The document discusses the development of a numerical model to simulate coupled heat and water flow processes during soil freezing and thawing. It introduces the governing equations, including Richards' equation to model unsaturated water flow and equations for mass and energy conservation to account for phase changes during freezing. A finite volume scheme is presented to discretize and solve Richards' equation, while closure relationships are required to define water and ice content as functions of pressure and temperature during freezing conditions. The overall goal is to simulate realistic soil temperatures and phase changes in the ground.

1. Introduction Water flow in soil Freezing soil Numerical method
FREEZING-THAWING PROCESSES STUDY
WITH NUMERICAL MODEL
Niccolò Tubini
Università degli Studi di Trento
6th October 2016

2. Introduction Water flow in soil Freezing soil Numerical method
Contents
1 Introduction
2 Water flow in soil
Darcy’s equation
Darcy-Buckingham’s equation
Richards’ equation
3 Freezing soil
Mass conservation law
Energy conservation law
Ground energy budget
4 Numerical method

3. Introduction Water flow in soil Freezing soil Numerical method
What is the purpose?
The aim of my Master’s thesis is to develop a solver of
Richards’s equation 3D plus freezing soil with the Nested
Newton method.

4. Introduction Water flow in soil Freezing soil Numerical method
Why studing the influence of coupled heat and water flow in
soils?
studies have shown that proper frozen soil schemes help
improve land surface and climate model simulation
(e.g. Viterbo et al., 1999 and Smirnova et al., 2000);

5. Introduction Water flow in soil Freezing soil Numerical method
Why studing the influence of coupled heat and water flow in
soils?
to simulate more realistic soil temperature
(Luo et al., 2003);

6. Introduction Water flow in soil Freezing soil Numerical method
Why studing the influence of coupled heat and water flow in
soils?
phase change in the ground;

7. Introduction Water flow in soil Freezing soil Numerical method
Darcy’s equation
R.Rigon
Darcy’s experiment:
Jv =
Q
A
∝
∆h
L

8. Introduction Water flow in soil Freezing soil Numerical method
Darcy’s equation
R.Rigon
Darcy’s experiment:
Jv =
Q
A
∝
∆h
L
Mathematically Darcy’s law:
Jv = −Ks
∂h
∂z

9. Introduction Water flow in soil Freezing soil Numerical method
Darcy’s equation
Go to in detail...
Physical quantities
Unit of
measurement
h = z +
p
ρw g
Hydraulic head [L]

10. Introduction Water flow in soil Freezing soil Numerical method
Darcy’s equation
Go to in detail...
Physical quantities
Unit of
measurement
h = z +
p
ρw g
Hydraulic head [L]
z Elevation head [L]

11. Introduction Water flow in soil Freezing soil Numerical method
Darcy’s equation
Go to in detail...
Physical quantities
Unit of
measurement
h = z +
p
ρw g
Hydraulic head [L]
z Elevation head [L]
ψ =
p
ρw g
Pressure head [L]

12. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equationPhukan,1985
Water
content:
θw =
Vw
Vc

13. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Vadose zone
Infiltration often involves unsaturated flow through porous
media.
As a result:
capillary pressure arise;
cross-sectional area of the water conducting region is
reduced.

14. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Capillary pressure
Pressure is determined by the tesion and
curvature of air-water interface as given:
pw − pA = −γwa
2
r

15. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Capillary pressure
Pressure is determined by the tesion and
curvature of air-water interface as given:
pw − pA = −γwa
2
r
pA = 0
pw = −γwa
2
r
= −gρw z

16. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Soil Water Retention Curve: θw = θw (ψ)https://www.researchgate.net/

17. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Soil Water Retention Curve: θw = θw (ψ)
Hypotesis
solid matrix is rigid;
hydraulic hysteresis is ignored.

18. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Parametric form of the SWRC
Equation Authors
θ = θr + (θs − θr )(ψm/ψe)λ
Brooks and Corey
θ = θr + (θs − θr ) [1 + (αψm)n
]−m
Van Genuchten

19. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Reduction of cross-sectional area
Darcy’s law is independent of the size of particles or the state
of packing:
Ks → K(θw )

20. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Parametric form of the capillary conductivity
Mualem, 1976
K(Se) = KsSν
e
f (Se)
f (1)
2
f (Se) =
Se
0
1
ψ(x)
dx
Se =
θ(ψ) − θr
θs − θr

21. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Parametric form of the capillary conductivity
Choosing Van Genuchten’s parametric SWRC
K(Se) = KsSν
e 1 − 1 − S1/m
e
m 2
m = 1 − 1/n
or
K(ψ) =
Ks 1 − (αψ)mn
[1 + (αψ)n
]
−m 2
[1 + (αψ)n
]
mν m = 1 − 1/n

