Water movements in frozen soils Gilles Lefebvre Samir Kaced Saïd Makhlouf. Ventilation Biological treatment Thermal desorption...

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1 Water movements in frozen soils Gilles Lefebvre Samir Kaced Saïd Makhlouf 1.On site treatment 2. Out site treatment Physical Chemical Biological In situ treatment Excavation Ventilation Biological treatment Thermal desorption... 1/

2 Slow freezing Moving freezing front Displacement of impurities, polluants Concentration of polluants, impurities in reduced soil volumes Polluants Frozen soil Non frozen soil 2/

3 Context Proposal Problem Model Heat and mass transfer : a complex problem Coupling between heat, dry air, vapor, water and ice Non linear equations Conservation Transport Huge number of state variables and equations (pde!) There is liquid water at temperature lower than 0 C Bad knowledge of the microscopical phenomena Many parameters value required (direct measurement not always possible or easy) 3/ gilles.lefebvre@u-pec.fr

4 Model hyp Hypothesis Local thermal, physical and chemical species and phases equilibrium Pores are spherical and are homogeneously filled by the ice The behaviour of the frozen porous media is the one of a usual porous media whose porosity and permeabilities depend on the ice content Air and water vapor are perfect gas Transport is diffusive (Darcy, Fick) Non deformable porous dry soil 4/ gilles.lefebvre@u-pec.fr

5 Model struct. Model structure : three fluids and two solids model 5 species: Liquid water Water vapor Ice Solid dry soil Dry air State variables Liquid and gases: pressure, density, temperature Ratio of frozen water... 5/ gilles.lefebvre@u-pec.fr

6 Equations 1 Mass conservation for reach i t + div specie w i i = a i Water conservation a el + a es + a eg = 0 Energy conservation Energy balance U t + div h aw a + h eg w eg + h el w el + q = 0 U = r a u a + r el u el + r eg u eg + r es u es + r m u m v el = w el Liquid water flow (Darcy) = k ρ l grad p el el Water vapor and air flows (Darcy+Fick) v i = w i = k ρ g grad p g i p g p i Dfgrad p i p g Conductive heat flux (Fourier) q = λgradt 6/ gilles.lefebvre@u-pec.fr

7 Equations 2 Air and water vapor are perfect gases p i = ρ irt M i p eg p sat eg T Thermodynamical gas-liquid p el = equilibrium RT p atm + ρ (Clapeyron) el ln M e Mechanical gas-liquid equilibrium (Kelvin-Laplace S l = exp αχβ + empirical) p c = p g p el = 2σ T χ Liquid water concentration in soils Ice content versus temperature (empirical) siθ < 0alorsω = ae θ c siθ > 0alorsω = b + 1 b 1 e θ d oùθ = T 273,15 W (%) silt clay sand ( C) 7/ gilles.lefebvre@u-pec.fr

8 Equations 3 Water superficial stress σ = 0, T 1, ,625 1 T T = T T c T c = 647,096K Liquid and gas permeabilities k l = kk rl μ l k g = kk rg μ g k rl = S l 1 1 S l 1 m m 2 1 m 2m k rg = S l 1 S l Water vapor partial pressure lnp sat eg T = 68, ,64 6,2973lnT T Equivalent conductivity λ = λ 1 Φ m λ es Φ Φ Φ S λ l Φ S g el λg + closing equation 8/ gilles.lefebvre@u-pec.fr

9 Numerical Control volume method Scilab implementation N 1 N 2 N 3 N 4 N n Figure (2) maillage uniforme 9/ gilles.lefebvre@u-pec.fr

10 Numerical Homogeneous soil : clay Figure (3) Evolution de la température imposée en surface du sol 10/ gilles.lefebvre@u-pec.fr

11 Soil temperature evolution Top (node 1) 0.2m (node 3) 0.4m (node 5) 0.6m (node 7) 0.8m (node 9) 1 m (node 11) Results Liquid water apparent density ev θ ( C) time (days) ρ wl (W.m -3 ) time (days) 11/ gilles.lefebvre@u-pec.fr

12 Velocity (m/s) Apparent liquid water velocity evolution Depth Top (node 1) 0.2m (node 3) 0.4m (node 5) 0.6m (node 7) 0.8m (node 9) 1 m (node 11) Results time (days) = Bottom-up velocity m/s Water flow (m Every day (m Every 5 days (m 12/ gilles.lefebvre@u-pec.fr

13 Top freezing Top thawing Results frozen cryosuccion not frozen 13/