# master thesis presentation by niccolò tubini

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Department of Civil, Environmental and Mechanical Engineering

Master of Science inEnvironmental and Land Engineering

THEORETICAL PROGRESS INFREEZING – THAWING PROCESSES STUDY

Supervisor StudentProf. Riccardo Rigon Niccolò Tubini

Co-supervisorsProf. Stephan GruberDr. Francesco Serafin

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

This work is licensed under a CreativeCommons “Attribution-ShareAlike 4.0International” license.

Niccolò Tubini Theoretical progress in freezing – thawing processes study

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 newinterpretation of modeling freezing soils.

Niccolò Tubini Theoretical progress in freezing – thawing processes study 2 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Why studing the influence of coupled heat and water flowin soils?

Freezing – thawing processes entail a hugeexchange of heat;

To simulate more realistic soil temperature(Luo et al., 2003).

Studies have shown that proper frozen soilschemes 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

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Why studing the influence of coupled heat and water flowin soils?

Freezing – thawing processes entail a hugeexchange of heat;

To simulate more realistic soil temperature(Luo et al., 2003).

Studies have shown that proper frozen soilschemes 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

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Why studing the influence of coupled heat and water flowin soils?

Freezing – thawing processes entail a hugeexchange of heat;To simulate more realistic soil temperature(Luo et al., 2003).

Studies have shown that proper frozen soilschemes 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

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

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

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

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Unsaturated soils

Young – Laplace equation

r γaw

α pw = pa −2γaw cosα

r

Niccolò Tubini Theoretical progress in freezing – thawing processes study 7 / 34

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

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 beselective processes.

Niccolò Tubini Theoretical progress in freezing – thawing processes study 9 / 34

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 thedefinition of the relationship between

θ–ψ and K–ψ

Niccolò Tubini Theoretical progress in freezing – thawing processes study 10 / 34

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 tothe soil texture and other soil properties

Despite their usfulness they do not emphasize thephysical significance of their empirical parameters

Niccolò Tubini Theoretical progress in freezing – thawing processes study 11 / 34

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 tothe soil texture and other soil properties

Despite their usfulness they do not emphasize thephysical significance of their empirical parameters

Niccolò Tubini Theoretical progress in freezing – thawing processes study 11 / 34

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 fromthe pore-size distribution:

f (r) := dθdr

Niccolò Tubini Theoretical progress in freezing – thawing processes study 12 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Unsaturated soil hydraulic properties

Lognormal distribution model (Kosugi, 1996)

r0

50

100

150

f(r)

R

Water

f (r) = θs − θr√2π σr

exp

−[ln( r

rm

)]22σ2

θ(R) = θr +∫ R

0f (r)dr

Niccolò Tubini Theoretical progress in freezing – thawing processes study 13 / 34

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 thepore-size distribution into the capillary pressure

distribution function

g(ψ) = f (r) drdψ

Niccolò Tubini Theoretical progress in freezing – thawing processes study 14 / 34

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 cosαg ρw R

Niccolò Tubini Theoretical progress in freezing – thawing processes study 15 / 34

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

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

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

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 thethermodynamic equilibrium

Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34

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 thethermodynamic equilibrium

Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34

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 thethermodynamic equilibrium

Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34

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 cosαρw ` r + πw Tm

ρw `

Capillary effectDissolved solutes

Niccolò Tubini Theoretical progress in freezing – thawing processes study 20 / 34

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:

r̂(T ) := −2 γaw Tm cosαρw `(T − Tm) for T < Tm∗

Niccolò Tubini Theoretical progress in freezing – thawing processes study 21 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Water and ice content

Let us define

r ∗ :=R if r̂ ≥ R or T ≥ Tm

r̂ otherwise

∂r ∗∂t :=

∂R∂t if r̂ ≥ R or T ≥ Tm

∂ r̂∂t otherwise

Niccolò Tubini Theoretical progress in freezing – thawing processes study 22 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Water and ice content

r0

50

100

150

f(r)

Rr̂ = r ∗

WaterIce

θw = θr +∫ r∗

0f (r)dr

θi =∫ R

r∗f (r)dr

Niccolò Tubini Theoretical progress in freezing – thawing processes study 23 / 34

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

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Water and ice content

The phase change rate

r0

20

40

60

80

100

120f(

r)

R(t) R(t + δt)r̂

WaterIce at time tIce at time t + δt

Niccolò Tubini Theoretical progress in freezing – thawing processes study 25 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Water and ice content

