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- 1. Department of Civil, Environmental and Mechanical Engineering Master of Science in Environmental and Land Engineering THEORETICAL PROGRESS IN FREEZING – THAWING PROCESSES STUDY Supervisor Student Prof. Riccardo Rigon Niccolò Tubini Co-supervisors Prof. Stephan Gruber Dr. Francesco Seraﬁn
- 2. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions This work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license. Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 3. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions What is the purpose? The aim of my Master’s thesis is to develop a new interpretation of modeling freezing soils. Niccolò Tubini Theoretical progress in freezing – thawing processes study 2 / 34
- 4. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Why studing the inﬂuence of coupled heat and water ﬂow in soils? Freezing – thawing processes entail a huge exchange of heat; Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
- 5. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Why studing the inﬂuence of coupled heat and water ﬂow in soils? To simulate more realistic soil temperature (Luo et al., 2003). Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
- 6. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Why studing the inﬂuence of coupled heat and water ﬂow in soils? Studies have shown that proper frozen soil schemes help improve climate model simulation (Viterbo et al., 1999 and Smirnova et al., 2000). Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
- 7. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soils Some deﬁnitions Air gas Liquid water Soil particle Va Vw Vs Vc Niccolò Tubini Theoretical progress in freezing – thawing processes study 4 / 34
- 8. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soils Some deﬁnitions Soil porosity φ := Vs Vc Water content θ := Vw Vc Niccolò Tubini Theoretical progress in freezing – thawing processes study 5 / 34
- 9. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soils Some deﬁnitions Assuming the rigid soil scheme θs := φ 0 < θr ≤ θ ≤ θs < 1 Niccolò Tubini Theoretical progress in freezing – thawing processes study 6 / 34
- 10. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soils Young – Laplace equation r γaw α pw = pa − 2γaw cos α r Niccolò Tubini Theoretical progress in freezing – thawing processes study 7 / 34
- 11. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soils Young – Laplace equation pa ← 0 Let us deﬁne suction as ψ := pw gρw Niccolò Tubini Theoretical progress in freezing – thawing processes study 8 / 34
- 12. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soil hydraulic properties Mualem’s assumption Wetting and drying processes are assumed to be selective processes. Niccolò Tubini Theoretical progress in freezing – thawing processes study 9 / 34
- 13. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soil hydraulic properties Water – retention – hydraulic – conductivity models Dealing with unsaturated soils requires the deﬁnition of the relationship between θ–ψ and K–ψ Niccolò Tubini Theoretical progress in freezing – thawing processes study 10 / 34
- 14. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soil hydraulic properties Empirical curve-ﬁtting models Parameters of these models have been related to the soil texture and other soil properties Niccolò Tubini Theoretical progress in freezing – thawing processes study 11 / 34
- 15. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soil hydraulic properties Empirical curve-ﬁtting models Parameters of these models have been related to the soil texture and other soil properties Despite their usfulness they do not emphasize the physical signiﬁcance of their empirical parameters Niccolò Tubini Theoretical progress in freezing – thawing processes study 11 / 34
- 16. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soil hydraulic properties Lognormal distribution model (Kosugi, 1996) The idea is to derive the water retention curve from the pore-size distribution: f (r) := dθ dr Niccolò Tubini Theoretical progress in freezing – thawing processes study 12 / 34
- 17. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soil hydraulic properties Lognormal distribution model (Kosugi, 1996) r 0 50 100 150 f(r) R Water f (r) = θs − θr √ 2π σr exp − ln r rm 2 2σ2 θ(R) = θr + R 0 f (r)dr Niccolò Tubini Theoretical progress in freezing – thawing processes study 13 / 34
