Mathematics of non-Darcy CO2 injection
into saline aquifers
Ana Mijic
Imperial College London
Department of Civil and Environmental Engineering
PMPM Research Network
2015 Annual Meeting, Edinbourgh
Thanks to
Tara LaForce and Ann Muggeridge
Simon Mathias
Sebastian Geiger
Grantham Institute for Climate Change
John Archer Fund
ORSAS Fund
Problem background
Context
•
•
•
•
CO2 injection is limited by the formation fracture
pressure
Injected CO2 can induce the resident saline water’s
evaporation resulting in salt deposition
CO2 is a compressible gas whose properties vary
with the change of the reservoir pressure
At high injection rates inertial effects may become
significant
Analytical solutions to CO2-brine displacement
Inaccurate predictions at high injection rates
(Lu et al., 2009)
Nordbotten et al. (2005)
Nordbotten and
Celia (2006)
Segregated flow
approach
Vilarrasa
et al. (2010)
Compositional
displacement
Phase
Compressibility
Mathias et al. (2009)
Non-Darcy flow
Burton et al. (2008)
Zeidouni et al. (2009)
Diffuse flow
approach
Analytical solutions to CO2-brine displacement
Analysis of critical processes
in the near-well region
Nordbotten et al. (2005)
Nordbotten and
Celia (2006)
Segregated flow
approach
Vilarrasa
et al. (2010)
Compositional
displacement
Phase
Compressibility
Mathias et al. (2009)
Non-Darcy flow
Burton et al. (2008)
Zeidouni et al. (2009)
Diffuse flow
approach
Mathematics
Buckley-Leverett model
•
•
•
Linear oil-water
system
Momentum equation
by Darcy’s law
Shock front by
Method of
Characteristics
(MOC)
Modelling approach
•
•
•
Radial convection equation for a gas-liquid system
Gas flow governed by the Forchheimer equation
Fractional flow function depending both on saturation
and radial distance from the well
•
•
Solution for saturation by the modified MOC
Solution for pressure by numerical integration
Radial convection equation for gas phase
Gas = phase 1
Liquid =
phase 2
radial distance r
Fractional flow function
Two-phase extension
of the Forchheimer equation
Two-phase extension
of the Darcy’s law
Fractional flow function
Two-phase extension
of the Forchheimer equation
Darcy fraction of
the gas velocity, q10
Two-phase extension
of the Darcy’s law
Non-Darcy scaling
factor, b ≥ 1.0
Non-Darcy fractional
flow function
Non-Darcy gas phase velocity
Darcy fraction of
the gas velocity
Forchheimer
flow parameter
Quadratic
representation
of gas phase
velocity
Forchheimer
factor
Non-Darcy flow calculation algorithm
Solution for saturation by modified MOC
Equal area rule
Solving for pressure
•
•
Transient solution (Theis, 1935)
Impact of the boundary condition:
•
•
Open aquifer (Thiem, 1906)
Closed aquifer (Dake, 1983)
More details can be found in:
Mijic, A., and T. C. LaForce (2012),
Water Resour. Res., 48, W09503, doi:10.1029/2011WR010961.
Miscible displacement
F1
Da
r
cyf
lo
1
F1(
C1,r2)
F1(
C1,r1)
For
ch
h
ei
merf
lo
s1c
1
C1
Solutions for saturation and
pressure as extension of the
immiscible model!
Salt precipitation
•
Salt saturation
•
Permeability
reduction
Model from Zeidouni et al. (2009)
Correction for gas compressibility (1)
Correction for gas compressibility (2)
Solved iteratively using using
the mean flux as a
convergence criterion
More details can be found in:
Mijic, A., T. C. LaForce, and A. H. Muggeridge (2014),
Water Resour. Res., 50, 4163–4185, doi:10.1002/2013WR014893.