The phase change rate

r0

20

40

60

80

100

120f(

r)

Rr̂(t)r̂(t + δt)

WaterIce formed in δtIce at time t

Niccolò Tubini Theoretical progress in freezing – thawing processes study 26 / 34

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

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Mass conservation equation

θ

J ET

~Jw

∂

∂t (ρwθw + ρiθi) = −ρw∇ · ~Jw

Water flux:~Jw = −K (ψ∗) ~∇(ψ∗ + z)

Niccolò Tubini Theoretical progress in freezing – thawing processes study 28 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Mass conservation equation

θ

J ET

~Jw

∂

∂t (ρwθw + ρiθi) = −ρw∇ · ~Jw

Water flux:~Jw = −K (ψ∗) ~∇(ψ∗ + z)

Niccolò Tubini Theoretical progress in freezing – thawing processes study 28 / 34

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

∂tNiccolò Tubini Theoretical progress in freezing – thawing processes study 29 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Energy conservation equation

ε

J

HRn

ET

~Jw~Jg

∂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

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Energy conservation equation

ε

J

HRn

ET

~Jw~Jg

∂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

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

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Energy conservation equation: if ice occurs

ψ∗ := ψ̂∂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

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 + ρwcwθw

Niccolò Tubini Theoretical progress in freezing – thawing processes study 33 / 34

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 + ρwcwθw

Niccolò Tubini Theoretical progress in freezing – thawing processes study 33 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

To take home

Freezing=drying and rigid soil schemeassumptions are useful when freezing-inducedmechanical deformations are not considered;

Freezing/thawing processes do not occur at thethermodynamic equilibrium (Kurylyk, 2013).

Kosugi retention model has the benefit to bestraightforward extended to freezing soils caseby making use of Gibbs – Thomson equation;

Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

To take home

Freezing=drying and rigid soil schemeassumptions are useful when freezing-inducedmechanical deformations are not considered;

Freezing/thawing processes do not occur at thethermodynamic equilibrium (Kurylyk, 2013).

Kosugi retention model has the benefit to bestraightforward extended to freezing soils caseby making use of Gibbs – Thomson equation;

Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

To take home

Freezing=drying and rigid soil schemeassumptions are useful when freezing-inducedmechanical deformations are not considered;Freezing/thawing processes do not occur at thethermodynamic equilibrium (Kurylyk, 2013).

Kosugi retention model has the benefit to bestraightforward extended to freezing soils caseby making use of Gibbs – Thomson equation;

Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

To take home

This formulation allows to take into account ofdissolved solutes;

It is possible to solve the mass and energyequation in a decoupled way;

Niccolò Tubini Theoretical progress in freezing – thawing processes study 35 / 34

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

To take home

This formulation allows to take into account ofdissolved solutes;

It is possible to solve the mass and energyequation in a decoupled way;

Niccolò Tubini Theoretical progress in freezing – thawing processes study 35 / 34

Ottaw

aRiver,17

thDec

2016

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

References

I K. Kosugi, Lognormal distribution model for unsaturatedsoil hydraulic properties, Water Resources Research, vol. 32,no. 9, pp. 2697–2703, 1996.

I J. T. Acker et al., Intercellular ice propagation:experimental evidence for ice growth through membranepores, Biophysical journal, vol. 81, no. 3, pp. 1389–1397,2001.

I M. Dall’Amico et al., A robust and energy-conservingmodel of freezing variably-saturated soil, The Cryosphere,vol. 5, no. 2, p. 469, 2011.

I M. Dall’Amico, Coupled water and heat transfer inpermafrost modeling, Ph.D. dissertation, University ofTrento, 2010.Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

References

I E. J. Spaans and J. M. Baker, The soil freezingcharacteristic: Its measurement and similarity to the soilmoisture characteristic, Soil Science Society of AmericaJournal, vol. 60, no. 1, pp. 13–19, 1996.

I R. D. Miller, Phase equilibria and soil freezing, vol. 287, pp.193–197, 1965.

I B. L. Kurylyk and K. Watanabe, The mathematicalrepresentation of freezing and thawing processes invariably-saturated, non-deformable soils, Advances in WaterResources, vol. 60, pp. 160–177, 2013.

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

References

I 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.

I T. G. Smirnova, J. M. Brown, S. G. Benjamin, and D. Kim,Parameterization of cold-season processes in the mapsland-surface scheme, Journal of Geophysical Research:Atmospheres, vol. 105, no. D3, pp. 4077– 4086, 2000.