22. Introduction Water flow in soil Freezing soil Numerical method
Darcy-Buckingham’s equation
Darcy-Buckingham’s law
In vadose zone, specif discharge can be written as:
Jv = K(θ(ψ)) (h)

23. Introduction Water flow in soil Freezing soil Numerical method
Richards’ equation
Equation of continuity for capillary flow
Hypotesis
It is assumed that no phase transition takes place;
density of water is constant.
∂θ
∂t
= · Jv (ψ)

24. Introduction Water flow in soil Freezing soil Numerical method
Richards’ equation
To sum up
C(ψ)
∂ψ
∂t
= · K(θ) (z + ψ)
C(ψ) =
∂θ
∂ψ
Se = [1 + (−αψ)m
]
−n
Se =
θ − θr
θs − θr
K(Se) = Ks
√
Se 1 − (1 − Se)1/m
m 2

25. Introduction Water flow in soil Freezing soil Numerical method
Phukan,1985
Ice content:
θi =
Vi
Vc

26. Introduction Water flow in soil Freezing soil Numerical method
What is needed to study freezing soil?
water can be both in liquid and solid phase;

27. Introduction Water flow in soil Freezing soil Numerical method
What is needed to study freezing soil?
water can be both in liquid and solid phase;
freezing/thawing processes involve energy fluxes;

28. Introduction Water flow in soil Freezing soil Numerical method
What is needed to study freezing soil?
water can be both in liquid and solid phase;
freezing/thawing processes involve energy fluxes;
soil temperature.

29. Introduction Water flow in soil Freezing soil Numerical method
Hypotesis
rigid soil scheme ⇒ ρw = ρi ;
"Freezing = drying" (Miller, 1965; Spaans and Baker,
1996).

30. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
We need of new closure equations
θw = θw (ψ, ?)
θi = ?

31. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Pressure and temperature under freezing condition
Dall’Amico,2010
Air-water interface
pw0 = pa − 2γaw /r0

32. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Pressure and temperature under freezing conditionDall’Amico,2010
Air-ice interface
pi = pa − 2γai /r0
Ice-water interface
pw1 = pi − 2γiw /r1

33. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Pressure and temperature under freezing condition
Dall’Amico,2010
Air-water interface
pw1 = pa − 2γaw /r1

34. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Clapeyron’s equation
ρw Lf
dT
T
= dpw
Integrating
T∗
Tm
Lf
dT
T
=
pw0
0
dpw

35. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Clapeyron’s equation
ρw Lf
dT
T
= dpw
Integrating
T∗
Tm
Lf
dT
T
=
pw0
0
dpw
T∗
Tm
Lf
dT
T
= Lf ln
T∗
Tm
≈ Lf
T∗
− Tm
Tm

36. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Integrating
T∗
Tm
Lf
dT
T
=
pw0
0
dpw

37. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Integrating
T∗
Tm
Lf
dT
T
=
pw0
0
dpw
Melting temperature at unsatured conditions
T∗
= Tm +
gTm
Lf
ψw0
If the soil is unsaturated, the surface tension at water-air
interface
decreases the water melting temperature to a value T∗
< Tm.

38. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Integrating
T
Tm
Lf
dT
T
=
pw
pw0
dpw

39. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Integrating
T
Tm
Lf
dT
T
=
pw
pw0
dpw
Melting temperature at unsatured conditions
ψ(T) = ψw0
gTm
Lf
(T − T∗
)
Water pressure depends on the intensity of freezing condition
provided by T.

40. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Therefore



ψ(T) = ψw0 +
Lf
gT∗
(T − T∗
) T < T∗
ψ(T) = ψw0 T ≥ T∗

41. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Therefore



ψ(T) = ψw0 +
Lf
gT∗
(T − T∗
) T < T∗
ψ(T) = ψw0 T ≥ T∗



K = K(θw )10−ωq
T < T∗
K = K(θw ) T ≥ T∗

42. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Total water content, liquid water content and ice content
According to the Van Genuchten model:
Θv = θr + (θs − θr ) · {1 + [−αψw0]n
}
−m

43. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Total water content, liquid water content and ice content
According to the Van Genuchten model:
Θv = θr + (θs − θr ) · {1 + [−αψw0]n
}
−m
θw = θr + (θs − θr ) · {1 + [−αψ(T)]n
}
−m

44. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Total water content, liquid water content and ice content
According to the Van Genuchten model:
Θv = θr + (θs − θr ) · {1 + [−αψw0]n
}
−m
θw = θr + (θs − θr ) · {1 + [−αψ(T)]n
}
−m
θi = Θv (ψw0) − θw [ψ(T)]

45. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
SWC and SFCDall’Amico,2010

46. Introduction Water flow in soil Freezing soil Numerical method
Mass conservation law
Richards’equation in freezing soil
∂Θm(ψw0, T)
∂t
+ · Jv (ψw0, T) + Sw = 0
Jv = −K (z + ψ)
Θm = θw +
ρi
ρw
θi

47. Introduction Water flow in soil Freezing soil Numerical method
Energy conservation law
U
The energy content of the soil is represented by the internal
energy U [Jm−3
].
U = Usp + Ui + Uw

48. Introduction Water flow in soil Freezing soil Numerical method
Energy conservation law
Energy conservation law
∂U
∂t
+ · (G + J) + Sen = 0

49. Introduction Water flow in soil Freezing soil Numerical method
Energy conservation law
Go in details
Sen
It represents a sink term due to energy losses. [Wm−3
]

50. Introduction Water flow in soil Freezing soil Numerical method
Energy conservation law
Go in details
G
It is the conduction flux through the volume boundaries.
[Wm−2
]
G = −λT T

51. Introduction Water flow in soil Freezing soil Numerical method
Energy conservation law
Go in details
J
It is the heat advected by flowing water. [Wm−2
]
J = ρw · [Lf + cw (T − Tref ] · Jv

52. Introduction Water flow in soil Freezing soil Numerical method
Ground energy budget
Ground energy budget
Cp
∂T
∂t
+
k∈{w,i}
ρkhk = R − λeET − H

53. Introduction Water flow in soil Freezing soil Numerical method
Richards’ equation 1D
∂θ(ψ)
∂t
=
∂
∂x
K(ψ)
∂ψ
∂x
+ K(ψ)

54. Introduction Water flow in soil Freezing soil Numerical method
Finite volume scheme: FTCS
θn+1
i = θn
i +
∆t
∆x
Kn
i+1/2
ψn+1
i+1 − ψn+1
i
∆x
+ Kn
i+1/2 −
∆t
∆x
Kn
i−1/2
ψn+1
i − ψn+1
i−1
∆x
− Kn
i−1/2

55. Introduction Water flow in soil Freezing soil Numerical method
Finite volume scheme: FTCS
θn+1
i = θn
i +
∆t
∆x
Kn
i+1/2
ψn+1
i+1 − ψn+1
i
∆x
+ Kn
i+1/2 −
∆t
∆x
Kn
i−1/2
ψn+1
i − ψn+1
i−1
∆x
− Kn
i−1/2
System in matrix form
θ + Tψ = rhs

56. Introduction Water flow in soil Freezing soil Numerical method
Newton-Raphson method
The system is non linear so it must be solved by iteration.
The moisture content is in general a nonlinear function of the
pressure head: it’s derivative isn’t nondecreasing nonincreasing
function.

57. Introduction Water flow in soil Freezing soil Numerical method
Jordan decomposition
If the derivative of the functions is nondecreasing, then
Newton’s method converge.

58. Introduction Water flow in soil Freezing soil Numerical method
Jordan decomposition
If the derivative of the functions is nondecreasing, then
Newton’s method converge.
The idea is to find two nondecreasing function whose difference
approximate θ.
θ(ψ) = θ1(ψ) − θ2(ψ)

59. Introduction Water flow in soil Freezing soil Numerical method
Nested Newton method
Splitting θ
θ1(ψn+1
) − θ2(ψn+1
) + Tψn+1
− rhs
n
= 0

60. Introduction Water flow in soil Freezing soil Numerical method
Nested Newton method
Splitting θ
θ1(ψn+1
) − θ2(ψn+1
) + Tψn+1
− rhs
n
= 0
θ1 linearization
−θ2(Tn
) + M + θ1(Tn
) · Tn+1,k
+
θ1(Tn
) − θ1(Tn
) Tn
− bn+1
= 0

61. Introduction Water flow in soil Freezing soil Numerical method
Nested Newton method
Splitting θ
θ1(ψn+1
) − θ2(ψn+1
) + Tψn+1
− rhs
n
= 0
θ1 linearization
−θ2(Tn
) + M + θ1(Tn
) · Tn+1,k
+
θ1(Tn
) − θ1(Tn
) Tn
− bn+1
= 0
θ2 linearization
M + θ1(Tn
) − θ2(Tn
) · Tn+1,k,l
+
θ1(Tn
) − θ2(Tn
)−
θ1(Tn
) − θ2(Tn
) Tn
− bn+1
= 0