Freezing-thawing processes study with numerical models

Introduction Water flow in soil Freezing soil Numerical method

FREEZING-THAWING PROCESSES STUDYWITH NUMERICAL MODEL

Niccolò Tubini

Università degli Studi di Trento

6th October 2016


Contents

1 Introduction

2 Water flow in soilDarcy’s equationDarcy-Buckingham’s equationRichards’ equation

3 Freezing soilMass conservation lawEnergy conservation lawGround energy budget

4 Numerical method


What is the purpose?

The aim of my Master’s thesis is to develop a solver ofRichards’s equation 3D plus freezing soil with the NestedNewton method.


Why studing the influence of coupled heat and water flow insoils?

studies have shown that proper frozen soil schemes helpimprove land surface and climate model simulation(e.g. Viterbo et al., 1999 and Smirnova et al., 2000);

to simulate more realistic soil temperature(Luo et al., 2003);phase change in the ground;



studies have shown that proper frozen soil schemes helpimprove land surface and climate model simulation(e.g. Viterbo et al., 1999 and Smirnova et al., 2000);

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

phase change in the ground;



studies have shown that proper frozen soil schemes helpimprove land surface and climate model simulation(e.g. Viterbo et al., 1999 and Smirnova et al., 2000);to simulate more realistic soil temperature(Luo et al., 2003);

phase change in the ground;


Darcy’s equationR.

Rigo

n

Darcy’s experiment:

Jv = QA ∝

∆hL

Mathematically Darcy’s law:

Jv = −Ks∂h∂z


Darcy’s equationR.

Rigo

n

Darcy’s experiment:

Jv = QA ∝

∆hL

Mathematically Darcy’s law:

Jv = −Ks∂h∂z


Darcy’s equation

Go to in detail...

Physical quantities Unit ofmeasurement

h = z + pρwg Hydraulic head [L]

z Elevation head [L]

ψ = pρwg Pressure head [L]


Darcy’s equation

Go to in detail...






Darcy’s equation

Go to in detail...






Darcy-Buckingham’s equationPh

ukan,198

5

Watercontent:

θw = Vw

Vc


Darcy-Buckingham’s equation

Vadose zoneInfiltration often involves unsaturated flow through porousmedia.As a result:

capillary pressure arise;

cross-sectional area of the water conducting region isreduced.



Capillary pressurePressure is determined by the tesion andcurvature of air-water interface as given:

pw − pA = −γwa2r

{ pA = 0pw = −γwa

2r = −gρwz



Capillary pressurePressure is determined by the tesion andcurvature of air-water interface as given:

pw − pA = −γwa2r

{ pA = 0pw = −γwa

2r = −gρwz



Soil Water Retention Curve: θw = θw (ψ)https://

www.re

searchgate.net/



Soil Water Retention Curve: θw = θw (ψ)

Hypotesissolid matrix is rigid;

hydraulic hysteresis is ignored.



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



Reduction of cross-sectional areaDarcy’s law is independent of the size of particles or the stateof packing:

Ks → K (θw )



Parametric form of the capillary conductivity

Mualem, 1976

K (Se) = KsSνe[

f (Se)f (1)

]2

f (Se) =∫ Se

0

1ψ(x)dx

Se = θ(ψ)− θr

θs − θr



Parametric form of the capillary conductivity

Choosing Van Genuchten’s parametric SWRC

K (Se) = KsSνe[1−

(1− S1/m

e

)m]2m = 1− 1/n

or

K (ψ) =Ks{1− (αψ)mn [1 + (αψ)n]−m}2

[1 + (αψ)n]mν m = 1− 1/n



Darcy-Buckingham’s lawIn vadose zone, specif discharge can be written as:

~Jv = K (θ(ψ))~∇(h)


Richards’ equation

Equation of continuity for capillary flow

HypotesisIt is assumed that no phase transition takes place;density of water is constant.

∂θ

∂t = ~∇ · ~Jv (ψ)


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


Phuk

an,1

985

Ice content:

θi = Vi

Vc


What is needed to study freezing soil?water can be both in liquid and solid phase;

freezing/thawing processes involve energy fluxes;

soil temperature.




soil temperature.




soil temperature.


Hypotesisrigid soil scheme ⇒ ρw = ρi ;

"Freezing = drying" (Miller, 1965; Spaans and Baker,1996).


Mass conservation law

We need of new closure equationsθw = θw (ψ, ?)

θi = ?



Pressure and temperature under freezing conditionDall’A

mico,

2010

Air-water interfacepw0 = pa − 2γaw/r0




mico,

2010

Air-ice interfacepi = pa − 2γai/r0

Ice-water interfacepw1 = pi − 2γiw/r1




mico,

2010

Air-water interfacepw1 = pa − 2γaw/r1



Clapeyron’s equation

ρwLfdTT = dpw

Integrating ∫ T∗

TmLf

dTT =

∫ pw0

0dp′w

∫ T∗

TmLf

dTT = Lf ln

(T ∗Tm

)≈ Lf

T ∗ − Tm

Tm



Clapeyron’s equation

ρwLfdTT = dpw


TmLf

dTT =

∫ pw0

0dp′w

∫ T∗

TmLf

dTT = Lf ln

(T ∗Tm

)≈ Lf

T ∗ − Tm

Tm




TmLf

dTT =

∫ pw0

0dp′w

Melting temperature at unsatured conditions

T ∗ = Tm + gTm

Lfψw0

If the soil is unsaturated, the surface tension at water-airinterface

decreases the water melting temperature to a value T ∗ < Tm.