I P. Viterbo, A. Beljaars, J.-F. Mahfouf, and J. Teixeira, Therepresentation of soil moisture freezing and its impact onthe stable boundary layer, Quarterly Journal of the RoyalMeteorological Society, vol. 125, no. 559, pp. 2401–2426,1999.Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Freezing=drying assumptionDall’A

mico,

2010

pa

pw (R)

R

r

Air-water interfacepw (R) = pa −

2 γaw cosαR

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Freezing=drying assumptionDall’A

mico,

2010

pa

pi

pw (r)

R

r

Air-ice interfacepi = pa −

2 γai cosαR

Ice-water interfacepw (r) = pi −

2 γiw cosαr

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Freezing=drying assumptionDall’A

mico,

2010

pa

pi ≡ pa

pw (r)

R

r

Air-water interfacepw (r) = pa −

2 γaw cosαr

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

r0

20

40

60

80

100

120f(

r)

R = r ∗ r̂

Water

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

The phase change rate

Thanks to the Young-Laplace equation

ψ∗ :=

Ψ if r̂ ≥ 2 γaw cosα

ρw g Ψ or T ≥ Tm

ψ̂ = ψ(r̂) otherwise

Niccolò Tubini Theoretical progress in freezing – thawing processes study

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 r̂ ≥ 2γaw cosα

ρwgΨ or T ≥ Tm

∂ψ̂

∂t otherwise

Niccolò Tubini Theoretical progress in freezing – thawing processes study

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:

dTdpw

= Tρw `

T ∗ = Tm + g Tm

`ψw0

ψ(T ) = ψw0 + `

g T ∗ (T − T ∗)H(T ∗ − T )

Niccolò Tubini Theoretical progress in freezing – thawing processes study

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:

dTdpw

= Tρw `

T ∗ = Tm + g Tm

`ψw0

ψ(T ) = ψw0 + `

g T ∗ (T − T ∗)H(T ∗ − T )

Niccolò Tubini Theoretical progress in freezing – thawing processes study

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

∂tNiccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Mass conservation equation: if ice occurs

ψ∗ := ψ̂∂ψ∗

∂t := `

g Tm

∂T∂t

∂θ

∂t = ∂ψ̂

∂t g(Ψ) = ∇ · [K (ψ̂)~∇(ψ̂ + z)]

∂θi

∂t = ∂Ψ∂t g(Ψ)− ∂ψ̂

∂t g(ψ̂)

∂θw

∂t = ∂θ

∂t −∂θi

∂tNiccolò Tubini Theoretical progress in freezing – thawing processes study

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 − ρi l

������

������

����: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 = 0Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Energy conservation equation: if there is no ice

CT∂T∂t + ρwcw~Jw · ~∇T + ρwgz∇ · ~Jw −∇ · ~Jg = 0

CT := ρscs(1− θs) + ρiciθi + ρwcwθw

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Energy conservation equation: if there is no ice

CT∂T∂t + ρwcw~Jw · ~∇T + ρwgz∇ · ~Jw −∇ · ~Jg = 0

CT := ρscs(1− θs) + ρiciθi + ρwcwθw

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Numerical scheme for unfrozen soils

The mass conservation equation ⇒ Nested Newtonmethod (Casulli and Zanolli, 2010).

The energy consevation equation ⇒ Implicit upwindmethod

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Numerical scheme for frozen soils

The mass conservation equation becomes∂θ

∂t = ∇ ·(K (ψ̂) ~∇(ψ̂ + z)

)

Nested Newton method (Casulli and Zanolli, 2010).should be extended for equations of two variables

The energy consevation equation ⇒ Implicit upwindmethod

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Numerical scheme for frozen soils

The mass conservation equation becomes∂θ

∂t = ∇ ·(K (ψ̂) ~∇(ψ̂ + z)

)

Nested Newton method (Casulli and Zanolli, 2010).should be extended for equations of two variables

The energy consevation equation ⇒ Implicit upwindmethod

Niccolò Tubini Theoretical progress in freezing – thawing processes study

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions

Numerical scheme for frozen soils

The mass conservation equation becomes∂θ

∂t = ∇ ·(K (ψ̂) ~∇(ψ̂ + z)

)

Nested Newton method (Casulli and Zanolli, 2010).should be extended for equations of two variables

The energy consevation equation ⇒ Implicit upwindmethod

Niccolò Tubini Theoretical progress in freezing – thawing processes study