freezing soil for the class of environmental modelling

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

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Niccolò Tubini Theoretical progress in freezing – thawing processes study

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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


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).



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).



Unsaturated soils

Some definitions

Air gas

Liquid water

Soil particle

Va

Vw

Vs

Vc



Unsaturated soils

Some definitions

Soil porosity

φ := Vs

Vc

Water content

θ := Vw

Vc



Unsaturated soils

Some definitions

Assuming the rigid soil scheme

θs := φ

0 < θr ≤ θ ≤ θs < 1



Unsaturated soils

Young – Laplace equation

r γaw

α pw = pa −2γaw cosα

r



Unsaturated soils

Young – Laplace equation

pa ← 0

Let us define suction as

ψ := pw

gρw



Unsaturated soil hydraulic properties

Mualem’s assumption

Wetting and drying processes are assumed to beselective processes.




Water – retention – hydraulic – conductivity models

Dealing with unsaturated soils requires thedefinition of the relationship between

θ–ψ and K–ψ




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




Lognormal distribution model (Kosugi, 1996)

The idea is to derive the water retention curve fromthe pore-size distribution:

f (r) := dθdr





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





Young-Laplace equation allows to transform thepore-size distribution into the capillary pressure

distribution function

g(ψ) = f (r) drdψ





θ(Ψ) = θr +∫ Ψ

−∞g(ψ)dψ

Ψ = −2γaw cosαg ρw R



Some definitions

Air gas

Ice

Liquid water

Particle soil

Va

Vi

Vw

Vs

Vc



Some definitions

Liquid water content

θw := Vw

Vc

Ice content

θi := Vi

Vc



Some definitions

Total water content

θ := θw + θi

0 < θr ≤ θ ≤ θs < 1



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



Freezing point depression

Gibbs-Thomson equation (Acker et al., 2001)

Tm − T ∗ = 2 γaw Tm cosαρw ` r + πw Tm

ρw `

Capillary effectDissolved solutes



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∗



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




r0

50

100

150

f(r)

Rr̂ = r ∗

WaterIce

θw = θr +∫ r∗

0f (r)dr

θi =∫ R

r∗f (r)dr




The phase change rate

θi =∫ R

r∗f (r)dr

∂θi

∂t = ∂R∂t f (R)− ∂r ∗

∂t f (r ∗)





r0

20

40

60

80

100

120f(

r)

R(t) R(t + δt)r̂

WaterIce at time tIce at time t + δt





r0

20

40

60

80

100

120f(

r)

Rr̂(t)r̂(t + δt)

WaterIce formed in δtIce at time t





∂θi

∂t = ∂Ψ∂t g(Ψ)− ∂ψ∗

∂t g(ψ∗)



Mass conservation equation

θ

J ET

~Jw

∂

∂t (ρwθw + ρiθi) = −ρw∇ · ~Jw

Water flux:~Jw = −K (ψ∗) ~∇(ψ∗ + z)



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


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



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



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



The apparent heat capacity

Cph := CT + ρi [` + (cw − ci)(T − Tm)] `

g Tm

CT := ρscs(1− θs) + ρiciθi + ρwcwθw



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;



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;



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;


Ottaw

aRiver,17

thDec

2016


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


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.



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


Freezing=drying assumptionDall’A

mico,

2010

pa

pw (R)

R

r

Air-water interfacepw (R) = pa −

2 γaw cosαR




mico,

2010

pa

pi

pw (r)

R

r

Air-ice interfacepi = pa −

2 γai cosαR

Ice-water interfacepw (r) = pi −

2 γiw cosαr




mico,

2010

pa

pi ≡ pa

pw (r)

R

r

Air-water interfacepw (r) = pa −

2 γaw cosαr



r0

20

40

60

80

100

120f(

r)

R = r ∗ r̂

Water




Thanks to the Young-Laplace equation

ψ∗ :=

Ψ if r̂ ≥ 2 γaw cosα

ρw g Ψ or T ≥ Tm

ψ̂ = ψ(r̂) otherwise




Thanks to the Young-Laplace equation

∂ψ∗

∂t :=

∂Ψ∂t if r̂ ≥ 2γaw cosα

ρwgΨ or T ≥ Tm

∂ψ̂

∂t otherwise



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 )



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


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


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


Energy conservation equation: if there is no ice

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

CT := ρscs(1− θs) + ρiciθi + ρwcwθw



Numerical scheme for unfrozen soils

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

The energy consevation equation ⇒ Implicit upwindmethod



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


freezing soil for the class of environmental modelling

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