A cold air temperature pushes a freezing front down into wet soil. How deep does it go in a week, and how much does the road lift?
Cold snap: -10 °C at the surface. The red line marks the depth of frozen soil; blue = frozen, orange = unfrozen.
Seasonal swing = 0 keeps the surface constant. With a swing the surface follows mean − swing × sin(2πt / 1 year). Changing a parameter restarts the run. "Heave drawn" only exaggerates the picture; readouts are real (the drawn lift is capped at 1 m).
The exact Neumann solution is solved separately (Newton iteration for λ, error function) and never uses the grid. The energy residual compares stored enthalpy with the heat that crossed the boundaries.
Runs seven uniform soils (θ = 10–40 %) for 20 days at −10 °C and compares each numerical depth with the exact Neumann depth.
Heat conduction in a freezing soil is a Stefan problem: ∂H/∂t = ∂/∂z ( k ∂T/∂z ) with the volumetric enthalpy H(T): H = Cf T for T < 0, 0 ≤ H ≤ L at T = 0 (water turning to ice) and H = L + Cu T above. The latent heat is L = θ ρw Lf with Lf = 334 kJ/kg, so wetter soil needs far more heat removed per metre of frost. The code uses an explicit finite-volume scheme with 1 cm cells and a 15 s step.
Exact solution used as the check. For a uniform half-space starting at 0 °C with the surface held at −ΔT, the front is X(t) = 2λ√(αf t) where λ eλ² erf(λ) = St/√π and St = Cf ΔT / L (Neumann). The simple Stefan estimate √(2 kf ΔT t / L) neglects the heat capacity of the frozen layer and overestimates by a few percent.
Frost heave. Water expands by about 9 % on freezing: in-place heave 0.09 θ X. Frost-susceptible soils also pull water to the freezing front, where it forms ice lenses. The segregation-potential idea gives a heave rate 1.09 · SP · ∇T (SP in mm²/(day·K), ∇T near the front in K/mm). The SP values on the slider are illustrative orders of magnitude, not measurements of a particular soil; the lens is assumed to thaw in proportion to the frozen thickness.
Simplifications: 1-D vertical conduction only; no unfrozen water below 0 °C, no moisture migration beyond the SP term, no snow, no pavement layers (the surface temperature is applied directly to the soil), conductivities and heat capacities from rough formulas of θ (typical textbook magnitudes), no overburden pressure on the lens. Real frost heave depends strongly on soil grain size and water table; this is a teaching model.