1. About EOS for Äspö granite Klaus-Peter Kröhn, GRS Joint meeting of the Task Forces EBS and GWFTS 28 th November 2012

2. Overview 1. Problem at hand 2. Theoretical background 3. Model 4. Alternative Equations of state (EOS)  series 1: original EOS  series 2: scaled retention curves  series 3: equivalent Brooks-Corey approach 5. Latest data on EOS Introduction2

3. Preparing an uptake test Impact of flow resistance on water uptake by bentonite  Äspö granite between water supply and bentonite Problem: optimum length of the rock piece? Optimisation targets  allowing for significant uptake via water vapour  limiting breakthrough time of water at the opposite side 1. Problem at hand3 water granite bentonite

4. Simplified treatment of unsaturated flow As suggested at the TF EBS meeting in Rosersberg 2009  starting point: full set of two-phase flow equations  assumptions: neglecting gas flow density changing effects (compressibility, thermal expansion, etc.) the influence of gravity, matrix deformation ( → constant porosity) water sinks and sources  result: formally Fick’s second law independent variable: water saturation saturation-dependent „diffusion coefficient“  relative permeability-saturation relation (RPS)  capillary pressure-saturation relation (CPS) 2. Theoretical background4

5. Permeability: 10 -20 m² Porosity: 0.5 % Viscosity:10 -3 Pa s modelling until saturation front reaches the right hand side boundary → breakthrough time Model description 3. Model5 S=1 S=0 S=0.01 4 cm

6. EOS for granite 4. Alternative EOS: series 1 - original EOS6 Model AModel BModel CModel E source/FIN 95//BÖR 99//THO 03//GUO 06/* RPSVGpower lawVG CPSVG emp. relation locationGrimselÄspöURL * adapted

7. EOS for granite 4. Alternative EOS: series 1 - original EOS7 Model AModel BModel CModel E source/FIN 95//BÖR 99//THO 03//GUO 06/* RPSVGpower lawVG CPSVG emp. relation locationGrimselÄspöURL * adapted

8. EOS for granite 4. Alternative EOS: series 1 - original EOS8 Model AModel BModel CModel E source/FIN 95//BÖR 99//THO 03//GUO 06/* RPSVGpower lawVG CPSVG emp. relation locationGrimselÄspöURL * adapted

9. EOS for granite 4. Alternative EOS: series 1 - original EOS9 Model AModel BModel CModel D source/FIN 95//BÖR 99//THO 03//GUO 06/* RPSVGpower lawVG CPSVG emp. relation locationGrimselÄspöURL * adapted

10. Capillary pressure and relative permeability 4. Alternative EOS: series 1 - original EOS10

11. “Diffusion” coefficient 4. Alternative EOS: series 1 - original EOS11

12. Results for original EOS 4. Alternative EOS: series 1 - original EOS12 Model A /FIN 95/ VG Grimsel b.t. at 93 h Model D /GUO 06/* VG emp. relation URL b.t. at 198 h Model C /THO 03/ VG URL b.t. at 515 h Model B /BÖR 99/ power law VG Äspö b.t. at 22 h

13. Scaling of retention curves Topological similarity of the pore space → Leverett‘s J-function Different porosities and permeabilities at different sites Scaling retention curves for EOS  from Grimsel  from the URL to and at Äspö 4. Alternative EOS: series 2 – scaled retention curves13 GrimselURLÄspö permeability[m²]5.13 10 -19 5.00 10 -20 1.00 10 -20 porosity[-]0.010.005

14. Results for original EOS 4. Alternative EOS: series 2 – scaled retention curves14 Model A /FIN 95/ VG Grimsel b.t. at 93 h Model D /GUO 06/* VG emp. relation URL b.t. at 198 h Model C /THO 03/ VG URL b.t. at 515 h Model B /BÖR 99/ power law VG Äspö b.t. at 22 h

15. Results for series 2 4. Alternative EOS: series 2 – scaled retention curves15 Model F /FIN 95/ VG (Grimsel) b.t. at 18 h Model H /GUO 06/* VG emp. relation (URL) b.t. at 140 h Model G /THO 03/ VG (URL) b.t. at 364 h Model B /BÖR 99/ power law VG Äspö b.t. at 22 h

16. Brooks-Corey approach Most common approaches  van Genuchten 1980 (VG)  Brooks and Corey 1964 (BC) Difference: treatment of the air-entry pressure the BC-approach  physics closer to reality  in case of granite the BC-approach is not so dearly-loved 4. Alternative EOS: series 3 – equivalent Brooks-Corey approach16

17. Consequences EOS define „diffusion coefficient“ :  BC: finite value at S=1  VG: singularity as S → 1 4. Alternative EOS: series 3 – equivalent Brooks-Corey approach17

18. Switching from VG toBC 4. Alternative EOS: series 3 – equivalent Brooks-Corey approach18 Model FModel IModel J source/FIN 95/ RPSVG BC CPSVGBC location(Grimsel)

19. Flow parameters 4. Alternative EOS: series 3 – equivalent Brooks-Corey approach19 capillary pressure relative permeability “diffusion coefficient”

20. Results for series 3 4. Alternative EOS: series 3 – equivalent Brooks-Corey approach20 Model FModel IModel J source/FIN 95/ RPSVG BC CPSVGBC location(Grimsel) breakthrough18 h59 h50 h

21. Latest data from Åsa 5. Latest data21

22. Data fits 5. Latest data22

23. 5. Latest data23

24. 5. Latest data24

25. 5. Latest data25

26. 5. Latest data26

27. Gesteinseigenschaften27 „diffusion coefficient“ [m²/s]

28. Results for the latest data 5. Latest data28 Model EModel E2Model E3 source/BOC 12/ RPSVGVG (ad hoc fit)VG (from /FIN 95/) CPSVGVG (ad hoc fit)VG locationÄspö(Äspö) breakthrough1144 h165 h4 h

29. Conclusions for the 4 cm piece of granite Calculated breakthrough times between 4 and 1144 hours → no way of predicting breakthrough theoretically Reasons for failure  Transfer of EOS between similar materials can be strongly misleading  Both – CPS and RPS – are essential  RPS for the liquid hard to measure directly not necessarily given by measured CPS → choice of RPS a powerful calibration tool → imperative to determine the EOS for the investigated material directly Conclusions29

30. Conclusions concerning modelling Äspö granite Choice between van Genuchten and Brooks Corey approach  equivalent parameters still significantly different behaviour at high saturation values (> 80 %) moisture transport faster using VG than using BC  BC more convincingly describing the air entry pressure New retention data for Task 8  a minimum degree of saturation in the rock between 20 % and 30 % is indicated by Åsa‘s retention data in combination with tunnel humidity data  modelling unsaturated flow in Äspö granite presently thus relies on the extrapolated part of the measured retention data Conclusions30

31. 31 Thank you for your attention!