- 18. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soil hydraulic properties Lognormal distribution model (Kosugi, 1996) Young-Laplace equation allows to transform the pore-size distribution into the capillary pressure distribution function g(ψ) = f (r) dr dψ Niccolò Tubini Theoretical progress in freezing – thawing processes study 14 / 34
- 19. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Unsaturated soil hydraulic properties Lognormal distribution model (Kosugi, 1996) θ(Ψ) = θr + Ψ −∞ g(ψ)dψ Ψ = − 2γaw cos α g ρw R Niccolò Tubini Theoretical progress in freezing – thawing processes study 15 / 34
- 20. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Some deﬁnitions Air gas Ice Liquid water Particle soil Va Vi Vw Vs Vc Niccolò Tubini Theoretical progress in freezing – thawing processes study 16 / 34
- 21. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Some deﬁnitions Liquid water content θw := Vw Vc Ice content θi := Vi Vc Niccolò Tubini Theoretical progress in freezing – thawing processes study 17 / 34
- 22. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Some deﬁnitions Total water content θ := θw + θi 0 < θr ≤ θ ≤ θs < 1 Niccolò Tubini Theoretical progress in freezing – thawing processes study 18 / 34
- 23. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Model assumptions Model assumptions rigid soil scheme Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
- 24. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Model assumptions Model assumptions rigid soil scheme freezing = drying (Miller, 1965; Spaans and Baker, 1996) Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
- 25. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Model assumptions Model assumptions rigid soil scheme freezing = drying (Miller, 1965; Spaans and Baker, 1996) the phase change is assumed to occur at the thermodynamic equilibrium Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
- 26. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Freezing point depression Gibbs-Thomson equation (Acker et al., 2001) Tm − T∗ = 2 γaw Tm cos α ρw r + πw Tm ρw Capillary eﬀect Dissolved solutes Niccolò Tubini Theoretical progress in freezing – thawing processes study 20 / 34
- 27. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Freezing point depression Gibbs-Thomson equation (Acker et al., 2001) The ice-water interface occurs at: ˆr(T) := − 2 γaw Tm cos α ρw (T − Tm) for T < Tm∗ Niccolò Tubini Theoretical progress in freezing – thawing processes study 21 / 34
- 28. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Water and ice content Let us deﬁne r∗ := R if ˆr ≥ R or T ≥ Tm ˆr otherwise ∂r∗ ∂t := ∂R ∂t if ˆr ≥ R or T ≥ Tm ∂ˆr ∂t otherwise Niccolò Tubini Theoretical progress in freezing – thawing processes study 22 / 34
- 29. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Water and ice content r 0 50 100 150 f(r) Rˆr = r∗ Water Ice θw = θr + r∗ 0 f (r)dr θi = R r∗ f (r)dr Niccolò Tubini Theoretical progress in freezing – thawing processes study 23 / 34
- 30. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Water and ice content The phase change rate θi = R r∗ f (r)dr ∂θi ∂t = ∂R ∂t f (R) − ∂r∗ ∂t f (r∗ ) Niccolò Tubini Theoretical progress in freezing – thawing processes study 24 / 34
- 31. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Water and ice content The phase change rate r 0 20 40 60 80 100 120 f(r) R(t) R(t + δt)ˆr Water Ice at time t Ice at time t + δt Niccolò Tubini Theoretical progress in freezing – thawing processes study 25 / 34
- 32. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Water and ice content The phase change rate r 0 20 40 60 80 100 120 f(r) Rˆr(t)ˆr(t + δt) Water Ice formed in δt Ice at time t Niccolò Tubini Theoretical progress in freezing – thawing processes study 26 / 34
- 33. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Water and ice content The phase change rate ∂θi ∂t = ∂Ψ ∂t g(Ψ) − ∂ψ∗ ∂t g(ψ∗ ) Niccolò Tubini Theoretical progress in freezing – thawing processes study 27 / 34
- 34. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Mass conservation equation θ J ET Jw ∂ ∂t (ρw θw + ρiθi) = −ρw · Jw Niccolò Tubini Theoretical progress in freezing – thawing processes study 28 / 34
- 35. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Mass conservation equation θ J ET Jw ∂ ∂t (ρw θw + ρiθi) = −ρw · Jw Water ﬂux: Jw = −K(ψ∗ ) (ψ∗ + z) Niccolò Tubini Theoretical progress in freezing – thawing processes study 28 / 34
- 36. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Mass conservation equation Setting ρw = ρi ∂θ ∂t = ∂Ψ ∂t g(Ψ) = · [K(ψ∗ ) (ψ∗ + z)] ∂θi ∂t = ∂Ψ ∂t g(Ψ) − ∂ψ∗ ∂t g(ψ∗ ) ∂θw ∂t = ∂θ ∂t − ∂θi ∂t Niccolò Tubini Theoretical progress in freezing – thawing processes study 29 / 34
