Estimation of characteristic relations
for unsaturated flow through rock fractures
Jerker Jarsjö
Department of Physical Geography and
Quaternary Geology, Stockholm
University, 106 91 Stockholm, Sweden.
jerker.jarsjo@natgeo.su.se
Areas of fundamental research
(1) Characteristics of multiphase flow in fractured rock
under different ambient conditions
(2) Dependence on quantifiable fracture characteristics
(aperture distribution, connectivities)
(3) Multiphase flow in soil and fractured rock: Similarities
and differences.
-Are parameter translations of characteristic curves possible?
Relevance?
Useful for prediction of…
• Conductivity of gas or non-aqueous phase liquids
(NAPLS) in fractured media
• Immobilization and trapping of NAPLS in fractured media
Application examples:
• Storage of waste /oil in bedrock
• Storage of carbon dioxide storage in deep saline aquifers
and potential return flows
• Movement of accidental oil spills in fractured media
(Granite, karst, glaciers)
Experimental determination of pressure –
saturation (- conductivity) – relations
in soil
Step A-E:
A
B
C
Water
saturation,
S*
* S = Vw/Vtot
(Vw=vol. water,
Vtot=total vol.)
D
E
succesively
increased
underpressure
(-)
=0: atmospheric
pressure
Experimental determination of pressure –
saturation (- conductivity) – relations
in soil
Step A-E:
A
B
C
D
Water
saturation,
S* S =n A
0
succesively
increased
underpressure
(-)
=0: atmospheric
pressure
E
B
C
D
E
The water saturation is a function of the
underpressure, i.e. S= S().
Straightforward to determine experimentally
underpressure (-, m.water column)
Empirical vG relation


1  d h n 1 1  d h n  11 / n  




K ( h) = K s
n m 11 / n 
1  d h 

K(h)=Ks

2
for h<0
for h0
h =  pc / g , pc=capillary pressure
where
d, n, m = fitting parameters
Empirical vG relation
Related to bubble pressure


1  d h n 1 1  d h n  11 / n  




K ( h) = K s
n m 11 / n 
1  d h 

K(h)=Ks

2
for h<0
Related to width of soil psd
m=0.5 usually assumed
for h0
h =  pc / g , pc=capillary pressure
where
d, n, m = fitting parameters
The cubic law for water flow in a fracture
•
Single fractures: relation between aperture (a)
and fracture transmissivity T:
g 3
T=
a
12
”Cubic
law”
( =density, och µ =viscosity,
and g=graviational constant)
Direction of
flow
Direction of
a
a
flow
Cubic law: exact relation
Cubic law: approximately true
Fracture aperture relation
1h
5h
Darker areas=wider aperture; gas=white
48 h
(SKB TR-98-17 & 01-13 )
The fracture aperture distribution (and the mean aperture) can
be measured in situ or in the lab
Distribution of water and air in a fracture
probability density function
fln(a)
cut-off aperture
(ac) assumption
water
ac =2w/ pc
gas
air (gas)
a cu
aperture, a
Water occupies the tighter parts, and air the wider
parts. Similar to the porous medium case
Fracture aperture relation
For unsaturated fracture flow
Predict relative fracture transmissivity through consideration
of the cubic law (TR-98-17)
Tus (w)
ac
Trel ( pc ;  ln a ,  ln a ) =
3
a
 f ln (a; ln a ,  ln a )da
0

a
3
f ln (a;  ln a ,  ln a )da
0
 Ts
us=unsaturated
s=saturated
w=water
Fitting procedure
Fitted v an Genuchten parameter:
n_fit = 2.242
d_fit = 1.966
Trel and Krel [-]
1
Fracture based relation (eq. 5)
0.8
0.6
0.4
van
Genuchtenrelation (eq. 6)
0.2
Trel=0.05
0
2
4
Capillary pressure, pc [kPa]
6
Considered T-data
T-values estimated from hydraulic testing
(R-07-48)
5-1
RFMxxx
RFM029
RFM029
RFM029
RFM029
RFM029
RFM029
RFM029
RFM029
RFM029
RFM029
ZFMxxxx orSec-up
FFMxx Sec-low
FFM02
102
203
FFM01
203
216
Possible(G)
216
224
FFM01
224
267
ZFMENE1192
267
285
FFM01
285
386
ZFMENE1192
386
412
FFM01
412
639
ZFMENE2254
639
684
FFM01
684 1,001
Elev-up
–98
–199
–212
–220
–262
–280
–380
–406
–630
–674
Elev-low
–199
–212
–220
–262
–280
–380
–406
–630
–674
–982
No.PFL-f ST:PFL-F
23
1.92E-07
0
0
0
0
2
7.09E-10
2
7.79E-10
7
4.76E-09
0
0
0
0
0
0
0
0
Estimation of corresponding
hydraulic aperture
and mean aperture
Hydraulic aperture*, ah
Mean aperture**, a
*assuming one single fracture **assuming a/ah=1.7
(cubic law)
6.1719E-05
0.10492228
0
0
0
0
9.53968E-06
0.016217456
9.84383E-06
0.016734515
1.79965E-05
0.030594002
0
0
0
0
0
0
0
0
(b)
(b)
Conclusions
• Simple patterns emerge from the matching of
seemingly complex curves
• Fracture roughness related to the n-value of the
van Genuchten-formulation: the rougher the
fracture, the lower the matching n-value
• Implies that characteristic curves derived from
measurable aperture statistics can be described
with soil-based van Genuchten parameters
(standard description in most computer codes)
Geological storage in deep saline aquifers
Cap rock:
confining unit –
low permeability
Storage formation:
high permeability
high porosity
Feasable if return flows are sufficiently small (min 95% retained after 100 years)
Storage potential in Sweden and investigation site
Target: sandstone aquifer at 1670 m depth
Representation in the TOUGH2 code
Stratigraphic
uncertainty
Parameter value uncertainty
…confidence interval for k
Uncertainties addressed through scenario analyses
Considered scenarios:
A) Base case
B) No upper barrier (thin claystone layer not continuous)
C) High permeability (95% confidence limit)
D) Combination B+C
+ simulations for different injection pressures
Resulting plume migration (1000 days)
Volumetric
gas saturation [-]
Salt precipitation – injectivity effects
Permeability
reduction
factor k/k0 [-]
Summary of plume behaviour
Summary of plume behaviour
Conclusions




Stratigraphic uncertainty leads to large
differences in predicted CO2 storage in target
formation
Parameter uncertainty (permeability) has small
impact on CO2 storage predictions but affects
injectivity
Salt precipitation at the border of the target
formation affects CO2 injectivity
At low injection rates, salt precipitates within the
target formation, decreasing its storage ability
Journal reference:
Chasset, C., Jarsjö, J., Erlström, M., Cvetkovic, V. and Destouni, G., 2011. Scenario
simulations of CO2 injection feasibility, plume migration and storage in a saline aquifer,
Scania, Sweden. International Journal of Greenhouse Gas Control, 5(5), 1303-1318.
March 15, 2012
Airplane crash and kerosene spill
on top of Kebnekaise mountain (Rabots glacier)
Sweden
/
2096.3 m
2101.3 m
PROCESSES DETERMINING THE FATE OF THE HYDROCARBON POLLUTION
Sampling of water 1/week + passive
13-14 July 160 mm precipitation (TRS)
15, 18 July traced og naftalen & PAH in
Rabot jokk