Egu2017 pico

New insights in permafrost modelling
Niccolò Tubini, Francesco Serafin, Stephan Gruber, Vincenzo Casulli, and Riccardo Rigon
EGU General Assembly, 23-28 April 2017
Secessionbuilding,Wien

!2
3.2 water in soils 27
0 2 4 6 8
r [cm] 10−3
0
20
40
60
80
100
120
f(r)[cm−1]
R
Figure 7: The pore-size distribution is described by a two-parameters lognormal
distribution [50] for θs = 0.4, θr = 0.1, σ = 0.6, rm = 2.1 10−3[cm].
According with Mualem’s assumption water fulﬁlls pores with radius r
R.
where θr is the residual water content. Furthermore, the relation between
Se and R is
Se =
1
θs − θr
R
−0
f(r)dr (14)
At the beginning
Rigon et al.
Is just the pore size distribution

!3
Rigon et al.
What happens when we decrease temperature
Just the largest pore water freezes. The equation that connects the real
freezing temperature with the pore size is:
F r e e z i n g
Temperature Pore radius
Full story here (section 4.3): New insights in permafrost modelling, by N. Tubini

!4
What happens when we decrease temperature
Just the largest pore water freezes. The equation that connects the real
freezing temperature with the pore size is:
F r e e z i n g
Temperature of
capillary water
Pore radius
Full story here (section 4.3): New insights in permafrost modelling, by N. Tubini
F r e e z i n g
Temperature of
free water
s p e c i f i c
volume of
water
o s m o t i c
pressurec o n t a c t
angle
energy for unit
a r e a o f t h e
w a t e r - a i r
interface
entalphy of
fusion
Rigon et al.

!5
consistently with the fact that both the water and ice occur.
Conversely in Fig.(14), according with Eq.(108) the water-ice interface r∗ is
set to coincides with R and therefore Eq.(109) and Eq.(110) becomes respec-
tively
θw = θr +
R
0
f(r)dr θi =
R
R
f(r)dr = 0 (112)
consistently with the state of the system.
0 2 4 6 8
r [cm]
10−3
0
20
40
60
80
100
120
f(r)[cm−1]
Rˆr = r∗
Water
Ice
distribution [50] for θs = 0.4, θr = 0.1, σ = 0.6, rm = 2.1 10−3[cm]. Let
us assume that ρw = ρi. According with Eq.(108) water in pores with
radius smaller than r∗ remains liquid, whilst that in pores with radius
r∗ r R turns into ice.
According with Kosugi’s water retention model we have that
So, the situation inside the control volume is:
Rigon et al.
Smallest pore are free of ice and largest pores are frozen

!6
1. How the hell did you obtain the formula ?
2. How this can be used in the water mass
budget ?
3. How can this be coupled with the energy
budget (and, BTW, how it is the energy
budget) ?
4. When you have the two equations how can
you solve them ?
Questions
Rigon et al.

!7
Find this presentation at
http://abouthydrology.blogspot.com
Ulrici,2000?
Other material at
Links
http://abouthydrology.blogspot.co.at/2017/04/new-insights-in-permafrost-modelling.html
Rigon et al.

New insight in permafrost modelling:
Continue for the PICO-ers
Niccolò Tubini, Francesco Serafin, Stephan Gruber, Vincenzo Casulli, and Riccardo Rigon
EGU General Assembly, 23-28 April 2017
Secessionbuilding,Wien

!9
Life beyond oral (in a Pico session)
How the hell did you obtain the formula ?
A clean description of it can be found in New insights in permafrost
modelling, by N. Tubini, 2017 (click on the links ;-)
However, he derives it using the Thermodynamics notation in: Matteo
Dall’Amico, Coupled water and heat transfer in permafrost modelling,
2010  
Moreover he possibly follows the treatment by: Acker, J. A. Elliott, and L. E.
McGann, Intercellular ice propagation: experimental evidence for ice growth
through membrane pores, Biophysical journal, vol. 81, no.3, 1389–1397,
2001
Rigon et al.

!10
How this can be used in the water mass budget ?
Newinsightsinpermafrostmodelling,byN.Tubini,
Rigon et al.
Mass Budget - dashed lines connect to boundary conditions

!11
This representation is obtained by means a graphical representation
called Time Continuous Petri Net (TCPN), which is fully explained in
this AboutHydrology post.
Rigon et al.
Time continuos Petri Nets

Hydraulic conductivity !12
Suction Gravity
Rigon et al.
TCPN corresponds to differential equations

!13
Nothing special so far … just the usual form of Richards’ equation
(REq). However, if we introduce the pore size distribution f(r):
pore size distribution
largest pore size
filled
r e s i d u a l w a t e r
content
Rigon et al.
Mass Budget - connections between pore size distribution and volumetric water content

!14
becomes:
And:
After the equilibrium thermodynamical hypothesis that
smallest pores fill first (and largest pores empty first)
Rigon et al.
Mass Budget - connections between pore size distribution and volumetric water content