- 37. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Energy conservation equation ε J HRn ET JwJg ∂ε ∂t = − · (Jw + Jg) Niccolò Tubini Theoretical progress in freezing – thawing processes study 30 / 34
- 38. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Energy conservation equation ε J HRn ET JwJg ∂ε ∂t = − · (Jw + Jg) Advective ﬂux: Jw = Jw ρw [ + cw (T − Tm)] Heat conduction: Jg = −λ T Niccolò Tubini Theoretical progress in freezing – thawing processes study 30 / 34
- 39. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Energy conservation equation Setting ρw = ρi CT ∂T ∂t − ρi ∂Ψ ∂t g(Ψ) − ∂ψ∗ ∂t g(ψ∗ ) −ρi(cw − ci)(T − Tm) ∂Ψ ∂t g(Ψ) − ∂ψ∗ ∂t g(ψ∗ ) +ρicw Jw · T + ρigz · Jw − · Jg = 0 Niccolò Tubini Theoretical progress in freezing – thawing processes study 31 / 34
- 40. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Energy conservation equation: if ice occurs ψ∗ := ˆψ ∂ψ∗ ∂t := g Tm ∂T ∂t Cph ∂T ∂t − ρi[ + (cw − ci)(T − Tm)] ∂Ψ ∂t g(Ψ) + ρicw Jw · T + ρigz · Jw − · Jg = 0 Niccolò Tubini Theoretical progress in freezing – thawing processes study 32 / 34
- 41. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions The apparent heat capacity Cph := CT + ρi[ + (cw − ci)(T − Tm)] g Tm Niccolò Tubini Theoretical progress in freezing – thawing processes study 33 / 34
- 42. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions The apparent heat capacity Cph := CT + ρi[ + (cw − ci)(T − Tm)] g Tm CT := ρscs(1 − θs) + ρiciθi + ρw cw θw Niccolò Tubini Theoretical progress in freezing – thawing processes study 33 / 34
- 43. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions To take home Freezing=drying and rigid soil scheme assumptions are useful when freezing-induced mechanical deformations are not considered; Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
- 44. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions To take home Freezing/thawing processes do not occur at the thermodynamic equilibrium (Kurylyk, 2013). Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
- 45. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions To take home Kosugi retention model has the beneﬁt to be straightforward extended to freezing soils case by making use of Gibbs – Thomson equation; Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
- 46. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions To take home This formulation allows to take into account of dissolved solutes; Niccolò Tubini Theoretical progress in freezing – thawing processes study 35 / 34
- 47. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions To take home It is possible to solve the mass and energy equation in a decoupled way; Niccolò Tubini Theoretical progress in freezing – thawing processes study 35 / 34
- 48. OttawaRiver,17th Dec2016
- 49. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions References K. Kosugi, Lognormal distribution model for unsaturated soil hydraulic properties, Water Resources Research, vol. 32, no. 9, pp. 2697–2703, 1996. J. T. Acker et al., Intercellular ice propagation: experimental evidence for ice growth through membrane pores, Biophysical journal, vol. 81, no. 3, pp. 1389–1397, 2001. M. Dall’Amico et al., A robust and energy-conserving model of freezing variably-saturated soil, The Cryosphere, vol. 5, no. 2, p. 469, 2011. M. Dall’Amico, Coupled water and heat transfer in permafrost modeling, Ph.D. dissertation, University of Trento, 2010. Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 50. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions References E. J. Spaans and J. M. Baker, The soil freezing characteristic: Its measurement and similarity to the soil moisture characteristic, Soil Science Society of America Journal, vol. 60, no. 1, pp. 13–19, 1996. R. D. Miller, Phase equilibria and soil freezing, vol. 287, pp. 193–197, 1965. B. L. Kurylyk and K. Watanabe, The mathematical representation of freezing and thawing processes in variably-saturated, non-deformable soils, Advances in Water Resources, vol. 60, pp. 160–177, 2013. Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 51. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions References L. Luo et al., Eﬀects of frozen soil on soil temperature, spring inﬁltration, and runoﬀ: Results from the PILPS 2 (d) experiment at Valdai, Russia, Journal of Hydrometeorology, vol. 4, no. 2, pp. 334–351, 2003. T. G. Smirnova, J. M. Brown, S. G. Benjamin, and D. Kim, Parameterization of cold-season processes in the maps land-surface scheme, Journal of Geophysical Research: Atmospheres, vol. 105, no. D3, pp. 4077– 4086, 2000. P. Viterbo, A. Beljaars, J.