recent improvements to the pore-morphology method for ... · containing gas and liquid with a...
TRANSCRIPT
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© 2016 Math2Market GmbH 1
Sven LindenAndreas Wiegmann
Jens-Oliver Schwarz
Erik Glatt
GeoCT Workshop
University of Bremen
Recent Improvements to the Pore-Morphology Method for Capillary Pressure and Relative Permeability Simulations
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© 2016 Math2Market GmbH 2
The Math2Market GmbH is a society of limited liability.
Formerly a group of the department Flow and Material Simulations (SMS) of the Fraunhofer Institute for Industrial Mathematics, ITWM, we spun-off on September 21st, 2011, to become Math2Market GmbH.
Math2Market GmbH is privately owned by 3 operative founders, formerly at Fraunhofer ITWM 2 separate owners
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© 2016 Math2Market GmbH 3
Math2MarketServices provided to our clients
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GeoDictMaterial Characterization & Engineering
Lab
Properties
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GeoDictMath2Market Digital Rock Physics Portfolio
Geometrical parameters
Flowparameters
ElectricalParameters
Mechanical parameters
Porosity
Pore size distribution
Percolation
Surface area
Tortuosity
Absolute permeability
Multi-scale flow
Two-phase flow
Relative permeability
Cap. pressure curve
Formation factor
Resistivity index
Saturation exponent
Cementation
exponent
Elastic moduli
Stiffness
In-Situ conditions
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© 2016 Math2Market GmbH 7
GeoDictMath2Market Digital Rock Physics Portfolio
Geometrical parameters
Flowparameters
ElectricalParameters
Mechanical parameters
Porosity
Pore size distribution
Percolation
Surface area
Tortuosity
Absolute permeability
Multi-scale flow
Two-phase flow
Relative permeability
Cap. pressure curve
Formation factor
Resistivity index
Saturation exponent
Cementation
exponent
Elastic moduli
Stiffness
In-Situ conditions
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© 2016 Math2Market GmbH 8
Two-phase flow “In fluid mechanis, two-phase flow occurs in a system
containing gas and liquid with a meniscus separating the two phases.” [Wiki]
One fluid wets the surface, the less affinity is non-wetting SatuDict uses the Pore Morphology method to determine
the distribution of the two phases inside the porous media. Here Imbibition: wetting fluid displaces non-wetting fluid
Drainage: non-wetting fluid displaces wetting fluid
SatuDictTwo Phase Flows
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Young – Laplace equation Describes the capillary pressure difference
across interfaces between two fluids Relates pressure difference to the
shape of the surface
#######𝑝𝑝𝑐𝑐 = 𝛾𝛾 𝑑𝑑𝑑𝑑𝑑𝑑 𝑛𝑛#######𝑝𝑝𝑐𝑐 = 2 𝛾𝛾𝛾𝛾#######𝑝𝑝𝑐𝑐 = 𝛾𝛾 𝜅𝜅1 + 𝜅𝜅2#######𝑝𝑝𝑐𝑐 = 𝛾𝛾
1𝑟𝑟1
+ 1𝑟𝑟2
Assumption: cylindrical pores with radius 𝑟𝑟
#######𝑝𝑝𝑐𝑐 =2𝛾𝛾 cos 𝛽𝛽
𝑟𝑟
capillary pressure ⇔ pore radius
SatuDictCapillary Pressure
β: contact angle
non-wetting fluid
wetting fluid
r
β
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© 2016 Math2Market GmbH 11
Use this relation between pore size and capillary pressure to predict the distribution of the phases
Advantage: No partial differential equation (PDE) is solved Purely geometrical operations Very low runtime and memory requirements
Assumption: Quasi-stationary phase distribution Cylindrical pores
SatuDictPore Morphology MethodBasic Idea
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Connectivity of NWP to reservoir
Move in spheres Start: completely wet Start: large radius
(i.e. small pc) Steps: smaller radius
(higher pc) No residual WP
SatuDictDrainage I non-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP
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Connectivity of NWP to reservoir
Move in spheres Start: completely wet Start: large radius
(i.e. small pc) Steps: smaller radius
(higher pc) No residual WP
SatuDictDrainage I non-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP
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Connectivity of NWP to reservoir
Move in spheres Start: completely wet Start: large radius
(i.e. small pc) Steps: smaller radius
(higher pc) No residual WP
SatuDictDrainage I non-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP
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Connectivity of NWP to reservoir
Move in spheres Start: completely wet Start: large radius
(i.e. small pc) Steps: smaller radius
(higher pc) No residual WP
SatuDictDrainage I non-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP
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Connectivity of NWP to reservoir
Move in spheres Start: completely wet Start: large radius
(i.e. small pc) Steps: smaller radius
(higher pc) No residual WP
