about eos for Äspö granite klaus-peter kröhn, grs joint meeting of the task forces ebs and gwfts...

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







* adapted

EOS for granite


Model A Model B Model C Model D





* 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



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 D, based on /GUO 06/; pc

model D, based on /GUO 06/; kr

“Diffusion” coefficient



'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


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


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


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


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


D̂

Switching from VG toBC


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



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


'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


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


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









5. Latest data 23


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 E, /FRA 12/; pc

model E, /FRA 12/; kr

5. Latest data 24


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 E2, based on /FRA 12/; pc

model E2, based on /FRA 12/; kr

5. Latest data 25


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















5. Latest data 26


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!

about eos for Äspö granite klaus-peter kröhn, grs joint meeting of the task forces ebs and gwfts...

Documents

alternative eos

original eos series

eos introduction2 slide

h slide

adapted slide

original eos11 slide

original eos10 slide

original eos12 model