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About EOS for Äspö granite
Klaus-Peter Kröhn, GRS
Joint meeting of the Task Forces EBS and GWFTS
28th November 2012
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
Introduction 2
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 hand 3
wat
er granite bentonite
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 background 4
0ˆ
x
SD
xt
S ww
w
cwr
w S
pk
kD
ˆ
D̂
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. Model 5
S=1 S=0S=0.01
4 cm
EOS for granite
4. Alternative EOS: series 1 - original EOS 6
Model A Model B Model C Model E
source /FIN 95/ /BÖR 99/ /THO 03/ /GUO 06/*
RPS VG power law VG VG
CPS VG VG VG emp. relation
location Grimsel Äspö URL URL
* adapted
EOS for granite
4. Alternative EOS: series 1 - original EOS 7
Model A Model B Model C Model E
source /FIN 95/ /BÖR 99/ /THO 03/ /GUO 06/*
RPS VG power law VG VG
CPS VG VG VG emp. relation
location Grimsel Äspö URL URL
* adapted
EOS for granite
4. Alternative EOS: series 1 - original EOS 8
Model A Model B Model C Model E
source /FIN 95/ /BÖR 99/ /THO 03/ /GUO 06/*
RPS VG power law VG VG
CPS VG VG VG emp. relation
location Grimsel Äspö URL URL
* adapted
EOS for granite
4. Alternative EOS: series 1 - original EOS 9
Model A Model B Model C Model D
source /FIN 95/ /BÖR 99/ /THO 03/ /GUO 06/*
RPS VG power law VG VG
CPS VG VG VG emp. relation
location Grimsel Äspö URL URL
* adapted
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
rela
tive
pe
rme
ab
iity
[-]
0.2 0.4 0.6 0.810-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
model A, /FIN 95/; pc
model A, /FIN 95/; kr
model B, /BOE 99/; pc
model B, /BOE 99/; kr
model C, /THO 03/; pc
model C, /THO 03/; kr
Capillary pressure and relative permeability
4. Alternative EOS: series 1 - original EOS 10
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
rela
tive
pe
rme
ab
iity
[-]
0.2 0.4 0.6 0.810-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
model A, /FIN 95/; pc
model A, /FIN 95/; kr
model B, /BOE 99/; pc
model B, /BOE 99/; kr
model C, /THO 03/; pc
model C, /THO 03/; kr
model D, based on /GUO 06/; pc
model D, based on /GUO 06/; kr
“Diffusion” coefficient
4. Alternative EOS: series 1 - original EOS 11
effective saturation [-]
'diff
usi
on
coe
ffici
en
t'[m
²/s]
0.2 0.4 0.6 0.810-17
10-15
10-13
10-11
10-9
10-7
model A; /FIN 95/model B; /BOE 99/model C; /THO 03/model D; based on /GUO 06/
Results for original EOS
4. Alternative EOS: series 1 - original EOS 12
Model A
/FIN 95/
VG
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
VG
URL
b.t. at 515 h
Model B
/BÖR 99/
power law
VG
Äspö
b.t. at 22 h
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
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 curves 13
kpSJ c
)(
%5.0 ²10 20 mk
Grimsel URL Äspöpermeability [m²] 5.13 10-19 5.00 10-20 1.00 10-20
porosity [-] 0.01 0.005 0.005
Results for original EOS
4. Alternative EOS: series 2 – scaled retention curves 14
Model A
/FIN 95/
VG
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
VG
URL
b.t. at 515 h
Model B
/BÖR 99/
power law
VG
Äspö
b.t. at 22 h
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
Results for series 2
4. Alternative EOS: series 2 – scaled retention curves 15
Model F
/FIN 95/
VG
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
VG
(URL)
b.t. at 364 h
Model B
/BÖR 99/
power law
VG
Äspö
b.t. at 22 h
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
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 approach 16
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
0 0.2 0.4 0.6 0.8 110-2
10-1
100
pc (BC)pc (VG)
eff. saturation
'diff
usi
on
coe
ffici
en
t'[m
²/s]
0 0.2 0.4 0.6 0.8 110-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
EOS: VGEOS: BC
Consequences
EOS define „diffusion coefficient“ : BC: finite value at S=1 VG: singularity as S → 1
4. Alternative EOS: series 3 – equivalent Brooks-Corey approach 17
D̂
Switching from VG toBC
4. Alternative EOS: series 3 – equivalent Brooks-Corey approach 18
Model F Model I Model J
source /FIN 95/ /FIN 95/ /FIN 95/
RPS VG VG BC
CPS VG BC BC
location (Grimsel) (Grimsel) (Grimsel)
Flow parameters
4. Alternative EOS: series 3 – equivalent Brooks-Corey approach 19