-F. Mahfouf, and J. Teixeira, The representation of soil moisture freezing and its impact on the stable boundary layer, Quarterly Journal of the Royal Meteorological Society, vol. 125, no. 559, pp. 2401–2426, 1999. Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 52. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Freezing=drying assumptionDall’Amico,2010 pa pw (R) R r Air-water interface pw (R) = pa − 2 γaw cos α R Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 53. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Freezing=drying assumptionDall’Amico,2010 pa pi pw (r) R r Air-ice interface pi = pa − 2 γai cos α R Ice-water interface pw (r) = pi − 2 γiw cos α r Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 54. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Freezing=drying assumptionDall’Amico,2010 pa pi ≡ pa pw (r) R r Air-water interface pw (r) = pa − 2 γaw cos α r Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 55. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions r 0 20 40 60 80 100 120f(r) R = r∗ ˆr Water Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 56. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions The phase change rate Thanks to the Young-Laplace equation ψ∗ := Ψ if ˆr ≥ 2 γaw cos α ρw g Ψ or T ≥ Tm ˆψ = ψ(ˆr) otherwise Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 57. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions The phase change rate Thanks to the Young-Laplace equation ∂ψ∗ ∂t := ∂Ψ ∂t if ˆr ≥ 2γaw cos α ρw gΨ or T ≥ Tm ∂ ˆψ ∂t otherwise Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 58. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Comparison with Dall’Amico model (Dall’Amico et al., 2011) By making use of Clausius – Clapeyron equation: dT dpw = T ρw Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 59. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Comparison with Dall’Amico model (Dall’Amico et al., 2011) By making use of Clausius – Clapeyron equation: dT dpw = T ρw T∗ = Tm + g Tm ψw0 ψ(T) = ψw0 + g T∗ (T − T∗ )H(T∗ − T) Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 60. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Mass conservation equation: if there is no ice ψ∗ := Ψ ∂ψ∗ ∂t := ∂Ψ ∂t ∂θ ∂t = ∂Ψ ∂t g(Ψ) = · [K(Ψ) (Ψ + z)] ∂θi ∂t = :0 ∂Ψ ∂t g(Ψ) − ∂ψ∗ ∂t g(ψ∗ ) ∂θw ∂t = ∂θ ∂t Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 61. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Mass conservation equation: if ice occurs ψ∗ := ˆψ ∂ψ∗ ∂t := g Tm ∂T ∂t ∂θ ∂t = ∂ ˆψ ∂t g(Ψ) = · [K( ˆψ) ( ˆψ + z)] ∂θi ∂t = ∂Ψ ∂t g(Ψ) − ∂ ˆψ ∂t g( ˆψ) ∂θw ∂t = ∂θ ∂t − ∂θi ∂t Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 62. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Energy conservation equation: if there is no ice ψ∗ := Ψ ∂ψ∗ ∂t := ∂Ψ ∂t CT ∂T ∂t − ρil :0 ∂Ψ ∂t g(Ψ) − ∂ψ∗ ∂t g(ψ∗ ) − ρi(cw − ci)(T − Tm) :0 ∂Ψ ∂t g(Ψ) − ∂ψ∗ ∂t g(ψ∗ ) + ρicw Jw · T + ρigz · Jw − · Jg = 0 Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 63. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Energy conservation equation: if there is no ice CT ∂T ∂t + ρw cw Jw · T + ρw gz · Jw − · Jg = 0 Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 64. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Energy conservation equation: if there is no ice CT ∂T ∂t + ρw cw Jw · T + ρw gz · Jw − · Jg = 0 CT := ρscs(1 − θs) + ρiciθi + ρw cw θw Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 65. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Numerical scheme for unfrozen soils The mass conservation equation ⇒ Nested Newton method (Casulli and Zanolli, 2010). The energy consevation equation ⇒ Implicit upwind method Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 66. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Numerical scheme for frozen soils The mass conservation equation becomes ∂θ ∂t = · K( ˆψ) ( ˆψ + z) Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 67. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Numerical scheme for frozen soils The mass conservation equation becomes ∂θ ∂t = · K( ˆψ) ( ˆψ + z) Nested Newton method (Casulli and Zanolli, 2010). should be extended for equations of two variables Niccolò Tubini Theoretical progress in freezing – thawing processes study
- 68. Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions Numerical scheme for frozen soils The mass conservation equation becomes ∂θ ∂t = · K( ˆψ) ( ˆψ + z) Nested Newton method (Casulli and Zanolli, 2010). should be extended for equations of two variables The energy consevation equation ⇒ Implicit upwind method Niccolò Tubini Theoretical progress in freezing – thawing processes study

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