!15
Suction itself, via the Young-Laplace equation, can be seen as a function
of the largest pore size (according to Kosugi et al., 1999):
Or, viceversa, the pore size occupied can be related to a suction
From which it is easy to obtain the usual formulation of REq as
function of suction (via hydraulic capacity)
acceleration due to gravity
a convenient constant
Rigon et al.
Which is the independent variable ? … It depends

!16
This managing of the REq as function of pore size becomes useful when
we want to understand the mechanism of soil freezing. The equation
below:
can, in fact, be seen as revealing which pore size contains frozen water
given the control volume temperature
In this case, the smallest pores frozen are given by:
Rigon et al.
Arguing about the freezing point depression temperature

!17
becomes respectively
θw = θr +
r∗
0
f(r)dr θi =
R
r∗
f(r)dr (111)
consistently with the fact that both the water and ice occur.
Conversely in Fig.(14), according with Eq.(108) the water-ice interface r∗ is
set to coincides with R and therefore Eq.(109) and Eq.(110) becomes respec-
tively
θw = θr +
R
0
f(r)dr θi =
R
R
f(r)dr = 0 (112)
consistently with the state of the system.
0 2 4 6 8
r [cm]
10−3
0
20
40
60
80
100
120
f(r)[cm−1]
Rˆr = r∗
Water
Ice
distribution [50] for θs = 0.4, θr = 0.1, σ = 0.6, rm = 2.1 10−3[cm]. Let
Which is visualised by the following figure.
Rigon et al.
Mass Budget - connections between pore size distribution and volumetric water and ice content

!18
A better definition for r* is:
meaning that it either coincides with R (the largest pore filled by liquid or
solid water) or just depends on temperature.
Then, the water content can be written as:
The ice content can be written as:
Rigon et al.
r*

Suction relative to water. This, in turn, is calculated
according to the freezing = drying assumption (Miller,
1965), meaning that the usual soil water retention curves
are used with water content set to the liquid water content.
!19
Before going back to a modified version of REq, let’s assume that only the
liquid water moves, while frozen water movements can be considered
negligible. Thus we have to consider just the energetic status of liquid water
now.
Therefore (details in Tubini’s Thesis), the (total) mass conservation is :
Rigon et al.
Richards ++

!20
Usually, in these derivation is also assumed
meaning that an error of ten per cent is acceptable with these
simplifications. Ice content is given instead by
Rigon et al.
Richards ++

!21
If dealing with radius was useful, it would be important to know what happens
when soil saturates.
It turns out that r varies from 0 to such an rmax, pressure can vary from
to
where negative pressures means that we are in vadose conditions and null or
positive pressures means that we are in the saturation range.
Therefore, using the Young-Laplace equation is possible to get back to the usual
form of Richards equation where soil water content is a function of pressure
(e.g. Kosugi et al., 1999). We can define then:
distribution of pressure (suctions)
Rigon et al.
Richards ++: going back to the suction as independent variable

!22
In this case, corresponding to
there is:
Rigon et al.

!23
And the extended Richards equation in term of suction is:
Rigon et al.
The good of this formulation is that it can be extended to positive pressure
specific storativity
And can be solved with a new numerics, called Nested Newton algorithm.

!24
I omitted here some details on hydraulic
conductivity
Mualem’s theory allows to derive relative hydraulic conductivity
from soil water retention curves. At the same time is well known that
saturated hydraulic conductivity depends on viscosity and through it
from temperature. So the only warn I put here is: “take into account
it !”
permeability
k i n e m a t i c
viscosity
d y n a m i c
viscosity
Kestin, J., Sokolov, M., & Wakeham, W. A. (1978). Viscosity of Liquid Water in the Range -8 C to 150 C. J. Phys.
Chem. Ref. Data, 7(3), 941–048.
Rigon et al.
Richards: some variatiation on the theme

!25
Treating explicitly pore sizes is a productive idea in any case.
See for instance a different application in
Brangari et al., WRR 2017
Rigon et al.
Richards: some variatiation on the theme

!26
How can this be coupled with the energy budget
(and, BTW, how it is the energy budget) ?
Rigon et al.
The Energy budget

!27
Rigon et al.
The Energy budget

!28
To make the long story short, this is the final
energy budget equation
Thermal capacity
Apparent thermal capacity
Advective fluxes
C o n v e c t i v e
fluxes
Full story here : New insights in permafrost modelling, by N. Tubini
Rigon et al.
The Energy budget in brief

!29
When you have the two equations how can you
solve them ?
Rigon et al.
Work in progress
Be patient for the last step

!30
However the idea is to use the Casulli and Zanolli ,
2010 nested Newton method for the extended
Richards equation (Richards ++).
And an implicit upwind method for the Energy
budget (well maybe we could think to put the
equation in a conservative form).
Hints
Rigon et al.

!31
Stay Tuned
Rigon et al.Rigon et al.
It will not take a long time

!32
Find this presentation at
http://abouthydrology.blogspot.com
Ulrici,2000?
Other material at
There is another further slide besides this ;-)
http://abouthydrology.blogspot.co.at/2017/04/new-insights-in-permafrost-modelling.html
Rigon et al.

!33
http://abouthydrology.blogspot.co.at/2013/12/the-abouthydrology-mailing-list-at.html
Rigon et al.

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