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
rela
tive
pe
rme
ab
iity
[-]
0.2 0.4 0.6 0.810-3
10-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
model F; /FIN 95/, corrected after Leverettrel. permeabilitycapillary pressuremodel I; /FIN 95/, corrected, CPS after BCrel. permeabilitycapillary pressuremodel J; /FIN 95/, corrected, CPS and RPS after BCrel. permeabilitycapillary pressure
effective saturation [-]
'diff
usi
on
coe
ffici
en
t'[m
²/s]
0 0.2 0.4 0.6 0.8 110-17
10-15
10-13
10-11
10-9
10-7
model F; /FIN 95/, corrected after Leverettmodel I; /FIN 95/, corrected, CPS after BCmodel J; /FIN 95/, corrected, CPS and RPS after BC
capillary pressurerelative permeability
“diffusion coefficient”
Results for series 3
4. Alternative EOS: series 3 – equivalent Brooks-Corey approach 20
Model F Model I Model J
source /FIN 95/ /FIN 95/ /FIN 95/
RPS VG VG BC
CPS VG BC BC
location (Grimsel) (Grimsel) (Grimsel)
breakthrough 18 h 59 h 50 h
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
Latest data from Åsa
5. Latest data 21
saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
0 0.2 0.4 0.6 0.8 110-2
10-1
100
101
102
103
granitediorite
saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
0 0.2 0.4 0.6 0.8 110-2
10-1
100
101
102
103
granitediorite/BOC 12/
saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
0 0.2 0.4 0.6 0.8 110-2
10-1
100
101
102
103
granitediorite/BOC 12/ad hoc alternative
Data fits
5. Latest data 22
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
rela
tive
pe
rme
ab
iity
[-]
0.2 0.4 0.6 0.810-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
model A, /FIN 95/; pc
model A, /FIN 95/; kr
model B, /BOE 99/; pc
model B, /BOE 99/; kr
model C, /THO 03/; pc
model C, /THO 03/; kr
model D, based on /GUO 06/; pc
model D, based on /GUO 06/; kr
5. Latest data 23
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
rela
tive
pe
rme
ab
iity
[-]
0.2 0.4 0.6 0.810-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
model A, /FIN 95/; pc
model A, /FIN 95/; kr
model B, /BOE 99/; pc
model B, /BOE 99/; kr
model C, /THO 03/; pc
model C, /THO 03/; kr
model D, based on /GUO 06/; pc
model D, based on /GUO 06/; kr
model E, /FRA 12/; pc
model E, /FRA 12/; kr
5. Latest data 24
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
rela
tive
pe
rme
ab
iity
[-]
0.2 0.4 0.6 0.810-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
model A, /FIN 95/; pc
model A, /FIN 95/; kr
model B, /BOE 99/; pc
model B, /BOE 99/; kr
model C, /THO 03/; pc
model C, /THO 03/; kr
model D, based on /GUO 06/; pc
model D, based on /GUO 06/; kr
model E, /FRA 12/; pc
model E, /FRA 12/; kr
model E2, based on /FRA 12/; pc
model E2, based on /FRA 12/; kr
5. Latest data 25
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
rela
tive
pe
rme
ab
iity
[-]
0.2 0.4 0.6 0.810-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
model A, /FIN 95/; pc
model A, /FIN 95/; kr
model B, /BOE 99/; pc
model B, /BOE 99/; kr
model C, /THO 03/; pc
model C, /THO 03/; kr
model D, based on /GUO 06/; pc
model D, based on /GUO 06/; kr
model E, /FRA 12/; pc
model E, /FRA 12/; kr
model E2, based on /FRA 12/; pc
model E2, based on /FRA 12/; kr
model E3, based on /FRA 12/; pc
model E3, based on /FRA 12/; kr
5. Latest data 26
effective saturation [-]
cap
illa
ryp
ress
ure
[MP
a]
0.2 0.4 0.6 0.810-17
10-15
10-13
10-11
10-9
10-7
model A, /FIN 95/model B, /BOE 99/model C, /THO 03/model D, based on /GUO 06/model E, /FRA 12/model E2, based on /FRA 12/model E3, based on /FRA 12/
Gesteinseigenschaften 27
„diff
usio
n co
effic
ient
“ [m
²/s]
Results for the latest data
5. Latest data 28
Model E Model E2 Model E3
source /BOC 12/
RPS VG VG (ad hoc fit) VG (from /FIN 95/)
CPS VG VG (ad hoc fit) VG
location Äspö (Äspö) (Äspö)
breakthrough 1144 h 165 h 4 h
position [m]
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
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
Conclusions 29
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
Conclusions 30
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
position [m]
eff.
satu
ratio
n
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
0.2
0.4
0.6
0.8
1
31
Thank you for your attention!