SatuDictDrainage I non-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP
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Connectivity of NWP to reservoir
Move in spheres Start: completely wet Start: large radius
(i.e. small pc) Steps: smaller radius
(higher pc) No residual WP
SatuDictDrainage I non-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP
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Connectivity of NWP to reservoir
Move in spheres Start: completely wet Start: large radius
(i.e. small pc) Steps: smaller radius
(higher pc) No residual WP
SatuDictDrainage I non-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP
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Ahrenholz et al. 2008 WP must be connected to
reservoir Residual WP (orange)
SatuDictDrainage IInon-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP Res. WP NWP
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Ahrenholz et al. 2008 WP must be connected to
reservoir Residual WP (orange)
SatuDictDrainage IInon-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP Res. WP NWP
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Ahrenholz et al. 2008 WP must be connected to
reservoir Residual WP (orange)
SatuDictDrainage IInon-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP Res. WP NWP
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Ahrenholz et al. 2008 WP must be connected to
reservoir Residual WP (orange)
SatuDictDrainage IInon-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP Res. WP NWP
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Ahrenholz et al. 2008 WP must be connected to
reservoir Residual WP (orange)
SatuDictDrainage IInon-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP Res. WP NWP
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Ahrenholz et al. 2008 WP must be connected to
reservoir Residual WP (orange)
SatuDictDrainage IInon-wetting displaces wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP Res. WP NWP
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WP must be connected to reservoir
NWP must be connected to NWP reservoir
Move out spheres Start: completely dry Start: small radius
(i.e. large pc) Steps: larger radius
(smaller pc)
SatuDictImbibition IIIwetting displaces non-wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP Res. NWP
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WP must be connected to reservoir
NWP must be connected to NWP reservoir
Move out spheres Start: completely dry Start: small radius
(i.e. large pc) Steps: larger radius
(smaller pc)
SatuDictImbibition IIIwetting displaces non-wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP Res. NWP
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WP must be connected to reservoir
NWP must be connected to NWP reservoir
Move out spheres Start: completely dry Start: small radius
(i.e. large pc) Steps: larger radius
(smaller pc)
SatuDictImbibition IIIwetting displaces non-wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP Res. NWP
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WP must be connected to reservoir
NWP must be connected to NWP reservoir
Move out spheres Start: completely dry Start: small radius
(i.e. large pc) Steps: larger radius
(smaller pc)
SatuDictImbibition IIIwetting displaces non-wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP Res. NWP
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WP must be connected to reservoir
NWP must be connected to NWP reservoir
Move out spheres Start: completely dry Start: small radius
(i.e. large pc) Steps: larger radius
(smaller pc)
SatuDictImbibition IIIwetting displaces non-wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP Res. NWP
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WP must be connected to reservoir
NWP must be connected to NWP reservoir
Move out spheres Start: completely dry Start: small radius
(i.e. large pc) Steps: larger radius
(smaller pc)
SatuDictImbibition IIIwetting displaces non-wetting phase
Non-wetting phase reservoir
Wetting phase reservoir
Solid WP NWP Res. NWP
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Drainage experiment Air (NWP) drains brine (WP) 40° contact angle NWP reservoir at the top and WP reservoir at the bottom
SatuDictExample: Capillary Pressure forBerea Sandstone
Air drains brine and we visualize air saturations of 25%, 50% and 75%.
0
20
40
60
80
100
120
140
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0.00 20.00 40.00 60.00 80.00 100.00
Cap
illar
yP
ress
ure
[kP
a]
Brine Saturation [%]
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Limitations Contact angle is always zero for computed phase distributions Constant contact angle for the whole domain
SatuDictPore Morpholoy Method
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When does the NWP (air) enter a cylindrical capillary?
