numerical modelling of capillary transition zones geir terje eigestad, university of bergen, norway...
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![Page 1: Numerical Modelling of Capillary Transition zones Geir Terje Eigestad, University of Bergen, Norway Johne Alex Larsen, Norsk Hydro Research Centre, Norway](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ed15503460f94be0697/html5/thumbnails/1.jpg)
Numerical Modelling of Capillary Transition zones
Geir Terje Eigestad, University of Bergen, Norway
Johne Alex Larsen, Norsk Hydro Research Centre, Norway
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Acknowledgments
Svein Skjaeveland and coworkers:
Stavanger College, Norway
I. Aavatsmark, G. Fladmark, M. Espedal:
Norsk Hydro Research Centre/
University of Bergen, Norway
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Overview
• Capillary transition zone: Both water and oil occupy pore-space due to capillary pressure when fluids are immiscible
• Numerical modeling of fluid distribution
• Consistent hysteresis logic in flow simulator
• Better prediction/understanding of fluid behavior
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Skjaeveland’s Hysteresis Model
• Mixed-wet reservoir
• General capillary pressure correlation
• Analytical expressions/power laws
• Accounts for history of reservoir
• Arbitrary change of direction
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Capillary pressure functions
• Capillary pressure for water-wet reservoir:
• Brooks/Corey:
• General expression: water branch + oil branch
• c’s and a’s constants; one set for drainage, another for imbibition
• Swr[k], Sor[k] adjustable parameters
( )1
w
w
aw wr
wr
cPc
S SS
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Hysteresis curve generation
• Initial fluid distribution; primary drainage for water-wet system
• Imbibition starts from primary drainage curve
• Scanning curves• Closed scanning loops
Pc
Sw
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Relative permeability
• Hysteresis curves from primary drainage
• Weighted sums of Corey-Burdine expressions
• Capillary pressure branches used as weights
kro
krw
Sw
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Numerical modelling
• Domain for simulation discretized
• Block center represents some average
• Hysteresis logic apply to all grid cells
• Fully implicit control-volume formulation:
1n n n n nj
j
m m t f Q
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Numerical issues
• Discrete set of non-linear algebraic equations
• Use Newtons method
• Convergence: Lipschitz cont. derivatives
• Assume monotone directions on time intervals
• ‘One-sided smoothing’ algorithm
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Numerical experiment
• Horizontal water bottom drive
• Incompressible fluids
• Initial fluid distribution; water-wet medium
• Initial equilibrium gravity/capillary forces
• Given set of hysteresis-curve parameters
• Understanding of fluid (re)distribution for different rate regimes
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Initial pressure gradients
• • OWC: Oil water contact• FWL: Free water level• Threshold capillary
pressure, wdc
Pc gh
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Low rate: saturation distribution
• Production close to equilibrium
• Steep water-front; water sweeps much oil
• Small saturation change to reach equilibrium after shut off
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Low rate: capillary pressure
• Almost linear relationship cap. pressure-height
• Low oil relative permeability in lower part of trans. zone
• Curve parameters important for fronts
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Medium rate: saturation distribution
• Same trends as for lowrate case
• Water sweeps less oil in lower part of reservoir
• Redistribution after shut- off more apparent
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Medium rate: capillary pressure
• Deviation from equilibrium
• Larger pressure drop in middle of the trans. zone
• Front behaviour explained by irreversibility
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High rate: saturation distribution
• Front moves higher up in reservoir
• Less oil swept in flooded part of transition zone
• Front behaviour similar to model without capillary pressure
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High rate: capillary pressure
• Large deviation from equilibrium
• Bigger pressure drop near the top of the transition zone
• Insignificant effect for saturation in top layer
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Comparison to reference solution
• Compare to ultra-low rate • Largest deviation near
new FWL• Same trends for compressed
transition zone
Relative deviations from ultra-low rate
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Comparison to Killough’s model
• Killough’s model in commercial simulator
• More capillary smoothing with same input data
• Difference in redistribution in upper part
• Scanning curves different for the models
• Convergence problems in commercial simulator
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What about the real world?
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Conclusions
• Skjaeveland’s hysteresis model incorporated in a numerical scheme
• ‘Forced’ convergence
• Agreement with known solutions
• Layered medium to be investigated in future
• Extension to 3-phase flow