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Egu2017 pico

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These are the slides presented at EGU 2017 General Meeting, the Pico session was entlited: Monitoring and modelling flow paths, supply and quality in a changing mountain cryosphere

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Egu2017 pico

  1. 1. 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. 2. !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. 3. !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. 4. !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. 5. !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 Figure 13: 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]. 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. 6. !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. 7. !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.
  8. 8. 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. 9. !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. 10. !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. 11. !11 Newinsightsinpermafrostmodelling,byN.Tubini, 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
  12. 12. Hydraulic conductivity !12 Newinsightsinpermafrostmodelling,byN.Tubini, Suction Gravity Rigon et al. TCPN corresponds to differential equations
  13. 13. !13 Newinsightsinpermafrostmodelling,byN.Tubini, 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. 14. !14 Newinsightsinpermafrostmodelling,byN.Tubini, 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. 15. !15 Newinsightsinpermafrostmodelling,byN.Tubini, 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. 16. !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. 17. !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 Figure 13: 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]. 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. 18. !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*
  19. 19. 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. 20. !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. 21. !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. 22. !22 In this case, corresponding to there is: Rigon et al. Richards ++: going back to the suction as independent variable
  23. 23. !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. Richards ++: going back to the suction as independent variable
  24. 24. !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. 25. !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. 26. !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. 27. !27 Rigon et al. The Energy budget
  28. 28. !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. 29. !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. 30. !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. 31. !31 Stay Tuned Rigon et al.Rigon et al. It will not take a long time
  32. 32. !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. 33. !33 http://abouthydrology.blogspot.co.at/2013/12/the-abouthydrology-mailing-list-at.html Rigon et al.

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