TmLf

dTT =

∫ pw0

0dp′w


T ∗ = Tm + gTm

Lfψw0

If the soil is unsaturated, the surface tension at water-airinterface

decreases the water melting temperature to a value T ∗ < Tm.



Integrating ∫ T

TmLf

dTT =

∫ pw

pw0dp′w


ψ(T ) = ψw0gTm

Lf(T − T ∗)

Water pressure depends on the intensity of freezing conditionprovided by T .



Integrating ∫ T

TmLf

dTT =

∫ pw

pw0dp′w


ψ(T ) = ψw0gTm

Lf(T − T ∗)

Water pressure depends on the intensity of freezing conditionprovided by T .



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 ∗



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 ∗



Total water content, liquid water content and ice contentAccording 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 )]




Θv = θr + (θs − θr ) · {1 + [−αψw0]n}−m

θw = θr + (θs − θr ) · {1 + [−αψ(T )]n}−m





Θv = θr + (θs − θr ) · {1 + [−αψw0]n}−m

θw = θr + (θs − θr ) · {1 + [−αψ(T )]n}−m




SWC and SFCDall’A

mico,

2010



Richards’equation in freezing soil∂Θm(ψw0,T )

∂t + ~∇ · ~Jv (ψw0,T ) + Sw = 0

~Jv = −K ~∇(z + ψ)

Θm = θw + ρi

ρwθi


Energy conservation law

UThe energy content of the soil is represented by the internalenergy U [Jm−3].

U = Usp + Ui + Uw



Energy conservation law∂U∂t + ~∇ · (~G + ~J) + Sen = 0



Go in details

Sen

It represents a sink term due to energy losses. [Wm−3]



Go in details

~GIt is the conduction flux through the volume boundaries.[Wm−2]

~G = −λT ~∇T



Go in details

~JIt is the heat advected by flowing water. [Wm−2]

~J = ρw · [Lf + cw (T − Tref ] · ~Jv


Ground energy budget

Ground energy budget

Cp∂T∂t +

∑k∈{w ,i}

ρkhk = R − λeET − H


Richards’ equation 1D∂θ(ψ)∂t = ∂

∂x

[K (ψ)∂ψ

∂x + K (ψ)]


Finite volume scheme: FTCS

θn+1i = θn

i + ∆t∆x

[K n

i+1/2ψn+1

i+1 − ψn+1i

∆x + K ni+1/2

]−

∆t∆x

[K n

i−1/2ψn+1

i − ψn+1i−1

∆x − K ni−1/2

]

System in matrix form~θ + T~ψ = ~rhs


Finite volume scheme: FTCS

θn+1i = θn

i + ∆t∆x

[K n

i+1/2ψn+1

i+1 − ψn+1i

∆x + K ni+1/2

]−

∆t∆x

[K n

i−1/2ψn+1

i − ψn+1i−1

∆x − K ni−1/2

]

System in matrix form~θ + T~ψ = ~rhs


Newton-Raphson methodThe system is non linear so it must be solved by iteration.

The moisture content is in general a nonlinear function of thepressure head: it’s derivative isn’t nondecreasing nonincreasingfunction.


Jordan decompositionIf the derivative of the functions is nondecreasing, thenNewton’s method converge.

The idea is to find two nondecreasing function whose differenceapproximate θ.

θ(ψ) = θ1(ψ)− θ2(ψ)


Jordan decompositionIf the derivative of the functions is nondecreasing, thenNewton’s method converge.

The idea is to find two nondecreasing function whose differenceapproximate θ.

θ(ψ) = θ1(ψ)− θ2(ψ)


Nested Newton method

Splitting ~θ~θ1(~ψn+1)− ~θ2(~ψn+1) + T~ψn+1 − ~rhs

n= 0

~θ1 linearization−~θ2(T n) +

[M + ~θ1(T n)′

]· ~T n+1,k +[

~θ1(T n)− ~θ1(T n)′ ~T n − ~bn+1]

= 0

~θ2 linearization[M + ~θ1(T n)′ − ~θ2(T n)′

]· ~T n+1,k,l +[

~θ1(T n)− ~θ2(T n)−(~θ1(T n)′ − ~θ2(T n)′

)~T n − ~bn+1

]= 0




n= 0


[M + ~θ1(T n)′

]· ~T n+1,k +[

~θ1(T n)− ~θ1(T n)′ ~T n − ~bn+1]

= 0


]· ~T n+1,k,l +[

~θ1(T n)− ~θ2(T n)−(~θ1(T n)′ − ~θ2(T n)′

)~T n − ~bn+1

]= 0




n= 0


[M + ~θ1(T n)′

]· ~T n+1,k +[

~θ1(T n)− ~θ1(T n)′ ~T n − ~bn+1]

= 0


]· ~T n+1,k,l +[

~θ1(T n)− ~θ2(T n)−(~θ1(T n)′ − ~θ2(T n)′

)~T n − ~bn+1

]= 0

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Freezing-thawing processes study with numerical models

Education