Tv Hdr² / cv, doubling the drainage path makes consolidation four times slower: this is why a 20 mm laboratory specimen finishes in an hour and a 6 m clay layer needs years.
4. Settlement. For a normally consolidated clay the final settlement is sf = Cc H / (1 + e0) log10((σ'0 + Δσ)/σ'0) and s(t) = U(t) sf. The simulation uses the secant mv = sf / (H Δσ) for the load step, so cv follows from the permeability k you choose. The local effective stress is σ' = σ'0 + Δσ − u.
5. Loading history. If the load is raised linearly over a construction time tc, the equation gets a source term (dΔσ/dt) and the peak pore pressure stays below the full load; the settlement curve is the superposition of step responses (Schiffman). Non-uniform initial pressures (triangular) are solved with the same method.
6. The oedometer test. A 20 mm clay disc is loaded in steps and the dial gauge is read against time. Two classic constructions find cv: Taylor plots the reading against √t; the straight early part is extended back to d0, a line with abscissae 1.15 times larger crosses the curve at t90 and cv = 0.848 Hdr² / t90. Casagrande plots the reading against log t; d0 comes from the parabola rule (d(4t) − d(t) ≈ d(t) early on), d100 from the end, then cv = 0.197 Hdr² / t50. Both are performed in the page on the simulated dial readings.
Numerics. The layer is split into 80 cells; time stepping is Crank-Nicolson (second order) with a Rannacher start of four backward-Euler steps, which removes the oscillation caused by the discontinuity between the loaded interior and the drained face. The tridiagonal system is solved with the Thomas algorithm at every step; sealed faces use a mirror ghost node.
Numerical checks run in the page. (1) Finite-difference pore pressure against the eigenfunction series for the current layer, drainage and shape; (2) the textbook points U(0.197) = 50 % and U(0.848) = 90 %; (3) second-order convergence of the grid; (4) ramp loading against the superposition integral; (5) cv recovered by the Taylor and Casagrande methods from the simulated dial readings; (6) single drainage takes four times as long as double drainage.
Simplifications. One-dimensional flow and strain; cv, k and mv are constant during the load step (in a real clay k falls and the stiffness changes as e decreases, so the real curve is not exactly this one); no secondary compression (creep), which in practice adds a slow tail after t100; no lateral drainage or vertical drains (they make consolidation much faster); a wide loaded area, not the pressure bulb under a small footing; the soil presets are typical illustrative values, not data of a particular site. In the drawing, compression is exaggerated so that it can be seen; the readouts give the true values.
Sources to spot-check (from memory, flagged): Terzaghi 1925 (Erdbaumechanik); Terzaghi and Peck 1948; Taylor 1948 (Fundamentals of Soil Mechanics); Casagrande 1936 (1st ICSMFE); Schiffman 1958 (ramp loading); ASTM D2435 (one-dimensional consolidation test).