Master thesis presentation by Niccolò Tubini

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

Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
This work is licensed under a Creative
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International” license.
Niccolò Tubini Theoretical progress in freezing – thawing processes study

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

Why studing the inﬂuence of coupled heat and water ﬂow
in soils?
Freezing – thawing processes entail a huge
exchange of heat;

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

in soils?
Studies have shown that proper frozen soil
schemes 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 cos α
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 be
selective processes.

Water – retention – hydraulic – conductivity models
Dealing with unsaturated soils requires the
deﬁnition of the relationship between
θ–ψ and K–ψ

Empirical curve-ﬁtting models
Parameters of these models have been related to
the soil texture and other soil 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

Lognormal distribution model (Kosugi, 1996)
The idea is to derive the water retention curve from
the pore-size distribution:
f (r) :=
dθ
dr

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

Young-Laplace equation allows to transform the
pore-size distribution into the capillary pressure
distribution function
g(ψ) = f (r)
dr
dψ

θ(Ψ) = θr +
Ψ
−∞
g(ψ)dψ
Ψ = −
2γaw cos α
g 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

Model assumptions
Model assumptions
rigid soil scheme
freezing = drying (Miller, 1965; Spaans and Baker, 1996)

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

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

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∗

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

r
0
50
100
150
f(r)
Rˆr = r∗
Water
Ice
θw = θr +
r∗
0
f (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∗
)

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

r
0
20
40
60
80
100
120
f(r)
Rˆr(t)ˆr(t + δt)
Water
Ice formed in δt
Ice at time t

∂θi
∂t
=
∂Ψ
∂t
g(Ψ) −
∂ψ∗
∂t
g(ψ∗
)

Mass conservation equation
θ
J ET
Jw
∂
∂t
(ρw θw + ρiθi) = −ρw · Jw

θ
J ET
Jw
∂
∂t
(ρw θw + ρiθi) = −ρw · Jw
Water ﬂux:
Jw = −K(ψ∗
) (ψ∗
+ z)

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

Energy conservation equation
ε
J
HRn
ET
JwJg
∂ε
∂t
= − · (Jw + Jg)

ε
J
HRn
ET
JwJg
∂ε
∂t
= − · (Jw + Jg)
Advective ﬂux:
Jw = Jw ρw [ + cw (T − Tm)]
Heat conduction:
Jg = −λ T

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∗
:= 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

The apparent heat capacity
Cph := CT + ρi[ + (cw − ci)(T − Tm)]
g Tm

The apparent heat capacity
Cph := CT + ρi[ + (cw − ci)(T − Tm)]
g Tm
CT := ρscs(1 − θs) + ρiciθi + ρw cw θw

To take home
Freezing=drying and rigid soil scheme
assumptions are useful when freezing-induced
mechanical deformations are not considered;

To take home
Freezing/thawing processes do not occur at the
thermodynamic equilibrium (Kurylyk, 2013).

To take home
Kosugi retention model has the beneﬁt to be
straightforward extended to freezing soils case
by making use of Gibbs – Thomson equation;

To take home
This formulation allows to take into account of
dissolved solutes;

To take home
It is possible to solve the mass and energy
equation in a decoupled way;

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.

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.

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.

Freezing=drying assumptionDall’Amico,2010
pa
pw (R)
R
r
Air-water interface
pw (R) = pa −
2 γaw cos α
R

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

pa
pi ≡ pa
pw (r)
R
r
Air-water interface
pw (r) = pa −
2 γaw cos α
r

r
0
20
40
60
80
100
120f(r)
R = r∗
ˆr
Water

Thanks to the Young-Laplace equation
ψ∗
:=



Ψ if ˆr ≥
2 γaw cos α
ρw g Ψ
or T ≥ Tm
ˆψ = ψ(ˆr) otherwise

Thanks to the Young-Laplace equation
∂ψ∗
∂t
:=



∂Ψ
∂t
if ˆr ≥
2γaw cos α
ρw gΨ
or T ≥ Tm
∂ ˆψ
∂t
otherwise

Comparison with Dall’Amico model (Dall’Amico et al., 2011)
By making use of Clausius – Clapeyron equation:
dT
dpw
=
T
ρw

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)

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

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

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

CT
∂T
∂t
+ ρw cw Jw · T + ρw gz · Jw − · Jg = 0

CT
∂T
∂t
+ ρw cw Jw · T + ρw gz · Jw − · Jg = 0
CT := ρscs(1 − θs) + ρiciθi + ρw cw θw

Numerical scheme for unfrozen soils
The mass conservation equation ⇒ Nested Newton
method (Casulli and Zanolli, 2010).
The energy consevation equation ⇒ Implicit upwind
method

Numerical scheme for frozen soils
The mass conservation equation becomes
∂θ
∂t
= · K( ˆψ) ( ˆψ + z)

∂θ
∂t
= · K( ˆψ) ( ˆψ + z)
Nested Newton method (Casulli and Zanolli, 2010).
should be extended for equations of two variables

∂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

Master thesis presentation by Niccolò Tubini

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