Fractional
flow
module
Computational scheme
Saturation
profile
module
Pressure
profile
module
Compressibility
correction
module
Results
Relative permeability
1
CO modelled
2
0.9
Brine modelled
CO2 experimental
•
Corey (1954) and
van Genuchten (1980)
models as a function of
the end-point relative
permeability
Relative permeability (−)
0.8
Brine experimental
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
CO2 saturation (−)
0.6
0.7
0.8
Exp. data from Krevor et al. (2012)
Forchheimer coefficient for the gas phase
•
14
−1
Forchheimer parameter for the gas phase (m )
10
Wong (1970)
Evans and Evans (1988)
Janicek and Katz (1955)
Geertsma (1974)
Kutasov (1993)
Frederick and Graves (1994)
12
10
10
•
10
8
Carbonates
•
10
6
10
4
10
Ottawa sand
2
10
−14
10
−13
10
10
−12
10
−11
−10
10
2
Effective permeability (m )
10
−9
Limited experimental
data
Modified Janicek and
Katz (1955)
formulation
Range of W between
3.2 10-9 and 3.2 10-7
m1.5
Non-Darcy fractional flow curves
1
CO2
0.9
0.2 m
1m
10 m
0.8
100 m
2
CO fractional flow (−)
Darcy
0.7
r
0.6
0.5
•
0.4
0.3
•
0.2 m
1m
10 m
100 m
Darcy
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
CO saturation (−)
2
0.6
0.7
0.8
rd
The most significant
influence of the non-Darcy
effect is near the well
For a given saturation
value the inertial losses
significantly slow down the
gas phase flow
Non-Darcy saturation profiles
0.8
0.8
Q =30 kg s
Darcy flow, W=0
−8
1.5
Non−Darcy flow, W=3.2 10 m
−7 1.5
Non−Darcy flow, W=3.2 10 m
0.7
Q =100 kg s−1
0.7
m
Q =120 kg s
0.6
CO saturation (−)
0.5
0.4
−1
m
0.6
0.5
0.4
2
2
CO saturation (−)
−1
m
0.3
0.3
0.2
0.2
0.1
0.1
0
0
5
10
15
20
Radial distance (m)
25
30
Effects of non-Darcy flow
35
0
0
5
10
15
20
Radial distance (m)
25
Effects of injection rate
30
Non-Darcy pressure distribution
a) H=50 m and k=10
−13
m
2
b) H=200 m and k=10
100
24
80
70
60
50
40
30
22
Pressure (MPa)
Darcy flow, t=12 h
Non−Darcy flow, t=12 h
Darcy flow, t=5 d
Non−Darcy flow, t=5 d
Darcy flow, t=50 d
Non−Darcy flow, t=50 d
90
Pressure (MPa)
−13
20
18
16
14
20
0
10
2
10
Radial distance (m)
12
0
10
2
10
Radial distance (m)
Effects of non-Darcy flow and formation properties
m
2
Effects of partial miscibility
Overall fractional flow curves in single- and two-phase regions
Permeability reduction
Effects of the Forchheimer coefficient variability in
incompressible displacement on permeability reduction
Incompressible displacement
Compressible displacement
1
1
Darcy
non−Darcy const. b
1
1
2
0.6
0.4
0.2
0
1
0.6
0.4
0
10
20
30
Radial distance (m)
40
0
50
21
21
20
20
19
19
18
17
16
15
15
13
2
10
Radial distance (m)
20
30
Radial distance (m)
16
13
0
10
17
14
10
0
18
14
12
non−Darcy var. b
0.2
d) Pressure (MPa)
c) Pressure (MPa)
Incompressible
and
compressible
saturation
and pressure
profiles
1
0.8
non−Darcy var. b
b) CO saturation (−)
2
a) CO saturation (−)
0.8
Effects of
compressibility
Darcy
non−Darcy const. b
12
0
10
2
10
Radial distance (m)
40
50
Comparison with ECLIPSE 300
1
20
Simulator
Incompressible Analytical
Compressible Analytical
18
b) Pressure (MPa)
2
a) CO saturation (−)
0.8
19
0.6
0.4
17
16
15
14
0.2
13
0
0
50
100
Radial distance (m)
150
12
0
10
2
10
Radial distance (m)
Excellent agreement between analytical and simulation results.
However, the comparison is with constant values of the
Forchheimer coefficient!
Implications of non-Darcy effects (1)
The error associated with neglecting the non-Darcy flow
increases significantly with the injection rate
and decreased formation permeability
Implications of non-Darcy effects (2)
In non-Darcy displacement CO2 injectivity is limited by the
pressure increase at high rates and highly sensitive to
formation permeability
Conclusions
The application example showed significant influence of
non-Darcy effects in low permeability formations
when CO2 is injected at high rates:
•
•
•
•
CO2 displacement efficiency improvement
Additional pressure increase at the well
Salt precipitation reduction
Alteration of injectivity function
Where this can be taken
•
•
•
•
•
Estimation of Forchheimer coefficient for the gas
phase in formations suitable for CO2 injection
Correction of pressure estimation due to salt
precipitation in near-well region
Analytical modelling of
effects of rock dissolution
Multiple well analysis in
two-phase displacement
Reservoir simulator with
spatially varying saturationdependent Forchheimer
flow
Where this can be taken
•
•
•
•
•
Estimation of Forchheimer coefficient for the gas
phase in formations suitable for CO2 injection
Correction of pressure estimation due to salt
precipitation in near-well region
Analytical modelling of
effects of rock dissolution
Multiple well analysis in
two-phase displacement
Reservoir simulator with
spatially varying saturationdependent Forchheimer
flow
Ana Mijic
ana.mijic@imperial.ac.uk
@leiastarspear