𝑝𝑝 =2𝛾𝛾𝑟𝑟
𝑝𝑝 differential pressure𝑟𝑟 pore radius𝛾𝛾 surface tension
complete wetting 𝛽𝛽 = 0
SatuDictSaturation with Variable Wettability
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When does the NWP (air) enter a cylindrical capillary?
𝑝𝑝 =2𝛾𝛾𝑟𝑟
cos𝛽𝛽
𝑝𝑝 differential pressure𝑟𝑟 pore radius𝛾𝛾 surface tension𝛽𝛽 contact angle
partial wetting 0° < 𝛽𝛽 < 90°
SatuDictSaturation with Variable Wettability
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Idea (Schulz et al, 2014)
dilate by 𝑟𝑟 cos𝜃𝜃 (pore radius) erode by 𝑟𝑟 (sphere radius)
Result: contact angle 𝜃𝜃 on pore wall
Young-Laplace: 𝑝𝑝 = 2𝛾𝛾𝑟𝑟
SatuDictNew Idea:Can we have variable contact angles?
𝑟𝑟 cos𝜃𝜃𝑟𝑟
V.P. Schulz, E. A. Wargo, E. Kumbur, Pore-Morphology-Based Simulation of Drainage in Porous Media Featuring a Locally Variable Contact Angle, Transport in Porous Media, 2014.
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SatuDictMultiple Contact Angles
Material 1 Material 2 NWP WP
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© 2016 Math2Market GmbH 50
SatuDictMultiple Contact Angles
1. Dilate material 1
Material 1 Material 2 NWP WP
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SatuDictMultiple Contact Angles
1. Dilate material 12. Dilate material 2
Material 1 Material 2 NWP WP
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SatuDictMultiple Contact Angles
1. Dilate material 12. Dilate material 23. Check
connectivity
Material 1 Material 2 NWP WP
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SatuDictMultiple Contact Angles
1. Dilate material 12. Dilate material 23. Check
connectivity4. Erode
Material 1 Material 2 NWP WP
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SatuDictMultiple Contact Angles
1. Dilate material 12. Dilate material 23. Check
connectivity4. Erode 5. Final result
Material 1 Material 2 NWP WP
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SatuDict2D Example
Contact angle 0°Contact angle 40°Water (non-wetting) Air (wetting)
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SatuDict2D Example
Contact angle 0°Contact angle 40°Water (non-wetting) Air (wetting)
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SatuDict2D Example
Contact angle 0°Contact angle 40°Water (non-wetting) Air (wetting)
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SatuDict2D Example
Contact angle 0°Contact angle 40°Water (non-wetting) Air (wetting)
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SatuDict2D Example
Contact angle 0°Contact angle 40°Water (non-wetting) Air (wetting)
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SatuDict2D Example
Contact angle 0°Contact angle 40°Water (non-wetting) Air (wetting)
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SatuDict2D Example
Contact angle 0°Contact angle 40°Water (non-wetting) Air (wetting)
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Choose Saturation level and Wetting model
Find Oil distribution (black) and Brine distribution
Solve Stokes equations in Brine phase and treat oil phase as solid Oil phase and treat brine phase as solid
SatuDictRelative PermeabilityBasic Idea
0
50
100
150
0 25 50 75 100
Per
mea
bili
ty [
mD
]
Saturation NWP [%]
Relative Permeability
WP
NWP
Oil (NWP) Brine (WP)
Stokes Flow in Wetting Phase
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Challenges DRP Parameter that is most expensive to compute Low saturation states are most expensive but results are the smallest
Improvements Speedup of solvers Restart of computations Compute permeability from highest to lowest saturation state Use result from previous computation to speed up the next one
New stopping criterion Relative difference to the permeability of the full saturated state
SatuDictRelative Permeability
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Benchmark Structure: Berea Sandstone Compute Relative Permeability for WP Compare old and improved computations
SatuDictRelative PermeabilityExample: Berea Sandstone
0
10
20
30
40
50
0 20 40 60 80 100
Per
mea
bil
ity
[mD
]
WP Saturation [%]
Relative Permeability
Old
New
0
500
1000
1500
2000
2500
0 20 40 60 80 100
Ru
nti
me
[s]
WP Saturation [%]
Runtime
Old
New
Segmented Berea Sandstone with256³ voxels and 0.74 µm voxel length
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Thank you for your attention!
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